topics in nonlinear dynamics, volume 3: proceedings of the 30th imac, a conference on structural...

Conference Proceedings of the Society for Experimental Mechanics Series

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D. Adams • G. Kerschen • A. Carrella

Editors

Topics in Nonlinear Dynamics,Volume 3

Proceedings of the 30th IMAC, A Conference on StructuralDynamics, 2012

EditorsD. AdamsPurdue UniversityWest Lafayette, IN, USA

G. KerschenUniversity of LiegeBelgium

A. CarrellaLMS InternationalLeuven, Belgium

ISSN 2191-5644 e-ISSN 2191-5652ISBN 978-1-4614-2415-4 e-ISBN 978-1-4614-2416-1DOI 10.1007/978-1-4614-2416-1Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2012936657

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Preface

Topics in Nonlinear Dynamics represents one of six volumes of technical papers presented at the 30th IMAC, A Conference

and Exposition on Structural Dynamics, 2012 organized by the Society for Experimental Mechanics, and held in

Jacksonville, Florida, January 30–February 2, 2012. The full proceedings also include volumes on Dynamics of Civil

Structures; Substructuring and Wind Turbine Dynamics; Model Validation and Uncertainty Quantification; and Modal

Analysis, I & II.

Each collection presents early findings from experimental and computational investigations on an important area within

Structural Dynamics. Nonlinearity is one of these areas. The vast majority of real engineering structures behave nonlinearly.

Therefore, in order to go From the Laboratory to the Real World it is necessary to include nonlinear effects in all the steps ofthe engineering design: in the experimental analysis tools (so that the nonlinear parameters can be correctly identified) and inthe mathematical and numerical models of the structure (in order to run accurate simulations). In so doing, it will be possibleto create a model representative of the reality which (once validated) can be used for better predictions. This volume

addresses theoretical and numerical aspects of nonlinear dynamics (covering rigorous theoretical formulations and robust

computational algorithms) as well as experimental techniques and analysis methods.

The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in

this track.

West Lafayette, IN, USA D. Adams

Belgium G. Kerschen

Leuven, Belgium A. Carrella

Contents

1 Application of the Restoring Force Surface Method to a Real-life Spacecraft Structure . . . . . . . . . . . . . . . . . . . . . . . 1J.P. Noel, G. Kerschen, and A. Newerla

2 Nonlinear Dynamic Model and Simulation of Morphing Wing Profile

Actuated by Shape Memory Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Cassio T. Faria, Carlos De Marqui Jr., Daniel J. Inman, and Vicente Lopes Jr.

3 Environmental Testing and Data Analysis for Non-linear Spacecraft Structures:

Impact on Virtual Shaker Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Simone Manzato, Bart Peeters, Raphael Van der Vorst, and Jan Debille

4 Using Impact Modulation to Detect Loose Bolts in a Satellite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Janette Jaques and Douglas E. Adams

5 Nonlinear Modal Analysis of the Smallsat Spacecraft. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45L. Renson, G. Kerschen, and A. Newerla

6 Filter Response to High Frequency Shock Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Jason R. Foley, Jacob C. Dodson, and Alain L. Beliveau

7 Simplified Nonlinear Modeling Approach for a Bolted Interface Test Fixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Charles Butner, Douglas Adams, and Jason R. Foley

8 Transmission of Guided Waves Across Prestressed Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Jacob C. Dodson, Janet Wolfson, Jason R. Foley, and Daniel J. Inman

9 Equivalent Reduced Model Technique Development for Nonlinear

System Dynamic Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Louis Thibault, Peter Avitabile, Jason R. Foley, and Janet Wolfson

10 Efficient Computational Nonlinear Dynamic Analysis Using Modal

Modification Response Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Tim Marinone, Peter Avitabile, Jason R. Foley, and Janet Wolfson

11 Spectral Domain Force Identification of Impulsive Loading in Beam Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Pooya Ghaderi, Andrew J. Dick, Jason R. Foley, and Gregory Falbo

12 Free-Pendulum Vibration Absorber Experiment Using Digital Image Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167Richard Landis, Atila Ertas, Emrah Gumus, and Faruk Gungor

13 Suppression of Regenerative Instabilities by Means of Targeted Energy Transfers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181A. Nankali, Y.S. Lee, and T. Kalmar-Nagy

14 Force Displacement Curves of a Snapping Bistable Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191Alexander D. Shaw and Alessandro Carrella

15 Characterization of a Strongly Nonlinear Vibration Absorber for Aerospace Applications. . . . . . . . . . . . . . . . . . . . 199Sean A. Hubbard, Timothy J. Copeland, D. Michael McFarland,Lawrence A. Bergman, and Alexander F. Vakakis

vii

16 Identifying and Computing Nonlinear Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209A. Cammarano, A. Carrella, L. Renson, and G. Kerschen

17 Nonlinear Identification Using a Frequency Response Function With the Jump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217A. Carrella

18 Nonlinear Structural Modification and Nonlinear Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225Taner KalaycIoglu and H. Nevzat Ozg€uven

19 Nonlinear Dynamic Response of Two Bodies Across an Intermittent Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239Christopher Watson and Douglas Adams

20 Application of Continuation Methods to Nonlinear Post-buckled Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245T.C. Lyman, L.N. Virgin, and R.B. Davis

21 Comparing Measured and Computed Nonlinear Frequency Responses

to Calibrate Nonlinear System Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255Michael W. Sracic, Shifei Yang, and Matthew S. Allen

22 Identifying the Modal Properties of Nonlinear Structures Using Measured

Free Response Time Histories from a Scanning Laser Doppler Vibrometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269Michael W. Sracic, Matthew S. Allen, and Hartono Sumali

23 Nonlinear System Identification of the Dynamics of a Vibro-Impact Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287H. Chen, M. Kurt, Y.S. Lee, D.M. McFarland, L.A. Bergman, and A.F. Vakakis

24 Modeling of Subsurface Damage in Sandwich Composites Using

Measured Localized Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301Sara S. Underwood and Douglas E. Adams

25 Parametric Identification of Nonlinearity from Incomplete FRF Data

Using Describing Function Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311Murat Aykan and H. Nevzat Ozg€uven

26 Finding Local Non-linearities Using Error Localization from Model Updating Theory . . . . . . . . . . . . . . . . . . . . . . . . 323Andreas Linderholt and Thomas Abrahamsson

viii Contents

Chapter 1

Application of the Restoring Force Surface Method

to a Real-life Spacecraft Structure

J.P. Noel, G. Kerschen, and A. Newerla

Abstract Many nonlinear system identification methods have been introduced in the technical literature during the last 30

years. However, few of these methods were applied to real-life structures. In this context, the objective of the present paper is

to demonstrate that the Restoring Force Surface (RFS) method can provide a reliable identification of a nonlinear spacecraft

structure. The nonlinear component comprises an inertia wheel mounted on a support, the motion of which is constrained by

eight elastomer plots and mechanical stops. Several adaptations to the RFS method are proposed, which include the

elimination of kinematic constraints and the regularization of ill-conditioned inverse problems. The proposed methodology

is demonstrated using numerical data.

Keywords Nonlinear system identification • Space structure • Restoring force surface method

1.1 Introduction

Nonlinear structural dynamics has been studied for a relatively long time, but the first contributions to the identification of

nonlinear structural models date back to the 1970s. Since then, numerous methods have been developed because of the

highly individualistic nature of nonlinear systems [1]. A large number of these methods were targeted to Single-Degree-Of-

Freedom (SDOF) systems, but significant progress in the identification of Multi-Degree-Of-Freedom (MDOF) lumped

parameter systems has been realized during the last 10 or 20 years. To date, simple continuous structures with localized

nonlinearities are within reach. Among the well-established methods, there exist

• Time-domain methods such as the Restoring Force Surface (RFS) and Nonlinear Auto-Regressive Moving Average with

eXogeneous input (NARMAX) methods [2, 3];

• Frequency-domain methods such as the Conditioned Reverse Path (CRP) [4] and Nonlinear Identification through

Feedback of the Output (NIFO) methods [5];

• Time-frequency analysis methods such as the Wavelet Transform (WT) [6].

The RFS method, introduced in 1979 by Masri and Caughey [7], constitutes the first attempt to identify nonlinear

structures. Since then, many improvements of the RFS method were introduced in the technical literature. Without being

comprehensive, we mention the replacement of Chebyschev expansions in favour of more intuitive ordinary polynomials

[8], the design of optimized excitation signals [9] or the direct use of the state space representation of the restoring force as

nonparametric estimate [10].

In theory [11], the RFS method can handle MDOF systems. However, a number of practical considerations diminish this

capability and its scope is, in fact, bound to systems with a few degrees of freedom only. For example, Al-Hadid and Wright

J.P. Noel (*) • G. Kerschen

Space Structures and Systems Laboratory (S3L), Structural Dynamics Research Group Department of Aerospace

and Mechanical Engineering, University of Liege, Belgium

e-mail: [email protected]; [email protected]

A. Newerla

European Space Agency (ESTEC), Noordwijk, The Netherlands

e-mail: [email protected]

D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,

DOI 10.1007/978-1-4614-2416-1_1, # The Society for Experimental Mechanics, Inc. 2012

1

mailto:[email protected]



[12] studied a T-beam structure with well-separated bending and torsion modes. Another extensively studied system of this

kind is the automotive damper [13, 14].

The objective of the present paper is to demonstrate the usefulness of the RFS method in the particular case of a real-life

nonlinear spacecraft structure: the SmallSat spacecraft from EADS-Astrium. Starting from a review of the required

ingredients for a RFS-based identification, we will propose solutions to the inherent limits of the method. First, we will

simplify the kinematics of the nonlinear device of the spacecraft, termed WEMS, in order to explicitely formulate its

dynamic equations. This formulation will be based on the necessary use of the coordinates of its center of gravity.

Eventually, we will discuss why our estimation of coefficients is ill-conditioned and how to circumvent this final issue.

The whole procedure will then be demonstrated using numerical data.

1.2 The SmallSat Spacecraft and Its Finite Element Modelling

The SmallSat structure has been conceived as a low cost structure for small low-earth orbit satellite [15]. It is a monocoque

tube structure which is 1.2 m long and 1 m large. It incorporates eight flat faces for equipment mounting purposes, creating

an octagon shape, as shown in Fig. 1.1a. The octagon is manufactured using carbon fibre reinforced plastic by means of a

filament winding process. The structure thickness is 4.0 mm with an additional 0.25 mm thick skin of Kevlar applied to both

the inside and outside surfaces to provide protection against debris. The interface between the spacecraft and launch vehicle

is achieved through four aluminium brackets located around cut-outs at the base of the structure. The total mass including the

interface brackets is around 64 kg.

The SmallSat structure supports a telescope dummy composed of two stages of base-plates and struts supporting various

concentrated masses; its mass is around 140 kg. The telescope dummy plate is connected to the SmallSat top floor via three

shock attenuators, termed SASSA (Shock Attenuation System for Spacecraft and Adaptator) [16], the behaviour of which is

considered as linear in the present study. The top floor is a 1 m2 sandwich aluminium panel, with 25 mm core and 1 mm

skins. Finally, as shown in Fig. 1.1c a support bracket connects to one of the eight walls the so-called Wheel Elastomer

Mounting System (WEMS) device which is loaded with an 8 kg reaction wheel dummy. The purpose of this device is to

isolate the spacecraft structure from disturbances coming from reaction wheels through the presence of a soft interface

between the fixed and mobile parts. In addition, mechanical stops limit the axial and lateral motion of the WEMSmobile part

during launch, which gives rise to nonlinear dynamic phenomena. Figure 1.1d depicts the WEMS overall geometry, but

details are not disclosed for confidentiality reasons.

The Finite Element (FE) model in Fig. 1.1b was created in Samcef software and is used in the present study to conduct

numerical experiments. The comparison with experimental measurements revealed the good predictive capability of this

model. The WEMS mobile part (the inertia wheel and its cross-shaped support) was modeled as a flexible body, which is

connected to the WEMS fixed part (the bracket and, by extension, the spacecraft itself) through four nonlinear connections,

labeled NC 1–4 in Fig. 1.1d. Black squares signal such connections. Each nonlinear connection possesses

• A nonlinear spring (elastomer in traction plus 2 stops) in the axial direction,

• A nonlinear spring (elastomer in shear plus 1 stop) in the radial direction,

• A linear spring (elastomer in shear) in the third direction.

The spring characteristics (piecewise linear) are listed in Table 1.1 and are displayed in Fig. 1.1e. We stress the presence

of two stops at each end of the cross in the axial direction. This explains the corresponding symmetric bilinear stiffness

curve. In the radial direction, a single stop is enough to limit the motion of the wheel. For example, its +x motion is

constrained by the lateral stop number 2 while the connection 1�x limits the opposite �x motion. The corresponding

stiffness curves are consequently asymmetric. Dissipation is introduced in the FEM through proportional damping and local

dampers to model the elastomer plots.

Sine-sweep excitation was applied locally at the bracket in different directions. The frequency band of interest spans the

range from 5 to 50 Hz and the sweeping rate is chosen equal to four octaves per minute. This frequency range encompasses

the local modes of the WEMS device and some elastic modes of the structure. More precisely, around 11 Hz, the WEMS

vibrates according to two symmetric bending modes (around x and y axis). Around 30 Hz, two other symmetric modes

appear combining bending (around x and y axis) and translation (along x and y axis). A mode involving the WEMS and the

bracket is also present around 30 Hz. The first lateral bending modes and the first axial mode of the structure finally appear

between 30 and 50 Hz.

2 J.P. Noel et al.

Axial nonlinearity

In-plane nonlinearities

NC 1 (-x)

NC 2 (+x)

NC 3 (-y)

NC 4 (+y)

Inertiawheel

SmallSat

Inertia wheel

Bracket

Metalliccross

Filteringelastomer plot

Metallicstop

a b

c d

e

Fig. 1.1 SmallSat structure. (a) Real structure without the WEMSmodule; (b) finite element model; (c) WEMSmodule mounted on a bracket and

supporting a dummy inertia wheel; (d) close-up of the WEMS mobile part (NC stands for nonlinear connection) and (e) graphical display of the

nonlinear restoring forces

1 Application of the Restoring Force Surface Method to a Real-life Spacecraft Structure 3

1.3 Methodology of Identification

In this paper, we address the identification of nonlinear mechanical systems whose nonlinearities are supposed to be

localized and for which there exists an underlying linear regime of vibration. The amplitude, the direction and the frequency

content of the excitation determine in which regime the structure vibrates. For example, we exclude from our scope

distributed or essential nonlinearities.

Such nonlinear systems are modelled through the equations

M€qðtÞ þ C _qðtÞ þKqðtÞ þXn

j¼1WjgjðtÞ ¼ pðtÞ (1.1)

where M, C and K are the mass, damping and stiffness matrices, respectively, q is the generalized displacement vector and

p is the external force vector. The n nonlinear forces acting on the structure are described through some weighted basis

functions gj. Our interest lies in the estimation of the weights, or the coefficients, introduced in these nonlinear expressions

and denoted Wj.

We now give a practical introduction to the RFS method. Other ways to get onto this method can be found in [11, 17, 18].

First, (1.1) is recast into

M€qðtÞ þ fnlðtÞ ¼ pðtÞ (1.2)

where fnl(t) contains all the restoring forces of the system. This offers a simple way to assess the coefficients Wj. Indeed, if

we know the modelling gj(t) of the nonlinear forces hidden in fnl(t), the excitation force p(t), an estimate of the mass matrix

M and the kinematic signals q(t), _qðtÞ and €qðtÞ, the matrixWj (along with K and C) can be estimated, for instance, in a least

squares sense.

For simplicity, we restrict ourselves to the underlying conservative system: neither the damping C of the underlying

linear structure nor the localized dissipation of the elastomer plots of the WEMS will be identified. Concerning the

characterizing functions gj, the nature of the nonlinearity can guide us to its functional form. This is the case of the

WEMS where a bilinear model is obvious.

The access to the excitation signal p(t) can appear to be trivial. However, space structures are universally tested under a

base excitation for which the actual force produced by the shaker is often unknown. In our case, since the exciting force is

applied locally and not directly onto the nonlinear connections, it will not complicate our identification procedure.

The practical knowledge of the mass matrix and of the kinematic signals is more questionable and will entail the whole

methodology developed in this paper. Note that, in practice, only acceleration signals only are recorded. Integration and/or

differentiation are then used to compute displacement q(t) and velocity _qðtÞ [19].

1.3.1 An Illustrative Example

It is interesting to examine the access to these two pieces of information (M and €qðtÞ) in the case of a simple continuous

structure comprising one lumped nonlinearity. The structure of interest is here a linear clamped-free beam with a cubic

nonlinear spring at its free end. This numerical set-up models the geometrical nonlinearity induced by a thin beam part

positioned at the main beam free end as in [20]. Figure 1.2 displays the Finite Element Model (FEM) of the structure where

ten 2D beam elements are considered. Each element possesses a translational (vertical) and a rotational Degree Of Freedom

(DOF) denoted yi and yi, respectively. They are both numbered from 1 to 10, the “nonlinear” DOF being y10.

Table 1.1 Nonlinear spring

characteristics (adimensional

values for confidentiality)Spring Clearance

Stiffness of the

elastomer plot

Stiffness of the

mechanical stop

Axial caxial ¼ 1 1 13.2

Lateral cradial ¼ 1.27 0.26 5.24

4 J.P. Noel et al.

The nonlinear vibrations of that DOF are governed by the equation (assuming that there is no excitation applied at beam tip)

M½y10;y9� €y9ðtÞ þ M½y10;y9�€y9ðtÞ þ M½y10;y10� €y10ðtÞ þ M½y10;y10�

€y10ðtÞ þ . . .

C½y10;y9� _y9ðtÞ þ C½y10;y9�_y9ðtÞ þ C½y10;y10� _y10ðtÞ þ C½y10;y10�

_y10ðtÞ þ . . .

K½y10;y9� y9ðtÞ þ K½y10;y9�y9ðtÞ þ K½y10;y10� y10ðtÞ þ K½y10;y10�y10ðtÞ þ knl y10ð Þ3 ¼ 0: (1.3)

We group the restoring forces together to finally obtain

M½y10;y9� €y9ðtÞ þ M½y10;y9�€y9ðtÞ þ M½y10;y10� €y10ðtÞ þ M½y10;y10�

€y10ðtÞ þ fnlðtÞ ¼ 0: (1.4)

This equation shows that the computation of the restoring forces fnl(t) requires the knowledge of the mass matrix M and

of the accelerations measured at the translational DOF’s y9 and y10 and at the rotational DOF’s y9 and y10. Without either

resorting to a FEM or complicating the experimental procedure, the access to a reliable estimate ofM is a first serious issue.

In addition, in practice, the measurement of rotational DOF’s, such as y9 and y10, is not usually carried out.

This example immediately reveals why there exists almost no application of the RFS method to large-scale structures in

the literature. Most often [21, 22], (1.3) is truncated and adopted under the form

M½y10;y10� €y10ðtÞ þ C½y10;y10� _y10ðtÞ þ K½y10;y10� y10ðtÞ þ knl y10ð Þ3 ¼ 0: (1.5)

The scope of the method is then reduced to qualitative information, i.e. nonlinearity characterization, where it proves to

be a useful tool. It is, however, no longer capable of assessing parameters. In the next subsection, we show how it is possible

to perform RFS-based system identification of the SmallSat spacecraft under an assumption concerning the WEMS

kinematics.

1.3.2 Assumption of a Rigid WEMS Device

One should observe that rotational DOF’s, such as yi in the previous subsection, are central to describe the kinematics of a

flexible body. For instance, the bending of the beam elements in Fig. 1.2 is linked, by essence, to the rotation of their ends.

On the contrary, rotations can be avoided in the description of the motion of a rigid bar element. More generaly, it is possible

to completely define the kinematics of a rigid body through the measure of six translations only, without entailing rotations.

It thus appears that a rigid body assumption is a way to prevent the use of unmeasurable rotational DOF’s.

However, such an assumption is not applicable in the case of the clamped-free beam with cubic nonlinearity. Indeed, this

latter is caused by large deflections (or deformations) of the beam and thus needs flexibility to be activated. On the other

hand, several types of nonlinearity do not resort to such a flexibility in their dynamics. We can cite the geometrical

nonlinearities due to large displacements which always arise in fully rigid multibody systems. This is also the case for the

nonlinearities that are lumped in essence or, in other words, that are caused by localized mechanisms (e.g., friction in a

bolted connection or a damper in an automotive suspension). Alternatively stated, such nonlinearities are not denatured

whether the masses they connect are taken to be rigid. That is not to say that the physics of the structure itself is not modified

(see next subsection).

The WEMS case belongs to this latter class. Its bilinear behaviour in stiffness is indeed localized since it originates from

the combination of lumped elements that are the elastomer plots and the stops. We consequently assume the rigidity of the

inertia wheel and of its cross-shaped support. We simplify further our model by reducing the inertia wheel to a point mass

whose inertia properties are allocated to the center of the cross.

1 2 3 4 5 6 7 8 9 10

knl × y3 [N/m3]

x

y

Fig. 1.2 FEM of the

nonlinear beam


To assess this rigidity assumption, we propose in Table 1.2 the first six elastic frequencies of the WEMS mobile part

alone. Their magnitude gives sense to our approach. We will come back later on the verification of this assumption. Indeed,

in the case of large vibrations, the shocks between the cross and the stops can give rise to flexible effects that have to be

monitored.

The fourth point of the required information listing established above is fulfiled: the kinematic signals q(t), _qðtÞ and €qðtÞare within reach. Furthermore, rigidity gives an easy access to an analytical computation of the mass matrixM. The different

elements of the WEMS as we model it were already displayed in Fig. 1.1d. The rigid metallic cross and the point mass

inertia wheel can thereof be seen.

1.3.3 Kinematic Constraints

The three x–y–z displacements of each end of the cross (twelve in total) naturally describe the kinematic of the WEMS.

Since this description requires the knowledge of six coordinates only, six of them turn out to be redundant. If the vector q

collects this set as

qT ¼ x1 y1 z1 x2 y2 z2 x3 y3 z3 x4 y4 z4ð Þ; (1.6)

it can thus be split into two subsets qI and qD, the former containing six user-chosen independent coordinates.

As for the kinematic constraints, they express the invariance of the distance between any two points of the WEMS

module. In general, this leads to a set of

r!

2! r � 2ð Þ! (1.7)

relationships. In the case of the WEMS, six distance invariants result from the last formula in agreement with the sizes of the

sets q, qI and qD.

Considering the scheme in Fig. 1.3, they can be formulated as

r1 � r2k k2 ¼ d2

r3 � r4k k2 ¼ d2

r2 � r3k k2 ¼ c2

r3 � r1k k2 ¼ c2

r1 � r4k k2 ¼ c2

r4 � r2k k2 ¼ c2

8>>>>>>>>>><>>>>>>>>>>:

(1.8)

where d and c are the length of an arm of the cross and the distance between two of its adjacent ends, respectively. These six

relations indicate that the four ends of the cross are, in fact, the vertices of a square. They intrinsically express the

invariability of the lengths of the two arms, their perpendicularity and their common Center of Gravity (CoG).

It is actually possible, but herein skipped, to demonstrate that the set of constraints (1.8) can be transformed into the more

intuitive relationships of length, orthogonality and coincidence.

Table 1.2 Six first elastic

frequencies of the WEMS

mobile part in free–free

conditions

Natural frequencies (Hz)

1 3,109

2 3,163

3 3,175

4 3,176

5 6,624

6 7,036

6 J.P. Noel et al.

r1 � r2k k2 ¼ d2

r3 � r4k k2 ¼ d2

ðr1 � r2Þ � ðr3 � r4Þ¼ 0

r1 þ r2ð Þ=2 ¼ r3 þ r4ð Þ=2:

8>>>><>>>>:

(1.9)

Already at this stage, we want to underline the mathematical complexity of these relations. Their impossibility to

explicitely define qD, no matter its definition, will play a major role further in this paper.

In the previous subsection, we gave in a first argument in favour of the rigid body assumption (in terms of elastic

frequencies). We can herein make use of the geometrical conditions of rigidity to control on-line the quality of our

assumption. This is central as our approach would loose its suitability in case of flexibility effects. Indeed, significant

terms in (1.1) would then be erroneously neglected.

To that purpose, we formulate the aforementioned geometrical constraints in terms of relative errors and seek potential

deviations from rigidity during the increase of the excitation frequency:

r1 � r2k k2=d2 � 1 ¼ 0

r3 � r4k k2=d2 � 1 ¼ 0

cos�1 r1 � r2ð Þ � r3 þ r4ð Þr1 � r2k k2 r3 þ r4k k2

!2

p� 1 ¼ 0

r1 þ r2ð Þ= r3 þ r4ð Þ � 1 ¼ 0:

8>>>>>>><>>>>>>>:

(1.10)

In Fig. 1.4, we propose a first example of this verification means (z excitation on the bracket at 300 N). At this excitation

level, the system is nonlinear. As intuitively expected, the perpendicularity is almost exactly verified. The 4�z stop is

actually reached and this is visible in the deviations of the second and sixth constraints (explained by impacts on the stops).

The influence of the resonances of the structure are also clearly detectable on these six plots.

We can inspect a second set of constraints under x excitation at 300 N (for which the system is now linear) in Fig. 1.5.

Their verification is improved mainly on the sixth constraint. This highlights the role of the impacts in the relevancy of the

rigid body assumption.

In conclusion, we see that our geometrical verification approach provides a qualitative measure of the confidence in our

identification strategy, and therefore in the subsequently estimated coefficients.

1.3.4 Explicit Formulation of the WEMS Dynamics

We already explained in Sect. 1.3.1 that a rigorous and thorough writing of Newton’s law of motion is crucial to the RFS

method. It is worth pointing out that an unconstrained form of these equations is also obviously sought. Whereas the writing

of such a form is direct for classical vibrating structures, the situation gets more complicated in the presence of kinematic

constraints. This problem is addressed in the present Subsection.

1

2

3 4

d

c

y

xr3

S/C

Fig. 1.3 Top view of the

square-shaped WEMS

mobile part


0 20 40−0.03

0.03

Swept frequency (Hz)

Len

gth

r 1−

r 2

0 20 40−1

1


Len

gth

r 3−

r 4

0 20 40−1

1x 10

−6


Per

pend

icul

arity

0 20 40−1.5

1.5x 10

−5


CoG

x

0 20 40−1

1x 10

−4


CoG

y

0 20 40−0.02

0.02


CoG

zFig. 1.4 On-line verification (in percents) of the geometrical conditions of rigidity at 300 N (z excitation)

0 20 40−0.2

0.2


Len

gth

r 1−

r 2

0 20 40−0.4

0.4


Len

gth

r 3−

r 4

0 20 40−1.5

1.5x 10

−7


Per

pend

icul

arity

0 20 40−1

1x 10

−4


CoG

x

0 20 40−3

3x 10

−5


CoG

y

0 20 40−5

5x 10

−6


CoG

z

Fig. 1.5 On-line verification (in percents) of the geometrical conditions of rigidity at 300 N (x excitation)

8 J.P. Noel et al.

First, we explicitely introduce the elastic restoring forces in the system through the trilinear form (for which we assume

the asymmetry of the WEMS)

fnlðqiÞ ¼ðk0i þ k1i Þqri þ k1i d

1i if qri<� d1i

k0i qri if � d1i � qri � d2i

ðk0i þ k2i Þqri � k2i d2i if qri>d2i

8>><>>: (1.11)

where

• qi is the ith component of q,

• ki0, ki

1 and ki2 are the stiffnesses of the elastomer and of the stops, respectively,

• di1 and di

2 are the associated clearances.

Figure 1.6 displays this stiffness curve. Note that we use the notation qir rather than qi to remind that the force in the

springs is linked to the relative motion of their ends. The relative displacement qir thus designates the difference between two

opposite displacements of the fixed and mobile parts of the WEMS. We also draw attention to the connections noted 1�y,2�y, 3�x and 4�x that are linear and for which we simply write

fnlðqiÞ ¼ k0i qri : (1.12)

In general, the expression of the elastic forces in the WEMS can be shortened following

fnlðqiÞ ¼ kiqri þ k�

i (1.13)

where ki and ki∗ are piecewise constant.

The requirement for reaching unconstrained equations of motion is the possibility to free them from qD by substitution.

In other words, the requirement is an explicit knowledge of the relation

qD ¼ F qI� �

: (1.14)

Such a relation is out of reach because of the complexity of the set of constraints (1.9). This can be clarified by

considering the writing of the potential energy in the system. This energy, stored in the twelve linear and nonlinear

stiffnesses, has the form

V ¼ VðqÞ ¼X12

i¼1

1

2ki q

ri

� �2 þ k�i q

ri : (1.15)

qri

fnl(qi)

d2i

d1i

k0i

k2i

k1i

Fig. 1.6 Bilinear (or here

trilinear since asymmetry

is supposed) force to

be identified


After elimination of qD, we should reach its unconstrained expression, depending on qI only,

V� ¼ V� qI� �

: (1.16)

More particularly, we are interested in the elastic forces which require its gradient, in the Lagrange’s formalism,

@V�

@qI¼ @V

@q

@q

@qI: (1.17)

We can detail this latter as

@V�

@qIj¼X

i:qi2qI@V

@qidij þ

Xi:qi2qD

@V

@qi

@qi@qIj

(1.18)

where it becomes obvious that the last derivate cannot be computed. Indeed, it needs the explicit relationship (1.14) while we

only possess its implicit definition (1.9) of the form

C qI; qD� � ¼ 0: (1.19)

We are consequently compelled to choose a new describing set of independant coordinates and we naturally turn to the

CoG of the WEMS mobile part. We formulate the new definition

qI ¼ xCoG yCoG zCoG a b gð ÞT (1.20)

where xCoG, yCoG and zCoG measure the translation of the CoG in the x–y–z frame and a, b and g parametrize its rotation. We

also redefine qD as

qD ¼ x1 y1 z1 x2 y2 z2 x3 y3 z3 x4 y4 z4ð ÞT (1.21)

and q as qIS

qD.

The immediate advantage of this choice is the possibility to write the transformation qD ¼ F(qI) as

xi

yi

zi

0BB@

1CCA ¼

xCoG

yCoG

zCoG

0BB@

1CCAþ Rða; b; gÞ

Xi

Yi

Zi

0BB@

1CCA; i ¼ 1; . . . ; 4 (1.22)

where Xi Yi ZiT is the initial (undeformed) position of one end of the cross and R is a rotation matrix.

In the parametrization of the WEMS rotation, we opt for Bryant angles because of their intuitive interpretation. The roll

and pitch angles directly quantify rotations around the x and y arms of the cross. As for the yaw angle, it corresponds to a

linear torsion of the WEMS around z axis.

One can go back to the computation of the elastic forces and write, since the potential energy V (see (1.15)) is independant

of the CoG coordinates,

@V

@qIj¼X

i:qi2qD@V

@qi

@qi@qIj

: (1.23)

It thus appears that the restoring force ∂V / ∂qjI can be computed from the product between the gradient ∂V / ∂qi of V

and the jth column of the Jacobian J associated with the set of (1.22).

In addition, the kinetic energy in the system takes the simple form

T ¼ 1

2m _x2CoG þ m _y2CoG þ m _z2CoG þ Ix _a2 þ Iy _b

2 þ Iz _g2� �

(1.24)

and leads to the diagonal mass matrix

10 J.P. Noel et al.

M ¼

m

m 0

m

Ix

0 Iy

Iz

0BBBBBBBB@

1CCCCCCCCA: (1.25)

Under the rigid body assumption, the following unconstrained equation describes the motion of the WEMS rigourously

and is suited to a RFS-based parameter estimation:

Mj€qIj þ ~rV � Jj ¼ 0 (1.26)

where the coefficients to be assessed are hidden in ~rV and whereMj and Jj designate the jth diagonal term ofM and column

of J, respectively. It can be shown that the first three scalar (1.26) give the access to the estimation of the twelve stiffnesses of

the system since they write

m€xCoG þ fnlðx1Þ þ fnlðx2Þ þ fnlðx3Þ þ fnlðx4Þ ¼ 0

m€yCoG þ fnlðy1Þ þ fnlðy2Þ þ fnlðy3Þ þ fnlðy4Þ ¼ 0

m€zCoG þ fnlðz1Þ þ fnlðz2Þ þ fnlðz3Þ þ fnlðz4Þ ¼ 0:

8>><>>: (1.27)

We will consequently restrict our results to the use of these three equations. Rigidity rises a last issue, discussed in the

following subsection, and linked to the rank deficiency of the matrix ~rV � J.

1.3.5 Identification of a Rigid Body: An Ill-Conditioned Problem

Let us consider the identification of the WEMS stiffnesses in the x direction and in linear regime. The first equation of (1.27)

then becomes

m€xCoG þ k0x1 xr1 þ k0x2 xr2 þ k0x3 x

r3 þ k0x4 x

r4 ¼ 0: (1.28)

For the purpose of the identification, this equation is written as a least squares problem:

xr1 xr2 xr3 xr4� �|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

A

k0x1 k0x2

k0x3 k0x4

� �T|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

x

¼ �m€xCoG|fflfflfflffl{zfflfflfflffl}b

:(1.29)

In practice, matrixA turns out to be badly conditioned preventing an accurate estimation of the coefficients k0xi . More precisely,

thismatrix appears to be not of full rank. The explanation is twofold. On the one hand, and according the fourth and last equation

of constraint (1.9), there exists a linear relationship between the displacements of theWEMSmobile part considered direction per

direction. On the other hand, since theWEMS fixed part (actually the bracket) is almost at rest in our frequency band of interest

(5–50 Hz), this linear dependance is not altered when moving to relative displacements as in matrix A.

As noted in the literature [23], without perturbations and rounding errors, the solution to the rank-deficient system of

equations A x ¼ b is straightforward. Indeed, if we introduce the Singular Value Decomposition (SVD) of A

A ¼Xn

i¼1ui si vTi (1.30)

this solution writes

xideal ¼XrankðAÞ

i¼1

uTi b

sivi: (1.31)


However, practically speaking, A is never exactly rank deficient since it has one or several small but non-zero singular

values [23]. It is said to be numerically rank deficient and causes the aforementioned least squares problem to be ill-

conditioned, i.e. its solution is dominated by the errors. A simple regularization strategy consists in truncating the singular

value spectrum of A and thus replacing its smallest elements with exact zeros. In other words, A is seen as a noisy

representation of the mathematically rank deficient matrix Ak defined as

Ak ¼Xk<n

i¼1ui si vTi : (1.32)

The stiffness coefficients of the WEMS are then computed in a numerically stable way through

x ¼ Ayk b ¼

Xk

i¼1

uTi b

sivi (1.33)

where { denotes the inverse of a rectangular matrix in a least squares sense. This approach is known as the Truncated

Singular Value Decomposition (TSVD). Its difficulty lies in the choice of k, i.e. the number of sufficiently large singular

values, that is to say in the definition of what a small singular value is.

1.4 Identification Results

We first consider an axial (z) excitation applied to the WEMS bracket at 200 N. At this excitation level, no mechanical stop

is reached and the dynamics thus remains linear. Our interest lies in the estimation of the axial stiffnesses since the relative

displacements in the lateral directions are negligible. Figure 1.7 presents our results, summarized in Table 1.3. The

estimation is of high quality as proved by a simple visual inspection of the force-displacement curves.

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

Relative displacement

Res

tori

ng for

ce

Nonparametric RFS estimate

Fitted stiffness curve

Exact stiffness curve

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1x 10−3

x 10−3x 10−3

Res

tori

ng for

ce


−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1


Res

tori

ng for

ce

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Res

tori

ng for

ce


Fig. 1.7 Parametric estimation of the stiffnesses 1�z (top left), 2�z (top right), 3�z (bottom left) and 4�z (bottom right)

12 J.P. Noel et al.

Figure 1.8 shows a qualitative application of the RFS method. As explained in Sect. 1.3.1, this simplified approach

constitutes a powerful characterization tool. However, it cannot be used for parameter estimation purposes since it is based

on truncated equations of motion, as obvious in this figure.

As a last result in the linear case, we prove the relevancy of a TSVD regularization in Table 1.4. The four singular values of

thematrixA to be inverted in our least squares resolution present a large gap between the second and the third ones. It reveals a

numerical rank equal to 2. The consequent truncation of this singular value spectrum led us to the results in Fig. 1.7.

Table 1.3 Summary of the identification results in the axial direction. Each estimate is given with a relative deviation from its exact value

Identification case 1 � z 2 � z 3 � z 4 � z

200 N Linear est. 1.00 1.06 1.03 1.02

(linear estim.) 0.26% 5.50% 2.55% 2.12%

300 N Nonlinear est. – – – 12.93

(only nonlinear estim.) 2.08%

300 N Linear est. 1.66 0.84 0.82 0.80

(full estim.) 66.29% 16.41% 17.79% 20.03%

Nonlinear est. – – – 12.90

2.32%

400 N Nonlinear est. – – 12.26 12.29

(only nonlinear estim.) 7.14% 6.92%

500 N Nonlinear est. – – 13.92 13.10

(only nonlinear estim.) 5.41% 0.78%

Table 1.4 The third and fourth singular values should be equal to zero. They differ

because of rounding errors

200 N axial case Singular value (%)

1 55.6

2 43.3

3 0.6

4 0.5

−1 −0.5 0 0.5 1−4

−2

0

2

4x 10−3


Res

tori

ng for

ce

Qualitative RFS

Exact RFS

Fig. 1.8 Qualitative application of the RFS method to the 4�z connection. The different WEMSmodes in the frequency band of excitation appear

with various and inexact slopes


Prior to addressing a nonlinear case, we anticipate a discussion about displacement coupling tackled later in this paper.

The confidence factor [17] shown in Fig. 1.9 shows that there is no contribution of the lateral motion in the fitting of the

axial restoring forces.

We now move to a 300 N excitation level. The axial nonlinear connection numbered 4 � z is activated when sweeping

the modes of vibration around 30 Hz. This activation is clear on the corresponding time series (Fig. 1.10) where a jump

phenomenon is highlighted.

We chose to assess the 4 � z nonlinear stiffness using the four linear estimates obtained in the previous step (Fig. 1.7).

This approach turns out to be the most accurate. Our result is depicted in Fig. 1.11 and also listed in Table 1.3. Here again,

we note an excellent agreement with the exact result (relative error of 2%). We can however try to simultaneously fit both the

0

5

10

15

20

25

30

Con

fiden

cefa

ctor

z1 z2 z3 z4 x1 x2 x3 x4 y1 y2 y3 y4

Fig. 1.9 The recourse to lateral contributions appears to be useless in axial identification

27 29 31 33 35 37 39Frequency (Hz)

FR

Fm

agni

tude

Linear

Nonlinear Jump

0 10 20 30 40 50−2

0

2

Swept frequency(Hz)

Rel

ativ

e 4–

z di

spla

cem

ent

Fig. 1.10 Jump phenomenon caused by the distortion of the corresponding FRF

14 J.P. Noel et al.

four linear and the nonlinear forces (next case in Table 1.3). The nonlinear estimate remains of high quality whereas the

linear parameters get worse. This gives sense to our decoupled estimation.

We finally address 400 N and 500 N excitations and focus on the first modes of the WEMS around 11 Hz. Indeed,

considering their shapes, the modes of vibration around 30 Hz cannot lead to the activation of several nonlinearities. The

reach of the 4 � z nonlinear regime actually prevents other stops from being impacted. This explains our interest in the first

modes which are activated at higher energy but involve two nonlinear restoring forces. Here again, we exploit our

aforementioned linear estimates in order to focus on a nonlinear estimation problem. The estimates take place in Table 1.3

and a graphical display at 500 N is given in Fig. 1.12. As one may expect, the higher the level of activation of the

nonlinearities, the more accurate their estimation.

The estimation of the lateral stiffnesses (under a lateral excitation) turns out to be much more challenging and will clearly

underline the limitations of our RFS-based identification strategy. Their explanation lies in the existence of kinematic

couplings.

To detail this issue, we now excite the structure along an oblique direction with respect to the x and y reference axis (at the

linear 200 N regime) and try to estimate the y-stiffness coefficients. The result provided in Table 1.5 is of unsatisfactory

quality. This quality can be enhanced by considering the existence of kinematic couplings. To that purpose, we include in

our fitting basis all the available x, y and z restoring forces as potential candidates. The estimation is depicted in Fig. 1.13

and summarized in Table 1.5. Its quality appears to be acceptable and the analysis of the stiffness curves confirms that

judgement. The need for an extended fitting panel is obvious in Fig. 1.14. In particular, the 3 � z and 4 � z contributions aremore relevant than the y stiffnesses themselves. Finally, we stress that a much improved quality would require additional

fitting terms (mainly modelling dissipation) as demonstrated by the remaining disturbances on the force-displacement curves

(Fig. 1.13).

Another proof of interdependence between the directions of motion of the WEMS is given in Fig. 1.15 (x excitation).

This restoring force curve reveals a trilinear behaviour. Such a form is actually caused by the simultaneous reach of an axial

and a lateral stop. Their appearance on a single force-displacement curve is indeed an additional proof of coupling.

An experimental campaign was also performed on the SmallSat spacecraft, and the results are currently analysed.

A preliminary result in Fig. 1.16 clearly highlights the piecewise linear nature of the WEMS device.

−1.5 −1 −0.5 0 0.5 1 1.5−7.5

−5

−2.5

0

2.5

5

7.5x 10−3


Res

tori

ng for

ce


Fitted stiffness curveExact stiffness curve

Fig. 1.11 Excellent estimation of the 4 � z nonlinear force


−1.5 −1 −0.5 0 0.5 1 1.5−7.5

−5

−2.5

0

2.5

5

7.5x 10−3


Res

tori

ng for

ce




−1.5 −1 −0.5 0 0.5 1 1.5−7.5

−5

−2.5

0

2.5

5

7.5x 10−3


Res

tori

ng for

ce

Fig. 1.12 Simultaneous identification of the 3�z (left) and 4�z (right) restoring forces

−0.25 0 0.25 −1

0


Res

tori

ng for

ce

−0.25 0 0.25−1

0

1x 10−4


Res

tori

ng for

ce

−0.25 0 0.25−1

0

1x 10−4


Res

tori

ng for

ce

−0.25 0 0.25−1

0

1x 10−4


Res

tori

ng for

ce




Fig. 1.13 Parametric estimation of the stiffnesses 1�y (top left), 2�y (top right), 3�y (bottom left) and 4�y (bottom right)

Table 1.5 Summary of the identification results in the y lateral direction. Each estimate is given with a relative deviation from its exact value

Identification case 1 � y 2 � y 3 � y 4 � y

200 N Linear est. � 0.73 0.96 0.12 0.12

(linear estim.) 379.42% 268.95% 55.18% 55.37%

200 N Linear est. 0.32 0.24 0.28 0.28

(extended basis) 22.83% 6.75% 7.00% 8.00%

0

2

4

6

8

10

12

14

16

18

Con

fiden

cefa

ctor

y1 y2 y3 y4 x1 x2 x3 x4 z1 z2 z3 z4

Fig. 1.14 The whole fitting basis is relevant in the case of the y identification

−1.5 −1 −0.5 0 0.5 1 1.5−5

−2.5

0

2.5

5x 10−4


1–z

rest

orin

gfo

rce

Fig. 1.15 The reach of two perpendicular stops induces a coupled trilinear behaviour


1.5 Conclusions

This paper aimed at applying the Restoring Force Surface method to a real-life space structure: the SmallSat spacecraft. The

elimination of kinematic constraints and the regularization of ill-conditioned inverse problems are the two main

contributions of the study. To date, the identification results are promising. A particular attention should be devoted to

damping contributions in a further work. A possible approach would be a non-physical modelling through Fourier series. In

addition, the SmallSat identification from experimental data is also in progress.

Acknowledgements This paper has been prepared in the framework of the ESA Technology Research Programme study “Advancement of

Mechanical Verification Methods for Non-linear Spacecraft Structures (NOLISS)” (ESA contractNo.21359/08/NL/SFe).

The author J.P. Noel would like to acknowledge the Belgian National Fund for Scientific Research (FRIA fellowship) for its financial support.

The authors finally thank Astrium SAS for sharing information about the SmallSat spacecraft.

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−2.5

0

2.5

5x 10−3


3–x

rest

orin

g fo

rce

Fig. 1.16 A first qualitative

restoring force computed

on experimental

measurements

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Chapter 2

Nonlinear Dynamic Model and Simulation of Morphing

Wing Profile Actuated by Shape Memory Alloys

Cassio T. Faria, Carlos De Marqui Jr., Daniel J. Inman, and Vicente Lopes Jr.

Abstract Morphing aircraft have the ability to actively adapt and change their shape to achieve different missions

efficiently. The development of morphing structures is deeply related with the ability to model precisely different designs

in order to evaluate its characteristics. This paper addresses the dynamic modeling of a sectioned wing profile (morphing

airfoil) connected by rotational joints (hinges). In this proposal, a pair of shape memory alloy (SMA) wires are connected to

subsequent sections providing torque by reducing its length (changing airfoil camber). The dynamic model of the structure is

presented for one pair of sections considering the system with one degree of freedom. The motion equations are solved using

numerical techniques due the nonlinearities of the model. The numerical results are compared with experimental data and

a discussion of how good this approach captures the physical phenomena associated with this problem.

Keywords Morphing wing • Shape memory alloy • Nonlinear dynamics

2.1 Introduction

Since the beginning of self propelled flight in 1903 the design of aircraft have been based on recreate nature’s ability of

flight. However, the recent design trend lines shifted towards a more rigid structure powered by engines, a different approach

compared to the one developed in nature during the course of evolution. The advances achieved by this different approach

were magnificent, creating machines that could flight faster and higher then any other bird or flying animal in nature.

However, these advances, came with improved efficiencies limited to specific flight conditions for a single vehicle.

Inspired by birds that adapt their wing aerodynamics (change in shape) to improve their performance when executing a

specific task [1], a new concept in wing design has been developed, called “morphing wing”. Several authors have been

developing these concepts, such as in [2] where small and continuous adjustments in the wing shape were made, or like in [3]

where small changes in the flow field around the wing improved the aircraft control. A more complete description of the

evolution of this concept is presented in [4].

This paper addresses the modeling of a novel morphing wing configuration that uses shape memory alloys (SMA) as

actuators to provide the desired change in shape [5]. By sectioning a wing profile into two or more pieces and connecting

them back with rotational joints (hinges), the main structure of the proposed morphing wing is built, such as illustrated in

Fig. 2.1a. One pair of SMA wires is positioned over each joint to allow the structure to be actuated in both directions.

Figure 2.1b shows the motion created when the top wire is activated and Fig. 2.1c illustrates the actuation of the bottom wire.

The actuation mechanism consists of one pair of deformed shape memory alloy wires positioned over each hinge. The

shape memory effect [6] is activated by heating one wire, that reduce its length or, if restricted, apply a force. This reduction

in length apply a torque in the structure. To provide a good aerodynamic response an elastic skin has to be designed in order

C.T. Faria (*) • D.J. Inman

Virginia Polytechnic Institute and State University, 310 Durham Hall, 24061 Blacksburg, VA, USA


C. De Marqui Jr.

Engineering School of Sao Carlos, Uni of Sao Paulo, Av. Trabalhador Sancarlense 400, 13566-590 Sao Carlos, SP, Brazil

V. Lopes Jr.

Uni Estadual Paulista – UNESP, Av. Brasil Nº56, 15385-000 Ilha Solteira, SP, Brazil



21


to eliminate the discontinuities. The skin has to be compliant to reduce the load on the actuator, but has to be stiff enough to

support aerodynamic loads, as presented by [7].

In practice such structure needs to be controlled in order to maintain or achieve a desired shape, to do so an appropriate

dynamic model has to be derived. During this process two sources of nonlinearities were identified, one is related with

geometrical problem, since the force is always applied in the direction of the wire, and another one is related with the

hysteretic nature of the shape memory alloy transformations. Both explored and treated further in this paper.

2.2 Modeling of the Proposed Morphing Wing

Consider a wing profile sectioned and connected by a rotational joint (hinge) at midchord position. The leading edge section

is attached to an inertial reference frame (I) XY, presented in Fig 2.2a, while the other rotating half is linked to a second

mobile frame (B1) X1Y1. Other important points are illustrated in Fig. 2.2b, such as the center of mass (G), the rotational

point (O), the connection points of the upper actuator (A and B) and the connection points of the lower actuator (C and D). It

is important to notice that the dimension presented for points A and C are related to the inertial frame I, for points B, D and G

they are described in terms of the mobile frame B1.The transformation matrix that relates a vector from the inertial frame (I) to the mobile frame (B1) can be simply derived

considering a rotation angle y.

TB1 ¼cos y sin y 0

� sin y cos y 0

0 0 1

264

375 (2.1)

One can calculate the acceleration of the center of mass (G) of the second body based on the position vector ( IrOG),

considering that, for this case, there is no relative velocity ( IVrel) or acceleration ( Iarel) between the two reference frames.

IaG ¼ IaO þ Iw� Iw� IrOG þ I _w� IrOG þ 2 Iw� IVrel þ Iarel ¼�€y L1 siny� _y

2L1 cos y

€y L1 cosy� _y2L1 sin y

0

264

375 (2.2)

The angular velocity vector ( Iw) is defined as the time derivative of the angular position (y) with orientation defined by

the right hand rule, being positive in the z direction when y moves counter clockwise. The over-dot denotes differentiation

with respect to time.

The distance vector between A-B and C-D can be defined in the inertial frame as a linear operation between the position

vectors ( IrOA, B1rOB, IrOC and B1rOD), such that:

IrAB ¼ � IrOA þ TTB1B1rOB ¼

b1 cos y� b2 sin yþ a1b1 sin yþ b2 cos y� a2

0

24

35 (2.3)

aP1 P2 P3

q

q

P4

b

c

Fig. 2.1 (a) Proposed model

of morphing wing with four

sections with a pair of

actuators between P2 and P3.

(b) Actuation of the top wirein the model. (c) Actuation of

the bottom wire

22 C.T. Faria et al.

IrCD ¼ � IrOC þ TTB1B1rOD ¼

d1 cos yþ d2 sin yþ c1d1 sin y� d2 cos y� c2

0

24

35 (2.4)

It is important to notice that the subscript located to the left of the vector symbol indicates the frame where it is defined,

and for position vectors the subscript index to the right indicates the initial and final point were it is defined, respectively.

By activating the shape memory effect on the top wire/actuator (SMA1), the force with direction A–B is applied in both

bodies ( IFAB and IFBA) but with different orientation, a similar effect can be described for the bottom actuator (SMA2) thatwill apply a force in the C-D direction in both bodies ( IFCD and IFDC).

IFAB ¼ FSMA1IrAB

IrABk k ¼ � IFBA (2.5)

IFCD ¼ FSMA2IrCD

IrCDk k ¼ � IFDC (2.6)

Both parameters FSMA1 and FSMA2 are functions of the angle y and the electrical current in each wire. These forces are

defined in the next section based on the constitutive relation for shape memory alloys. Euller’s principle applied in the

second (moving) body at the origin point (O) is:

XMO ¼ I2 I _wþ Iw� I2 Iwþ m2 IrOG � IaG ¼

0

0

IZZ€yþ m2L21€y

24

35 (2.7)

where I2 is the inertial matrix (for this problem it is a diagonal matrix), m2 is the total mass of the body and IrOG is the

distance vector between point O and G. The external moments applied at point O in the second body are calculated based on

the free body diagram presented by Fig. 2.3, where aerodynamic forces (FAER), weight forces (FP), reaction forces (RX and

RY) and actuator forces (FBA and FDC) are acting on the body.

External moments are calculated by the cross product between the position vector of the application point and the force

vector, for the moments applied by the actuators one can get:

IMOB ¼ IFBA � IrOB ¼0

0

�FSMA1K1 sin yþK2 cos yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

K3þK1 cos y�K2 sin yp

24

35 (2.8)

Y1 Y

a1

a

b

c1

c2

A B

DG

C

a2 b2

d2

d1

A

CO

OB

DG

L1b1

X

X1

θ

Fig. 2.2 (a) Frame of references illustration and positive angle measurement. (b) Structures points of interest

2 Nonlinear Dynamic Model and Simulation of Morphing Wing Profile Actuated by Shape Memory Alloys 23

IMOD ¼ IFDC � IrOD ¼0

0

�FSMA2H1 sin y�H2 cos yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

H3þH2 cos yþH1 sin yp

24

35 (2.9)

where

K1 ¼ a1b1 � a2b2

K2 ¼ a1b2 þ a2b1

K3 ¼ a21 þ a22 þ b21 þ b22

(2.10)

H1 ¼ c1d1 � c2d2

H2 ¼ c1d2 þ c2d1

H3 ¼ c21 þ c22 þ d21 þ d22

(2.11)

By setting the weight and aerodynamic forces to zero one can sum equations (2.8) and (2.9), and insert them into equation

(2.7), resulting in the following expression in the Z direction.

IZZ þ m2L21

� �€yþ FSMA1

K1 sin yþ K2 cos yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK3 þ K1 cos y� K2 sin y

p þ FSMA2H1 sin y� H2 cos yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

H3 þ H2 cos yþ H1 sin yp ¼ 0 (2.12)

Equation (2.12) is the nonlinear dynamic equation of themorphingwingwith a single degree of freedom.Nonlinear terms can

be isolated by looking at the expression that multiply the force applied by each actuator. These expressions can be interpreted as

the moment arm for each actuator force and they are functions of the angular position. A second source of non-linearity arises

from the hysteretic behavior of the shape memory alloys that is defined for each actuator force (FSMA1 and FSMA2).

FSMA1 ¼ ASMA1sSMA1 (2.13)

where ASMA1 represent the cross section area of the first actuator and sSMA1 the tension associated with it. A similar expression

can be written for the second actuator. The tension in each actuator is a function of the deformation of the wire and the

temperature, as described in detail by [6]. The deformation in each wire can be related with the angular position, such that:

eSMA1 ¼ IrABk k � L0SMA1

L0SMA1

(2.14)

where L0SMA1 is the initial (undeformed) length of the top actuator. Similar expression can be written for the second actuator.

The temperature in the actuator is the main mechanism to produce actuation by the SMA wire. The change in temperature

will induce a phase transformation in the material microstructure, as illustrated in Fig. 2.4a. Another important effect is the

microstructural transformation induced by stress, illustrated in Fig. 2.4b. More details about these transformation process

and modeling of these phenomena can be found in [6].

To relate the input electric current with a wire temperature [8] presents the following relation:

T ¼ T1 þ R

hCAC1� e�t=th

� �I2 þ ðT0 � T1Þ e�t=th (2.15)

Fig. 2.3 Free body diagram

of the second (moving) body


where R is electrical resistance of the wire/actuator, T0 the temperature in the beginning of the process, I the input current, hCthe convection heat transfer coefficient, AC is the heat transfer area, T1 the environment temperature and th is the time

constant defined by:

th ¼ rASMAcPhCAC

(2.16)

where cP is the specific heat of the wire and r the SMA density.

2.3 Simulations and Experimental Verification

The solution of the proposed model that includes nonlinearities, such as the geometrical relation and hysteresis in the phase

transformation, was obtained through numerical approach [6]. Equation (2.12) is rewritten using the state space format.

_y€y

� �¼ 0 1

0 0

� �y_y

� ��

0FSMA1

IZZ þ m2L21� � K1 sin yþ K2 cos yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

K3 þ K1 cos y� K2 sin yp þ FSMA2

IZZ þ m2L21� � H1 sin y� H2 cos yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

H3 þ H2 cos yþ H1 sin yp

24

35 (2.17)

A fifth order Runge–Kutta-Fehlberg method was used to solve the numerical method [9]. The method was implemented

using a Matlab® code with variable time step. A new time step (smaller) is defined if the local error, i.e. the difference

between the solutions provided by a higher order solution, is bigger then a threshold value. This technique was implemented

to increase the code speed and keep the solution accurate in regions with fast dynamics.

Table 2.1 presents all the parameters used in the simulations. It is important to notice that both actuators are considered to

have the same dimensions and properties. One may notice the large number of variables that have to be defined in the

proposed model once the system dynamics arises from the coupling of different physical phenomena. This fact decreases

model accuracy once that all uncertainties in parameter estimation comes into play.

Table 2.2 presents the initial conditions applied to the model. These conditions correspond to the ones observed during

the experimental tests described in detail further in this paper.

Input electrical current applied to the problem was a step current with amplitude of 1.98A, as illustrated by Fig. 2.5.

The experimental setup, Fig. 2.6, consists of the prototype presented by Fig. 2.1, where section P3 and P4 were rigidly

connected with each other, and section P1 and P2 connected to an inertial table. The relative rotation motion could only be

possible (by this configuration) between section P2 and P3, where a pair of SMA actuators were positioned. Angular

information was collected by a linear potentiometer (TRIMER 3386cw), and the resistance measured was related to a

specific angle by previous calibration. This information was filtered by an analog second order Butterworth filter with cutting

frequency of 1.5 Hz.

A data acquisition system from DSpace® (DS1103) outputted a control signal to the current source (WORNDER BOX

from LORD®) which applied the desired current in one of the actuators as shown by Fig. 2.5. This DAQ was also responsible

for collecting the filtered data from the potentiometer with a sampling frequency of 10 Hz.

It is important to notice that the system had to be actuated a number of times before achieving a constant replicable cycle.

Figure 2.7 presents the experimental results as well as the simulation results, both being actuated as described by Fig. 2.5,

and with the same initial conditions.

Fig. 2.4 (a) Hysteretic

transformation of martensite

into austenite for high

temperatures, b represents the

total martensite fraction in

the material microstructure.

(b) Creation of martensite

induced by stress


2.4 Discussions and Conclusions

The proposed model has two different types of nonlinearities, one is associated with the geometric feature that is functional

of the angle and the other one is associated with the hysteretic nature of SMA transformations. The first one can be linearized

for small angles around the origin and even though it is not possible to obtain a linear expression of the moment arm as

a function of the angle (y). Based on the geometrical parameters presented in Table 2.1, one can easily plot these moment

arm functions for different angles. The maximum difference between these functions and a linear expression that interpolate

Table 2.2 Initial conditions

applied in the numerical

simulation

y sSMA1 TSMA1 bS SMA1 bT SMA1 ISMA1

�0.82 rad 0 Pa 301 K 100% 0% 0 A

o sSMA2 TSMA2 bS SMA2 bT SMA2 ISMA2

0 rad/s 0 Pa 301 K 0% 0% 0A

Fig. 2.5 Current input pattern

for experimental and

numerical simulations

Fig. 2.6 Experimental

prototype

Table 2.1 Model parameters applied in the numerical simulation

a1 b2 d1 ASMA CP MF AS CM L1 IZZ54 mm 13 mm 50 mm 0.203 mm2 857 J/kg K 313 K 328 K 8 MPa/K 130 mm 18.636 kg mm2

a2 c1 d2 AC L SMA1 MS AF CA m2 hC

17 mm 54 mm 13 mm 1595 mm2 103 mm 323 K 343 K 13 MPa/K 0.021 kg 30 W/K m2

b1 c2 s SCRIT s F

CRIT EA EM T1 R eL r50 mm 17 mm 100 MPa 170 MPa 67 GPa 26 GPa 301 K 3.75 O 2.4% 6450 kg/m3


the initial and final moment arm value were less then 5% of the maximum moment arm. In practice this nonlinearity will not

be important and can be easily substituted by a linear expression. This nonlinearity can be used to increase model precision,

if necessary.

Hysteretic nonlinearity will play an important role in this problem. This phenomenon will be responsible for the change

in the system equilibrium points, resulting in a new camber without additional input (electrical current) after the desired form

is achieved. In this case, if the applied external forces and temperatures change does not exceed the transformation limits of

the actuators, both will maintain their new shape, shifting the equilibrium point to a new angular position. Dynamics

associated with the transformation process is also an important system feature once they will dictate the transition between

equilibrium points. This process is function of the stress state in the actuators and temperature (or indirectly it is function of

the electrical current).

Another important issue with the presented model is the large number of parameters, for the simulated problem 30

physical properties were used, after considering symmetry between actuators. With such a large number of parameters the

errors associated with the estimation of each one can distort the simulation result. Estimation problems are mainly

concentrated in the actuator properties, once they require a series of specialized equipment. Overcome of our limitation

was possible by executing another experiment, were a SMA wire was connected to a linear spring, in such conditions the

alloy properties could be determined by parameters adjustments to fit the SMA model to the experimental data. Further

investigation is necessary in other to identify the effects of perturbations in the model parameters.

Figure 2.7 is the comparison between experimental data and numerical simulation. In such problem a series of energy

conversions from different domains occur, i.e. electrical to thermal and thermal to mechanical, and also hysteretic behavior

are present. Modeling the interaction of these phenomena is not an easy task. The presented model was capable to capture all

these effects, getting the simulated model to a new equilibrium point. It is important to notice that both curves have a very

similar slope during the transformation process between 3 and 7 seconds, indicating a good agreement between model and

experiment in that case.

The mismatch between both curves can be explained by problems in parameter estimation. For example, if the

transformation temperature (AS) were smaller and the maximum recoverable strain (eL) was bigger both curves would

match during the transformation period (between 2 and 5 s) and the steady value. Numerical simulation also presents another

issue, a reverse transformation, such effect occurred by the cooling of the actuator and after the 22nd second a small recovery

can be noticed. Such phenomenon does not occur in practice, and this discrepancy can be explained by problem in the

parameters used during the simulation. Some noise can also be noticed in the experimental data despite the filtering process

applied.

It is worth to mention that the presented model can capture the most important phenomena associated with the dynamics

of the proposed morphing wing. Such design has an exciting potential to be applied in aerodynamic structures that demand a

Fig. 2.7 Experimental and

simulation results of the

morphing wing being actuated

in one direction


low frequency change in shape to adapt itself for different stages during the mission. The main advantage is the fact that a

new equilibrium point is achieved by activating the shape memory effect, demanding none (or small amount) energy to keep

the aerodynamic shape.

Acknowledgments The authors are thankful to CNPq and FAPEMIG for partially funding the present research work through the INCT-EIE.

References

1. Akl W, Poh S, Baz A (2007) Wireless and distributed sensing of the shape of morphing structures. Sensor Actuat A Phys Amst 140(1):94–102

2. Hall JM (1989) Executive summary AFTI/F-111 mission adaptive wing. Ohio, Wright Research and Development Center Technical Report. ID.

WRDC-TR-89-2083

3. Natarajan A, Kapania RK, Inman DJ (2004) Aeroelastic optimization of adaptive bumps for yaw control. J Aircraft NY 41(1):175–185

4. Seigler TM, Neal DA, Bae JS, Inman DJ (2007) Modeling and flight control of large-scale morphing aircraft. J Aircraft NY 44(4):1077–1087

5. Faria CT (2010) Controle da variacao do arqueamento de um aerofolio utilizando atuadores de memoria de forma. Dissertation Uni Estadual

Paulista – UNESP. Print

6. Brinson LC (1993) One dimensional constitutive behavior of shape memory alloys: themomechanical derivation with non constant material

functions and redefined martensite internal variable. J Intell Mater Syst Struct Va 4(2):229–242

7. Gandhi F, Anusonti-Inthra P (2008) Skin design studies for variable camber morphing airfoils. Smart Mater Struct NY 17(1):15–25

8. Leo DJ (2007) Engineering analysis of smart material systems. Wiley, New York, p 331

9. Mathews JH, Fink KK (2004) Numerical methods using matlab, 4th edn. Prentice-Hall, New Jersey, pp 497–499


Chapter 3

Environmental Testing and Data Analysis for Non-linear

Spacecraft Structures: Impact on Virtual Shaker Testing

Simone Manzato, Bart Peeters, Raphael Van der Vorst, and Jan Debille

Abstract This paper reports on the results of the environmental testing and data analysis that was performed on a satellite

structure that incorporates some typical structural non-linearities present in actual flight hardware. The project coordinator

EADS Astrium provided the bread-board satellite model and LMS International was responsible for the execution of the

environmental tests including sine and random tests at various vibration levels and in multiple directions. Next to a

presentation of the test results with an emphasis on the non-linear behaviour, advanced experimental modal estimation

technique were applied on the data.

Keywords Non-linear • Environmental testing • Vibration control • Modal analysis

3.1 Introduction

The European Space Agency (ESA) launched a research project on the advancement of mechanical verification methods for

non-linear spacecraft [1]. For the purpose of spacecraft structure design development and verification by analysis, the

structures are generally assumed to behave linear. However, experience has shown that various non-linearities might exist in

spacecraft structures and the consequences of their dynamic effects can significantly affect the design verification

procedures, in particular (1) the evaluation of flights loads with linearized models employed in launcher/satellite coupled

dynamic loads analyses (CLA); (2) the performance of dynamic verification tests; and (3) the demonstration that non-linear

dynamic effects have been well covered by the satellite verification tests.

In most cases the encountered structural non-linearities are associated with the presence of non-linear damping and/or

stiffness characteristics. Several sources of such non-linearities are well known: backlash, joint gapping, slippage when

friction forces are overcome, rattling, non-linear material characteristics, and discontinuities in force–deflection curves when

pre-loads are exceeded. ESA [1] shows a summary of typical spacecraft structural non-linearities associated with damping

and stiffness characteristics and their consequences on qualification testing (Table 3.1).

In the frame of this research project, EADS Astrium provided a bread-board satellite structure, incorporating some typical

structural non-linearities. LMS International was responsible for the execution of the environmental tests including sine and

random tests at various vibration levels and in multiple directions. This paper reports on these tests, the experimental study

of the non-linear behaviour, and the application of advanced experimental modal parameter estimation technique.

3.2 Structure and Tests

In this section, the representative structure will be introduced as well as the vibration test programme. For cost-saving

reasons, it was a requirement specified in the statement of work of the research project, to use existing hardware to the largest

extent possible. Therefore, Astrium proposed the SMALLSAT structure (Fig. 3.1 left), originally developed in 2000, to

S. Manzato (*) • B. Peeters • R. Van der Vorst • J. Debille

LMS International, Interleuvenlaan 68, B-3001 Leuven, Belgium




29


which non-linear elements representative of typical space hardware were added specifically for the present research project

[2]. The following non-linearities have been included:

• Dummy Instrument on Top of SMALLSAT: i.e. a non-linearity involving a large mass and mainly revealing itself as input-

level dependent damping for the first lateral modes.

• Instrument Isolation: The main non-linear effect is introduced by the isolation system located at the interface where this

element is connected to the rest of the structure, i.e. the so-called SASSA (Shock Attenuation System for Spacecraft and

Adaptor [3]) modules; see Fig. 3.1 right.• Suspended Actuator: A local non-linearity was added using an actuator dummy suspended by elastomeric elements,

called WEMS (Wheel Elastomer Mounting System). This element acts by varying the frequency at resonance for low-

levels input, while at higher levels it works as a mechanical stop.

The purpose of this test is not to qualify either the SMALLSAT structure or the non-linear devices which have already

been tested in previous sub-system test campaigns, statically and dynamically. The test will consist of two axis, vertical (Z)

Table 3.1 Overview of typical structural non-linearities in spacecraft structures and their consequences for the test [1]

Physical source of non-linearity Effect on dynamic responses Characterisation criteria

Consequences on

qualification process

Weak non-linearity (sliding at

interface)

Variation of damping

factor

Transmissibility variations Potential need for intermediate

run(s)

Small or no effect on

eigenfrequencies

Estimation of qualification levels

possible

Small gap or material nonlinearity Eigenfrequency shift Eigenfrequency variations Intermediate run is required

Large variations of amplitude Close monitoring of qualification

levels is required

Large gap or nonlinear stiffness Large perturbations of dynamic

test runs

Differences between “global”

and “fundamental” levels

Difficult control of excitation

input

High frequency content (“shocks”) Successful qualification is

jeopardized

Fig. 3.1 Smallsat installed on the shaker table at theAstrium testing site at Stevenage,UK (left) and SASSAmodule introducing non-linearities (right)

30 S. Manzato et al.

and lateral (X) direction, sine and random tests with the objectives to identify the effects of non-linear behaviour and to

identify at what level the non-linear effects impact the spacecraft behaviour. Following tests took place:

• Z-direction increasing and decreasing sine sweep 5–100 Hz: low level (0.1 g), intermediate level (0.4 g), intermediate

level (0.6 g), qualification level (1 g). After each level, in addition a low level (0.1 g) test took place allowing comparing

results with initial low-level test and checking integrity of structure. All tests took place at sweep rate of 2 Oct/min;

except for qualification level in which sweep rate of 4 Oct/min was used.

• Z-direction narrowband sine sweep 15–25 Hz at 0.6 g and sweep rate of 0.5 Oct/min for detailed non-linearity assessment.

• Z-direction random 0–2,000 Hz, 0.001 g2/Hz.

• X-direction increasing and decreasing sine sweep 5–100 Hz: low level (0.1 g), intermediate level (0.2 g), intermediate

level (0.4 g), qualification level (0.6 g). After each level, in addition a low level (0.1 g) test took place allowing comparing

results with initial low-level test and checking integrity of structure. All tests took place at sweep rate of 2 Oct/min.

Multiple entities took part in the tests: Astrium Ltd (test facility, mechanical engineering, quality inspection), Astrium

SAS Satellites (mechanical engineering, quality inspection), LMS (driving test facility, data post-processing), University of

Liege (data post-processing for non-linearity assessment), ESA (project sponsor). In [4], simulation data from the same

structure is used to verify non-linearity assessment methods.

3.3 Modal Parameter Estimation

The purpose of the processing is to apply experimental modal analysis estimation techniques to the test data, both for sine

and random excitation and for different level of excitation, with the following objectives:

• Extract the modal parameters using different processing techniques.

• Identify from the processed data at what level the non-linear effects impact the spacecraft.

There is quite some literature available on the topic of using qualification test data for modal parameter estimation. For

instance, F€ullekrug and Sinapius of the German Aerospace Center (DLR) carried out thorough investigations on the problem

of modal parameter identification from base-driven vibration data [5]. If only the outputs are measured, i.e. accelerations at

the interface between the shaker and the structure and accelerations of the structure, the eigenfrequencies, damping ratios

and mode shapes of the fixed interface structure can be obtained. If a Force Measurement Device (FMD) is used to measure

also the six DOF force input between shaker and structure, both the free and fixed interface modes can be obtained from

(multi-axial) base excitation test (Table 3.2).

In [6], the possibilities to integrate both the modal survey and the vibration qualification test are investigated. Among

other things the use of Frequency Response Functions (FRFs), transmissibilities or spectra as primary data in the modal

parameter estimation process is discussed (Table 3.3). In [7], an approach to non-linear Experimental Modal Analysis was

proposed that starts from base-excitation data. Some recent evolutions in the field of modal parameter estimation are

described in [8, 9]. In this section, this new technology will be applied to the data from the bread-board model tests. The

additional challenge, next to the fact that modal parameters need to be extracted from a test which was not designed to be a

modal test, is that the tested structure exhibits non-linearities.

Table 3.2 Relation between measured quantities and interpretation (boundary conditions) of the identified modal parameters

Measured quantities Fixed interface structure Free structure

Interface and structure accelerations (output-only) Eigenfrequencies, damping ratios, mode shapes –

+ Interface forces (+ inputs) + Modal participation factors All modal parameters

Table 3.3 Interpretation of resonances of FRFs, transmissibilities and cross-spectra in terms of structural modes

Type of base excitation

Measurement functions

FRFs Transmissibilities Cross spectra

White noise (flat force spectrum) Free modes Fixed interface modes Free modes

Shaped noise such that the control acceleration has a flat spectrum Free modes Fixed interface modes Fixed interface modes

3 Environmental Testing and Data Analysis for Non-linear Spacecraft Structures: Impact on Virtual Shaker Testing 31

Usually, modal methods require force measurements that are combined with structural response measurements to obtain

Frequency Response Functions. The issue of not having any information about the actual load conditions during the carried

out environmental tests can be solved by properly processing the acquired data. The results, in terms of transmissibilities

(ratio between output and input accelerations), can then be used in standard modal analysis tools to obtain the modal

parameters of the structure. A first method to obtain the transmissibilities is to use the directly available data during the sine

sweep test. When defining the measurement process, four different amplitude estimators can be selected: peak, RMS,

average and harmonic. In order to obtain also phase information, the latter method is here applied. This filter method, which

works completely in the time domain, offers the best estimate of the amplitude of the fundamental frequency and provides

an excellent harmonic rejection. It is also to be used when noise or harmonic acceleration levels should be filtered out as

much as possible.

During the measurement, also the time histories were acquired with an oversampling factor of 5. For most of the

measurement runs, this led to a sampling frequency of 1,600 Hz. The transmissibilities are computed from the time data

using the so-called H1-estimator, which boils down in this case to the ratio of the cross spectrum between response and input

accelerations SyuðoÞ and the power spectrum of the input acceleration SuuðoÞ:

H1ðoÞ ¼ SyuðoÞSuuðoÞ ; SyuðoÞ ¼ 1

Nb

XNb

b¼1

YðbÞðoÞ UðbÞðoÞ� ��

(3.1)

where Nb is the number of blocks and superindex b denotes the block index. The so-called weighted averaged periodogram

(also known as modified Welch’s periodogram) method allows obtaining the cross- and power spectra from time histories. It

is traditionally used in combination with random data, but may also be used in case of sine sweep data. Welch’s method

consists of cutting the time data in possibly overlapping blocks, applying a weighting (e.g. a Hanning window) to reduce

leakage, and then average the product of the Fourier transform of each block UðbÞðoÞ; YðbÞðoÞ with complex conjugated

Fourier transforms; see (3.1). Figure 3.2 compares online and post-processed transmissibilities from the sine control test.

The quality of the transmissibilities obtained by applying the Welch’s method to sine sweep measurement is typically much

higher than those estimated online.

Also in the case of a random test a distinction can be made between online data and post-processed data. Although in both

cases, the transmissibility estimation is the same (H1 estimator), there are some differences: (1) in the online method, the

cross-spectra estimations are based on a combination of linear and exponential averages, whereas typically for the post-

processing of the time data, just linear averages are used and (2) there is a difference in frequency resolution of the obtained

transmissibilities. Indeed, to have a faster control update, a coarse spectral resolution is typically defined in the online

process: faster control update means lower acquisition time means coarser frequency resolution. On the other hand, for

further (modal) processing, a finer frequency resolution may be desired, and therefore, the transmissibilities acquired online

are not optimal. Figure 3.3 compares online and post-processed transmissibilities from the random control test. The post-

processed results have a much finer resolution.

FRF (Harmonic) 211Z: + Z/013Z:+Z Online - upFRF (Harmonic) 211Z: + Z/013Z:+Z Online - downFRF 211Z:+Z/013Z:+Z H1 estimator

−180.00

180.00

0.00

2.20

Pha

seI

Am

plitu

de

80.005.00 Hz

Fig. 3.2 Sine control—

comparison between

transmissibility estimates:

online during run-up, online

during run-down, H1

estimator from post-processed

time data


The PolyMAX modal parameter estimation method is applied to all the available data. Depending on the direction of

application of the load, the load level, the sweep direction and transmissibilities calculation method, modal parameter

estimates may differ. A detailed discussion of all the results can be found in [10]. In this paper, a synthesis of the results is

presented.

An example of a PolyMAX stabilization diagram can be found in Fig. 3.4. Figure 3.5 shows some typical mode shapes

retrieved from the sine control data and Fig. 3.6 shows the agreement between the measured transmissibilities and the

synthesized ones (i.e. obtained from curve-fitting the data with a modal model). In general, the modal parameter estimation

results are of high quality: a clear stabilization diagram is obtained, the mode shape animations look physically sound,

frequency and damping ratio estimates are within (FE) expectations, and the data is well fitted by the modal model.

5.00−70.00

20.00

l dB

500.00Hz

FRF 123Y:-Y/013Z:+Z H1 OnlineFRF 123Y:-Y/013Z:+Z H1 Post-processing

Fig. 3.3 Random control—comparison between transmissibility estimate online versus post-processing of the time data

1.46

232e-35.00 Linear

Hz94.5

43424140393837363534333231302928272625242322212019181716151413121110987

/A

mpl

itude

Fig. 3.4 PolyMAX stabilization diagram for the X-direction 0.4 g sweep data


However, some of the modes that are clearly present at a certain excitation level are not present at another level. This is

probably caused by the following reasons: (1) in some cases the level of excitation is not high enough to properly excite the

mode and (2) due to the presence of non-linearities, as will be explained more in detail in the following, the parameter

estimation method increases the number of identified modes around such a non-linearity in an attempt to better fit the

measured transmissibilities. In Fig. 3.6 (bottom), it can be observed that the data between 20 and 30 Hz is not fitted very well.In this frequency region and for this excitation level (0.6 g in Z-dir), the transmissibility estimates are heavily distorted due

to non-linearity in the data.

Following concluding remarks can be made on the modal parameter estimation:

• The results from the online estimation are of sufficient quality to perform a quick check of the modal properties

immediately after performing a vibration control test on the structure.

• As a general rule, however, it can be observed that the results obtained from post-processing the time histories (i.e. using

the transmissibility H1 estimates) are better than those obtained using the online processing (i.e. transmissibilities

estimated as ratio of harmonic spectra).

• Some differences between the sweep up and down can be immediately observed, but they will be discussed more in detail

in the following section as these differences can be seen as indicators of non linear behavior.

Fig. 3.5 Some typical mode shapes obtained using sine sweep data from both X and Z direction: (left) WEMS mode; (middle) top instrument

mode; (right) global torsion mode

Fig. 3.6 Quality assessment of modal curve fitting: comparison of sum of measured and synthesized transmissibilities from online run-up data:

(top) Z-dir 0.4 g; (bottom) Z-dir 0.6 g. Frequency range from 5 till 100 Hz is shown


3.4 Non-linearity Assessment

After the modal parameters have been extracted, some further processing techniques can be applied to the data in order to

highlight the presence of non-linear behaviour. It can also be useful to understand at which excitation the non-linearities

become apparent. The following processing will be applied to check for non-linearities in the acquired sine sweep data:

• Visual analysis of time histories and spectral maps.

• Comparison between transmissibilities measured at the same point for positive and negative sweep.

• Comparison between transmissibilities at same point for different excitation level.

• Tracking eigenfrequencies identified at different excitation levels.

One of the easiest methods to assess the presence of non-linearities is to analyze the time histories to check for example,

the presence of peaks in the signal when the excitation level increases. Another possibility is to perform a time-frequency

analysis and present the results as a spectrogram: the activation of non-linearities can be recognized from the presence of

higher order harmonics of the excitation frequency or from time intervals in which the whole frequency band is excited. The

example given in the following concerns the second WEMS mode around 20–22 Hz. Non-linear behavior is expected due to

the presence of the elastomeric support and the mechanical stop. In Fig. 3.7, the spectrograms of the control channel at the

shaker–structure interface and of a sensor at the satellite side of the WEMS are represented. Both an intermediate (0.4 g) and

the qualification level are shown. The range of the dB scale of the colormaps is adapted to the excitation as to better highlight

the relative importance of higher-order harmonics.

As can be seen in Fig. 3.7 (top), the non-linearities do not have a major impact on the control channel. However,

occurrence of non-linearity is clear from WEMS sensor (Fig. 3.7, bottom). At the time instant when the excitation reaches

roughly 22 Hz, an horizontal line appears, meaning that at that instant all frequency are excited. In particular, this

phenomenon became clearer as the excitation level increases and is related to the elastomers in the WEMS, which are no

more able to absorb enough energy and impacts are transmitted to the structure. Moreover, in the frequency region

Fig. 3.7 Spectrogram with time-frequency analysis of Z sine sweep data: (top) control accelerometer at shaker table; (bottom) response

accelerometer at WEMS; (left) intermediate level 0.4 g; (right) qualification level 1 g


between 8 and 10 Hz, the higher order harmonics of the excitation frequency are clearly excited (Fig. 3.7, bottom-right).The behavior of the structure excited at 22 Hz can be further investigated by analyzing directly the time response at the

different excitation levels (Fig. 3.8). By looking at the plots, it is immediately clear how, as the excitation level increases,

the response is distorted and, at 0.6 g, sharp peaks appear in the response. Moreover, it can be observed how, starting from

the 0.4 g level, the acceleration are distorted and no more centered around zero. According to [11], such a response can be

related to a bilinear stiffness characteristic of the element with some initial preloading (in this case the weight of the

suspended actuator module).

Another indicator of non-linearity in the system is to check for a “jump” in the spectral response of some degrees of

freedom when sweeping through some resonances with different sweep directions. This effect was for instance detected

when observing the WEMS degrees of freedom during the sine control test in X direction. In Fig. 3.9, the run-up and run-

down transmissibilities for different excitation level are presented, to be able to identify which excitation level activates this

effect. At low excitation the transmissibilities from both directions coincide, whereas with increasing levels, the discrepancy

grows.

Linear behaviour can also be observed when comparing transmissibilities at different excitation level. Indeed, for

linear mechanical structures, scaling the magnitude of the excitation force will result in the same scaling of the responses

and hence, “linear” transmissibilities are independent from the excitation level. However, by observing the response of

sensors located on the WEMS or SASSA module, it is clear that, as the level increases, there is both a shift in the

frequency and the amplitude of peaks corresponding to natural frequencies (Figs. 3.10, 3.11), as the level increases, the

peaks are shifted to lower values of frequencies and amplitudes. Non-linearities are apparent for the modes around 10, 36,

55 and 89 Hz. It can be verified that these frequencies correspond to identified modes related to the top instrument and

SASSA module.

Figure 3.11 represents a sensor location at the WEMS: for most of the modes, increasing the input level corresponds to a

decrease in the frequency and the amplitude. An exception to this rule is the peak at 8 Hz, where, as the level increases, the

peak appear to be shifted at higher frequency: this can be explained by assuming that the elastomeric element supporting the

component has a hardening characteristic and, as the excitation level increases, acts as a mechanical stop.

2.10

67.58 68.65s

69:213Y:+Y 0.1 g69:213Y:+Y 0.4 g69:213Y:+Y 0.6 g

Rea

lg

−2.10

Fig. 3.8 Time histories from WEMS channel around the resonance at 22 Hz for different Z-dir excitation levels

15.00 15.00

5.00 5.00 5.00Hz Hz30.00 30.00

time run-uptime run-down



Hz 30.00

−35.00 −35.00

/ dB

/ dB

15.00

−35.00

/ dB

Fig. 3.9 WEMS transmissibilities of X-dir sine sweep at different excitation levels: (left) 0.2 g; (middle) 0.4 g; (right) 0.6 g. Sweep up and sweepdown directions are compared with each other


Finally, the effect of frequency shifting due to non-linearities can also be visualized by comparing the estimated

eigenfrequencies at different excitation levels. Figure 3.12 shows the frequency values for the same modes at different

excitation levels. It is clear how, for the analyzed modes, the frequency is decreasing as the level of excitation increases,

indicating a non-linear behavior of the local stiffness characteristics.

15.00

11.07

11.07

5.00

36.32 56.45

56.45

FRF 321Y:-Y/013Z:+Z time run- 01ZFRF 321Y:+Y/013Z:+Z time run- 02ZFRF 321Y:+Y/013Z:+Z time run- 04ZFRF 321Y:+Y/013Z:+Z time run- 06Z

88.56

88.56

100.00Hz

36.32

180.00

−180.00

−40.00

/ dBP

hase

Fig. 3.10 Transmissibilities for SASSA channel at different excitation levels in Z direction

70

60

50

40

30

20

10

00.1g

57.988 55.373 55.26554.115 32.895

32.623

31.308

30.428

32.59433.9935.07837.062

0.4g 0.6g

Excitation level

Evolution of natural frequency with load level Evolution of natural frequency with load level

Fre

qu

ency

[H

z]

Fre

qu

ency

[H

z]

Excitation level

1g 0.1g 0.4g 0.6g 1g

33.5

33

32.5

32

31.5

31

30.5

30

29.5

29

Fig. 3.12 Evolution of natural frequency with excitation level. (left) Two modes identified from processing of the Z-dir sine sweep transmissi-

bilities; (right) mode from X-dir sine sweep

20.00

5.00

8.67 20.50 33.92 45.10

FRF 221Y:-Y/013X:-X time run-up 01XFRF 221Y:-Y/013X:-X time run-up 02XFRF 221Y:-Y/013X:-X time run-up 04XFRF 221Y:-Y/013X:-X time run-up 06X

45.1033.9220.508.67

Hz 100.00

180.00

−180.00

Pha

se

−70.00

/ dB

Fig. 3.11 Transmissibilities for WEMS channel at different excitation level in X direction


3.5 Conclusions

In this paper, advancedmodal parameter identification techniques have been applied to the acquired data from the bread-board

model tests performed at the Astrium facility in Stevenage, UK. Usually, modal parameter estimation methods require

measurements of all applied forces (the inputs to the system) in order to compute FrequencyResponse Functions. In the present

case, no interface forces could be measured, and transmissibilities between satellite responses and shaker interface

accelerations have been used as primary data in the identification. For both sine sweep and random excitation, two methods

to compute the transmissibilities are compared: one that is performed online during the control test and the otherwhich is based

on a post-processing of the time histories that have been acquired in parallel with the same measurement system during the

control test. It is observed that the post-processed transmissibilities provide clearer stabilization diagrams and amore accurate

estimation of the parameters. It should be noted however, that the online estimated data are accurate enough to provide a very

fast verification of the behavior of the system immediately after the test is performed. Finally, an overview is given of

methodologies to assess non-linear behavior. By combining the results of the proposed methods, it is possible to clearly

identify a non-linearity, understand some of its properties and identify the level of excitation at which it is activated. In general,

for this processing, sine sweep data are used.

Interesting to mention is that the observed satellite non-linearities presented in this paper can also be predicted using a so-

called “virtual shaker testing” approach, consisting of a coupled electro-mechanical shaker model, a vibration controller

software model, and a structural model with linear sub-components that may be interconnected with non-linear elements.

Betts et al. [12] and Ricci et al. [13] discuss this virtual shaker testing approach and demonstrate the high degree of realism

that can be obtained from such coupled shaker-controller-structure simulations.

References

1. ESA European Space Agency directorate of technical and quality management (2007) Advancement of mechanical verification methods for

non-linear spacecraft structures. Statement of work, reference TEC-MCS/2007/1558/ln/AN, June 20072. Russell AG (2000) Thick skin, faceted, CFRP, monocoque tube structure for smallsats. In: Proceedings of the European conference on

spacecraft structures, materials and mechanical Testing, ESTEC, Noordwijk, 2000

3. Camarasa P, Kirylenko S (2009) Shock attenuation system for spacecraft adaptors (SASSA) final verification. In: Proceedings of

theWworkshop on spacecraft shock environments and verification, ESTEC, Noordwijk, 2009

4. Kerschen G, Soula B, Vergiaud JB, Newerla A (2011) Assessment of non-linear system identification method using the SmallSat spacecraft

structure. In: Proceedings of IMAC 29, Jacksonville, 2011

5. Sinapsius JM (1996) Identification of free and fixed interface normal modes by base excitation. In: Proceedings of IMAC 14, Dearborn, 1996

6. Peeters B, Van der Auweraer H, Guillaume P (2003) Modal survey testing and vibration qualification testing: the integrated approach. J IEST

46:110–118

7. Link M, B€oswald M, Laborde S, Weiland M, Calvi A (2010) An approach to non-linear experimental modal analysis. In: Proceedings of IMAC

28, Jacksonville,2010

8. Peeters B, van der Auweraer H, Guillaume P, Leuridan J (2004) The polyMAX frequency-domain method: a new standard for modal parameter

estimation? Shock Vib 11:395–409

9. LMS International (2011) LMS test lab modal analysis. Leuven, www.lmsintl.com, 2011

10. Manzato S, Peeters B (2010) WP4240: application of advanced experimental modal estimation techniques to test data. Technical report issued

under ESA(2007), LMS International, Leuven, 2010

11. Boeswald M, Link M, Meyer S, Weiland M (2002) Investigation on the non-linear behaviour of a cylindrical bolted casing joint using high

level base excitation test. In: Proceedings of the ISMA 2002, Leuven, 2002

12. Betts JF, Vansant K, Paulson C, Debille J (2008) Smart testing using virtual vibration testing. In: Proceedings of the 24th aerospace testing

seminar (ATS), Manhattan Beach, 2008

13. Ricci S, Peeters B, Fetter R, Boland D, Debille J (2009) Virtual shaker testing for predicting and improving vibration test performance.

In: Proceedings of IMAC 2009, Orlando, 2009


http://www.lmsintl.com

Chapter 4

Using Impact Modulation to Detect Loose Bolts in a Satellite

Janette Jaques and Douglas E. Adams

Abstract Quickly-assembled, on-demand satellites are being developed to meet the needs of responsive space initiatives.

The short testing times and rapid assembly procedures associated with these satellites create the need for an efficient method

to verify the satellite’s structural integrity. In particular, the ability to identify loose bolts within the satellite structure is of

interest. In this work, Impact Modulation is explored as a possible means of detecting loose bolts. A Torque Index metric is

developed which was able to identify the presence of loose bolts within a satellite panel without the use of historical data by

using a dot product analysis to quantify the difference in response amplitudes at the natural frequencies and those at the

sideband frequencies across an array of impact locations.

4.1 Introduction

Today’s satellites are often designed to perform specific tasks and can take months, even years, to develop, assemble, test,

and launch [1]. New efforts are underway to develop modular satellites that can accomplish a wide range of tasks and can be

ready for launch within days of when their need is established [2]. These quickly-built satellites have unique issues due to

time constraints and the variability of their geometry which are not typically associated with traditional satellites. One such

issue is the ability to quickly assure the structural integrity of the satellite after rapid assembly in order to make certain that it

will survive the launch environment.

The goal of this work is to develop a method to diagnose the condition of the bolted joints of the satellite. One requirement

for this method is that the method be insensitive to changes in the geometry of the satellite, because the satellites configuration

of each satellite depends on the requirements of the mission. Methods that require extensive baseline readings are not

applicable because changes in the geometry of the satellite would require the time consuming task of collecting new baseline

data. One method that meets this criteria is Impact Modulation. Impact Modulation (IM) is a nonlinear, vibrations-based

method which uses a combination of low and high frequency excitations to interrogate structures for damage. This work seeks

to use IM testing to identify the presence of loose bolts in a satellite structure without the use of baseline, or historical, data.

Other researchers have also addressed the issue of loose bolt detection using methods that range from laser vibrometry [3]

to wave propagation [4, 5], although the majority of these works depend on detailed analytical models or access to extensive

baseline data sets. A small number of works in the literature develop methods which do not rely on baseline data or an

analytical model to assess the level of torque on the bolts. Milanese et al. [6] developed a method which looked for frequency

content in the measured strain response of a test specimen above the maximum excitation frequency to indicate that the bolt

was loose. A damage index was developed based on probabilistic analysis that was proven effective in identifying loose bolts.

In the method presented by Nichols et al. [7], surrogate baseline data was generated from the response of a structure by using

the iterative amplitude adjusted Fourier transform method (IAAFT). IAAFT operates under the assumption that a healthy

structure is linear and a damaged structure is nonlinear. The linear part of the response is extracted from the full data and

compared to the full data itself. The level of nonlinearity present in the response is used as an indicator of the presence of loose

bolts within the system. Finally, in [8], Amerini and Meo use a technique similar to IM called Vibro-Acoustic Modulation

(VM). For a two-plate, one-bolt structure, they were able to detect torque loss by measuring the difference in the amplitude of

J. Jaques (*) • D.E. Adams

Purdue Center for Systems Integrity, Purdue University, 1500 Kepner Road, IN 47905, Lafayette, USA




39


response at the actuator frequency and the average amplitude of the first sidebands. To quantify the torque loss, they fit a

hyperbolic tangent curve to the data. There are numerous examples in the literature of using IM to detect cracks in various

materials and structures, but up to this point, IM has not been applied to loose bolt detection. The work presented in this paper

demonstrates the applicability of IM to detect loose bolts in a satellite structure.

4.2 Experimental Setup and Test Procedure

Testing was performed on a prototype Plug and Play satellite panel provided by the Air Force Research Laboratory (AFRL)

in Albuquerque, New Mexico. The aluminum panel, shown in Fig 4.1a, is a 31 34inch square and has threaded bolt holes

evenly spaced at 2 in. intervals in both directions. While the top side of the panel is smooth and plate-like, the underside is

quite complex. Figure 4.1b shows that the underside has been designed with channels for cable management and with space

for interior components.

The goal of the experiments that were performed on the satellite panel was to determine if IM could identify loose bolts in

the connection of an external component without reference data. To simulate an external component, a 4� 4� 38in.

aluminum plate with four through holes drilled to match the bolt hole pattern on the panel was machined. This plate was

bolted on to the face of the satellite panel, as shown in Fig. 4.1a. Four 8–32 bolts were used to secure the plate to the panel.

Throughout the experiments presented here, the torques on all four bolts were kept at the same level. Three torque levels

were used: 24, 2, and 1 in �lb. According to data provided by the AFRL, 24 in �lbs is considered proper bolt torque for

external component connections.

The panel was equipped with ten PCB356A32 100 mV/g triaxial accelerometers and a PI P-010.10P piezo stack actuator

whose locations are shown in Fig. 4.2. An 8 �8 grid of impact locations was marked with 4 in. spacing between points.

A PCB 086C01 impact hammer was used to impact the satellite and an Agilent E8408A VXI data acquisition system used to

drive an actuator and collect sensor data. In addition a power amplifier was added to the setup to increase the amplitude range

of the actuator. Four bolts that were screwed into bolt holes at each corner on the underside of the panel provided support to

the panel during testing, which created pin-like boundary conditions. IM was performed by impacting the panel at each

impact location while simultaneously exciting the panel with a high frequency (7,500 Hz) signal produced by the actuator.

Acceleration time histories for each of the 64 impacts were collected from each of the ten sensors. The time data was

windowed using a Tukey window with a ratio of tapered section to constant section of 0.5. The windowed data was then

transformed into the frequency domain for analysis via the Discrete Fourier Transform algorithm in MATLAB. In the

frequency domain, the results of the interaction between the modal response, or the response due to the impact, and the high

frequency response, or the response due to the actuator input, are response peaks at frequencies which are linear

combinations of the actuator frequency and the natural frequencies. These peaks are called sidebands. The analysis

procedure which is presented below is based on the amplitudes of the sidebands in the response spectra.

The first step in the analysis was to pick out the mode of vibration to analyze. After analyzing the response specrta (not

shown here) from initial IM tests at several impact locations, it was determined that the 72 Hz mode would be used for the

analysis because of its strong response at both the low frequency and the corresponding sideband frequencies. The sidebands

Fig. 4.1 Satellite panel. (a) Top view of the satellite panel. (b) Underside of the satellite panel

40 J. Jaques and D.E. Adams

that correspond to the 72 Hz mode occur at 7,428 Hz (7,500–72 Hz) and at 7,572 Hz (7,500 + 72 Hz). Next, the amplitudes

of the response at 72, 7,428, and 7,572 Hz were recorded for each of the 64 IM tests (one test per impact location). Figure 4.3

shows the shapes that result when these amplitudes are plotted as a function of impact location for the case when the bolts are

tightened to 24 in �lbs. Qualitatively, the shapes appear to correlate very well. During initial testing, it was noted that when

the bolts were loosened, the shape of the response at the sideband frequencies changed dramatically while the shape of the

response at the natural frequency showed little change. Based on this observation, a metric was developed to quantify the

correlation between the shape of the response at the natural frequency and the shapes at the sideband frequencies. Strong

correlation between the shapes indicates that the bolts are tight. Weak correlation between the shapes indicates that loose

bolts are present. The metric, called the Torque Index, (TI) is calculated by averaging the sum of the dot products of the

derivatives (slopes) of each row and column of the response matrices as follows:

TI ¼ TIL þ TIR2

where

TIL ¼XNrow

ii¼1

ðMrowiiÞslope � ðSBLrowii

ÞslopekðMrowii

ÞslopekkðSBLrowiiÞslopek

þXNcol

jj¼1

ðMcoljjÞslope � ðSBLcoljjÞslopekðMcoljjÞslopekkðSBLcoljjÞslopek

" #=ðNrow þ NcolÞ (4.1)

where M is the amplitude of the response at the natural frequency, SBL is the amplitude of the response at the left sideband

frequency, and Nrow and Ncol are the number of rows and columns in the response matrices. TIR is calculated using SBR, theamplitude of response at the right sideband frequency, in place of SBL. As shown in [9], the dot product of the slope of two

x

xxxxxxxx

xxxxxxxxxxxxxxxx

xxxxxxxx

xxxxxxxx

xxxxxxx xxxxx

xxxxxxxx

xxxxxxxx1 2 3 4 5 6 7 8

9 10 12 13 14 15 16

17

25

33

41

49

57

= Impact Location

= Sensor Location

= Actuator Location

= Component Location

##

11

18 21 22 23 24

26 28 29 30 31 3227

34 36 37 38 39 4035

42 44 45 46 47 4843

50 52 53 54 55 5651

58 60 61 62 63 6459

2019

Impact Amplitude: 600-675 N Actuator Frequency: 7,500 HzActuator Amplitude: 15 V Boundary Conditions: PinnedBolt Torque: Case All Bolts

24 24 in·lbs2 2 in·lbs1 1 in·lb

Fig. 4.2 Setup and parameters for the satellite panel IM testing

Response atNatural Frequency

Response atMode Sideband Frequencies

1

Fig. 4.3 Scaled response amplitudes versus impact location for IM tests for Case 24

4 Using Impact Modulation to Detect Loose Bolts in a Satellite 41

mode shapes better accentuates differences in the shapes as compared to using the shapes themselves. Note that a normalized

dot product is a measure of the orthogonality of two vectors and can take on a value between 0 and 1. A value of 1 indicates

that the vectors are very well correlated. A value of 0 indicates that the vectors have no correlation. Therefore, a TI valuenear 1 indicates thatMslope and SBLslope and SBRslope are nearly identical, indicating that all the bolts in the structure are tight.

A lower value would indicate that one or more of the bolts is loose.

After performing an IM test at each of the 64 impact locations, TIwas calculated. This procedure was repeated for each ofthe three torque cases.

4.2.1 Results

Figure 4.4 shows TI for the three torque cases. The values for Case 1 (0.764), Case 2 (0.826), and Case 24 (0.924) show the

expected trend of a decrease in the TI value with a decrease in torque. In addition, the value of TI for Case 24 can be

considered close to 1, the maximum possible TI value. The other TI values can be considered much less than 1. These

distinctions indicate the possibility of using IM to detect loose bolts in a structure without the use of historical data, because

the upper threshold for TI is independent of the structure being evaluated.

4.3 Conclusions

In this work, the effectiveness of using IM to detect loose bolts on a satellite structure was demonstrated. A metric called the

Torque Index, TI, was defined to quantify the difference between the shape of the response amplitudes across impact

locations at a natural frequency and the shapes at the corresponding sideband frequencies. An important characteristic of TIis that its value is limited to the range between zero and one, eliminating the need for a historical reference data. It was shown

that the value of TI was relatively close to one when all the bolts within the satellite structure were tight. The TI valuedropped over 10% when four connection bolts were loosened.

Case

Tor

que

Inde

x(T

I)

24 in·lb 2 in·lb 1 in·lb0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 4.4 Torque index for each of the three torque cases

42 J. Jaques and D.E. Adams

References

1. Arritt BJ, Buckley SJ, Ganley JM, Welsh JS, Henderson BK, Lyall ME, Williams AD, Prebble JC, DiPalma J, Mehle G, Roopnarine R (2008)

Development of a satellite structural architecture for operationally responsive space. In: Proceedings of the international society for optical

engineering (SPIE), vol 6930, 2008

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biological systems 2010, 7650(1):76500E, San Diego, 2010

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4 Using Impact Modulation to Detect Loose Bolts in a Satellite 43

Chapter 5

Nonlinear Modal Analysis of the Smallsat Spacecraft

L. Renson, G. Kerschen, and A. Newerla

Abstract Non-linear elements are present in practically all spacecraft structures. The assumption of a (quasi-)linear

structure is nevertheless adequate for structural analyses and design verification purposes in those cases where these

structural non-linearities are relatively weak or not substantially activated by the mechanical environments encountered

during the launch or during ground testing. However, when significant non-linear effects in spacecraft structures are no

longer negligible then linear modal analysis will not be able to handle non-linear dynamical phenomena in an adequate

manner: the development of a non-linear analogue of linear modal analysis becomes an urgent and important issue. The

objective of this paper is to show that nonlinear normal modes (NNMs) represent a useful and practical tool in this context.

A full-scale spacecraft structure is considered and is modeled using the finite element method. Its NNMs are computed using

advanced numerical algorithms, and the resulting dynamics is then carefully analyzed. Nonlinear phenomena with no linear

counterpart including nonlinear modal interactions are also highlighted.

Keywords Nonlinear dynamics • Modal analysis • Nonlinear normal modes • Space structure

5.1 Introduction

Spacecraft structures are subjected to severe dynamic environments during the launch phase. In order to ensure the structural

integrity of the spacecraft (SC) and the payload (PL) items minimum frequency requirements are usually defined for the SC

in order to avoid dynamic coupling between the main frequency ranges of the launch vehicle (LV) excitations and the SC

fundamental linear normal modes (LNMs), i.e., those modes where the effective modal masses are important. The launcher

authority might however accept a non-compliance with the requirements in those cases where the SC eigenmode represents a

“localized” dynamic effect with only a small effective mass involved.

From a linear point of view, these requirements avoid potentially disastrous coupling and energy exchanges between the

LV and the SC. However, for nonlinear structures, this article will show on a representative SC structure developed by

EADS Astrium, the SmallSat, that these requirements might not be sufficient. In particular, the excitation of global SC

structure modes and PL modes involving local nonlinear effects is presented.

The paper is organized as follows. In Sect. 5.2, a brief review of nonlinear normal modes (NNMs) is achieved. In Sect. 5.3

the SC structure and its finite element model are described. Themodeling of the nonlinearities is also discussed. In Sect. 5.4, a

linear modal analysis of the SC is performed and employed as an introduction to the nonlinear modal analysis of Sect. 5.6.

Section 5.5 shortly presents the algorithm used for the computation of nonlinear normal modes. Finally, Sect. 5.6 presents and

discuss different NNMs of the structure.

L. Renson • G. Kerschen

Space Structures and Systems Laboratory (S3L), Structural Dynamics Research Group, Department of Aerospace

and Mechanical Engineering, University of Liege, Liege, Belgium


A. Newerla

European Space Agency (ESTEC), Noordwijk, The Netherlands



45


5.2 Review of Normal Modes for Nonlinear Systems

A detailed description of NNMs and of their fundamental properties (e.g., frequency-energy dependence, bifurcations, and

stability) is given in [1, 2] and is beyond the scope of this paper. For completeness, the main definition of a conservative

NNM is briefly reviewed in this section.

The free response of discrete mechanical systemswithN degrees of freedom (DOFs) is considered, assuming that continuous

systems (e.g., beams, shells, or plates) have been spatially discretized using the FE method. The equations of motion are

M €xðtÞ þKxðtÞ þ fnl xðtÞf g ¼ 0 (5.1)

where M and K are the mass and stiffness matrices, respectively; x and €x are the displacement and acceleration vectors,

respectively; fnl is the nonlinear restoring force vector.

Targeting a straightforward nonlinear extension of the concept of LNMs, Rosenberg defined an NNM motion as a

synchronous periodic oscillation. This definition requires that all material points of the system reach their extreme values

and pass through zero simultaneously and allows all displacements to be expressed in terms of a single reference

displacement. At first glance, Rosenberg’s definition may appear restrictive in two cases:

1. In the presence of internal resonances, an NNM motion is no longer synchronous, but it is still periodic. This is why an

extended definition was considered in [2, 3]; an NNM motion was defined as a (non-necessarily synchronous) periodicmotion of the undamped mechanical system.

2. The definition cannot be easily extended to nonconservative systems. However, as shown in [2], the damped dynamics

can be interpreted based on the topological structure of the NNMs of the underlying conservative system, provided that

damping has a purely parasitic effect.

For illustration, the system depicted in Fig. 5.1 and governed by the equations

€x1 þ ð2x1 � x2Þ þ 0:5 x31 ¼ 0

€x2 þ ð2x2 � x1Þ ¼ 0 (5.2)

is considered. The NNMs corresponding to in-phase and out-of-phase motions are represented in the frequency-energy plot

(FEP) of Fig. 5.2. An NNM is represented by a point in the FEP, which is drawn at a frequency corresponding to the minimal

period of the periodic motion and at an energy equal to the conserved total energy during the motion. A branch, represented

by a solid line, is a family of NNM motions possessing the same qualitative features (e.g., in-phase NNM motion).

5.3 The SmallSat Spacecraft and Its Finite Element Modelling

The SmallSat structure has been conceived as a low cost structure for small low-earth orbit satellite [4]. It is a monocoque

tube structure which is 1.2 m long and 1 m large. It incorporates eight flat faces for equipment mounting purposes, creating

an octagon shape, as shown in Fig. 5.3a. The octagon is manufactured using carbon fibre reinforced plastic by means of a

filament winding process. The structure thickness is 4.0 mm with an additional 0.25 mm thick skin of Kevlar applied to both

the inside and outside surfaces to provide protection against debris. The interface between the spacecraft and launch vehicle

is achieved through four aluminium brackets located around cut-outs at the base of the structure. The total mass including the

interface brackets is around 64 kg.

The SmallSat structure supports a telescope dummy composed of two stages of base-plates and struts supporting various

concentrated masses; its mass is around 140 kg. The telescope dummy plate is connected to the SmallSat top floor via three

shock attenuators, termed SASSA (Shock Attenuation System for Spacecraft and Adaptator) [5], the behaviour of which is

1 1

1 1 1

0.5 x1 x2

Fig. 5.1 Schematic

representation of the 2DOF

system example

46 L. Renson et al.

considered as linear in the present study. The top floor is a 1 square meter sandwich aluminium panel, with 25 mm core and

1 mm skins. Finally, as shown in Fig. 5.3c, a support bracket connects to one of the eight walls the so-called Wheel

Elastomer Mounting System (WEMS) device which is loaded with an 8 kg reaction wheel dummy. The purpose of this

device is to isolate the spacecraft structure from disturbances coming from reaction wheels through the presence of a soft

interface between the fixed and mobile parts. In addition, mechanical stops limit the axial and lateral motion of the WEMS

mobile part during launch, which gives rise to nonlinear dynamic phenomena. Figure 5.3d depicts the WEMS overall

geometry, but details are not disclosed for confidentiality reasons.

The Finite Element (FE) model in Fig. 5.3b was created in Samcef software and is used in the present study to conduct

numerical experiments. The comparison with experimental measurements revealed the good predictive capability of this

model. The WEMS mobile part (the inertia wheel and its cross-shaped support) was modeled as a flexible body, which is

connected to the WEMS fixed part (the bracket and, by extension, the spacecraft itself) through four nonlinear connections,

labeled NC 1–4 in Fig. 5.3d. Black squares signal such connections. Each nonlinear connection possesses:

• A nonlinear spring (elastomer in traction plus 2 stops) in the axial direction,

• A nonlinear spring (elastomer in shear plus 1 stop) in the radial direction,

• A linear spring (elastomer in shear) in the third direction.

The spring characteristics (piecewise linear) are listed in Table 5.1 and are displayed in Fig. 5.3e. We stress the presence

of two stops at each end of the cross in the axial direction. This explains the corresponding symmetric bilinear stiffness

curve. In the radial direction, a single stop is enough to limit the motion of the wheel. For example, its +x motion is

constrained by the lateral stop number 2 while the connection 1 limits the opposite -x motion. The corresponding stiffness

curves are consequently asymmetric.

5.3.1 Nonlinearities Modeling

From a computational standpoint, the use of piecewise linear stiffnesses requires special numerical treatments, which

increase the computational burden and complicate convergence processes. Therefore, piecewise behaviors are usually

regularized.

One can avoid the introduction of piecewise-linear stiffnesses in replacing themby polynomials (e.g., using a single nonlinear

term as in (5.3)). Despite its simplicity, this approach has the disadvantage to consider a nonlinear behavior from the origin.

Fr ¼ klinxþ knlxn (5.3)

10−5 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Energy (J, log scale)

Fre

quen

cy(H

z)

Fig. 5.2 Frequency-energy

plot of system (5.2).

NNM motions depicted

in the configuration space

are inset

5 Nonlinear Modal Analysis of the Smallsat Spacecraft 47

Axial nonlinearity

In-plane nonlinearities

NC 1 (-x)

NC 2 (+x)

NC 3 (-y)

NC 4 (+y)

Inertiawheel

SmallSat

Inertia wheel

Bracket

Metalliccross

Filteringelastomer plot

Metallicstop

a b

c d

e

Fig. 5.3 SmallSat structure. (a) real structure without the WEMS module; (b) finite element model; (c) WEMS module mounted on a bracket and

supporting a dummy inertia wheel; (d) close-up of the WEMS mobile part (NC stands for nonlinear connection) and (e) graphical display of the

nonlinear restoring forces

48 L. Renson et al.

In order to avoid such approximation, the regularization technique developed in this paper uses Hermite polynomials to

smooth the transition between both linear stiffnesses. A regularization area a� D; aþ D½ � is considered where a is the

transition point between linear regimes and where 2D is the size of the regularization area (Fig. 5.4). This approach has the

advantage to keep the restoring force behavior purely linear out of the regularization area (Fig. 5.4).

The nonlinear force is now given by (5.4) and displayed in Fig. 5.4.

f nlðxÞ ¼signðxÞðk1aþ k2ð xj j � aÞÞ xj j � aþ D

p�ðtðxÞÞ aþ D> xj j>a� D

k1x a� D � xj j � 0

8>>><>>>:

(5.4)

where t(x) is defined by (5.5) and is used in the definition of the Hermite interpolation polynomials p � (t(x)) (5.6).

tðxÞ ¼ x� xkxkþ1 � xk

(5.5)

p�ðtÞ ¼ h00ðtÞpk þ h10ðtÞðxkþ1 � xkÞmk þ h01ðtÞpkþ1 þ h11ðtÞðxkþ1 � xkÞmkþ1 (5.6)

where pk and pk + 1 are the values of the restoring force at the points xk and xk + 1. xk ¼ signðxÞða� DÞ and

xkþ1 ¼ signðxÞðaþ DÞ. mk and mk + 1 are the values of the derivatives at the same points. For the piecewise-linear stiffness,

mk ¼ k1 and mkþ1 ¼ k2. The hij(t) are given by (5.7)–(5.10).

h00ðtÞ ¼ 2t3 � 3t2 þ 1 (5.7)

h10ðtÞ ¼ t3 � 2t2 þ t (5.8)

h01ðtÞ ¼ �2t3 þ 3t2 (5.9)

h11ðtÞ ¼ t3 � t2 (5.10)

Table 5.1 Nonlinear spring characteristics (adimensional values for confidentiality)

Spring Clearance Stiffness of the elastomer plot

Stiffness of the

mechanical stop

Axial caxial ¼ 1 1 13.2

Lateral cradial ¼ 1. 27 0.26 5.24

Displacement Displacement

Res

toring

for

ce

Res

toring

for

ce

2Δ

a b

2Δ

Fig. 5.4 Piecewise-linear stiffness (�) and regularized stiffness (��). (a) Restoring force overview. (b) Zoom on regularization areas


5.4 Linear Modal Analysis

Before the computation of NNMs (Sect. 5.6), a linear modal analysis is performed. The frequency range of interest for this

study is 0–100 Hz. Table 5.2 presents the resonance frequencies of the LNMs of the structure included in this frequency

range. Among these LNMs, four are considered of particular interest for the nonlinear modal analysis of this paper. The first

one is the first mode (Fig. 5.5a) which presents a local WEMS motion with a concave trajectory about the axis A (Fig. 5.6).

Table 5.2 Summary of the

linear normal mode frequencies– Frequency (Hz)

Mode 1 10.66

Mode 2 11.00

Mode 3 28.12

Mode 4 28.38

Mode 5 30.18

Mode 6 30.50

Mode 7 31.60

Mode 8 32.65

Mode 9 37.42

Mode 10 38.26

Mode 11 43.30

Mode 12 52.59

Mode 13 71.36

Mode 14 75.73

Mode 15 80.44

Mode 16 84.35

Mode 17 90.52

Mode 18 95.41

Mode 19 101.56

Mode 20 101.93

Fig. 5.5 First (a), third (b), seventh (c), and ninth (d) LNM modal shapes of the SmallSat

50 L. Renson et al.

The second mode considered is the third LNM (Fig. 5.5b) which again presents a local WEMS motion with convexe

trajectory about A. The seventh LNM involving a SASSAmode with alternate compression of springs 1, 2, and 3 (Fig. 5.7) is

considered. This mode also includes a vertical motion of the WEMS. Finally, the ninth mode which presents a local SASSA

mode without WEMS motion is considered.

5.4.1 Reduced-Order Model

As presented in Sect. 5.3, the finite element model contains more than 65,000 dofs. It appears that the computation of the

NNMs for such a large number of dofs is not currently feasible in a reasonable amount of time. Therefore, a reduced-order

model (ROM) was created using the Craig-Bampton technique [6].

This method consists in describing the system in terms of some retained DOFs and internal vibration modes. By

partitioning the complete system in terms of nR remaining xR and nC ¼ n� nR condensed xC DOFs, the n governing

equations of motion of the global finite element model are written as

A

B

Fig. 5.6 WEMS local axis

SASSA 2

SASSA 1

SASSA 3

D

C

Fig. 5.7 SASSA local axis


MRR MRC

MCR MCC

� �€xR€xC

� �þ KRR KRC

KCR KCC

� �xRxC

� �¼ gR

0

� �(5.11)

The Craig-Bampton method expresses the complete set of initial DOFs in terms of: (1) the remaining DOFs through the static

modes (resulting from unit displacements on the remaining DOFs) and (2) a certain number m < nC of internal vibration

modes (relating to the primary structure fixed on the remaining nodes). Mathematically, the reduction is described by relation

xRxC

� �¼ I 0

�K�1CCKCR Fm

� �xRy

� �¼ R

xRy

� �(5.12)

which defines the n �(nR + m) reduction matrix R. y are the modal coordinates of the m internal linear normal modes

collected in the nC �m matrix Fm ¼ [f(1). . .f(m)]. These modes are solutions of the linear eigenvalue problem

corresponding to the system fixed on the remaining nodes

KCC � o2ðjÞMCC

� �fðjÞ ¼ 0 (5.13)

The reduced model is thus defined by the ðnR þ mÞ � ðnR þ mÞ reduced stiffness and mass matrices given by

M ¼ R�MR

K ¼ R�KR (5.14)

where star denotes the transpose operation. After reduction, the system configuration is expressed in terms of the reduced

coordinates (i.e., the remaining DOFs and the modal coordinates). The initial DOFs of the complete model are then

determined by means of the reduction matrix using relation (5.12).

Table 5.3 summarizes the features of the different ROMs investigated. The eight nodes involved in the superelement

definition are the minimum ones required to define the different nonlinearities of the WEMS.

Before proceeding to nonlinear analysis, the accuracy of the reduced-order linear model is assessed. To this end, the linear

normal modes of the initial complete finite element model are compared to those predicted by the reduced model. The

deviation between the mode shapes of the original model x(o) and of the reduced model x(r) is determined using the Modal

Assurance Criterion (MAC)

MAC ¼x�ðoÞxðrÞ�� 2

x�ðoÞxðoÞ�� x�ðrÞxðrÞ�� (5.15)

MAC values range from 0 in case of no correlation to 1 for a complete coincidence. The minimum correlation criteria are a

maximum relative error on frequencies of 1% andMAC values above 0. 9 in the frequency range 0–200 Hz. Both criteria are

displayed for each ROM in Fig. 5.8 while Table 5.4 presents the frequency range accuratly covered by the different models.

ROMs with 100 and 500 internal modes both satisfy accuracy requirements. However the selection of the appropriate

ROM for Sect. 5.6 is not trivial. A ROM with numerous internal modes provides the best chances to observe the modal

interactions but this larger model increases the computation time. In addition, the number of modal interactions observed

tends to become prohibitive and dramatically increases the computation time too. Therefore, in the following parts of this

study, the reference ROM used for the computation of NNMs is “ROM85”.

Table 5.3 Features of the

different reduced-order

models created

Model Nodes Internal modes

ROM84 8 50

ROM85 8 100

ROM86 8 500

52 L. Renson et al.

5.5 Numerical Computation of NNMs

The numerical method proposed here for the NNM computation relies on two main techniques, namely a shooting technique

and the pseudo-arclength continuation method. A detailed description of the algorithm is given in [7].

5.5.1 Shooting Method

The equations of motion of system (5.1) can be recast into state space form

_z ¼ gðzÞ (5.16)

where z ¼ x� _x�½ �� is the 2n-dimensional state vector, and star denotes the transpose operation, and

gðzÞ ¼ _x

�M�1 Kxþ fnlðx; _xÞ½ �

� �(5.17)

is the vector field. The solution of this dynamical system for initial conditions zð0Þ ¼ z0 ¼ x�0 _x�0 �

is written as z(t) ¼z(t, z0) in order to exhibit the dependence on the initial conditions, z(0, z0) ¼ z0. A solution zp(t, zp0) is a periodic solutionof the autonomous system (5.16) if zpðt; zp0Þ ¼ zpðtþ T; zp0Þ, where T is the minimal period.


different ROMs performancesModel Valid modes Frequency range covered

ROM84 1–18 0–95.4 Hz

ROM85 1–50 0–248.2 Hz

ROM86 1–246 0–1020.8 Hz

0 50 100 150 200 250 300 350 400 450 500 5500

1

2

3

4

5

Mode number [−]

Rel

.er

ror

onfr

eq.

[%]

0 50 100 150 200 250 300 350 400 450 500 5500

0.5

1

Mode number [−]

MA

C

Fig. 5.8 Relative error on frequencies and MAC for the different ROMs investigated


The NNM computation is carried out by finding the periodic solutions of the governing nonlinear equations of motion

(5.16). In this context, the shooting method is probably the most popular numerical technique. It solves numerically the two-

point boundary-value problem defined by the periodicity condition

Hðzp0; TÞ � zpðT; zp0Þ � zp0 ¼ 0 (5.18)

Hðz0; TÞ ¼ zðT; z0Þ � z0 is called the shooting function and represents the difference between the initial conditions and the

system response at time T. Unlike forced motion, the period T of the free response is not known a priori.

The shooting method consists in finding, in an iterative way, the initial conditions zp0 and the period T that realize a

periodic motion. To this end, the method relies on direct numerical time integration and on the Newton-Raphson algorithm.

Starting from some assumed initial conditions zp0(0), the motion zp

(0)(t, zp0(0)) at the assumed period T (0) can be obtained

by numerical time integration methods (e.g., Runge-Kutta or Newmark schemes). In general, the initial guess (zp0(0), T (0))

does not satisfy the periodicity condition (5.18). A Newton-Raphson iteration scheme is therefore to be used to correct an

initial guess and to converge to the actual solution. The corrections Dzp0(k) and DT (k) at iteration k are found by expanding the

nonlinear function

H zðkÞp0 þ DzðkÞp0 ; T

ðkÞ þ DTðkÞ� �

¼ 0 (5.19)

in Taylor series and neglecting higher-order terms (H.O.T.).

The phase of the periodic solutions is not fixed. If z(t) is a solution of the autonomous system (5.16), then z(t + Dt) isgeometrically the same solution in state space for any Dt . Hence, an additional condition, termed the phase condition, has tobe specified in order to remove the arbitrariness of the initial conditions. This is discussed in detail in [7].

In summary, an isolated NNM is computed by solving the augmented two-point boundary-value problem defined by

Fðzp0; TÞ �Hðzp0; TÞ ¼ 0

hðzp0Þ ¼ 0

((5.20)

where h(zp0) ¼ 0 is the phase condition.

5.5.2 Continuation of Periodic Solutions

Due to the frequency-energy dependence, the modal parameters of an NNM vary with the total energy. An NNM family,

governed by (5.20), therefore traces a curve, termed an NNM branch, in the (2n + 1)-dimensional space of initial conditions

and period (zp0, T). Starting from the corresponding LNM at low energy, the computation is carried out by finding

successive points (zp0, T) of the NNM branch using methods for the numerical continuation of periodic motions (also

called path-following methods) [8, 9]. The space (zp0, T) is termed the continuation space.

Different methods for numerical continuation have been proposed in the literature. The so-called pseudo-arclength

continuation method is used herein.

Starting from a known solution (zp0, (j), T(j)), the next periodic solution ðzp0;ðjþ1Þ; Tðjþ1ÞÞ on the branch is computed using a

predictor step and a corrector step.

5.5.2.1 Predictor Step

At step j, a prediction ð~zp0;ðjþ1Þ; ~Tðjþ1ÞÞ of the next solution ðzp0;ðjþ1Þ; Tðjþ1ÞÞ is generated along the tangent vector to the

branch at the current point zp0, (j)

~zp0;ðjþ1Þ~Tðjþ1Þ

� �¼ zp0;ðjÞ

TðjÞ

� �þ sðjÞ

pz;ðjÞpT;ðjÞ

� �(5.21)

54 L. Renson et al.

where s(j) is the predictor stepsize. The tangent vector p(j) ¼[pz, (j)∗ pT, (j)]

∗ to the branch defined by (5.20) is solution of the

system

@H

@zp0

��ðzp0;ðjÞ;TðjÞÞ

@H

@T


@h

@zp0

��ðzp0;ðjÞÞ

0

266664

377775

pz;ðjÞpT;ðjÞ

" #¼ 0

0

" #(5.22)

with the condition pðjÞ�� ¼ 1. The star denotes the transpose operator. This normalization can be taken into account by

fixing one component of the tangent vector and solving the resulting overdetermined system using the Moore-Penrose matrix

inverse; the tangent vector is then normalized to 1.

5.5.2.2 Corrector Step

The prediction is corrected by a shooting procedure in order to solve (5.20) in which the variations of the initial conditions

and the period are forced to be orthogonal to the predictor step. At iteration k, the corrections

zðkþ1Þp0;ðjþ1Þ ¼ z

ðkÞp0;ðjþ1Þ þ DzðkÞp0;ðjþ1Þ

Tðkþ1Þðjþ1Þ ¼ T

ðkÞðjþ1Þ þ DTðkÞ

ðjþ1Þ (5.23)

are computed by solving the overdetermined linear system using the Moore-Penrose matrix inverse

@H@zp0

��ðzðkÞ

p0;ðjþ1Þ;TðkÞðjþ1ÞÞ

@H@T

��ðzðkÞ


@h@zp0

��ðzðkÞ

p0;ðjþ1ÞÞ0

p�z;ðjÞ pT;ðjÞ

2666664

3777775

DzðkÞp0;ðjþ1Þ

DTðkÞðjþ1Þ

264

375 ¼

�HðzðkÞp0;ðjþ1Þ;TðkÞðjþ1ÞÞ

�hðzðkÞp0;ðjþ1ÞÞ

0

266664

377775 (5.24)

where the prediction is used as initial guess, i.e., zð0Þp0;ðjþ1Þ ¼ ~zp0;ðjþ1Þ and T

ð0Þðjþ1Þ ¼ ~Tðjþ1Þ. The last equation in (5.24)

corresponds to the orthogonality condition for the corrector step.

This iterative process is carried out until convergence is achieved. The convergence test is based on the relative error of

the periodicity condition:

Hðzp0; TÞ��

zp0�� ¼ zpðT; zp0Þ � zp0

�� zp0

�� <E (5.25)

where e is the prescribed relative precision.

5.5.3 Sensitivity Analysis

Each shooting iteration involves the time integration of the equations of motion to evaluate the current shooting residue

H zðkÞp0 ; T

ðkÞ� �

¼ zðkÞp ðTðkÞ; zðkÞp0 Þ � z

ðkÞp0 . As evidenced by (5.24), the method also requires the evaluation of the 2n �2n

Jacobian matrix

@H

@z0z0;Tð Þ ¼ @zðt; z0Þ

@z0

��t¼T

� I (5.26)


where I is the 2n �2n identity matrix.

The classical finite-difference approach requires to perturb successively each of the 2n initial conditions and integrate thenonlinear governing equations of motion. This approximate method therefore relies on extensive numerical simulations and

may be computationally intensive for large-scale finite element models.

Targeting a reduction of the computational cost, a significant improvement is to use sensitivity analysis for determining

∂z(t, z0) / ∂z0 instead of a numerical finite-difference procedure. The sensitivity analysis consists in differentiating the

equations of motion (5.16) with respect to the initial conditions z0 which leads to

d

dt

@z t; z0ð Þ@z0

� �¼ @gðzÞ

@z

��zðt;z0Þ

@zðt; z0Þ@z0

� �(5.27)

with

@zð0; z0Þ@z0

¼ I (5.28)

since z(0, z0) ¼ z0. Hence, the matrix ∂z(t, z0) / ∂z0 at t ¼ T can be obtained by numerically integrating over T the initial-

value problem defined by the linear ordinary differential equations (ODEs) (5.27) with the initial conditions (5.28).

In addition to the integration of the current solution z(t, x0) of (5.16), these two methods for computing ∂z(t, z0) / ∂z0require 2n numerical integrations of 2n-dimensional dynamical systems, which may be computationally intensive for large

systems. However, (5.27) are linear ODEs and their numerical integration is thus less expensive. The numerical cost can be

further reduced if the solution of (5.27) is computed together with the solution of the nonlinear equations of motion in a

single numerical simulation [10].

The sensitivity analysis requires only one additional iteration at each time step of the numerical time integration of the

current motion to provide the Jacobian matrix. The reduction of the computational cost is therefore significant for large-scale

finite element models. In addition, the Jacobian computation by means of the sensitivity analysis is exact. The convergence

troubles regarding the chosen perturbations of the finite-difference method are then avoided. Hence, the use of sensitivity

analysis to perform the shooting procedure represents a meaningful improvement from a computational point of view.

As the monodromy matrix ∂zp(T, zp0) / ∂zp0 is computed, its eigenvalues, the Floquet multipliers, are obtained as a by-

product, and the stability analysis of the NNM motions can be performed in a straightforward manner.

5.5.4 Algorithm for NNM Computation

The algorithm proposed for the computation of NNM motions is a combination of shooting and pseudo-arclength continua-

tion methods, as shown in Fig. 5.9. It has been implemented in the MATLAB environment. Other features of the algorithm

such as the step control, the reduction of the computational burden and the method used for numerical integration of the

equations of motion are discussed in [7].

So far, the NNMs have been considered as branches in the continuation space (zp0, T). An appropriate graphical depictionof the NNMs is to represent them in a frequency-energy plot (FEP). This FEP can be computed in a straightforward

manner: (1) the conserved total energy is computed from the initial conditions realizing the NNM motion; and (2) the

frequency of the NNM motion is calculated directly from the period.

5.6 Nonlinear Modal Analysis

As presented in Sect. 5.3, the nonlinearities are located at WEMS ends. These nonlinearities are activated when

displacements are large enough to hit the mechanical stops. Due to the flexibility of WEMS attachment, large displacements

are observed for the majority of LNMs. However, according to the linear study of Sect. 5.4, the nonlinear investigations are

restricted to four modes.

The energy range of interest for the continuation is determined by the displacements observed at WEMS ends. Indeed,

experimental observation demonstrated that displacements are limited. Therefore, relative displacements larger than the

observed values are not representative of the real physical behavior of the structure.

56 L. Renson et al.

Among the NNMs presented, one can readily distinguish two categories of modes, namely the energy-dependent (e.g.,

Fig. 5.11) and the energy-independent modes (e.g., Fig. 5.10). The latter correspond to the nonlinear extension of linear

modes that do not involve WEMS motion. An example is provided by the ninth mode (Fig. 5.10) where the deformation is

localized at the SASSA. As the energy increases, the modal shape (Fig. 5.5d) presented in the linear study is unchanged and

the mode remains linear.

Figure 5.11 presents the continuation of the first LNM. This LNM correspond to a local WEMS motion and is therefore

sensitive to nonlinearities present in the model. For low energies, the structural behavior remains purely linear and the

resonance frequency does not depend on the energy. However, as energy increases, a steep modification of the frequency-

energy dependence appears. At this transition, one can observe that relative displacements at WEMS ends enter in the

regularization area. Beyond this transition, a plateau appears and interactions between the first LNM and other LNMs are

achieved. The frequency content of the periodic solution evolves with the energy and includes third, fifth, and higher-order

harmonics (up to the 17th order).

Fig. 5.9 Algorithm for NNM

computation


Figure 5.12 displays the frequency-energy dependence of the third NNM. It highlights the presence of a tongue, revealing

the existence of an 3:1 internal resonance between the third (Fig. 5.5b) and the sixteenth LNM (Fig. 5.13b). The latter

corresponds to an octagonal structure panels mode. Along the tongue (e.g., point b in Fig. 5.12), the modal shape evolves

from the third to the sixteenth LNM and is therefore a special combination of both modes (Fig. 5.13a). This mode has no

linear counterpart and highlights the possibility of interactions between local and global SC modes due to the presence of

nonlinearities.

The frequency-energy dependence of the seventh NNM is presented in Fig. 5.14. The presence of a tongue again

highlights modal interactions between the seventh LNM (Fig. 5.5c) and a higher-order LNM. Here, it is interesting to

observe that the seventh LNM mainly involves a motion of the SASSA. However, due to the nonlinearities of the WEMS,

nonlinear couplings between this SASSA mode and global structural modes are achieved.

10−2 100 102 10437.41

37.412

37.414

37.416

37.418

37.42

37.422

37.424

37.426

37.428

Energy [J]

Freq

uenc

y[H

z]

Fig. 5.10 Frequency-energy

dependence of the

ninth NNM

10−6 10−4 10−2 100 102 104 10610.65

10.66

10.67

10.68

10.69

10.7

10.71

10.72

10.73

10.74

Energy [J]

Freq

uenc

y[H

z]Fig. 5.11 Frequency-energy

dependence of the

first NNM

58 L. Renson et al.

Fig. 5.13 (a) Modal shape of the third NNM combining the third and the sixteenth LNMs (b in Fig. 5.12). (b) Modal shape at the internal

resonance equivalent to the sixteenth LNM (WEMS remains quiescent)(c in Fig. 5.12)

10−5 10−4 10−3 10−2 10−1 100 10128.1

28.15

28.2

28.25

28.3

28.35

28.4

28.45

28.5

Energy [J]

Freq

uenc

y[H

z]

(a) (b)

(c)

Fig. 5.12 Frequency-energy dependence of the third NNM. (a) Low-energy point. (b) Point in the tongue describing the 3:1 internal resonance.

(c) Bifurcation point

10−3 10−2 10−1 100 101 102 10331.58

31.6

31.62

31.64

31.66

31.68

31.7

31.72

31.74

31.76

Energy [J]

Freq

uenc

y[H

z]

Fig. 5.14 Frequency-energy dependence of the seventh NNM


5.7 Conclusions

In this paper, the fundamental concepts regarding undamped nonlinear normal modes and their numerical computation were

reviewed. A new regularization procedure was presented and revealed to be accurate for the modeling of piecewise linear

restoring forces.

Targeting the computation of the nonlinear modes, a linear modal analysis was presented and some interesting modes

were identified for further investigations. A reduced-order model accurate in the (0–200 Hz) range was employed to reduce

the computational burden.

Finally, the nonlinear normal modes of the spacecraft were presented. Internal resonances highlighted the possibility of

mode interactions between local (WEMS) and global structural modes.

Acknowledgements This paper has been prepared in the framework of the ESA Technology Research Programme study “Advancement of

Mechanical Verification Methods for Non-linear Spacecraft Structures (NOLISS)” (ESA contract No.21359/08/NL/SFe).

The authors would like to thank Dr. Maxime Peeters for all the constructive discussions. The author L. Renson would like to acknowledge the

Belgian National Fund for Scientific Research (FRIA fellowship) for its financial support.

References

1. Vakakis AF, Manevitch LI, Mikhlin YV, Pilipchuk VN, Zevin AA (1996) Normal modes and localization in nonlinear systems. Wiley,

New York

2. Kerschen G, Peeters M, Golinval JC, Vakakis AF (2009) Nonlinear normal modes, part I: a useful framework for the structural dynamicist.


3. Lee YS, Kerschen G, Vakakis AF, Panagopoulos PN, Bergman LA, McFarland DM (2005) Complicated dynamics of a linear oscillator with a

light, essentially nonlinear attachment. Phy D-Nonlinear Phenom 204(1–2):41–69

4. Russell AG (2000) Thick skin, faceted, CFRP, monocoque tube structure for smallsats. European conference on spacecraft structures,

materials and mechanical testing, Noordwijk

5. Camarasa P, Kiryenko S (2009) Shock attenuation system for spacecraft and adaptor (SASSA). European conference on spacecraft structures,

materials and mechanical testing, Noordwijk

6. Craig R, Bampton M (1968) Coupling of substructures for dynamic analysis. AIAA J 6:1313–1319

7. Peeters M, Viguie R, Serandour G, Kerschen G, Golinval JC (2009) Nonlinear normal modes, part II: toward a practical computation using

numerical continuation techniques. Mech Syst Signal Process 23(1):195–216

8. Seydel R Practical bifurcation and stability analysis, from equilibirum to chaos, 2nd edn. Springer, New York (1994)

9. Nayfeh AH, Balachandran B (1995) Applied nonlinear dynamics: analytical, computational, and experimental methods. Wiley, New York

10. Br€uls O, Eberhard P (2006) Sensitivity analysis for dynamic mechanical systems with finite rotations. Int J Numer Meth Eng 1:1–29

60 L. Renson et al.

Chapter 6

Filter Response to High Frequency Shock Events

Jason R. Foley, Jacob C. Dodson, and Alain L. Beliveau

Abstract A variety of effects can introduce nonlinearities into filter response when measuring shock signals: small- versus

large-signal frequency response, electronic nonlinearities, spurious electrical noise, etc. This paper examines the analytic

response of ideal filters in response to a variety of analytic shock-like signals, including instantaneous pulses of varying rise

rate, frequency content and duration as well as pulse trains with variable timing and duty cycle. The maximum permitted

slew rate, or instantaneous rate, of the shock signal is shown to be a function of the filter type and order. Instantaneous

slew rate is also discussed as an indicator of impulse response, i.e., spectral frequency content that is higher than the filter

cutoff frequency.

Keywords Shock • High frequency • Slew rate • Filter response • Data acquisition • Signal conditioning • Analog filter •

Digital filter • Impulse response

Nomenclature

d Delta function

s Attenuation/neper frequency

t Pulse width/duration

o Angular/radian frequency

F Phase response (frequency domain)

i Imaginary number

s Complex frequency (Laplace variable)

t Time

x Input function (time domain)

G Frequency response or gain function

H Transfer function (frequency domain)

SR Slew rate

T Period or pulse train timing

X Input function (frequency domain)

Y Output function (frequency domain)

J.R. Foley (*)

Air Force Research Laboratory, AFRL/RWMF, 306 W. Eglin Blvd, Bldg. 432, Eglin AFB, FL 32542-5430, USA


J.C. Dodson

Air Force Research Laboratory, AFRL/RWMF, Eglin AFB, FL, USA

A.L. Beliveau

Applied Research Associates, Inc., Niceville, FL, USA



61


6.1 Introduction

Acquisition of valid shock and vibration data requires an in-depth understanding of both the physical system dynamics and

the response of every component of the data acquisition system. For example, sigma-delta (SD) analog-to-digital converters(ADC’s) are commonly used in data acquisition systems for digitization of voltage signals [1, 2]. Signal conditioners are

typically used to isolate, filter, convert (to voltage), and/or amplify the output from sensors [3]. To avoid aliasing and

minimize high frequency noise, low pass filters are routinely included in the systems, either integrated into the signal

conditioners or as standalone components. Additionally, the frequency domain characteristics of cables, interfaces, and other

physical components must be considered in precision measurements. Figure 6.1 shows an idealized and simplified block

diagram in the frequency domain of a typical sensor-to-data-acquisition experiment expanded into the various components.

Highly transient events are common in experimental measurements of the dynamics of structures due to shock and

vibration. These impulsive signals are characterized by step-like high voltage levels with correspondingly large frequency

content which often far exceeds the linear bandwidth of individual electronic components. This has the potential to

subsequently induce nonlinear responses in these components [4]. Understanding and diagnosing any possible nonlinearities

is therefore critical to collecting valid shock data. For example, slew induced distortion [5] is possible if frequency content

and instantaneous rates exceed the linear range of amplifiers in the data acquisition and/or signal conditioning systems. This

paper examines analytic filter models to determine the expected dependence on key parameters, such as maximum slew rate,

of their output due to a variety of inputs. This information is critical for reviewing experimental shock data for validity.

6.2 Theory

In practical shock experimentation, a variety of waveforms can be expected for the physical input (X). An impulse function

[6] is perhaps the most basic descriptor. The analytic form of a finite impulse [7] is

xðtÞ ¼ dðtÞ ¼ 1

2p

ð1�1

eiotdo: (6.1)

Equation (6.1) can be used as a convolution kernel to obtain a variety of other possible input functions; these are

categorized and presented in Table 6.1.

The response yðtÞ of a system due to a time input xðtÞ is defined [8] as the convolution integral

yðtÞ ¼ xðtÞ � hðtÞ ¼ð1�1

x tð Þh t� tð Þdt; (6.2)

where hðtÞ is the impulse response (h t� tð Þ is the time domain response at time t from an instantaneous input at t ¼ t). Otherforms of input, such as a rectangular pulse or step response, can obtained by integrating the impulse response over the

duration of the step.

Physicalinput

X

Sensorresponse

Hs

Cable response

Hc1

Front endresponse

Hf

Cable response

Hc2

Data acq.response

Hd

Digitized outputY

Physicalinput

X

Systemresponse

H

Digitizedoutput

Y

Fig. 6.1 Simplified block diagram of transfer functions involved in practical data acquisition

62 J.R. Foley et al.

Table 6.1 Analytic input functions and nomenclature of waveforms commonly encountered in shock and vibration

Case name

Amplitude

PolaritySmall signal finite Large signal finite Infinite

Impulse

t

x(t)

t0

A

SSF

t

x(t)

t0

A

LSF

t

x(t)

t0

II

t

x(t)

t0

P(ositive) or N(egative)

Impulse train

t

x(t)

t0

A

T

SSFIT

t

x(t)

t0

A

T

LSFIT

t

x(t)

t0

T

IIT

t

x(t)

t0

T

[P(ositive)/N(egative)]1

Pulse

tt0

A

τ

SSFP

tt0

A

τ

LSFP

N/A

t

x(t)t0

A

τ


Pulse train

t

x(t)

t0

A

τ

T

SSFPT

t

x(t)

t0

A

τ

T

LSFPT

NAt

x(t)

t0

-A

τ

AT

[P(ositive)/N(egative)]1

Step

t

x(t)

t0

A

SSFS

t

x(t)

t0

A

LSFS

N/A

t

x(t)t0

-A


Multi-step

t

x(t)

t

A1

A2

t

SSFMS

t

x(t)

A1

A2

t1 t2

LSFMS

N/A

t

x(t)A1

A2

t1 t2

P(ositive)/N(egative)

6 Filter Response to High Frequency Shock Events 63

The slew rate of an electronics component (generally amplifiers) is the maximum rate where the device response will

remain linear. The analytic expression for the instantaneous slew rate of a signal y is simply its instantaneous time rate of

change, i.e.,

dy

dt¼ y

0 ðtÞ: (6.3)

Combining (6.1) through (6.3) while using the convolution property [7] of a derivative of the delta function,

dy

dt¼ d

dtdðtÞ � hðtÞ½ � ¼ d

0 ðtÞ � hðtÞ ¼ð1�1

d0t� tð Þh tð Þdt ¼ h

0 ðtÞ; (6.4)

we find the impulse slew rate is indeed the derivative of the impulse response. The impulse response of two commonly used

low-pass filter types are now considered: Butterworth and Chebyshev.

6.3 Butterworth Filter Response

The nth-order Butterworth filter [9] is given by the frequency domain transfer function [10] as

HðsÞ ¼ 1Qnj¼1 s� sj

� � (6.5)

where sj are the filter poles. The corresponding frequency response (or gain) function for the low-pass Butterworth filter withunity gain in the passband is

G oð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH ioð Þj j2

q¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ ooc

� �2nr (6.6)

and the phase response is

F oð Þ ¼ arg H iooc

� �� : (6.7)

The impulse response of the Butterworth filter is shown in Fig. 6.2a. The pulse response is also shown in Fig. 6.2b for

comparison.

0.010.0102

0.01040.0106

0.0108 0

2

4

6

80

0.2

0.4

0.6

0.8

1

a b

Filter order

Butterworth Filter (fc = 10 kHz) Applied to Finite Impulse

Time [s]

Am

plitu

de

0.010.0102

0.01040.0106

0.0108 0

2

4

6

8

0

0.2

0.4

0.6

0.8

1

Filter order

Butterworth Filter (fc = 10 kHz) Applied to Pulse

Time [s]

Am

plitu

de

Fig. 6.2 (a) Finite impulses and (b) pulses filtered with a Butterworth filter (cutoff at 10 kHz) with number of poles from 1 to 8


The maximum slew rate is then determined by taking the maximum slope of the impulse response, i.e., the

maximum of (6.4). The maximum allowed slew rate for Butterworth filters is plotted versus cutoff frequency and order

in Fig. 6.3.

6.4 Chebyshev (Type I) Filter Response

The Chebyshev (type I) transfer function is given by

HðsÞ ¼ HnðeÞQnj¼1 s� sj

� � (6.8)

where HnðeÞ is a function of the filter order and the passband ripple, e [10]. The impulse and pulse response is shown in

Fig. 6.4a,b, respectively. The cutoff frequency and order dependence is shown in Fig. 6.5a,b for two different ripple values

(3 dB and 0.75 dB).

6.5 Discussion

The maximum allowable slew rate is proportional to the cutoff frequency, i.e., max h0 ðtÞ / oc. This linear dependence

with cutoff frequency is readily seen in Figs. 6.3 and 6.5. The decrease slew rate with respect to filter order is found by using

a Heaviside expansion [8] of the impulse response of a Butterworth filter, given by

hðtÞ ¼ L�1 HðsÞ½ � ¼Xnr¼1

Krest; (6.9)

102 103 104 105 106 107 108102

103

104

105

106

107

108

109

Filter Cutoff Frequency [Hz]

Max

imum

Allo

wed

Sle

w R

ate

(dx/dt

) [s

-1]

Maximum Allowed Slew Rate vs. Filter Cutoff Frequency for Butterworth Filter

0

12

3

4

5

67

8

Filter Order

Fig. 6.3 Maximum allowed slew rate for Butterworth filters of order 1 through 8


where Kr ¼ s� srs� s1ð Þ s� s2ð Þ . . . s� snð Þ

��s¼sr

: Correspondingly, the slew rate is then given by

h0 ðtÞ ¼

Xnr¼1

sKrest: (6.10)

Therefore, as the pole order increases, the coefficients Kr decrease as a power of s2�n. The dependence of the maximum

rate thus is monotonically decreasing versus filter order r, as shown in Fig. 6.6 below.

Together, the dependence of the maximum slew rate on cutoff frequency and filter order establishes an upper limit to

expected outputs from a given filter design. This upper limit is hypothesized to be a criterion for diagnosing both impulse

inputs (input spectral frequency content higher than the filter cutoff frequency) as well as possible nonlinearities in

measurement systems due to high frequency/high amplitude sensor output in shock experiments.

6.6 Future Work

Experiments to validate the linear and nonlinear response of analog filters in data acquisition components are ongoing. These

experiments will have a schematic layout as shown in Fig. 6.7. Oscilloscopes will generally have a>100� higher sample

rate and bandwidth [11] at the expense of decreased vertical resolution when compared with a data acquisition system (see,

for example, Refs. [12, 13]). The scope is used to capture “truth data” on the output waveform in the pulse generator into an

equivalent impedence data so that the maximum rate generated by the pulse generator can be verified.

Future work also includes analyzing “black box” filters as an inverse parametric identification problem. Full system

models will be implemented to model nonlinear responses, such as slew-rate distortion. Additionally, the filter

characteristics will then be used to estimate the impulsive nature of an observed mechanical response.

6.7 Summary

The analytic response of ideal filters was calculated for a variety of analytic signals typical of shock. The maximum

permitted slew rate of the shock signal was shown to be a monotonic function of the filter type and order. Instantaneous slew

rate was proposed as an indicator for impulse response and possible nonlinear filter response.

0.010.0102

0.01040.0106

0.0108 0

2

4

6

80

0.2

0.4

0.6

0.8

1

a b

Filter order

Chebyshev Type I Filter (fc = 10 kHz) Applied to Impulse

Time [s]

Am

plitu

de

0.010.0102

0.01040.0106

0.0108 0

2

4

6

8-0.2

0

0.2

0.4

0.6

0.8

1

Filter order

Chebyshev Type I Filter (fc = 10 kHz) Applied to Pulse

Time [s]

Am

plitu

de

Fig. 6.4 (a) Finite impulses and (b) 10 ms pulses filtered with a Chebyshev type I filter (cutoff at 10 kHz) with number of poles from 1 to 8


102

103

104

105

106

107

108

102

103

104

105

106

107

108

109


Max

imum

Allo

wed

Sle

w R

ate

(dx/dt

) [s

-1]

Maximum Allowed Slew Rate vs. Filter Cutoff Frequency for Chebyshev Type I Filter

0

12

3

4

5

67

8

Filter Order

102

103

104

105

106

107

108

102

103

104

105

106

107

108

109


Max

imum

Allo

wed

Sle

w R

ate

(dx/dt

) [s

-1]

Maximum Allowed Slew Rate vs. Filter Cutoff Frequency for Chebyshev Type I Filter

01

2

3

45

6

78

Filter Order

a

b

Fig. 6.5 Maximum allowed slew rate for Chebyshev type I filters of order 1 through 8 with passband ripple of (a) 3 dB and (b) 0.75 dB


Acknowledgements The authors would like to thank AFOSR (Program Manager: Dr. David Stargel) for supporting this research effort via

research task 09RW01COR. Opinions, interpretations, conclusions, equipment selections, and recommendations are those of the authors and are

not necessarily endorsed by the United States Air Force.

References

1. Boser BE, Wooley BA (1988) The design of sigma-delta modulation analog-to-digital converters. Solid-State Circuit IEEE J 23(6):1298–1308

2. van de Plassche R (1978) A sigma-delta modulator as an A/D converter. Circuit Syst IEEE Trans 25(7):510–514

3. Wilson JS (2005) Sensor technology handbook. Elsevier, Amsterdam

4. Walter PL (1978) Limitations and corrections in measuring dynamic characteristics of structural systems. Sandia National Laboratories

technical report, SAND-78-1015

5. Allen P (1978) A model for slew-induced distortion in single-amplifier active filters. Circuit Syst IEEE Trans 25(8):565–572

6. Walter P, Nelson H (1979) Limitations and corrections in measuring structural dynamics. Exp Mech 19(9):309–316

t

x(t)

t0

A

T

PulseGenerator

DAQ

Oscilloscope

Fig. 6.7 Experiment schematic to monitor slew response of DAQ

1 2 3 4 5 6 7 8 9 10104

105

106

107

108

Filter Order

Max

imum

Sle

w R

ate y

'(t)

Fig. 6.6 Maximum slew rate versus filter order for a Butterworth filter (cutoff frequency is 100 MHz)


7. Bracewell RM (1965) The Fourier transform and its applications. McGraw-Hill, New York, pp 6–7, 244–250

8. Rorabaugh CB (1999) DSP primer. McGraw-Hill, New York

9. Butterworth S (1930) On the theory of filter amplifiers. Exp Wirel Wirel Eng 7:536–541

10. Bateman VI, Hansche BD, Solomon OM (1995), Use of a laser doppler vibrometer for high frequency accelerometer characterizations. Sandia

National Laboratories technical report SAND-95-1041C

11. —— (2010) Digital Phosphor Oscilloscope TDS5034B/TDS5054B/TDS5104B Data Sheet, Tektronix, Beaverton

12. —— (2003) NI PXI-5122 Specifications, National Instruments, Austin

13. —— (2003) NI PXI-6133 Specifications, National Instruments, Austin


Chapter 7

Simplified Nonlinear Modeling Approach for a Bolted

Interface Test Fixture

Charles Butner, Douglas Adams, and Jason R. Foley

Abstract The sensitivity of the response characteristics of a bolted interface to the bolt preload level used in the joint can

often cause the dynamic properties of a system to be difficult to predict. A bolted interface test fixture was fabricated to

investigate the effects of preload changes on the system dynamic response characteristics, and experimental results indicated

that increases in bolt preload led to increases in modal frequency and decreases in modal damping. Furthermore, the system

demonstrated a nonlinear behavior that resulted in the increase in modal frequency due to increases in impact amplitude

when preload levels were low. The experimental results motivated the creation of a simplified low order nonlinear system

model to represent the two dominant modes of the system. A model was used to describe the relationship between static

stiffness and preload to account for the changes in initial bolt preload, and cubic stiffness terms were included to account for

the amplitude dependent nonlinearity that was observed. The resulting model was able to accurately simulate system

frequencies and general trends, but was unable to match some response characteristics of the system due to the lack of force

information for the high amplitude loading used in the experiments.

Keywords Nonlinear dynamics • Bolted interface • Modal frequency • Modal damping

7.1 Introduction and Background

The uncertainty that is introduced into a measurement based on the mounting condition of a sensor is a problem that often

influences the validity of structural vibration tests. Even when a sensor can be bolted to the test specimen, the amount of

preload in this joint can significantly change the response of the sensor. In a particular problem being studied by the Air

Force Research Lab Fuzes Branch, a triaxial sensor mount is attached to a more massive body through a preloaded interface,

and the preload used in the joint is a known source of uncertainty in the measurement. To study this particular interaction, a

test fixture was designed and built to simulate the interface. The fixture design is illustrated in Fig. 7.1.

The first component of the fixture is a large circular plate with a diameter of 460 mm and a thickness of 20 mm.

The second component of the fixture is a smaller square plate with a 180 mm height and width and a 20 mm thickness.

The square plate also contains three spherical standoffs of 20 mm height that localize the contact area between the two plates.

Both plates were machined from 4140 Alloy Steel. The fixture is assembled by bolting the two components together with M-

16 bolts and instrumented load washers to measure the static and dynamic preload in the bolts.

The study of the fixture began with experiments that subjected the bolted interface fixture to impulsive loading. The true

load path of the system being simulated by the fixture passes through the circular plate and into the square plate.

The experimental phase began with a low force amplitude modal test, in which the fixture dynamic behavior was

characterized using modal impact testing. In the second phase of testing, high amplitude impacts were applied to the test

fixture with a Hopkinson bar in order to approach a more realistic loading scenario, since the structure being modeled was

known to undergo very large forces.

C. Butner (*) • D. Adams

Purdue University, Center for Systems Integrity, 1500 Kepner Drive, Lafayette, IN 47905, USA


J.R. Foley

Air Force Research Laboratory Munitions Directorate, Fuzes Branch, Eglin Air Force Base, Eglin AFB, FL, USA



71


The results of the experiments conducted on the fixture led to the creation of a nonlinear three degree of freedom system

model that can be used to predict certain response characteristics of the system. The modal testing was used to estimate

modal parameters of the system for the two dominant modes present, while the high amplitude test results were used to

determine a relationship between bolt preload and natural frequency for the system.

Previous work by Adams et al. [1], Butner et al. [2] and Butner in [3] details the past experimental work that was

conducted on this fixture and the interface that it represents. The results of the work demonstrated an amplitude dependent

nonlinearity in the fixture at low preloads that was less severe as preload was increased. The work also noted that forces

where amplified across the interface at high preload levels, and that a preload relaxation event took place during the impacts,

indicating that the system response characteristics could vary due to an impact.

A review article by Ibrahim and Pettit [4] included a discussion of a vast range of bolted joint characteristics including

energy dissipation, bolt relaxation, natural frequency dependence, and bolt modeling approaches. Many uncertainties

associated with a bolted interface were also described. The accepted modeling techniques for bolted joint properties such

as joint stiffness, which is modeled as a constant value in many simplistic approaches, were found to be inadequate as

described by Grosse and Mitchell in [5] as well as in the work by Lehnhoff and Bunyard in [6]. The lack of analytical

relationships to accurately model these properties has led researchers to use experimental data and finite element models to

understand how bolted joints respond for different preload levels and excitation forces. An article written by Moo-Zung Lee

[7] explained how the preload in a bolted joint could change the stiffness properties. When a joint has no preload, an applied

force that attempts to separate the members will be applied entirely to the bolt. In this case, the stiffness of the joint is equal

to the stiffness of the bolt alone. When preload is applied to the bolt, the members are preloaded together, and the joint

stiffness becomes a function of the bolt and member stiffness values. This relationship indicated that a change in bolt preload

could also lead to nonlinearity in the system response. The Budynas and Nisbett text [8] presented an approach to model the

clamped member stiffness by using a conical change in pressure throughout the bolted joint. This approach motivates the use

of a nonlinear cubic stiffness model for joint stiffness.

Studies have also been performed on the changes in modal properties of multibody systems as a function of bolt preload.

A study was performed by Caccese et al. in [9] that attempted to detect loose bolts in a hybrid composite/metal bolted

interface through several approaches; one of which observed changes in modal parameters. The test setup consisted of

16 bolts around the perimeter of a square plate that clamped a composite panel to the square metallic plate. When only one

bolt was loosened slightly, little change in the fundamental frequency of the structure was detected, but when the bolt

was completely loosened, a drop in frequency was detected. The fundamental frequency of the assembly dropped more

quickly when all of the bolts were loosened. This work demonstrated that there was a large change in frequency for the

completely loose bolts case and partially tightened bolt case, but much less of a change between partially tightened bolts and

completely tightened bolts. This result was consistent with the behavior described about bolted joint stiffness in [7]. Stiffness

increases significantly when the bolt is initially preloaded, which results in a large upward shift in fundamental frequency,

but continued tightening results in a smaller change in stiffness, and thus a smaller increase in fundamental frequency.

A study done by Peairs et al. in [10] demonstrated similar behavior. A general trend of increasing modal frequency with an

increase in bolt preload was seen for many modes.

7.2 Experimental Approach and Key Results

A detailed description of the experiments conducted on the test fixture can be found in [2, 3], but since the purpose of this

paper is to describe the modeling approach used to simulate the system, only the experimental results that motivated the

modeling approach will be discussed. As mentioned previously, the experimental component of this research took place in

two phases. The first phase consisted of characterizing the system dynamics through the use of modal impact testing.

Fig. 7.1 Bolted interface test

fixture used for experiment

72 C. Butner et al.

The second phase consisted of applying large amplitude impacts to the fixture by using a Hopkinson bar in order to approach

a more realistic loading scenario to characterize the dynamic response for these higher loading levels. The test configuration

for the modal test and Hopkinson bar test can be seen in Figs. 7.2 and 7.3, respectively.

To characterize the system dynamics initially, four complete modal impact tests were conducted on the fixture.

Two variables were adjusted between the tests. The modal impact tests were conducted with the static preload in the

bolts tuned at levels between 1 kN and 20 kN, and the amplitude range of the impacts were held between 10 lbf and 20 lbf in

one set of tests or between 200 lbf and 300 lbf for a second set of tests. The goal of the test was to characterize the nonlinear

response characteristics due to impact amplitude as the preload in the bolts was adjusted.

A second experiment was conducted using a Hopkinson Bar to impact the test fixture. The Hopkinson bar applies an

impact through a transfer bar. This method was chosen because a one inch diameter aluminum transfer bar could be used in

order to prevent deformation at the impact location on the much harder steel test fixture. These tests were conducted by

suspending the test fixture vertically from an engine hoist and preloading a transfer bar against the test fixture. The bolts were

tightened to a pre-defined torque with a torque wrench, and each load cell’s static value was recorded. The transfer bar was

then impacted by a projectile that was fired by a gas gun. The impact force was stepped up by monitoring the air pressure that

was used to launch the projectile. For this experiment, six PCB 350C02 shock accelerometers were used to measure

acceleration at both ends of each of the three bolts. The load washer dynamic data was also recorded. During this round of

testing, preload levels of hand tight, 50 ft-lb, and fully tight, 100 ft-lb, were tested with impacts at 5 psi, 10 psi, 15 psi, and

20 psi. Due to the excitation type used, impact forces could not be measured in this experiment. The aim of this experiment

was to achieve an excitation that better reflected the realistic loading scenario that was expected.

Fig. 7.2 (a) Instrumented test fixture and (b) impact hammer with load washer signal conditioners

Fig. 7.3 Hopkinson bar test

setup

7 Simplified Nonlinear Modeling Approach for a Bolted Interface Test Fixture 73

The results of the four modal tests revealed several key trends in the system that would later be used in the model

formulation. First, a general trend of increasing natural frequency with increasing bolt preload was seen, which was

consistent with the results in literature. The modal damping in the system was seen to decrease with an increase in preload,

due to the reduction in the slapping and/or slipping across the interface. Damping increased with an increase in impact

amplitude as well. The system was also shown to behave more linearly with respect to impact amplitude as preload was

increased.

The high amplitude Hopkinson bar testing also demonstrated an increase in natural frequency with preload, and helped to

provide a relationship between static bolt preload in ft-lbs with the modal frequencies of interest. This experiment also

demonstrated a force amplification effect across the interface at high preload. These experiments also demonstrated an

amplitude dependence of the system natural frequencies when preload was low, as shown in Fig. 7.4. The natural frequency

increases during an impact event because as the joint is forced together by the impact, a stiffening effect occurs. This effect is

not seen in the higher preload cased because the impact does not overcome the preload force.

7.3 Three Degree of Freedom Nonlinear System Model

The analysis of the experimental results for the test fixture identified several interesting phenomena pertaining to the

effects of preload on the dynamic response of the two coupled bodies. Some of these effects could be qualitatively

explained using a linear system model, such as the amplification effects across the bolted interface and modal behavior

that was observed in the shock test experiments. The changes in natural frequency and damping as a function of the initial

static preload could be modeled by introducing changes in the linear connecting springs and dampers. Some of the

experimental results could not be explained with a linear model. The most prevalent nonlinear effect of interest was the

dependence of the natural frequency on the impact force amplitude. This result was observed most clearly in the shock

test data that was plotted in Fig. 7.4. A low order nonlinear system model was developed to explain the changes in natural

frequency as a function of impact force level. This model would be useful for predicting the behavior of future system

configurations, and gaining a better understanding of the behavior of the system in general for use in interpreting measured

data in operation.

7.3.1 Modeling Approach

The decision to use a low order modeling approach was motivated by the results of the Hopkinson bar shock loading

experiments. In this data, only two modes were dominant in the low frequency range. The reason for this was that the input

force was applied at the center of the circular plate, which was a node of vibration for many of the system modes of vibration.

500 550 600 650 700 750 800 850 9000

2

4

6

8

10

12

14

Frequency (Hz)

Am

plitu

de o

f Spe

ctru

m (

m/s

^2/H

z) 5psi Impact10psi Impact15psi Impact20psi Impact

Fig. 7.4 First mode of vibration for test fixture with hand-tight preload

74 C. Butner et al.

Since the force was being applied to a common node of vibration, it did not excite modes that shared this node to a significant

level. The two dominant modes that were excited had an anti-node at the center of the circular plate. In reality, the load path

of the system that was emulated using the test fixture lies along the edge of the circular plate. It was reasoned that if a

uniform distributed force was applied around the circumference of the circular plate, the same modes would be excited as if

a force was applied at the center of the plate due to the circular symmetric nature of the modes of vibration. The low

frequency modes for which the center of the circular plate was a node of vibration also consisted of non-uniform motion at

the outer edge of the circular plate, while those that had an anti-node at the center of the circular plate had uniform motion

at the edge of the circular plate. Because of these observations, the modeling approach was deemed to be valid for uniform

forces applied around the circular plate’s edge, which was the load path for this structure in practice.

While only two dominant modes were observed in the shock testing data, and many aspects of the system lent themselves

to being represented by a two degree of freedom model, this configuration would not represent the system accurately. In the

data, the first mode that was excited was a flexible body mode of the circular plate and rigid body motion of the square plate.

The second excited mode exhibited asynchronous motion between the two plates as the primary motion. Taking this

observation into account, the system for these two modes of vibration was modeled using three degrees of freedom as shown

in Fig. 7.5. This model included an additional degree of freedom and, therefore, resulted in the presence of a third mode of

vibration. This third mode was tuned to a very low frequency by using soft ground springs and allowing the entire mass of the

test fixture to oscillate as a rigid body. This motion was an accurate representation of the test fixture because the three modes

of vibration allowed for the flexible body mode of the circular plate, the asynchronous mode between the plates, and a low

frequency rigid body mode that simulated the test fixture at it was swung on its flexible supports. Linear, adjustable springs

and dampers were placed between the degrees of freedom in order to allow for system adjustments for different static

preload values. Additional preload would result in increased joint stiffness and decreased damping as discussed in the

literature in [7], as well as in previously described experimental results. Nonlinear elements were also required to allow

the system’s modal frequencies to increase with impact force amplitude. An asymmetric cubic stiffness nonlinearity was

chosen for this purpose.

The decision to use a cubic stiffness term was motivated by several factors. The primary reason for choosing this type of

nonlinearity was the observations made in the experimental data. When the fixture was forced with impacts of increasing

amplitude, the modal frequencies increased. This result could be accomplished with any spring that stiffens with deflection.

In the case of the test fixture, however, the system behaved linearly in cases where the impact amplitude was small or the

preload was high. These cases both corresponded to conditions where deflection across the bolted interfaces was low.

A cubic stiffness term had a small effect for small deflections, which allowed the system response to remain mostly linear for

this case. When the static preload was lowered or the impact amplitude was increased, the increased deflections resulted in

an increase in natural frequency. The cubic stiffness was introduced for increased compression across the bolted interface,

M1 M2 M3

Circular Plate Square Plate

Cubic Springs

Linear Adjustable Springs and Dampers

To Account for Static Preload Changes

x1 x2 x3

f

C1,K1 C2,K2 C3,K3

Fig. 7.5 Three degree

of freedom nonlinear

system model


however, because it was thought that large impacts momentarily increased the preload between members by forcing them

together, thereby increasing the joint stiffness. It was for these reasons that a cubic stiffness term was considered to be

acceptable.

7.3.2 Determination of System Parameters

The formulation of the system model began with the construction of a linear system of equations that accurately

approximated the system for a given preload value and impact amplitude. This model could then be adjusted to reflect

changes in initial preload and the cubic stiffness terms could be included to account for nonlinear effects. The first step in this

process was to determine the modal mass, stiffness, and damping for each mode from the experimental data. The model was

meant to approximate motion on either side of the bolted interface, so data was used that captured the motion on either side

of the interface. Frequency response functions were used from accelerometers on either side of the bolted interface near one

of the bolts, when applying an impact at the accelerometer location on the circular plate. This measurement approach

provided a driving point and a cross point measurement. While only a driving point measurement was required for the

estimation of modal parameters, the cross point measurement was needed to obtain accurate estimates of the modal vectors.

The lr and Apqr values for the two modes of interest were calculated from the system simultaneously using the Local

Least Squares Algorithm, which is a low order frequency domain technique that operates on a single degree of freedom

assumption. For systems without closely spaced modes, the data near the peak can be considered to behave as a single degree

of freedom system corresponding to the parameters from that particular mode. Nearby modes could contribute residual

values to these peaks and reduce the accuracy of these assumptions, but for the test fixture data this was not the case for the

modes of interest.

Once the lr and Apqr values were obtained from the experimental data and verified for both the driving point and cross

point measurements, the modal mass, stiffness and damping parameters could be estimated. The modal parameters that were

calculated all had small imaginary portions with respect to the real portion, so the values were approximated using only the

real portion for model simplicity. Before these values could be reduced to a system of equations of motion to represent the

system, a rigid body mode was created. For this mode, a modal mass was chosen that was approximately equal to the weight

of the entire fixture. A modal stiffness value was chosen to match the 1.5 Hz mode of the test fixture at it was swung on its

supports. A modal damping value was chosen to be proportional to the other modes. Table 7.1 listed the modal mass,

damping, and stiffness values that were used for the final model. The addition of a third degree of freedom required the

modal vectors to be reformulated. The rigid body mode consisted of all masses moving together. The second mode, which

corresponded to the flexible body mode for the circular plate, was represented by the first and second masses moving relative

to one another, with no deflection in the spring between the second and third masses. The third mode consisted of no

deflection between the first two masses, and asynchronous motion between the second and third masses. The modal vectors

chosen for the final form of the model were listed in Table 7.2.

In order to complete the linear component of the system model, the modal mass, stiffness, and damping matrices were

converted from modal coordinates to absolute coordinates. The modal vector matrix was used to transform between

coordinates. In order verify the accuracy of the linear system of equations, the FRFs were calculated from the equations

and compared to experimental data. The driving point position versus force FRF for this simple linear system was calculated

from the linear system representation by matrix inversion. This measurement was compared to the experimental data as

shown in Fig. 7.6. It was seen that the model approximated the experimental FRF well despite the simplifications that were

made throughout the estimation process. The artificial rigid body mode also matched the low frequency content well. Note

that the system only estimates the two dominant modes from the symmetric loading case, where the loading used to generate

the experimental FRF contains more modal content.

7.3.3 Model Adjustments to Account for Preload and Nonlinear Behavior

The linear system of equations that was formulated for the system were found to be an accurate representation of the

experimental data, but the model in this initial form only had the capacity to recreate one set of test conditions. Previous

examination of experimental data indicated that these two modal frequencies showed the tendency to shift upward with

increases in preload. The damping was also known to reduce as preload was increased. A model was developed to describe

the stiffness increase as a function of initial preload.

76 C. Butner et al.

http://dx.doi.org/10.1007/978-1-4614-2416-1_5

http://dx.doi.org/10.1007/978-1-4614-2416-1_5

After reviewing the literature, it was found that the modeling approach for bolted joint stiffness varied from source to

source. While some textbook references modeled bolted joint stiffness as a constant value, finite element modeling

approaches in [2] determined that joint stiffness was a function of applied load. Since no analytical formula for joint

stiffness as a function of bolt preload was readily available, it was decided to examine experimental data for a trend that

would reveal the proper form for the model. The expression for the natural frequency of a single degree of freedom system in

terms of stiffness and mass was given as follows:

on ¼ffiffiffiffiffi

K

M

r

: (7.1)

After solving this expression for the stiffness term K, the following expression was obtained:

K ¼ Mo2n: (7.2)

It was then assumed that the modal mass underwent minimal change with changes in preload; therefore, the stiffness was

found to be proportional to the natural frequency squared.

It was expected that changes in bolt preload would affect both of the modes of vibration of interest in different ways.

While the increase of preload would result in an increase in modal frequency for both modes, the ratio of stiffness change

would not be identical for both modes, since the type of motion exited was different. For this reason an expression was

calculated for the stiffness of the spring that dominated each mode separately. The K2 spring in Fig. 7.5 was the primary

spring that participated (deflected to a large degree) in the first mode of vibration, while the K3 spring dominated the second

mode. The model for joint stiffness was determined in the same manner for each mode. First, the known preload values were

0 500 1000 1500 2000 2500 300010-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

Frequency (Hz)

|FR

F| (

m/N

)

Fig. 7.6 Comparison of driving point position versus force FRF from experimental data (blue) and three degree of freedom system of equations

(green) (color figure online)

Table 7.1 Modal parameter

estimates used for three degree

of freedom system model

Mode Mr (kg) Cr (N s/m) Kr (N/m)

1 50 624.9 4441.3

2 40.7 312.4 7.385e+08

3 18.3 328.1 5.609e+09

Table 7.2 Modal vectors

used for three degree

of freedom system model

– Mode 1 Mode 2 Mode 3

Mass 1 1.0000 0.8626 �0.2316

Mass 2 1.0000 1.0000 �0.2316

Mass 3 1.0000 1.0000 1.0000


plotted with their corresponding natural frequency squared for each mode, and a power function was fit to this data. These

power functions where then scaled appropriately for each mode by taking the ratio of the corresponding spring stiffness and

natural frequency from the linear system equations and scaling the power functions by that ratio. The result was a curve that

enabled the spring stiffness estimate to be calculated for any preload value in ft-lbs, where the hand-tight preload was

estimated as 1 ftlb. The stiffness curves for the K2 and K3 springs were plotted in Figs. 7.7 and 7.8, respectively. The shapes

of these curves indicated that when preload was adjusted from a very low value to a mid-range value, the stiffness and natural

frequency changed significantly, but changed little for continued increases in preload. This data was consistent with the

results predicted by Moo-Zung Lee in [7], and was experimentally observed by Caccese et al. in [9]. A decrease in damping

was also seen in the system with increased preload as discussed previously. This trend was not as clear, and the proper

frequency response function data that would be required to develop a curve for the damping values was not available from

the shock testing results, since the input forces could not be measured. For this reason, the damping values were changed

proportionally to the stiffness values, but in the opposite direction. As stiffness was increased, damping was decreased

proportionally because of the nature of the experimentally observed results that indicated the damping decreased with

increases in static preload level.

The final portion of the model that needed to be determined was the cubic stiffness nonlinearity. This nonlinearity was

chosen because of its use in the literature, as well as its ability to allow for the stiffening behavior that was observed

0 50 100 1502

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4 x 1010

Preload (ft-lb)

Con

nect

ing

Spr

ing

K2

Stif

fnes

s (N

/m)

Fig. 7.7 K2 spring stiffness versus preload level

0 50 100 1502

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4 x 109

Preload (ft-lb)

Con

nect

ing

Spr

ing

K3

Stif

fnes

s (N

/m)

Fig. 7.8 K3 spring stiffness versus preload level

78 C. Butner et al.

experimentally. The goal of this nonlinear term was to produce an increase in stiffness for either very small initial preloads or

very large impacts. This nonlinearity was a result of the impact force that enabled the motions of the masses on either side of

the bolt to overcome the initial preload across the interface, and effectively increased the stiffness in the joint. For this

reason, a cubic spring was implemented that only contributed stiffness to the system when the relative motion between the

two bodies it connected deflected in a way that caused a compression of the spring. When the compression was relaxed,

the cubic spring would not cause a stiffness lower than the initial joint stiffness. The model set the contribution of the cubic

stiffness to zero when elongation of the joint was detected in the numerical simulation.

A larger force resulted in a larger deflection, which would make the effect of a cubic spring more evident. The stiffness

coefficients for the nonlinear terms were tuned so that small deflections would not result in a detectable effect from the cubic

spring and the system would behave linearly. On the other hand, a large deflection would cause an appreciable effect from the

nonlinear spring. When examining the response of the linear system, it was determined that relative deflection values between

the masses had overlapping regions between high and low preload levels for the given impact amplitude range. Since the cubic

stiffness nonlinearity should have only affected the low preload level substantially for the impact amplitude range being tested,

it was determined that a scale factor was needed to achieve this effect. Since the system became more resistant to separation at

higher preloads, it was decided to model this scale factor inversely to the stiffness model. The lowest preload level tested in the

model as 1 ft-lb. For this value the relative deflections between masses was scaled by one, or held the same. As preload was

increased slightly, the scale factor decreased quickly in magnitude, but decreased less for further increases in preload.

This scale factor allowed for the system to be more resistant to nonlinear behavior at higher preloads.

7.3.4 Numerical Solution Approach for Nonlinear Modal

With the addition of the cubic stiffness terms into the model, the simple solution methods that were utilized previously were

no longer sufficient. Newmark’s method was chosen to solve the nonlinear form of the model. The solution procedure was

taken from the Geradin and Rixen textbook [11]. This method consisted of the calculation of the displacement, velocity, and

acceleration vectors at each time step for a given forcing function. The estimates were adjusted through an iterative

procedure until a specified convergence criterion had been achieved. In the model, the forcing function that was used was

an impulse that lasted for only one time step. This was an idealized approximation of a shock impact. The peak amplitude

was scaled accordingly. Once the displacement, velocity, and acceleration time histories had been calculated, FRFs were

created and compared. Preloads of 1 ft-lb, 50 ft-lb, and 100 ft-lb were simulated, with force levels of 2,500 lbf, 5,000 lbf,

7,500 lbf, and 10,000 lbf. The force estimates used where incremented proportionally to the levels used in the shock testing,

but the actual force inputs where unknown.

7.3.5 Results of Model Simulation

Once the model solution had been obtained for all of the test parameters, the resulting FRFs were examined to determine if

the model accurately represented the behavior of the test fixture. First, the changes in natural frequency and damping with

preload value were examined. The results for all three preload levels and the 2,500 lbf impact were plotted in Fig. 7.9.

This force was low enough to minimize the nonlinear effects for the low preload case. It was seen in the plot that the model

contained the same frequencies that were observed in the experimental data for each preload level. A trend of increasing

natural frequencies with preload was also evident in the plot. The damping reduction was also evident, as the peaks became

more narrow and higher in amplitude as preload was increased.

After it was determined that the model properly reflected changes in modal frequency and damping due to changes in

initial preload, the FRFs were examined to determine if the cubic stiffness nonlinearity did a proper job of modeling the

system response to large forces at low preload levels. Figure 7.10 shows the FRFs for all impact amplitudes for the low

preload level zoomed in on mode one. It was seen that the nonlinearity was obviously excited as the FRFs were not all the

same. For a linear system, the FRFs would not change as a function of impact amplitude. The peaks shifted upward in

frequency, while they also broadened and decreased in amplitude. This indicated that the modal frequency increased as

expected from the shock testing results, and the damping increased as seen in the earlier results from the modal tests.

The only issue with these results was the peak distortion and harmonics introduced by the cubic stiffness term. This was not

seen in the experimental data, but all other aspects of the model forced response matched the experimental data.

Once the low preload results from the model were determined to be sufficiently accurate, the 50–100 ftlb cases were

examined. Figure 7.11 shows the FRFs for all impact amplitudes at the 50 ftlb preload level, also zoomed in on mode one.


It can be seen from these plots that the model response was nearly linear for the higher preload ranges for the impact

amplitudes that were tested. This result was consistent with what was noted in experimental results. The resistance to

separation caused by the high preload levels suppressed the effects of the nonlinear cubic stiffness terms.

Once the performance of the model was tested, the results of the simulation were compared to the experimental data from

the shock testing experiments. Since accurate input force information was not available for the shock input tests, FRFs where

not available for comparison. For this reason, the acceleration spectra from the model and shock experiments where

compared. Figure 7.12 shows the hand tight preload level for all impact air pressures used. This data is what revealed the

increasing natural frequency with increasing impact amplitude behavior for high amplitude shock loading. When the model

acceleration traces from the driving point measurements for each of the impact amplitudes were first examined, it was

determined that the responses were much more lightly damped than those seen in the experimental data. This is because

despite the fact that the shock data trends where used to tune the nonlinear aspects of the model, the base linear

characteristics where taken from a low amplitude modal test. The slapping damping mechanism was not activated for

very low impact amplitudes, so damping behavior was expected to differ for high amplitude tests. The model damping was

increased in order to achieve a similar frequency range to those seen in the experimental data. The result of the increased

damping model is shown in Fig. 7.13. With the increase in damping, the general shapes of both the model results and the data

matched very well. In both cases, a common backbone is shared by the different responses, but the increased amplitude

0 500 1000 1500 2000 2500 300010-12

10-11

10-10

10-9

10-7

10-6

Frequency (Hz)

|FR

F| (

m/N

)10-8

Fig. 7.9 Model position FRFs for hand-tight preload (blue), 50 ftlb preload (green), and 100 ftlb preload (red) for the 2,500 lbf impact (color

figure online)

600 605 610 615 620 625

10-8

10-7

Frequency (Hz)

|FR

F|(

m/N

)

Fig. 7.10 First mode of model position FRFs for 2,500 lbf impact (blue), 5,000 lbf impact (green), 7,500 lbf impact (red), and 10,000 lbf impact

(cyan) for the hand-tight preload level (color figure online)

80 C. Butner et al.

640 650 660 670 680 690 700 710 720

10-8

10-7

Frequency (Hz)

|FR

F| (

m/N

)

Fig. 7.11 First mode of model position FRFs for 2,500 lbf impact (blue), 5,000 lbf impact (green), 7,500 lbf impact (red), and 10,000 lbf impact

(cyan) for the 50 ftlb preload level (color figure online)

500 550 600 650 700 750 800 850 9000

2

4

6

8

10

12

14

Frequency (Hz)

Am

plitu

de o

f Spe

ctru

m (

m/s

^2/H

z)

5psi Impact10psi Impact15psi Impact20psi Impact

Fig. 7.12 Accelerometer one frequency spectra for hand tight preload level and all impact amplitudes

500 550 600 650 700 750 800 850 9000

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Frequency (Hz)

Am

plitu

de o

f Spe

ctru

m (

m/s

^2/H

z)

Fig. 7.13 Driving point frequency spectra from model with increased damping for hand tight preload level and 2,500 lbf (blue) 5,000 lbf (green)7,500 lbf (red) and 10,000 lbf (cyan) impact amplitudes (color figure online)


led to a larger frequency excitation range. This confirmed that the cubic stiffness nonlinearity was fairly accurate.

The major difference in the plots however was the difference in scale. The experiential data was excited to acceleration

levels that were two orders of magnitude above the model results.

The large difference in amplitude was the result of several contributing factors. The first of these was that the model only

consisted of two modes, where the experimental data had contributions from many more high frequency, and weakly excited

low frequency modes of vibration. Each of these modes contributed a small residual that increased the amplitude of the

spectra. This alone cannot explain such a large difference. The model results could be scaled to similar levels to those seen in

the experiments by simply reducing the values of the mass, stiffness, and damping matrices by two orders of magnitude.

Reducing these matrices is equivalent to increasing the force levels by two orders of magnitude, so the large difference in

values could be explained partially by inaccurate forcing levels. The force levels used in the model where chosen to be close

to the approximate force estimations from the shock testing. The force estimates where known to be inaccurate due to system

nonlinearity, but another important factor in the accuracy of the estimates was how the training data was taken for these

approximations. The impacts where applied with a modal hammer to the fixture through a transfer bar. This means that the

predictions made using this data were for the force applied to the transfer bar, not the force applied to the circular plate.

The transfer bar introduced its own frequency response function and dynamics, so these two forces were not equal.

The transfer bar was necessary however to avoid damaging the fixture. The forces applied to the fixture could have been

much larger than the estimates indicated. Finally, it is known that the system dynamic properties changed with impact

amplitude range. This change in dynamics could have also resulted in changes to the coefficient matrices, which could have

also resulted in the change in response amplitude.

7.4 Conclusions

After the analysis of the nonlinear system model was complete, it was determined that the model performance was sufficient

at approximating the system response. However, the model was limited in several ways. The low order approach was based

on the assumption that the excitation was applied uniformly to the outer edge of the circular plate, or to the center of the

circular plate. This was only the case if the loads were applied exactly as desired in the use of the actual interface being

simulated in this research. The peak distortion and harmonics that were seen in the model results were also an undesirable

effect, but these effects were minimal if the excitation amplitude was not extremely large. One effect of the system that was

not included in this model was the preload relaxation effect. While this effect was known to exist, it was not known for

certain how this affected the system during an impact. It would certainly lead to a lower initial preload for the system

subsequent to an impact, but it was unknown how this would affect the modal response during a single impact. The primary

reason for this lack of information was that the shock data consisted of a very short time window length and lacked sufficient

frequency resolution to differentiate between frequency components from early and late portions of the response time

history. The model also resulted in inaccurate damping and response amplitude levels when compared to the shock data. This

was due to the differences in system dynamics for the shock data, and the data used to create the model. The model was

sufficient for many realistic simulations, and could be used to reasonably predict system response characteristics for

conditions that have yet to be tested.

References

1. Adams DE, Yoder N, Butner CM, Bono R, Foley J, Wolfson J (2010) Transmissibility analysis for state awareness in high bandwidth structures

under broadband loading conditions. In: Proceedings of IMAC XXVII, Jacksonville

2. Butner CM, Adams DE, Foley JR (2011) Understanding the effect of preload on the measurement of forces transmitted across a bolted

interface. In: Proceedings of IMAC XXIX, Jacksonville

3. Butner CM (2011) Investigation of the effects of bolt preload on the dynamic response of a bolted interface. Thesis, Purdue University

4. Ibrahim RA, Pettit CL (2005) Uncertainties and dynamic problems of bolted joints and other fasteners. J Sound Vib 279(3–5):857–936

5. Grosse IR, Mitchell LD (1990) Nonlinear axial stiffness characteristics of axisymmetric bolted joints. J Mech Des 112(3):442–449

6. Lehnhoff TF, Bunyard BA (2001) Effects of bolt threads on the stiffness of bolted joints. J Press Vessel Technol 123(2):161–165

7. Kenneth JK, Moo-Zung L (2010) Modeling the effects of bolt preload

8. Budynas RG, Nisbett JK (2008) Shigley’s mechanical engineering design, 8th edn. McGraw-Hill, New York

9. Caccese V, Mewer R, Vel SS (2004) Detection of bolt load loss in hybrid composite/metal bolted connections. Eng Struct 26(7):895–906

10. Inman DJ, Peairs DM, Park G (2001) Investigation of self-monitoring and self-healing bolted joints. In: Proceedings of 3rd international

workshop on structural health monitoring, Stanford, pp 430–439

11. Gradin M (1997) Mechanical vibrations. Wiley, New York

82 C. Butner et al.

Chapter 8

Transmission of Guided Waves Across Prestressed Interfaces

Jacob C. Dodson, Janet Wolfson, Jason R. Foley, and Daniel J. Inman

Abstract Prestresses can be both applied and/or environmentally generated and change the dynamics of structures. The

preload can change due environmental factors, such as temperature variations, and operational excitation including

impulsive loading. This study uses a novel Hopkinson bar configuration, the Preload Interface Bar, and investigates the

effect of prestress on the stress wave propagation across interfaces between the incident and transmission bars. The non-ideal

interface with a partial gap is experimentally investigated and analyzed. The preloaded partial gap causes generation of

flexural waves at the interface. The interface is modeled as a preload dependent elastic joint in which flexural and

longitudinal waves can be transmitted and reflected. The analytical and experimental energy transfer and mode conversion

from reflection and transmission will be examined in the spectral domain. The analysis shows that as the preload increases

the interface becomes stiffer and converges to a near-perfect 1-D interface.

8.1 Nomenclature

x longitudinal degree of freedom r reflection amplitude ratio Superscripts

y vertical degree of freedom t transmission efficiency A extensional strain

k wavenumber x reflection efficiency B bending stain

o angular frequency K joint stiffness Total total

A cross sectional area �K normalized joint stiffness transmission/reflection

h height u(x) axial displacement

c wave velocity v(x) vertical displacement Subscripts

E modulus of elasticity F(x) axial force L longitudinal

I area moment of inertia V (x) shear force B bending

r mass density M(x) moment n near field

n Poisson’s ratio y(x) angle of rotation 1 motion/force of bar 1

U wave amplitude s(x, t) stress 2 motion/force of bar 2

t transmission amplitude ratio, time exx(x, t) axial strain S motion/force of

interfacial spring

x property in x direction

y property in y direction

y property in y direction

J.C. Dodson (*)

Department of Mechanical Engineering, Virginia Polytechnic Institute and State University,

310 Durham Hall, Mail Code 0261 Blacksburg, VA


J. Wolfson • J.R. Foley

Air Force Research Laboratory AFRL/RWMF, 306 W. Eglin Blvd., Bldg. 432, Eglin AFB, FL 32542–5430,

D.J. Inman

Department of Aerospace Engineering, University of Michigan Ann Arbor, MI 48109,



83


8.2 Introduction

To predict the dynamics of a complex structure the dynamics of the individual components and the interfaces between them

must be understood. In an environment which prestresses the mechanical system and its components, the preloaded

interfaces are a crucial part that is not currently understood. Initial investigations on how the preload affects transmission

are very recent [1]. Older research shows that preload induces nonlinear effects on the response of structures such as

beams [2]. Due to the complexities of interfaces, they are often assumed to be ideal and they are modeled with the

compatibility conditions of the displacements and stresses being equal on both sides of the interface. While this approach

simplifies the problem it does not address lack of understanding of interfacial dynamics.

In the cases where non-ideal, or partial, interfaces occur a common approach is to use a contact stiffness model which

assumes an elastic force between the interfaces [3, 4]. Alternatively another method of modeling the partial interface

problem is to model the initial gap at the interface. Barber et al. examined the dynamics of an initial gap which closes

during the wave transmission between two semi-infinite media [5]. Daehnke and Rossmanith also addressed the case in

which a gap is initially open but the wave displacement closes the gap and allow transmission for the highest amplitude

part of the pulse [4]. Both papers generate piecewise displacement and force relations. While this method may be more

accurate in the time domain, modeling of the angled gap between two finite bars becomes difficult quickly and may be

addressed in a later paper. For this paper we will only use the contact stiffness method to model the preloaded imperfect

interface.

The longitudinal and shear waves propagating over various types of interfaces or joints in semi-infinite medium have

been examined in the geophysics field for waves due to mining explosives and seismic activity [4]. Daehnke and Rossmanith

discuss the reflection, transmission, and conversion of longitudinal and shear waves at; ideal interfaces, interfaces with

normal and shear stiffness values, and prestressed interfaces among others. The preload dependent stiffness values have been

empirically determined for various types of rocks, the authors hypothesize that as the preload increases the joints become

stiffer allowing more energy to propagate across the interface.

Guided waves are waves constructed of the reflections and refractions of the longitudinal and shear waves off the stress-

free boundaries of a finite structure. Guided waves propagate along the length of the structure and can be described by the

motion of the waves. Two of the primary wave modes are longitudinal and flexural, or bending. The mechanism for

converting and transmitting the wave modes across the interfaces in bars, beams, and plates are similar to the plane wave

interaction with the interfaces in a semi-infinite medium. The primary difference is the extra degree of freedom given to the

finite structures. In the 2-D semi-infinite medium the interface has only two directions it is able to move, while in the finite

bar, beam, and plate the interfaces at the end of the structure can also rotate. When Daehnke and Rossmanith modeled the

interface with stiffness values, they used a transverse and normal stiffness to transmit the shear and longitudinal

components [4]. Leung and Pinnington have investigated the transmission and reflection of incident longitudinal and

flexural waves on a right angle joint of two plates using linear spring and dashpot models utilizing three degrees of freedom:

axial displacement, vertical displacement and rotational motion [6, 7]. In order to accurately model the system, we have to

capture both axial and flexural motion of the joint. We will follow Leung and Pinnington’s approach and model the interface

as a parallel combination of linear and rotational springs whose values are preload dependent and relate to the longitudinal

and flexural modes in the bars.

This paper discusses the modeling and experimentation conducted on a imperfect interface under various preloads. The

novel Hopkinson bar configuration used, the Preload Interface Bar, has been previously discussed in an investigation

examining the effect of grease on the interface on the stress wave propagation across interfaces between the incident and

transmission bars [8]. The preload affects the transmission of axial waves and generation of flexural waves at the interface.

The interface is modeled with a contact stiffness model with three degrees of freedom and attempts to capture the dynamics

present in the experiment.

8.3 Theory

8.3.1 Axial Wave Propagation

The propagation of strain pulses along a uniform right cylindrical bar is a well-studied area of mechanics [9] that provides a

starting point for data analysis. The partial differential equation for elementary rod theory is given as [10, 11]

84 J.C. Dodson et al.

E@2uðx; tÞ@x2

� r@2uðx; tÞ

@t2¼ 0 (8.1)

where E is the elastic modulus and r is the density of the bar material. For an arbitrary function h(x, t), the time-space

Fourier transform is h(k, o) and is defined by

hðk;oÞ ¼Z 1

�1

Z 1

�1hðx; tÞe�iðkxþotÞdxdt (8.2)

and the inverse is given by

hðx1; tÞ ¼ 1

4p2

Z 1

�1

Z 1

�1hðk;oÞeiðkxþotÞdkdo (8.3)

where x is the spacial dimension, k is the spatial wavenumber, o is the angular frequency, and t is time. Applying the time-

space Fourier transform (8.2) to the elementary rod equation (8.1) yields

ð�Ek2 þ ro2Þuðk;oÞ ¼ 0: (8.4)

Solving for k give us the wavenumber relation for the bar

kbar ¼ oc

(8.5)

where c is the longitudinal elastic wave speed in a bar and is defined as

c ¼ffiffiffiE

r

s: (8.6)

In the development of the elementary rod differential equation (8.1) the assumed displacement relation is

�uðx; y; tÞ ¼ uðx; tÞ (8.7)

where �u(x, y, t) is the total longitudinal displacement field and u(x, t) is the longitudinal displacement at the midpoint of the

beam (y ¼ 0). For a longitudinal wave traveling down the bar the associated axial, or extensional, strain ExxA(x, t) can be

written as

EAxxðx; tÞ ¼@uðx; tÞ@x

: (8.8)

8.3.2 Flexural Wave Propagation

The propagation of bending, or flexural, waves can be typically modeled as satisfying the Euler-Bernoulli Beam or

Timoshenko Beam equation [10]. For simplicity of the argument we will use the Euler-Bernoulli Beam differential equation

given as

EI@4vðx; tÞ@x4

þ rA@2vðx; tÞ

@t2¼ 0 (8.9)

8 Transmission of Guided Waves Across Prestressed Interfaces 85

Applying the time-space Fourier transform (8.2) to the elementary rod equation (8.9) yields

ðEIk4 � rAo2Þvðk;oÞ ¼ 0 (8.10)

solving for the wavenumber gives

k2 ¼ �ffiffiffiffiffiffiffiffiffiffiffio2rAEI

r(8.11)

In the development of the Euler-Bernoulli beam differential equation (8.9) the assumed displacement relation is

�uðx; y; tÞ ¼ �y@vðx; tÞ@x

(8.12)

�vðx; y; tÞ ¼ vðx; tÞ (8.13)

where �vðx; y; tÞ is the total vertical displacement field and v(x, t) is the vertical displacement at the midpoint of the beam

y ¼ 0. The associated axial strain due to bending (ExxB) is

EBxxðx; y; tÞ ¼@�uðx; y; tÞ

@x¼ �y

@2vðx; tÞ@x2

(8.14)

8.3.3 Separation of Measured Strain

Using measured strain pairs on the surfaces of the beam we can decouple the strain due to bending and the strain due to

extension. If we assume a coupled elementary rod and Euler-Bernouilli beam model then our axial displacement would be

�uðx; y; tÞ ¼ uðx; tÞ � y@vðx; tÞ@x

(8.15)

and using the definitions of axial strain due to longitudinal and flexural waves from (8.8) and (8.14) our axial strain on the top

and bottom of the beam would be

Exxðx; h2; tÞ ¼ EAxxðx; tÞ � EBxxðx; tÞ (8.16)

Exxðx;� h

2; tÞ ¼ EAxxðx; tÞ þ EBxxðx; tÞ: (8.17)

Adding the measured strain on top (8.16) and bottom (8.17) will cancel the bending strain, while subtraction of the measured

strains will cancel the axial strain. From the strain measurements on the top and bottom of the beam we can decouple the

strain due to axial motion and the strain due to flexural motion. The decoupled stains can be written as

EAxxðx; tÞ ¼Exxðx; h2 ; tÞ þ Exxðx;� h

2; tÞ

2(8.18)

EBxxðx; tÞ ¼Exxðx;� h

2; tÞ � Exxðx; h2 ; tÞ2

: (8.19)

8.3.4 Elastic Joint Model

Leung and Pinnington have investigated the transmission and reflection of incident longitudinal and flexural waves on a right

angle joint using linear spring and dashpot models [6, 7]. This analysis sets up the problem in the same manner and models


the interface with elastic springs. The functional form of the transmitted and reflected waves are assumed and the amplitude

components are solved for using the force relations at the interface. Here we model the non-ideal interface as an elastic joint

with three degrees of freedom: axial displacement, vertical displacement and rotation. The model of the joint and the force

diagram can be seen in Fig. 8.1a, b, respectively.

In this analysis the elementary rod theory is assumed for axial motion and the Euler-Bernoulli beam theory is assumed for

transverse motion and rotation. The longitudinal and flexural waves are assumed to travel uncoupled along the semi-infinite

bars and coupling only takes place at the joint. We assume an longitudinal wave is incident into the joint.

Consider a longitudinal incident wave of amplitudeU(o) propagating into the joint in bar 1. The axial displacement in the

frequency domain u(x, o) is composed of the incident wave and the reflecting wave and can be written using the longitudinal

displacement as

u1ðx;oÞ ¼ UðoÞðe�ikbarðoÞx þ rLðoÞeikbarðoÞxÞ (8.20)

where i ¼ ffiffiffiffiffiffiffi�1p

, kbar(o) is the corresponding bar wavenumber as a function of frequency, and rL(o) is the amplitude of the

reflected longitudinal component. The transmitted longitudinal wave can be written as

u2ðx;oÞ ¼ UðoÞtLðoÞe�ikbarðoÞx (8.21)

where tL(o) is the amplitude of the transmitted longitudinal component. The reflected flexural wave can be written using the

vertical displacement as

v1ðx;oÞ ¼ UðoÞðrBðoÞeik1ðoÞx þ rnðoÞek2ðoÞxÞ (8.22)

where k1(o) and k2(o) are the corresponding beam wavenumbers as a function of frequency, rB(o) is the amplitude of the

reflected flexural component, and rn(o) is the amplitude of the reflected near field evanescent flexural component. The

transmitted flexural wave is

v2ðx;oÞ ¼ UðoÞðtBðoÞe�ik1ðoÞx þ tnðoÞe�k2ðoÞxÞ (8.23)

tB(o) is the amplitude of the transmitted flexural component, and tn(o) is the amplitude of the transmitted near field

evanescent flexural component. Using the force diagram in Fig. 8.1b the force, moment and displacement relations for the

elementary rod and Euler-Bernoulli beam can be written as [10]

M1ðx;oÞ ¼ EI@2v1ðx;oÞ

@x2M2ðx;oÞ ¼ EI

@2v2ðx;oÞ@x2

(8.24)

V1ðx;oÞ ¼ �EI@3v1ðx;oÞ

@x3V2ðx;oÞ ¼ �EI

@3v2ðx;oÞ@x3

(8.25)

F1ðx;oÞ ¼ �EA@u1ðx;oÞ

@xF2ðx;oÞ ¼ �EA

@u2ðx;oÞ@x

(8.26)

y1ðx;oÞ ¼ @v1ðx;oÞ@x

y2ðx;oÞ ¼ @v2ðx;oÞ@x

: (8.27)

∞ ∞

u1 u2v2v1

V2V1

VSFSMSF1 F2

θ1 θ2M1 M2

Ka

b

y

Kx

Kθ

∞ ∞

1

21

2

Fig. 8.1 The modeled

interface (a) spring models (b)

force diagram at the interface

of the bars


At the joint between the two rods we assume three linear elastic springs, which correspond to the three degrees of freedom in

the joint. The force and moment boundary conditions are applied at the joint (x ¼ 0) and are

M1ð0;oÞ ¼ M2ð0;oÞ ¼ �Kyðy2ð0;oÞ � y1ð0;oÞÞ (8.28)

V1ð0;oÞ ¼ V2ð0;oÞ ¼ �Kyðv2ð0;oÞ � v1ð0;oÞÞ � Kyxðu2ð0;oÞ � u1ð0;oÞÞ (8.29)

F1ð0;oÞ ¼ F2ð0;oÞ ¼ �Kxðu2ð0;oÞ � u1ð0;oÞÞ � Kxyðv2ð0;oÞ � v1ð0;oÞÞ (8.30)

where Ky, Ky, Kx, Kxy, and Kyx are the rotational, shear, axial, and cross stiffness values. Using (8.20)–(8.30) and assuming

that the area (A), elastic modulus (E) and area moment of inertia (I) are the same for both bars the unknown reflection and

transmission coefficients rB, rL, tB, and tL can be solved for. For each frequency the unknown coefficients can be written as

rL ¼ � kbarðk21k22 þ 2ik1 �Ky þ 2k2 �KyÞk21k

22ðkbar þ 2i�KxÞ þ k1ð2ikbar �Ky � 4�Kx

�Ky þ 4�Kxy�KyxÞ þ 2k2ðkbar �Ky þ 2ið�Kx

�Ky � �Kxy�KyxÞÞ

(8.31)

tL ¼ � 2ik21k22�Kx þ 4ik2ð�Kx

�Ky � Kxy�KyxÞ þ k1ð�4�KxKy þ 4�Kxy

�KyxÞk21k

22ðkbar þ 2i�KxÞ þ k1ð2ikbar �Ky � 4�Kx

�Ky þ 4�Kxy�KyxÞ þ 2k2ðkbar �Ky þ 2ið�Kx

�Ky � �Kxy�KyxÞÞ

(8.32)

rB ¼ �2k22kbar�Kyx

ð�ik1 þ k2Þðk21k22ðkbar þ 2i�KxÞ þ k1ð2ikbar �Ky � 4�Kx�Ky þ 4�Kxy

�KyxÞ þ 2k2ðkbar �Ky þ 2ið�Kx�Ky � �Kxy

�KyxÞÞÞ(8.33)

tB ¼ �rB (8.34)

where the dimensionless terms �Kx ¼ Kx

AE ,�Ky ¼ Ky

EI ,�Kyx ¼ Kyx

EI , and�Kxy ¼ Kxy

AE are the stiffness ratios. The ratios �Ky and �Kyx are

the ratios of the respective interfacial stiffness to the bending stiffness of the beam (EI) while �Kx and �Kxy are the ratios of the

respective interfacial stiffness to the axial stiffness of the bar (AE). Note that Ky does not appear in the solution so we can set

Ky ¼ 0. Note that the cross stiffness term Kyx governs the axial to flexural wave conversion. The preload has not been

explicitly modeled in the interface, the stiffness values �Kx, �Ky, �Kyx, and �Kxy are a function of the preload on the bars and will

be empirically determined.

The transmission and reflection efficiency are related measures with respect to the vibrational power [6, 7]. For materials

of the same cross-section and material properties the transmission efficiency t and the reflection efficiency x are

t ¼ transmitted vibrational power

incident vibrational power(8.35)

x ¼ reflected vibrational power

incident vibrational power: (8.36)

For the given problem we can write the transmission and reflection efficiencies for the transmitted and reflected longitudinal

waves as

tLðoÞ ¼ jtLðoÞj2 xLðoÞ ¼ jrLðoÞj2 (8.37)

and the efficiencies for the bending or flexural waves as

tBðoÞ ¼ jtBðoÞj2 xBðoÞ ¼ jrBðoÞj2 (8.38)

To test the solution we will examine a few simple cases. If we assume no connections between the rods choose �Kx ¼ �Ky ¼�Kyx ¼ �Kxy ¼ 0 then the transmission and reflection efficiencies become

xLðoÞ ¼ 1 tLðoÞ ¼ 0 xBðoÞ ¼ 0 tBðoÞ ¼ 0


for all frequencies o, which are the efficiencies for the ideal free end of the rod, full axial reflection and no transmission or

flexural conversion. If we assume only axial motion through a joint choose �Ky ¼ �Kyx ¼ �Kxy ¼ 0 and let �Kx ! 1 then the

transmission and reflection efficiencies become

xLðoÞ ¼ 0 tLðoÞ ¼ 1 xBðoÞ ¼ 0 tBðoÞ ¼ 0

for all frequencies o, which are the efficiencies for a stiff joint at the intersection of the rods, we have full axial transmission

and no reflection or flexural conversion. One more thing to note is that the transmitted and reflected efficiencies of the

flexural wave are always equal

xBðoÞ ¼ tBðoÞ (8.39)

this implies the ability to generate transmitted and reflected flexural waves of equal amplitudes at the interface that will

propagate away from the interface in both bars.

8.3.5 Total Transmission and Reflection

The reflection and transmission efficiencies were developed to be a function of frequency. In order to compare the

transmission and reflection of various preloads, the total transmission and reflection efficiency will be used and is the sum

of the efficiencies over frequency and can be written as

tTotal ¼Po transmitted vibrational powerPo incident vibrational power

(8.40)

xTotal ¼Po reflected vibrational powerPo incident vibrational power

: (8.41)

The efficiencies tTotal and xTotal are the total energy transmitted or reflected across the interface.

8.4 Apparatus

At the Air Force Research Laboratory, Munitions Directorate, Fuzes Branch (ARFL/RWMF), Dr. Jason Foley, Dr. Alain

Beliveau, and Dr. Janet Wolfson have developed the Preload Interface Bar (PIB) located at Eglin AFB [8]. The preload

interface apparatus puts two bars in compression and introduces a dynamic signal to propagate across the interface. The PIB

setup can be seen in Fig. 8.2a.

Interface

Hydraulic PressBackstop

Transmission Bar4' Steel Bar - 1.5 Ø

Incident Bar8' Steel Bar - 4' at 1.5Ø + 4' at 1.25" Ø Strain Gages

24" 18" 18"12"

Dynamicloading

ImpactHammer

a

b

Fig. 8.2 The preload interface bar (a) experimental setup (b) static and dynamic loading on the bar


Both incident and transmission bars in the preload interface bar setup are hardened AISI 1566 steel (r ¼ 7.8 g/cm3,

E ¼ 210 GPa, n ¼ 0.29). The transmission bar is 4 ft long and has a diameter of 1.5 in., while the incident bar is 8 ft long

with two 4 ft sections. One section has a diameter of 1.5 in. and the other has a diameter of 1.25 in.. This second bar is set up

such that the smaller diameter runs through a backstop. The two 1.5 in. diameter sections are compressed against each other

using a hydraulic ram. This causes the interface to be under a compressional load. A dynamic load is imparted on the bar on

the 1.25 in. diameter section and it propagates through the backstop, into the 1.5 in. diameter section and across the

interface. Because the bars were not match finished, with no preload the ends of the bars are aligned, but not square with one

another. This can be seen in Fig. 8.3.

8.4.1 Instrumentation

Load cells, both dynamic and static are used to input and record the forces imparted on the preload interface bar. The

dynamic response of the incident and transmission bars were recorded using strain gages along the bars at various preload

values. We also used pressure sensitive film to monitor the interface between the bars as the preload changes.

Multiple uniaxial strain gages aremounted on both the incident and transmission bar along the two 4 ft sections that are under

preload. The axial distribution of the gages allows the tracking of the stress wave propagating through the system. The gages

locations are measured from the backstop and are located at 24, 42, 54, and 72 in.. The 24 and 72 in. locations have four strain

gages placed axially on the bar, one every 90 ∘ . The 42 and 54 in. locations only have two gages distributed axially, one every

180 ∘ . The axial distribution of the gages is to use the diametrically opposed pairs to allow bending and extensional cancellation

for the observation of either only the axial or flexural mode respectively as shown in (8.18) and (8.19). Uniaxial semiconductor

strain gages were used in this setup for their fast response time ( � 10 ns) and correspondingly large bandwidth ( � 10 MHz).

The gage length is typically � 1 mm and the gage factor is around 150 [12], providing orders-of-magnitude of improvement in

sensitivity over typical foil gages with gage factors of 2. One common disadvantage of semiconductor gages is a strong

temperature dependence, this is not a concern for the dynamic tests run due to the short duration ( � 10 ms).

Two different load cells were used in the experiment, the first was a dynamic impact hammer and the second was a static

load cell. A PCB model 086C04 impact hammer was used to record the impulse force input to the system. A static load cell

was used to determine the amount of preload the hydraulic press put on the system. Tests were run with the preload from the

hydraulic press at 0–5,500 lb force. Also medium range (1,400–7,000 psi) Fujifilm Prescale pressure sensitive film was used

to monitor the loading of the interface with increments of 1,000 lb of preload on the system up to 5,000 lb.

The data acquisition system used was a National Instruments PXIe-1065 chassis with a NI PXIe-8130 Controller using

two PXI 6133 cards. Triggering is based on the analog channel with the reference force hammer. Signal conditioning and

sensor power is accomplished via a Precision Filter 28000 chassis with 28144A Quad-Channel Wideband Transducer

Conditioner with voltage and current excitation cards. The data from the strain gages and force hammer was recorded with a

sampling rate of 2.5 Mhz and Precision Filter system run with open bandwidth (� 800 kHz).

8.5 Results and Discussion

The several sets of tests were conducted using the preload interface bar. Sets of five tests were run with the hydraulic press at

15 different preload values ranging from 0 to 5,500 lb force. Assuming the conservation of mechanical energy over the

interface, the incoming longitudinal wave can be transmitted or reflected, and in doing so part of it may be converted to a

flexural wave.

Fig. 8.3 Partial gap between

the aligned bars with no

preload


The ends of the bars are aligned, but not square with one another. Pressure sensitive film was used to investigate how

uniform the load distribution is over the cross-section of the bar. The film was put between the bars at increments of 1,000 lb

pf preload. The medium range (1,400–7,000 psi) pressure sensitive film was used. The dark pink has a stress of 7,000 psi or

greater, while the white areas have stress values of less than 1,400 psi. The varying color saturation between white and dark

pink corresponds to a linear scale of stress values between 1,400 and 7,000 psi. The film can be seen in Fig. 8.4. Due to the

type of finishing used on the ends of the bars, the the interface is only loaded in the outer 0.25 in. annulus of the bar as can be

seen in Fig. 8.4a–e. As the preload increases the gap closes, this is evident with the pressure sensitive film.Experimental

stress transmission and reflection coefficients were calculated and can be seen in Fig. 8.5. At low preloads (0–2,000 lb),

when the interface is partially open it can be seen that flexural waves are generated at the interface and transmitted and

reflected at near equal amplitudes. A point of interest is that the amplitude of the flexural transmission and reflection

efficiencies (tBtotal and xB

total) are around 3, much greater than the theoretical maximum of 1. This indicates that we are not

accounting for energy that is contributing to the flexural waves. We hypothesize that this unaccounted energy may be due to

the preload on the system, the incident energy used was only the incident longitudinal wave. Over the entire preload range

the longitudinal transmission of energy increases as the preload increases due to the stiffening of the interface. The

transmission acts much similar to a smoothed step function, only allowing a partial longitudinal transmission until the

critical preload range, between 1,000 and 2,000 lb, is reached which then the interface is closed and allows full transmission

of the longitudinal wave and no generation of the flexural wave. Notice that the change from generating flexural waves to

transmission of pure longitudinal waves is gradual over the preload range of 1,000–2,000 lb.

Time histories of the axial and flexural strain at representative low and high preload values of 100 and 4,500 lb can be

seen in Fig. 8.6. The longitudinal wave has a reflection at the interface for when the preload is 100 lb but no reflection in the

4,500 lb preload case. There is close to perfect axial transmission for a preload of 4,500 lb, and a small flexural wave that

travels the length of the bar generated with the impact hammer. When the preload is 100 lb a flexural wave is generated at

the interface and superimposed on the small flexural wave, we know this because the gages at 42 and 54 in. first see the large

flexural wave and then the gages at 24 and 72 in. see it close to 200 ms later.

Fig. 8.4 Pressure sensitive

film at the interface at various

preloads: (a) 1,000 lb

(b) 2,000 lb (c) 3,000 lb

(d) 4,000 lb (e) 5,000 lb

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500

0

0.5

1

1.5

2

2.5

3

Preload [lbs]

Tra

nsm

ission

and

Ref

lect

ion

Effic

ienc

ies τ

L

Total

ξL

Total

τB

Total

ξB

Total

Fig. 8.5 Experimental total

transmission and reflection

efficiencies as a function of

preload


Using (8.31)–(8.34) we constructed a numerical model and tried to match the transmission and reflection efficiencies to

the experimental data. A nonlinear least squares algorithm was used to minimize the error between the experimental data and

theory on the frequency interval of 0–10 kHz. The upper frequency bound of 10 kHz was chosen to allow for enough

frequency excitation amplitude to give us meaningful experimental values, the excitation amplitude dies off around 15 kHz.

We used 500 randomly generated initial dimensionless stiffness values for each preload value to find the stiffness values thatminimize the error. The stiffness values for each preload value can be seen in Fig. 8.7.

For the low preload regime (0–1,000 lb) the transverse stiffness ratio �Ky is at the upper bound (10,000), the transverse

cross stiffness ratio �Kyx is gradually increasing with preload, the longitudinal cross-stiffness ratio �Kxy remains constant

1200 1400 1600 1800 2000 2200-20

-10

0

10

20Longitudinal Strain time histories, Interface Bar - 4 ft Steel: Preload 100 lb

Time t [μs] Time t [μs]

Time t [μs] Time t [μs]

Mic

rost

rain

e(t)

[μm

/m]

24-0042-0054-0072-00

1200 1400 1600 1800 2000 2200-20

-10

0

10

20Flexural Strain time histories, Interface Bar - 4 ft Steel: Preload 100 lb

Mic

rost

rain

e(t)

[μm

/m]

1200 1400 1600 1800 2000 2200-20

-10

0

10

20Longitudinal Strain time histories, Interface Bar - 4 ft Steel: Preload 4500 lb

Mic

rost

rain

e(t)

[μm

/m]

24-0042-0054-0072-00

1200 1400 1600 1800 2000 2200-20

-10

0

10

20Flexural Strain time histories, Interface Bar - 4 ft Steel: Preload 4500 lb

Mic

rost

rain

e(t)

[μm

/m]

a b

Fig. 8.6 Strain response to impulse hammer with a preload of (a) 100 lb (b) 4,500 lb

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 55000

500

1000

1500

2000

Preload [lb]

Optimized Stiffness Values

Dim

ension

less

Stiffne

ss

Ky

Kx

Kxy

Kyx

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 55000

2000

4000

6000

8000

10000

Preload [lb]

Optimized Stiffness Values

Dim

ension

less

Stiffne

ss

Ky

Kx

Kxy

Kyx

a b

Fig. 8.7 Numerical dimensionless stiffness values calculated as a function of preload (a) full value range (b) closer view of dimensionless

stiffness ratios in the range from 0 to 2,000


around 80, and the longitudinal stiffness ratio �Kx is low ( < 10). In the transition preload regime (1,000–2,000 lb) the

transverse stiffness ratio �Ky and the longitudinal cross-stiffness ratio �Kxy drops to a near 0 values, the transverse cross

stiffness ratio �Kyx jumps to the upper bound (10,000), and the longitudinal stiffness ratio �Kx increases with preload to around

500. For the high preload regime ( � 2,000 lb) the only value that changes is the longitudinal stiffness ratio �Kx which

generally increases as the preload in the system increases. This implies that the interface is stiffing with the preload and

agrees with the general trend of Daehnke and Rossmanith’s empirical-based model [4].

The transmission and reflection efficiencies of 100 and 4,500 lb preloads are shown in Fig. 8.8. The dimensionless stiffness

values used to generate the analytical model at 100 lb preload shown in Fig. 8.8a are: �Kx ¼ 8, �Ky ¼ 10; 000, �Kxy ¼ 76 and,�Kyx ¼ 1; 420. For the 4,500 lb preload shown in Fig. 8.8b the dimensionless stiffnesses are: �Ky ¼ 0, �Kx ¼ 930, �Kxy ¼ 0,�Kyx ¼ 10; 000. Themodeling of the efficiencies for the 4,500 lb of preloadmatches verywellwith a least square error residual of

0.26, this is due to the near perfect interface created by the preload over the frequency range of 0–10 kHz. The model behavior

for preloads below 2,000 lb do not match the experimental data very well, the least square error residual is 30.34. Themismatch

of the model and the experimental data can be seen in Fig. 8.8a. This is due to the fidelity of the model. The elementary rod

theory and Euler-Bernoulli beam theory were assumed in the model development, these models do have the correct degrees of

freedom, but do not accurately capture the spectral dynamics of the interface. The model moderately captures the general trend,

but needs to be refined to more accurately capture the dynamics for the cases where preload is less than 2,000 lb.

8.6 Conclusion

The effect of the partial gap interface on the transmission, reflection, and conversion of waves was experimentally and

analytically analyzed. It was experimentally shown that at low preloads (0–2,000 lb) the partial gap generated near-equal

amplitude transmitted and reflected flexural waves, and a small amount of the energy was longitudinally transmitted. At high

1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

Frequency [kHz]

Transmission and Reflection Efficiencies Interface Bar - 4 ft Steel: Preload 100 lb

ξ L

TheoryExperiment

1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

Frequency [kHz]

ξ L

1 2 3 4 5 6 7 8 9 100

1

2

Frequency [kHz]

τ B

1 2 3 4 5 6 7 8 9 100

1

2

Frequency [kHz]

ξ B

ξ Lξ L

τ Bξ B

1 2 3 4 5 6 7 8 9 100

1

1.5

0.5

0

1

1.5

0.5

Frequency [kHz]

Transmission and Reflection Efficiencies Interface Bar - 4 ft Steel: Preload 4500 lb

TheoryExperiment

1 2 3 4 5 6 7 8 9 10Frequency [kHz]

1 2 3 4 5 6 7 8 9 100

1

2

Frequency [kHz]

1 2 3 4 5 6 7 8 9 100

1

2

Frequency [kHz]

a b

Fig. 8.8 Transmission and reflection efficiencies with a preload of and stiffness coefficients of (a) 100 lb with �Kx ¼ 8, �Ky ¼ 10; 000, �Kxy ¼ 76,�Kyx ¼ 1; 420 (b) 4,500 lb �Ky ¼ 0, �Kx ¼ 930, �Kxy ¼ 0, �Kyx ¼ 10; 000


preloads ( � 2,000 lb) the system allowed near 100% londitudinal transmission and no flexural wave generation at the

interface. An analytical model was developed modeling the partial gap interface as an elastic stiffness with cross terms. The

analytical model did not adequately predict the low preload transmission and reflection due to lack of spectral fidelity, but

did capture the high preload transmission. The model shows that as the preload increases above 2,000 lb, the stiffness of the

interface increases. Future work includes refining the model to accurately caputure the spring gap behavior at low preloads.

This analysis gives us some insight to see how preloaded intefaces act under dynamic loading, taking us one step closer to

fully understanding a complex prestressed mechanical system.

Acknowledgements J. Dodson would like to acknowledge support from the Department of Defense SMART (Science, Mathematics, And

Research Transformation) scholarship program. The authors also wish to thank the Air Force Office of Scientific Research (PM: Dr. David Stargel)

for supporting this project. Opinions, interpretations, conclusions and recommendations are those of the authors and are not necessarily endorsed

by the United States Air Force.

References

1. Adams D, Yoder N, Butner C, Bono R, Foley J, Wolfson J (2011) Transmissibility analysis for state awareness in high bandwidth structures

under broadband loading conditions. Nonlinear modeling and applications, vol. 2. In: Proulx T (ed) Vol. 11 of Conference proceedings of the

society for experimental mechanics series, Springer, New York, pp 137–148

2. Chattopadhyay S (1993) Dynamic response of preloaded joints. J Sound Vib 163(3):527–534

3. Offterdinger K, Waschkies E (2004) Temperature dependence of the ultrasonic transmission through electrical resistance heated imperfect

metal-metal interfaces. NDT & E Int 37(5):361–371

4. Daehnke A, Rossmanith H.P (1997) Reflection and refraction of plane stress waves at interfaces modelling various rock joints. Fragblast: Int J

Blasting Fragm 1(2):111–231

5. Barber J, Comninou M, Dundurs J (1982) Contact transmission of wave motion between two solids with an initial gap. Int J Solids Struct 18

(9):775–781

6. Leung R, Pinnington R (1990) Wave propagation through right-angled joints with compliance-flexural incident wave. J Sound Vib 142

(1):31–48

7. Leung R, Pinnington R (1992) Wave propagation through right-angled joints with compliance: longitudinal incidence wave. J Sound Vib 153

(2):223–237

8. Foley JR, Dodson JC, McKinion CM, Luk VK, Falbo GL (2010) Split Hopkinson bar experiments of preloaded interfaces. In: Proceedings of

the IMPLAST 2010 conference, SEM, 2010

9. Kolsky H, (1963) Stress waves in solid. Clarendon Press, Oxford

10. Doyle JF (1997) Wave propagation in structures : spectral analysis using fast discrete Fourier transforms, 2nd edn. Springer, New York

11. Inman DJ (2007) Engineering Vibration, Pearson, Upper Saddle River

12. Kulite Semiconductor Products, Kulite Strain Gage Manual, 2001


Chapter 9

Equivalent Reduced Model Technique Development

for Nonlinear System Dynamic Response

Louis Thibault, Peter Avitabile, Jason R. Foley, and Janet Wolfson

Abstract The dynamic response of structural systems commonly involves nonlinear effects. Often times, structural

systems are made up of several components, whose individual behavior is essentially linear compared to the total

assembled system. However, the assembly of linear components using highly nonlinear connection elements or contact

regions causes the entire system to become nonlinear. Conventional transient nonlinear integration of the equations of

motion can be extremely computationally intensive, especially when the finite element models describing the components

are very large and detailed.

In this work, the Equivalent Reduced Model Technique (ERMT) is developed to address complicated nonlinear contact

problems. ERMT utilizes a highly accurate model reduction scheme, the System Equivalent Reduction Expansion Process

(SEREP). Extremely reduced order models that provide dynamic characteristics of linear components, which are

interconnected with highly nonlinear connection elements, are formulated with SEREP for the dynamic response evaluation

using direct integration techniques. The full-space solution will be compared to the response obtained using drastically

reduced models to make evident the usefulness of the technique for a variety of analytical cases.

Keywords Nonlinear analysis • Forced response • Linear components for nonlinear analysis • Reduced order modeling •

Modal analysis

Nomenclature

Symbols

½Xn� Full set displacement vector

½Xa� Reduced set displacement vector

½Xd� Deleted set displacement vector

Ma½ � Reduced mass matrix

Mn½ � Expanded mass matrix

Ka½ � Reduced stiffness matrix

Kn½ � Expanded stiffness matrix

Ua½ � Reduced set shape matrix

Un½ � Full set shape matrix

Ua½ �g Generalized inverse

L. Thibault (*) • P. Avitabile

Structural Dynamics and Acoustic Systems Laboratory, University of Massachusetts Lowell, One University Avenue,

Lowell, MA 01854, USA


J.R. Foley • J. Wolfson

Air Force Research Laboratory, Munitions Directorate, Fuzes Branch, Eglin Air Force Base, 306 W. Eglin Blvd., Bldg 432,

Eglin AFB, FL 32542-5430, USA



95


T½ � Transformation matrix

TU½ � SEREP transformation matrix

½p� Modal displacement vector

M½ � Physical mass matrix

C½ � Physical damping matrix

K½ � Physical stiffness matrix

½F� Physical force vector

½€x� Physical acceleration vector

½ _x� Physical velocity vector

½x� Physical displacement vector

a Parameter for Newmark integration

b Parameter for Newmark integration

Dt Time step

U12½ � Mode contribution matrix

9.1 Introduction

Nonlinear response analysis typically involves significant computation, especially if the system matrices for the full

analytical model are used to obtain the forced nonlinear response solution. Due to the significant computational resources

required for these types of nonlinear problems, the analyst may often be unable to investigate specific nonlinear scenarios in

depth, particularly if the nonlinear elements are characterized with a set of performance characteristics related to tempera-

ture, preload, deflection, etc. Thus, there is significant motivation to develop several reduced order models that can

accurately predict nonlinear system response with substantially reduced computation time.

A particular area of interest is the dynamic response of systems with nonlinear connections. These systems are typically

made up of several components, whose individual behavior is essentially linear compared to the total assembled system.

Local regions where component interconnections exist cause the entire system to become nonlinear. The components that

make up the system may be linear but the response of the system is nonlinear due to the nature of the nonlinear component

interconnection. The technique employed in this paper, the Equivalent Reduced Model Technique (ERMT), was developed

to address this class of nonlinear problem.

ERMT [1] is implemented in this work using the System Equivalent Reduction Expansion Process (SEREP) [2], which

allows for the formulation of a dramatically reduced model that accurately preserves the full analytical model dynamics with

very few degrees of freedom (DOFs). Discrete nonlinear connection elements are then assembled into the reduced model in

the local regions where component interconnections occur. Using the reduced models that are developed with ERMT in

conjunction with direct integration techniques allows computationally efficient forced response solutions to be obtained.

These techniques can also be easily extended to experimental components if the system matrices are updated using any of the

direct model updating techniques such as those identified in [3].

This approach was first presented by Avitabile and O’Callahan [4] where a detailed overview of the applicable theory was

provided, along with a simple analytical example. Friswell et al. [5] looked at reducing models with local nonlinearities

using several different reduction schemes for a periodic solution. Lamarque and Janin [6] looked at modal superposition for

simple single-DOF and two-DOF systems with impact and concluded that modal superposition had limitations due to

difficulties in developing the general formulas with the nonlinear impacts. Ozguven and Kuran [7] converted nonlinear

Ordinary Differential Equations (ODE’s) into a set of nonlinear algebraic equations, which could be reduced by using linear

modes. This technique was found to provide the best reduction in computation time when the structure was excited at a

forcing frequency that corresponded to a resonance of the structure. An alternative approach that has been studied uses

Nonlinear Normal Modes (NNMs) which are formulated by Ritz vectors [8, 9]. This approach seeks to extend the concept of

linear orthogonal modes to nonlinear systems.

This paper presents the analytical time response results of a cantilevered beam system subjected to an input force pulse.

Four cases are studied: single beam with no contact, single beam with single contact, two beams with single contact, and two

beams with two contacts. For each of these cases, two types of contact stiffness are considered, a soft contact representing a

rubber/isolation material, and a hard contact representing a metal on metal contact. For all cases, the time response results of

the full-space model will be used as the reference solution.

96 L. Thibault et al.

9.2 Theory

9.2.1 Equivalent Reduced Model Technique (ERMT)

TheEquivalentReducedModel Technique (ERMT) is based on concepts related tomodel reduction,which are summarized herein.

9.2.1.1 General Reduction Techniques

Model reduction is typically performed to reduce the size of a large analytical model to develop a more efficient model for

further analytical studies. Most reduction or condensation techniques affect the dynamic characteristics of the resulting

reduced model. Model reduction is performed for a number of reasons, but the technique is used primarily as a mapping

technique for expansion. In general, a relationship between the full set of finite element DOFs and the reduced set of DOFs

needs to be formed as

Xnf g ¼ Xa

Xd

� �¼ T½ � Xaf g (9.1)

The ‘n’ subscript denotes the full set of finite element DOFs, the ‘a’ subscript denotes the active set of DOFs (sometimes

referred to as master DOFs), and the subscript ‘d’ denotes the deleted DOFs (sometimes referred to as omitted DOFs); the

[T] transformation matrix relates the mapping between these two sets of DOFs.

The reduced mass and stiffness matrices are related to the full-space mass and stiffness matrices using congruent matrix

operations as

Ma½ � ¼ T½ �T Mn½ � T½ � and Ka½ � ¼ T½ �T Kn½ � T½ � (9.2)

What is most important in model reduction is that the eigenvalues and eigenvectors of the original system are preserved as

accurately as possible in the reduction process. If this is not maintained then the matrices are of questionable value.

The eigensolution is then given by

Ka½ � � l Ma½ �½ � Xaf g ¼ 0f g (9.3)

Because reduction schemes such as Guyan Condensation [10] and Improved Reduced System Technique [11] are based

primarily on the stiffness of the system, the eigenvalues and eigenvectors will not be exactly reproduced in the reduced

model. However, the System Equivalent Reduction Expansion Process (SEREP) [2] exactly preserves the eigenvalues

and eigenvectors in the reduced model.

9.2.1.2 System Equivalent Reduction Expansion Process (SEREP)

The SEREP modal transformation relies on the partitioning of the modal equations representing the system DOFs relative to

the modal DOFs using

Xnf g ¼ Xa

Xd

� �¼ Ua

Ud

� �pf g (9.4)

Using a generalized inverse, this can be manipulated to give

pf g ¼ Ua½ �T Ua½ �� 1

Ua½ �T Xaf g ¼ Ua½ �g Xaf g (9.5)

which is then used to relate the ‘n’ DOFs to the ‘a’ DOFs as

Xnf g ¼ Un½ � Ua½ �g Xaf g ¼ TU½ � Xaf g (9.6)

with

TU½ � ¼ Un½ � Ua½ �g (9.7)

9 Equivalent Reduced Model Technique Development for Nonlinear System Dynamic Response 97

Equation (9.7) represents the SEREP transformation matrix that is used in the reduction of the finite element mass and

stiffness matrices as described in (9.2).

SEREP relies heavily on a “well developed” finite element dynamic model from which an ‘n’ dimensional eigensolution

is obtained. In addition, the quality of the SEREP reduced model depends on the selection of active ‘a’ DOFs used in the

formulation of the generalized matrix inverse of the ‘a’ DOFs modal vector partition. Both conditions affect the rank and

matrix conditioning required to define a good SEREP transformation matrix needed to develop well-behaved reduced system

matrices. The reduced models developed in this work are for the equivalent condition, where the number of modes used is

equal to the number of DOFs retained in the reduction process.

Since the transformation matrix is formed from the eigenvectors of the full-space finite element model, the reduced

matrices preserve the eigenvalues and eigenvectors of the full-space model. This implies that any collection of desired

eigenvectors can be retained in an exact sense for the reduced model. This fact is significant in terms of the development of

efficient models from large finite element models used for forced response studies, especially those that contain discrete

nonlinear effects that are typical in joints and connections for many structural systems.

An additional model reduction approach [12] can be used where a highly accurate reduced model is formulated by using

Guyan reduction in conjunction with analytical model improvement. The advantages of this technique are that the desired

mode shapes are preserved in an exact sense while the fully ranked and well-conditioned system matrices obtained from

Guyan reduction are maintained in the process.

9.2.1.3 Mode Contribution Identification

To ensure that the reduced models are minimally affected by modal truncation when assembling system models using

multiple reduced component models, the contribution of component modes to the assembled system is computed using

Mode Contribution ¼ UAn

� UB

n

� � �TMAB

n

� UAB

n

� (9.8)

where [UA] and [UB] are the unmodified component modal matrices that are organized into a partitioned matrix, [MAB] is the

modified system mass matrix, and [UAB] is the modified system modal matrix. The mode contribution matrix is computed

for all possible system configurations using full-space component models.

The resulting mode contribution matrix is the key to identifying the necessary set of modal vectors to accurately obtain

the final modified system modes. This is similar to using the [U12] matrix that is computed in Structural Dynamic

Modification (SDM) [13] to identify the contributions of component modes in the assembled system modes. The [U12]

matrix contains the scaling coefficients needed to form the final modified set of modal vectors [U2] from the initial

unmodified set of modal vectors [U1]. Figure 9.1 illustrates how the [U12] matrix is used in forming the final modified set

of modes [U2], where ‘m’ modes of the [U12] matrix are used, and ‘n-m’ modes are excluded.

Fig. 9.1 Mode contribution identification using [U12] matrix from SDM


9.2.1.4 Response Analysis Technique

The reduced component mass and stiffness matrices described in (9.2) are used in a normal system assembly to connect the

linear components with highly nonlinear connection elements. Nonlinear direct integration of the equations of motion is then

performed to obtain the system response. The ERMT process is shown schematically in Fig. 9.2.

9.2.1.5 Newmark Direct Integration Technique

In this work, the Newmark Method [14] is used to perform the direct integration of the equations of motion for the ERMT

solution process. From the known initial conditions for displacement and velocity, the initial acceleration vector is computed

using the equation of motion and the applied forces as

€~x0 ¼ M½ ��1 ~F0 � C½ � _~x0 � K½ �~x0� �

(9.9)

where€~x0 ¼ initial acceleration vector_~x0 ¼ initial velocity vector

x0 ¼ initial displacement vector~F0 ¼ initial force vector

Choosing an appropriate Dt, a, and b, the displacement vector is

~xiþ1 ¼ 1

a Dtð Þ2 M½ � þ baDt

C½ � þ K½ �" #�1

~Fiþ1 þ M½ � 1

a Dtð Þ2~xi þ1

aDt_~xi þ 1

2a� 1

�€~xi

!(

þ C½ � baDt

~xi þ ba� 1

�_~xi þ b

a� 2

�Dt2€~xi

�)(9.10)

FORMULATE PHYSICALDATA BASE

PERFORM NUMERICAL INTEGRATION FOR NEXT ΔT

CHECK FOR GAPS ORNONLINEARITY

ANY CHANGE IN THE CURRENT LINEAR PHYSICAL STATE?

SUBSTITUTE M, C, K MATRICES FOR APPLICABLE

CONTACT STATE

DETERMINE FORCE AND/ORINITIAL CONDITIONS

NO YES

Fig. 9.2 Schematic for ERMT process


The values chosen for a and b were ¼ and ½, respectively. This assumes constant acceleration and the integration process is

unconditionally stable, where a reasonable solution will always be reached regardless of the time step used. However, the time

step should be chosen such that the highest frequency involved in the system response can be characterized properly to avoid

numerical damping in the solution. The time step should be chosen to be at least 10 times smaller than the period of the highest

frequency involved in the system response. The time step used for the analytical cases studied in this paper was 0.0001 s.

Following the displacement vector calculation, the acceleration and velocity vectors are computed for the next time

step using

_~xiþ1 ¼ _~xi þ 1� bð ÞDt€~xi þ bDt€~xiþ1 (9.11)

€~xiþ1 ¼ 1

a Dtð Þ2 ~xiþ1 �~xið Þ � 1

aDt_~xi � 1

2a� 1

�€~xi (9.12)

This process is repeated at each time step for the duration of the time response solution desired.

9.2.2 Time Response Correlation Tools

In order to quantitatively compare two different time solutions, two correlation tools were employed: The Modal Assurance

Criterion (MAC) and the Time Response Assurance Criterion (TRAC).

9.2.2.1 Modal Assurance Criterion (MAC)

The Modal Assurance Criterion (MAC) [15] is widely used as a vector correlation tool. In this work, the MAC was used to

correlate all DOF at a single instance in time. The MAC is written as

MACij ¼X1if gT X2j

� � 2X1if gT X1if g�

X2j� T

X2j� h i (9.13)

where X1 and X2 are displacement vectors. MAC values close to 1.0 indicate strong similarity between vectors, where

values close to 0.0 indicate minimal or no similarity.

9.2.2.2 Time Response Assurance Criterion (TRAC)

The Time Response Assurance Criterion (TRAC) [3] quantifies the similarity between a single DOF across all instances in

time. The TRAC is written as

TRACji ¼X1j� T

X2if gh i2

X1j� T

X1j� h i

X2if gT X2if g� (9.14)

where X1 and X2 are displacement vectors. TRAC values close to 1.0 indicate strong similarity between vectors, where

values close to 0.0 indicate minimal or no similarity.

9.3 Model Description and Cases Studied

This section presents the analytical models developed as well as the cases studied. The full-space time solution is used as the

reference solution for all cases.


9.3.1 Linear Component Models: Beam A and Beam B

Two planar element beam models were generated using MAT_SAP [16], which is a finite element modeling (FEM) program

developed for MATLAB [17], and were used for all of the cases studied. Figure 9.3 shows the two beams assembled into

the linear system, where the red points are the active DOFs in the reduced order models, and the black arrow denotes the

force pulse input location (DOF 105). Note that 3 in. of each beam are clamped for the cantilevered boundary condition

that was applied.

Table 9.1 lists the characteristics of the beam models and Table 9.2 lists the natural frequencies for the first 10 modes of

each beam component model. The mode shapes for the unmodified beam components are provided in Appendix A. Damping

was assumed 1% of critical damping for all unmodified component modes as well as for all modified system modes in all of

the cases studied.

The force pulse input to the system is an analytic force pulse designed to be frequency band-limited, exciting modes up to

1,000 Hz while minimally exciting higher order modes. Using this force pulse, the number of modes involved in the response

can be determined easily, as modes above 1,000 Hz can be considered to have negligible participation in the response.

Modes 1–5 will be primarily excited in Beam A, while modes 1–4 will be primarily excited in Beam B. Figure 9.4 shows the

analytical force pulse in the time and frequency domain.

The full-space linear component models were reduced down to ‘a’ and ‘aa’ space using SEREP. The active DOFs and

modes retained in the reduced component models are listed in Table 9.3. Note that only translational DOF were used in the

reduced component models.

F

Full ‘n’ Space

‘a’ Space

‘aa’ Space

F

F

Beam A

Beam A

Beam A

Beam B

Beam B

Beam B

Fig. 9.3 Schematic of linear

beam models with force pulse

input location

Table 9.1 Beam model

characteristicsProperty Beam A Beam B

Length (in.) 18 16

Width (in.) 2 4

Thickness (in.) 0.123 0.123

# of elements 72 64

# of nodes 73 65

# of DOF 146 130

Node spacing (in.) 0.25 0.25

Material Aluminum Aluminum

Mass density (lb/in.3) 2.54E�4 2.54E�4

Young’s modulus (Msi) 10 10


9.4 Case 1: Single Beam with No Contact

For the first case, the system is a single beam (Beam A) with no contact. This system is linear and there is no change in the

state of the system, where the number of modes needed is only a function of the input force spectrum. Based on the input

force spectrum seen previously in Fig. 9.4 and the natural frequencies for Beam A in Table 9.2, five modes are needed to

accurately compute the system time response. Therefore, the smallest reduced model that can be used to prevent the effects

of mode truncation is five DOF to maintain the SEREP condition, as discussed previously. To determine if this assumption is

accurate, the system response was plotted in the time and frequency domain for ‘aa’ space with comparison to the full ‘n’

space solution, as shown in Fig. 9.5. The response for the ‘a’ space model was not shown, due to the accurate results

observed from the ‘aa’ space model response.

Table 9.2 Natural frequencies

of unmodified beam component

models

Mode # Beam A Beam B

1 12.91 22.62

2 84.12 141.56

3 252.34 396.60

4 519.59 776.92

5 806.16 1,284.71

6 1,256.55 1,918.28

7 1,682.96 2,678.33

8 2,201.36 3,563.89

9 2,755.52 4,572.70

10 3,510.01 5,707.04

0 500 1000 1500-110

-105

-100

-95

-90

-85

-80

-75

-70

-65

-60

Frequency (Hz)

dB F

orce

(lb

f)

FFT of Analytical Force Pulse

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-20

-15

-10

-5

0

5

Time (sec)

For

ce (

lbf)

Analytical Time Domain Force Pulse

Fig. 9.4 Analytical force pulse in the time domain (left) and frequency domain (right)

Table 9.3 Active DOFs and modes retained in the reduced component models

Model # of DOF Retained modes Active DOF

Beam A – ‘a’ space 17 1–17 33, 41, 49, 57, 65, 73, 81, 89, 97, 101, 105, 113, 121, 129, 137, 141, 143

Beam A – ‘aa’ space 5 1–5 57, 81, 101, 105, 141

Beam B – ‘a’ space 14 1–14 3, 5, 9, 17, 25, 33, 41, 45, 49, 57, 65, 73, 81, 89

Beam B – ‘aa’ space 5 1–5 5, 25, 45, 65, 89


The system response in Fig. 9.5 shows that the reduced model is able to accurately capture the response of the system due

to the inclusion of the modes primarily excited by the input force pulse. To quantify the similarity of the reduced model

results with the full-space solution, the MAC and TRAC were computed between the full model and the reduced model time

responses, which were then averaged, as listed in Table 9.4. The solution time for each model is also listed to show the

decrease in computation time when reduced models are used.

Table 9.4 provides further confirmation that the ‘aa’ space reduced model is sufficient for accurately computing the time

response for this particular case. In addition, the solution time for the full-space model is over 30 s in contrast to the reduced

‘aa’ space model, which is less than 1 s. ERMT was shown in this case to provide significant reduction in computation time

for a linear system.

This first case demonstrates that an accurate time response solution can be obtained using a drastically reduced model

with limited number of modes, if the primary modes excited by the structure are retained in the reduced model. However, if

the retained modes are selected incorrectly, an accurate time response will not be obtained, regardless of how many

additional modes and active DOFs are retained in the reduced model.

The following cases will show the application of ERMT for contact situations, which causes the system to become

nonlinear. In addition, both soft and hard contacts will be studied to show the effect that different contact stiffness has on the

accuracy of the reduced model results. The soft stiffness case will be studied first, as this contact stiffness is unlikely to excite

a higher frequency range than the input force pulse applied to the models.

9.5 Case 2: Soft Contact

9.5.1 Case 2-A: Single Beam with Single Soft Contact

This case consists of the tip of Beam A coming into contact with a fixed object once the beam has displaced a known gap

distance of 0.05 in., as shown in Fig. 9.6 for the full-space and reduced space models.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Time (sec)

Dis

plac

emen

t (in

)DOF 141 Time Response - Single Beam No Contact

n Space-146 DOF

aa Space-5 DOF

0 500 1000 1500-200

-180

-160

-140

-120

-100

-80

-60

-40

Frequency (Hz)

dB D

ispl

acem

ent (

in)

FFT of DOF 141 Time Response - Single Beam No Contact

n Space-146 DOF

aa Space-5 DOF

Fig. 9.5 Comparison of ‘n’ and ‘aa’ spacemodels for single beamwith no contact for DOF 141 – time response (left) and FFT of time response (right)

Table 9.4 Average MAC and TRAC for reduced models and solution times for single beam with no contact

Model # of DOF Average MAC Average TRAC Solution time (s)

‘n’ space 146 1 1 34.6

‘a’ space 17 0.99999687 0.99999999 1.9

‘aa’ space 5 0.99993281 0.99999992 0.7


The contact is represented as an additional spring stiffness that is added when the contact location on Beam A closes the

specified gap distance. This is performed by exchanging the physical system matrices from the initial no contact state to

the new contact state until the contact is separated. The contact stiffness of 10 lb/in. will be used to represent a soft contact,

which is typically seen in a damper or isolation mount. For the purposes of ERMT, the spring stiffness was applied to the

full-space physical model, prior to model reduction. In addition, the contact is represented as a spring stiffness in only the

translational DOF, and not in the rotational DOF as well. The contact location is at DOF 141 of Beam A. Table 9.5 lists the

first ten natural frequencies of the system with the soft contact stiffness applied. Figure 9.7 shows the mode contribution

matrix, which indicates the unmodified component modes that participate in the modified system modes. The mode

contributions are computed using full-space models so that modal truncation is not a concern. The various box colors

indicate the amount that each of the unmodified component modes contributes to a modified system mode; the actual

contribution ranges for each color are shown. The mode shapes for the modified system are provided in Appendix A.

Table 9.5 and Fig. 9.7 show that the addition of the soft spring has a pronounced effect on the first two lower order modes,

with modes 3 and higher remaining relatively unaffected. Figure 9.7 indicates that both modes 1 and 2 of the unmodified

component are needed to obtain either mode 1 or 2 of the modified system. The frequencies for modes 3 and higher of the

unmodified component are minimally affected for the modified system. Therefore, additional unmodified component mode

shapes are not needed to form the modified system modes above mode 3 (as indicated by the red main diagonal). For a soft

contact, the input force pulse primarily dominates the frequency spectrum excited. Therefore, the modes required to

accurately predict the system response are modes 1–5 for the unmodified component as well as for the modified system.

To confirm this, the system response was plotted in the time and frequency domain for ‘aa’ space with comparison to the full

‘n’ space solution, as shown in Fig. 9.8. The response for the ‘a’ space model was not shown, due to the accurate results

observed from the ‘aa’ space model response.


for single beam with single soft

contact

Mode # Unmodified Single soft contact

1 12.91 26.09

2 84.12 86.02

3 252.34 252.54

4 519.59 519.65

5 806.16 806.18

6 1,256.55 1,256.55

7 1,682.96 1,682.96

8 2,201.36 2,201.36

9 2,755.52 2,755.53

10 3,510.01 3,510.02

Full ‘n’ Space

‘a’ Space

‘aa’ Space

F

F

F

Fig. 9.6 Diagram of single

beam with single contact for

full-space and reduced space

models


The system response in Fig. 9.8 shows that the reduced model accurately captures the response of the system due to the

inclusion of the modes primarily excited by the input force pulse. To quantify the similarity of the reduced model results with

the full-space solution, the MAC and TRAC were computed between the full model and the reduced model time responses,

which were then averaged, as listed in Table 9.6. The solution time for each model is also listed to show the decrease in

computation time when reduced models are used.

Table 9.6 provides further confirmation that the ‘aa’ space reduced model is sufficient for accurately computing the time

response for this particular case. In addition, the solution time for the full-space model is over 30 s in contrast to the reduced

‘aa’ space model, which is less than 1 s. ERMT was shown in this case to provide significant reduction in computation time

for a nonlinear system that consists of a single component with a single soft contact.

Table 9.6 Average MAC and TRAC for reduced models and solution times for single beam with single soft contact


‘n’ space 146 1 1 34.8

‘a’ space 17 0.99999687 0.99999999 1.9

‘aa’ space 5 0.99993256 0.99999979 0.7

0 500 1000 1500-200

-150

-100

-50

Frequency (Hz)

dB D

ispl

acem

ent (

in)

FFT of DOF 141 Time Response - Single Beam Single Soft Contact

n Space-146 DOFaa Space-5 DOF

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-0.1

-0.05

0

0.05

0.1

0.15

Time (sec)

Dis

plac

emen

t (in

)

DOF 141 Time Response - Single Beam Single Soft Contact

n Space-146 DOFaa Space-5 DOF

Fig. 9.8 Comparison of ‘n’ and ‘aa’ space models for single beam with single soft contact for DOF 141 – time response (left) and FFT of

time response (right)

MODEModified System Mode Shapes

Single 10 lb/in Contact

1 2 3 4 5 6 7 8 9 10

Unm

odifie

d B

eam

A M

ode

Sha

pes 1

2

3

4

5

6

7

8

9

10

Bar Color Min Value Max Value

Black 0.005 0.1

Blue 0.1 0.2

Green 0.2 0.3

Cyan 0.3 0.5

Magenta 0.5 0.7

Yellow 0.7 0.9

Red 0.9 1.0

Fig. 9.7 Mode contribution matrix for single beam with single soft contact


This second case demonstrated that the mode contribution matrix should be examined to determine the number of

unmodified component modes that are needed to obtain the modified system modes of interest. The input force pulse may

excite only a few modified system modes, but several unmodified component modes may be required to produce the

modified system modes. Failure to include the contributing modes will result in an incorrect time response, regardless of how

many additional modes and active DOFs are retained in the reduced model. When the mode contribution matrix is examined

and the correct unmodified component modes are used in the reduced model, an accurate nonlinear time response solution

can be obtained using a drastically reduced model.

9.5.2 Case 2-B: Two Beams with Single Soft Contact

This case consists of the tip of Beam A coming into contact with Beam B once the specified gap distance of 0.05 in. between

Beams A and B is closed, as shown in Fig. 9.9 for the full-space and reduced space models.

The same soft contact stiffness of 10 lb/in. was used to represent the contact of the beams as explained in Case 2-A.

The contact occurs at DOF 141 of Beam A and DOF 45 of Beam B. Table 9.7 lists the natural frequencies for the modified

system as well as for the unmodified components. The mode shapes for the modified system are provided in Appendix A.

Figure 9.10 shows the mode contribution matrix used for identifying the unmodified component modes that contribute in

the modified system modes.

Due to the contact between the two beams, the first two modes from both unmodified components are needed to obtain the

first mode of the modified system, which increases the number of modes needed in the reduced models. Based on the input

force pulse, which excites up to 1,000 Hz, the first 10 modes of the modified system are expected to be excited, which can be

seen in Table 9.7. Examining Fig. 9.10 shows that for the first 10 modes of the modified system, the first 6 modes of Beam A

and the first 4 modes of Beam B are needed to obtain the first 10 modes of the modified system. For a soft contact, the input

force pulse primarily dominates the frequency spectrum excited. Therefore, the modes required to accurately predict the

system response are those observed in the mode contribution matrix, as stated previously. To show the effects of mode

truncation, the system response was plotted in the time and frequency domain for ‘a’ space and ‘aa’ space with comparison

to the full ‘n’ space solution in Fig. 9.11, where the ‘aa’ space model is missing a key contributing mode in the modified

system response – mode 6 of Beam A.

The ‘a’ space model response in Fig. 9.11 can be observed to have very good correlation with the full-space solution;

however, the ‘aa’ space model response indicates that not including mode 6 of Beam A degrades the results obtained.

Full ‘n’ Space

‘a’ Space

‘aa’ Space

F

F

F

Beam A

Beam A

Beam A

Beam B

Beam B

Beam B

Fig. 9.9 Diagram of two

beams with single contact for

full-space and reduced space

models for configuration 1


To quantify the similarity of the reduced model results with the full-space solution, the MAC and TRAC were computed

between the full model and the reduced model time responses, which were then averaged, as listed in Table 9.8a. The solution

time for each model is also listed to show the decrease in computation time when reduced models are used.

MODEModified System Mode Shapes -10 lb/in Contact -Configuration 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Unm

odifie

d B

eam

A M

ode

Sha

pes

1

2

3

4

5

6

7

8

9

10

11

Unm

odifie

d B

eam

B M

ode

Sha

pes 1

2

3

4

5

6

7

8

9

10


Black 0.005 0.1

Blue 0.1 0.2

Green 0.2 0.3

Cyan 0.3 0.5

Magenta 0.5 0.7

Yellow 0.7 0.9

Red 0.9 1.0

Fig. 9.10 Mode contribution matrix for two beams with single soft contact for configuration 1

Table 9.7 Natural frequencies for two beams with single soft contact for configuration 1

Mode # Two beams, single soft contact

Unmodified components

Beam A Beam B

1 20.35 12.91 22.62

2 29.58 84.12 141.56

3 86.01 252.34 396.60

4 142.43 519.59 776.92

5 252.54 806.16 1,284.71

6 396.71 1,256.55 1,918.28

7 519.65 1,682.96 2,678.33

8 777.00 2,201.36 3,563.89

9 806.18 2,755.52 4,572.70

10 1,256.55 3,510.01 5,707.04

11 1,284.80 – –

12 1,682.96 – –

13 1,918.28 – –

14 2,201.36 – –

15 2,678.39 – –

16 2,755.53 – –

17 3,510.02 – –

18 3,563.89 – –

19 3,948.55 – –

20 4,572.72 – –


To improve the ‘aa’ space model results, the addition of DOF 143 and mode 6 from Beam A was included in the reduced

model. From the mode contribution matrix in Fig. 9.10, mode 6 of Beam A is the primary contributing mode that is needed to

obtain mode 10 of the modified system. The reduced system response that uses 11 DOF and 11 modes is plotted in the time

and frequency domain with comparison to the full ‘n’ space solution in Fig. 9.12.

The ‘aa’ space model time response in Fig. 9.12 shows significant improvement in the correlation with the full-space

model when an additional DOF and key contributing mode of Beam A is included in the reduced model. In Table 9.8b, the

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-0.1

-0.05

0

0.05

0.1

0.15

Time (sec)

Dis

plac

emen

t (in

)DOF 141 Time Response - Two Beam Single Soft Contact

n Space-276 DOF

a Space-31 DOF

0 500 1000 1500-200

-150

-100

-50

Frequency (Hz)

dB D

ispl

acem

ent (

in)

FFT of DOF 141 Time Response - Two Beam Single Soft Contact

n Space-276 DOF

a Space-31 DOF

0 500 1000 1500-200

-150

-100

-50

0

50

Frequency (Hz)

dB D

ispl

acem

ent (

in)


n Space-276 DOF

aa Space-10 DOF

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-4

-3

-2

-1

0

1

2

3

4

5

6

Time (sec)

Dis

plac

emen

t (in

)

DOF 141 Time Response - Two Beam Single Soft Contact

n Space-276 DOF

aa Space-10 DOF

Fig. 9.11 Comparison of ‘n’, ‘a’, and ‘aa’ space models for two beams with single soft contact for DOF 141 – time response (top) and FFT of time

response (bottom)

Table 9.8 Average MAC and TRAC for reduced models and solution times for two beams with single soft contact


(a) Incorrect number of modes used in ‘aa’ space model

‘n’ space 276 1 1 87.9

‘a’ space 31 0.99999686 0.99999952 4.8

‘aa’ space 10 0.67818808 0.10422630 1.1

(b) Suitable number of modes used in ‘aa’ space model

‘n’ space 276 1 1 87.9

‘a’ space 31 0.99999686 0.99999952 4.8

‘aa’ space 11 0.99996327 0.99999980 1.1


MAC is improved from 0.678 to 0.999 and the TRAC is improved from 0.104 to 0.999 with essentially no change in

the solution time when DOF 143 and mode 6 from Beam A are included in the reduced model. This shows that significant

errors can result when key contributing component modes are not included in the assembled system for reduced models.

The improved MAC and TRAC results further confirm that the 11 DOF reduced model is sufficient for accurately computing

the time response for this particular case. In addition, the solution time for the full-space model is approximately 90 s in

contrast to the ‘aa’ space reduced model, which is only 1 s. ERMT was shown in this case to provide significant reduction

in computation time for a nonlinear multiple component system with a single soft contact while maintaining a highly

accurate time response solution.

9.5.3 Case 2-C: Two Beams with Multiple Soft Contacts

This case consists of Beam A coming into contact with Beam B in three different configurations at two possible contact

locations with a specified gap of 0.05 in., as shown in Fig. 9.13. Note that each system is a potential configuration of the two

components depending on the relative displacements of the two beams; no contact is also a possible configuration.

The same soft contact stiffness of 10 lb/in. was used to represent the contact of the beams as explained in Case 2-A.

The contact for configuration 1 occurs at DOF 141 of Beam A and DOF 45 of Beam B. The contact for configuration

2 occurs at DOF 101 of Beam A and DOF 5 of Beam B. The contact for configuration 3 occurs when both contacts for

configurations 1 and 2 are closed simultaneously. Table 9.9 lists the natural frequencies for the modified system

configurations as well as for the unmodified components. The mode shapes for the modified system configurations are

provided in Appendix A. Figure 9.14 shows the mode contribution matrices used for identifying the unmodified component

modes that participate in the modified system modes for the three possible system configurations.

For this case, three potential modified system configurations exist, which results in three separate mode contri-

bution matrices. Based on the input force pulse, which excites up to 1,000 Hz, the first 10 modes of the modified system

for all three configurations are expected to be excited, which can be seen in Table 9.9. Examining Fig. 9.14 indicates that for

the first 10 modes of the modified system for all three configurations, the first six modes of Beam A and the first four modes

of Beam B are needed to obtain the first 10 modes of the modified system for all potential configurations. For a soft contact,

the input force pulse primarily dominates the frequency spectrum excited. Therefore, the unmodified component modes

required to accurately predict the system response are those observed in the mode contribution matrices, as stated previously.

Table 9.9 and Fig. 9.14 also show that not only does the additional spring stiffness affect the number of unmodified

component modes needed, but the location of the spring affects the combination of modes needed for each configuration as

well. Depending on whether the spring is at the tip of Beam A or the tip of Beam B, the mode shapes and frequencies change

0 500 1000 1500-200

-150

-100

-50

Frequency (Hz)

dB D

ispl

acem

ent (

in)


n Space-276 DOF

aa Space-11 DOF

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-0.1

-0.05

0

0.05

0.1

0.15

Time (sec)

Dis

plac

emen

t (in

)DOF 141 Time Response - Two Beam Single Soft Contact

n Space-276 DOF

aa Space-11 DOF

Fig. 9.12 Comparison of ‘n’ and ‘aa’ space models for two beams with single soft contact for DOF 141 – time response (left) and FFT of time

response (right)


noticeably. For example, the second mode of Beam B is needed to obtain the first mode of the modified system for

configurations 1 and 3, but is not needed in configuration 2. To show the effects mode truncation has on the solution, the

system response was plotted in the time and frequency domain for ‘a’ space and ‘aa’ space with comparison to the full ‘n’

space solution in Fig. 9.15, where the ‘aa’ space model is missing a key contributing mode in the modified system response –

mode 6 of Beam A.


however, the ‘aa’ space model response indicates that not including mode 6 of Beam A degrades the results obtained.

To quantify the similarity of the reduced model results with the full-space solution, the MAC and TRAC were computed

between the full model and the reduced model time responses, which were then averaged, as listed in Table 9.10a.

The solution time for each model is also listed to show the decrease in computation time when reduced models are used.

Table 9.9 Frequencies for two beams with multiple soft contact configurations

Mode #

Modified system – soft contacts Unmodified components

Config. 1 Config. 2 Config. 3 Beam A Beam B

1 20.35 14.78 21.08 12.91 22.62

2 29.58 33.04 39.23 84.12 141.56

3 86.01 85.71 87.24 252.34 396.60

4 142.43 142.98 143.82 519.59 776.92

5 252.54 252.61 252.82 806.16 1,284.71

6 396.71 396.96 397.07 1,256.55 1,918.28

7 519.65 519.61 519.68 1,682.96 2,678.33

8 777.00 777.04 777.12 2,201.36 3,563.89

9 806.18 806.28 806.30 2,755.52 4,572.70

10 1,256.55 1,256.56 1,256.56 3,510.01 5,707.04

11 1,284.80 1,284.76 1,284.85 – –

12 1,682.96 1,682.99 1,682.99 – –

13 1,918.28 1,918.29 1,918.30 – –

14 2,201.36 2,201.37 2,201.37 – –

15 2,678.39 2,678.34 2,678.39 – –

16 2,755.53 2,755.53 2,755.53 – –

17 3,510.02 3,510.02 3,510.03 – –

18 3,563.89 3,563.89 3,563.89 – –

19 3,948.55 3,948.54 3,948.55 – –

20 4,572.72 4,572.70 4,572.72 – –

Configuration 1

Configuration 2

Configuration 3

Beam A

Beam A

Beam A

Beam B

Beam B

Beam B

Fig. 9.13 Diagram of two

beams with multiple contacts

for configurations 1, 2, and 3


0 500 1000 1500-200

-150

-100

-50

0

50

Frequency (Hz)

dB D

ispl

acem

ent

(in)

FFT of DOF 141 Time Response - Two Beam Two Soft Contacts

n Space-276 DOF

aa Space-10 DOF

0 500 1000 1500-200

-150

-100

-50

Frequency (Hz)

dB D

ispl

acem

ent

(in)


n Space-276 DOF

a Space-31 DOF

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-0.1

-0.05

0

0.05

0.1

0.15

Time (sec)

Dis

plac

emen

t (in

)

DOF 141 Time Response - Two Beam Two Soft Contacts

n Space-276 DOF

a Space-31 DOF

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-4

-3

-2

-1

0

1

2

3

4

5

6

Time (sec)

Dis

plac

emen

t (in

)DOF 141 Time Response - Two Beam Two Soft Contacts

n Space-276 DOF

aa Space-10 DOF

Fig. 9.15 Comparison of ‘n’, ‘a’, and ‘aa’ space models for two beams with multiple soft contacts for DOF 141 – time response (top) and FFT of

time response (bottom)

MODEModified System Mode Shapes -10 lb/in Contact - Configuration 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Unm

odifi

ed B

eam

A M

ode

Shap

es

1234567891011

Unm

odifi

ed B

eam

B M

ode

Shap

es 12345678910

Bar Color Min Value Max ValueBlack 0.005 0.1Blue 0.1 0.2Green 0.2 0.3Cyan 0.3 0.5Magenta 0.5 0.7Yellow 0.7 0.9Red 0.9 1.0

MODEModified System Mode Shapes -10 lb/in Contact - Configuration 2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Unm

odifi

ed B

eam

A M

ode

Shap

es

1234567891011

Unm

odifi

ed B

eam

B M

ode

Shap

es 12345678910

MODEModified System Mode Shapes - 10 lb/in Contact - Configuration 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Unm

odifi

ed B

eam

A M

ode S

hape

s 1234567891011

Unm

odifi

ed B

eam

B M

ode S

hape

s 12345678910

Fig. 9.14 Mode contribution matrix for two beams with soft contacts for configurations 1, 2, and 3

As seen in Case 2-B, the mode contribution matrix provides a good understanding of the effects mode truncation has on

the solution obtained. Therefore, to improve the ‘aa’ space results, the addition of DOF 143 and mode 6 from Beam A was

included in the reduced model. From the mode contribution matrices in Fig. 9.14, mode 6 of Beam A is the primary

contributing mode that is needed to obtain mode 10 of the modified system in all three configurations. The reduced system

response that uses 11 DOF and 11 modes is plotted in the time and frequency domain with comparison to the full ‘n’ space

solution in Fig. 9.16.

The ‘aa’ space model time response in Fig. 9.16 shows significant improvement in the correlation with the full-space

model when an additional DOF and key contributing mode of Beam A is included in the reduced model. In Table 9.10b, the

MAC is improved from 0.678 to 0.999 and the TRAC is improved from 0.104 to 0.999 with essentially no change in

the solution time when DOF 143 and mode 6 from Beam A are included in the reduced model. The improved MAC and

TRAC results further confirm that the 11 DOF reduced model is sufficient for accurately computing the time response for

this particular case. In addition, the solution time for the full-space model is approximately 90 s in contrast to the ‘aa’

space reduced model, which is only 1 s. ERMT was shown in this case to provide significant reduction in computation

time for a nonlinear multiple component system with multiple soft contacts while maintaining a highly accurate time

response solution.

The system response for this case is the same as in Case 2-B, which only has a single potential contact location, rather

than two as in this case. In the cases to follow where hard contact stiffness is studied, the second contact location does come

into contact, however for the soft contact stiffness, contact did not occur at all of the possible locations. This case was

included to maintain continuity with the hard contact stiffness cases studied in following section. The soft contact models

analyzed in Case 2 demonstrate how the mode participation matrix can be used to identify the component modes that are

needed to generate accurate time response solutions using reduced models. Case 2 also shows that ERMT can provide highly

accurate results with significantly reduced computation time for nonlinear systems with soft contacts.

0 500 1000 1500-200

-150

-100

-50

Frequency (Hz)

dB D

ispl

acem

ent (

in)


n Space-276 DOF

aa Space-11 DOF

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-0.1

-0.05

0

0.05

0.1

0.15

Time (sec)

Dis

plac

emen

t (in

)

DOF 141 Time Response - Two Beam Two Soft Contacts

n Space-276 DOF

aa Space-11 DOF

Fig. 9.16 Comparison of ‘n’ and ‘aa’ space models for two beams with multiple soft contacts for DOF 141 – time response (left) and FFT of time

response (right)

Table 9.10 Average MAC and TRAC for reduced models and solution times for two beams with multiple soft contacts


(a) Incorrect number of modes used in ‘aa’ space model

‘n’ space 276 1 1 89.7

‘a’ space 31 0.99999686 0.99999952 4.9

‘aa’ space 10 0.67818808 0.10422630 1.1

(b) Suitable number of modes used in ‘aa’ space model

‘n’ space 276 1 1 89.7

‘a’ space 31 0.99999686 0.99999952 4.9

‘aa’ space 11 0.99996327 0.99999980 1.1


9.6 Case 3: Hard Contact

For the soft contact stiffness models discussed in Case 2, the contact stiffness was soft enough that the contact did not excite

a frequency bandwidth beyond the input force spectrum. Thus, the number of modes needed in the time response was

controlled by the forcing function, and the contributing unmodified component modes needed to obtain the modified system

modes were based on the mode contribution matrix. As long as the frequency bandwidth excited by the contact stiffness is

below the highest frequency excited by the input force pulse, the procedure for identifying the number of modes needed

works well, as in Case 2. However, for a harder contact stiffness situation, there is a possibility that the contact can excite a

frequency bandwidth beyond the input spectrum. Under this scenario, the number of unmodified component modes needed

in the modified system response would not be a function of the input spectrum, but of the contact stiffness. In order to

examine this possible scenario in detail, the same cases were studied as in Case 2, but with 1,000 lb/in. contact stiffness.

9.6.1 Case 3-A: Single Beam with Single Hard Contact


distance of 0.05 in., as explained previously in Case 2-A. Table 9.11 lists the first 10 natural frequencies of the modified

system with the hard contact stiffness applied, while Fig. 9.17 shows the mode contribution matrix. The mode shapes for the

modified system are provided in Appendix A.

Table 9.11 and Fig. 9.17 indicate that the first 6 modes of the unmodified component are needed to obtain the first five

modes of the modified system. This is expected, as the mode shapes with the harder spring attached look less like the original

model and therefore requires more mode mixing to form the modified system modes. The frequencies of the unmodified

component for mode 6 and higher are minimally affected for the modified system, which do not require additional

unmodified component mode shapes to form the modified system modes (as indicated by the red main diagonal). Therefore,

six active DOF and the first six modes of the unmodified component are needed to accurately obtain the modified system

MODEModified System Mode Shapes

1000 lb/in Contact1 2 3 4 5 6 7 8 9 10

Unm

odifie

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eam

A M

ode

Sha

pes 1

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Black 0.005 0.1

Blue 0.1 0.2

Green 0.2 0.3

Cyan 0.3 0.5

Magenta 0.5 0.7

Yellow 0.7 0.9

Red 0.9 1.0

Fig. 9.17 Mode contribution

matrix for single beam with

single hard contact


for single beam with single hard

contact

Mode # Unmodified Single hard contact

1 12.91 66.68

2 84.12 223.55

3 252.34 326.04

4 519.59 529.68

5 806.16 808.16

6 1,256.55 1,256.73

7 1,682.96 1,682.97

8 2,201.36 2,201.52

9 2,755.52 2,755.76

10 3,510.01 3,510.46


response. In order to confirm this, the system response was plotted in the time and frequency domain for ‘a’ space and ‘aa’

space with comparison to the full ‘n’ space solution in Fig. 9.18. The response for the ‘aa’ space model is shown to illustrate

when a key contributing mode (mode 6 of Beam A) is not included in the reduced model for the modified system.


however, in contrast to Case 2-A the ‘aa’ space model response indicates that not including mode 6 of Beam A degrades the

results obtained. The first six modes of the unmodified component are needed to obtain the first five modes of the modified

system as indicated in Fig. 9.17, which were not able to be adequately represented using only modes 1–5 for the ‘aa’ space

model. Even though the input force spectrum was limited primarily to the first five modes of the unmodified component, the

first six modes are needed for the modified system. The hard contact does not appear to contribute significant energy beyond

1,000 Hz when contact occurs, which shows that the number of modes needed to accurately compute the time response for

the reduced models is governed by the forcing function and not the contact stiffness for this case. To quantify the similarity

0 500 1000 1500-160

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FFT of DOF 141 Time Response - Single Beam Single Hard Contact

n Space-146 DOF

aa Space-5 DOF

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FFT of DOF 141 Time Response - Single Beam Single Hard Contact

n Space-146 DOF

a Space-17 DOF

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.2

0

0.2

0.4

0.6

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Dis

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t (in

)DOF 141 Time Response - Single Beam Single Hard Contact

n Space-146 DOF

a Space-17 DOF

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.2

0

0.2

0.4

0.6

0.8

1

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Time (sec)

Dis

plac

emen

t (in

)

DOF 141 Time Response - Single Beam Single Hard Contact

n Space-146 DOF

aa Space-5 DOF

Fig. 9.18 Comparison of ‘n’, ‘a’, and ‘aa’ space models for single beam with single hard contact for DOF 141 – time response (top) and FFT of

time response (bottom)

Table 9.12 Average MAC and TRAC for reduced models and solution times for single beam with single hard contact


‘n’ space 146 1 1 35.1

‘a’ space 17 0.99999686 0.99999980 1.9

‘aa’ space 5 0.99854646 0.98781940 0.7


of the reduced model results with the full-space solution, the MAC and TRAC were computed between the full model and

the reduced model time responses, which were then averaged, as listed in Table 9.12. The solution time for each model is

also listed to show the decrease in computation time when reduced models are used.

The system response for the hard contact is observably more nonlinear compared to the soft contact case, where the

modes of the system are more difficult to identify in the FFT of the response. The FFT of the time response for both the soft

and hard contact is shown in Fig. 9.19 for comparison.

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FFT of DOF 141 Time Response - Single Beam Single Contact

Hard Contact

Soft Contact

Fig. 9.19 Comparison of the

FFT for single beam with

single soft and hard contact

Table 9.13 Natural frequencies for two beams with single hard contact for configuration 1

Mode # Two beams, single hard contact

Unmodified components

Beam A Beam B

1 21.25 12.91 22.62

2 67.74 84.12 141.56

3 127.04 252.34 396.60

4 236.02 519.59 776.92

5 338.31 806.16 1,284.71

6 424.26 1,256.55 1,918.28

7 532.04 1,682.96 2,678.33

8 786.59 2,201.36 3,563.89

9 809.25 2,755.52 4,572.70

10 1,256.69 3,510.01 5,707.04

11 1,294.66 – –

12 1,682.97 – –

13 1,918.72 – –

14 2,201.53 – –

15 2,683.90 – –

16 2,755.78 – –

17 3,510.46 – –

18 3,564.14 – –

19 3,948.79 – –

20 4,575.19 – –


The favorable MAC and TRAC results in Table 9.12 further confirm that the 17 DOF ‘a’ space model is sufficient for

accurately computing the time response for this particular case. In addition, the solution time for the full-space model is over

30 s in contrast to the reduced ‘a’ space model, which is less than 2 s. This case showed that an accurate time response

solution can be obtained using ERMT with drastically reduced models for a single component with a single hard contact.

9.6.2 Case 3-B: Two Beams with Single Hard Contact

This case consists of the tip of Beam A coming into contact with Beam B once the specified gap distance of 0.05 in. between

Beams A and B is closed, as discussed previously in Case 2-B. The same hard contact stiffness of 1,000 lb/in. was used to

represent the contact of the beams as explained in Case 3-A. Table 9.13 lists the natural frequencies for the modified system

as well as for the unmodified components. Figure 9.20 shows the mode contribution matrix for identifying the unmodified

component modes that contribute in the modified system modes. The mode shapes for the modified system are provided

in Appendix A.

Based on the input force pulse, which excites up to 1,000 Hz, the first 10 modes of the modified system are expected to be

excited, which can be seen in Table 9.13. Examining Fig. 9.20 shows that for the first 10 modes of the modified system, the

first six modes of Beam A and the first seven modes of Beam B are needed to obtain the first 10 modes of the modified

system. For a soft contact, the input force pulse primarily dominates the frequency spectrum excited, however this may not

always be true for a hard contact scenario. The modes required to accurately predict the system response are provided in the

mode contribution matrix; however, there may be additional higher order modes excited by the hard contact that cannot be

identified based solely on the input force spectrum. For hard contact situations, care must be taken when generating reduced

models such that higher order modes that participate in the system response are included. To show the effects of severe mode

truncation, the system response was plotted in the time and frequency domain for ‘a’ space and ‘aa’ space with comparison

to the full ‘n’ space solution in Fig. 9.21, where the ‘aa’ space model is missing several key contributing modes in the

modified system response.

Note that in contrast to the soft contact, the hard contact excited modes well above the 1,000 Hz bandwidth of the input

force pulse (up to approximately 3,000 Hz). Accordingly, the number of unmodified component modes identified using the

mode contribution matrix to accurately obtain the modified system time response is no longer governed by the input force

spectrum, but by the frequency bandwidth excited by the contact stiffness impact. To quantify the similarity of the reduced

model results with the full-space solution, the MAC and TRAC were computed between the full model and the reduced

model time responses, which were then averaged, as listed in Table 9.14a. The solution time for each model is also listed to

show the decrease in computation time when reduced models are used.

MODEModified System Mode Shapes -1000 lb/in Contact -Configuration 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Unm

odifi

ed B

eam

A M

ode

Shap

es

1234567891011

Unm

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eam

B M

ode

Shap

es 12345678910

Bar Color Min Value Max ValueBlack 0.005 0.1Blue 0.1 0.2Green 0.2 0.3Cyan 0.3 0.5Magenta 0.5 0.7Yellow 0.7 0.9Red 0.9 1.0

Fig. 9.20 Mode contribution matrix for two beams with single hard contact for configuration 1


The transient portion of the ‘a’ space model response in Fig. 9.21 can be observed to have very good correlation with the

full-space solution. However, in contrast to Case 3-A the hard contact appears to contribute noticeable energy beyond

the 1,000 Hz input force spectrum, which was not anticipated when identifying the unmodified component modes that

contribute in the modified system modes. In Fig. 9.21, the ‘aa’ space model produced less desirable results due to severe

mode truncation. The ‘a’ space model provides very good results for the initial transient portion of the system response,

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FFT of DOF 141 Time Response - Two Beam Single Hard Contact

n Space-276 DOF

aa Space-10 DOF

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FFT of DOF 141 Time Response - Two Beam Single Hard Contact

n Space-276 DOF

a Space-31 DOF

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-2

0

2

4

6

8

10

12

14

Time (sec)

Dis

plac

emen

t (in

)

DOF 141 Time Response - Two Beam Single Hard Contact

n Space-276 DOF

aa Space-10 DOF

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.2

-0.1

0

0.1

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0.3

0.4

0.5

Time (sec)

Dis

plac

emen

t (in

)DOF 141 Time Response - Two Beam Single Hard Contact

n Space-276 DOF

a Space-31 DOF

Fig. 9.21 Comparison of ‘n’, ‘a’, and ‘aa’ space models for two beams with single hard contact for DOF 141 – time response (top) and FFT of time

response (bottom)

Table 9.14 Average MAC and TRAC for reduced models and solution times for two beams with single hard contact


(a) MAC and TRAC for full time response (0–0.4 s)

‘n’ space 276 1 1 87.9

‘a’ space 31 0.99684042 0.97628620 4.9

‘aa’ space 10 0.40375102 0.12895379 1.1

(b) MAC and TRAC for transient portion of time response (0–0.2 s)

‘n’ space 276 1 1 87.9

‘a’ space 31 0.99995692 0.99958379 4.9

‘aa’ space 10 0.45859557 0.11002554 1.1


however degraded correlation can be observed after the initial transient. This is due to higher order modes that are missing in

the reduced model not being excited until a high force impact occurs at the hard stiffness contact location. This behavior will

be different depending on the location and level of the input excitation, due to the nonlinear characteristics of the system.

The number of modes included in the ‘a’ space model is sufficient when only the forces pulse is considered. However, due to

the hard contact, more modes are needed to accurately compute the system response beyond the initial transient portion.

To observe the high frequency content excited by the hard contact, the FFT of the system response is shown in Fig. 9.22 over

a 5,000 Hz band with comparison to the soft contact case.

Often times for nonlinear systems, the initial transient portion of the time response is of interest, not the entire solution.

When the ‘a’ space solution is evaluated for only the transient portion of the response (the first 0.2 s), the MAC improves

from 0.997 to 0.999 and the TRAC improves from 0.976 to 0.999, as shown in Table 9.14b.

The ‘a’ space model for the two beam system with a single hard contact produced very good results for the initial transient

portion of the response, which is generally the portion of the system response of interest for nonlinear systems. However, less

desirable results were obtained using the ‘aa’ space model, due to not enough modes being included to obtain the modified

system response. As seen in Fig. 9.20, the first 10 modes of the modified system requires the first six modes of Beam A and

the first seven modes of Beam B, which were not able to be adequately represented using the ‘aa’ space model. The number

of modes included in the ‘a’ space model is sufficient when only the force pulse is considered. However, due to the hard

contact stiffness, more modes are needed to accurately compute the system response beyond the initial transient portion.

This shows that throughout the nonlinear system time response, hard contacts can potentially excite higher frequencies than

the input excitation. Therefore, a reduced model with greater than 31 DOF and 31 modes is needed to obtain an accurate time

response solution beyond the initial transient for this case. The favorable MAC and TRAC results in Table 9.14b show that

the 31 DOF ‘a’ space model is sufficient for accurately computing the initial transient portion of the time response for this

particular case. In addition, the solution time for the full-space model is almost 90 s in contrast to the reduced ‘a’ space

model, which is approximately 5 s.

ERMT was shown in this case to provide significant reduction in computation time for a nonlinear multiple component

system with a single hard contact while maintaining a highly accurate initial transient time response solution. However, care

must be taken when generating reduced models for nonlinear systems with hard contact stiffness, due to the higher

frequencies that can potentially be excited in the system.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-300

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Frequency (Hz)

dB D

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(in)

FFT of DOF 141 Time Response - Two Beam Single Contact

Hard Contact

Soft Contact

Fig. 9.22 Comparison of the

FFT for two beams with single

soft and hard contact


9.6.3 Case 3-C: Two Beams with Multiple Hard Contacts

This case consists of BeamAcoming into contact with BeamB in three different configurationswith hard contact stiffness at two

possible contact locations with a specified gap distance of 0.05 in., as discussed previously in Case 2-C. Table 9.15 lists the

natural frequencies for the modified system as well as for the unmodified components. Figure 9.23 shows the mode contribution

matrices used for identifying the unmodified component modes that participate in the modified system modes for the three

possible system configurations. The mode shapes for the modified system configurations are provided in Appendix A.

Table 9.15 and Fig. 9.23 show that for the input force pulse applied, ten systemmodes are excited and modes 1–8 of Beam

A and modes 1–7 of Beam B are needed. However, Case 3-B showed that additional modes beyond the input excitation

spectrum are activated by the hard contact stiffness. Therefore, larger reduced models are needed for the inclusion of


Black 0.005 0.1

Blue 0.1 0.2

Green 0.2 0.3

Cyan 0.3 0.5

Magenta 0.5 0.7

Yellow 0.7 0.9

Red 0.9 1.0


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Unm

odifie

d B

eam

A M

ode

Sha

pes

1

2

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

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1

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

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odifie

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eam

A M

ode Sha

pes

1

2

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7

8

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ode

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7

8

9

10

Fig. 9.23 Mode contribution matrix for two beams with hard contacts for configurations 1, 2, and 3

Table 9.15 Natural frequencies for two beams with multiple hard contact configurations

Mode #

Modified system – hard contacts Unmodified components

Config. 1 Config. 2 Config. 3 Beam A Beam B

1 21.25 15.50 36.22 12.91 22.62

2 67.74 69.52 113.69 84.12 141.56

3 127.04 119.79 228.57 252.34 396.60

4 236.02 234.14 312.68 519.59 776.92

5 338.31 325.85 339.40 806.16 1,284.71

6 424.26 467.09 504.56 1,256.55 1,918.28

7 532.04 527.49 532.41 1,682.96 2,678.33

8 786.59 786.04 790.33 2,201.36 3,563.89

9 809.25 828.78 833.74 2,755.52 4,572.70

10 1,256.69 1,257.17 1,257.22 3,510.01 5,707.04

11 1,294.66 1,290.03 1,300.10 – –

12 1,682.97 1,686.00 1,686.01 – –

13 1,918.72 1,920.05 1,920.48 – –

14 2,201.53 2,202.67 2,202.84 – –

15 2,683.90 2,678.85 2,684.38 – –

16 2,755.78 2,756.01 2,756.28 – –

17 3,510.46 3,511.34 3511.79 – –

18 3,564.14 3,563.98 3564.23 – –

19 3,948.79 3,948.54 3948.79 – –

20 4,575.19 4,572.70 4575.19 – –


additional modes that participate in the modified system response to obtain an accurate time response solution. The system

response was plotted in the time and frequency domain for ‘a’ space and ‘aa’ space with comparison to the full ‘n’ space

solution in Fig. 9.24, where the ‘aa’ space model is missing several key contributing modes in the modified system response.

The transient portion of the ‘a’ space model response in Fig. 9.24 can be observed to have very good correlation with the

full-space solution. As seen in Case 3-B, the hard contact contributes significant energy beyond the 1,000 Hz input force

spectrum. In Fig. 9.24, the ‘aa’ space model produced less desirable results, due to severe mode truncation. The ‘a’ space

model provides very good results for the initial transient portion of the response, however degraded correlation is observed

after the initial transient. The number of modes included in the ‘a’ space model is sufficient when only the forces pulse is

considered. However, due to the hard contact, more modes are needed to accurately compute the system response beyond the

initial transient portion, which was discussed in Case 3-B. To observe the high frequency content excited by the hard contact,

the FFT of the system response is shown in Fig. 9.25 over a 5,000 Hz band with comparison to the soft contact case.

As observed in Case 3-B, the frequency excited by the hard contact stiffness was well above the bandwidth of the input

force pulse. The mode contribution matrix provided a good indication of the unmodified component modes that participate

in the modified system modes for the soft contact stiffness scenario. However, higher order modes beyond the input force

spectrum are excited by the hard contact stiffness and the modes identified in the mode contribution matrix are governed by

the frequency bandwidth excited by the hard contact impact. To quantify the similarity of the reduced model results with the

full-space solution, the MAC and TRAC were computed between the full model and the reduced model time responses,

which were then averaged, as listed in Table 9.16a. The solution time for each model is also listed to show the decrease in

computation time when reduced models are used.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.2

-0.1

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Dis

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)DOF 141 Time Response - Two Beam Two Hard Contacts

n Space-276 DOF

a Space-31 DOF

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FFT of DOF 141 Time Response - Two Beam Two Hard Contacts

n Space-276 DOF

aa Space-10 DOF

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FFT of DOF 141 Time Response - Two Beam Two Hard Contacts

n Space-276 DOF

a Space-31 DOF

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-10

-5

0

5

10

15

20

25

Time (sec)

Dis

plac

emen

t (in

)

DOF 141 Time Response - Two Beam Two Hard Contacts

n Space-276 DOF

aa Space-10 DOF

Fig. 9.24 Comparison of ‘n’, ‘a’, and ‘aa’ space models for two beams with multiple hard contacts for DOF 141 – time response (top) and FFT

of time response (bottom)


Often times for nonlinear systems, the initial transient portion of the time response is of interest, not the entire solution.

When the ‘a’ space solution is evaluated for only the transient portion of the response (the first 0.2 s), the MAC improves

from 0.890 to 0.999 and the TRAC improves from 0.574 to 0.999, as shown in Table 9.16b.

The ‘a’ space model for the two beam system with multiple hard contacts produced very good results for the initial transient

portion of the response, which is generally the portion of the system response of interest for nonlinear systems. However, less

desirable results were obtained using the ‘aa’ space model, due to not enough modes being included to obtain the modified

system response. As seen in Fig. 9.23, the first 10modes of the modified system requires the first eight modes of BeamA and the

first sevenmodes ofBeamB,whichwere not able to be adequately represented using the ‘aa’ spacemodel. The number ofmodes

included in the ‘a’ space model is sufficient when only the forces pulse is considered. However, due to the hard contact, more

modes are needed to accurately compute the system response beyond the initial transient portion, as discussed in Case 3-B.

Therefore, a reduced model with greater than 31 DOF and 31 modes is needed to obtain an accurate time response solution

beyond the initial transient for this case. The favorable MAC and TRAC results in Table 9.16b show that the 31 DOF ‘a’ space

model is sufficient for accurately computing the initial transient portion of the time response for this particular case. In addition,

the solution time for the full-space model is almost 90 s in contrast to the reduced ‘a’ space model, which is approximately 5 s.

ERMT was shown in this case to provide significant reduction in computation time for a nonlinear multiple component

system with multiple hard contacts where significant component interaction occurs. In addition, a highly accurate initial

transient time response solution was obtained using drastically reduced models.

Comparing the soft and hard contact cases shows that the number of modes needed in the reduced models can be

predicted accurately as long as the input spectrum bandwidth defines the frequency range that is excited, as seen in the soft

Table 9.16 Average MAC and TRAC for reduced models and solution times for two beams with hard contacts


(a) MAC and TRAC for full time response (0–0.4 s)

‘n’ space 276 1 1 88.5

‘a’ space 31 0.88955510 0.57445998 4.9

‘aa’ space 10 0.45974452 0.07460348 1.1

(b) MAC and TRAC for transient portion of time response (0–0.2 s)

‘n’ space 276 1 1 88.5

‘a’ space 31 0.99995692 0.99958379 4.9

‘aa’ space 10 0.53251049 0.24627142 1.1

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-250

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dB D

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FFT of DOF 141 Time Response - Two Beam Multiple Contact

Hard Contact

Soft Contact

Fig. 9.25 Comparison of the FFT for two beams with multiple soft and hard contacts


contact cases. However, once this condition is no longer true, determining the number of modes needed requires additional

knowledge of the frequency bandwidth excited by the contacts ahead of time. Care must be taken when generating reduced

models for nonlinear systems with hard contact stiffness, due to the higher frequencies that can potentially be excited in the

system. For both soft and hard contact cases, ERMT was shown to provide a substantial reduction in computation time while

maintaining highly accurate time response solutions, which demonstrates the usefulness of this technique.

9.7 Time Step Selection Effect on Solution

For the time step of 0.0001 s used in the cases studied, Raleigh Criteria and Shannon’s Sampling Theorem state that the

maximum frequency range that can be observed is 5 kHz, which is well above the 1,000 Hz frequency bandwidth excited by

the input force pulse. Although this time resolution may be fine enough to accurately capture the time response for the linear

system, this resolution may not be adequate when the response becomes nonlinear. In addition, the contact stiffness of the

nonlinear system also affects the frequency bandwidth excited, as discussed previously in Case 3. Since the time step chosen

affects the instance in time that contact occurs, the system may respond differently depending on whether the time duration

of the impact is short (soft contact) or long (hard contact). For a soft contact, the system remains in contact for a longer time

duration, and the time response should be relatively consistent regardless of the time step chosen. For the hard contact,

however, the system may experience high frequency impact chatter, which may not be captured when a coarse time step is

used. The system response was computed for the single beam with soft and hard contact using a time step of 0.00005 s and

the previously used time step of 0.0001 s, which are compared in Fig. 9.26 for the first 0.1 s of the time response solutions.

The left plot in Fig. 9.26 shows that for the soft contact, the smaller time step minimally effected the time response

solution. The results compare very well for the soft contact, due to the system remaining in contact with the soft spring for a

long time duration. For the hard contact in the right plot of Fig. 9.26, the system comes into contact and then immediately

bounces off. Since the time step directly affects the acceleration, there is a divergence in the solution immediately after the

first contact occurs. Although this study shows that the time step selected was not sufficiently small enough for the hard

contact cases, the time step of 0.0001 s was used for all of the previous cases in order to demonstrate the main principles and

computational advantages of ERMT. Further study is needed to determine the time step required when the contact stiffness

dictates the spectral energy content that participates in the nonlinear system response.

9.8 Comparison to Large Finite Element Models

ERMT was shown to provide substantial reduction in computation time for the simple beam models studied in this paper.

The computational advantages of the technique would be further emphasized when used for a detailed finite element model,

which are typically found in industry. Figure 9.27 shows a very detailed model of a helicopter/missile/wing configuration.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-0.1

0

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Dis

plac

emen

t (in

)

Single Beam Single Hard Contact Response at DOF 141, Sampling Rate Comparison

dt1 = 0.0001 sec

dt2 = 0.00005 sec

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emen

t (in

)Single Beam Single Soft Contact Response at DOF 141, Sampling Rate Comparison

dt1 = 0.0001 sec

dt2 = 0.00005 sec

Fig. 9.26 Comparison of system response using different time steps for soft contact (left) and hard contact (right)


These detailed models are used in dynamic simulations to compute the transient dynamics that occur during a missile

firing under a variety of conditions. Due to the system being very complex with many detailed components, the time

response solutions require significant computation (on the order of hours to days). However, utilizing reduced order models

of the various components allows for substantial reduction in computation time. In addition, multiple system configurations

can be efficiently studied in detail to provide the analyst with the wealth of information needed to improve the system design.

9.9 Conclusion

A computationally efficient technique – Equivalent Reduced Model Technique (ERMT) – that utilizes reduced linear

component models assembled with discrete nonlinear connection elements to perform nonlinear forced response analysis is

presented. Four cases of increasing complexity are studied. The mode contribution matrix is used to identify the modes

required to form accurate reduced order models. The technique was shown to yield accurate results when compared to the

reference solution and provided significant improvement in computational efficiency for the analytical nonlinear forced

response cases studied.

9.10 Future Work

Future workwill be performed using ERMT to demonstrate the usefulness of the technique with experimental data validation.

For comparison to experimental data, several additional considerations are needed to yield accurate results. First, the

underlying linear system will need to be a highly accurate model in order to predict the system time responses correctly.

Care will need to be taken in modeling and updating the component models to test data to ensure that both the model and

measurements are reflective of the physical structure. Second, the damping will have to be measured experimentally for the

linear system for as many modes as possible in order to have high correlation in the time domain. In addition, efforts will be

needed to determine the correct damping for the physical system models, as the damping may change once the component(s)

are in the state of contact. Third, the stiffness of the contact will have to be determined to accurately model the system and

compute the time response. In addition, the impact force from the beam contact may need to be accounted for in the analytical

models, where this additional force input to the structure may potentially affect the time response. Finally, variation in the

time step used was found to affect the results obtained and therefore, additional work is needed to remedy this item of concern.

Acknowledgements Some of the work presented herein was partially funded by Air Force Research Laboratory Award No. FA8651-10-1-0009

“Development of Dynamic Response Modeling Techniques for Linear Modal Components”. Any opinions, findings, and conclusions or

recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the particular funding agency.

The authors are grateful for the support obtained.

Fig. 9.27 Detailed FEM of

helicopter/missile/wing

configuration


Appendix A: Component and System Mode Shapes

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Unmodified Mode Shapes – Beam B

1 – 22.62 Hz 2 –141.56 Hz 3 – 396.60 Hz

4 –776.92 Hz 5 –1284.71 Hz 6 –1918.28 Hz

7 –2678.33 Hz 8 –3563.89 Hz 9 –4572.70 Hz

10 –5707.04 Hz 11 – 6956.91 Hz 12 –8324.25 Hz

Unmodified Mode Shapes –Beam A

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4 –519.59 Hz 5 –806.16 Hz 6 –1256.55 Hz

7 –1682.96 Hz 8 –2201.36 Hz 9 –2755.52 Hz

10 –3510.01 Hz 11 –3948.54 Hz 12 –5076.34 Hz

Fig. A.1 Mode shapes for unmodified components A and B

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Single Beam Single Contact Mode Shapes –10 lb/in Spring

1 – 26.09 Hz 2 – 86.02 Hz 3 – 252.54 Hz

4 – 519.65 Hz 5 – 806.18 Hz 6 – 1256.55 Hz

7 – 1682.96 Hz 8 – 2201.36 Hz 9 – 2755.53 Hz

10 – 3510.02 Hz 11 – 3948.55 Hz 12 – 5076.34 Hz

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Single Beam Single Contact Mode Shapes –1000 lb/in Spring

1 – 66.68 Hz 2 – 223.55 Hz 3 – 326.04 Hz

4 – 529.68 Hz 5 – 808.16 Hz 6 – 1256.73 Hz

7 – 1682.97 Hz 8 – 2201.52 Hz 9 – 2755.76 Hz

10 – 3510.46 Hz 11 – 3948.79 Hz 12 – 5076.36 Hz

0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18

0 2 4 6 8 10 12 14 16 180 2 4 6 8 10 12 14 16 18

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0 2 4 6 8 10 12 14 16 180 2 4 6 8 10 12 14 16 18

0 2 4 6 8 10 12 14 16 180 2 4 6 8 10 12 14 16 180 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18

0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18

0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18

0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18

0 2 4 6 8 10 12 14 16 18

Fig. A.2 Mode shapes for single beam with single soft (left) and single hard (right) contact


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Multiple Beam Contact Mode Shapes - Configuration 1 – 10 lb/in Spring

1 – 20.35 Hz 2 – 29.58 Hz 3 – 86.01 Hz

4 – 142.43 Hz 5 – 252.54 Hz 6 – 396.71 Hz

7 – 519.65 Hz 8 – 777.00 Hz 9 – 806.18 Hz

10 – 1256.55 Hz 11 – 1284.80 Hz 12 – 1682.96 Hz

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Multiple Beam Contact Mode Shapes – Configuration 2 – 10 lb/in Spring

1 – 14.78 Hz 2 – 33.04 Hz 3 – 85.71 Hz

4 – 142.98 Hz 5 – 252.61 Hz 6 – 396.96 Hz

7 – 519.61 Hz 8 – 777.04 Hz 9 – 806.28 Hz

10 – 1256.56 Hz 11 – 1284.76 Hz 12 – 1682.99 Hz

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1 – 21.08 Hz 2 – 39.23 Hz 3 – 87.24 Hz

4 – 143.82 Hz 5 – 252.82 Hz 6 – 397.07 Hz

7 – 519.68 Hz 8 – 777.12 Hz 9 – 806.30 Hz

10 – 1256.56 Hz 11 – 1284.85 Hz 12 – 1682.99 Hz

Fig. A.3 Mode shapes for two beam system for multiple soft contact configurations


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1 – 36.22 Hz 2 – 113.69 Hz 3 – 228.57 Hz

4 – 312.68 Hz 5 – 339.40 Hz 6 – 504.56 Hz

7 – 532.41 Hz 8 – 790.33 Hz 9 – 833.74 Hz

10 – 1257.22 Hz 11 – 1300.10 Hz 12 – 1686.01 Hz

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1 – 21.25 Hz 2 – 67.74 Hz 3 – 127.04 Hz

4 – 236.02 Hz 5 – 338.31 Hz 6 – 424.26 Hz

7 – 532.04 Hz 8 – 786.59 Hz 9 – 809.25 Hz

10 – 1256.69 Hz 11 – 1294.66 Hz 12 – 1682.97 Hz

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1 – 15.50 Hz 2 – 69.52 Hz 3 – 119.79 Hz

4 – 234.14 Hz 5 – 325.85 Hz 6 – 467.09 Hz

7 – 527.49 Hz 8 – 786.04 Hz 9 – 828.78 Hz

10 – 1257.17 Hz 11 – 1290.03 Hz 12 – 1686.00 Hz

Fig. A.4 Mode shapes for two beam system for multiple hard contact configurations


References

1. Avitabile P, O’Callahan JC, Pan EDR (1989) Effects of various model reduction techniques on computed system response. In: Proceedings of

the seventh international modal analysis conference, Las Vegas, February 1989

2. O’Callahan JC, Avitabile P, Riemer R (1989) System equivalent reduction expansion process. In: Proceedings of the seventh international

modal analysis conference, Las Vegas, February 1989

3. Van Zandt T (2006) Development of efficient reduced models for multi-body dynamics simulations of helicopter wing missile configurations.

Master’s thesis, University of Massachusetts Lowell, 2006

4. Avitabile P, O’Callahan JC (2009) Efficient techniques for forced response involving linear modal components interconnected by discrete

nonlinear connection elements. Mech Syst Signal Process 23(1):45–67, Special Issue: Non-linear Structural Dynamics

5. Friswell MI, Penney JET, Garvey SD (1995) Using linear model reduction to investigate the dynamics of structures with local non-linearities.


6. Lamarque C, Janin O (2000) Modal analysis of mechanical systems with impact non-linearities: limitations to a modal superposition. J Sound

Vib 235(4):567–609

7. Ozguven H, Kuran B (1996) A modal superposition method for non-linear structures. J Sound Vib 189(3):315–339

8. Al-Shudeifat M, Butcher E, Burton T (2010) Enhanced order reduction of forced nonlinear systems using new Ritz vectors. In: Proceedings of

the twenty-eighth international modal analysis conference, Jacksonville, February 2010

9. Rhee W Linear and nonlinear model reduction in structural dynamics with application to model updating. PhD dissertation, Texas Technical

University

10. Guyan RJ (1965) Reduction of stiffness and mass matrices. AIAA J 3(2):380

11. O’Callahan JC (1989) A procedure for an improved reduced system (IRS) model. In: Proceedings of the seventh international modal analysis

conference, Las Vegas, February 1989

12. Marinone T, Butland A, Avitabile P (2012) A reduced model approximation approach using model updating methodologies. In: Proceedings of

the thirtieth international modal analysis conference, Jacksonville, February 2012

13. Avitabile P (2003) Twenty years of structural dynamic modification – a review. Sound Vib Mag 37:14–27

14. Rao S (2004) Mechanical vibrations, 4th edn. Prentice Hall, New Jersey, pp 834–843

15. Allemang RJ, Brown DL (2007) A correlation coefficient for modal vector analysis. In: Proceedings of the first international modal analysis

conference, Orlando, February 2007

16. O’Callahan J (1986) MAT_SAP/MATRIX, A general linear algebra operation program for matrix analysis. University of Massachusetts

Lowell, 1986

17. MATLAB (R2010a) The MathWorks Inc., Natick, M.A.


Chapter 10

Efficient Computational Nonlinear Dynamic Analysis

Using Modal Modification Response Technique

Tim Marinone, Peter Avitabile, Jason R. Foley, and Janet Wolfson

Abstract Generally, structural systems contain nonlinear characteristics in many cases. These nonlinear systems require

significant computational resources for solution of the equations of motion. Much of the model, however, is linear where the

nonlinearity results from discrete local elements connecting different components together. Using a component mode

synthesis approach, a nonlinear model can be developed by interconnecting these linear components with highly nonlinear

connection elements.

The approach presented in this paper, the Modal Modification Response Technique (MMRT), is a very efficient technique

that has been created to address this specific class of nonlinear problem. By utilizing a Structural Dynamics Modification

(SDM) approach in conjunction with mode superposition, a significantly smaller set of matrices are required for use in the

direct integration of the equations of motion. The approach will be compared to traditional analytical approaches to make

evident the usefulness of the technique for a variety of test cases.

Keywords Nonlinear analysis • Forced nonlinear response • Linear components for nonlinear analysis • Modal analysis •

Mode superposition

Nomenclature

Symbols

M½ � Physical mass matrix

C½ � Physical damping matrix

K½ � Physical stiffness matrix�M1½ � Modal mass matrix for state 1

D �M12½ � Modal mass change matrix�K1½ � Modal stiffness matrix for state 1

D �K12½ � Modal stiffness change matrix

DM12½ � Physical mass change matrix

DK12½ � Physical stiffness change matrix

U½ �g Generalized Inverse

U1½ � Mode shapes for state 1

U12½ � Mode contribution matrix

U2½ � Mode shapes for state 2

T. Marinone (*) • P. Avitabile

Structural Dynamics and Acoustic Systems Laboratory, University of Massachusetts Lowell, One University Avenue,

Lowell, MA 01854, USA


J.R. Foley • J. Wolfson

Air Force Research Laboratory, Munitions Directorate, Fuzes Branch, Eglin Air Force Base, 306 W. Eglin Blvd, Bldg 432,

Eglin AFB, FL 32542-5430, USA



129


Ff g Physical force

€p1f g Modal acceleration

_p1f g Modal velocity

p1f g Modal displacement

€x1f g Physical acceleration

_x1f g Physical velocity

x1f g Physical displacement

10.1 Introduction

Generally, any nonlinear response analysis involves significant computation, especially if the full analytical model system

matrices are used for the forced response problem. These nonlinearities can be broadly broken down into two categories:

global and local, with significant effort expended on research of both. Due to the significant computational time required for

these nonlinear cases, the analyst may often be unable to investigate the nonlinearities in depth, especially if a set of

performance characteristics related to temperature, preload, deflection, etc. characterize the nonlinear connection elements.

Thus, there is significant motivation to develop a set of reduced order models that can accurately predict nonlinear response

at a substantially reduced computation time.

One area of interest involves the dynamic response of systems with nonlinear connections. These systems are typically

linear, but the introduction of the local nonlinearity causes the system to become highly nonlinear. The approach taken in this

paper is to treat the system as a linear system, but to model the nonlinear component as a change in the linear system by

utilizing a Structural Dynamic Modification. Accordingly, direct integration schemes in conjunction with mode superposi-

tion are employed in order to maximize the efficiency of the model.

This approach was first presented by Avitabile and O’Callahan [1] where a detailed overview of the theory and a simple

analytical example were provided. Friswell et al. [2] looked at reducing models with local nonlinearities with several

different reduction schemes for a periodic solution. Lamarque and Janin [3] looked at modal superposition using a 1-DOF

and 2-DOF system with impact and concluded that modal superposition had limitations due to difficulties in modeling

impact. Ozguven [4] converted non-linear ODE’s into a set of non-linear algebraic equations, which could be reduced by

using linear modes. This technique was found to provide the best reduction in computational time when the structure was

excited at a forcing frequency that corresponded to a resonance of the structure. An alternative approach that has been

studied uses Non-linear Normal Modes (NNMs) which are formulated by Ritz vectors [5, 6]. This approach seeks to extend

the concept of linear orthogonal modes to nonlinear systems.

This paper presents the analytical time results of a beam system subjected to a force impulse. Four cases are studied:

single beam with no contact, single beam with contact, multiple beams with single contact, and multiple beams with multiple

contacts. For each of these cases two types of contact stiffness will be studied; a soft contact representing a rubber/isolation

material, and a hard contact representing a metal on metal contact. For all cases, the time results of the full space model will

be used as a reference.

10.2 Theory

10.2.1 Modal Modification Response Technique (MMRT)

The MMRT technique is based on the Structural Dynamic Modification process and mode superposition method. From the

modal data base of an unmodified system, structural changes can be explored using the modal transformation to project

changes from physical space to modal space of the unmodified system; this results in a heavily coupled set of equations

which are drastically less than those of the physical model and can be written as

. ..

�M1

. ..

2664

3775þ D �M12½ �

2664

3775 €p1f g þ

. ..

�K1

. ..

2664

3775þ D �K12½ �

2664

3775 p1f g ¼ 0½ � (10.1)

where

130 T. Marinone et al.

D �M12½ � ¼ U1½ �T DM12½ � U1½ �D �K12½ � ¼ U1½ �T DK12½ � U1½ �

(10.2)

The solution of the modified set of modal equations in modal space produces an eigensolution and the resulting

eigenvectors of this are the [U12] matrix; this contains the scaling coefficients necessary to form the final modal vectors

from the starting modal vectors. This [U12] matrix is the key to identifying the necessary set of modal vectors to accurately

predict the final modified set of modes [7]. Figure 10.1 illustrates the contribution of the [U12] matrix in forming the final

modified set of modes [U2], where “m” modes of the [U12] matrix are used, and “n-m” modes are excluded.

Mode superposition is executed in a piecewise linear fashion depending on the “state” of the nonlinear connection

element. Once the linear state changes, a structural modification is performed to update the characteristics of the system

along with updated initial conditions to proceed on with the numerical integration. This equation is written as

IAm� �

IBm� ��

€pA� �€pBf g

� þ

O2A

m

h iO2B

m

h i24

35þ U½ �T KTIE½ � U½ �

24

35 pA

� �pBf g

� ¼ UA½ �T f A

� �UB½ �T f Bf g

( )(10.3)

where the initial conditions for the updated state are

pf gj ¼ U½ �gj xf gði�1Þ

_pf gj ¼ U½ �gj _xf gði�1Þ (10.4)

and the generalized inverse can be written as either a pseudo inverse or mass weighted inverse

U½ �gj ¼ Uj

� �TUj

� �h i�1

Uj

� �Tor

U½ �gj ¼ �Mj

� �Uj

� �TMj

� � (10.5)

Figure 10.2 shows the schematic of this technique where the state of the system is checked at each time step by computing

the physical response from mode superposition.

10.2.2 Direct Integration of the Equations of Motion

The direct integration of the equations of motion used here are that of the Newmark [8] method commonly used.

From the known initial conditions for displacement and velocity, the initial acceleration is

€~x0 ¼ M½ ��1 ~F0 � C½ � _~x0 � K½ �~x0 �

(10.6)

Fig. 10.1 Schematic for SDM process using [U12]

10 Efficient Computational Nonlinear Dynamic Analysis Using Modal Modification Response Technique 131

where

€~x0 ¼ initial acceleration vector_~x0 ¼ initial velocity vector

x0 ¼ initial displacement vector~F0 ¼ initial force vector

Choosing an appropriate Dt, a and b (the values chosen for the analytical case studies were 0.0001 s, 0.25 and 0.5 in order

to satisfy sampling parameters and to give constant acceleration), the displacement vector is

~xiþ1 ¼ 1

a Dtð Þ2 M½ � þ baDt

C½ � þ K½ �" #�1

~Fiþ1 þ M½ � 1

a Dtð Þ2~xi þ1

aDt_~xi þ 1

2a� 1

� €~xi

!(

þ C½ � baDt

~xi þ ba� 1

� _~xi þ b

a� 2

� Dt2€~xi

� (10.7)

With the displacement vector known, the acceleration and velocity vectors are

_~xiþ1 ¼ _~xi þ 1� bð ÞDt€~xi þ bDt€~xiþ1 (10.8)

€~xiþ1 ¼ 1

a Dtð Þ2 ~xiþ1 �~xið Þ � 1

aDt_~xi � 1

2a� 1

� €~xi (10.9)

Normal rules regarding integration of the equations of motion are utilized here and are not further discussed.

10.2.3 Time Response Correlation Tools

In order to quantitatively compare two different time solutions, two correlation tools (TRAC and MAC) will be used [9]. Xniand Xnj are the two compared displacement vectors.

TRAC (Time Response Assurance Criterion) – Correlates single DOF across all instances in time.

TRACji ¼Xnj� �T

Xnif gh i2

Xnj� �T

Xnj� �h i

Xnif gT Xnif g� � (10.10)

Formulate Modal Data Base

Determine Force and/orInitial Conditions

Compute Modal Force andModal Initial Conditions

Perform Numerical Integrationfor Next ΔT

Compute Physical ResponseFrom Mode Superposition

Check For Gaps orNonlinearity

Any Change in the CurrentLinear Modal State?

Perform Structural Dynamic Modification to

Reflect Changes

Update Modal Data Base

No Yes

Fig. 10.2 Schematic

for MMRT


MAC (Modal Assurance Criterion) – Correlates all DOF at single instance in time.

MACij ¼Xnif gT Xnj

� �� 2Xnif gT Xnif g� �

Xnj� �T

Xnj� �h i (10.11)

Both MAC and TRAC values close to 1.0 indicate strong similarity between vectors, where values close to 0.0 indicate

minimal or no similarity.

10.3 Model Description and Cases Studied

This section presents the analytical models developed as well as the cases studied. The full-space time solution is used as the

reference solution for all cases.

10.3.1 Model: Beam A and Beam B

Two planar element beam models created using MAT_SAP [10] (a FEM program developed for MATLAB [11]) are used

for all the cases studied. Figure 10.3 shows the two beams assembled into the linear system, where the red points are the

accelerometer measurements, and the blue location is the point of force applied to the system (point 14). Note that 3 in. of

each beam are clamped for the cantilevered boundary condition.

Table 10.1 lists the beam characteristics, while Table 10.2 lists the natural frequencies for the first 10 modes.

The force pulse is an analytic force pulse designed to be frequency band-limited, exciting modes up to 1,000 Hz while

minimally exciting the higher order modes. Using this force pulse, the number of modes that will be involved in the response

can be determined easily, as modes above 1,000Hz can be considered of negligible importance in the response.Modes 1–5will

be primarily excited in BeamA, while modes 1–4 will be primarily excited for BeamB. Figure 10.4a, b show the force pulse in

the time and frequency domain. Damping was assumed 1% of critical damping for all modes (both unmodified and system).

10.3.2 Case 1: Single Beam with No Contact

For the first case, the system is a single beam (Beam A) with no contact. As this system is linear and because there is no SDM

performed, the number of modes needed is only a function of the input excitation. Based on the input force spectrum seen

+

Beam A Beam B

F1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1716

Fig. 10.3 Schematic of impact and response points for beam system

Table 10.2 Model beam frequencies

Mode # 1 2 3 4 5 6 7 8 9 10

Beam A 12.91 84.12 252.34 519.59 806.16 1,256.55 1,682.96 2,201.36 2,755.52 3,510.01

Beam B 22.62 141.56 396.6 776.92 1,284.71 1,918.28 2,678.33 3,563.89 4,572.7 5,707.04

Table 10.1 Model beam characteristics

Beam

Length

(in.)

Width

(in.)

Thickness

(in.)

Num.

of elements

Num.

of nodes

Num.

of DOF

Node spacing

(in.) Material

Density

(lb/in.3)

Young’s modulus

(Msi)

A 18 2 0.123 72 73 146 0.25 Aluminum 2.54E-04 10

B 16 4 0.123 64 65 130 0.25 Aluminum 2.54E-04 10


previously in Fig. 10.4b, the time response needs five modes. In order to determine if this assumption is accurate, the FFT of the

response of systemwas plotted using an increasing number ofmodes and is compared to the full solution (using all 146modes) in

Fig. 10.5.

The FFT in Fig. 10.5 shows that modes 1–3 have a large magnitude, in contrast to modes 4 and 5, which are minimally

excited. Figure 10.5 further shows that a much smaller mode set is required for the time response, as the force pulse has only

excited the first few modes of the system. Based on these results, a minimum of three modes should be used in order to

approximate the full space model accurately. In order to confirm this, the time response at point 14 will be computed using

modes 1 through 5 and will be compared to the full time solution in Fig. 10.6a–e. In addition, Fig. 10.6f will be computed

using all modes (excluding mode 1) to show the effect if a key mode (one with significant magnitude) is not included.

As seen in Fig. 10.6a, b, using only one or two modes is not sufficient to accurately reproduce the full space model, as the

input force spectrum excited additional higher order modes. Once the third mode was added, however, the response overlays

almost perfectly and the further addition of modes 4 and 5 have a negligible effect. Finally, even though 145 of the 146 modes

were used in Fig. 10.6f, the exclusion of mode 1 prevents the accurate reconstruction of the full time response. In order to

further show the effect of mode truncation, the MAC and TRAC of the time responses were averaged as listed in Table 10.3.

Table 10.3 indicates that TRAC is more sensitive than MAC for determining the accuracy of time responses, as MAC is

weighted by DOFs with large responses. Table 10.3 also provides further confirmation that the first three beam modes are the

most critical modes for this time response, as the TRAC improves significantly when adding modes 2 and 3, but has a

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-20

-15

-10

-5

0

5

Time (sec)

For

ce (

lbf)

Analytical Time Domain Force Pulse

0 500 1000 1500-110

-105

-100

-95

-90

-85

-80

-75

-70

-65

-60

Frequency (Hz)

dB F

orce

(lb

f)

FFT of Force PulseAnalytical Time Domain Force Pulse

Fig. 10.4 Analytical force pulse in the time (a) and frequency (b) domain

0 500 1000 1500-200

-150

-100

-50dB

Res

pons

e (g

)

Frequency (HZ)

Analytical FFT of Single Beam With No Contact at Point 14

1 Mode2 Modes3 Modes4 Modes5 Modes146 Modes

Fig. 10.5 FFT of time

response for single beam no

contact using varying number

of modes


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Response at Pt. 14 due to Impact at Pt.14 for Single Beam With No Contact

Time [s]

Dis

plac

emen

t [in

]

146 Modes

1 Mode

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1


Time [s]

Dis

plac

emen

t [in

]

146 Modes

2 Modes

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1


Time [s]

Dis

plac

emen

t [in

]

146 Modes

3 Modes

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1


Time [s]

Dis

plac

emen

t [in

]

146 Modes

4 Modes

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1


Time [s]

Dis

plac

emen

t [in

]

146 Modes

5 Modes

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1


Time [s]

Dis

plac

emen

t [in

]

146 Modes

No Mode 1

a b

c d

e f

Fig. 10.6 Comparison of full solution results for single beam with no contact to results using modes 1 (a), 1–2 (b), 1–3 (c), 1–4 (d), 1–5 (e) and all

modes but mode 1 (f)


minimal improvement when adding the higher order modes. Finally, the solution time for the full system is close to 100 s in

contrast to the modal model, which is less than 5 s. MMRT is shown to have noticeable improvements in computational time

even for the linear system.

In order to demonstrate that the number of modes needed is controlled by the forcing function, the same beam

configuration was run where the analytical force input was changed to excite the first 250 Hz. As fewer modes are excited

over this new frequency bandwidth, the response is affected by fewer modes. Comparing the previous results for using one

and two modes in Table 10.3, the TRAC improves to 0.826 when using one mode, and improves to 0.993 when using two

modes. These results show that decreasing the bandwidth of the forcing function reduces the effect of mode truncation, as the

impact spectrum does not distribute energy to the higher frequency modes.

This first case demonstrated that an accurate time solution could be obtained using a limited number of modes, if the

primary modes excited by the structure are included in the modal database. If the wrong selection of modes is used, the

analyst will not obtain the correct time response, regardless of how many other modes are used.

The following cases will show the application of the MMRT when there is contact and the system becomes nonlinear.

In addition, both soft and hard contacts will be studied to show the effect of different contact stiffness on the accuracy of the

modal models. The soft stiffness case will be studied first, as this contact stiffness is unlikely to excite a high frequency range.

10.4 Case A: Soft Contact

10.4.1 Case A-2: Single Beam with Contact


distance (0.05 in.) shown in Fig. 10.7.

The contact is represented as an additional spring stiffness that is added as soon as Beam A closes the gap. The contact

stiffness of 10 lb/in. will be used to represent a soft contact, such as typically seen in a damper or isolation mount. For the

purposes of MMRT, the spring stiffness was applied to the original model as a SDM. In addition, the contact is only

represented as a spring stiffness in the translational DOF, not in the rotational DOF as well. Table 10.4 lists the frequency

values of the SDM system, while Fig. 10.8 shows the [U12] matrix, or the contribution of the original modes in the SDM (a

highlighted box indicates that the mode of the original model contributes to the SDM model with a magnitude greater than

1.2% – the actual range for each color is shown below).

Table 10.4 and Fig. 10.8 shows that the addition of the spring has a pronounced effect on the lower order modes, with higher

ordermodes remaining relatively unaffected. The soft spring has a noticeable effect on the first twomodes, asmodes 3 and up are

approximately at the same frequency and require no additional mode shape to be added to form the SDM modal matrix (as

indicated by the redmain diagonal). FromCase 1 (single beamwith no contact), themodes needed in theMMRT process are the

modes which are excited by the input force spectrum. For the forcing function used, these should be the first three modes of the

system at a minimum, with additional higher order modes having minimal effect. Examining Fig. 10.8 shows that only modes 1

and 2 are used in the SDM process, so using only three modes should provide a high correlation to the full response solution.

In order to confirm this, the time response at point 14 will be computed using modes 1 through 5 and will be compared to

the full time solution in Fig. 10.9a–e for the soft spring. In addition, Fig. 10.9f will show the time response when mode 3 is

not included when forming the SDM modal database.

As the first three modes of the SDM database (the primary modes excited by the forcing function) only required the first

three modes of the unmodified modal database with the soft spring, three modes were adequate to represent the full system

model with a high degree of accuracy. As seen previously, the addition of modes 4 and 5 had minimal effect due to the input

force spectrum drops off at the higher frequencies. Figure 10.9f showed that the exclusion of mode 3 from the modal

database had a small negative effect. As mode 3 did not contribute to any SDM modes other than mode 3, however, the

response was reasonable comparable to the full space solution (average TRAC ¼ 0.884).

Table 10.3 Average MAC

and TRAC versus # of modes

used for single beam

no contact solution

# of modes Average MAC Average TRAC Solution time (s)

1 0.9439 0.8098 1.67

2 0.9942 0.9044 2.13

3 0.9985 0.9862 2.81

4 0.9986 0.9876 3.52

5 0.9994 0.9991 4.51

146 1.0000 1.0000 99.25


TheMAC and TRAC of the time responses were averaged as listed in Table 10.5 to show the effect of the additional modes.

As explained previously, the first three modes were sufficient to approximate the soft spring solution with a high degree of

accuracy. Finally, using MMRT causes a drastic reduction in solution time as seen comparing the solution time from the full

solution to the time using only a few modes.

This second case demonstrated that the analyst should examine the [U12] matrix to determine the number of modes that

are used in the SDMmodal. Even though the forcing input may only excite a few modes, the SDMmodal database may need

additional modes. Failure to include these modes will produce an incorrect time response regardless of how many other

modes are used.

10.4.2 Case A-3: Multiple Beams with Single Contact

This case consists of the tip of Beam A coming into contact with Beam B once Beam A has displaced a known gap distance

(0.05 in.) shown in Fig. 10.10.

The model uses the same soft stiffness contact of 10 lb/in. to represent the contact of the beams as explained in Case A-2.

Table 10.6 lists the frequency values of the SDM system along with which beam is excited, while Fig. 10.11 shows the

contribution of the original modes in the SDM.

Due to the contact between the two beams, the first mode of the SDM system now consists of modes from both beams,

which increases the number of modes needed for the SDM modal database. Based on the force input, which excites up to

1,000 Hz, the first 10 modes of the system should be excited. Examining Fig. 10.11 shows that for the first 10 modes of the

Table 10.4 Frequencies

for single beam with contact

for soft spring

Mode # Unmodified Soft spring – SDM

1 12.91 26.09

2 84.12 86.02

3 252.34 252.54

4 519.59 519.65

5 806.16 806.18

6 1,256.55 1,256.55

7 1,682.96 1,682.96

8 2,201.36 2,201.36

Fig. 10.8 Mode contribution matrix for single beam with soft single contact

Fig. 10.7 Diagram of single

beam with contact


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

a b

c d

e f

0.1Response at Pt. 14 due to Impact at Pt.14 for Single Beam with Soft Contact

Time [s]

Dis

plac

emen

t [in

]

146 Modes1 Mode

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08


Time [s]

Dis

plac

emen

t [in

]

146 Modes2 Modes

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08


Time [s]

Dis

plac

emen

t [in

]

146 Modes3 Modes

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08


Time [s]

Dis

plac

emen

t [in

]

146 Modes4 Modes

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08


Time [s]

Dis

plac

emen

t [in

]

146 Modes5 Modes

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08


Time [s]

Dis

plac

emen

t [in

]

146 ModesNo Mode 3

Fig. 10.9 Comparison of full solution results for single beam with soft contact to results using modes 1 (a), 1–2 (b), 1–3 (c), 1–4 (d), 1–5 (e) and

all modes but mode 3 (f)


SDM system, the soft spring modal database requires modes 1 through 6 of Beam A and modes 1 through 4 of Beam B.

Note that the number of modes used in the time response is controlled by the frequency bandwidth of the input forcing

function, while the number of modes used in the SDM database is controlled by the [U12] matrix.

The time solution using the expected number of modes in the time response (10) with the required number of modes for

the SDM (6 from Beam A and 4 from Beam B) will be computed and compared to the full solution in Fig. 10.12a.

In addition, the effect of including incorrect modes in the time response will be shown by using the same modes in the time

response, where the SDM modal database contains all modes of Beam A but no modes of Beam B in Fig. 10.12b.

Figure 10.12a shows that including the necessary modes (6 from Beam A and 4 from Beam B) needed to form the [U2]

matrix based on the number of modes (10) excited by the input forcing function is enough to accurately reproduce the time

solution. If not all of the necessary modes are included as seen in Fig. 10.12b where no modes from Beam B were included,

using all of the modes of Beam A is not sufficient to accurately compute the response.

In contrast to the model of the single beam, more than five modes were needed in order to obtain an accurate time solution

of the system, due to the need for mode shapes from both beams. The MAC and TRAC of the time responses were averaged

as listed in Table 10.7.



used for single beam contact

solution

# of modes

Soft

Average MAC Average TRAC Solution time (s)

1 0.8816 0.2884 1.68

2 0.9670 0.8858 1.98

3 0.9910 0.9699 2.54

4 0.9659 0.9677 3.78

5 0.9893 0.9860 4.63

Full 1.0000 1.0000 186.25

Fig. 10.10 Diagram

of multiple beam with

single contact

Table 10.6 Frequencies for

multiple beams with single

contact for soft spring

Mode # Unmodified SDM – soft spring

1 Beam A – 12.91 20.35

2 Beam B – 22.62 29.58

3 Beam A – 84.12 86.01

4 Beam B – 141.56 142.43

5 Beam A – 252.34 252.54

6 Beam B – 396.6 396.71

7 Beam A – 519.59 519.65

8 Beam B – 776.92 777.00

9 Beam A – 806.16 806.18

10 Beam A – 1,256.55 1,256.55

11 Beam B – 1,284.71 1,284.80

12 Beam A – 1,682.96 1,682.96

13 Beam B – 1,918.28 1,918.28

14 Beam A – 2,201.36 2,201.36

15 Beam B – 2,678.33 2,678.39

16 Beam A – 2,755.52 2,755.53

17 Beam A – 3,510.01 3,510.02

18 Beam B – 3,563.89 3,563.89

19 Beam A – 3,948.54 3,948.55

20 Beam B – 4,572.70 4,572.72


The third case demonstrated that the mode shapes of both components affect the modeling of nonlinear systems.

The forcing function bandwidth governs the number of modes needed to compute the time response and may be different

from the number of modes needed to compute the SDMmodal database. The [U12] matrix is critical in order to determine the

number of modes needed to form the system mode shapes, and the exclusion of modes used in the matrix can noticeably

degrade the correlation. For the soft spring contact, using the predicted number of mode shapes based on the forcing function

and [U12] matrix provided an accurate analytical time solution.

Bar Color Min. ValueMaxValue

Black 0.0120 0.0200

Blue 0.0200 0.0500

Green 0.0500 0.1000

Cyan 0.1000 0.2000

Magenta 0.2000 0.5000

Yellow 0.5000 0.8000

Red 0.8000 1.0000

System Mode Shapes -Soft Spring

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Bea

m A

Unm

odifie

d M

ode

Sha

pes

12345678910

Bea

m B

Unm

odifie

d M

ode

Sha

pes

12345678910

Fig. 10.11 Mode contribution matrix for multiple beams with single contact for soft spring

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Response at Pt. 14 due to Impact at Pt.14 for Multiple Beams with Soft Single Contact

Time [s]

Dis

plac

emen

t [in

]

10 Modes in Time Response (146 Beam A and 0 Beam B - U2)276 Modes

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Response at Pt. 14 due to Impact at Pt.14 for Multiple Beams with Soft Single Contact

Time [s]

Dis

plac

emen

t [in

]


a b

Fig. 10.12 Comparison of results using 10 modes in the time response with the [U2] matrix formed from 6 Beam A and 4 Beam B modes (a) and

146 Beam A and 0 Beam B modes (b) for multiple beam single contact with soft spring

Table 10.7 Average MAC and TRAC versus # of modes used for multiple beam single contact solution

Soft spring


10 – time (6 Beam A + 4 Beam B) – [U2] 0.9998 0.9916 6.78

10 – time (146 Beam A + 0 Beam B) – [U2] 0.8146 0.6167 6.78

276 1.0000 1.0000 1137.74


10.4.3 Case A-4: Multiple Beams with Multiple Contacts

This case consists of the tip of Beam A coming into contact with the tip of Beam B shown in Fig. 10.13a–c. Note that each

system is a potential configuration of the beam depending on the relative displacements of the two beams.

The model uses the same soft spring stiffness value as described in Case A-3. Because the system has multiple possible

configurations, multiple SDMs are required in order to represent the different possible configurations. The modal database

must therefore contain all of the necessary modes for all of the configurations in order to obtain an accurate model.

Table 10.8 lists the frequency values of the SDM system, while Fig. 10.14 shows the contribution of the original modes in

the SDM.

Table 10.8 and Fig. 10.14 shows that not only does the stiffness of the spring affect the number of modes needed, but the

location of the spring affects the modes as well. Depending on whether the spring is at the tip of Beam A or tip of Beam B,

the mode shapes and frequencies change noticeably. For example, the second mode of Beam A is used for the first mode of

the soft spring SDM for configurations 1 and 3, but is not needed in configuration 2. Thus, if only configuration 2 came into

play this mode could be neglected in the modal database; due to also using configurations 1 and 3 that require this second

mode, this mode should be used in the database.

Examining Fig. 10.14 shows that for a forcing function which excites the first 10 modes of the system, the soft spring

configuration requires modes 1–6 of Beam A and modes 1–4 of Beam B. Figure 10.15 compares the full solution to the

solution using 10 modes for the time solution with 6 modes from Beam A and 4 modes from Beam B for the [U2] matrix for

the soft spring.

a b c

Fig. 10.13 Diagram of multiple beam with multiple contact for configurations 1(a), 2 (b) and 3 (c)



contact for soft spring Mode # Unmodified

Soft spring

Config. 1 Config. 2 Config. 3

1 Beam A – 12.91 20.35 14.78 21.08

2 Beam B – 22.62 29.58 33.04 39.23

3 Beam A – 84.12 86.01 85.71 87.24

4 Beam B – 141.56 142.43 142.98 143.82

5 Beam A – 252.34 252.54 252.61 252.82

6 Beam B – 396.6 396.71 396.96 397.07

7 Beam A – 519.59 519.65 519.61 519.68

8 Beam B – 776.92 777 777.04 777.12

9 Beam A – 806.16 806.18 806.28 806.3

10 Beam A – 1,256.55 1,256.55 1,256.56 1,256.56

11 Beam B – 1,284.71 1,284.80 1,284.76 1,284.85

12 Beam A – 1,682.96 1,682.96 1,682.99 1,682.99

13 Beam B – 1,918.28 1,918.28 1,918.29 1,918.30

14 Beam A – 2,201.36 2,201.36 2,201.37 2,201.37

15 Beam B – 2,678.33 2,678.39 2,678.34 2,678.39

16 Beam A – 2,755.52 2,755.53 2,755.53 2,755.53

17 Beam A – 3,510.01 3,510.02 3,510.02 3,510.03

18 Beam B – 3,563.89 3,563.89 3,563.89 3,563.89

19 Beam A – 3,948.54 3,948.55 3,948.54 3,948.55

20 Beam B – 4,572.70 4,572.72 4,572.70 4,572.72


As seen in Case 3, the [U12] matrix provides a good understanding of the soft spring model. Even with a more complex

model with multiple possible configurations, the computational savings gained from the MMRT technique are substantial.

This case used all of the lessons learned from the previous three cases in order to compute an accurate time solution using

MMRT with a subset of the modes and demonstrate the usefulness of the technique on a complex structure with multiple

potential interactions.

10.5 Case B: Hard Contact

For the soft stiffness contact cases shown above, the contact stiffness was soft enough that the contact did not excite a

frequency bandwidth beyond the excitation force spectrum. Thus, the forcing function controls the number of modes needed

in the time response, and the number of modes needed in the SDM modal database is based on those modes. As long as the

excitation bandwidth due to the contact stiffness impact is below the input bandwidth, the procedure to identify the number

of modes needed works well as seen in Cases A-2 through A-4.

For a harder stiffness contact case, however, there is a possibility that the contact stiffness would excite a frequency

bandwidth beyond the excitation force spectrum. Under this scenario, the number of modes needed in the time response

would be a function of not only the forcing function, but also of the contact stiffness. In order to examine this possible

scenario in detail, the contact stiffness of 1,000 lb/in. is used for the same cases.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

10

Bea

m A

Unm

odifie

d M

ode

Sha

pes

Bea

m B

Unm

odifie

d M

ode

Sha

pes

System Mode Shapes - Soft Spring - Configuration 1 System Mode Shapes - Soft Spring - Configuration 2 System Mode Shapes - Soft Spring - Configuration 3

1.00000.8000Red

0.80000.5000Yellow

0.50000.2000Magenta

0.20000.1000Cyan

0.10000.0500Green

0.05000.0200Blue

0.02000.0120Black

Max Value

Min. ValueBar Color

1.00000.8000Red

0.80000.5000Yellow

0.50000.2000Magenta

0.20000.1000Cyan

0.10000.0500Green

0.05000.0200Blue

0.02000.0120Black

Max Value

Min. ValueBar Color

Fig. 10.14 Mode contribution matrix for multiple beams with multiple contacts for soft spring

276 1 1 915.21

Soft Spring

# of Modes

0.9884 0.9774 11.7510 – Time

(6 Beam A + 4 Beam B) – [U2]

AverageMAC

AverageTRAC

SolutionTime (s)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1Response at Pt. 14 due to Impact at Pt.14 for Multiple Beams with Soft Multiple Contact

Time [s]

Dis

plac

emen

t [in

]


Fig. 10.15 Comparison of results in the time domain for multiple beams with multiple contacts with soft spring


10.5.1 Case B-2: Single Beam with Contact


distance (0.05 in.) as explained previously in Case A-2. Figure 10.16 shows the [U12] matrix and lists the natural frequencies

of the first eight modes.

Figure 10.16 shows that in order to accurately obtain the first five system modes, the first five modes of the unmodified

system are required. This is expected, as the mode shapes with the harder spring attached look less like the original model

and therefore require more modes in order to form the SDM. As Fig.10.16 shows that modes 1–5 of the unmodified system

are involved in the first three modes of the SDM, five modes should be used to provide a high correlation to the full response

solution. In order to confirm this, the time response at point 14 will be computed using 1 through 5 modes and compared to

the full time solution in Fig. 10.17a–e for the hard spring. In addition, Fig. 10.17f will show the time response when mode 3

is not included when forming the SDM modal database.

In contrast to Case A-2, using the hard spring produced poor results using modes 1–4. As seen in Fig. 10.16, modes 2 and

3 of the SDM database require modes 1–5 of the unmodified modal database and were not able to be adequately represented

using only modes 1–4. Even though the forcing function was limited primarily to the first three modes of the system, the first

3 modes of the SDM required more than three modes of the unmodified modal database. Figure 10.17f showed that the

exclusion of mode 3 from the modal database decreased the accuracy of the results. Examining Fig. 10.16 shows that mode 3

had a substantial contribution to SDM modes 1–4, and thus the SDM modes could not be fully represented without the

missing mode. The response is poor compared to the full space solution (average TRAC ¼ 0.439).

In order to show the effect of the modes, the MAC and TRAC of the time responses were averaged as listed in Table 10.9.

As the hard spring solution required five modes of the unmodified modal database, the response remained poor until all

five of the modes were included as seen in the sudden increase in TRAC (0.6615–0.9949).

Finally, the FFT of the time response for both the soft and hard spring is shown in Fig. 10.18. Note that the hard spring

FFT is more nonlinear in contrast to the soft spring, where the main frequencies of the system can still be identified clearly.

The hard spring does not contain significant energy beyond 1,000 Hz, showing that for this case, the forcing function and not

the contact stiffness govern the number of modes needed in the time response.

10.5.2 Case B-3: Multiple Beams with Single Contact

This case consists of the tip of Beam A coming into contact with the tip of Beam B once Beam A has displaced a known gap

distance (0.05 in.) as explained previously in Case A-3. Table 10.10 lists the frequency values of the SDM system along with

which beam is excited, while Fig. 10.19 shows the [U12] matrix.

As seen in Case A-3, the forcing function excited the first 10 modes of the system. Examining Fig. 10.19 shows that for

the first 10 modes of the SDM system, the hard spring modal database requires modes 1–6 of Beam A and modes 1–5 of

Beam B. The time solution using the expected number of modes in the time response (10) with the required number of modes

for the SDM (6 from Beam A and 5 from Beam B) will be computed and compared to the full solution in Fig. 10.20.

The expected number of modes used for the time response (10) with the modes used in the SDM modal database (6 from

Beam A and 5 from Beam B), does not produce an accurate solution as seen in Fig. 10.20. In contrast, Case A-3 with the soft

Mode #

1 12.91 66.681

12

3

4

5

6

78

2 3 4 5 6 7 8 9 10

223.55

326.04

529.68

808.16

84.12

252.34

519.59

806.16

1256.55 1256.73

Bea

m A

Unm

odifie

d M

ode

Shap

es

1682.96 1682.97

2201.36 2201.52

2

3

4

5

6

7

8

UnmodifiedHard Spring-

SDMSystem Mode Shapes-Hard Spring

Bar Color

Black 0.0120 0.02000.0500

0.10000.2000

0.5000

0.8000

1.0000

0.0200

0.0500

0.1000

0.2000

0.5000

0.8000

Blue

GreenCyan

Magenta

Yellow

Red

Min.Value

Max.Value

Fig. 10.16 Mode contribution matrix for single beam with single contact for soft (a) and hard (b) spring


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Response at Pt. 14 due to Impact at Pt.14 for Single Beam with Hard Contact

Time [s]

Dis

plac

emen

t [in

]

146 Modes1 Mode

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3


Time [s]

Dis

plac

emen

t [in

]

146 Modes2 Modes

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1

0

0.1

0.2

0.3

0.4

0.5


Time [s]

Dis

plac

emen

t [in

]

146 Modes3 Modes

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3


Time [s]

Dis

plac

emen

t [in

]

146 Modes4 Modes

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3


Time [s]

Dis

plac

emen

t [in

]

146 Modes5 Modes

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1

0

0.1

0.2

0.3

0.4

0.5


Time [s]

Dis

plac

emen

t [in

]

146 ModesNo Mode 3

a b

c d

e f

Fig. 10.17 Comparison of full solution results for single beam with hard contact to results using modes 1 (a), 1–2 (b), 1–3 (c), 1–4 (d), 1–5 (e) and

all modes but mode 3 (f)


spring did produce an accurate solution using the same mode set as seen in Fig. 10.12a. Figure 10.21 shows the time response

using 23 modes for the time response, where 20 modes from Beam A and 15 modes from Beam B formed the SDM database.

Figure 10.21 shows that additional modes are needed beyond the expected number of modes (based only on the input

force) in order to compute the accurate time response of the hard spring system. To investigate why using the expected

number of modes did not produce an accurate time response for the hard spring stiffness, the FFT of the time response for

both springs is shown in Fig. 10.22.

Note that in contrast to the soft spring, the hard spring contact excited modes above 1,000 Hz (almost up to 3,000 Hz),

which is above the input force spectrum. Accordingly, the number of modes needed in the time response is no longer

governed by the forcing function bandwidth, but by the contact stiffness bandwidth. This explains why Fig. 10.21 required

additional modes in order to obtain an accurate answer, as the contact stiffness was exciting modes that were not included

when the first 10 modes of the system formed the database. Since these unused higher order modes were excited by the

contact stiffness, they needed to be included to obtain a correct time solution. In order to show the effect of the modes, the

MAC and TRAC of the time responses were averaged as listed in Table 10.11.

10.5.3 Case B-4: Multiple Beams with Multiple Contacts

This case consists of the tip of Beam A coming into contact with the tip of Beam B once the relative displacements of the

beams are within a specified contact tolerance (0.001 in.) as described previously in Case A-4. Table 10.12 lists the

frequency values of the SDM system, while Fig. 10.23 shows the contribution of the original modes in the SDM.

Figure 10.23 shows that the hard spring configuration requires modes 1–6 of Beam A and modes 1–5 of Beam B. As Case

B-3 showed that additional modes beyond the modes excited by the forcing function are required when using the hard spring,

only the time solution where accurate results are obtained is shown. Using the number of modes based on the forcing



used for single beam contact

solution

# of modes

Hard

Average MAC Average TRAC Solution time (s)

1 0.8672 0.3591 1.65

2 0.8586 0.5056 2.29

3 0.8569 0.5479 3.01

4 0.8935 0.6615 3.73

5 0.9992 0.9949 4.27

Full 1.0000 1.0000 298.91

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-220

-200

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-160

-140

-120

-100

-80

-60

-40

-20

Frequency (Hz)

dB R

espo

nse

(g)

Analytical FFT at Point 14 - Single Beam with Contact

Soft SpringHard Spring

Fig. 10.18 FFT comparison

between the soft and hard

spring for single beam with

contact


function provides an average TRAC of 0.3625, showing that there are an insufficient number of modes used as expected.

Figure 10.24 compares the full solution to the solution using 23 modes for the time solution with 20 modes from Beam A and

15 modes from Beam B for the [U2] matrix for the hard spring.

As seen in Cases A-3 and B-3, the [U12] matrix provides a good understanding of the soft spring model, but a poor

understanding of the hard spring model. In order to show the effect of the modes, the MAC and TRAC of the time responses

were averaged as listed in Table 10.13.

Finally, the FFT of the time response was computed for both the soft and hard spring shown in Fig. 10.25.

As seen in Case B-3, the hard contact has excited modes above the forcing function bandwidth of 1,000 Hz. Since the

contact stiffness now governs the number of modes needed in the time response, more than the 10 modes excited by the

forcing function are required.

Comparing the soft and hard spring cases shows that the number of modes needed can be predicted accurately as long as

the forcing function bandwidth defines the frequency range that is excited as seen in the soft spring cases. Once that

condition is no longer true, however, determining the number of modes needed requires some knowledge of the frequency

Table 10.10 Frequencies

for multiple beams with single

contact for hard spring

Mode # Unmodified SDM – hard spring

1 Beam A – 12.91 21.25

2 Beam B – 22.62 67.74

3 Beam A – 84.12 127.04

4 Beam B – 141.56 236.02

5 Beam A – 252.34 338.31

6 Beam B – 396.6 424.26

7 Beam A – 519.59 532.04

8 Beam B – 776.92 786.59

9 Beam A – 806.16 809.25

10 Beam A – 1,256.55 1,256.69

11 Beam B – 1,284.71 1,294.66

12 Beam A – 1,682.96 1,682.97

13 Beam B – 1,918.28 1,918.72

14 Beam A – 2,201.36 2,201.53

15 Beam B – 2,678.33 2,683.90

16 Beam A – 2,755.52 2,755.78

17 Beam A – 3,510.01 3,510.46

18 Beam B – 3,563.89 3,564.14

19 Beam A – 3,948.54 3,948.79

20 Beam B – 4,572.70 4,575.19

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201234567891012345678910

Bea

m A

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ode

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Bea

m B

Unm

odifie

d M

ode

Sha

pes

System Mode Shapes - Hard Spring

Bar Color Min. ValueMax Value

Black 0.0120 0.0200

Blue 0.0200 0.0500

Green 0.0500 0.1000

Cyan 0.1000 0.2000

Magenta 0.2000 0.5000

Yellow 0.5000 0.8000

Red 0.8000 1.0000

Fig. 10.19 Mode contribution matrix for multiple beams with single contact for hard spring


bandwidth of contact ahead of time. For both cases, however, the analyst can obtain an accurate modal solution with

significant computational savings provided the used modes satisfy the force spectrum and the contact stiffness spectrum.

10.6 Contact Time Step Study

For the chosen Dt of 0.0001 s, Raleigh Criteria and Shannon’s Sampling Theorem state that the maximum frequency range

that can be observed is 5 kHz, well above the forcing function frequency range. Although this time resolution may be fine

enough to accurately capture the time response for the linear system, this resolution may not be adequate when the response

becomes nonlinear. Since the Dt chosen affects when the system first comes into contact, the system may respond differently

depending on whether the impact is slow or abrupt (i.e. soft or hard spring). For a slow spring, the system remains in contact

for a longer period of time, and the time response should remain consistent regardless of the time step chosen. For the hard

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1

-0.05

0

0.05

0.1

0.15Response at Pt. 14 due to Impact at Pt.14 for Multiple Beams with Hard Single Contact

Time [s]

Dis

plac

emen

t [in

]


Fig. 10.20 Comparison of

results using 10 modes in the

time response with the [U2]

matrix formed from 6 Beam A

and 5 Beam B modes for

multiple beam single contact

with hard spring

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Response at Pt. 14 due to Impact at Pt.14 for Multiple Beams with Hard Single Contact

Time [s]

Dis

plac

emen

t [in

]



results using 23 modes in the

time response with the [U2]

matrix formed from 20 Beam

A and 15 Beam B modes for

multiple beam single contact

with hard spring


0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-240

-220

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-100

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-60

Frequency (Hz)

dB R

espo

nse

(g)

Analytical FFT at Pt. 14 - Multiple Beam with Single Contact




spring for multiple beam with

single contact

Table 10.11 Average MAC and TRAC versus # of modes used for multiple beam single contact solution

Hard spring


10 – time (6 Beam A + 5 Beam B) – U2 0.7888 0.5232 6.87


276 1.0000 1.0000 1240.13



contact for soft and hard springs Mode # Unmodified

Hard spring

Config. 1 Config. 2 Config. 3

1 Beam A – 12.91 21.25 15.5 36.22

2 Beam B – 22.62 67.74 69.52 113.69

3 Beam A – 84.12 127.04 119.79 228.57

4 Beam B – 141.56 236.02 234.14 312.68

5 Beam A – 252.34 338.31 325.85 339.4

6 Beam B – 396.6 424.26 467.09 504.56

7 Beam A – 519.59 532.04 527.49 532.41

8 Beam B – 776.92 786.59 786.04 790.33

9 Beam A – 806.16 809.25 828.78 833.74

10 Beam A – 1,256.55 1,256.69 1,257.17 1,257.22

11 Beam B – 1,284.71 1,294.66 1,290.03 1,300.10

12 Beam A – 1,682.96 1,682.97 1,686.00 1,686.01

13 Beam B – 1,918.28 1,918.72 1,920.05 1,920.48

14 Beam A – 2,201.36 2,201.53 2,202.67 2,202.84

15 Beam B – 2,678.33 2,683.90 2,678.85 2,684.38

16 Beam A – 2,755.52 2,755.78 2,756.01 2,756.28

17 Beam A – 3,510.01 3,510.46 3,511.34 3,511.79

18 Beam B – 3,563.89 3,564.14 3,563.98 3,564.23

19 Beam A – 3,948.54 3,948.79 3,948.54 3,948.79

20 Beam B – 4,572.70 4,575.19 4,572.70 4,575.19


spring, however, the system may experience a high frequency impact chatter, which may not be seen with a large time step.

Figure 10.26a, b show the time response for the first 0.1 s for the soft and hard spring respectively, where a time step of

0.00005 s and the previously used time step of 0.0001 s are compared.

Figure 10.26a shows that for the soft spring system, the reduced time step had minimal effect on the time solution. Since

the system remains in contact with the soft spring for an extended period, the results compare very well. For the hard spring

in Fig. 10.26b, the system comes into contact and then immediately bounces off. Since the time step duration directly affects

the acceleration, there is a divergence in the solution.

Although this study shows that the chosen time step was not sufficiently fine for the hard spring, the time step of 0.0001 s

was used for all of the previous cases in order to demonstrate the main principles of MMRT. Further study is required to

determine the required time step when the contact stiffness dominates the response.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

10

System Mode Shapes - Hard Spring - Configuration 1 System Mode Shapes - Hard Spring - Configuration 2 System Mode Shapes - Hard Spring - Configuration 3

sepahSedo

Mde ifido

m nU

Ama e

Bsep ahS

ed oM

de ifid omn

UB

ma eB

1.00000.8000Red

0.80000.5000Yellow

0.50000.2000Magenta

0.20000.1000Cyan

0.10000.0500Green

0.05000.0200Blue

0.02000.0120Black

Max Value

Min. ValueBar Color

1.00000.8000Red

0.80000.5000Yellow

0.50000.2000Magenta

0.20000.1000Cyan

0.10000.0500Green

0.05000.0200Blue

0.02000.0120Black

Max Value

Min. ValueBar Color

Fig. 10.23 Mode contribution matrix for multiple beams with multiple contacts for hard spring

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Response at Pt. 14 due to Impact at Pt.14 for Multiple Beams with Hard Multiple Contact

Time [s]

Dis

plac

emen

t [in

]



results in the time domain for

multiple beams with multiple

contacts with hard spring

Table 10.13 Average MAC and TRAC versus # of modes used for multiple beams multiple contact solution

Hard spring



276 1.0000 1.0000 1196.69


10.7 Comparison to Large DOF Models

Although the computational time savings using MMRT is significant, the effect is not dramatically seen until comparing to a

large FEM model as typically seen in industry. Figure 10.27 shows the FEM of a missile rack system with a fine mesh

resolution.

Analysts use this model in dynamic simulations to compute the transient dynamics during a missile firing under a variety

of condition. As the system is a complex system with many detailed components, the time response models require

significant computation time (typically days). The computational time can be dramatically reduced by using the mode

shapes and frequencies of the various components. This allows analysts to study multiple configurations in detail, providing

further information in order to improve the design.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-220

-200

-180

-160

-140

-120

-100

-80

-60

-40

Frequency (Hz)

dB R

espo

nse

(g)

Analytical FFT at Pt. 14 - Multiple Beam with Multiple Contact




spring for multiple beam with

multiple contact

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08


Time [s]

Dis

plac

emen

t [in

]

dt = 0.0001 sdt = 0.00005 s

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-0.05

0

0.05

0.1

0.15

0.2

0.25


Time [s]

Dis

plac

emen

t [in

]

dt = 0.0001 sdt = 0.00005 s

a b

Fig. 10.26 Comparison of results with two different time steps for soft (a) and hard (b) contact


10.8 Conclusion

A proposed technique – Modal Modification Response Technique (MMRT) – for computing the time response of a nonlinear

system is described. For linear systems that are connected by a nonlinear local connection, mode superposition and structural

dynamic modification can be used to approximate the nonlinear response of the system at significant computational savings.

An analytical study of this technique is shown using a linear beam systemwith contact due to impact. Four cases of increasing

complexity are studied; observations on the number of modes needed for each case aremade based on the forcing function and

SDM mode contribution matrix. For all cases, MMRT yields accurate results with substantially less computation time.

10.9 Further Work

Further work will be done using this technique to demonstrate the usefulness of this approach. For comparison to

experimental results, several additional items will need to be studied in order to yield accurate results.

First, the underlying linear system will need to be a very high accuracy model in order to accurately predict the time

responses. Care will need to be taken in modeling and updating the model to test data to ensure that both the model and

measurements are reflective of the physical structure. Second, the damping will have to be measured experimentally for the

linear system for as many modes as possible in order to have high time correlation. In addition, efforts will be needed to

determine the correct damping for the SDM models, as the damping may change once the beam(s) are in contact. Third, the

stiffness of the contact will have to be determined in order to accurately calculate the SDM. In addition, the impact force of

the beam contact may need to be included in the analytical model as this is an additional force input to the structure that

affects the time response. Finally, variation in the time step used was found to affect the results obtained and will require

further study.

Acknowledgements Some of the work presented herein was partially funded by Air Force Research Laboratory Award No. FA8651-10-1-0009

“Development of Dynamic Response Modeling Techniques for Linear Modal Components”. Any opinions, findings, and conclusions or

recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the particular funding agency.

The authors are grateful for the support obtained.

Fig. 10.27 Typical large

FEM model


Appendix A: Component and System Mode Shapes

0 2 4 6 8 10 12 14 16-60

-40

-20

0

20

40

60

0 2 4 6 8 10 12 14 16-60

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-20

0

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60

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0 2 4 6 8 10 12 14 16-60

-40

-20

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0 2 4 6 8 10 12 14 16-60

-40

-20

0

20

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0 2 4 6 8 10 12 14 16-60

-40

-20

0

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40

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0 2 4 6 8 10 12 14 16-60

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20

40

60

Unmodified Mode Shapes – Beam B

1 – 22.62 Hz 2 – 141.56 Hz 3 – 396.60 Hz

4 – 776.92 Hz 5 – 1284.71 Hz 6 – 1918.28 Hz

7 – 2678.33 Hz 8 – 3563.89 Hz 9 – 4572.70 Hz

10 – 5707.04 Hz 11 – 6956.91 Hz 12 – 8324.25 Hz

Unmodified Mode Shapes – Beam A

0 2 4 6 8 10 12 14 16 18-60

-40

-20

0

20

40

60

0 2 4 6 8 10 12 14 16 18-60

-40

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601 - 12.91 Hz 2 – 84.12 Hz 3 – 252.34 Hz

0 2 4 6 8 10 12 14 16 18-60

-40

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0 2 4 6 8 10 12 14 16 18-60

-40

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-40

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-40

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0

20

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60

4 – 519.59 Hz 5 – 806.16 Hz 6 – 1256.55 Hz

7 – 1682.96 Hz 8 – 2201.36 Hz 9 – 2755.52 Hz

10 – 3510.01 Hz 11 – 3948.54 Hz 12 – 5076.34 Hz

0 2 4 6 8 10 12 14 16 18-60

-40

-20

0

20

40

60

0 2 4 6 8 10 12 14 16 18-60

-40

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0 2 4 6 8 10 12 14 16 18-60

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0 2 4 6 8 10 12 14 16 18-60

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60

Single Beam Single Contact Mode Shapes – 10 lb/in Spring

1 – 26.09 Hz 2 – 86.02 Hz 3 – 252.54 Hz

4 – 519.65 Hz 5 – 806.18 Hz 6 – 1256.55 Hz

7 – 1682.96 Hz 8 – 2201.36 Hz 9 – 2755.53 Hz

10 – 3510.02 Hz 11 – 3948.55 Hz 12 – 5076.34 Hz

0 2 4 6 8 10 12 14 16 18-60

-40

-20

0

20

40

60

0 2 4 6 8 10 12 14 16 18-60

-40

-20

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20

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0 2 4 6 8 10 12 14 16 18-60

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0 2 4 6 8 10 12 14 16 18-60

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0 2 4 6 8 10 12 14 16 18-60

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0 2 4 6 8 10 12 14 16 18-60

-40

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0 2 4 6 8 10 12 14 16 18-60

-40

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0 2 4 6 8 10 12 14 16 18-60

-40

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0 2 4 6 8 10 12 14 16 18-60

-40

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0 2 4 6 8 10 12 14 16 18-60

-40

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0 2 4 6 8 10 12 14 16 18-60

-40

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20

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0 2 4 6 8 10 12 14 16 18-60

-40

-20

0

20

40

60

Single Beam Single Contact Mode Shapes – 1000 lb/in Spring

1 – 66.68 Hz 2 – 223.55 Hz 3 – 326.04 Hz

4 – 529.68 Hz 5 – 808.16 Hz 6 – 1256.73 Hz

7 – 1682.97 Hz 8 – 2201.52 Hz 9 – 2755.76 Hz

10 – 3510.46 Hz 11 – 3948.79 Hz 12 – 5076.36 Hz


0 5 10 15 20 25-60

-40

-20

0

20

40

60

0 5 10 15 20 25-60

-40

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60

Multiple Beam Contact Mode Shapes - Configuration 1 – 10 lb/in Spring

1 – 20.35 Hz 2 – 29.58 Hz 3 – 86.01 Hz

4 – 142.43 Hz 5 – 252.54 Hz 6 – 396.71 Hz

7 – 519.65 Hz 8 – 777.00 Hz 9 – 806.18 Hz

10 – 1256.55 Hz 11 – 1284.80 Hz 12 – 1682.96 Hz

0 5 10 15 20 25-60

-40

-20

0

20

40

60

0 5 10 15 20 25-60

-40

-20

0

20

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0 5 10 15 20 25-60

-40

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0 5 10 15 20 25-60

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0 5 10 15 20 25-60

-40

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0 5 10 15 20 25-60

-40

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20

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0 5 10 15 20 25-60

-40

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0

20

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0 5 10 15 20 25-60

-40

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0

20

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0 5 10 15 20 25-60

-40

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0

20

40

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0 5 10 15 20 25-60

-40

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0

20

40

60

0 5 10 15 20 25-60

-40

-20

0

20

40

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0 5 10 15 20 25-60

-40

-20

0

20

40

60

0 5 10 15 20 25-60

-40

-20

0

20

40

60


1 – 14.78 Hz 2 – 33.04 Hz 3 – 85.71 Hz

4 – 142.98 Hz 5 – 252.61 Hz 6 – 396.96 Hz

7 – 519.61 Hz 8 – 777.04 Hz 9 – 806.28 Hz

10 – 1256.56 Hz 11 – 1284.76 Hz 12 – 1682.99 Hz

0 5 10 15 20 25-60

-40

-20

0

20

40

60

0 5 10 15 20 25-60

-40

-20

0

20

40

60

0 5 10 15 20 25-60

-40

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20

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0 5 10 15 20 25-60

-40

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20

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0 5 10 15 20 25-60

-40

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0 5 10 15 20 25-60

-40

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0 5 10 15 20 25-60

-40

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20

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0 5 10 15 20 25-60

-40

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20

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0 5 10 15 20 25-60

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0 5 10 15 20 25-60

-40

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0 5 10 15 20 25-60

-40

-20

0

20

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0 5 10 15 20 25-60

-40

-20

0

20

40

60


1 – 21.08 Hz 2 – 39.23 Hz 3 – 87.24 Hz

4 – 143.82 Hz 5 – 252.82 Hz 6 – 397.07 Hz

7 – 519.68 Hz 8 – 777.12 Hz 9 – 806.30 Hz

10 – 1256.56 Hz 11 – 1284.85 Hz 12 – 1682.99 Hz


0 5 10 15 20 25-60

-40

-20

0

20

40

60

0 5 10 15 20 25-60

-40

-20

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20

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0 5 10 15 20 25-60

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0 5 10 15 20 25-60

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0 5 10 15 20 25-60

-40

-20

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0 5 10 15 20 25-60

-40

-20

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20

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0 5 10 15 20 25-60

-40

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20

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0 5 10 15 20 25-60

-40

-20

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20

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0 5 10 15 20 25-60

-40

-20

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20

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0 5 10 15 20 25-60

-40

-20

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20

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0 5 10 15 20 25-60

-40

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20

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0 5 10 15 20 25-60

-40

-20

0

20

40

60


1 – 36.22 Hz 2 – 113.69 Hz 3 – 228.57 Hz

4 – 312.68 Hz 5 – 339.40 Hz 6 – 504.56 Hz

7 – 532.41 Hz 8 – 790.33 Hz 9 – 833.74 Hz

10 – 1257.22 Hz 11 – 1300.10 Hz 12 – 1686.01 Hz

0 5 10 15 20 25-60

-40

-20

0

20

40

60

0 5 10 15 20 25-60

-40

-20

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20

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0 5 10 15 20 25-60

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0 5 10 15 20 25-60

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0 5 10 15 20 25-60

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0 5 10 15 20 25-60

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0 5 10 15 20 25-60

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0 5 10 15 20 25-60

-40

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0 5 10 15 20 25-60

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20

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0 5 10 15 20 25-60

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0 5 10 15 20 25-60

-40

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0 5 10 15 20 25-60

-40

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0

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60


1 – 21.25 Hz 2 – 67.74 Hz 3 – 127.04 Hz

4 – 236.02 Hz 5 – 338.31 Hz 6 – 424.26 Hz

7 – 532.04 Hz 8 – 786.59 Hz 9 – 809.25 Hz

10 – 1256.69 Hz 11 – 1294.66 Hz 12 – 1682.97 Hz

0 5 10 15 20 25-60

-40

-20

0

20

40

60

0 5 10 15 20 25-60

-40

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0 5 10 15 20 25-60

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0 5 10 15 20 25-60

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60


1 – 15.50 Hz 2 – 69.52 Hz 3 – 119.79 Hz

4 – 234.14 Hz 5 – 325.85 Hz 6 – 467.09 Hz

7 – 527.49 Hz 8 – 786.04 Hz 9 – 828.78 Hz

10 – 1257.17 Hz 11 – 1290.03 Hz 12 – 1686.00 Hz


References

1. Avitabile P, O’Callahan J (2009) Efficient techniques for forced response involving linear modal components interconnected by discrete

nonlinear connection elements. Mech Syst Signal Process 23(1):45–67

2. Friswell MI, Penney JET, Garvey SD (1995) Using linear model reduction to investigate the dynamics of structures with local non-linearities.


3. Lamarque C, Janin O (2000) Modal analysis of mechanical systems with impact non-linearities: limitations to a modal superposition. J Sound

Vib 235(4):567–609

4. Ozguven H, Kuran B (1996) A modal superposition method for non-linear structures. J Sound Vib 189(3):315–339

5. Al-Shudeifat M, Butcher E, Burton T (2010) Enhanced order reduction of forced nonlinear systems using new Ritz vectors. In: Proceedings of

the IMAC-XXVIII. 2010. Print

6. Rhee W (2000) Linear and nonlinear model reduction in structural dynamics with application to model updating. PhD dissertation. Texas Tech

University

7. Avitabile P (2003) Twenty years of structural dynamic modification – a review. Sound Vib 37(1):14–27

8. Rao S (2004) Mechanical vibrations, 4th edn. Prentice Hall, New Jersey, pp 834–843

9. Van Zandt T (2006) Development of efficient reduced models for multi-body dynamics simulations of helicopter wing missile configuration.

Master’s thesis, University of Massachusetts Lowell, April 2006

10. O’Callahan J (1986) MAT_SAP/MATRIX, A general linear algebra operation program for matrix analysis. University of Massachusetts

Lowell, 1986

11. MATLAB (R2010a) The MathWorks Inc., Natick, MA


Chapter 11

Spectral Domain Force Identification of Impulsive Loading

in Beam Structures

Pooya Ghaderi, Andrew J. Dick, Jason R. Foley, and Gregory Falbo

Abstract In this paper, an identification method is presented for calculating impulsive loads from the propagating

mechanical wave which are produced in beam-like structures. This method uses a spectral finite element method (SFEM)

model of a segment of the structure to calculate force information from the measured response. The SFEMmodel is prepared

from the Euler-Bernoulli beam equation in the frequency domain. The method is studied using simulated response data and

then applied to data collected from an experimental system. Excellent performance is observed for nominal conditions and a

parametric study is performed to determine how different factors affect accuracy. Factors studied include structure size,

loading location, and loading duration. When limited to only acceleration data, the use of finite differencing methods to

obtain the required slope response information is determined to provide the most significant source of error in the identified

force information.

Keywords SFEM • Structural finite element method • Impact force identification

11.1 Introduction

Direct measurement of the impact force applied to amechanical structure is not always possible due to the nature of the impact

or the complexity of the structure. Research efforts have focused on developing indirect methods to identify the applied

impact force. Inverse methods are a common technique for calculating the impact force from a measured response and an

accurate model of the system. Impact force identification has been studied for applications in various fields such as gas pipes

[1], composite structures [2], and health monitoring [3].

Numerous methods have been developed using inverse methods for impact force identification. Some common

techniques for force identification are the deconvolution method [4], state variable formulation [5], the sum of weighted

accelerations [6] and the spectral element method [7]. The popular method is the deconvolution technique which uses an

assumption of linear behavior of the response to allow for the application of the convolution integral in order to determine

the system response. When using this method, the applied force is obtained by extracting the impact force from the

relationship between the response and the convolution integral of the impact force and the system’s impulse response.

This technique has been applied in both the time domain (e.g. [4]) and the spectral domain (e.g. [8]).

In the work of Doyle [4], a time domain deconvolution technique was developed in order to experimentally obtain

dynamic contact laws. Response behavior was monitored by using strain gauges mounted onto a beam-like structure and the

impulsive load was applied by using a pendulous ball. Chang and Sun [9] calculated the applied impact force by using an

experimental Green’s function and the time domain signal deconvolution. The reconstructed force was found independent of

the location of the sensor on a composite beam-like structure.

P. Ghaderi • A.J. Dick (*)

Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX 77005, USA


J.R. Foley

Air Force Research Laboratory, Eglin Air Force Base, FL, USA

G. Falbo

LMS Americas, Inc, Detroit, MI, USA



157


The deconvolution technique has also been used in the spectral domain for solving force identification problems. Due to

the ease of the deconvolution calculation in the frequency domain, this form of the force identification method was more

easily implemented. In the work of Doyle [8], a spectral domain deconvolution method was developed for experimentally

determining contact laws. The process of determining the impact force by using experimental strain data was based on the

relationship between the contact force and the strain at different locations on the test structure. The contact force was

obtained in the spectral domain and extra preprocessing was done in order to prepare the experimental data for the force

identification process. Due to the realistic boundary conditions of the experimental system, reflections were present in the

collected data. In order to remove the reflections, the signal was appended with the theoretical solution of the infinite beam.

To address the periodic nature of the fast Fourier transform (FFT), zero padding was also used to improve the quality of the

identified force.

According to the complexity of a structure, finite element techniques are more compatible than analytical methods for

predicting a response. By implementing the finite element method in a spectral domain, a powerful tool was developed [10].

The spectral finite element method (SFEM) is highly effective for modeling wave propagation problems and used for

structural impact analysis. The SFEM is used in structural mechanics for applications such as force detection [7] and damage

detection [11, 1212]. In the work of Doyle [13], an FFT based SFEM was used to study wave propagation problems in

beams. In the work of Mitra and Gopalakrishnan [7], the discrete wavelet transform was used in the implementation of the

SFEM in a wavelet domain. In this work, the accuracy and computational costs of the wavelet domain SFEM were

determined to outperform an equivalent frequency domain SFEM when performing force identification in beams. However,

applying the wavelet transform to the system model requires rigorous mathematics while the frequency domain implemen-

tation is more straight-forward.

In this study, a frequency domain SFEM is used for identifying the impulsive force applied to beam-like structures.

Acceleration data from multiple locations on the structure are used to calculate displacement and slope information at these

positions. A subset of the response, both spatially and temporally, is identified in order to avoid reflections. In the spectral

domain, the inverse method is applied to the response data by using a simplified model which corresponds to the spatial

subset of the data to calculate the impulsive force. The simplified model utilizes semi-infinite elements or “throw-off”

elements in order to satisfy the lack of reflection observed in the subset of the response data.

The remainder of the paper is organized as follows. The specific test system on which the performance of the force

identification method is studied is presented in Sect. 2. The impulse force identification procedure is described in Sect. 3. The

numerical implementation and results of a parametric study are detailed in Sects. 4 and 5, respectively. Experimental

verification is presented in Sect. 6. Concluding remarks are included in Sect. 7.

11.2 Test System

The performance of the force identification method is studied both numerically and experimentally on a beam structure.

The experimental structure is a 6 ft (1.83 m) long aluminum beam with a square cross-section and width of 1 in. (25.4 mm) as

shown in Fig. 11.1. The structure is instrumented with 22 mounted linear accelerometers distributed evenly along its length.

The acceleration data is collected by the sampling frequency of 16.384 kHz. The beam is suspended from bungee cords in

Fig. 11.1 Photograph of test

system

158 P. Ghaderi et al.

order to provide free-free boundary conditions. An impact hammer is used for applying the impulsive force and the force is

measured by a force transducer for comparison with identified force information.

Acceleration data is initially obtained from numerical simulations with a full SFEM model of this system. Material

properties are selected based on the aluminum material of the test structure. Structural damping properties are determined by

comparing experimental and simulated responses. Section 6 describes the application of the force identification method

using experimentally obtained acceleration data.

11.3 Impulse Force Identification

In this study, a method using SFEM is developed to identify an impulse force applied to a beam structure. The acceleration

data from the test structure is transformed into the spectral domain and is converted to displacement. The slope information

required for the identification process is calculated by using a finite difference method. Zero-padding is applied to the

response data in order to address the assumption of periodicity intrinsic to the FFT and improve the final results. A spectral

finite element beam model is used to obtain the stiffness matrix for a segment of the beam around the location where the

impulse force is applied. The frequency domain response data and the stiffness matrix from the simplified model are used to

calculate the force in the frequency domain. The force identification procedure is illustrated in Fig. 11.2. Detailed

descriptions of each step are presented in the following subsections.

11.3.1 Acceleration Data

The process starts with acceleration data from multiple locations along the length of the beam system. The corresponding

deflections of the system are calculated by integrating the acceleration data. The integration of the acceleration is performed

in the frequency domain in order to improve accuracy and ensure compatibility with the frequency domain representation of

the system. The integration procedure is performed by dividing the spectral domain acceleration data by the imaginary form

of the frequency.

11.3.2 Deflection and Slope Calculation

Since the acceleration data only provides information regarding the translational motion at discrete points along the

structure, it is necessary to calculate the corresponding slope response at each location. This is accomplished by using a

Pade scheme finite differencing method [14]. By utilizing the relationship which the Pade scheme provides between the

position and slope of points along the structure, the rotation at each of the nodes is calculated. Due to the nature of the Pade

scheme, higher accuracy is achieved near the center of the beam and the slope information near the ends of the structure

suffer from lower accuracy. Therefore, this error in the slope calculations might influence the force identification procedure

which will be discussed further in Sect. 5.3.

Fig. 11.2 Force calculation

algorithm

11 Spectral Domain Force Identification of Impulsive Loading in Beam Structures 159

11.3.3 Signal Conditioning for the Force Calculations

The force identification procedure uses the propagating wave directly after the impact to identify the applied impulsive

force. Because the FFT algorithm assumes periodicity and because the time series deflections and rotations are truncated to

contain only the propagating signals, zero padding is applied in order to address convergence issues [8].

11.3.4 Spectral Finite Element Model

The wave propagation starts directly after the impact force is applied to the beam and it propagates out from this location

until it reaches the ends of the structure and is reflected. In this force identification method, the impulse force is calculated

from a subset of the data. This subset of data corresponds to a reduction of both the spatial and temporal dimensions around

the location where the propagated wave originated, as illustrated by Fig. 11.3.

The data subset is then used with a simplified model of the system to calculate the desired force information. The

simplified model corresponds to the segment of the original system resulting from the reduction of the spatial dimension.

This model also uses semi-infinite or throw-off elements, which allows the propagating wave to pass through a node without

any reflections. The throw-off elements are added at each end of the simplified model to eliminate reflections at the boundary

conditions. Although the simplified SFEM model differs considerably from the full SFEM model used for the simulations,

the response predicted by the simplified model agrees very well with the local response behavior of the full model as

illustrated in Fig. 11.4. The beam SFEM model is derived from the Euler-Bernoulli beam equation. The spectral finite

element model for the beams is detailed in the work by Doyle [15].

X (m)

time

(ms)

0 0.5 1 1.50

0.5

1

1.5

-2000

-1000

0

1000

2000Fig. 11.3 Experimentally

observed wave propagation

(acceleration) in a free-free

beam and the boxed area

corresponds to data selected

for analysis with simplified

model

X (m)

time

(ms)

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

-500

0

500

1000

1500Fig. 11.4 Simulated

mechanical wave propagation

(acceleration) using simplified

model with throw-off

elements at the ends to

eliminate reflection


11.4 Numerical Implementation

Results obtained by applying the force identification method described in the previous section to response behavior from

numerical simulations are presented in this section. The procedure is first applied to response behavior by using true

deflection and slope information produced by the simulation. This eliminates the potential error introduced by applying the

finite difference method to calculate slope information.

The calculated force information corresponds to the node locations which were included in the subset of data analyzed.

This initial work is focused on characterizing point forces which are applied at the location of one of the accelerometers. The

force values calculated from the simulated response are plotted in Fig. 11.5. The simulated beam is 18 ft (5.49 m) long

aluminum beam with a square cross-section and a width of 1 in. (25.4 mm). This length corresponds to three times that of the

beam described in Sect. 2. The increased length allows for a greater amount of time before the reflected wave returns to the

segment of the beam from which the subset of data is selected for analysis.

The force identified from the simulated response agrees very well with the true applied force. In order to quantify the

accuracy of the force identification method, a Root Mean Square (RMS) error is calculated by comparing the identified force

values to the true values used in the simulation. For the nominal case presented in Fig. 11.5, the RMS error is 2.74 N at

the location of loading during the impulse and averages 2.17 N for five locations considered over the remainder of the 2 ms

presented. This error is quite small when compared with the 600 N peak value of the impulsive load. The time series of the

applied force used in the simulation is based on the type of loading that the impact hammer produces. This loading data

includes an impulsive load with 0.4 ms duration.

11.5 Parametric Study

The geometric and loading properties utilized in the previous section were selected in order to provide nominal conditions

for performing force characterization. In this section, these properties are varied in order to study how they influence the

ability to accurately calculate the applied impulsive load. This includes varying the length of the structure, the effect of slope

calculation on the force identification results, the location of the applied force, and the duration of the impulsive load.

11.5.1 Structure Length

In order to study the influence of the length of the structure on the accuracy of the calculate force information, the process is

applied to response behavior simulated for a 6 ft (1.83 m) long beam. This length corresponds to the length of the test system

described in Sect. 2. All other conditions are maintained and the simulated displacement and slope information is used in the

force calculation. This allows for the separation of the influence of structure length and slope calculation on the accuracy of

the identified force information.

0 0.5 1 1.5 2-100

0

100

200

300

400

500

600

Time ms

For

ce (

N)

Neighboring nodesNeighboring nodesDetected forceNeighboring nodesNeighboring nodesActual force

Fig. 11.5 Force information

calculated for five nodes using

a simulated response for an

18 ft (5.49 m) long beam


The force information calculated for the shorter structure is plotted in Fig. 11.6. The force information agrees very well

with the true applied force. The RMS error is 4.01 N at the loading location during the impulse and averages 1.98 N over the

remainder of the 2 ms for this configuration. The calculated force values for the neighboring nodes have small non-zero

values, deviation from the true zero values. This is believed to result due to the shorter length of time before the reflected

wave returns to the position on the structure where the load was applied. This time between the initial propagating wave and

the reflection and how it is influenced by changing the structure size is illustrated in Fig. 11.7.

11.5.2 Slope Calculated by Using Pade Scheme

As the response of the structure in the experimental setup is measured by using standard accelerometers, only deflection data

can be calculated directly from the measured acceleration. The slope information is calculated by using a Pade scheme with

the simulated deflection data. The accuracy of the finite difference slope calculation is illustrated in Fig. 11.8 where the

magnitude of the difference between the simulated values and values calculated by using the Pade scheme is plotted against

time and position. The effect of including the finite difference calculation on the accuracy of the calculated force information

is illustrated in Fig. 11.9. The RMS error for this case is 31.7 N at the loading location during the impulse and averages

6.09 N during the remainder of the 2 ms. By using the finite differencing method to calculate the slope information, the error

in the identified impulsive force is significantly increased. The RMS error value for the force information after the impulse is

also increased by calculating the slope information but to a lesser extent.

0 0.5 1 1.5 2-100

0

100

200

300

400

500

600

Time (ms)

For

ce (

N)



calculated for five nodes using

a simulated response for a 6 ft

(1.83 m) long beam

0 2 4 6-2000

0

2000

Time (ms)

acc.

(m

/s2 )

0 2 4 6-2000

0

2000

Time (ms)

acc.

(m

/s2 )

Fig. 11.7 Acceleration time

series for the beam position

where the impulsive load is

applied for (top) a 18 ft

(5.49 m) beam and (bottom)an 6 ft (1.83 m) long beam


11.5.3 Loading Location

As discussed in the Sect. 3, the central differencing equations are of a higher order than those used for calculating derivatives

at the ends of the structure. All of the calculated force information previously presented corresponds to impulsive loading

near the center of the structure. In order to study how the reduced accuracy of the calculated slope information near the ends

of the structure influences the calculated force information, response behavior is simulated than analyzed when the

impulsive load is applied near the end of the structure. This force information is presented in Fig. 11.10. The RMS error

for the identified impulsive load, which is 31.2 N, is not significantly affected by this change is location. However, by

moving closer to the end, the decreased accuracy in the slope information is observed to significantly influence the period of

time after the impulse, increasing the RMS error to 94.3 N.

In addition to the reduced accuracy of the calculated slope information, by moving the position of loading toward the end

of the beam there is less time before the reflected waves return to this location. As a result, large discrepancies are observed

in the calculated force information after 1 ms of time has passed. While changes to the geometric or material properties of the

structure can increase this time, the limited accuracy of the calculated slope data is still expected to influence the

performance of the force characterization procedure.

11.5.4 Loading Duration

In order to further study the performance of the force identification procedure, the duration of the applied load is varied.

A system response is simulated for an impulsive load with twice the original duration. The simulation is performed with the

X (m)

time

(ms)

0 0.5 1 1.50

0.5

1

1.5

1

2

3

4

5

6

7

8

9

x 10-4Fig. 11.8 Magnitude of

difference between simulated

slope data and slope

information calculated from

displacement data versus time

and position

0 0.5 1 1.5 2-100

0

100

200

300

400

500

600

Time (ms)

For

ce (

N)



calculated from simulation

data when load is applied

at the middle of the beam


model of the 18 ft (5.49 m) long beam and the simulated slope data is used to calculate the force information. This force

information is presented in Fig. 11.11 and good agreement is seen. The RMS errors for the impulse and the period after the

pulse are 6.54 and 1.29 N, respectively. Although the duration of the impulsive load is increased, the time for the mechanical

wave to reflect back to the loading location remains the same. The longer duration of the load only decreases the time

between the initial wave and the reflection by the 0.4 ms that the duration of the load is extended.

11.6 Experimental Verification

Experimental data is collected from the system described in Sect. 2. The force information obtained by applying the force

identification method to the acceleration data is presented in Fig. 11.12. While the location of the loading is successfully

identified along with the qualitative characteristics of the impulsive load, errors exist between the force data measured with

the force transducer of the impact hammer and the calculated values. The RMS errors for the impulse and the period shown

in the figure after the impulse are 67.3 and 106.2 N, respectively. When compared to the force information calculated for

nominal conditions, the experimental structure is shorter, the load is applied away from the center, and the slope response

information is calculated by using the finite differencing method. Each of these differences contribute to the reduced

accuracy of the calculated for information. Based on the results of the parametric study, the main source of error is believed

to be due to the use of the calculated slope information.

0 0.5 1 1.5 2-200

0

200

400

600

800

Time (ms)

For

ce (

N)




data when load is applied near

one end of the beam

0 1 2 3-100

0

100

200

300

400

500

600

Time (ms)

For

ce (

N)




data for impulsive loading

with twice duration


11.7 Concluding Remarks

In this study, the impulse force identification procedure using the SFEM for a beam structure is presented. The procedure is

applied to simulation and experimental data for propagating mechanical waves resulting from the application of an impulse

force. Excellent agreement is observed for nominal conditions. However, different sources of error are identified through a

parametric study. These sources of error are observed to influence the identification process when it is applied to

experimental data. The most significant source of error is determined to be the use of a finite difference method to calculate

slope response information. Significant improvement in the accuracy of the identified force information would be achieved

by the direct measurement of the slope information, possible through the use of coupled accelerometer–gyro sensors.

Acknowledgements The support of this work from the Air Force Research Laboratory under Cooperative Agreement FA8651-10-2-0006 and

from the Air Force Office of Scientific Research under Grant FA9550-11-1-0108 is gratefully acknowledged.

References

1. Kim M-S, Lee S-K, Kim S-J (2008) Identification of impact force on the gas pipe based on analysis of acoustic wave. Int J Mod Phys B

22(9–11):1039–1044

2. Yan G, Zhou L (2009) Impact load identification of composite structure using genetic algorithms. J Sound Vib 319(3–5):869–884

3. Hu N, Fukunaga H (2005) A new approach for health monitoring of composite structures through identification of impact force. J Adv Sci

17(1–2):82–89

4. Doyle JF (1984) An experimental method for determining the dynamic contact law. J Exp Mech 24(1):10–16

5. Hollandworth PE, Busby HR (1989) Impact force identification using the general inverse technique. Int J Impact Eng 8(4):315–322

6. Bateman VI, Carne TG, Gregory DL, Attaway SW, Yoshimura HR (1991) Force reconstruction for impact tests. J Vib Acoust 113(2):192–200

7. Mitra M, Gopalakrishnan S (2005) Spectral formulated wavelet finite element for wave propagation and impact force identification in

connected 1-D waveguides. Int J Solids Struct 42(16–17):4695–4721

8. Doyle JF (1984) Further developments in determining the dynamic contact law. J Exp Mech 24(4):265–270

9. Chang C, Sun CT (1989) Determining transverse impact force on a composite laminate by signal deconvolution. J Exp Mech 29(4):414–419

10. Patera AT (1984) A spectral element method for fluid dynamics: laminar flow in a channel expansion. J Comput Phys 54(3):468–488

11. Ostachowicz WM (2007) Damage detection of structures using spectral finite element method. J Comput Struct 86(3–5):454–462

12. Krawczuk M (2002) Application of spectral beam finite element with a crack and iterative search technique for damage detection. J Finite Elem

Anal Des 38(6):537–548

13. Doyle JF, Farris TN (1990) Spectrally formulated element for wave propagation in 3-D frame structures. Int J Anal Exp Modal Anal

5(4):223–237

14. Lele SK (1992) Compact finite difference schemes with spectral-like resolution. J Comput Phys 103(1):16–42

15. Doyle JF (1997) Wave propagation in structures: spectral analysis using fast discrete Fourier transforms. Springer, New York

0 0.2 0.4 0.6 0.8 1-400

-200

0

200

400

600

800

1000

Time (ms)

For

ce (

N)

Neighboring nodes

Neighboring nodes

Detected force

Neighboring nodes

Neighboring nodes

Actual force


calculated from experimental

data


Chapter 12

Free-Pendulum Vibration Absorber ExperimentUsing Digital Image Processing

Richard Landis, Atila Ertas, Emrah Gumus, and Faruk Gungor

Abstract Using image processing and analysis, the dynamic behavior of the beam-free-pendulum system under low

and high sinusoidal excitation was investigated. The system responses were investigated experimentally in the neighborhood

of primary resonance condition. The results exhibited autoparametric interaction between the beam and the free pendulum

when the primary resonance condition was satisfied. Experiments were conducted for two different pendulum weights

under two different shaker forcing amplitudes, and the results were compared. Experimental data were obtained by sweeping

between the frequencies that contain the resonance condition under investigation. The results of experiments for

different beam-tip mass and pendulum mass ratios indicate that more powerful absorption action can be achieved when

the smaller mass ratios are used.

Keywords Nonlinear vibration • Autoparametric vibration absorber • Internal resonance • Free pendulum absorber

12.1 Introduction

There is widespread interest in pendulum modeling and the use of the pendulum as a vibration absorber. This interest ranges

from the dynamics of Josephson’s Junction in solid state physics [1] to the rolling motion of ships [2] and the rocking motion

of buildings and structures under earthquakes [3]. Vibration mitigation has found extensive usage in aerospace structures,

civil engineering structures, and mechanical machinery. A comprehensive survey on vibration suppression devices was

given by Sun et al. [4]. They reviewed the current developments in passive absorbers, adaptive absorbers, and active

absorbers. Passive tuned vibration absorbers are also referred to as dynamic vibration absorbers [5] or tuned mass dampers.

Much of the analytical work done on the inverted spherical pendulum for undamped systems was done by Lowenstern

[6], Hemp and Sethna [7] and by Moran [8] for damped systems. These three papers dealt with the stability criteria of the

inverted spherical pendulum. Recently, Ertas and Garza investigated the dynamics and bifurcations of an impacting

spherical pendulum with large angle and parametric forcing. The pendulum system was studied with nine different bobs

and two different base configurations with an external frequency of 24.6–24.9 Hz. Comparative analysis was performed at

low and high Coulomb damping values for the inverted, impacting pendulum [9].

Passive and active vibration absorbers were used by many researchers to reduce the vibration level of flexible structures.

Miwa et al. reported the case involving an active mass damper (AMD) system installed on the roof of a building to investigate

the vibration characteristics of multi-story houses built on soft ground near vibration sources such as railways and expressways

[10]. Muller et al. studied the modeling and control techniques of an active vibration isolation system. They compared

experimental findings with simulated data and discussed the results [11]. Holt and Singh investigated the active/passive

vibration control of continuous systems by zero assignment. They reported that the results of their study would lead to the

developments in the control strategies for complex structures and implementation of piezoelectric actuators and sensors for

vibration control [12]. Viguie and Kerschen, used the concept of nonlinear energy sink (NES) to reduce the vibration level of

multi-degree-of-freedom linear structures. They reported that this approach requires the development of an efficient NES

design procedure. Their research presented such a procedure based upon bifurcation analysis using the softwareMatCont [13].

R. Landis • A. Ertas (*) • E. Gumus • F. Gungor

Mechanical Engineering Department, Texas Tech University, Lubbock, TX 79409, USA




167


Many mechanical structures can be modeled as flexible beams with tip mass attached along the span. For example,

as a model for an “autoparametric” vibration absorber, Haxton and Barr [14] studied a flexible column with tip mass fixed to

a heavy block undergoing parametric vibration. The Autoparametric Vibration Absorber is a device designed to absorb

the energy from the primary mass (main mass) at conditions of combined internal and external resonance. Autoparametric

resonance is a special case of parametric vibration and is said to exist if the condition at the internal resonance and

external resonance are met simultaneously due to external force [15–18].

Autoparametric resonance may occur in nonlinear systems with two or more degrees of freedom, and if normal mode

frequencies of the corresponding linearized systems are governed by the linear relationship

Xn

i¼1

kioi ¼ 0

where ki are integers, n is the number of degrees of freedom, and oi are the system’s natural frequencies. When the linear

spring–mass–damper system is coupled to a pendulum, the resulting system possesses quadratic nonlinearities due to inertial

coupling with the rotational motion of the pendulum. If the system has quadratic nonlinearity in two degree of freedom, then

system internal resonance occurs when

o2 ¼ O and o2 ¼ 2o1

where O is the excitation frequency,o1 is the lower mode frequency, and o2 is the higher mode frequency. This is called 1:2

internal resonance condition and leads to nonlinear modal coupling between two modes.

The Autoparametric Vibration Absorber has received considerable attention since the mid-1980s, and researchers

published many interesting papers. Struble and Heinbockel [19] used asymptotic methods to investigate the energy transfer

observed by Selvin [20]. Struble [21] used the perturbation method for a parametrically excited pendulum and obtained

resonant solutions when the excitation frequency was nearly twice the natural frequency of the system. Hatwal et al. [22]

investigated nonlinear vibrations of a spring-mass-damper system with a parametrically excited pendulum. The harmonic

balance method was used to solve the system response. Performance of the system as an autoparametric vibration absorber

was studied. Cuvalci and Ertas [23] investigated a simple pendulum as a vibration absorber for flexible structures. Through

experimental and theoretical studies, they reported that a simple pendulum can be effectively used as a vibration absorber for

flexible structures.

Baja et al. [24] studied forced, weakly nonlinear oscillations of a two degree-of-freedom autoparametric vibration

absorber system at resonant excitations. The authors used a method of averaging to obtain first-order approximations

to the response of the system. They studied the bifurcation when the pendulum was lucked and also observed that the

coupled-mode response can undergo Hopf bifurcation to limit cycle motions when the two linear modes are mistuned away

from the exact internal resonance condition. Vyas and Bajaj [25] studied the dynamics of a resonantly excited single-degree-

of-freedom linear system coupled to an array of non-linear autoparametric vibration absorbers. They investigated

the stability and bifurcations of equilibria of the averaged equations. In their paper, the effect of various parameters on

the performance of the Autoparametric Vibration Absorber (AVA) is discussed. A nonlinear adaptive vibration absorber to

control the vibrations of flexible structures at a two-to-one internal resonance was investigated by Ashour and Nayfeh [26].

Through a proposed model, energy was exchanged between the structure and the controller and, near resonance, the

structure’s response almost diminished.

Authors of this paper, as many other distinguished researchers, investigated the possibility of the pendulum for the

vibration absorber [27–32]. In these studies, a digital angular measurement system for measuring the full 360� angular

displacement of the pendulum was used [33]. At that time, there was no reliable device available to measure the full

pendulum response. In this paper, image processing technique was used for measuring the angular displacement of the

pendulum to analyze the nonlinear dynamic response of an Autoparametric Vibration Absorber of a free-pendulum system.

12.2 Experimental Setup and Procedure

The experiments were performed to obtain the frequency response curves for the different pendulum weights (one ball and

two balls) under two different shaker forcing amplitudes. For the detail dynamics of the free pendulum, the experiments were

conducted at the specific forcing frequencies and time histories to plot the FFTs and the phase plains. The complete experimental

168 R. Landis et al.

system is shown in Fig. 12.1. The model is excited by the shaker table, the amplitude and frequency of which controlled by the

vibration control system.

The experimental model used in this research study has a beam with appendages consisting of a mass-free-pendulum

system. A high-speed camera was used to capture the position of the beam/tip-mass/free pendulum system through the

duration of the experiment. The beam/tip-mass/free pendulum system consists of a flexible beam that is rigidly clamped at

the base. The tip of the beam consists of an appendage, which consists of a lumped mass and a free pendulum. Both systems

are shown in Figs. 12.2 and 12.3, respectively.

The experiments were performed for 1.00 Hz, 2.00 Hz, 3.00 Hz, 3.50 Hz, 3.75 Hz, 3.85 Hz, 3.90 Hz, 3.95 Hz, 4.00 Hz,

4.05 Hz, 4.10 Hz, 4.25 Hz, 4.30 Hz, 4.40 Hz, 4.50 Hz, and 5.00 Hz for both the one and two ball free pendulum systems.

Experimental data were obtained by sweeping between the frequencies (o) that contain the resonance condition

under investigation obean=pendulum�system ¼ 4:0 Hz and: opendulum ¼ 2:0 Hz� �

. The data (response amplitudes of beam

and the free pendulum) were recorded, and it was assumed that the system reached a steady state within each increment

of the frequency. In other words, for all frequency sweep experiments, at each frequency interval, the system was allowed to

dwell for 60 s in order to reach stability before the data were taken. In the experiment, detuning was set to be

opendulum

obeam=pendulum�system¼ 0:5

Experiments were performed for two amplitudes of f0 ¼ 1:88 mm excitation and f0 ¼ 2:3 mm for non-impact

and for impact cases, respectively. Since test procedures can vary, experiments should begin in a specific manner to

avoid the inclusion of too many variables into the experimental test, hence causing uncertainty in the decision-making of

the test results. The beam material and length, the tip mass, the damping coefficient of the beam, and the pendulum were

taken to be constant for all experiments. As mentioned before, the experiment was investigated for two main parameters:

the excitation amplitude and the free pendulum weight.

Fig. 12.1 Experimental setup

Fig. 12.2 One ball beam/tip-mass/free pendulum system

12 Free-Pendulum Vibration Absorber Experiment Using Digital Image Processing 169

For beam and free pendulum responses, measurement at a high speed image collection of up to 500 frames per second

with a 1280 � 1024 image resolution camera was used. Due to the frame grabber limitations of 1 GB of memory,

the maximum recording time for all the experimental cases was approximately 26 s at a frame rate of 200 frames per

second. The experiment duration was long enough to observe all possible dynamics of the system under consideration.

The collected frames were captured into memory and then stored as tiff files onto the hard drive. Then the files were

converted to JPG format RealJPEG Pro and analyzed using MatlabR2006r2 basic and imagery toolboxes.

Many of the techniques of digital image processingwere developed in the early 1960s. Since then, image processing has been

used for satellite imaging, medical imaging, videophone, character recognition and many other applications. However, the cost

of image processingwas quite highwith the computing equipment of that era. In the 1970s, digital image processing proliferated

when the cost of computers decreased and they became faster. Recently, emerging video processing techniques demonstrated

their potential applications in engineering. In order to effectively use image screening and analysis techniques for beam/

tip-mass/free pendulum experiments, camera and environment setup and verification was critical in reducing or eliminating

image distortion, misalignment, tonal misadjustment and image processing complexity between sequential image pairs.

12.3 Results and Discussions

The experimental analysis was performed for three cases, and the following diagrams are plotted for detailed dynamics:

Frequency response curves.

Time history.

FFT.

Phase plain.

12.3.1 Case-I

In this case, the mass ratio of the beam/tip-mass and the pendulum mass (one ball) was taken to be approximately 1/18 and

the shaker excitation amplitude was set approximately to f0 ¼ 1:88 mm. The natural frequency of the beam/free pendulum

system was set approximately to obeam system ¼ 4:0 Hz and to maintain the condition of autoparametric interaction,

the frequency of the pendulum was set approximately to opendulum ¼ 2:0 Hz. As shown in Fig. 12.4, the first experiment

was performed to investigate the beam response curves without the pendulum being in action. For this experiment, two balls

Fig. 12.3 Two balls beam/

tip-mass/free pendulum

system


that will be used as a damper later were locked on the tip mass (housing). The forcing frequency was increased in a

stepwise manner until the necessary regions of interest were covered. Figure 12.4 demonstrates that the maximum beam

response occurred at the excitation frequency of 4.05 Hz. Observing Fig. 12.4, the beam was oscillating approximately 25�

peak-to-peak when the primary resonance condition occurred, namely when the natural frequency of the system reached

approximately obeam system ¼ 4:05. Note that when the pendulum was lucked on the tip mass (housing track), the natural

frequency of the system turned out to be slightly higher than the tuning frequency of 4.00 Hz (see Fig. 12.4). It was difficult

to adjust the exact tuning ratio. To create Fig. 12.4, experiments were performed at 3.00 Hz, 3.50 Hz, 3.75 Hz, 3.85 Hz,

3.90 Hz, 3.95 Hz, 4.00 Hz, 4.05 Hz, 4.10 Hz, 4.25 Hz, 4.50 Hz, and 5.00 Hz for the lucked beam/tip-mass/pendulum system.

When an absorbing pendulum was free in the housing track and the frequency of the free pendulum was tuned to one half

of the beam-mass system frequency, the free pendulum had a sudden amplitude increase at the start of the autoparametric

region whereas the beam response decreased to approximately 0.9� at the first response peak (Fig. 12.5a, point P1) and 2.25�


frequency response curves

without absorber

Frequency (Hz)

ϕ (d

eg)

Frequency (Hz)

Autoparametric Region

Complete EnergyExchange

θ (d

eg)

a

b


frequency response curves

(a) beam/tip-mass response

with absorber, (b) free-pendulum response for

f0 ¼ 1.88 mm and mass

ratio ¼ 1/18 (one ball)


at the second response peak of the beam (Fig. 12.5a, point P2). The motion of the beam/tip-mass was almost diminished and

the whole energy was absorbed by the free pendulum when the primary resonance case was reached (Fig. 12.5a, b). In other

words, the complete energy transfer between two modes occurred when the beam/tip-mass frequency was twice the free

pendulum frequency and the forcing frequency was equal to beam/tip-mass frequency. To create Fig. 12.5a, b, experiments

were performed at 3.00 Hz, 3.50 Hz, 3.75 Hz, 3.85 Hz, 3.90 Hz, 3.95 Hz, 4.00 Hz, 4.05 Hz, 4.10 Hz, 4.25 Hz, Hz, 4.30 Hz,

4.35 Hz, 4.40 Hz, 4.50 Hz, and 5.00 Hz with the shaker excitation amplitude of f0 ¼ 1:88 mm.

Second peak observed in the beam/tip-mass response was due to the beating phenomenon and will be explained later in

this section. As shown in Fig. 12.5b, point A was the starting point of the autoparametric region. Energy transfer from beam

to pendulum continued to increase until point B and then took the shape of a bathtub curve, which consists of three

responses, namely, a decreasing response followed by an approximately constant response and then an increasing response

until point C. The beating phenomenon was observed when the excitation frequency reached 4.25 Hz (between points C and

C0). After point C’ until point D (frequencies at 4.30–4.35 Hz), the beating phenomenon ceased to exist. Points B and C on

the pendulum response curve were important as they define the complete energy exchange region. As shown in this figure,

the autoparametric region ended at point E.

From Fig. 12.6a, it is evident that the oscillation of the beam was reduced to almost zero and the complete energy was

absorbed by the pendulum. Some insight into the characteristics of the limit-cycle oscillation can be obtained by examining

the phase portrait of pendulum as shown in Fig. 12.6f. At the neighborhood of the tuning frequency (3.95 Hz), the pendulum

was having purely sinusoidal oscillation and exhibited elliptical phase orbit. Beam and pendulum FFT diagrams showed

strong peaks where the beam frequency was twice the pendulum frequency (see Fig. 12.6c,d). Complete autoparametric

interaction between beam and pendulum and one to two frequency relationship obeam ¼ 2opendulum and Oforcing ¼ obeam

� �

60a

e f

b c

d

g

40

20

Deg

ree

Deg

ree

Time (sec)

Theta (degree) vs Time (sec) Phi (degree) vs Time (sec) FFT of the Beam

FFT of the Pendulum

Phi vs Theta

Wp (Hz)

Wb (Hz)

Phi

dPhi

/dt

The

taf(

Wp)

f(W

b)

dthe

ta/d

t

PhiTheta

Theta dot Vs Theta Phi dot vs Phi

Time (sec)

0

-20

-40

-60

60

40

20

0

-20

-40

-600

800

600

400

200

-200

-400

-800

-600

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

2500

2000

1500

1000

500

-500

-1000

-1500

-2000

-2500-50 -40 -30 -20 -10 0 10 20 30 40 50 -50

-0.6

-0.4

-0.2

0

0.2

1.4

0.6

0.8

1

-40 -30 -20 -10 0 10 20 30 40 50

0

1

0

1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.51

1.52

2.53

3.54

4.5

0

10

20

30

40

50

60

6x102

Fig. 12.6 Detail beam/tip-mass/pendulum system dynamics for the forcing frequency of 3.95 Hz, f0 ¼ 1:88 mm


between beam and pendulum can also be verified by the phase plane shown in Fig. 12.6g. The loop shown in this figure is the

evidence for one to two frequency relationships between beam and pendulum.

The data taken during the experiment at 4.25 Hz frequency are shown in Fig. 12.7. An FFT was performed to give

the spectra observed in Fig. 12.7c,d. Figure 12.7 shows an interesting behavior of the beam/tip-mass/-pendulum system.

Since the forcing frequency of 4.25 Hz was close enough to the natural frequency of the system (approximately 4.0 Hz),

a phenomenon known as beating occurred. As shown in Fig. 12.7c, besides the peaks with periods around 4.25 Hz

which correspond with the shaker excitation, a peak with a period around 3.95 Hz was observed. As seen from the FFT,

the fluctuating beam response having peaks at these two mode frequencies yield the beating phenomenon. Although the

mode frequency of 4.25 Hz within the complete energy exchange region, Fig. 12.7g shows a very distinct pattern instead of

the regular loop, indicating the o1 ¼ 2o2 relationship. As seen from Fig. 12.7a, the amplitude of the beam response built up

and then diminished in a regular pattern. Figure 12.7e shows that the beam data was scattered. This is not only because beam

oscillations do not exactly repeat themselves from one cycle to the next but also because of the beating effect. Figure 12.7f

shows that the pendulum was having multi-limit-cycle with the response of approximately 60�, which was larger than

the pendulum response cases in the neighborhood of exact autoparametric interaction region (around 3.95–4.0 Hz).

Of course, this response increased due to beating effect.

The beating phenomenon may occur in machines, flexible structures, twin-engine propeller airplanes, electric power

houses, bells, human ears, guitar strings, etc. In some cases, the beating phenomenon is desirable, but in some cases,

it may not be desirable. For this application, complete energy exchange region shown in Fig. 12.5 should be free from

beating action as much as possible. In this design application, the beating phenomenon is an undesirable effect because

energy can be transferred from the secondary system (i.e., free pendulum) back to the primary system (beam). Hence, the

investigation of frequency of the beating phenomenon existence and associated harmonics within the design range of energy

exchange is important.

10a

e f

b c

d

g

8

6

4

2

Deg

ree

Deg

ree

Time (sec)

Theta (degree) vs Time (sec) Phi (degree) vs Time (sec) FFT of the Beam

FFT of the Pendulum

Phi vs Theta

Wp (Hz)

Wb (Hz)

Phi

dPhi

/dt

The

ta

f(W

p)f(

Wb)

dthe

ta/d

t

PhiTheta

Theta dot Vs Theta Phi dot vs Phi

Time (sec)

0

-2

-4

-8

-40

-10

60

40

20

0

-20

-40

-600

800

600

400

200

-200

-400

-800

-600

-2.5 -2 -1.5 -1 -0.5 0 1 1.5 2 2.50.5

2500

2000

1500

1000

500

-500

-1000

-1500

-2000

-2500-80 -60 -40 -20 0 20 40 60

-80

2.5

1.5

1

0.5

-0.5

-1

-1.5

-2

-2.5

0

2

-60 -40 -20 0 20 40 50

00

1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 100

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.51

1.52

2.5

33.5

44.5

0

10

20

30

40

50

60

6x102

Fig. 12.7 Detail beam/tip-mass/pendulum system dynamics for the forcing frequency of 4.25 Hz, f0 ¼ 1:88 mm


12.3.2 Case II

The aim of this part of the experimentation was to assess and compare two experimental models, namely, with one ball

and two balls (mass ratio of the beam-tip-mass and the pendulum mass (two balls) is taken to be approximately 2/17)

by experimenting with the same parameters used in Case I. Namely, the shaker excitation amplitude was set

approximately to f0 ¼ 1:88 mm. The natural frequency of the beam/free pendulum system was approximately

obeam system ¼ 4:00 Hz, and to maintained the condition of autoparametric interaction, the frequency of the pendulum

was set approximately to opendulum ¼ 2:00 Hz. Figure 12.8 compares one and two ball free pendulum models responses

over the interval from 3.00 Hz to 5.0 Hz. The broken line represents the two ball free pendulum model results while the

solid line represents the one ball free pendulum model results. In comparing both models, the following observations

were made. As seen from Fig. 12.8, the autoparametric region for the one ball free pendulum model started at point A1

and died at point E1 while the autoparametric region for the two ball free pendulum model started at A2 and died at E2.

The delay in the start of the two ball free pendulum models was observed because larger mass required more energy

level to swing. In both models, the energy level continued to increase as both systems approached the exact

autoparametric energy level shown by points B1 and B2. At these points, the autoparametric interaction condition

(obeamsystem ¼ 2 � opendulum) was satisfied. In comparing the one and two ball free pendulum responses, the two ball

system had a smaller angle of swing than one ball system since the larger mass absorbed more energy without having

bigger angle of swing. From Fig. 12.8, it is clear that both pendulum model responses exhibited quite different energy

exchange trends between points B and C. Both models showed the beating phenomenon exactly at the same frequency

of 4.25 Hz.

Figure 12.9 shows the comparison of experimental frequency response curves for one ball free pendulum system for

the excitation amplitude of, f0 ¼ 1:88 mm. As shown in this figure, the beam beating phenomenon was observed when the

excitation frequency reached 4.25 Hz. This figure clearly shows that the beating was distorted and ceased to exist at

frequency 4.30 Hz and frequency 4.35 Hz, respectively. However, at these frequencies, because of the effect of the beating,

the pendulum response still fluctuated with high amplitude. At the frequency of 4.40 Hz, the pendulum oscillation almost

died out. Beam FFT plots shown in Fig. 12.9 are another indication of how the beating phenomenon ceased out. When

forcing frequency reached 4.30 Hz, the peak that causes the beating phenomenon at the natural frequency of the system

disappeared (see Fig. 12.9c for 4.35 Hz).

Figure 12.10 shows the comparison of experimental frequency response curves for the two ball free pendulum system

for the excitation amplitude of f0 ¼ 1:88 mm. As in the case of the one ball model, this model also experienced the beating

phenomenon exactly at the same excitation frequency of 4.25 Hz. It is interesting to note that immediately after the

beating frequency of 4.25 Hz, the effect of the beating disappeared and, consequently, the pendulum oscillation died out

at the excitation frequency of 4.30 Hz. This shows that an absorber with a larger mass can eliminate the region C01 – D1

shown in Fig. 12.8.

ϕ (d

eg)

Frequency (Hz)

Fig. 12.8 Comparison of experimental frequency response curves for one ball and two balls free pendulum systems for f0 ¼ 1:88 mm


Fig. 12.9 Comparison of experimental frequency response curves for one ball free pendulum systems for f0 ¼ 1:88 mm


12.3.3 Case III

The objective of this part of the experimentation was to observe the response of the beam and free pendulum when the shaker

excitation amplitude was set approximately to f0 ¼ 2:3 mm peak to peak. As in the previous cases, the natural frequency

of the beam/free pendulum system was set approximately to obeam system ¼ 4:00 Hz, and to maintain the condition of

autoparametric interaction, the frequency of the pendulum was set approximately to opendulum ¼ 2:00 Hz. Many researchers

investigated the use of impact dampers to achieve efficient structural damping [34–40]. They have studied the effect of

certain parameters and their control on the damping effect of the vibration absorber. In this case, by setting a higher shaker

excitation amplitude, the phenomenon of impact was created to investigate whether impact would change the characteristics

of the pendulum damping and the system response behavior. In this case, the experiments were performed by using one and

two pendulum balls at the forcing frequency of 4.05 Hz, 4.25 Hz, and 4.30 Hz. Figure 12.11 shows the position of the free

pendulum when impact occurs. Considering the reference points in image processing calculations, the impact angle for the

one ball case was approximately 65� with � 2� error possibility. In the two ball free pendulum case, the impact angle was

54� with � 3� error. Our observations revealed that the clear impact case occurred at 4.05 Hz for both one ball and two ball

beam pendulum systems. A comparison of the response behavior of the one ball beam pendulum system for excitation inputs

1.88 mm and 2.3 mm is given in Fig. 12.12. Figure 12.12 clearly reveals that the impact of the pendulum did not reduce the

Fig. 12.10 Comparison of experimental frequency response curves for two ball free pendulum systems for f0 ¼ 1:88 mm


Fig. 12.11 Position of impact angles for free pendulum

Fig. 12.12 Comparison of one ball beam pendulum system response for excitation amplitude of 1.88 mm and 2.3 mm at 4.05 Hz

oscillation of the beam; instead it increased the response amplitude considerably. When the system was excited with

1.88 mm, the beam’s maximum response was approximately 3� peak to peak. Whereas, when the excitation amplitude

increased to 2.3 mm, the beam response amplitude increased to 49� peak to peak (see Fig. 12.12). When the system was

excited with lower amplitude, it had strong periodic motion and satisfied the autoparametric interaction for vibration

absorption (see Fig. 12.12c, d for amplitude of 1.88 mm). However, the higher excitation amplitude caused many irregular

frequencies as it lost its periodicity (see Fig. 12.12c, d for amplitude of 2.3 mm).

Figure 12.13 compares the response behavior of the two ball pendulum system with excitation amplitudes of 1.88 mm and

2.3 mm at 4.05 Hz. This figure illustrates that many distinct frequencies existed in the system response when the excitation

amplitude was 2.3 mm. Also, a comparison of Fig. 12.13a for both amplitudes (no impact at 1.88 mm excitation and impact

at 2.3 mm excitation) supports that the impact of the pendulum increased beam oscillation. However, at a higher excitation

amplitude, the pendulum with larger mass created autoparametric interaction, hence coupling between two modes.

This coupling can only reduce the maximum beam oscillation amplitude from 49� to 37.9�.More insight into the characteristics of the impact response of the system can be obtained by examining Fig. 12.14.

As shown in Fig. 12.14, the ball impacted at point A when the pendulum angle reached 54.8�. During this time, the beam was

swinging upward. It is interesting to note that after the impact, the ball remained approximately at the same position until it

reached point B. Basically, the downward swing of the beam was holding the impacted ball at the top of the pendulum track

Fig. 12.13 Comparison of two ball beam pendulum system response for excitation amplitude of 1.88 mm and 2.3 mm at 4.05 Hz


from point A to B until it reached its bottom position. After point B, the ball traveled with very high velocity to the other side

of the track and impacted at point C. Similarly, it remained at the same position until it reached point D. Note the separation

at angle �50.5� between the two balls, which creates a valley in the ball response.

12.4 Conclusions

Using image processing and analysis, the dynamic behavior of beam free pendulum system under low and high sinusoidal

excitation was investigated for the external resonance condition of Oexcitation ¼ obeam with the primary resonance

obeam ¼ 2opendulum. The system responses were investigated experimentally in the neighborhood of primary resonance

condition. The results exhibited autoparametric interaction between the beam and the free pendulum when the primary

resonance condition was satisfied. The results of experiments for different beam-tip mass and pendulum mass ratios

indicates that more powerful absorption action can be achieved when the smaller mass ratios are used. For this model,

either one ball or two ball free pendulum, creating impact will not reduce the oscillation of the beam; instead it will increase

the response amplitude of the beam. Although image processing is relatively new in vibration measurements and analysis,

it is a cost effective and valuable tool.

Acknowledgment We wish to thank Dr. Murat M. Tanik for his assistance and help.

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4. Sun JQ, Jolly MR, Norris MA (1995) Passive, adaptive and active tuned vibration absorbers – a survey. J Mech Des 117:234–242

5. Jordanov IN, Cheshankov BI (1988) Optimal-design of linear and non-linear dynamic vibration absorbers. J Sound Vib 123(1):157–170

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A B

-55-45-35-25-15

-55

1525354555

0 0.2 0.4 0.6Time (Seconds)

Deg

rees

49o 54.8o 54.8o -45o -50.5o -53o

C D

LegendPendulumBeam

Fig. 12.14 Two ball beam pendulum system response for excitation amplitude of 2.3 mm at 4.05 Hz


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Control 1172:218–225

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29. Ekwaro-Osire S, Ertas A (1995) Response statistics of a beam- mass oscillator under combined harmonic and random excitation. J Vib Control

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Chapter 13

Suppression of Regenerative Instabilities by Means of Targeted

Energy Transfers

A. Nankali, Y.S. Lee, and T. Kalmar-Nagy

Abstract Regenerative effects in machining arise from the fact that the cutting force exerted on a tool is influenced not only

by the current position but also by that in the previous revolution. Hence, the equation of motion for the tool appears as a delay

differential equation, and the regenerative instability results in steady-state periodic motions, called limit cycle oscillations.

We study targeted energy transfers for suppressing regenerative instabilities by applying a nonlinear energy sink (NES) to a

single-degree-of-freedom machine tool model. A series of bifurcation analysis by means of numerical continuation

techniques demonstrate that there are three distinct suppression mechanisms; that is, recurrent burstouts and suppressions,

and partial and complete suppressions of regenerative instabilities.We characterize each suppressionmechanism numerically

by means of wavelet and Hilbert transforms and analytically by means of the complexification-averaging (CX-A) technique.

Furthermore, we extend the CX-A analysis to perform asymptotic analysis by introducing a reduced-order model and

partitioning slow-fast dynamics. The resulting singular perturbation analysis yields parameter conditions and regions for

the three suppression mechanisms, which exhibit good agreement with the bifurcations sets obtained from numerical

continuation methods. The results will help design NESs for passively controlling regenerative instabilities in machine tools.

Keywords Targeted energy transfer • Regenerative instability • Nonlinear energy sink • Machine tool dynamics

13.1 Introduction

We study targeted energy transfers for suppressing regenerative instabilities in a single-degree-of-freedom (SDOF) machine

tool model by coupling an ungrounded nonlinear energy sink (NES). Regenerative effects in machining arises from the fact

that the cutting force exerted on a tool is influenced not only by the current position but also by that in the previous

revolution. Hence, the equation of motion for the tool appears as a delay differential equation, which renders even an SDOF

dynamical system to be infinite-dimensional (See, for example, Dombovari et al. [1] and Nayfeh and Nayfeh [2] for recent

studies on machine tool dynamics).

Kalmar-Nagy et al. [3] analytically proved, by means of center manifold theorem (See, e.g., [4]), the existence of

subcritical Hopf bifurcation in an SDOF machine tool model with the regenerative cutting force being retained up to the

cubic order. Furthermore, practical stability limit in turning process was investigated by considering contact loss issues in

the regenerative cutting force [5], which can predict stable, steady-state periodic tool vibrations (or limit cycle oscillations –

LCOs). Such LCOs would create adverse effects on machining quality, and various passive and active means have been

considered to improve machining stability boundary (e.g., see [6–11]). In particular, direct use or variations of linear/

nonlinear tuned mass damper (TMD [8, 9]) are probably the most popular approach to passive chatter suppression.

However, even if the TMD is initially designed (tuned) to eliminate resonant responses near the eigenfrequency of a

primary system, the mitigating performance may become less effective over time due to aging of the system, temperature or

A. Nankali • Y.S. Lee (*)

Department of Mechanical and Aerospace Engineering, New Mexico State University, 1040 S. Horseshoe St, Las Cruces, NM 88003, USA


T. Kalmar-Nagy

Department of Aerospace Engineering, Texas A&M University, College Station, TX 77845, USA




181




humidity variations and so forth, thus requiring additional adjustment or tuning of parameters. It is only recently that

passively controlled spatial (hence dynamic) transfers of vibrational energy in coupled oscillators to a targeted point where

the energy eventually localizes were studied by utilizing an NES (See Vakakis et al. [12] for the summary of up-to-date

developments); and this phenomenon is simply called targeted energy transfer (TET). The NES is basically a device that

interacts with a primary structure over broad frequency bands; indeed, since the NES possesses essential stiffness non-

linearity, it may engage in (transient) resonance capture [13] with any mode of the primary system. It follows that an NES

can be designed to extract broadband vibration energy from a primary system, engaging in transient resonance with a set of

most energetic modes. In particular, Lee et al. [14–16] applied an ungrounded NES to an aeroelastic system, and numerically

and experimentally demonstrated that a well-designed NES can even completely eliminate aeroelastic instability. Three

suppression mechanisms were identified; that is, recurrent burstouts and suppressions, intermediate and complete elimina-

tion of self-excited instability in the aeroelastic system. Such mechanisms were investigated by means of bifurcation

analysis and complexification-averaging (CxA) technique [17].

In this workwe present study of TETmechanisms in suppressing regenerative chatter instability in a turning process. For this

purpose, we first review nonlinear dynamics of a SDOFmachine tool model; then, perform a linear stability analysis to explore

the effects of NES parameters on the occurrence of Hopf bifurcation (i.e., stability boundary on the plane of cutting depth and

rotational speed of a workpiece). To properly understand the suppressionmechanisms that appear similar to those in the previous

aeroelastic applications, numerical bifurcation analysis is performed by utilizing DDEBIFTOOL [18]. The CxA technique is

also utilized for analytical study of TET mechanisms (i.e., resonance captures). Finally, we perform asymptotic analysis for

regenerative instability suppression, which reveals the domain of attractions for the three suppression mechanisms [19, 20].

13.2 Dynamics of an SDOF Tool Model

We review of an SDOF machine tool dynamics model [3, 5], and then introduce an ungrounded NES to the machine tool

model. First, neglecting the NES in Fig. 13.1a, we can write the equation of motion for the SDOF machine tool as

€xþ 2zon _xþ o2nx ¼ �DFx=m (13.1)

where on is the (linearized) natural frequency; z ¼ c=ð2monÞ, the damping factor; and DFx, the cutting force variation. The

cutting force Fxð f Þ is frequently modeled as a power law by curve-fitting data obtained from quasisteady cutting tests [3, 5];

that is, we write

Fxð f Þ ¼ 0 for f � 0;Fxð f Þ ¼ Kwf a for f>0 (13.2)

where a is the cutting force exponent (a ¼ 0:75will be used in this work); w, the cutting width; f , the chip thickness; K, atest-related parameter assumed to be constant.

Tool

Workpiece

+

NES

5

4.5

4

3

3.5

2.5

2

1.5

1

0.5

00.5 1 1.5

Unstable region

a b

Stable region

2 2.5 3cs ms

m

k

ks

x

yc

n = 4n = 3

n = 2

n = 1

pp m

in

u3

Ω

Ω

Fig. 13.1 Machine tool model coupled to an ungrounded NES (a) and stability chart (b)

182 A. Nankali et al.

The cutting force variation can be expressed as

DFxð f Þ ¼ Fx � Fxð f0Þ ¼ �Fxðf0Þ for Df � � f0Kwð f a � f a0 Þ for Df>� f0

�(13.3)

where f0 refers to the nominal chip thickness at steady-state cutting; and Df , the chip thickness variation, which can be

expressed as Df ¼ f � f0 ¼ xðtÞ � xðt� tÞ � x� xt where t ¼ 2p=O is the delay or the period of revolution of the

workpiece with constant angular velocity O. Then, (13.1) can be rewritten as

€xþ 2zon _xþ o2nx ¼

k1f0ma

for Df � � f0

k1f0ma

1� f=f0ð Þa½ � for Df>� f0

8>><>>: (13.4)

where the cutting force coefficient k1 is introduced, k1 ¼ @Fx=@f jf¼f0¼ aKwf a�1

0 , which is the slope of the power-law curve

at the nominal chip thickness. Introducing the rescaling introduced in [5], we finally obtain the fully-nondimensional

equations of motion as

€xþ 2z _xþ x ¼pð2� aÞ

3afor Dx� 2� a

3

pð2� aÞ3a

1� 1� 3

2� aDx

� �a� �for Dx<

2� a3

:

8>><>>: (13.5)

where p ¼ k1=ðmo2nÞ is the nondimensional chip thickness; and Dx ¼ min xt � x; ð2� aÞ=3ð Þ due to the multiple regenera-

tive effects (or the contact loss for periods longer than a revolution of the workpiece). Since contact loss occurs when the

amplitude of tool vibrations is sufficiently large [5], a permanent contact can be assumed for sufficiently small amplitudes.

Taylor-expanding the cutting force about the nominal chip thickness and retaining the nonlinear terms up to the cubic order,

we obtain the equation of motion for a permanent contact model

€xþ 2z _xþ x ¼ pDx� pdðDx2 þ Dx3Þ (13.6)

where Dx ¼ xt � x and d ¼ 32ða� 1Þ=ð2� aÞ. Figure 13.1b depicts the stability chart in ðp;OÞ domain for a ¼ 0:75 so that

d ¼ 0:3, and z ¼ 0:1 so that pmin ¼ 2zð1þ zÞ ¼ 0:22 (see, e.g., [3]).

13.3 Suppression of Regenerative Instability by Means of TET

13.3.1 Nonlinear Energy Sink and Stability Enhancement

Now we apply an ungrounded nonlinear energy sink (NES) to the SDOF machine tool model (cf. Fig. 13.1a). Then, the

nondimensional equations of motion in state-vector form can be written as

_x ¼ Axþ Rxt þ fðx; xtÞ (13.7)

where x ¼ fx1; x2; x3; x4gT , x1 ¼ x; x2 ¼ y; x3 ¼ _x; x4 ¼ _y;

A ¼

0 0 1 0

0 0 0 1

�1� p 0 �2ðzþ z1Þ 2z10 0 2z1=ò �2z1=ò

26664

37775; R ¼

0 0 0 0

0 0 0 0

p 0 0 0

0 0 0 0

26664

37775; f ¼

0

0

�Cðx1 � x2Þ3 þ pd½ x1 � x1tð Þ2 � x1 � x1tð Þ3��Cðx2 � x1Þ3=ò

8>>><>>>:

9>>>=>>>;

where ò;C, and z1 respectively denote the mass ratio, coupling stiffness, and the damping factor of the NES.

13 Suppression of Regenerative Instabilities by Means of Targeted Energy Transfers 183

Assuming and substituting the solution of (13.7) to be xðtÞ ¼ expðltÞX, then we obtain the eigenvalue problem typical for

a delay-differential system.

lI� A� Re�lt� �X ¼ 0 (13.8)

where I is an identity matrix. For a nontrivialX, we derive the characteristic equation as |lI � A � e�ltR| ¼ 0. Substitution

l ¼ jo where j2 ¼ �1 and separation of real and imaginary parts yield

�2ðzþ z1 þ z1=òÞo2 þ 2z1ð1þ pÞ=ò ¼ ð2z1p=òÞ cosotþ po sinot� o3 þ ð1þ pþ 4zz1=òÞo¼ po cosot� ð2z1p=òÞ sinot (13.9)

By squaring and summing both sides of the two equations in (13.9), we obtain

pðoÞ ¼ GðoÞ=FðoÞ (13.10)

where the numerator and denominator can be written as

GðoÞ ¼ o6 � 2ð1� 2z21 � 2z21=ò2 � 4z21=ò� 4zz1 � 2z2Þo4 þ ð1� 8z21=ò

2 � 8z21=òþ 16z2z21=ò2Þo2 þ 4z21=ò

2

FðoÞ ¼ 2o4 � 2ð1� 4z21=ò2 � 4z21=òÞo2 � 8z21=ò

2(13.11)

Also, noting that 1� cosot ¼ 2sin2ðot=2Þ and sinot ¼ 2 sinðot=2Þ cosðot=2Þ, we rearrange (13.9) as

�2ðzþ z1 þ z1=òÞo2 þ 2z1=ò ¼ 2pR sinðot=2Þ cosðot=2þ fÞ� o3 þ ð1þ 4zz1=òÞo ¼ �2pR sinðot=2Þ sinðot=2þ fÞ (13.12)

where R ¼ ½ð2z1=òÞ2 þ o2�1=2 and f ¼ tan�1½2z1=ðòoÞ�. Then, we compute

tan ot=2þ fð Þ ¼ ½o3 � ð1þ 4zz1=òÞo�=½�2ðzþ z1 þ z1=òÞo2 þ 2z1=ò�KðoÞ (13.13)

Since t ¼ 2p/O, the rotational speed O can be derived as

OðoÞ ¼ po=½npþ tan�1KðoÞ � f� (13.14)

where n is the order of the lobe in the stability chart.

Figure 13.2 (left) depicts the changes of the stability boundary by varying the mass ratio ò and fixing the other two NES

parameters. Also, stability enhancement due to the application of anNES can bemeasured by directly calculating the point-wise

shift amount as Dp ¼ (p0 � p)/p � 100 (%), where p and p0 denote the values at the stability boundary with respect to each

5

100

50

Δp (%

)pp m

in pp m

inΔp

(%

)

10−1 10−1100 100

0

4

3

2

No NESNo NES=0.02 =0.1

Ω Ω

=0.2 =0.3 =0.6ζ1=0.02 ζ1=0.06 ζ1=0.1 ζ1=0.2

1

0

5

100

50

0

4

3

2

1

0

Fig. 13.2 Stability charts: (left) effects of NES mass ratio (ò) for z1 ¼ 0:1 and C ¼ 0:5; (right) effects of ENS damping factor (z1) for ò ¼ 0:1 andC ¼ 0:5


O without and with an NES, respectively. Upward shift of the stability boundary occurs more significantly near the valley than

near the cusp points of the lobes, which will be useful in practical applications of chatter suppression. The shifting amount of the

transition curve does not appear to be significant with a small NES mass (about 5% improvement near valley of the lobes);

however, the upward shift becomes increasing monotonically as the mass ratio increases. The ranges of the eigenfrequencies at

the transition curves tend to become lower as the mass ratio increases; and above certain mass ratio the eigenfrequency intervals

are shifted upward (cf. ò ¼ 0:6).The changes of stability boundary with respect to the NES damping factor with the other NES parameters fixed are

depicted in Fig. 13.2 (right). Although by increasing z1 the transition curve is shifted upward, stability enhancement by

increasing the damping factor is less prominent compared to that by increasing the NES mass. Rather, if z1 increases toohigh, Dp decreases, from which we conclude that adding more damping to a system does not always result in delay

occurrence of Hopf bifurcation. The eigenfrequency ranges on the transition curves shift upward as z1 increases initially, butthen shift downward for higher damping factors. We remark that such delay occurrence of Hopf bifurcations due the NES

parameters is absolutely independent of the magnitude of the essential nonlinearity C in the NES. This is because (13.10) and

(13.14) do not contain any terms associated with the essential nonlinearity. Indeed, this behavior was already observed in the

bifurcation analysis considered in Lee et al. [16] for understanding of the robustness enhancement of aeroelastic instability

suppression by means of SDOF and MDOF NESs.

The evaluation of overall stability enhancement can be discussed in terms of the amount of the upward shift at the valleys

of the lobes (i.e., the minimum value of pðoÞwhich is independent of the lobe order). We introduce the following quantity as

a measure of such shift at the valleys of the transition curve.

Dpmin ¼ ðp0min � pminÞ=pmin � 100ð%Þ (13.15)

where pmin ¼ 2zð1þ zÞ and p0min are the minimum values of pðoÞ without and with an NES, respectively.

Figure 13.3 depicts the contour plot of Dpmin with respect to ðò; z1Þ, which clearly illustrates that the optimal stability

enhancement (i.e., maxDpmin) occurs at a certain nonlinear relation between ò and z1 (the thick line which can be approximated

as the function, z1 � 0:35òq where q ¼ 1=1:3 from minimizing the mean square errors between the two curves).

13.3.2 Bifurcation Analysis and TET Mechanisms

We apply the numerical continuation technique for delay-differential equations (DDEBIFTOOL [18]) to study bifurcation

behaviors of the trivial equilibrium and the limit cycles. For example, Fig. 13.4a compares the bifurcation diagram for the

tool amplitudes by means of numerical continuation for the case when no NES is applied (thick dashed line) and when an

NES is involved (solid lines). As in the previous aeroelastic applications [14], three distinct TET mechanisms are identified

1250(%)

200

150

100

50

0

0.8

0.9

0.7

0.6

0.5ζ 1

0.4

0.3

0.2

0.1

00 0.2

max Δpmin

0.4 0.6 0.8 1

Fig. 13.3 NES mass ratio and

damping factor for optimal

stability enhancement


in suppressing regenerative chatter instability; that is, recurrent burstouts and suppression, intermediate and complete

elimination of regenerative instability (cf. See Fig. 13.4b for typical time history for each suppression mechanism).

The first suppression mechanism is characterized by a recurrent series of suppressed burstouts of the tool response,

followed by eventual complete suppression of the regenerative instabilities. The beating-like (quasiperiodic) modal

interactions observed during the recurrent burstouts turn out to be associated with Neimark-Sacker bifurcations of a periodic

solution (cf. Fig. 13.4a) and to be critical for determining domains of robust suppression [16]. To investigate a more detail of

this mechanism, Fig. 13.5 depicts the displacements of both the tool and NES and their wavelet transforms. Also, rigorous

energy exchanges between the two modes are evidenced in Fig. 13.5, through which a series of 1:1 transient resonance

captures and escapes from resonance occurs.

The second suppression mechanism is characterized by intermediate suppression of LCOs, and is commonly observed

when there occurs partial LCO suppression. The initial action of the NES is the same as in the first suppression mechanism.

Targeted energy transfer to the NES then follows under conditions of 1:1 TRC, followed by conditions of 1:1 PRC where the

tool mode attains constant (but nonzero) steady-state amplitudes. We note that, in contrast to the first suppression

mechanism, the action of the NES is nonrecurring in this case, as it acts at the early phase of the motion stabilizing the

tool and suppressing the LCO. The third suppression mechanism involves energy transfers from the tool to the NES through

1

0.8

a b

0.6

0.4

Dis

plac

emen

ts

0.2

Tool amplitudew/o NES NES amplitude

NS2

H

NS1

LPC2

LPC1

Tool amplitudew/ NES

0

1 2 3p/pmin

4 5 6 0−0.2

−0.5

0.5

−1

1

−1

0.2

0x(t)

x(t)

x(t)

1

0

0

50 100 150Time, (s)

200

3rd Suppression Mechanism

2nd Suppression Mechanism

1st Suppression Mechanism

250 300

Fig. 13.4 (a) Bifurcation diagram for the tool and NES amplitudes for (13.6) (O ¼ 2:6; e ¼ 0:2; z1 ¼ 0:1;C ¼ 0:5): H, LPC and NS denote Hopf,

limit point cycle, and Neimark-Sacker bifurcation points, respectively. (b) Typical tool displacements for the three suppression mechanisms

1

−1

−1

1

0

0

0 00

1

2

3

40

1

2

3

4

50

NE

S

Fre

quen

cy

Too

l

100 150 100Time Time

200 200250 300 300

Fig. 13.5 Displacements and their wavelet transform spectra for a typical first suppression mechanism


nonlinear modal interactions during 1:1 RCs. The displacements for the tool and the NES exhibit exponential decrement

finally resulting in complete elimination of LCOs.

In order to analytically study the underlying TETmechanisms,we employ theCxAmethod first introduced byManevitch [17].

We introduce the new complex variables in the following.

C1ðtÞ ¼ _xðtÞ þ joxðtÞ � ’1ðtÞe jot;C2ðtÞ ¼ _yðtÞ þ joyðtÞ � ’2ðtÞe jot (13.16)

where j2 ¼ �1. Then, denoting by ðÞ the complex conjugate, we can express the original real variables in terms of the new

complex ones

xðtÞ ¼ 1

2joðC1 �C

1Þ ¼1

2joð’1e

jot � ’1e

�jotÞ; xðt� tÞ ¼ 1

2joð’1ðt� tÞe joðt�tÞ � ’

1ðt� tÞe�joðt�tÞÞ

_xðtÞ ¼ 1

2ðC1 þC

1Þ ¼1

2ð’1e

jot þ ’1e

�jotÞ; €xðtÞ ¼ ð _’1 þ jo’1Þe jot � jo2ð’1e

jot þ ’1e

�jotÞ(13.17)

and similar expressions can be obtained for the NES variables. Substituting into the equations of motion and averaging out

the fast dynamics over e jot, we obtain a set of two complex-valued modulation equations governing the slow-flow dynamics,

_’ ¼ Fð’; ’tÞ (13.18)

where ’ ¼ f’1; ’2gT . Expressing the slow-flow amplitudes in polar form, ’kðtÞ ¼ akðtÞe jbkðtÞ, where akðtÞ; bkðtÞ 2 R;k ¼ 1; 2, we obtain the set of real-valued slow-flow equations such that

_a1 ¼ f1ða1; a2;fÞ; _a2 ¼ f2ða1; a2;fÞ; _f ¼ gða1; a2;fÞ (13.19)

where f � b1 � b2.Figure 13.6 directly compares the approximate and exact solutions of the tool displacement for the case of the first

suppression mechanism, which demonstrates a good agreement; furthermore, the non-time-like patterns (i.e., formation of

multiple loops) of the phase difference f depicts that the underlying TET mechanism involves a series of 1:1 transient

resonance captures and escapes from resonance.

Finally, we note that the numerical and analytical studies for TET mechanisms above are valid only for vibrations with

small amplitudes (i.e., before contact loss occurs); in particular, the permanent contact model with truncated nonlinear terms

cannot predict any stable steady-state periodic vibrations of high amplitudes. That is, the truncated nonlinearity in the

regenerative cutting force will not predict the existence of a saddle-node bifurcation point right after contact loss occurs.

The details of machine tool dynamics can be found in [5], where stable periodic motions are predicted. By applying the

ungrounded NES, we can still observe the three distinct TET mechanisms, as depicted in Fig. 13.7. Similar arguments can be

made for nonlinear modal interactions between the tool and NES as in the case of permanent contact.

1

0

0 50 100 150

direct numerical simulation analytical approximation

Time, t Phase difference f

f•

200 250 300 350 −2.5 −1.5 −0.5−2 0−1

−0.5

0.5T

ool D

ispl

acem

ent

1.5

0.5

−0.5

−1

1

0

−1

Fig. 13.6 Comparison of the approximate solution from the CxA analysis with the (numerically) exact solution for a typical first suppression

mechanism (on the left); demonstration of non-time-like phase difference (on the right)


13.4 Asympotic Analysis of Regenerative Instability Suppression

In this section, we perform an asymptotic analysis [19, 20] for the three suppression mechanisms to estimate the domains of

attraction in the parameter space. For this purpose, we introduce the coordinate transformation

v ¼ xþ ey;w ¼ x� y (13.20)

where v and w are the physical quantities for the center of mass (with a factor of 1 + e) and the relative displacement,

respectively. Then, the equations of motion (13.7) become

€vþ 2z_vþ e _w1þ e

þ vþ ew1þ e

¼ pvt þ ewt

1þ e� vþ ew

1þ e

� �

€wþ 2z_vþ e _w1þ e

þ 2z1 _wþ vþ ew1þ e

þ 4

31þ eð Þw3 ¼ p

vt þ ewt

1þ e� vþ ew

1þ e

� � (13.21)

where proper rescaling conditions are applied [19]. Since a single fast-frequency dominates for the three suppression

mechanisms, we introduce the complexification similar to (13.16).

_vðtÞ þ jovðtÞ � ’1ðtÞe jot; _wðtÞ þ jowðtÞ � ’2ðtÞe jot (13.22)

Substituting into (13.21) and performing averaging over the fast component ejot, we obtain the slow-flow equation

_’1 ¼ F1ð’1; ’2; ’1t; ’2tÞ; _’2 ¼ F2ð’1; ’2; ’1t; ’2tÞ (13.23)

Introducing polar form to the slow variables, ’1 ¼ V exp½jy1�; ’2 ¼ W exp½jy2�, we can derive real-valued slow-flow

dynamics

_V ¼ eF1ðV;W;Vt;Wt;fÞ; _W ¼ F2ðV;W;Vt;Wt;fÞ;_f ¼ G1ðV;W;Vt;Wt;fÞ þ eG2ðV;W;Vt;Wt;fÞ

(13.24)

2

−2

−1

−10 50 100 150 200 250

Tool Displacements

w/o NES

1st Supp Mech

2nd Supp Mech

3rd Supp Mech

w/ NES

Time (s)

300 350 400 450 500

0

1

1

0

0x(t)

x(t)

x(t)

Fig. 13.7 Typical time

responses for the three

suppression mechanisms

for the contact loss model

in (13.5)


where V;W 2 R;f ¼ y1 � y2;F1;F2;G1;G2 ¼ Oð1Þ, and the details of the right-hand sides are omitted because of their

complexity. For sufficiently small e, we can write (13.24) as

_V ¼ OðeÞ ) V ¼ VðetÞ; _W ¼ F2ðV;W;Vt;Wt;fÞ;_f ¼ G1ðV;W;Vt;Wt;fÞ þ OðeÞ

((13.25)

Considering the equilibrium points of (13.24), we can derive the slow-invariant manifold (SIM) for equilibrium and super

slow-flow (SSF) dynamics, respectively, as

SIM : F2ðV0;W0;f0Þ ¼ 0 and G1ðV0;W0;f0Þ ¼ 0 ) HðV0;W0Þ ¼ 0

SSF : @V=@ðetÞ ¼ F1ðV0;W0Þ(13.26)

The intersections between SIM and SSF equations provide the number of equilibrium points for the slow-flow dynamics

(13.24) and their stability [19]. Figure 13.8 presents a result for asymptotic analysis for a typical first suppression mechanism,

where amplitude modulations can be observed due to relaxation oscillations (strongly-modulated response [19]).

13.5 Concluding Remarks

Suppression of regenerative instability in a single-degree-of-freedom (SDOF) machine tool model was studied by means of

one-way, passive, broadband targeted energy transfers (TETs). Two models were considered for the tool dynamics:

Permanent contact model and contact loss model. Stability and bifurcation analysis were carried out for both models.

An ungrounded nonlinear energy sink (NES) was coupled to the SDOF tool, by which biased energy transfers from the tool

to the NES. Shifts of the stability boundary (i.e., Hopf bifurcation point) with respect to chip thickness were examined for

various NES parameter conditions. It was shown that there should be an optimal value of damping for a fixed mass ratio to

shift the stability boundary for stably cutting more material off by increasing chip thickness. Also, magnitude of NES

nonlinear stiffness does not have any effect on stability boundary while increasing mass ratio improves stability. The limit

cycle oscillation (LCO) due to the regenerative instability in a tool model which appeared as being subcritical for permanent

contact model were (locally) eliminated or attenuated at a fixed rotational speed of a workpiece (i.e., a delay period) by TETs

to the NES. Contact loss model depicted supercritical LCOs at relatively high displacement of the tool. Utilizing NES for

contact loss model shifted bifurcation diagram of tool displacement such a way to improve stability. Three suppression

mechanisms have been identified as was in the previous aeroelastic applications, and each suppression mechanism was

investigated numerically by time histories of displacements, and wavelet transforms and instantaneous modal energy

exchanges. The analytical means by complexification-averaging technique showed that resonance captures are the underlying

dynamical mechanism for TETs.

3

2.5

= 0.02; p = 0.7

SIM : H (V0,W0) = 0

SSF : F1(V0,W0) = 0

1.5

0.5

2

1

00 2 4

x(t)

6 8 0 200 400TimeW0

2( t)

V02 (

t)

600 800 1000

1

0.5

−0.5

−1

0

Fig. 13.8 Asymptotic analysis of the first suppression mechanism and corresponding tool response


Acknowledgments This material is based upon work supported by the National Science Foundation under Grant Numbers CMMI-0928062 (YL)

and CMMI-0846783 (TK).

References

1. Dombovari Z, Barton DAW, Wilson RE, Stepan G (2011) On the global dynamics of chatter in the orthogonal cutting model. Int J Non Lin

Mech 46:330–338

2. Nayfeh AH, Nayfeh NA (2011) Analysis of the cutting tool on a lathe. Nonlinear Dyn 63:395–416

3. Kalmar-Nagy T, Stepan G, Moon FC (2001) Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations. Nonlinear

Dyn 26:121–142

4. Namachchivaya NS, Van Roessel HJ (2003) A center-manifold analysis of variable speed machining. Dyn Syst 18(3):245–270

5. Kalmar-Nagy T (2009) Practical stability limits in turning (DETC2009/MSNDC-87645). In: ASME 2009 international design engineering

technical conferences and computers and information in engineering conference, San Diego, California, 30 August–2 September, 2009

6. Lin SY, Fang YC, Huang CW (2008) Improvement strategy for machine tool vibration induced from the movement of a counterweight during

machining process. Int J Mach Tool Manuf 48:870–877

7. Khasawneh FA, Mann BP, Insperger T, Stepan G (2009) Increased stability of low-speed turning through a distributed force and continuous

delay model. J Comput Nonlin Dyn 4:041003-1-12

8. Moradi H, Bakhtiari-Nejad F, Movahhedy MR (2008) Tunable vibration absorber design to suppress vibrations: an application in boring

manufacturing process. J Sound Vibrat 318:93–108

9. Wang M (2011) Feasibility study of nonlinear tuned mass damper for machining chatter suppression. J Sound Vibrat 330:1917–1930

10. Ganguli A (2005) Chatter reduction through active vibration damping. Ph.D. dissertation, Universite Libre de Bruxelles

11. Ast A, Eberhard P (2009) Active vibration control for a machine tool with parallel kinematics and adaptronic actuator. J Comput Nonlin Dyn

4:0310047-1-8

12. Vakakis AF, Gendelman O, Bergman LA, McFarland DM, Kerschen G, Lee YS (2008) Passive nonlinear targeted energy transfer in

mechanical and structural systems: I and II. Springer, Berlin/New York

13. Arnold VI (ed) (1988) Dynamical systems III. Encyclopaedia of mathematical sciences. Springer, Berlin/New York

14. Lee YS, Vakakis AF, Bergman LA, McFarland DM, Kerschen G (2007) Suppression of aeroelastic instability by means of broadband passive

targeted energy transfers, part I: Theory. AIAA J 45(3):693–711

15. Lee YS, Kerschen G, McFarland DM, Hill WJ, Nichkawde C, Strganac TW, Bergman LA, Vakakis AF (2007) Suppression of aeroelastic

instability by means of broadband passive targeted energy transfers, part II: Experiments. AIAA J 45(10):2391–2400

16. Lee YS, Vakakis AF, Bergman LA, McFarland DM, Kerschen G (2008) Enhancing robustness of aeroelastic instability suppression using

multi-degree-of-freedom nonlinear energy sinks. AIAA J 46(6):1371–1394

17. Manevitch L (2001) The description of localized normal modes in a chain of nonlinear coupled oscillators using complex variables. Nonlinear

Dyn 25:95–109

18. Engelborghs K, Luzyanina T, Roose D (2002) Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM

Trans Math Softw 28(1):1–21

19. Gendelman OV, Vakakis AF, Bergman LA, McFarland DM (2010) Asymptotic analysis of passive nonlinear suppression of aeroelastic

instabilities of a rigid wing in subsonic flow. SIAM J Appl Math 70(5):1655–1677

20. Gendelman OV, Bar T (2010) Bifurcations of self-excitation regimes in a van der Pol oscillator with a nonlinear energy sink. Physica D

239:220–229


Chapter 14

Force Displacement Curves of a Snapping Bistable Plate

Alexander D. Shaw and Alessandro Carrella

Abstract Bistable structures are characterised by rich dynamics, because they necessarily include regions of negative

stiffness between their stable configurations. The presence of this region results in marked nonlinear behaviour, with

a response that can range from periodically stable to chaotic. However, this region can also be exploited to tailor

force-displacement curves.

A possible application of an ad-hoc load deflection curve is a vibration isolator with High Static Low Dynamic Stiffness

(HSLDS). The idea is to confer to the isolator a low dynamic stiffness whilst the high static stiffness maintains a high load

bearing capacity. Therefore coupling this apparatus with mass on a conventional anti-vibration mount demonstrates that

it can be used to reduce the natural frequency of this system, thereby increasing the isolation region of the mount.

This study presents the load-displacement curve of a bistable composite plate, which is loaded transversely at its centre,

whilst its corners are free to rotate and move laterally. Both numerical and experimental results are presented, and it is shown

that the response is highly directional and hysteretic, and that the force is also influenced by velocity as well as displacement.

The geometrical sources of these effects are considered.

Keywords Bistable • Nonlinear • Composite • Isolator

14.1 Introduction

Bistable composite plates can occupy two different stable configurations, between which they may ‘snap’ when forced [1].

They are attracting considerable interest in the field of morphing structures, and for their potential use in actuators, due to

their ability to form multiple shapes with no ongoing power consumption [2, 3]. They also are attracting interest from

dynamics researchers, as their multiple potential wells can lead to highly nonlinear and chaotic responses to excitation [4].

A consequence of having two stable configurations is a region of negative transverse stiffness occurring between these states;

this region is encountered during a snap between one stable state and another. This negative stiffness can be exploited to tailor

force displacement curves, in particular to create a High Static LowDynamic Stiffness (HSLDS) anti-vibrationmount. This is a

device that has high static stiffness, to provide it with good load bearing capacity. However, near its equilibrium point it features

a region of low stiffness, which therefore lowers the natural frequency and increases the isolation region of the mount [5].

In order to create an HLSDS mount exploiting the negative stiffness of a composite bistable plate, we must investigate

the transverse force displacement curve of such plates. This work presents an experimental and numerical investigation into

the force displacement curve of a thermally formed bistable plate, extending the findings of a previous study by Potter et al.

[7] to consider the effect of repeated and reversed displacement cycles. The study shows that the response is rate dependant,

direction dependant and therefore highly hysteretic.

A.D. Shaw (*)

Advanced Composites Centre for Innovation and Science (ACCIS), University of Bristol, Queens Building,

University Walk, Bristol BS8 1TR, UK


A. Carrella

Faculty of Engineering, Bristol Laboratory for Advanced Dynamic Engineering (BLADE), LMS International,

Queens Building, University Walk, Bristol BS8 1TR, UK



191


14.2 Experimental Study

14.2.1 Experimental Design

The plate was made from eight plies of IM7/8552 Intermediate Modulus Carbon Fibre pre-preg in the stacking sequence

[0�4; 90�4]T . It was vacuum bagged using a tool plate on both sides, to ensure the most symmetrical resin distribution.

It was cured using the manufacturers recommended cycle in an autoclave at 180�C for 2 h. The ‘flat’ plate dimensions are

280 � 280 mm, in line with the fibre direction. A 5 mm hole was drilled at the centre, to accommodate the bolt for the load

tester. Holes 12 mm in diameter were drilled 10 mm in from each corner, to accommodate pivot joints described below.

To restrict the degree to which moisture absorption could affect the material properties, the plate was stored in a sealed

cupboard with desiccant whenever it was not in use.

The plate was supported by its corners, on apparatus designed to provide boundary conditions which do not restrain

the snap through. These corner boundary conditions are vertical pins i.e. they allow free out of plane rotation, zero vertical

displacement and free lateral translation. To achieve the first of these conditions, the corners were fitted with spherical

bearings that were bonded into holes drilled through the plate, that permitted pivoting in any direction. By providing a small

angle to the bearing casing relative to the plate, these bearings allowed unrestricted motion to the full range of angles that

the corners of the plate would adopt between each of its stable states. To simultaneously achieve the second and third

boundary conditions, the bearings were mounted on tall slender steel posts (250 mm long, 3 mm in diameter), which were

rigidly attached to an adjustable base. Vertically these provided stiffness greatly in excess of the plate’s transverse stiffness.

Horizontally, the posts acted as soft cantilever springs. The maximum lateral displacement of the corners is approximately

5 mm so modelling the posts as a simple cantilever shows the maximum horizontal reaction of the corners would be of the

order of 0.75 N. The vertical deflection caused by this motion can also be seen to be negligible.

The entire apparatus was placed on the moving base of an Instron 1341 load tester, with the centre of the plated bolted to a

1 kN load cell. Therefore raising and lowering the base effectively applied displacements to the centre of the plate, allowing

the load displacement curve to be measured. The complete apparatus is shown in Fig. 14.1.

The displacement cycle applied to the plate consisted of increasing displacement at constant velocity over a range

including both stable positions of the plate, then returning at the same velocity to the starting point. This was performed

twice, at a velocity of 100 mm/min.


method

192 A.D. Shaw and A. Carrella

14.3 Results

Figure 14.2 shows the experimental force displacement curve. As can be seen, the force shows steady progression with

displacement, interrupted by two sudden changes in force, as shown by steep negative gradients followed by transients

on the graph. These correspond to sudden changes of the shape of the plate, from and to a ‘half snap’ configuration, as shown

in Fig. 14.3. These shape changes occur at different displacements, depending on the direction of the motion.

14.4 Numerical Analysis

14.4.1 Model

The numerical model was created in Abaqus 6.10, using the Dynamic/Implicit solver using nonlinear geometry to allow

the bistable properties to resolved and the quasi-static option to provide appropriate amounts of artificial dissipation

automatically. The mesh was constructed from four-node shell elements constructed with the appropriate composite lay

up. Figure 14.4 shows how the mesh was biased to provide greater resolution near the edges, where stress gradients are seen

to be higher. Vertically pinned, laterally free boundary conditions are applied the locations of the centres of the plate pivot

joints. At the centre, lateral translation and rotation about the vertical axis are restricted, and the vertical displacement is

controlled throughout the simulation.

Fig. 14.2 Experimental force displacement curve, two loading cycles at 100 mm/min. Arrows show direction of motion around cycle

Fig. 14.3 Upward snap sequence, showing initial, half snap and final configurations

14 Force Displacement Curves of a Snapping Bistable Plate 193

The layup and initial geometry of the plate was as described for the experimental design, using lamina properties as

described in Table 14.1. Small eccentricities of 0.25 and 0.5 mm in x and y respectively were added to the position of the

central point, to allow asymmetric shapes to form. It was found that varying these dimensions over a range similar to that

expected for manufacturing errors in the physical experiment did not have a major effect on results found.

The solver follows the following steps:

1. The initial step defines the flat plate at cure temperature.

2. The plate is cured to room temperature (20�C) and adopts one of its bistable shapes. A nominal load is applied to the

centre to ensure that the configuration resolved is consistently the same one of the two possible states.

3. The nominal load is removed.

4. The centre of the plate is displaced to starting position, 40 mm vertically above the original horizontal plate plane.

5. The centre vertical displacement is varied in a linear ramp to �40 mm, over a period of 100 s.

6. The previous step is reversed; the vertical displacement is varied in a linear ramp to 40 mm over 100 s.

14.5 Results

Figure 14.5 shows the results of the FEA simulation. It shows the snap through region characterised by multiple stages,

separated by three sudden shape-change events. Again, the displacement at which these events occur varies with the

direction of motion. Figure 14.6 shows the total strain energy reported by Abaqus against displacement, and it can be seen

that shape change events coincide with sharp drops in the strain energy.

Fig. 14.4 Mesh used for

FEA. Dots indicate whereboundary conditions are

applied

Table 14.1 Assumed lamina properties for IM7/8552 CFRP. Properties taken from

manufacturer’s data, with transverse isotropy used to estimate through-thickness quantities.

t is an average taken from multiple plate measurements

In plane Young’s Modulus, fibre direction, E1 164 GPa

In plane Young’s Modulus, transverse direction, E2 12 GPa

In plane Poisson ratio, n12 0.3

In plane shear modulus, G12 4.6 GPa

Through thickness shear modulus, fibre direction, G13 4.6 GPa

Through thickness shear modulus, transverse direction, G23 4.1 GPa

Ply thickness, t 0.122 mm


14.6 Discussion

14.6.1 Comparison of Experiment and Numerical Results

The numerical and physical experiments show some qualitative similarities, but some key differences remain. Firstly,

the experimental graph is asymmetrical, in that the minima near 10 mm displacement is smaller in magnitude and shows

smoother curves that the maxima near �5 mm displacement. It is believed this is because the plain bearing joints were

necessarily bonded in place when the plate was in one of its stable states. Hence when the plate is flipped to its other state, the

fillet of glue forms a residual stress that applies moments to the plate, reducing the snapping force and distorting the force

displacement graph. Secondly, maximum force is much smaller in the physical experiment than in FEA. It is thought this

is explained by the ingress of moisture from ambient humidity; atmospheric effects on such a magnitude are described

by Etches et al. [6]. Finally, over much of the graph where there is no obvious influence due to shape changes, there

is a small difference due to the direction of travel present in the experimental work that is not apparent in the FEA.

This is attributed to a small amount of frictional or viscous force within the experimental apparatus.

Fig. 14.5 Numerical force displacement curve, arrows indicate path followed

Fig. 14.6 Internal strain energy against displacement. Arrow indicates path followed when displacement moves from negative to positive.

Dashed lines show the assumed projection of the four continuous curves that intersect to form graph


Despite these shortcomings, the positive peak and much of the negative stiffness region of Fig. 14.2 suggest that

FEA captures many of the snapping plate, qualitatively if not quantitatively.

14.6.2 Validity of Quasi-static Assumption

A conventional force-displacement graph implicitly assumes that the system concerned is quasi-static; that motion

is sufficiently slow that dynamic effects may be neglected, and that each point on the graph therefore defines a point of

static equilibrium. For most of Fig. 14.2 and Fig. 14.5 this assumption is met; even where a negative slope is present, the

plate is held in a static equilibrium by the restraint at the centre. However, in the immediate vicinity of shape changes

the quasi-static assumption is invalid, because the plate is performing a rapid motion between a state that has become

unstable, and a new stable state, and is then subsequently oscillating for a short while after this change.

14.6.3 History Dependence; Irreversibility and Hysteresis

The system described is history dependent; for example the force at zero displacement in Fig. 14.5 may be positive if

displacement has moved directly from the negative equilibrium point, or negative if the opposite motion has just been

performed. Related to this is that the shape-change events are irreversible; once a shape change has occurred, returning

directly to the displacement just before the shape change will return to a different centre reaction force and plate shape than

previously existed.

To return to a given shape and displacement after a shape change, displacement must retract further until the opposite

shape change occurs, and then be advanced to the original position. However, this process forms a hysteresis loop that will

permanently dissipate energy input into the system.

14.6.4 Total Strain Energy

Figure 14.6 shows that the total strain energy of the plate can be used to model the behaviour. If we consider the

strain energy/displacement graph as the intersection of four curves (as arbitrarily extended in the figure) associated with

each shape, the plate generally follows the path that has the lowest total strain energy. However, when two energy curves

intersect, the graph follows its original course briefly before jumping down to the new configuration. This delay is explained

qualitatively below.

At the intersection, the two different shapes offer exactly the same total strain energy. Therefore there is no energy

gradient inducing any transformation from one to the other, indeed the intermediate states may demand higher strain energy.

Therefore, the initial configuration must progress until it has higher energy than the new configuration, and it becomes

unstable. Furthermore, since the initial speed of the shape change may be very slow due to a low energy gradient, the rate

of the controlling displacement may bias the apparent displacement at shape change, even at apparently quasi-static

displacement rates.

14.7 Conclusions and Future and Ongoing Work

This work has shown the transverse force-displacement curve for a thermally formed bistable composite plate, using both

numerical and experimental methods. It has demonstrated that the negative stiffness region is not smooth, but punctuated by

a number of unstable shape change events, which are irreversible and lead to a history dependant, hysteretic force

displacement curve. Theses shape changes are related to drops in strain energy of the plate.

Ongoing work is investigating dynamic effects on this curve, applying high frequency deformations to the plate and

investigating its effects when coupled to conventional vibration mounts. Future work will also develop mathematical shape

functions for the stable and half snapped configurations, which allow the shape to be modelled for any given displacement.

It will also investigate other fabrication techniques for bistable plates.


References

1. Dano M-L, Hyer MW (1998) Thermally-induced deformation behavior of unsymmetric laminates. Int J Solids Struct 35(17):2101–2120

2. Daynes S, Weaver PM, Trevarthen JA (2011) A morphing composite air inlet with multiple stable shapes. J Intell Mater Syst Struct

22(9):961–973

3. Diaconu CG, Weaver PM, Mattioni F (2008) Concepts for morphing airfoil sections using bi-stable laminated composite structures. Thin Wall

Struct 46(6):689–701

4. Arrieta A, Neild S, Wagg D (2009) Nonlinear dynamic response and modeling of a bi-stable composite plate for applications to adaptive

structures. Nonlinear Dyn 58:259–272

5. Ibrahim RA (2008) Recent advances in nonlinear passive vibration isolators. J Sound Vibrat 314(3–5):371–452

6. Etches J, Potter K, Weaver P, Bond I (2009) Environmental effects on thermally induced multistability in unsymmetric composite laminates.

Compos Part A Appl Sci Manuf 40(8):1240–1247, Special issue: 15th French national conference on composites - JNC15

7. Potter K, Weaver P, Seman AA, Shah S (2007) Phenomena in the bifurcation of unsymmetric composite plates. Compos Part A Appl Sci Manuf

38(1):100–106


Chapter 15

Characterization of a Strongly Nonlinear Vibration Absorber

for Aerospace Applications

Sean A. Hubbard, Timothy J. Copeland, D. Michael McFarland, Lawrence A. Bergman,

and Alexander F. Vakakis

Abstract We consider the identification of a nonlinear energy sink (NES) designed to limit the vibration of an aircraft wing

by attracting and dissipating energy before a transient response can build into a limit-cycle oscillation (LCO). The device

studied herein is the prototype of an NES intended to be mounted at the tip of a scale-model wing, housed in a winglet, and

capable of interacting dynamically with the wing over a broad frequency range. Because the stiffness of the NES is

essentially nonlinear (i.e., its force-displacement relation is nonlinearizable), it cannot be regarded as a perturbation of a

linear system. Furthermore, the action of the NES requires the presence of some amount of damping, here assumed to be

viscous. Both the nonlinear stiffness and the linear viscous damping have been evaluated using the restoring force surface

method (RSFM), and found to be repeatable across trials and across builds of the system. These findings are summarized and

used in simulations of the NES attached to the wing. The simulations are then compared to experiments (ground vibration

tests), revealing good agreement of transient responses and of frequency-energy dependence, the latter revealed by wavelet

transforms of the computed and measured time series.

Keywords Limit-cycle oscillations • LCO • Aircraft structures • Nonlinear energy sink • NES

15.1 Introduction

One of the most-developed applications of targeted energy transfer (TET) for vibration suppression is the stabilization of

limit-cycle oscillations (LCOs) of aircraft structures. While such self-excited oscillations would seem to be steady-state

phenomena, detailed analysis has shown the onset of LCO in a typical-section (rigid-airfoil) model in subsonic flow to be

characterized by a transient 1:1 resonance between the wing’s heave and pitch motions, followed by a sustained 3:1

resonance. If the initial 1:1 resonance is interrupted, the higher-energy 3:1 resonance, and thus the LCO, may be prevented

entirely. A properly designed passive attachment—a strongly nonlinear vibration absorber—can achieve this by coupling to

small, flow-induced motions of the primary structure early in its response.

Similar events lead to LCO in more realistic conditions, such as a plate-like structure in transonic flow, although the

response frequencies may vary more and more aeroelastic modes may participate. The vibration absorber now cannot be

tuned to a specific frequency, but must be inherently broadband. A nonlinear energy sink (NES), incorporating a small mass,

S.A. Hubbard

Aerospace Engineering Department, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA


T.J. Copeland (*)

m+p international, inc. 271 Grove Avenue, Bldg G, Verona, NJ 07044, USA


D.M. McFarland • L.A. Bergman

Aerospace Engineering Department, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA


A.F. Vakakis

Mechanical Science and Engineering Department, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA




199






an essentially nonlinear (i.e., nonlinearizable) coupling stiffness, and a (typically linear, viscous) damping element is ideal for

this application, and we consider such a device here in prototype form, as designed a fabricated for use with a uniform-plate

mock-up of a wind-tunnel model wing. This NES has a single degree of freedom, represented relative rotation of the NES

mass with respect to its mounting point at the wingtip. An essentially nonlinear spring provides restoring torque, and

dissipation occurs in the bearing supporting the mass and in the connections to the springs. The focus of this paper is on

the characterization of these dynamic elements, especially the nonlinear springs, to show that they can be accurately designed

and reliably and repeatably installed.

15.2 Rotary Nonlinear Energy Sink Design and Assembly

Figure 15.1 shows an exploded view of the NES components and how they are assembled, along with a view of the

assembled NES.The backing plate and anchor blocks were machined from 6061 aluminum alloy, and the shaft, anchor

clamps, NES mass, and posts were all machined out of steel. The wires used were straight stock steel wire, and the rotary

bearing was a standard type deep groove ball bearing typically used in automotive applications. The reverse side of the plate

features two ribs which fit over and bolt onto the tip of the uniform-thickness aluminum plate swept wing. The NES has

several features which allow for it to be examined under numerous configurations. The plate allows for one of the anchor

Fig. 15.1 (a) Exploded view

of NES components;

(b) Assembled NES

200 S.A. Hubbard et al.

blocks to be bolted into one of three positions which control the span of the wires, or the blocks can be moved to several

different positions on the circumference of the plate causing the direction of the wire span to rotate. The NES mass features

threaded holes that can be used to add mass, increasing its mass moment of inertia. The wire diameter is only limited by the

diameter of the wire channel in the anchor clamp, and the gaps and positioning of the posts are adjustable to some extent.

The aluminum alloy backing plates serves as a frame upon which the remaining components can be assembled. It also

provides the interface for the device to be attached to the plate wing. The plate is approximately 6 in. in diameter and is

shaped as such to accommodate several configurations. The pattern of 0.5 in holes through the plate serve no purpose other

than to remove unnecessary mass. As their names indicate, the anchor blocks and clamps provide the required boundary

conditions for the wire ends.

When the assembled device in Fig. 15.1b is free to oscillate, or active, the NES mass is allowed to pivot about the shaft.

As the mass pivots, the posts make contact with the wires causing them to be displaced transversely near the center of the

span. As mentioned already, this will produce a nonlinear response from the wires which will force the NES mass in the

reverse direction. This is the configuration that was considered for all that follows.

Some information important for the analyses that follow is listed here. The mass of the fully assembled device shown in

Fig. 15.1b is 0.687 kg, and its mass moment of inertia about the shaft is approximately 1. 27 �10 � 3kgm2 when the NES

mass is locked in place. The mass moment of inertia of the NES mass about the shaft is 2. 2 �10 � 4kgm2.

When assembled, the NES mass is allowed to pivot about the bearing such that the posts are in contact with the piano

wires which are clamped at both ends. The most critical stage of the assembly is the installation of the wires which must be

straight and without tension or compression. When assembled properly, the NES experiences negligible resistance to

rotation from the springs when it is close to its neutral position. Effective performance of the NES requires essentially-

nonlinear stiffness (i.e., nonlinearizable stiffness). As the angle of rotation increases, however, the reaction force due the the

wires, or nonlinear springs, scales with the displacement cubed due the geometric nonlinearity of the displaced wires.

Additional steps were taken to ensure that close clearances were achieved and impacts between various components,

especially the springs and posts, were limited so as to avoid their effects on the dynamics of the system.

15.3 Identification of System Parameters

To confirm that the NES design succeeded in achieving an essential stiffness nonlinearity, several tests were conducted to

estimate the properties of the system. These tests also provided data critical for the modeling of the interaction between the

NES and scale-model wing. The goal of the identification of the NES was to estimate the stiffness and damping of the

system. Each parameter was tested using two independent approaches, one static and one dynamic, which allowed for some

additional understanding of the properties of the device beyond the estimation of the parameters.

15.3.1 Static Test

The static stiffness approach was the most direct method employed for estimating the NES stiffness. It consisted of applying

a known torque and measuring the angular displacement. To accomplish this, the fully assembled NES was bolted to the

testing jig and then clamped in its upright position using a milling vice. Next, the NES mass was locked in its “zero” or

resting position using the locking bolt. In this context, the “zero” position refers to the NES mass position in which the wires

provide no restoring torque. For this test, it was assumed that the assembly of the NES and installation of the wires succeeded

in aligning the zero position with the position of the locked NES mass. Finally, an aluminum block was clamped onto the

plate so that it was offset from the NES mass and aligned parallel to the unloaded wires. This served as a reference position

from which measurements could be taken during testing. The setup is shown in Fig. 15.2.With this configuration, a known

torque could be applied by attaching one end of a string to a post on the NES mass and suspending a mass at the other end

over a pulley. With a torque applied, the displacement of some designated position on the NES mass could be measured with

respect to the reference block. This process was repeated by incrementing the mass and measuring displacement until some

designated maximum torque or displacement was achieved. Then the process was reversed and the displacement

measurements repeated as the torque was unloaded. Upon completion of the loading and unloading of the NES, the process

was repeated so as to induce displacements in the direction opposite that of the previously applied torque.

The data collected from this test can be used to evaluate the stiffness of the NES by assuming some form of the stiffness

function and then estimating the parameters of that function using a least-squares fit. This data was particularly useful for

evaluating the symmetry or asymmetry of the stiffness because of the separate data obtained for positive and negative rotations.

15 Characterization of a Strongly Nonlinear Vibration Absorber for Aerospace Applications 201

15.3.2 Dynamic Test

The restoring force surface method (RFSM), originally developed by Masri and Caughey [5], was used to simultaneously

estimate stiffness and damping. This method consists of measuring the dynamic input and response of a single degree of

freedom system and using that data to estimate the unknown system parameters. A basic explanation follows by considering

(15.1) where m is the mass, €x is the acceleration, pt is the external force in time, all known.

m€xþ f _x; xð Þ ¼ p tð Þ (15.1)

If f _x; xð Þ is some unknown function of the displacement and velocity, x and _x, then the equation of motion can be

rearranged to give

f _x; xð Þ ¼ p tð Þ � m€x|fflfflfflfflfflffl{zfflfflfflfflfflffl}restoringforce

:(15.2)


for static stiffness test


for RFSM test


If the form of the function f _x; xð Þ is known or assumed, then the parameters of the function may be estimated using a least

squares fit to the restoring force. Typically, amean-squared error of less than 5% is considered an accurate fit. The background

and details of the method are discussed by Kerschen and Golinval [2], Kerschen et al. [3, 4].

Figure 15.3 shows a photo of the experimental setup for the RFSM test. Similar to the static stiffness test already

described, the NES was bolted to the testing jig which was then clamped to the working surface in an upright position using a

milling vise. A PCB triaxial accelerometer was attached to the upper surface of the NES mass such that one axis of

measurement would intersect the center of rotation. This alignment positioned one of the other measurements axes in such a

way that it would detect only the tangential acceleration of the NES mass. With the accelerometer in place, the distance

between the center of rotation of the NES mass and the center of the transducer was measured. It was assumed that the

accelerometer detects the acceleration at its center, so this measurement allowed the angular acceleration to be determined

from the measured tangential acceleration.

Due to the difficulty of accurately measuring excitation of the rotary system, we consider only the transient response

in the parameter identification. Thus, the restoring force in (15.2) consists only of the inertia term. In practice, the NES

was excited using a PCB impact hammer. The hammer and triaxial accelerometer were powered by a VibPilot data

acquisition system from m+p international. The acquisition system was configured to trigger upon the detection of a

tangential acceleration. Data was sampled at 1,024 Hz for a period of 4 s beginning approximately 0.8 s before the

trigger. The impact hammer force pulse was recorded so that the forced and transient portions of the response could

be separated.

15.4 Results

Before estimating the NES parameters, we assume that the system can be modeled as

f ðx; _xÞ ¼ c _xþ klinxþ knlsgn ðxÞjxja; (15.3)

i.e., viscous dissipation with linear and exponential stiffness components. Weseek to find the values of the dissipation and

stiffness coefficients and the unknownexponent. Using RFSM, all of these parameters can be identified simultaneously.Only

the stiffness terms can be determined using the static test. To demonstratethe consistency of the system, the device was

assembled, tested and identified, andthen disassembled three times. The cycles will be referred to as cases one, two,and

three, respectively, and all results are for systems using 0.025 in. diameter steelwires.

15.4.1 Static Test Results

Data collected from static stiffness tests are summarized in Fig. 15.4a for each case, and Fig. 15.4b shows the result of fitting

f(x) ( _x has been omitted here) to the data.In each case the data indicate that the stiffness when the displacement is zero is

negligible; i.e., essentially nonlinear stiffness has been achieved. Table 15.1 summarizes the results of the parameter

identification with and without the linear-stiffness term.The difference in error between the two models is negligible,

indicating that the exponential term alone is adequate for modeling the system.

−0.1 −0.05 0 0.05 0.1−300

−200

−100

0

100

200

300

Displacement (rad)

Tor

que

(N m

m)

case 1case 2case 3

−0.1 −0.05 0 0.05 0.1−300

−200

−100

0

100

200

300

Displacement (rad)

Tor

que

(N m

m)

a b

Fig. 15.4 (a) Summary of data collected during static-stiffness tests; (b) f(x) fit to static-stiffness data in case three


15.4.2 Dynamic Test Results

Figure 15.5 shows one set of data collected using the procedure outlined by for the restoring force surface method. The solid

line represents the portion of the transient response that was considered when estimating the system parameters. Figure 15.6

shows the resulting transient velocity and displacement and Fig. 15.7 shows the measured and reconstructed restoring

torque. This particular data set was collected as part of the identification of NES assembly case two, but is qualitatively

similar to the other data sets collected.

For each of the three cases described in the static stiffness results section, four or five sets of data similar to those shown in

Fig. 15.5 were collected and processed. The system parameters were estimated from the combination of all of the data

simultaneously. This approach was selected over estimating parameters for each trial and then averaging the results for a

number of reasons. First, performing the identification on all trials simultaneously gave more weight to the trials which

collected more useful data. Otherwise, a trial which collected 1 s of useful data would be of the same importance as a trial

that collected 2 s of useful data. More significantly though, it has been shown that the coefficients of the stiffness function

are extremely sensitive to changes in the exponent. This does not allow for a simple linear averaging technique if the

exponent is different for each trial. Thus, considering all of the data sets simultaneously ensures that the best combination of

exponents and coefficients will be determined.

The results of the stiffness parameter estimation are listed in Table 15.2. Once again, the results support the observation

that the linear component of the NES stiffness is insignificant. There was no appreciable reduction in mean squared error

when the coefficient of the linear term, klin, was included in the estimation. Also, when klin was included, it was severalorders of magnitude smaller than the coefficient of the nonlinear term.

Results from RFSM tests for the stiffness in each case were relatively similar (see Fig. 15.8a), indicating that the system

is repeatable. As should be expected, the estimated stiffness was independent of the method used. Figure 15.8b demonstrates

this for case three.The RFSM test also provided an estimate for viscous damping coefficient, although dissipation in the

system is not exclusively viscous. Nonetheless, from case three, c was estimated to be 1.65Nmms/rad. This value is used in

subsequent simulations.


identified stiffness parameters

using the static stiffness method

fkx ¼ knlsgn (x)jxja f k xð Þ ¼ klinxþ knlsgn xð Þ xj ja

Case knl Nm

rada

� �a MSE(%) klin Nmm

rad

� �knl Nm

rada

� �a MSE(%)

1 193 3.10 3.24 2 193 3.10 3.24

2 695 3.77 1.35 195 3081 4.51 1.23

3 486 3.74 2.11 101 962 4.01 2.07

0 0.5 1 1.5 2 2.5 3 3.5 4−5000

0

5000

10000

15000

Time (s)

Acc

eler

atio

n( ra

ds2

)−5

0

5

10

15

20

For

ce (

mV

)Time historyTransient response

a

b

Fig. 15.5 Typical RFSM data

set


1 1.5 2 2.5 3−20

−10

0

10

20

Time (s)

Vel

ocity

( rad s

)1 1.5 2 2.5 3

−0.2

−0.1

0

0.1

0.2

Time (s)

Dis

plac

emen

t(r

ad)

a

b

Fig. 15.6 Example of

experimentally determined

time history: (a) Velocity;

(b) Displacement

1 1.5 2 2.5 3−1000

−500

0

500

1000

Time (s)

Res

tori

ng T

orqu

e (N

mm

)

MeasuredReconstructed

Fig. 15.7 Measured and

reconstructed restoring torque

Table 15.2 RFSM estimated

stiffness parameters fx ¼ knlsgn (x)jxja f xð Þ ¼ klinxþ knlsgn xð Þ xj ja

Case knl Nm

rada

� �a MSE(%) klin Nmm

rad

� �knl Nm

rada

� �a MSE(%)

1 77 2.75 2.32 153 237 3.28 1.94

2 58 2.67 3.93 130 184 3.21 3.52

3 260 3.43 4.17 75 354 3.61 4.06

−0.1 0 0.1

−400

−200

0

200

400

Displacement (rad)

Res

tori

ng T

orqu

e (N

mm

)

Case1Case2Case3

−0.1 0 0.1

−400

−200

0

200

400

Displacement (rad)

Res

tori

ng T

orqu

e (N

mm

)

RFSMSS

a bFig. 15.8 (a) RFSM

estimated stiffness curves;

(b) Static and RFSM

estimated stiffness curves

for case three


15.5 Experimental and Computational Model Validation

The NES was designed to be attached to the tip of a scale-model wing so that its effects could be studied, with the goal of

predicting and observing targeted energy transfer (TET), the one-way transfer of energy from a primary system to a

nonlinear attachment where it is dissipated. The model wing discussed herein is swept with a semispan of 1.35 m and

uniform-thickness aluminum 6061 alloy; it is pictured in Fig. 15.9.A computational model of the wing and NES was

developed using thin-plate finite-elements to which the NES equations of motion were coupled. Details of the computational

model are given by Hubbard [1]. With the NES attached to the wingtip and free to oscillate, the system was excited by a

hammer impact applied to the wing. The excitation and response was recorded at several locations on the wing using

accelerometers. The experimental excitation was then applied to the computational model to verify that was capable of

accurately predicting the nonlinear phenomena. The numerical model can only agree with the experimental results if the

NES model and identified parameters are accurate. We offer one example of many with good agreement between

experimental and numerical results.

The response of the wing with NES attachment was observed for a hammer impact at location “P” in Fig. 15.10a. This

location coincides approximately with the nodal line of the first torsional mode of the wing and the antinode of the second

Fig. 15.9 Photo of the

scale-model wing

0 1 2 3 4 5

0

0.5

1

1.5

Time (ms)

For

ce (

kN)

SimExp

a bFig. 15.10 (a) Locations

of the hammer impact “P” and

the positions of

accelerometers “LT” (leading

tip) and “TT” (trailing tip);

(b) Experimentally measured

force pulse and the

corresponding force pulse

used in simulations


bending mode, thus we expect that the second-bending mode comprises significant component of the response The profile of

the force pulse is shown in Fig. 15.10b, along with the corrected signal that was used in the corresponding simulation.

Figure 15.11a compares the experimentally-measured velocity at “TT” to the simulated velocity, showing very good

agreement.The most notable discrepancy between experiment and simulation is the difference in the fundamental frequency

which is due to error in the finite-element model of the wing. Figure 15.11b shows the frequency content of the simulated

velocity at “TT” as a function of time. It indicates that the energy that was initially in the second bending mode was mostly

dissipated within approximately 5 s. Recall that the excitation was provided near the antinode of the second bending mode

and the wing is lightly damped (aluminum alloy) so that, without an NES, the second bending mode should appear as a more

significant component of the response. Thus, the numerical model, using the identified NES parameters accurately predicted

a strongly nonlinear response.

15.6 Conclusion

A single-degree-of-freedom, rotary nonlinear energy sink has been designed and constructed on the scale of a wind-tunnel

model wing. The essentially nonlinear stiffness and the damping of this device have been identified, showing it to behave as

intended. These properties have been found to be robust, with response curves changing little over successive re-assemblies

of the components. The methods proven here will be used in the development of NESs for aeroelastic stability enhancement.

References

1. Hubbard SA (2009) Targeted energy transfer between a model flexible wing and a nonlinear energy sink: computational and experimental

results. Master’s thesis, University of Illinois at Urbana-Champaign

2. Kerschen G, Golinval JC (2001) Theoretical and experimental identification of a non-linear beam. J Sound Vib 244(4):597–613

3. Kerschen G, Lenaerts V, Golinval JC (2003) VTT benchmark: application of the restoring force surface method. Mech Syst Signal Pr

17(1):189–193

4. Kerschen G, Lenaerts V, Marchesiello S, Fasana A (2001) A frequency versus a time domain identification technique for nonlinear parameters

applied to wire rope isolators. J Dyn Syst Meas Control 123:645–650

5. Masri SF, Caughey TK (1979) Nonparametric identification technique for non-linear dynamic problems. J Appl Mech-T ASME 46(2):433–447

0 1 2 3 4−1

−0.5

0

0.5

1

Time (s)

Vel

.(m

/s)

SimExp

Time (s)

Freq

uenc

y(H

z)

1B

2B

1T

3B

2T4B

0 1 2 3 40

20

40

60

80a bFig. 15.11 (a) Comparison

of experimental and simulated

response at “TT”;

(b) Frequency content

as a function of time in the

simulated response


Chapter 16

Identifying and Computing Nonlinear Normal Modes

A. Cammarano, A. Carrella, L. Renson, and G. Kerschen

Abstract Non linear normal modes offer a rigorous framework, both mathematical and physical, for theoretical and

experimental dynamical analysis. Albeit still in its infancy, the concept of non linear normal modes has the potential of

providing to both the academic and the industrial establishment a powerful tool for the analysis of non linear dynamical

systems. However, in order to exploit the full potential of this theory (and its associated simulation capability), there is need

to integrate it with other branches of non linear structural dynamics: namely, in order for the non linear normal modes of a

real—physical—structure to be computed, there is need to identify and quantify its non linearity. In this paper, an

identification method based on the measured Frequency Response Function (FRF) is employed to identify and quantify

the system’s non linearity before computing the system’s non linear normal modes.

16.1 Introduction

The last decades havewitnessed a continuous demand for structures to becomemore light and efficient without loosing in safety

and durability. This approach, which is well known in the aerospace engineering, is slowly influencing other fields of the

automotive engineer as well as new branches of the civil engineer. In practice, the design process which relies on the theoretical

and numerical modelling of a system and on the experimental observation for the identification and validation of these models,

lacks methods which enables to account for the non linearities which occur in operational regime. In the last century the great

advance in computational science and in numerical methods provided indispensable tools for solving the complex system of

equations needed tomodel this type of structures where large displacements might not meet the hypothesis of linearity. This step

is absolutely crucial for their analysis, but nevertheless not sufficient. Themain problem is the definition of themodel itself. How

is it possible to associate a mathematical model with a given structure so that we are able to describe its dynamics? This science,

known as identification, is the real question that we are not able to answer yet. The typology of non-linearity is generally an

unknown of the problem as well as the parameters which characterize the equations. The identification of this information from

experimental data is not easy and requires, in general some assumptions on the non-linearity [1]. Even in that case, it is not

entirely clear which experiments are more useful for a full parameters’ identification and what data are strictly necessary for the

definition of a suitable mathematical model. This work aims to answer some of the questions still open in the world of non-linear

structures.With this purpose inmind, the authors simulated numerically the behaviour of a non-linear systems with three degree

of freedom. The data generated, i.e., the numerical FRFs are analysed with the method presented in [2], (also referred to as

CONCERTO). The main advantage of this approach is that the structure is simulated numerically and both the type of non-

linearity and the equation parameters are known. In this contest a combination of numerical and simulation procedures is used to

enhance the identification and thus the prediction capability. After a short description of the numerical method used to generate

A. Cammarano (*)

Department of Aerospace Engineering, University of Bristol, Unversity Walk, BS8 1TR, Bristol, UK


A. Carrella



L. Renson • G. Kerschen

Department of Aerospace and Mechanical Engineering, University of Liege, 1, Chemin des Chevreuils, Liege, B-4000, BE



209


the data and themethod behindCONCERTO, some results will be presented. In particular the testwill showhow this can be used

for the identification ofmulti degree of freedom (MDOF) systems , highlighting its advantages and its limitations. Finally, a few

considerations about the method will be presented and some possible directions for future works suggested.

16.2 Continuation of Forced, Periodic Response of Non Linear Systems

The forced response of discrete mechanical systems with n degrees of freedom (DOFs) is considered, assuming that

continuous systems (e.g., beams, shells or plates) have been spatially discretized using the finite element method. The

equations of motion are

M €xðtÞ þ C _xðtÞ þKxðtÞ þ fnl xðtÞ; _xðtÞf g ¼ fðtÞ (16.1)

whereM is the mass matrix; C is the damping matrix;K is the stiffness matrix; x, _x and €x are the displacement, velocity and

acceleration vectors, respectively; fnl is the non linear restoring force vector and f(t) is the external force vector.The numerical method proposed here for the computation of the forced periodic response of non linear systems relies on

the algorithm developed for the computation of non linear normal modes (NNMs), which are periodic responses of the

undamped, unforced system [3]. The NNM algorithm relies on two main techniques, namely a shooting technique and the

pseudo-arclength continuation method and is described below.

16.2.1 Shooting Method

The undamped, unforced equations of motion of system (16.1) can be recast into state space form

_z ¼ gðzÞ (16.2)

where z ¼ x� _x�½ �� is the 2n-dimensional state vector, and star denotes the transpose operation, and

gðzÞ ¼ _x�M�1 Kxþ fnlðx; _xÞ½ �

� �(16.3)

is the vector field. The solution of this dynamical system for initial conditions zð0Þ ¼ z0 ¼ x�0 _x�0� ��

is written as z(t) ¼ z(t, z0)in order to exhibit the dependence on the initial conditions, z(0, z0) ¼ z0. A solution zp(t, zp0) is a periodic solution of the

autonomous system (16.2) if zpðt; zp0Þ ¼ zpðtþ T; zp0Þ, where T is the minimal period.

The computation is carried out by finding the periodic solutions of the governing nonlinear equations of motion (16.2).

In this context, the shooting method is probably the most popular numerical technique. It solves numerically the two-point

boundary-value problem defined by the periodicity condition

Hðzp0; TÞ � zpðT; zp0Þ � zp0 ¼ 0 (16.4)

Hðz0; TÞ ¼ zðT; z0Þ � z0 is called the shooting function and represents the difference between the initial conditions and the

system response at time T. For forced motion, the period T of the response is known a priori.

The shooting method consists in finding, in an iterative way, the initial conditions zp0 and the period T that realize a

periodic motion. To this end, the method relies on direct numerical time integration and on the Newton-Raphson algorithm.

Starting from some assumed initial conditions zp0(0), the motion zp

(0)(t, zp0(0)) at the assumed period T(0) can be obtained by

numerical time integration methods (e.g., Runge-Kutta or Newmark schemes). In general, the initial guess (zp0(0), T(0)) does

not satisfy the periodicity condition (16.4). A Newton-Raphson iteration scheme is therefore to be used to correct an initial

guess and to converge to the actual solution. The corrections Dzp0(k) and DT(k) at iteration k are found by expanding the

nonlinear function

H zðkÞp0 þ DzðkÞp0 ; T

ðkÞ þ DTðkÞ� �

¼ 0 (16.5)

in Taylor series and neglecting higher-order terms (H.O.T.).

210 A. Cammarano et al.

The phase of the periodic solutions is not fixed. If z(t) is a solution of the autonomous system (16.2), then z(t + Dt) isgeometrically the same solution in state space for any Dt. Hence, an additional condition, termed the phase condition, has tobe specified in order to remove the arbitrariness of the initial conditions. This is discussed in detail in [3]. For forced motion,

the phase is fixed by the external forcing.

In summary, an isolated periodic solution is computed by solving the augmented two-point boundary-value problem

defined by

Fðzp0; TÞ �Hðzp0; TÞ ¼ 0

hðzp0Þ ¼ 0

((16.6)

where h(zp0) ¼ 0 is the phase condition.

16.2.2 Continuation of Periodic Solutions

Different methods for numerical continuation have been proposed in the literature. The so-called pseudo-arclength

continuation method is used herein.

Starting from a known solution (zp0, (j), T(j)), the next periodic solution ðzp0;ðjþ1Þ; Tðjþ1ÞÞ on the branch is computed using a

predictor step and a corrector step.

16.2.2.1 Predictor Step

At step j, a prediction ð~zp0;ðjþ1Þ; ~Tðjþ1ÞÞ of the next solution ðzp0;ðjþ1Þ; Tðjþ1ÞÞ is generated along the tangent vector to the

branch at the current point zp0, (j)

~zp0;ðjþ1Þ~Tðjþ1Þ

" #¼

zp0;ðjÞ

TðjÞ

" #þ sðjÞ

pz;ðjÞpT;ðjÞ

" #(16.7)

where s(j) is the predictor stepsize.The tangent vectorp(j) ¼ [pz,(j)∗ pT,(j)]

∗ to the branchdefinedby (16.6) is solutionof the system

@H

@zp0


@H

@T


@h

@zp0

��ðzp0;ðjÞÞ

0

266664

377775

pz;ðjÞpT;ðjÞ

" #¼ 0

0

" #(16.8)

with the condition kpðjÞk ¼ 1. The star denotes the transpose operator. This normalization can be taken into account by fixing

one component of the tangent vector and solving the resulting overdetermined system using the Moore-Penrose matrix

inverse; the tangent vector is then normalized to 1.

16.2.2.2 Corrector Step

The prediction is corrected by a shooting procedure in order to solve (16.6) in which the variations of the initial conditions

and the period are forced to be orthogonal to the predictor step. At iteration k, the corrections

zðkþ1Þp0;ðjþ1Þ ¼ z

ðkÞp0;ðjþ1Þ þ DzðkÞp0;ðjþ1Þ

Tðkþ1Þðjþ1Þ ¼ T

ðkÞðjþ1Þ þ DTðkÞ

ðjþ1Þ (16.9)

16 Identifying and Computing Nonlinear Normal Modes 211

are computed by solving the overdetermined linear system using the Moore-Penrose matrix inverse

@H

@zp0

��ðzðkÞ


@H

@T

��ðzðkÞ


@h

@zp0

��ðzðkÞ

p0;ðjþ1ÞÞ0

p�z;ðjÞ pT;ðjÞ

266666664

377777775

DzðkÞp0;ðjþ1Þ

DTðkÞðjþ1Þ

24

35 ¼

�HðzðkÞp0;ðjþ1Þ; TðkÞðjþ1ÞÞ

�hðzðkÞp0;ðjþ1ÞÞ0

26664

37775 (16.10)

where the prediction is used as initial guess, i.e, zð0Þp0;ðjþ1Þ ¼ ~zp0;ðjþ1Þ and T

ð0Þðjþ1Þ ¼ ~Tðjþ1Þ. The last equation in (16.10)

corresponds to the orthogonality condition for the corrector step.

This iterative process is carried out until convergence is achieved. The convergence test is based on the relative error of

the periodicity condition:

Hðzp0; TÞ

zp0 ¼ zpðT; zp0Þ � zp0

zp0

<e (16.11)

where e is the prescribed relative precision.

The system used for this work consists of three masses (namely M1, M2 and M3) connected between each other and to

the ground through springs and dashpots (as shown in Fig. 16.1). The dashpots and the spring characteristics are linear

excepting for the spring connecting the first mass to the ground, which responds to a movement of M1 with the elastic

force Fe1 given by

Fe1 ¼ Kl x1 þ Knl x31 (16.12)

where x1 is the displacement of the first mass in comparison with its equilibrium position, Kl is the linear stiffness of the

spring and Knl is the coefficient of the non linear term. The forcing term f(t) is a sinusoidal force with amplitude F and is

applied toM1. The system, when the non linear term of Fe1 is set to zero is linear and has three natural frequencies at 0.096,

0.226, and 0.257 Hz. The theory of non linear systems suggests that the position of the peaks in the frequency response is not

influenced by the non-linear term and that for small forcing amplitude the frequency response tends to the linear response. In

the case of non linear behaviour the amplitude of the frequency response function (FRF) has been evaluated considering the

maximum amplitude of time response over a period. It’s phase is computed using the Fourier Transform: the phase of the

applied force is subtracting from the phase of the response. The identification tool uses this information to evaluate the

characteristics of the system.

16.3 Identification Tool: Theory

The method used for the identification procedure is referred tp as CONCERTO and is described in [2], thus only a brief

summary is given here. CONCERTO is based on the assumption that each peak dominates the response in correspondence of

the resonant frequency and therefore the system can be thought as a single-degree-of-freedom (SDOF) system, with

amplitude-dependent damping and/or stiffness—which are the most common classes of non linearity in engineering

F

M1 M2 M3

C1 C2 C3 C4

K1 K2 K3 K4

Fig. 16.1 Schematic of the simulated system: the the arrow on the spring named K1 indicates its non linear stiffness


structures. The equation of motion for a SDOF is the same as (16.1) with the only difference that the coefficient of the

equation are now scalar terms

M €xðtÞ þ C _xðtÞ þ K xðtÞ þ f nl xðtÞ; _xðtÞf g ¼ f ðtÞ (16.13)

Assuming that the system responds at the same frequency as the excitation, the FRF is measured. The forcing term is a

sinusoidal excitation with constant amplitude and variable frequency. For the point of the FRF at any given response

amplitude, x, the functions fnl(x) in (16.13) are in effect constants. This implies that it is possible to linearise the system at

that specific response amplitude so that the systems FRF is given by

HðX;oÞ ¼ 1oo2ðXÞ � o2 þ joo

2ðXÞ�ðXÞ; (16.14)

where oo are the natural frequency and the modal loss factor at that given amplitude. It is important to note that the

linearisation must refer to a given value of amplitude of displacement. The functions oo(X) and Z(X) can be extracted from

the measured real and imaginary part of (16.14) as follows:

ooðXÞ ¼ ðR2 � R1ÞðR2o22 � R1o1

2Þ þ ðI2 � I1ÞðI2o22 � I1o1

2ÞðR2 � R1Þ2 þ ðI2 � I1Þ2

; (16.15)

�ðXÞ ¼ � ðI2 � I1ÞðR2o22 � R1o1

2Þ þ ðR2 � R1ÞðI2o22 � I1o1

2Þoo ðR2 � R1Þ2 þ ðI2 � I1Þ2

h i��

��; (16.16)

where R1 and R2 are the real parts of the FRF at the amplitude X and I1 and I2 its imaginary parts; o1 and o2 are the

frequencies at which a certain amplitude of displacement occurs: by definition they these frequency values are always

before and after the resonance frequency omegao. If the peak, due to non linear effect, is bent over itself so that a given

value of displacement occurs more than two times, the program consider the extreme values of the interval [omega1,omegan] and the method loose in validity. This effect will be discussed in the following section and will be supported by

graphical examples.

16.4 Results and Discussion

In this section some results will be presented. CONCERTO has been used to identify the backbones of the FRF peaks. In the

procedure the FRF relative to five different levels of excitation have been considered: 0.01, 0.03, 0.04, 0.05, and 0.1 N. For

each level of excitation, and each mass, the displacement values around one peak per time have been considered. The

procedure has been repeated for each level of excitation and than for each peak. The results are presented in Figs. 16.2, 16.3,

and 16.4. For each mass the details of the peak and their backbones are presented. The backbones are evealuated both with

the NNM algorithm described in Sect. 16.2 (red dashed lines) and with FRF based method described in Sect. 16.3 (black

solid lines). In the first case the backbones have been evaluated considering the free oscillations of the undamped system. In

the second case the natural frequency has been computed as function of the response amplitude according to (16.15). From

each case, it can be noticed that the curves evaluated with the FRF method are generally very close to those computed with

the simulation procedure. The error tends to be particularly small around the peak of the considered FRF and tend to diverge

for smaller values of displacement. In previous works the FRF method has been applied only to weakly non linear systems

where no unstable branch exists. In this work all the cases present both stable and unstable branches. An extension to this

identification method to FRF with jumps (i.e., unstable branches) is presented in [4]. It is clear from this study (see for

example Fig. 16.2d), that if the unstable branch of the FRF is provided the method works perfectly. On the other hand, the

strong non linearity highlights a weakness of the method. In presence of strongly non linear effects, e.g., if the peak bends on

itself (see for example Figs. 16.3d and 16.4d) there can be more than two frequencies at which a given amplitude is reached.

In this case CONCERTO is not able to follow the peak because it looks for the minimum and maximum frequency at which

the given displacement occurs. The results is that the found backbone heavily diverge from the real one as shown in the same

figures. However, although of great academic interest, this occurrence is rare in practical engineering structures and

therefore can be object of future studies.


16.5 Conclusion

The combination of a continuation procedure with an identification method based on the FRF have shown to be a valid

approach to study practical non linear structures. At this stage the data flow has been unidirectional from the simulation

tools to the identification tool. The final goal of this study is the creation of a more complex tool able to extrapolate the

parameter from experimental data and to iterate the passage between the identification program and the simulation tool to

converge toward a model which can predict the behaviour of the analysed system. There are several issues which have

still to be addressed but the good matching between the NNM method and the FRF based procedure is definitely an

interesting result for such an early stage. The next phase will be focussed also on understanding the interaction between

the the modes to see if it is possible to reduce the error in the identification procedure. One issue to tackle is the definition

of a suitable parameter domain for this study. Currently modal (natural frequency) and spacial (displacement) models are

(mis)used. In fact, although it can be considered generally acceptable, the backbones found by the identification tools

diverge if the displacement values are not close enough to the peak. Moreover there are some cases in which poorer

results have been obtained (see Fig. 16.4c). Nevertheless, unless strong non linear behaviours occur, the trend of the

backbone can be easily identified. The possibility to use weighted regression techniques will be strongly considered in

future works to create a complete backbone from a limited set of data. This information is crucial for the estimation of the

stiffness of the system.

0.1 0.15 0.2 0.25 0.30

0.5

1

1.5

2

2.5

Frequency

Displacement

0.09 0.1 0.11 0.120

0.2

0.4

0.6

0.8

1

Frequency

Displacement

0.21 0.22 0.23 0.24

0.2

0.4

0.6

0.8

1

Frequency

Displacement

0.26 0.28 0.3 0.32

0.5

1

1.5

2

Frequency

Displacement

Fig. 16.2 FRF of the displacement of the massM1 (gray solid lines) for different levels of eccitation and backbones line evaluated with the NNMcode (dashed lines) and with CONCERTO (black silid lines)


0.1 0.15 0.2 0.25 0.3

0.2

0.4

0.6

0.8

1

1.2

1.4

Frequency

Displacement

0.08 0.1 0.12

0.2

0.4

0.6

0.8

1

1.2

1.4

Frequency

Displacement

0.225 0.23 0.2350

0.05

0.1

0.15

0.2

Frequency

Displacement

0.26 0.28 0.3 0.32

0.1

0.2

0.3

0.4

0.5

Frequency

Displacement



References


Mech Syst Signal Pr 20(3):505–592

2. Carrella A, Ewins DJ (2010) Identifying and quantifying structural nonlinearities in engineering applications from measured frequency response

functions. Mech Syst Signal Pr

3. Peeters M, Viguie R, Serandour G, Kerschen G, Golinval JC (2009) Nonlinear normal modes, part ii: toward a practical computation using

numerical continuation techniques. Mech Syst Signal Pr 23(1):195–216

4. Carrella A (2012) Nonlinear identification using a frequency response functions with the jump. In: IMAC XXX, vol 11, Jacksonville, Feb 2012

0.1 0.15 0.2 0.25 0.30

0.5

1

1.5

Frequency

Displacement

0.08 0.1 0.120

0.2

0.4

0.6

0.8

1

Frequency

Displacement

0.215 0.225 0.235

0.2

0.4

0.6

0.8

1

1.2

1.4

Frequency

Displacement

0.26 0.28 0.3 0.320

0.2

0.4

0.6

Frequency

Displacement



Chapter 17

Nonlinear Identification Using a Frequency Response

Function With the Jump

A. Carrella

Abstract Recently, an identification method (referred to as CONCERTO) based on the measured Frequency Response

Function (FRF) has been proposed. Amongst its advantages there are its simplicity and its general applicability using

standard measurement techniques. This makes it particular suitable for use in practical applications and by the wider

engineering community. The method, however, also present some limitations: it applies to weakly-linear structures

(implements the Harmonic Balance method to a first order expansion and assumes that the system is linear at given

amplitude of the response), it is a single-degree-of-freedom (SDOF) method and it fails in the presence of an FRF with the

‘jump’. The aim of this paper is to present the latest findings that show that the nonlinear characteristic can be correctly

identified if the FRF of a system with cubic nonlinearity contains both stable and unstable branches: in order to do so, a

hybrid experimental-analytical approach is proposed.

17.1 Introduction

The majority of dynamical systems in practice behave in a nonlinear fashion. In order to create numerical (e.g. finite

element) models which are faithful of the dynamics of a structure, there is need to measure these nonlinearities. This branch

of engineering, called ‘Identification’, has been very active in producing more and more refined techniques for identification

of dynamic nonlinearities from measurements [1]. Recently, Carrella et al. have proposed a nonlinear identification method

based on the analysis of the measured Frequency Response Function (FRF) [2]. Amongst the advantages of this method there

is its mathematical simplicity, the applicability to standard measurement techniques (it does not require special sensors or

test-set up) and its capability of identifying both stiffness and damping nonlinearities without any a priori knowledge of thestructure. However, this techniques is still in its infancy and presents several shortfalls: is based on a linearisation and

therefore only provides a first degree approximation of the nonlinearity, it is a single-degree-of-freedom (SDOF) method and

hence it assumes that the mode being analysed dominates the response and it fails in the presence of a ‘jump’ in the response.

The aim of this letter is to present some later findings regarding this latter issue. It will be shown that the identification

method proposed in reference [2], also referred to as CONCERTO, is applicable also to the measured FRF of a system with

cubic stiffness excited with an amplitude such as to produce the characteristic ‘jump’. The jump is a well-known

phenomenon which is due to a bifurcation of the system, [3, 4]. At a particular frequency, there are three possible states

that the system can attain: two are stable and one is not. The ‘jump’ marks the passage from one stable solution to the other.

Therefore, in practice the unstable branch cannot be measured. Nonetheless, it is possible to use analytical solutions to

compute the frequencies at which the jump occurs: these expressions for the jump-down and jump-up frequencies have been

proposed in reference [5]. In particular, the jump-down frequency is a function of the damping ratio and the nonlinear

coefficient. It important to point out that, although the hybrid approach proposed in this work is of general validity, it is

limited by the need of having simplified analytical expressions given in [5] and which are based on the two assumptions:

(1) the damping of the system is that of the underlying linear system. In practice, also the energy dissipation mechanism can

be nonlinear. In this case the formulation becomes rather complex and closed form expression become no longer

A. Carrella (*)






217


advantageous if at all available, [6] and numerical methods are more suitable; (2) the measured frequency at which the

system passes from one stable state to the other is considered to be the jump frequency. In reality this is very sensitive to a

number of parameters, but mainly the step-change of the excitation frequency (during stepped-sine measurements) and its

accurate measurement is a rather challenging task [7]. An experimental validation of the proposed method goes beyond the

scope of this paper but it will be object of future studies.

In this article it is shown that CONCERTO can identify and quantify the system’s stiffness if the FRF analysed contains

both stable and unstable branches. In order to identify the system’s amplitude-dependent natural frequency (or backbone),

a hybrid experimental-analytical approach is proposed:

1. With a low-level excitation, i.e. without exciting the nonlinear behaviour, calculate the system’s damping;

2. Increase the level of excitation until the jump occurs;

3. Using the analytical expression for the jump-down, the nonlinear coefficient can be computed;

4. Regenerate, numerically, the FRF of the system for that level of excitation: this will contain both the stable and unstable

branches;

5. Apply CONCERTO to the numerical curve

17.2 Nonlinear Identification from Measured FRF

A detailed description of the method is presented in reference [2], therefore only its key aspects will be given in here. Briefly,

the identification procedure enables one to extract the amplitude dependent modal parameters (natural frequency and

damping ratio) provided that both FRF and excitation force are recorded (so to calculate the displacement). The algorithm

is based on the assumption that the mode analysed dominates the response (i.e. is a SDOF method). If the system has only

one degree of freedom, then the spatial characteristics (stiffness and damping) can also be computed.

The dynamic equation of a mass m suspended on a parallel combination of an amplitude-dependent damping, c(X), andstiffness, k(X), excited by a harmonic force of amplitude F0 at frequency fe, can be expressed by the following relation:

m€xþ cðXÞ _xþ kðXÞx ¼ F0 sinð2p fe tÞ (17.1)

Assuming that the system’s response is at the same frequency of the excitation force, for a given amplitude of the

response X, the frequency response function (for the mode being considered) can be expressed as:

HðoÞ ¼ XðoÞFðoÞ ¼

Ar þ jBr

o2r ðXÞ � o2 þ jo2

r ðXÞ�rðXÞ(17.2)

In particular, at the same level of response amplitude Xi, there correspond a pair of points on the FRF. Thus

H1ðo1Þ ¼ Xi

Fðo1Þ ¼A1r þ jB1r

o2r ðXiÞ � o2

1 þ jo2r ðXiÞ�rðXiÞ ¼ Re1 þ jIm1

H2ðo2Þ ¼ Xi

Fðo2Þ ¼A2r þ jB2r

o2r ðXiÞ � o2

2 þ jo2r ðXiÞ�rðXiÞ ¼ Re2 þ jIm2

8>><>>: (17.3)

By solving this system (with the unknowns Ar;Br;or; �r) for each value of displacement Xi, the natural frequency and

damping ratio as function of the response amplitude can be found as:

o2r ðXÞ ¼

ðR2 � R1ÞðR2o22 � R1o2

1Þ þ ðI2 � I1ÞðI2o22 � I1 o2

1ÞðR2 � R1Þ2 þ ðI2 � I1Þ2

(17.4)

�rðXÞ ¼ðI2 � I1ÞðR2o2

2 � R1o21Þ þ ðR2 � R1ÞðI2o2

2 � I1o21Þ

o2r ½ðR2 � R1Þ2 þ ðI2 � I1Þ2�

�� (17.5)

218 A. Carrella

One of the limitations of this approach is that it is necessary to measure the two points which have the same amplitude of

the response (one before, one after the peak). In the occurrence of a jump (e.g. the jump down which may occur for high level

of excitation in a Duffing oscillator as the frequency is increased), there are no measurable FRF points which corresponds to

those on the stable branch. In the next section, it will be shown that these can be numerically generated making the

identification technique applicable.

17.3 The Jump in the Duffing Oscillator

A classical example of a system that presents a bifurcation that yield the characteristic jump is the Duffing oscillator. This is

ubiquitous in the literature [4]. Consider the system with hardening cubic stiffness expressed by the equation of motion:

m€xþ c _xþ kxþ knlx3 ¼ F0 sin ð2pftÞ (17.6)

There are different techniques to solve (17.6). Numerical solutions can be sought by directly integrating the equation of

motion (e.g. using a Runge–Kutta algorithm); there are also approximate analytical expressions which provide a solution to

(17.6). Amongst these, the Harmonic Balance (HB) method offers a good first order approximation. In reference [5]

approximate analytical expressions for the jump-up and jump-down frequencies have been obtained following the HB

approach. It is noticeable that by applying the HB method it is possible to extract an equivalent stiffness function [3]:

keq ¼ k þ 3

4knlx

2 (17.7)

For the purpose of this paper the Duffing oscillator considered has the following value:

The FRF shown in Fig. 17.1. has been obtained by solving the equation of motion (6) with the built-in Matlab ODE45

solver, and then by computing the ratio of the Fourier coefficients of the response and the excitation. As it can be seen, for

very low levels of excitation, the jump does not occur and the system can be considered as linear (F0 ¼ 0.1 N); by increasing

the force level the nonlinearity is excited but the jump not occurs (F0 ¼ 0.6 N). For these low levels of excitation the jump

does not occur and the identification method, also referred to as CONCERTO, yields reliable results, as plotted with the

dotted and solid lines in Fig. 17.2. Furthermore, the validity of the identified nonlinearity is assessed by superimposing the

curve extracted with CONCERTO with that obtained using the analytical expression in (17.7) (dot-diamond line in

Fig. 17.2).

On the contrary, for a higher level of excitation (F0 ¼ 1 N), as shown by the dashed-circle line in Fig. 17.1, the FRF

presents the characteristic jump. When such a FRF is processed with using the identification algorithm described earlier, it is

obtained the dashed curve in Fig. 17.2 which is clearly not representative of the system being analysed. This is due to the lack

9.4 9.6 9.8 10 10.2 10.4 10.6 10.8 110

1

2

3

4

5

6

7

8

9x 10-3

Frequency [Hz]

|H|

F0=0.1 N

F0=0.6 N

F0=1 N

Fig. 17.1 Simulated

frequency response functions

(FRFs) of a dynamic system

with hardening cubic stiffness

(Duffing oscillator). At high

forcing level the jump occurs

(-o line)

17 Nonlinear Identification Using a Frequency Response Function With the Jump 219

of two physical points in the FRF with equal displacement response amplitude. As a result, the algorithm interpolates the

measured curve creating numerical points which have no physical meaning, hence the error of the identified curve.

In order to find a solution to this problem, and considering that the identification technique implements the HB to a first

order expansion, it’s reasonable to assume that the unstable branch would provide the necessary information to apply

CONCERTO algorithm. Because the unstable branch is not measurable, it is proposed a hybrid experimental-analytical

approach. The analytical basis can be found in reference [5] which provides some simple expressions for the jump-up and

jump-down frequencies. The equation of motion of the system, (17.6), can be re-written in a non-dimensional form as [5]:

€xþ 2z _xþ xþ ax3 ¼ cosðOtÞ (17.8)

where x is the non-dimensionalised displacement, z << 1 is the damping ratio and F0

^and Ω are respectively the non-

dimensional magnitude and frequency ratio of the excitation force, the · operator denotes differentiation with respect to non-

dimensional time t and

z ¼ c

2mono2

n ¼k

ma ¼ knlx

20

kO ¼ o

ont ¼ ont €xþ

_x

o2nx0

_x ¼ _x

o2nx0

x ¼ x

x0x0 ¼ F0

k

The application of the Harmonic Balance leads to the cubic equation in X2 (or a quadratic equation in O2) (17.9) which

represents the relation between the response, the amplitude and frequency of excitation:

9

16a2X6 þ 3

2ð1� O2ÞaX4 þ ½ð1� O2Þ2 þ 4z2O2�X2 ¼ 1 (17.9)

and the phase of the frequency response function:

’ ¼ tan�1 � 2zOX

ð1� O2ÞX þ 34aX

3

!(17.10)

Equations (17.9) and (17.10) are plotted in Fig. 17.3.

17.4 Nonlinear Identification Using Hybrid (Experimental-Analytical) Data

In order to build the approximate analytical FRF (using the HB formulation of (17.9)) there is need of the parameters a and z.For what concerns the latter, the damping is taken to be that of the underlying linear system and as such can be retrieved from a

low-level linear identification test. It is noteworthy that from the low-level linear test, also the linear stiffness can be computed.

0 2 4 6 8

x 10-3

10.05

10.1

10.15

10.2

10.25

10.3

10.35

10.4

10.45

10.5

Displacement [m]

Nat

ural

Fre

quen

cy [H

z]

F0=0.1 N

F0=0.6 N

F0=1 N

theoretical trend

Fig. 17.2 Natural frequency

as function of the

displacement amplitude

identified using CONCERTO

algorithm: when the FRF

contains the jump the

extracted natural frequency

does not compare well with

the analytical curve (-* line)

obtained using (17.7)

220 A. Carrella

On the other hand, the nonlinear coefficient of the stiffnessa can be extracted using a hybrid experimental-analytical approach.

From the experimental FRF with the jump the jump-down frequency can be measured. Using the simplified analytical

expression for the jump-down frequency proposed in reference [5],

Od ffi 1ffiffiffi2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 3a

4z2

svuut (17.11)

it is possible to calculate a as:

a ¼ 16

3O4

d � O2d

� �z2 (17.12)

or

knl ¼163

O4d � O2

d

� �z2k

F0

k

� �2 (17.13)

Having retrieved the nonlinear coefficient of the stiffness, it is possible to generate the analytical FRF of the system using

the approximate solution (9). The analytical FRF calculated using (17.9) with the nonlinear coefficient a identified from the

9 9.5 10 10.5 11 11.5 120

1

2

3

4

5

6

7

8

9x 10-3

Frequency [Hz]

|H|

F0=1 N

F0=2 N

F0=3 N

9 9.5 10 10.5 11 11.5 12-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Frequency [Hz]

φ

F0=1 N

F0=2 N

F0=3 N

a

b

Fig. 17.3 Approximate

analytical FRF, magnitude

(a) and phase (b), of a Duffing

oscillator


hybrid procedure described earlier is shown in Fig. 17.4. The figure also shows that now both stable and unstable branches

are available and approximate fairly accurately the results of the numerical FRF.

For the specific numerical example presented in this paper, given in Table 17.1, the ‘experimental’ (or measured) FRF

have provided the damping, z, and the natural frequency (i.e. the stiffness k) of the linear system (from the low-level FRF,

F0 ¼ 0.1 N) and the jump-down frequency, Od (from the high level FRF, F0 ¼ 1 N). With these values using (17.13), the

nonlinear coefficient has been calculated to be knl1 ¼ 6:85� 106 N/m (note that this value differs from the exact numerical

value given in Table 17.1 by 2%). With these values is now possible to compute the analytical FRF. Finally, CONCERTO

can be applied to the regenerated analytical curve. The result is shown in Fig. 17.5: it can be noticed that now, unlike the

previous results shown in Fig. 17.2, the method is able to identify the correct natural frequency as function of the

displacement as shown by the good match with the analytical curve.

9.4 9.6 9.8 10 10.2 10.4 10.6 10.8 110

1

2

3

4

5

6

7

8

9x 10-3

Frequency [Hz]

|X|

FRFHB

FRFnumerical

Fig. 17.4 Regenerated FRF:

the solid line is the FRF

reconstructed using the

nonlinear coefficient

calculated using the hybrid

approach and (17.12)

Table 17.1 Value of the system’s parameters used for the numerical simulations

Mass Stiffness Nonlinear stiffness Damping ratio Damping coefficient Force amplitude

m ¼ 1:5 kg k ¼ 6000 Nm knl ¼ 7 � 106 N

m3 z ¼ 0:01 c ¼ 2zffiffiffiffiffiffikm

pF0 ¼ 0:1N

F0 ¼ 0:6N

F0 ¼ 1N

0 2 4 6 8

x 10-3

10

10.05

10.1

10.15

10.2

10.25

10.3

10.35

10.4

Displacement [m]

Fre

quen

cy [H

z]

F0=1 N

Theoretical function

Fig. 17.5 Identified nonlinear

characteristic using

CONCERTO (dotted line):

unlike the plot in Fig. 17.2

this compare very well with

the analytical expression

(solid line)

222 A. Carrella

17.5 Conclusion

A recent nonlinear identification method was shown to be valid for nonlinear systems which do not present the jump

phenomenon. In fact, for the technique (also referred to as CONCERTO) to be applied there is need of measuring two points

at the same displacement amplitude at either side of the resonance. However, if in a system with cubic nonlinearity the jump

occurs there are no measurable points on one part of the FRF.

In this paper it has been shown that it is possible to identify the nonlinear characteristic correctly if CONCERTO is

applied to a FRF which contains also the unstable branch. However, because this cannot be measured it has been proposed to

use numerically generated data. In order to do so, simplified expression for the jump-down frequency has been used and a

hybrid experimental-analytical approach has been proposed to retrieve the nonlinear coefficient and produce a nonlinear

numerical FRF (which contains also the unstable branch). Future works will need to focus on validating this approach using

experimental data.

References

1. Kerschen G et al (2006) Past, present and future of nonlinear system identification in structural dynamics. Mech Syst Signal Process

20(3):505–592

2. Carrella A, Ewins DJ (2011) Identifying and quantifying structural nonlinearities in engineering applications from measured frequency response

functions. Mech Syst Signal Process 25(3):1011–1027

3. Worden K, Tomlison GR (2001) Nonlinearity in structural dynamics. Institute of Physics, UK

4. Jordan DW, Smith P (1999) Nonlinear ordinary differential equations, IIIth edn. Oxford Presss, New York

5. Brennan MJ et al (2008) On the jump-up and jump-down frequencies of the Duffing oscillator. J Sound Vib 23(4–5):1250–1261

6. Peeters M (2011) Theoretical and experimental modal analysis of nonlinear vibrating structures using nonlinear normal modes. Ph. D thesis,

University of Liege

7. Ravindra B, Mallik AK (1994) Stability analysis of a non-linearly damped Duffing oscillator. J Sound Vib 171(5):708–716


Chapter 18

Nonlinear Structural Modification and Nonlinear Coupling

Taner KalaycIoglu and H. Nevzat Ozg€uven

Abstract Structural modification methods were proved to be very useful for large structures, especially when modification

is local. Although there may be inherent nonlinearities in a structural system in various forms such as clearances, friction and

cubic stiffness, almost all of the structural modification methods are for linear systems. The method proposed in this work is

a structural modification/coupling method developed previously, and extended to systems with nonlinear modification and

coupling recently. It is based on expressing nonlinear internal force vector in a nonlinear system as a response level

dependent “equivalent stiffness matrix” (the so-called “nonlinearity matrix”) multiplied by the displacement vector, by quasi

linearizing the nonlinearities using Describing Function Method. Once nonlinear internal force vector is expressed as a

matrix multiplication then several structural modification and/or coupling methods can easily be used for nonlinear systems,

provided that an iterative solution procedure is employed and convergence is obtained. In this paper, formulations for each of

the following cases are given: nonlinear modification of a linear structure with and without adding new degrees of freedom,

and elastic coupling of a nonlinear substructure to a main linear structure with linear or nonlinear elements. Case studies for

three of those cases are given and an application of the method to a real life engineering problem is demonstrated.

Keywords Structural modification • Nonlinear structural modification • Vibration of nonlinear structures • Nonlinear

structural coupling

18.1 Introduction

Over the last five decades or so, the finite element (FE) method has established itself as the major tool for the dynamic

response analysis of engineering structures. However, for the dynamic reanalysis of large engineering structures modified

locally, constituting FE model each time is expensive and time consuming, especially when several alternatives are to be

studied. Therefore, it will be more practical to predict the dynamic behavior of a modified structure by using dynamic

response information of the original structure and dynamic properties of the modifying structure. Various structural

modification methods, focusing on the change of dynamic behavior of a structure due to modifications, have been developed

in order to reduce the effort involved in the dynamic reanalysis of such systems. Although the structural modification

methods based on linearity assumption are available in the literature, a review of which can be found in recent papers of

Hang et al. [1–3], these methods cannot be used when there is nonlinearity in the system.

During the past two decades, several structural modification/coupling methods have been suggested taking the nonlinear

effect into account. Watanabe and Sato [4] used first order describing function in order to linearize the nonlinear stiffness

of a beam structure and developed the so-called “Nonlinear Building Block” approach for coupling nonlinear structures

with local nonlinearity. C€omert and Ozg€uven [5] developed a method for calculating the forced response of linear

T. KalaycIoglu

Department of Mechanical Engineering, Middle East Technical University, 06800 Ankara, Turkey

MGEO Division, ASELSAN Inc., 06011 Ankara, Turkey


H.N. Ozg€uven (*)

Department of Mechanical Engineering, Middle East Technical University, 06800 Ankara, Turkey




225



substructures coupled with nonlinear connecting elements. Ferreira and Ewins [6] proposed a new Nonlinear Receptance

Coupling Approach for fundamental harmonic analysis based on describing functions. They suggested an approach that is

capable of coupling structures with local nonlinear elements whose describing functions are available considering just the

fundamental frequency. Then, Ferreira [7] extended the approach and introduced Multi-Harmonic Nonlinear Receptance

Coupling Approach. This approach is able to couple linear and nonlinear structures with different types of joints by

specifying the multi-harmonic describing functions for all nonlinear joints. Chong and Imreg€un [8] suggested an iterative

algorithm for coupling nonlinear systems with linear ones. Maliha et al. [9] coupled a nonlinear dynamic model of a spur

gear pair with linear FE models of shafts carrying them, and with discrete models of bearings and disks. Huang [10] worked

on dynamic analysis of assembled structures with nonlinearity.

In a recent work [11], the authors of this paper proposed a new approach for dynamic reanalysis of a large linear

structure modified locally with a nonlinear substructure. The method suggested is an extension of the method developed by

Ozg€uven [12] for structural modifications of linear systems. In the present study, the proposed approach is further extended

for dynamic reanalysis of linear structures coupled with nonlinear substructures by using linear and nonlinear coupling

elements. In this paper, applications of the method to modification analysis problems where a linear system is modified with

a nonlinear system for the following three cases are presented: structural modification with additional degrees of freedom

(DOFs), structural coupling with linear elements and structural coupling with nonlinear elements. Finally, a real life

structure modeled with FE method is used in order to show the applicability of the method to large ordered systems.

18.2 Theory

The structural modification method proposed by Ozg€uven [12] more than two decades ago can be used for modified linear

systems with or without additional DOFs. The method is for dynamic reanalysis of systems where there is a modification in

the mass, stiffness and/or damping of the system. The frequency response functions (FRFs) of a modified system are

calculated from those of the original system and the dynamic stiffness matrix representing the modifications in the system by

using the following equations [12]:

H�11

� � ¼ �½I� þ ½H11�½Z11��1½H11� (18.1)

H�12

� �T ¼ H�21

� � ¼ ½H21� ½I� � ½Z11� H�11

� �� (18.2)

H�22

� � ¼ ½H22� � ½H21�½Z11� H�12

� �(18.3)

where [H] and [H*] are receptance matrices of the original and modified systems, respectively. [Z] represents the dynamic

stiffness matrix of structural modifications, and superscripts 1 and 2 denote the coordinates on which a modification is

applied and the remaining coordinates, respectively.

The method has been extended in the same work for modifications which require additional DOFs, in other words, for

modifications which also causes coupling of another system to the original one. The resultant equations for such cases are as

given below [12]:

H�ba

� �

H�ca

� �" #

¼ ½I� ½0�½0� ½0�

� �þ ½Hbb� ½0�

½0� ½I��

:½Z�� 1 ½Hba�

½0��

(18.4)

H�bb

� �H�

bc

� �

H�cb

� �H�

cc

� �" #

¼ ½I� ½0�½0� ½0�

� �þ ½Hbb� ½0�

½0� ½I��

:½Z�� 1 ½Hbb�

½0�½0�½I�

� �(18.5)

H�aa

� � ¼ ½Haa� ��½Hab� ½0�

�� ½Z� H�ba

� �

H�ca

� �" #

(18.6)

��H�

ab� H�ac

� �� ¼ �½Hab� ½0�� ½I� � ½Z� H�

bb

� �H�

bc

� �

H�cb

� �H�

cc

� �" #" #

(18.7)

226 T. KalaycIoglu and H.N. Ozg€uven

where the subscript a represents the coordinates that belong to the original system only, the subscript b denotes

connection coordinates, and the subscript c represents coordinates that belong to modifying structure only. The

method is most useful when such modifications are on limited number of coordinates, that is, when modification is

local. Then the order of the matrix to be inverted will be reduced considerably, irrespective of the total size of the

original structure.

The method can be extended to modification of a linear system where modifying system has nonlinearity. Nonlinear

internal forces can be included in the analysis by considering an additional equivalent stiffness matrix in the calculations

which is a function of unknown response amplitudes [11]. Then, the dynamic stiffness matrix for the modifying system

showing nonlinear behavior will take the form

ZðXÞ½ � ¼ ½DK� � o2 ½DM� þ jo ½DC� þ j ½DD� þ DðXÞ½ � (18.8)

where [DK], [DM], [DC] and [DD] represent stiffness, mass, viscous and structural damping matrices of the modifying

structure, respectively, and {X} is the amplitude vector of harmonic response of the system. The formulation for [D(X)],named as “nonlinearity matrix”, was first introduced by Budak and Ozg€uven [13, 14] for certain types of nonlinearities, andlater extended by Tanrikulu et al. [15] for any type of nonlinearity by using Describing Function Method. The elements of

nonlinearity matrix are given [15] as

Dkk ¼Xn

m¼1

nkm (18.9)

Dkk ¼ �nkm; ðk 6¼ mÞ (18.10)

where subscripts k andm represent two engagement coordinates of a nonlinear element. Here, due to the nonlinearity matrix,

[Z(X)] will be response level dependent, and therefore solution can only be obtained by employing an iterative solution

procedure. The details of the above formulation can be found in [11].

18.2.1 Formulation for Nonlinear Structural Modification Without Additional DOFs

For nonlinear structural modifications without additional DOFs, (18.1)–(18.3) are used where dynamic stiffness matrix for

the modifying system is expressed as follows:

Z11 ðXÞ½ � ¼ ½DK11� � o2 ½DM11� þ jo ½DC11� þ j ½DD11� þ D11 ðXÞ½ � (18.11)

Here, the subscript 1 refers to the coordinates of the system where there is a modification. To be able to use (18.1)–(18.3),

renumbering of the coordinates of the original system may be necessary. It should be also noted that although there is

a matrix inversion in the formulation, the size of the matrix to be inverted is equal to the total DOFs of the modifying

system (size of the matrix [Z11(X)]). Therefore, the method is most useful when local modifications are applied to large

ordered systems.

18.2.2 Formulation for Nonlinear Structural Modification with Additional DOFs

When the modification is such that modifying nonlinear system does not only change the system properties of the original

system at some coordinates, but also couples another system to the original system, then the total DOF of the modified

system will be increased. Such a structural modification case is referred to as structural modification with additional

DOFs. In such applications, (18.4)–(18.7) are used for the receptance of modified system. [Z(X)] in these equations will

be as shown in (18.8).

18 Nonlinear Structural Modification and Nonlinear Coupling 227

18.2.3 Formulation for Nonlinear Structural Coupling with Linear Elements

The same formulation given in Sect. 18.2.2 can also be used for analyzing coupled systems by treating the problem as an

equivalent structural modification problem as shown in Fig. 18.1. That is, for each connection node on the original system a

massless node is added to the coupled subsystem.

Firstly, the stiffness matrix of the coupled subsystem, [DK], is expanded. For example, if p number of massless nodes are

added to the coupled subsystem where the DOF per node is q, p � q number of rows and columns are added to the stiffness

matrix of the coupled subsystem. Then, the stiffness values of the linear elastic coupling elements are inserted in proper

locations of added rows and columns of [DK]. The mass, nonlinearity, viscous and structural damping matrices of the

coupled subsystem should also be expanded in the same way. However, just zeros will be inserted in these rows and

columns. Lastly, by defining additional massless nodes as new rigid connection nodes of the coupled subsystem, the problem

can be taken as a nonlinear structural modification problem as defined in Sect. 18.2.2.

18.2.4 Formulation for Nonlinear Structural Coupling with Nonlinear Elements

Again, the same formulation given in Sect. 18.2.2 can be used for analyzing coupled systems by treating the problem as an

equivalent structural modification problem as shown in Fig. 18.2. That is, for each connection node on the original system a

massless node is added to the coupled subsystem as in the previous case. The only difference will be the nonlinear character

of the connection elements. Therefore, again; stiffness, mass, nonlinearity, viscous and structural damping matrices of the

coupled subsystem are expanded by adding p � q number of rows and columns, where p is the number of massless nodes

added to the coupled subsystem and q is the DOF per node.

Fig. 18.1 Structural coupling problem with linear elastic elements

Fig. 18.2 Structural coupling problem with nonlinear elements


However this time, the added rows and columns of the nonlinearity matrix [D(X)] will be filled with proper elements

representing the nonlinear connection elements. If there are linear stiffness counterparts of the connecting elements, these

values will also be properly inserted into the expanded rows and columns of the stiffness matrix of the coupled subsystem.

Lastly, by defining additional massless nodes as new rigid connection nodes of the coupled subsystem, the problem can

be taken as a nonlinear structural modification problem as defined in Sect. 18.2.2.

18.3 Case Studies

In this section applications of the proposedmethod firstly to discrete systems and then to real engineering structuremodeledwith

FEmethod will be given. The first three case studies illustrate modification and coupling analyses of two discrete subsystems in

three main categories, namely, nonlinear structural modification with additional DOFs, nonlinear structural couplingwith linear

elements and nonlinear structural coupling with nonlinear elements. An application for structural modification without

additional DOF is not given in this paper, since it was well investigated in our recent paper [11]. In the last case study, a real

life engineering problem with nonlinear structural modification will be considered.

18.3.1 Nonlinear Structural Modification with Additional DOFs

In this case study, nonlinear modification of a linear discrete system is considered. As can be seen from Fig. 18.3, the

modifying nonlinear system adds a new DOF to the original linear system.

In the same figure, there exists a cubic stiffness element between coordinates 4 and 5 showing hardening behavior.

Parameters of this nonlinear element and the properties of both subsystems are given as follows:

m1 ¼ m2 ¼ m3 ¼ 1 kg and m4 ¼ m5 ¼ 0:5 kg;

k1 ¼ k2 ¼ k3 ¼ k4 ¼ k5 ¼ k6 ¼ 1000 N=m;

nðx; _xÞNL ¼ K0xþ bx3 where K0 ¼ 0 N=m and b ¼ 2x106 N=m3 (18.12)

Structural damping with a loss factor of 0.0015 is assumed in the analysis for all linear elastic elements. Frequency

response of the modified system at the point where a harmonic force of magnitude 4 N is applied is shown in Fig. 18.4.

The results show that the nonlinearity in the modifying system becomes effective in all four modes of the modified system

which reveals the importance of including nonlinearity in this specific case. Although the nonlinearity changes the frequency

responses around resonances considerably by causing a jump, which is a typical response behavior due to cubic stiffness

Fig. 18.3 Structural modification with additional DOFs


element, no convergence problem is observed in the solution when the proposed method is used. Note that, since the

proposed method is an FRF based method, only the FRFs related with the DOFs we are interested and with the connection

DOFs are to be included in the calculation. Moreover, the size of the matrix to be inverted is 2 by 2 in this example, which

is the order of the modifying system (and therefore it would still be 2, even though the size of the original system were

much higher). These are the important features of the method which makes it more advantageous for large ordered systems

with local modification.

18.3.2 Nonlinear Structural Coupling with Linear Elements

As the second case study, nonlinear structural coupling analysis of the same linear and nonlinear discrete system coupled

with a linear elastic element, as shown in Fig. 18.5, is considered.

Here, the stiffness of the linear elastic element is taken as 200 N/m. Assuming structural damping with a loss factor of

0.0015 again for all linear elastic elements, frequency response of the modified system at the point where a harmonic force

of magnitude 4 N is applied, is obtained as shown in Fig. 18.6.

It can be seen from the figure that nonlinearity is more effective on second, third and fifth modes of the system compared

to the other two modes. Note that, since the proposed method is based on FRFs, in addition to the FRFs related

with the connection DOFs, it is sufficient to include only the FRFs related with the required DOFs in the calculations.

1

0

−1

−2

−3

Log(

Dis

plac

emen

t[m])

−4

−50 2 4 6

Frequency [Hz]

RESPONSE vs FREQUENCY

8 10

Non-linear response – Forward sweep

Non-linear response – Reverse sweepLinear response

12 14

Fig. 18.4 Frequency response of m3 after modification

Fig. 18.5 Structural coupling of linear and nonlinear systems with a linear element


Moreover, the size of the matrix to be inverted during calculations is again 2 � 2 in this example (which is the size of

the coupled subsystem). This saves considerable computational time especially in large ordered original systems as long as

the modification is of small order. This feature of the method makes it very desirable in parametric studies, for instance,

while investigating the effects of the linear elastic coupling element stiffness on system response. In Fig. 18.7, the effect

of different stiffness values of the linear elastic coupling element on the system response around third resonance

is examined in detail. It can be seen from the figure that increasing values of the linear elastic coupling element stiffness

will not only shift the third natural frequency to higher frequencies, but also will increase the effect of nonlinearity on

this mode.

1

0

−1

−2

−3

Log(

Dis

plac

emen

t[m])

−4

−5

−60 2 4 6

Frequency [Hz]


8 10



12 14

Fig. 18.6 Frequency response of m3 after coupling

−1

−2

−3

−47.2 7.4 7.6 7.8

Frequency [Hz]

RESPONSE vs FREQUENCYa

c

b


Frequency [Hz]


Frequency [Hz]8.2 8.4 8.68

7.2 7.4 7.6 7.8 8.2 8.48

7.2 7.4 7.6 7.8 8.2 8.4 8.68

−1.5

−2.5

Log(

Dis

plac

emen

t[m]}

Log(

Dis

plac

emen

t[m]}

Log(

Dis

plac

emen

t[m]}

−3.5

−4

−1

−2

−3

−0.5

−1.5

−2.5

−3.5

−4

−1

−2

−3

−0.5

−1.5

−2.5

−3.5



Non-linear response – Forward sweepNon-linear response – Reverse sweepLinear response


Fig. 18.7 Frequency responses around third resonance for different linear elastic coupling elements (a) kLC ¼ 200 N/m, (b) kLC ¼ 400 N/m,

(c) kLC ¼ 600 N/m


18.3.3 Nonlinear Structural Coupling with Nonlinear Elements

The third case study discusses nonlinear structural coupling analysis of the same linear and nonlinear discrete systems

with nonlinear elements. The linear coupling element used in the previous case study is kept the same, but an additional

nonlinear coupling element is considered as shown in Fig. 18.8.

As the nonlinear coupling element, a linear spring having a stiffness of magnitude 200 N/m and 0.02 m clearance on each

side is inserted between two coupling coordinates. Assuming structural damping with a loss factor of 0.0015 again for all

elastic elements, frequency response of the modified system at the point where the harmonic force of magnitude 4 N is

applied is obtained as shown in Fig. 18.9.

When Figs. 18.6 and 18.9 are compared with each other, it can be observed that nonlinear coupling element affects

first and third modes of the system more than it does the other modes. Again, the advantage of using only the FRFs related

with the required and connection DOFs, which is the FRF related with the third mass in this example, and inverting

a matrix only in the size equal to the DOF of the modifying system in this method, the method can be used in design

analyses where, for instance, the effects of using different nonlinear coupling elements on the system response are

investigated.

In Fig. 18.10, the effect of different stiffness values of nonlinear coupling element on the system response around third

resonance is examined in detail. It can be seen from Fig. 18.10 that typical response distortion due to clearance type of

nonlinearity is observed as an abrupt change in the frequency response at the point of transition where the response

Fig. 18.8 Structural coupling of linear and nonlinear systems with a linear and a nonlinear element

1

0

−1

−2

−3

Log(

Dis

plac

emen

t[m])

−4

−5

−60 2 4 6

Frequency [Hz]


8 10



12 14

Fig. 18.9 Frequency response of m3 after coupling with a linear and a nonlinear element


amplitude reaches to the value of clearance. As an expected result, the displacement value where this abrupt change

occurs differs depending on the value of the clearance (compare Fig. 18.10a and b). On the other hand, for nonlinear

spring elements having different stiffness values but the same clearance, this abrupt change occurs at the same

displacement value but the frequency responses after that point show different behaviors due to different additional

linear spring stiffnesses of the nonlinear coupling elements after the response amplitude reaches to the value of clearance

(compare Fig. 18.10a and c).

18.3.4 Nonlinear Structural Modification: A Real Life Engineering Problem

As the last case study, a real life engineering problem, a shaft and mirror plate assembly usually used in land platforms for

optical purposes, is considered (Fig. 18.11).

The mirror plate is a costly part due to its well machined reflective surface, so once it is designed it is not desired to

be modified further in the design optimization of the assembly depending on the vibration characteristics of the platform it

is mounted. Therefore, when it is used in a platform, it may be necessary to modify the shaft and/or bearings, in order to

minimize the vibration of the mirror plate so that its reflection performance is increased. In order to make a more precis

analysis, nonlinearity introduced by the bearings is included in the dynamic analysis, which can easily be handled by the

nonlinear structural modification analysis method suggested here.

Solid elements are used in the FE model of the mirror plate with three DOFs per node yielding 2,655 total DOFs

(Fig. 18.12). The shaft is also modeled by using solid elements with three DOFs per node resulting in 186 total DOFs. The

FE model of the shaft is also shown in the same figure.

Material properties of the mirror plate made of an aluminum alloy and of the shaft made of a structural steel are given in

Table 18.1.

−1

−2

−3

7 7.5Frequency [Hz]

RESPONSE vs FREQUENCYa

c

b


Frequency [Hz]


Frequency [Hz]8.58 7 7.5 8.58

7 7.5 8.58

−1.5

−2.5

Log(

Dis

plac

emen

t[m]}

Log(

Dis

plac

emen

t[m]}

Log(

Dis

plac

emen

t[m]}

−3.5

−1

−2

−3

−1.5

−2.5

−3.5

−1

−2

−3

−1.5

−2.5

−3.5

−4




Fig. 18.10 Frequency responses around third resonance for different nonlinear coupling elements (a) kNLC ¼ 200 N/m, d ¼ 0.02 m;

(b) kNLC ¼ 200 N/m, d ¼ 0.04 m; (c) kNLC ¼ 400 N/m, d ¼ 0.02 m


Here, mirror plate is taken as the original structure since it is not desired to be changed during the design phase of the

assembly. Shaft and bearing assembly on the other hand is taken as the nonlinear modifying structure where ball bearings at

the two ends of the shaft, shown in Fig. 18.11, are modeled as grounded nonlinear springs in horizontal and vertical

directions. The nonlinear behavior of the ball bearings can be taken to be cubic in nature [16]. The nonlinear parameters of

the ball bearings are taken as follows:

nðx; _xÞNL ¼ K0xþ bx3 where K0 ¼ 2x102 N=m and b ¼ 5x107 N=m3 (18.12)

YZ X

1 Elements

YZ

X

38

8 56

72

541

554

642

45

4624

23

22

21

20

3

48

56

52

54

44

47

49

51

53

43

57

16

15

14

13

1

9

10

17

a bAUG 3 2011

16:55:04

1 Elements

AUG 3 201116:50:18

Fig. 18.12 The FE model of (a) the mirror plate and (b) the shaft

Table 18.1 Material properties

of the mirror plate and the shaftMirror plate Shaft

Young’s modulus 71 GPa 200 GPa

Poisson’s ratio 0.33 0.3

Density 2,770 kg/m3 7,850 kg/m3

Fig. 18.11 The shaft and mirror plate assembly


In the analysis, firstly the receptances of the mirror plate are calculated for connection points and for any other point we

might be interested in (i.e., points of which response is required or a force is applied to, by using the standard modal

analysis). Then, the structural modification method is employed and the receptances of the required points on the modified

structure are calculated. As the response of corner points on the mirror have the major importance since they are more prone

to effect the reflection performance, the FRF related with a point near one of the corners of the mirror plate is calculated

when a harmonic force of magnitude 2 N is applied to the same point. The calculated frequency response is shown in

Fig. 18.13 with the linear FRF of the assembly without considering bearing nonlinearity. Using structural modification

method, it is very easy and fast to recalculate the response for any design change in the shaft and/or bearings.

Furthermore, the FRF values for the resulting nonlinear system will be a function of the amplitude of the applied

harmonic force, which will require the recalculation of the FRFs for each forcing amplitude level even though nothing

is changed in either of the subsystems. In such analyses the method suggested here, again, provides drastic computational

advantage, since the FRFs of the linear part of the structure (which is usually the major part of the system) are calculated

once and then used to find the FRFs for the nonlinear overall system. In this later phase of the computations, which

requires iterative solution, only the FRFs related with the points we are interested in (in addition to those of the modifying

structure) are used, rather than all DOFs (which would be the case if the coupled nonlinear system were to be analyzed with

standard approaches). In this case study, FRF related with a point near one of the corners of the mirror plate is calculated

at two more different forcing levels. The results are shown in Figs. 18.14 and 18.15. The effect of forcing level on the FRF

related with a point near one of the corners of the mirror plate after modification can easily be observed by comparing

Figs. 18.13, 18.14 and 18.15.

Fig. 18.13 The direct point FRF related with a point near one of the corners of the mirror plate for F ¼ 2 N



18.4 Discussion and Conclusions

The noble structural modification method originally developed for linear systems [12] almost two decades ago was recently

extended for dynamic reanalysis of linear structures modified locally with a nonlinear substructure [11]. In this paper, the

same approach is further applied for dynamic reanalysis of linear structures coupled with nonlinear substructures by using

linear and/or nonlinear coupling elements. In this approach, the frequency responses of the modified structure are calculated

from those of the original structure and the system matrices of the modifying nonlinear structure (which can be in the form of

a coupled nonlinear substructure). The formulation used in this approach is given for each of the four nonlinear modification

or coupling cases investigated. Case studies for three of those cases are presented in this paper. Finally, an application of the

method to a real life engineering problem is demonstrated.

The method is based on the computation of the FRFs of a modified system from those of the original system and the

dynamic stiffness matrix representing themodifications in the system. Due to the nonlinear behavior of the modifying system,

the dynamic stiffness matrix turns out to be response level dependant and therefore the solution requires an iterative approach.

The formulation is for rigid connection of the nodes of the original and modifying systems. For the cases where a

nonlinear subsystem is coupled to a linear system with elastic elements (linear or nonlinear), the problem is treated as an

equivalent structural modification problem where to each free end of a connecting elastic element a massless node is added

and that node is rigidly coupled to the main system. With numerical case studies, applications of the method are

demonstrated. Firstly, a discrete linear system modified with a discrete nonlinear system is considered. The same original

system with different types of modification is used in the first three case studies. The effects of modifications and/or coupling

elements are demonstrated. The iterative numerical solution was found to be successful as far as convergence to a solution is

concerned. It should be noted that since the modified system is a nonlinear system, the calculated FRFs are valid only for the

level of the force applied, and different FRFs are obtained when the amplitude of the external harmonic force is changed.

As the last case study, a real life engineering problem is considered in order to show the applicability of the method to real

structural systems modeled with FE method. In this problem, structural modification analysis of a mirror plate modified with

a shaft-bearing assembly, where bearings at the two ends of the shaft are taken as hardening stiffnesses, is studied.

It should be noted that since the proposed method is an FRF based method, only the FRFs of the original system related

with the DOFs we are interested in, in addition to the ones at the connection DOFs are to be included in the calculations.

Although the formulation includes a matrix inversion, the size of the matrix to be inverted is equal to the DOF of the

modifying system, and therefore the method is most advantages when the modification is local. Especially in the design of

large main structures which may need to be modified locally, the method is very useful and makes it possible for the designer

to try various possible design changes or to make a parametric study with minimum computational cost.

References

1. Hang H, Shankar K, Lai JCS (2008) Prediction of the effects on dynamic response due to distributed structural modification with additional

degrees of freedom. Mech Syst Signal Process 22:1809–1825

2. Hang H, Shankar K, Lai JCS (2009) Effects of distributed structural dynamic modification with reduced degrees of freedom. Mech Syst Signal

Process 23:2154–2177



3. Hang H, Shankar K, Lai JCS (2010) Effects of distributed structural dynamic modification with additional degrees of freedom on 3D structure.

Mech Syst Signal Process 24:1349–1368

4. Watanabe K, Sato H (1988) A modal analysis approach to nonlinear multi-degrees-of-freedom system. ASME J Vibrat Stress Reliabil Des

110:410–411

5. C€omert MD, Ozg€uven HN (1995) A method for forced harmonic response of substructures coupled by nonlinear elements. In: Proceedings of

the 13th international modal analysis conference, Nashville, pp 139–145, Feb 1995

6. Ferreira JV, Ewins DJ (1996) Nonlinear receptance coupling approach based on describing functions. In: Proceedings of 14th international

modal analysis conference, Dearborn, pp 1034–1040, 12–15 Feb 1996

7. Ferreira JV (1998) Dynamic response analysis of structures with non-linear components. Ph.D. thesis, Imperial College London, Department

of Mechanical Engineering, Dynamics Section

8. Chong YH, Imreg€un M (2000) Coupling of non-linear substructures using variable modal parameters. Mech Syst Signal Process 14

(5):731–746

9. Maliha R, Dogruer CU, Ozg€uven HN (2004) Nonlinear dynamic modeling of gear-shaft-disk-bearing systems using finite elements and

describing functions. J Mech Des 126(3):534–541

10. Huang S (2007) Dynamic analysis of assembled structures with nonlinearity. Ph.D. thesis, Imperial College London, University of London

11. Kalaycioglu T, Ozg€uven HN (2011) Harmonic response of large engineering structures with nonlinear modifications. In: Proceedings of the

8th international conference on structural dynamics, EURODYN 2011, Leuven, pp 3623–3629, 4–6 July 2011

12. Ozg€uven HN (1990) Structural modifications using frequency response functions. Mech Syst Signal Process 4(1):53–63

13. Budak E, Ozg€uven HN (1990) A method for harmonic responses of structures with symmetrical nonlinearities. In: Proceedings of the 15th

international seminar on modal analysis and structural dynamics, Leuven, vol 2, pp 901–915, 17–21 Sept 1990

14. Budak E, Ozg€uven HN (1993) Iterative receptance method for determining harmonic response of structures with symmetrical non-linearities.


15. Tanrikulu O, Kuran B, Ozg€uven HN, Imreg€unM (1993) Forced harmonic response analysis of non-linear structures using describing functions.

AIAA J 31(7):1313–1320

16. Tiwari R (2000) On-line identification and estimation of non-linear stiffness parameters of bearings. J Sound Vib 235(5):906–910


Chapter 19

Nonlinear Dynamic Response of Two Bodies Across

an Intermittent Contact

Christopher Watson and Douglas Adams

Abstract An Improvised Explosive Device models as an outer buffer layer and an inner layer of surrogate material. It is

hypothesized that the area in contact between these layers and pressure across this area determines in large part the

transmission of vibration energy from the outer layer to the inner layer. Even if these layers of material display linear

elastic behavior separately, the combined vibration properties of the coupled bodies will exhibit nonlinear elastic behavior.

To study these interactions, a two-dimensional numerical finite element plate model is developed of the buffer layer and the

interface is represented as a random array of transverse springs simulating the stiffness of the inner layer beneath the buffer

layer. Natural frequencies and corresponding modal deflection shapes are calculated and shown to change depending on the

nodes connected by these springs. Modal experiments are performed with a polycarbonate sheet resting on a polymeric plate.

Experiments for low and high amplitude impulsive forcing functions demonstrate nonlinear vibration behavior in the two

coupled bodies and that this nonlinear behavior is a function of the location of the forcing function as well as the location of

the sensor that is used to measure the outer layer’s response.

19.1 Introduction

Sixty to seventy percent of deaths of coalition forces in Iraq and Afghanistan have been attributed to roadside Improvised

Explosive Devices (IEDs). In particular, as the Afghanistan campaign has intensified in the last few years, total IED

incidents have increased from about 200 per month to over 1,000 per month as of April 2010.1 Current methods of detection,

including infrared imaging of roadside areas beneath which IEDs are placed, are effective to some degree but more

sensitivity in these measurements is desired to reduce the rate of false positives and negative detections. The goal of this

research is to develop a fundamental understanding of how vibrations pass through multi-body solid objects and to explore

the use of acoustic excitations to cause IEDs to vibrate and/or emit energy, and be subsequently detected using other

measurements like infrared imaging. One facet of this research focuses on developing a modeling capability for describing

the dynamic interactions between two or more structural components undergoing vibratory response. Such a vibratory

response may cause intermittent contact between these bodies leading to complex nonlinear dynamic behavior.An improvised explosive device like the one pictured in Fig. 19.1a is conceptualized as consisting of an outer buffer

(inert) layer and an inner target (energetic) material among other components. The material properties and geometry of the

outer layer determine in part the reception of acoustic energy into the outer layer of the device. The amount of area in contact

between the outer layer and inner material along with the pressure between these two materials also determines in part the

transmission of energy in the form of vibrations through the interface. For example, the hypothetical device shown in

Fig. 19.1a has a metallic material only partially in contact with an inner target material. Because of this intermittent contact

between the two bodies, the combined dynamic properties of these bodies will exhibit both linear and nonlinear changes:

(1) changes in the linear modal vibration response of the two bodies will occur as the contact area between the two bodies

changes causing the boundary conditions on each body to fluctuate and (2) changes in the forced vibration response of the

two bodies will occur as the two bodies interact at the interface.

C. Watson (*) • D. Adams

Purdue University, 1500 Kepner Drive, Lafayette, IN 47905, USA


1 Center for Strategic and International Studies, IED Metrics for Afghanistan January 2004–May 2010.



239


The nature of the interactions between the outer layer and inner (target) material is important from a standpoint of

detection because these interactions dictate how readily the target material can be stimulated and its response observed based

on surface measurements on the outer layer. Figure 19.1b illustrates a conceptualized view of the transmission path for

vibratory energy. A shield is included for generality in this figure but is not considered in the present analysis.

19.2 Methods and Procedures

In order to study the interactions between the buffer layer and the inner material, a two-dimensional numerical finite element

model is developed for each layer as well as for the interface between these layers. First, a plane stress finite element model

of an upper plate with three degrees of freedom per node was developed. This upper plate represents the buffer layer and was

assumed to be thinner than the target material. The material properties for the buffer layer were chosen to correspond to those

of polycarbonate (Young’s Modulus: 2.6 GPa; Bulk Modulus: 2.3 GPa; Poisson’s Ratio: 0.37). The dimensions of this plate

were set as 254 mm long by 178 mm wide by 4.8 mm thick. In this initial study, the inner target material was modeled using

an elastic foundation that exists under portions of the buffer layer. The target material was designed to mimic the mechanical

and molecular properties of common energetic materials without being energetic itself. A binder material of hydroxyl-

terminated polybutadiene (htpb) was impregnated with inert Ammonium Chloride (NH4Cl) crystals, which replace the

otherwise energetic components typical used (see Fig. 19.2).

The interface between the outer plate and the inner target was observed to contain gaps and voids due to the nature of the

surrogate material (see Fig. 19.3), so this interface is represented in the model as a random array of transverse springs, which

restrain the transverse vibrations of the plate (see Fig. 19.4). These springs were used to simulate the stiffness supplied by the

inner (target) material that is beneath the buffer layer. The natural frequencies and corresponding modal deflection shapes

were calculated and were observed to change depending on the nodes that were connected by these springs. The possible

modal properties for each mode of vibration of the body were calculated by randomly connecting nodes and then solving the

corresponding eigenvalue problems. The resulting set of modal properties defined the bounds of the natural frequencies and

modal deflection shapes for the coupled vibrating system. Modal experiments were performed with a polycarbonate sheet

resting on a surrogate plate to motivate the model approach.Modal testing with a modal impact hammer was performed in order to identify the modal frequencies and modal

deflection shapes of the polycarbonate outer plate and the combination of the polycarbonate plate sitting on the inner target

material. To perform these tests, the plates were subdivided to contain 70 nodes. Two accelerometers were attached to the

plate, and each node was excited with the modal impact hammer in order to gather data (Fig. 19.5).

The effect of dynamic coupling on the two bodies was investigated by placing the polycarbonate plate on top of the target

material. In order to investigate potential nonlinearities involved when coupling the outer plate and the inner target material,

modal tests were performed with two different magnitudes of force input. If the system was linear, it was expected that no

change in the magnitude or phase of the measured frequency response functions or modal properties would be observed

(Fig. 19.6).

The numerical model consisted of a finite element model of the polycarbonate outer plate. A random array of transverse

springs was then attached to the nodes of this model and then the other ends of the springs were attached to ground. The

number of randomly arrayed springs was adjusted to represent differing levels of contact between the outer plate and the

inner target material. The stiffness of the springs can also be adjusted in the model. The intent is to also introduce Rayleigh

damping, at some point, and be able to use experimental results to help tune the model.

Fig. 19.1 (a) Inert improvised explosive device. (b) Conceptual layering of materials for use in developing intermittent contact model to study

transmission of acoustic energy from one body into the other

240 C. Watson and D. Adams

Fig. 19.3 (a) Polycarbonate outer layer resting on surrogate target layer and (b) close up of edge indicating some gaps and voids between these

two layers of material

Fig. 19.4 Outer (e.g., buffer) layer ofmaterial couples to inner (e.g., target) layer ofmaterial thatwill be simulatedusing a randomized elastic foundation

Fig. 19.2 (a) Polycarbonate (buffer) layer and (b) surrogate (target) layer of materials

19 Nonlinear Dynamic Response of Two Bodies Across an Intermittent Contact 241

19.3 Results

Numerical modeling is still underway. Figure 19.7 shows the simulated natural frequencies for an undamped polycarbonate

plate and for an undamped plate with transverse springs running to ground. Figure 19.7 also shows the first four mode shapes

for the simulated polycarbonate plate.

Modal testing of the polycarbonate outer layer resting on the surrogate target layer at two different levels of excitation

yielded the frequency response functions shown below in Fig. 19.8.

19.4 Conclusions

At this point we have determined, in a qualitative sense, that the combined dynamic properties achieved when coupling the

two bodies together do have a nonlinear component to them. As seen in Fig. 19.8, the amplitude of the frequency response

functions does change with a change in the excitation force.

Fig. 19.6 Polycarbonate

outer layer resting on

surrogate target layer

undergoing modal impact

testing

Fig. 19.5 Polycarbonate outer layer undergoing modal impact testing


It should be noted that nodes 1 and 5 are at locations where there are gaps and voids, whereas nodes 6 and 11 are areas of

more substantial contact. Regardless of the level of excitation, the response seems to be quite different between these

two groups.

Continued development of the numerical model is proceeding to predict a range of modal frequencies that can account for

the random nature of the contact interface between the outer and inner layers.

Fig. 19.7 Frequency vs. mode number plots for polycarbonate only layer model and polycarbonate on random elastic foundation indicating a

change in the resulting natural frequencies as a result of the foundation provided the surrogate layer of material

19 Nonlinear Dynamic Response of Two Bodies Across an Intermittent Contact 243

101

101

100100

H(jw) vs FrequencyNode 1


Low ImpactHigh Impact



Frequency (Hz) Frequency (Hz)



101

100

100

h(jw

)h(

jw)

h(jw

)H

(jw)

10−1

10−1

10−1

10−2

10−2

100 200 300 400 500Frequency (Hz) Frequency (Hz)

600 700 800 900 1000

100 200 300 400 500 600 700 800 900 1000 100 200 300 400 500 600 700 800 900 1000

100 200 300 400 500 600 700 800 900 1000

Fig. 19.8 Frequency response functions for polycarbonate outer layer resting on surrogate target layer at two different levels of excitation at four

different locations


Chapter 20

Application of Continuation Methods to Nonlinear

Post-buckled Structures

T.C. Lyman, L.N. Virgin, and R.B. Davis

Abstract Continuation and path following methods have been applied to many nonlinear problems in mathematics and

physics. There is less widespread application of these methods, however, to structural systems. Since structural buckling and

stability problems are primarily concerned with system behavior as a control parameter (most often the load) varies, they are

particularly well suited for continuation methods and bifurcation analysis. In this work, the continuation package AUTO is

utilized to calculate post-buckled configurations, natural frequencies, and mode shapes of flat plates. Additionally, the

continuation analysis identifies bifurcation points and is also adapted to plate configurations that include slight initial

imperfections. Finally, the path following methods are also applied to track the unstable snap-through solution and natural

frequencies of post-buckled plates subject to a transverse load.

Keywords Buckling • Post-buckling • Continuation • Nonlinear • Snap through

20.1 Introduction

In modern engineering, there is a considerable interest in predicting the behavior of post-buckled structures. With current

lightweight aerospace and high performance applications, structural elements frequently operate beyond their buckled load.

This is especially true of plates, which are capable ofmaintaining stability at loads several times their critical buckling load [1].

Previous studies [2, 3] have investigated the post-buckled natural frequencies of plates by first linearizing the von Karman

plate equations about equilibrium points, and then solving the linear perturbation vibration problem. This method is

effective, although requires analytically determining the linearized equations of motion describing the behavior of small

perturbations around an equilibrium point.

Alternatively, given the equations of motion, natural frequencies and stability can be extracted from a numerical

linearization, taken at any given equilibrium point for the system. Continuation methods, specifically, take linearizations

at each step to aid in finding equilibrium solutions and stability information; however, this linearization can also be used to

find natural frequencies and mode shapes for dynamic mechanical systems. Continuation methods have been the basis of

study for a variety of dynamical systems in mathematics and physics [4], but to the extent of the authors’ knowledge have

been applied to problems in mechanics in a very limited sense [5].

The work presented herein introduces the basic steps and procedure behind continuation methods. Then a continuation

package, AUTO, is used to solve a Galerkin approximation applied to the von Karman plate equations. Both pre- and post-

buckled static and modal behavior is extracted. Finally, the methods are also applied to briefly look into the snap-through of

a transversely loaded, buckled, square plate.

T.C. Lyman (*)

Department of Mechanical Engineering and Materials Science, Duke University, Box 90300, Durham, NC 27708, USA


L.N. Virgin

Department of Mechanical Engineering and Materials Science, Duke University, Box 90300, Durham, NC 27708, USA

R.B. Davis

Aerospace Engineer, NASA Marshall Space Flight Center, Mail Code ER41, Huntsville, AL 35812, USA



245


20.2 Continuation Methods

Numerical continuation is a tool most often used to study nonlinear sets of equations, and is normally based on predictor-

corrector type methods. The main advantage of continuation methods is they handle solutions of nonlinear systems better

than conventional Newton methods, especially when turning points are encountered. Seydel [6, 7] lists that the predictor-

corrector continuation methods consist of four main components:

1. Predictor step,

2. Choice of parametrization,

3. Corrector iteration,

4. Step length control.

Figure 20.1 graphically shows the procedure of the predictor step. The predictor utilizes a tangent of the system at the

current solution point, x1, by numerically calculating a Jacobian of the system, and then takes a step towards the new

solution. The ‘predicted’ new point is illustrated in Fig. 20.1 by point x2P.

The next step in the continuation procedure is to choose a parameterization. The parameterization decides which set of

variables the corrector iteration will operate on to converge to a new solution. The parameterization can be done in many

different manners, several of which are shown in Fig. 20.2. It is implemented by adding a constraint equation to the system

of equations that will hold the increment of a certain parameter, or set of parameters, equal to a constant value. Traditionally

parameterizations based on holding the control parameter increment, Dl, constant are chosen. This choice encounters

problems if turning points are reached, in which case the system will have a vertical tangent and the corrector iteration will

not be able to successfully iterate to a solution. Another option is to parameterize the system via some combination of system

parameters, Dx, called a local parameterization. Although more robust, turning points can still arise and cause difficulty for

the corrector iterations. Finally, the most commonly used approach is to use an arc-length parameterization, where the

incremental arc-length of the overall solution from step to step is set and controlled. This method alleviates the problem of

navigating around turning points.

After the parameterization is chosen, a corrector iteration, usually Newton’s method, is applied to the system starting at

the predicted state, x2P, to find the solution at the next step, x2. Given the specific parameterization, it is likely that the Newton

iteration steps will correct in both the solution parameters, x, and the control parameter, l. Figure 20.1 shows corrector

iterations solely in x from point x2P to point x2, and another parameterization that corrects in both x and l parameters from

point x2P to point x2

0.The final step includes adjusting the step size that is taken in the predictor step. Using some type of metric, usually a check

on the number of Newton iterations needed for convergence, the step size of the next predictor is adjusted accordingly.

Fig. 20.1 Predictor step of

continuation method

246 T.C. Lyman et al.

Continuation methods are frequently researched in the mathematics and physics communities and several software

packages are freely available and maintained for solving problems in this manner. For the work presented here, the AUTO

continuation package, maintained by Eusebius Doedel of Concordia University and his collaborators, is used [8].

20.3 Application to Dynamic Mechanical Systems

Continuation methods are particularly well suited to solving dynamic systems with a changing control parameter, such as a

structure with an applied load. Second order mechanical systems can be written as a set of first order state space equations,

_xðtÞ ¼ fðx; lÞ; (20.1)

where x is a vector of state variables, f is a vector of linear or nonlinear equations and l is a control parameter. Continuation

methods not only solve the set of equations given by (20.1), but also calculate a Jacobian matrix for the system at each

equilibrium point in order to assist in finding the next incremented point. For dynamic systems, the eigenvalues of the

Jacobian will indicate the stability of the system [9]. Additionally, for dynamic mechanical systems, the eigenvalues will

actually take on a physical meaning and relate to the natural frequencies of the system while the eigenvectors will represent

the small perturbation vibration modes [10].

20.4 Plate Results

For finite deflections of a plate, the von Karman plate equations are used [11]. The von Karman plate equations are given in

dimensional form as [5]

Dr4ðw� w0Þ þ m@2w

@t2þ C

@w

@tþ Dp ¼ Fyywxx þ Fxxwyy � 2Fxywxy; (20.2)

and

r4F ¼ EhððwxyÞ2 � ðw0xyÞ2 � wxxwyy þ w0xxw0yyÞ: (20.3)

a b c

Fig. 20.2 Different types of variable parameterizations used in continuation methods, after [7]. (a) Parameterization of control parameter.

(b) Local parameterization. (c) Arc length parameterization

20 Application of Continuation Methods to Nonlinear Post-buckled Structures 247

The following non-dimensionalizations, also from [5], are used:

x ¼ a~x y ¼ a~y

w ¼ffiffiffiffiffiffi

D

Eh

r

~w F ¼ D ~F

Px ¼ D

a2~Px Py ¼ D

a2~Py

Pxy ¼ D

a2~Pxy m ¼ D

a4~m

C ¼ffiffiffiffiffiffiffi

Dmp

a2~C Dp ¼ D

a4

ffiffiffiffiffiffi

D

Eh

r

dp

t ¼ffiffiffiffiffiffiffiffi

a4m

D

r

t r ¼ b

a; (20.4)

where D ¼ Eh4=12ð1� n2Þ is the plate stiffness, m is the density per unit area, C is a damping coefficient, Dp is a pressure

applied to the plate, w is the out-of-plane deflection, F is the Airy stress function, and w0 is the initial out of plane

imperfection. Finally, the biharmonic operator, ∇4(�) is defined as

r4ð�Þ ¼ @4ð�Þ@x4

þ 2@4ð�Þ@x2@y2

þ @4ð�Þ@y4

: (20.5)

The Airy stress function, F, is defined as follows:

Fxx ¼ Py; (20.6)

Fyy ¼ Px; (20.7)

and

Fxy ¼ Pxy: (20.8)

Substituting the expressions from (20.4) into (20.2) and (20.3), results in the following set of non-dimensional von Karman

equations,

~r4ð~w� ~w0Þ þ ~w00 þ ~C~w0 þ dp ¼ ~F~y~y~w~x~x þ ~F~x~x~w~y~y � 2~F~x~y~w~x~y; (20.9)

and

~r4 ~F ¼ ~w2~x~y � ~w2

0~x~y � ~w~x~x~w~y~y þ ~w0~x~x~w0~y~y: (20.10)

For simply supported boundary conditions, Fourier solutions were assumed for both the out of plane displacement and the

Airy stress function,

~wðxÞ ¼X

m

X

namnðtÞfmðxÞcnðyÞ; (20.11)

and

~FðxÞ ¼X

m

X

nbmnðtÞfmðxÞcnðyÞ; (20.12)

where

fm ¼ sin mpxð Þ; (20.13)

and

cn ¼ sinnpyr

� �

: (20.14)


Equations 20.11 and 20.12 were substituted into (20.9) and (20.10). The series was truncated to a finite number of terms,

and then a Galerkin approximation was applied in order to minimize the errors associated with truncating the series. The

result is a set of nonlinear differential equations in terms of the Fourier coefficients, amn and bmn, that describe the deflectionof a plate with an applied axial load.

To solve the set of equations, a nine-term approximation was used with m, n ¼ {1, 2, 3} for a square plate, r ¼ 1.

A simple representation of the axially-loaded plate is shown in Fig. 20.3. The equations were solved using the AUTO

continuation package installed on Mac OS X. Using a uniaxially applied load, the bifurcation diagram showing non-

dimensional load versus the L2-norm of the Fourier coefficients is shown in Fig. 20.4. The solid lines in the figures indicate

stable branches, while the dotted lines indicate unstable branches.

The square markers along the abscissa axis indicate bifurcation points. The first square marker, at Px ¼ 1, indicates the

first linear elastic buckling load of the uniaxially compressed plate in the (m, n) ¼ (1, 1) mode. The following squares along

the axis are the linear buckling loads of the higher order modes. The buckling levels of the higher order modes correspond

with those predicted by energy methods [12].

As the axial load is applied to the plate, the natural frequencies will change with load and at least one frequency will drop

to zero as the plate approaches the buckling limit. As the plate follows a stable post-buckled equilibria, it will regain its

stiffness and, in turn, the natural frequencies will all be nonzero. Figure 20.5b shows the variation in natural frequency,

calculated by AUTO, of the plate, about the trivial, flat, solution from Px ¼ [0, 1] and then about the (m, n) ¼ (1, 1)

dominated equilibrium solution, shown in Fig. 20.5a, after buckling. The four lowest mode shapes, calculated from the

eigenvectors of the Jacobian matrix produced by AUTO, for the unloaded case are shown in Fig. 20.6.

Fig. 20.4 Bifurcation

diagram for a uniaxially

loaded square plate

Fig. 20.3 Schematic

representation of uniaxially

compressed square plate


0

00

1

2

3

4

5

6

1 2 3 4 5 6 7

10.8

0.60.4

0.20 0 0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

2.5

3

a b

Fig. 20.5 Static deflection and modal behavior of a uniaxially compressed square plate. (a) (m,n) ¼ (1,1) mode post-buckled static deflection,

Px / Pcritical ¼ 4.0, L2 norm ¼ 11.1933. (b) Variation of four lowest, linearized, natural frequencies for the main bifurcation branch of a square

plate

0.35

a

c d

b

0.4

0.3

0.2

0.1

−0.1

−0.2

−0.3

−0.4

0

0.4

0.3

0.2

0.1

−0.1

−0.2

−0.3

−0.410.9

0.80.70.6

0.50.40.3

0.20.1 0.1 0.2 0.3

0.40.5 0.6

0.7 0.8 0.91

0 0

0

10.90.8

0.70.6

0.50.4

0.30.2

0.10 0 0.1 0.2 0.3

0.4 0.5 0.6 0.7 0.8 0.9 1

0.4

0.3

0.2

0.1

−0.1

−0.2

−0.3

−0.4

0

10.90.8

0.70.6

0.50.40.3 0.2

0.1 0.1 0.20.3 0.4 0.5 0.6 0.7 0.8

0.9 1

0 0

0.25

0.15

0.05

0.1

00.9

0.80.7

0.60.50.4

0.30.2

0.10 0.1 0.2 0.3 0.4 0.5 0.6

0.7 0.8 0.9 1

0

1

0.3

0.2w

w

w

w

y

y

y

y

x

x

x

x

Fig. 20.6 Vibration mode shapes corresponding to the four lowest natural frequencies


Additionally, AUTO predicts that the secondary post-buckled equilibrium branch, which has the buckled mode shape

(m, n) ¼ (2, 1) shown in Fig. 20.7a, will stabilize after approximately Px/Pcritical ¼ 2.0 as shown in Fig. 20.4. Figure 20.7b

shows the natural frequencies of the secondary buckling branch after it regains stability.

20.5 Initial Imperfections

In order to model more physical systems, initial imperfections in the plate are taken into account since they are always

present in axially-loaded physical specimens. The continuation problem can be reformulated to start with a perfect plate,

x0 ¼ 0, and use the prediction correction steps until a desired imperfection level is reached. The plate was given an initial

imperfection in the (m, n) ¼ (1, 1) mode with a maximum central displacement equivalent to one plate thickness. The

imperfect plate bifurcation diagram and natural frequencies are shown in Fig. 20.8. Since there is no distinctive buckling

point (when compared with the perfect case) none of the frequencies for the main bifurcation branch drop to zero as before.

Additionally, with the initial imperfection present, the secondary branch still maintains stability through a portion of the

post-buckled regime as with the perfect case.

20.6 Snap Through

The final aspect that the continuation methods were applied to, and are particularly well suited to investigate, was the

snap through behavior of the post-buckled plate. Starting with a perfect plate, in the post-buckled region with an axial load of

Px/Pcritical ¼ 1.63, a transverse point load was applied to the center point of the plate. The standard bifurcation diagram, with

the pressure loading included is shown in Fig. 20.9a. The axial load is held constant while the continuation method varies

and follows the path of the transverse load needed for equilibrium. Figure 20.9b shows the equilibrium path as the pressure

load is applied. As the load is applied, the plate reaches a limit point where it becomes unstable and then eventually moves

dynamically, regains stability, and reaches the symmetric equilibrium condition on the opposite side. The points labeled 1–4

on Fig. 20.9a–c indicate the loading pattern. Figure 20.9c shows the lowest natural frequency as the central point load is

applied to the plate. The frequency drops as the load is applied, and eventually goes to zero as the equilibrium becomes

unstable. After the plate regains stability the lowest frequency becomes positive again and eventually reaches its original

starting point again as the plate reaches the symmetric equilibrium state.

1.5

0.5

−0.5

−1.5

−21

0.80.6

0.40.2

0 000

1

2

3

4

5

6ba

1 2 3 4 5 6 70.20.4

0.60.8

1

−1

0

1

2

Fig. 20.7 Static deflection and modal behavior of secondary buckling branch for a uniaxially compressed square plate. (a) (m,n) ¼ (2,1) mode

post-buckled static deflection, Px / Pcritical ¼ 4.0, L2 norm ¼ 6.6834. (b) Variation of four lowest, linearized, natural frequencies for the secondary

bifurcation branch of a square plate


a b

c

Fig. 20.8 Buckling and post-buckling behavior of an imperfect, uniaxially compressed square plate. (a) Bifurcation diagram of an initially

imperfect, uniaxially loaded square plate. (b) Variation of natural frequencies for main branch of an imperfect, uniaxially loaded square plate.

(c) Variation of natural frequencies for secondary branch of an imperfect, uniaxially loaded square plate


20.7 Conclusions and Future Work

Continuation methods were applied to a problem of plate buckling to determine both pre and post-buckled behavior. The

continuation method produced results that correspond well with previously published data. Additionally, the continuation

code, AUTO, produced information corresponding to the post-buckled stability, natural frequency, and mode shapes. AUTO

was also utilized to trace an initially unstable secondary buckling branch that was found to stabilize after a certain increment

in the axial load. Finally the behavior of an imperfect, post-buckled plate, and snap through behavior of a post-buckled plate

were investigated using the continuation methods. Experiments will need to be conducted as future work to validate several

of the results found using the continuation analytical methods. Most importantly, verifying that the post-buckled stable

region of the secondary buckling path for the square, simply supported plate is physically realizable.

r

a

c

Fig. 20.9 Snap through behavior of post-buckled plate. (a) Bifurcation diagram showing both main buckling branch and snap through branch.

(b) Bifurcation diagram showing the snap through equilibrium path. (c) Variation of lowest natural frequency of square plate for snap through

equilibrium path


Acknowledgements The authors would like to acknowledge the NASAGraduate Student Researchers Program (GSRP) grant NNX09AJ17H and

the Air Force Office of Scientific Research grant FA9550-09-1-0204 for support.

References

1. Murphy KD (1994) Theoretical and experimental studies in nonlinear dynamics and stability of elastic structures. PhD thesis, Duke University

2. Hui D, Leissa AW (1983) Effects of geometric imperfections on vibrations of biaxially compressed rectangular flat plates. J Appl Mech

50:750–756

3. Ilanko S (2002) Vibration and post-buckling of in-plane loaded rectangular plates using a multiterm Galerkin’s methods. J Appl Mech

69:589–592

4. Doedel E, Keller HB, Kernevez JP (1991) Numerical analysis and control of bifurcation problems (II): bifurcation in ininite dimensions.

Int J Bifurcat Chaos 1:745–772

5. Chen H, Virgin LN (2004) Dynamic analysis of modal shifting and mode jumping in thermally buckled plates. J Sound Vib 278:233–256

6. Seydel R (1991) Tutorial on continuation. Int J Bifurcat Chaos 1:3–11

7. Seydel R (2010) Practical bifurcation and stability analysis. Springer, New York

8. Doedel EJ, Oldeman BE (2009) AUTO-07P: continuation and bifurcation software for ordinary differential equations. Concordia University,

Oct 2009

9. Strogatz SH (2001) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Westview Press,

Cambridge,

10. Meirovitch L (2001) Fundamentals of vibrations. McGraw-Hill, Boston

11. Dowell EH (1975) Aeroelasticity of plates and shells. Noordhoff International, Leyden

12. Bulson PS (1970) The stability of flat plates. Chatto and Windus, London


Chapter 21

Comparing Measured and Computed Nonlinear Frequency

Responses to Calibrate Nonlinear System Models

Michael W. Sracic, Shifei Yang, and Matthew S. Allen

Abstract Many systems of interest contain nonlinearities that are difficult to accurately model from first principles, so it

would be preferable to characterize the system experimentally. For many nonlinear systems, it is now possible to measure

frequency response curves with stepped sine testing and to compute frequency response curves with numerical continuation.

Nonlinear frequency response curves are very sensitive to the system model and the nonlinearities and they provide a lot of

insight into the response of the system to a variety of inputs. This paper explores the feasibility of a nonlinear model updating

approach based on nonlinear frequency response and the experimental and analytical tools that are needed. For the

experiment, a cantilever beam with an unknown nonlinearity is driven with a harmonic force at various frequencies.

The steady-state response is measured and processed with the fast Fourier transform to obtain the frequency response

curve. Some subtle yet important details regarding how this is implemented are discussed. An analytical model is also

constructed and its frequency response computed using a recently developed technique. The measured and simulated

frequencies are then compared and used to tune the analytical model.

Keywords Nonlinear model updating • Frequency response • Stepped sine testing

21.1 Introduction

The frequency response function has long been used to characterize the dynamic response of linear systems. For nonlinear

systems, the nonlinear frequency response has also proven very useful, even with the additional complexities that are

introduced. Several different approaches have been taken to calculate frequency response from a known mathematical

model. A few studies combine the Harmonic Balance Technique with a numerical continuation algorithm to compute the

frequency response curves of torsional sub-systems with clearance nonlinearities [1] or nonlinear mesh phase interactions in

multi-gearbox drive systems [2]. This method is a semi-analytical approach since it uses the HBM formulation and closed-

form equations of motion. In another paper [3], Padmanabhan and Singh used a purely numerical approach based on

shooting methods to compute nonlinear frequency response curves and to characterize jump phenomena, subharmonic

responses, and chaos. However, as they discuss in [3], the algorithm seems to break down for higher-order nonlinear

systems. Ribeiro also used a numerical shooting approach to study frequency responses of geometrically nonlinear beams

and plates [4], but the algorithm used in that study doesn’t appear to be equipped to follow turning points [5] on branches that

lead to unstable periodic solutions. Finally, Gibert [6] used a numerical continuation approach based on nonlinear normal

modes (NNMs) to compute the nonlinear frequency responses of a beam model, and then used model updating method to

match those frequency responses to an actual beam. However, his method requires one to choose a specific number of NNMs

to use in the calculation, and this approximation may break down for certain nonlinearities or if an insufficient number of

nonlinear normal modes is used.

In this paper, the numerical algorithm presented by the authors in [7] is used to calculate nonlinear frequency responses.

The algorithm is based on shooting and continuation techniques that are very similar to those used in [8] to find the nonlinear

M.W. Sracic (*) • S. Yang • M.S. Allen

Department of Engineering Physics, University of Wisconsin-Madison, 535 Engineering Research Building,

1500 Engineering Drive, Madison, WI 53706, USA

e-mail: [email protected]; [email protected]; [email protected]



255




normal modes of unforced systems. Here, the system is excited with a sinusoidal input force and the response is found over

one period of the excitation. The initial conditions are then adjusted until a steady state response is obtained and numerical

continuation techniques are used to calculate a branch of periodic responses for different forcing frequencies (i.e. essentially

a frequency response). The algorithm used in this work is identical to the one presented in [7] except that in this work the

Jacobians that are required were computed numerically using finite differences, so the equations of motion do not need to be

known in closed form.

Once a model has been formed and its frequency response has been calculated, one can use experiments to validate and

update the model. In the paper by Ribeiro [4] this was done by calculating one nonlinear frequency response and comparing

it to a measured frequency response near one of the beam’s resonances. In [9], Carella and Ewins measure nonlinear

frequency response a system using a stepped-sine testing approach and extract a first order approximation (linear) model for

specific forcing amplitudes. This approach though is applied to a single mode of the system and doesn’t seem to consider any

higher harmonic information associated with the nonlinearity. In [6], Gibert compares frequency responses between an

analytical model of a beam and an actual beam near three resonances, and in this paper a similar approach will be employed.

A cantilever beam was constructed in the laboratory, and a geometric stiffness nonlinearity was created by attaching a small

strip of nylon between the beam’s free end and a fixture. The nonlinear frequency response was then estimated using

stepped-sine excitation. A numerical model of the beam was constructed using a Ritz-Galerkin approach and its frequency

responses were computed using a numerical continuation approach and compared to the measured frequency responses of

the actual beam. The information from the frequency responses and from the harmonic information contained in the periodic

time history responses are used to evaluate the accuracy of the model, and some initial is made to update the model to

correctly capture the beam’s dynamics.

Themethodsused in this paper shouldbe applicable to a broad range of systems.The nonlinear frequency responseof a system

can be computed over a wide range of frequencies using efficient shooting and continuation techniques [3, 7, 8] and they are not

reliant on any approximation, such as an assumed number of polynomial terms approximating the nonlinearity. The resulting

nonlinear frequency responses can be readily compared to measurements, paralleling the way in which linear finite element

models are sometimes validated by comparing their linear frequency response functions with measured ones [10].

21.2 Theoretical Development

Generally, a forced nonlinear system can be represented in state space with the following equation

_x ¼ f x; uð Þ (21.1)

where f is a function that describes how the time-dependent state of the system, x(t), and the time-dependent inputs applied to

the system, u(t), influence the dynamics of the system. The output response of the system yðtÞ is often simply a subset of the

states, but in general it can be a general nonlinear function of the state and input as follows.

y ¼ h x; uð Þ (21.2)

When the nonlinear system is subjected to a periodic input with period T (i.e. uðtþ TÞ ¼ uðtÞ), the state and output will

often be periodic with xðtþ TÞ ¼ xðtÞ and yðtþ TÞ ¼ yðtÞ. (In the most general case, the nonlinear system may respond with

a longer or shorter period or may respond chaotically even in the presence of periodic input [5, 11], but these issues will not

be addressed in this paper.) Thus, it shall be assumed that the period of the forcing will always equal the period of the

response. It is more convenient to work with frequencies rather than periods, oT¼ ð2pÞ T= , so they will be used in place of

the period in the rest of this paper.

21.2.1 Frequency Response Attributes

The nonlinear frequency response curves describe how the periodic response(s) of a nonlinear system change as the input

frequency is varied. For linear systems this is captured by the frequency response function, which relates the magnitude and

phase of the input to that of the output. Each point on the frequency response function curve describes a periodic orbit g,which is a trajectory in the state space that contains the state �x for every time t. A linear FRF typically has a large magnitude

256 M.W. Sracic et al.

peak near each natural frequency corresponding to high amplitude periodic orbits in the state space. Similarly, the nonlinear

frequency response has resonance peaks near its nonlinear normal mode frequencies [12], but there are a few other important

differences. Resonance peaks in nonlinear systems can occur away from the nonlinear normal modes (e.g. superharmonic

resonances [5]), resonance peaks can contain multi-valued regions where several periodic orbits are possible for single

forcing frequencies, certain branches on the nonlinear frequency response curve may contain unstable periodic responses,

and finally the law of superposition does not hold for nonlinear systems so frequency responses cannot be linearly scaled

when the input amplitude is scaled. Based on the latter point, a nonlinear frequency response is defined by the response alone

(i.e. there is no scaling by the input such as in the linear frequency response), and it is only valid for a specific forcing

function and amplitude.

Using some of the previous facts, nonlinear frequency response curves can be built from time domain periodic responses,

and it is advantageous to have time domain signals because they contain more information about the nonlinearity in the

system. For example, consider the Duffing oscillator equation of motion with o1 ¼ 1, o3 ¼ 0:5, and sinusoidal forcing at a

frequency oT ¼ 1.

€xþ 0:02 _xþ o21xþ o2

3x3 ¼ sinðoTtÞ (21.3)

The response of this system to initial conditions x; _x½ � ¼ �0:056; 1:594½ � is periodic. The displacement is plotted over

five cycles of the response in Fig. 21.1a, and the signal appears to be sinusoidal. The magnitude of the frequency spectrum of

this response is plotted in Fig. 21.1b revealing that this is not a pure sinusoid but that there are several frequencies present at

1, 3, 5, etc. rad/s (labeled with open circles). The magnitude of these frequency components diminishes with increasing

frequency. There are also sharp peaks that rise above the noise floor at 2, 4, 6, etc. rad/s.

The system is clearly nonlinear since the input was a single frequency sinusoid, yet the system responded at several

different frequencies. In particular, if the forcing frequency is designated as the m ¼ 1 harmonic then the odd harmonics of

the system (i.e. the m ¼ 3, 5, . . .) designate peaks at 3 rad/s, 5 rad/s, etc. The even harmonics also appear in the spectrum at

2 rad/s, 4 rad/s, etc., however, those peaks are more than ten orders of magnitude smaller than the dominant peak and may be

the result of numerical integration error. The harmonic information can be used to build a single point on the frequency

response curve. For example, one could record the amplitude of a certain harmonic term in the response for a number of

different forcing frequencies and then plot the magnitude of that harmonic term versus the forcing frequency. In some cases a

harmonic other than the m ¼ 1 (i.e. the driving frequency) harmonic might dominate the responses, so it might be beneficial

to choose a harmonic other than the driving frequency. One can also sum the complex amplitudes of all the harmonics at

each driving frequency and then plot the magnitude of the sum of the harmonics versus the forcing frequency. No matter

which method is chosen, one needs to repeat this process for each stable periodic response on the frequency response curve.

21.2.2 Simulated Frequency Responses

Numerical continuation techniques can be used to efficiently calculate all of the periodic responses of a system over a wide

forcing frequency range. In this paper, a Newton–Raphson correction technique is used to solve the following two-point

boundary value problem.

0 5 10 15 20 25 30-2

-1

0

1

2

time, s

Dis

p x

Periodic Response

0 5 10 15

10-10

100

Frequency, rad/s

|FF

T(x

) |

FFT of Periodic Responsea b

Fig. 21.1 Periodic time domain response and frequency domain response of the Duffing system

21 Comparing Measured and Computed Nonlinear Frequency Responses to Calibrate Nonlinear System Models 257

Hðx; TÞ ¼ xðTÞ � xð0Þ ¼ 0 (21.4)

In particular for a system that is forced at the frequency oT¼ ð2pÞ T= , if the response vector xðTÞ is equal to the initial

condition vector xð0Þ, then the response is periodic. This solution provides a single point on the frequency response curve.

Once a solution has been established, the following system of equations can be solved to make a prediction for the next

solution on the curve.

@H@x

��xðTÞ;Tð Þ

@H@T

��xðTÞ;Tð Þ

h iPf g ¼ 0f g

Pf g ¼ PTx PT

� �T (21.5)

The Jacobian matrices @H @x=½ � and @H @T=½ � can be calculated using closed form equations of motion and the methods

described in [7, 13] or by using numerical integration and finite difference equations. In this paper, the latter method is used.

The vector P is in the null space of the matrix defined by the Jacobian matrices, so it is tangent to the frequency response

curve by construction [7, 8]. Once P has been calculated, it can be used to calculate the initial conditions for the prediction

of the next periodic response: xjþ1ð0Þ ¼ xjð0Þ þ sPx and Tjþ1 ¼ Tj þ sPT , where j ¼ 0 defines the first periodic solution

(i.e. j ¼ 0, 1, 2, 3, . . .) and s is a step-size parameter which can be automatically changed to increase the efficiency of the

calculations [7, 8]. The new prediction may need to be corrected so that (21.4) is satisfied. Therefore, corrections can be

calculated from the following system of equations.

@H@x

��xðkÞðjþ1Þ; T

ðkÞðjþ1Þ

� � @H@T

��xðkÞðjþ1Þ; T

ðkÞðjþ1Þ

� �Pzf gT PT

24

35 DxðkÞðjþ1Þ

DTðkÞðjþ1Þ

( )¼ �H x

ðkÞðjþ1Þ; T

ðkÞðjþ1Þ

� �0

( )(21.6)

The solutions of the previous system have been constrained to be orthogonal to the tangent prediction vector. This was

enforced to increase the convergence rate of the calculations. The variable k is used to track the number of corrections that

are required until the (21.4) has converged. In this paper, the solution has converged when H x; Tð Þk k< xð0Þk k � 10�6.

The prediction-correction process is repeated for each periodic response within a desired forcing frequency range.

These equationswere incorporated into an algorithm that calculates the entire frequency response curve over a large frequency

range for one forcing amplitude. The algorithm is automated and efficient, and can be used for relatively high order systems.

21.3 Measuring Frequency Response Curves

Swept sine tests are often used to measure the frequency response of linear systems. A similar approach can be used for

nonlinear systems; the frequency of the forcing signal is changed step by step in the tests, while the forcing amplitude is the

same for all the forcing frequencies. Then, the system’s response at each forcing frequency is measured and the nonlinear

frequency response curves for this specific forcing amplitude can be constructed within the tested frequency range.

However, there are some important differences between swept sine tests for linear and nonlinear systems. A nonlinear

system generally has different frequency response curves for different forcing amplitudes. Furthermore, even with a fixed

forcing amplitude, as implied earlier, the system may have several periodic orbits in some frequency ranges. The higher

energy periodic orbits are often less stable, so when increasing/decreasing the forcing frequency the system may jump to a

lower energy orbit when a disturbance occurs, making the high energy branch more difficult to capture. The high energy

branch often contains some of the most valuable information, so it is desirable to capture as much of that branch as possible.

Therefore, it is necessary to keep the forcing amplitude constant during the whole test. Sometimes the force can be

monitored and controlled to reject disturbances or fluxuations by using a feedback control algorithm [9].

In this work, a National Instrument PXI system was used to generate the step-sine forcing signal and to acquire the

responses. A LabView program was designed to generate a continuous forcing signal that has a fixed number of samples per

period, ensuring that zero amplitude is obtained at the beginning and the end of each period. Because there are a fixed

number of samples per period, the generation rate of the forcing signal varies with the forcing frequency (or the period).

When the forcing frequency is stepped, the program waits until the end of a period (i.e. when the amplitude is zero) to change

the frequency so that the transient response (disturbance) incurred to the system is minimized. The input force and the

response are monitored and the responses are recorded with a fixed sampling rate after the transient responses disappear.


At each forcing frequency, the data acquisition program waits prescribed amount of time (typically a few seconds) for the

response to reach steady state prior to recording the response and exporting the time series to Matlab for analysis. The

magnitudes of the response versus frequency can then be computed using the procedure described in Sect. 21.2.1.

21.4 Updating Models Using Nonlinear Frequency Response Curves

Although nonlinear frequency response curves can be substantially more complicated than linear FRFs, general dynamics

principles can still be used to interpret the curves. Figure 21.2 shows several frequency response curves of the Duffing

system in (21.3), which were calculated using the numerical continuation technique that was summarized previously. The

results are plotted in terms of the magnitude of the periodic displacement. For each curve, a different set of values of linear

and nonlinear stiffness terms (i.e. o1 and o3, respectively) was used. All the curves have well known bent resonance shape

with a region where multiple solutions are possible for a single forcing amplitude and frequency. For some sets of the

parameters o1 and o3 this multi-valued solution region extends over a larger range of frequency than for others. When the

linear stiffness is increased, the whole resonance peak shifts to higher frequencies. For example, the base of the curve shown

with a solid line (i.e. o1 ¼ 1, o3 ¼ 0:5) seems to originate near 1 rad/s. When the linear stiffness is increased to o1 ¼ 2 the

resonance shifts upward, as shown by the curve with open circles. When the linear stiffness is held constant and the nonlinear

stiffness is varied, the shape of the peak changes. For example, the curve with a dashed line is for o3 ¼ 0:25, and is seen to

bend less severely than the curve for o3 ¼ 0:5.Figure 21.2 also shows that the linear and nonlinear stiffness terms have coupled effects on the shape of the shape of the

nonlinear FRF. For example, the shape of the frequency response curves can be seen to change when the linear stiffness is

varied, even though the nonlinear stiffness term is held constant at o3 ¼ 0:5. Furthermore, none of the curves in the figure

has the same peak magnitude, even thought the forcing amplitude was the same in all cases (i.e. it was equal to 1 in all cases).

Nevertheless, the nonlinear frequency response curves do change in a fairly straightforward way so it would appear that,

given a reasonable set of measurements, one could adjust the system model until it has the same frequency response as the

measurement. The dynamics of higher order nonlinear systems can be more complicated, but they often show this same

characteristic shape and hence these same principles could be used to tune a model until it reproduces each resonance

accurately.

0 2 4 6 8 1010-2

10-1

100

101

102

Frequency, rad/s

Mag

Dis

p

Duffing Frequency Responses

ω1=1, ω3=0.5

ω1=2, ω3=0.5

ω1=4, ω3=0.5

ω1=1, ω3=0.25

ω1=1, ω3=1

Fig. 21.2 Frequency

response curves of the Duffing

system for different linear

(o1) and nonlinear (o3)

stiffness values


21.5 Nonlinear Cantilever Beam System

In a few recent papers, the Sracic and Allen have worked with a cantilever beam that has a geometric nonlinearity at its free

end [14, 15]. Figure 21.3 below shows a top view photograph of the actual experimental setup. An aluminum 6061 alloy

beam is bolted to a fixture that approximates a fixed base. A small strip of nylon is bolted to the free end of the cantilever and

clamped to the fixture. The beam is oriented such that the bending axis is parallel to the plane of the table top. Figure 21.4

shows a close top and front view of the nylon strip that is clamped between the tip of the beam and the right hand side

support. Table 21.1 below provides the physical dimensions of the beam and the nylon strip in millimeters.

The nylon strip at the tip of the beam adds stiffness at that point that depends nonlinearly on the tip displacement. This

setup was originally proposed in [16], although their beam had a strip of spring steel instead of nylon, and other researchers

have studied similar beam setups [17–19].

Fig. 21.3 Top view of the experimental nonlinear beam setup

Fig. 21.4 Top view (a) and front view (b) of the spring steel connected to free end of the cantilever beam

Table 21.1 Dimensions, in millimeters, of the 6061 aluminum beam and the nylon strip, and the location of measurement sensors from

the approximated fixed end

Dimension Al 6061 beam Nylon 6/6 strip Sensor location

Length 1016 53.2 a1 ¼ 45

Width 25.4 25.4 a2 ¼ 508

Thickness 9.5 0.254 a3 ¼ 984


An analytical model of this system was created, as described in the schematic in Fig. 21.5. The beam is modeled as a

uniform, prismatic cantilever beam with material density r, elastic modulus Eb, cross sectional area Ab, bending area

moment of inertia I, and length L. The position along the length of the beam is given by the variable ‘x’. The deflection of thebeam is designated with the variable y. The beam has a nonlinear spring at its tip with stiffness knl. The equations of motion

of the analytical beam are derived in Appendix A.

In order to mimic the experimental system, the following parameters were used for the original model, which are based on

the nominal properties of the experimental hardware: r ¼ 2,700 kg/m3, Eb ¼ 68 GPa, Ab ¼ 3.23 � 10�4 m2, I ¼ 4.34

� 10�9 m4, L ¼ 1.016 m. Using these properties with the Ritz-Galerkin method, the two linear natural frequencies of the

system are o1/(2p) ¼ 9.97 Hz and o2/(2p) ¼ 62.51 Hz. Modal damping was added to the model assuming a coefficient of

critical damping of z ¼ 0.01 for all modes. The transverse stiffness of the nylon strip is approximated in the model as

k3 ¼ EnAn ð2l3nÞ�

, where En, An, and ln are the elastic modulus, cross sectional area, and length of the nylon strip. For the

original model, the nylon was assumed to have an elastic modulus of En ¼ 3.9 GPa, which makes the nonlinear stiffness

k3 ¼ 8.75 � 107 N/m3. A derivation of this approximation can be found in the appendix in [14], although in that work spring

steel was used at the beam’s free end instead of nylon.

The experiment was designed to produce a cubic nonlinearity. However, several practical design aspects could lead to

other forms of nonlinearity in the response. For example, Fig. 21.4a shows that the nylon strip is offset from the neutral

bending axis of the beam, which has the potential to produce a quadratic nonlinear contribution to the response. Addition-

ally, the fixed end of the cantilever is realized using bolts, and the effective length of the beam may change depending on the

direction of the deflection. The finite order model of the beam was developed assuming a perfect cantilever and a perfectly

cubic spring at the beam’s free end. Therefore, the goal is to use the frequency responses to improve the model so that it more

closely represents the actual beam.

21.5.1 Frequency Responses of the Nonlinear Beam

The procedure described in Sect. 21.3 was used to measure the periodic responses of the nonlinear beam in order to calculate

the beam’s frequency responses. Harmonic excitation was applied to the beam with a model 2100E11-100 lb Modal Shaker

from The Modal Shop, Inc. The beam was approximated as a having a fixed support, so the shaker was freely hung from a

lateral excitation stand, as recommended in [20]. A thin steel stinger was used to transmit the excitation from the shaker to

the beam. One end of the stinger was clamped inside the shaker armature and the other end was fixed to a force transducer,

model 208 C04 from PCB Piezotronics, Inc. (PCB), which was bolted to the beam at a location x ¼ 45 mm from the fixed

end of the beam. Harmonic forcing was provided by the National Instruments PXI system described in Sect. 21.3. The peak

force amplitude, which was measured by the force transducer, was 70 N. The response was measured with two Endevco

model 66A12 triaxial accelerometers (only the z-channels were used) located at x ¼ 45 mm (DOF 1; shaker location) and

x ¼ 508 mm (DOF 2; beam center) and with an Edevco model 256-100 isotron accelerometer located near the free end of the

beam at x ¼ 984 mm (DOF 3; beam tip). The degree-of-freedom locations are labeled with arrows in Fig. 21.3. All of the

accelerometers were secured to the beam with wax.

Initially, a fast sweep was performed using a large step size (frequency increment) in order to estimate the resonance

frequencies. The first three resonances were found to be near 14, 45, and 120 Hz. The program was then set to automatically

perform forward and backward frequency sweeps using a variable step size. Specifically, a step size of 0.1 Hz was used near

the resonances and 0.5 Hz in the regions away from the resonances. This approach decreased the testing time significantly

while still capturing all of the features of the nonlinear frequency response accurately.

After the measurements were obtained, the method described in Sect. 21.2.1 was used to calculate each point on the

frequency response curve from the measured periodic responses. In particular, the spectra of the measured accelerations

were calculated with the Fast Fourier Transform, and the amplitudes of all the harmonic peaks were collected (since the

knl = k3 y(L)3

xy

DOF 2 DOF 3 DOF 1

Fext = Aext sin(w Tt )

r, Eb, Ab, I, L

Fig. 21.5 Schematic of the

nonlinear beam


steady-state response is periodic, these are the coefficients of the Fourier Series description of the response). In order to

estimate displacement, each harmonic peak was integrated twice by dividing frequency squared. Then, the displacement

amplitudes of all the harmonics were summed to give the estimate for the magnitude of displacement.

Next, the frequency response of the model beam was calculated with the numerical continuation technique. The algorithm

was set to find the nonlinear frequency response between 2 and 150 Hz. For each periodic response, the instant at which

the displacement of DOF 3 (i.e. the tip displacement) was maximum was found and recorded, and the displacements of the

other DOF were recorded at the same instant. These amplitudes were used to plot the magnitude of the frequency response.

Note that this assumes that all the degrees-of-freedom reach their extreme displacement at the same instant, which may not

be the case for very nonlinear systems or for when there are complicated modal interactions. However, it will be shown that

the accuracy of the model can still be inferred using this approach.

Figure 21.6 shows the frequency response functions that were calculated from the measurements of the beam as well from

the numerical continuation simulations. The dashed curves with markers show the results from the experiment (open blue

squares-accelerometer a1, open green circles-accelerometer a2, red dots-accelerometer a3). The solid lines are the frequencyresponse curves that were calculated with the numerical continuation technique (blue-DOF 1, green-DOF 2, red-DOF 3).

There are three dominant resonance peaks in most of the curves except for the DOF 2 curves (green). The first two resonance

peaks in the model curves occur near 10 and 47 Hz and are bent to higher frequencies, while the third peak near 130 Hz

appears to be predominantly linear. In the curves from the experiment, the first two peaks occur near 14 and 44 Hz. They

seem to have a bent shape, but when they reach a certain frequency the curve sharply drops in magnitude, which many would

recognize as the well know jump-phenomenon for the multi-valued region (i.e. the response jumps from a large amplitude

periodic response to a small one). Additionally, the first resonance of the model seems to bend more significantly than the

curves from the experiment and the curves from the model contain several superharmonic resonances below 5 Hz. This

range was not tested in the experiment because the freely suspended shaker moved quite a bit in that frequency range causing

the measurements to be unreliable. Lastly, the amplitude of the resonance peaks in the model’s responses for DOF 1 (i.e. the

blue curve, especially for the peaks near 47 and 130 Hz) appear to be much smaller than those of the actual system.

The frequency response curves show that the model accurately characterizes the number of modes in this frequency band

of interest, and the fact that the nonlinearity is most apparent in the first and second modes. However, there are several

significant discrepancies. The amplitude of the peaks in the model’s curve for DOF 1 are too small, indicating that this

DOF’s amplitude is smaller than the actual hardware. This discrepancy may arise because the model has a perfect fixed base,

while the fixture in the experiment has finite compliance. Another important issue is that the frequencies where the

resonances occur differ between the model and the actual beam. The frequencies of the second two modes are too high in

the model. As in the single degree-of-freedom case, this property is influenced by the linear stiffness of the system.

Therefore, the elastic modulus of the beam can be reduced to decrease the linear stiffness and hence the frequency of all

the resonance peaks. Figure 21.7 shows the second comparison between the measured frequency responses and those from

0 50 100 15010-7

10-6

10-5

10-4

10-3

10-2

Frequency, Hz

Mag

nitu

de D

isp

Nonlinear Beam Frequency Response

Exp DOF 1

Exp DOF 2

Exp DOF 3

Mod DOF 1

Mod DOF 2

Mod DOF 3

Fig. 21.6 Comparison #1 of frequency response curves for the experiment (Exp) and the model (Mod) for model parameters: Eb ¼ 68 GPa,

En ¼ 3.9 GPa, k3 ¼ 8.75 � 107 N/m3


the model after reducing the elastic modulus of the beam to Eb ¼ 55.5 GPa (note that this value is still in the nominal range

for the aluminum). No other parameters in the model were changed.

With a lower elastic modulus, the location of the resonance peaks has changed to be more in line with the measurements,

but the shapes of the peaks has changed as well. The resonance of the third mode near 119 Hz seems to align very well with

the measurements, and the peaks seem to have the same magnitude except for DOF 1. The model’s second resonance peak

also seems to align more closely with the measurements. However the peaks in the simulated frequency responses bend over

more than the measurements and more than they did in Fig. 21.6. This is evidence that the linear and the nonlinear stiffness

of the model are coupled, since only the linear stiffness was changed but a nonlinear feature of the second resonance also

changed. The same seems to be true for the first resonance. However, one must use caution when comparing the simulated

frequency responses with the measurements, since the actual beam may have jumped to a lower amplitude response before

reaching the absolute peak of the frequency response curve. The phase of the response can be used to investigate this, since

the phase should be 90� at the extreme of the resonance curve. The phase of the response was calculated at the jump

frequency for the resonance near 14 Hz by computing the difference between the phase of the fundamental harmonic of the

response and the fundamental harmonic of the force. For DOF 3 at 14.3 Hz the phase difference was found to be �83.26�,suggesting that the experiment captured nearly the full nonlinear resonance curve for DOF 3 before the jump happened. In

contrast, the phase of DOF 3 at the 45.11 Hz jump frequency was 74.86�, revealing that portion of the second resonance

curve was probably missed in the experiment. However, the peaks in the model’s responses still seem to bend over larger

frequency ranges than those of the actual beam suggesting that the nonlinear stiffness of the model is too large. Hence, the

model was updated again by reducing the modulus of the nylon spring to En ¼ 0.24 GPa, which yields a nonlinear stiffness

coefficient of k3 ¼ 5.38 � 106 N/m3. The frequency responses of the model were recomputed and are shown in Fig. 21.8.

This has caused the shape of the model’s resonances to change so that the first two resonances do not bend as much towards

higher frequencies. Unfortunately, the peaks near 44 Hz now do not seem to bend as much as those in the measurements. The

peaks near 12 Hz have a larger maximummagnitude than the previous case and yet they still seem to bend more than those in

the measurements. It also seems apparent now that the model’s resonance curves are centered at a significantly lower

frequency than those in the measurements.

21.5.1.1 Discussion

By simply adjusting the modulus of the beam (i.e. in Fig. 21.7), the nonlinear frequency responses of the model were made to

agree much more closely with the actual measurements, at least for the second and third resonances. The shape of the

resonance curve for the second mode was also improved as well. However, the linear and nonlinear stiffness terms have a

coupled effect on the nonlinear FRFs so it was not trivial to bring the first mode into agreement nor to match the shape of that

resonance. It is likely that the current beam model cannot be updated to agree more closely with the measurements without

0 50 100 15010-7

10-6

10-5

10-4

10-3

10-2

Frequency, Hz

Mag

nitu

de D

isp


Exp DOF 1

Exp DOF 2

Exp DOF 3

Mod DOF 1

Mod DOF 2

Mod DOF 3

Fig. 21.7 Comparison #2 of frequency response curves for the experiment (Exp) and the model (Mod) for model parameters: Eb ¼ 55.5 GPa,

En ¼ 3.9 GPa, k3 ¼ 8.75 � 107 N/m3


relaxing certain assumptions in the model. For example, it may be important to consider the finite stiffness of the support at

the base of the cantilever, rather than treating it as a fixed support. Likewise, the nylon strip may actually impart a linear

stiffness to the beam tip and the mass of the bolts, accelerometers and/or load cell may be significant.

By construction, the methods in this paper can also be used to validate the terms that are used to describe the nonlinearity.

For example, Fig. 21.9 shows the Fourier coefficients of the estimated displacement response of the actual beam for the first

resonance. The coefficients of the first three harmonics (i.e.m ¼ 1, 2, and 3) are shown in subplots (a), (b), and (c), respectively.

12 12.5 13 13.5 14 14.5 15 15.5 1610-6

10-4

10-2 Fourier Coefficients: m=1

Frequency, Hz

|Dis

p|, m

DOF 1

DOF 2

DOF 3

12 12.5 13 13.5 14 14.5 15 15.5 16

10-5

Fourier Coefficients: m=2

Frequency, Hz

|Dis

p|, m

12 12.5 13 13.5 14 14.5 15 15.5 16

10-5

Fourier Coefficients: m=3

Frequency, Hz

|Dis

p|, m

a

b

c

Fig. 21.9 Fourier coefficients of the actual beam’s steady state forced response, extracted during the nonlinear frequency response calculation.

The first three harmonics are shown in (a), (b), and (c)

0 50 100 15010-7

10-6

10-5

10-4

10-3

10-2

10-1

Frequency, Hz

Mag

nitu

de D

isp


Exp DOF 1

Exp DOF 2

Exp DOF 3

Mod DOF 1

Mod DOF 2

Mod DOF 3

Fig. 21.8 Comparison #3 of frequency response curves for the experiment (Exp) and the model (Mod) for model parameters: Eb ¼ 55.5 GPa,

En ¼ 0.24 GPa, k3 ¼ 5.38 � 106 N/m3


The degrees-of-freedom are shown with open squares (DOF 1), open diamonds (DOF 2), and dots (DOF 3). Recall that for

a linear system only the coefficients in 9a would be nonzero. Hence, by viewing the responses in this way one can see that the

first resonance is dominated by the m ¼ 1 and m ¼ 3 responses, while the m ¼ 2 harmonic is significantly weaker. Other

harmonics are present but are not shown.

The harmonics in the response provide information on the type of nonlinearity that is present in the system. In particular,

the m ¼ 3 harmonic is associated with a cubic nonlinearity. The presence of a m ¼ 2 harmonic could be due to the existence

of a quadratic type nonlinearity in the system. So, one could continue to interrogate the harmonics of the responses to see

what terms provide a contribution, and ideally the responses of the model should also contain these harmonic terms. The

model in this paper was not designed with any quadratic nonlinear terms, but these observations suggest that a term of the

form k2yðLÞ2 � C1ðLÞ � � � CNðLÞ½ �T should be inserted into the equations of motion in Appendix A.

21.6 Conclusions

This paper has explored model updating for nonlinear systems by measuring the nonlinear frequency response of the system

and comparing those with the simulated frequency responses of a representative model. The experiment was performed with

a controlled stepped-sine test that measures the steady state, periodic response of the nonlinear system at each forcing

frequency. Then, the measured time histories can be processed to display the nonlinear frequency response over a range of

forcing frequencies. The nonlinear frequency response of the system model was found using a numerical continuation

technique. The algorithm employed a Newton–Raphson shooting and updating technique and finite differences gradients so

that the closed form equations of motion were not needed. One advantage of this approach is that the time histories used by

this algorithm could be supplied by a finite element analysis package, simplifying the modeling process. The frequency

responses contain many characteristics that are similar to linear frequency response functions (e.g. resonance peaks that are

related to the modes of the system), so many of the same principles from linear modal analysis can be used to evaluate and

update the system model. In addition, the nonlinear characteristics of these curves (e.g. the shape of bent resonance peaks),

can be used to validate the nonlinear parameters of the model.

This approach was evaluated using an actual cantilever beamwith a geometric nonlinearity at its tip. The beam’s nonlinear

frequency responses were measured (up to the limits of the modal shaker) and nonlinearity was clearly observed. A simple

Ritz-Galerkin model of the beam was created and its parameters were adjusted to more closely reproduce the three measured

resonance peaks. The locations of the resonance peaks were used to update the linear stiffness of the model, which brought the

model and measurements into much closer agreement. On the other hand, the model had difficulty reproducing the shapes of

the first two resonance curves for any value of the model parameters. The results suggest that other features of the model must

be adjusted to obtain better agreement. For example, the response of the beam was interrogated, revealing that a weak

quadratic term was present and should be added to the beam model in order to more faithfully reproduce the response.

Appendix A: Ritz-Galerkin Discrete Model

A Galerkin approach was used to create a finite-order model of the experimental structure. Assuming that the beam behaves

linear-elastically, mode shapes corresponding to transverse bending motion were used as shape functions to construct the

Ritz-Galerkin representation [21]. The displacement of the beam at a position x was approximated as

y x; tð Þ ¼XNm

r¼1

CrðxÞqrðtÞ (A.1)

where cr(x) is the rth Euler-Bernoulli beam mode shape for a cantilever, qr(t) is the rth generalized coordinate, and Nm is the

number of modes used. The system’s undamped equations of motion are provided in the following equation, where the

coordinates are the amplitudes of the basis functions.

rAbL M½ �€qþ EbI

L3K½ �q ¼ Q ¼

XfextCrðxf Þ (A.2)


For the generalized coordinates, a time derivative is denotedwith an over-dot (e.g. the generalized acceleration vector is €q).Modal damping was added to the system by performing an eigenvector analysis on the linear system and then using,

C½ � ¼ ðrAbLÞ2 M½ � fb½ � diag 2zrorð Þ½ � fb½ �T M½ � (A.3)

where [fb] is a matrix containing the eigenvectors in the columns, or is the rth circular natural frequency, and zr is the rthdesired damping ratio. The generalized force vectorQ is a sum of the product between all external forces and the value of the

shape functions at the point where the force is applied, xf. Therefore, Q includes the applied or external forces, Fext in

Fig. 21.5, as well as the nonlinear restoring force due to the spring [21]. The beam provides linear stiffness at the tip due to its

flexural rigidity, so the discrete spring’s stiffness was chosen to be purely nonlinear

knl ¼ k3yðLÞ2 (A.4)

where k3 is a stiffness constant associated with the nonlinear spring. The physical restoring force due to the spring is then

equal to fsp ¼ k3yðLÞ3. The generalized force vector then has components corresponding to the nonlinear spring located at

x ¼ L and the externally applied force located at x ¼ xf.

Qf g ¼ k3yðLÞ3c1ðLÞ

..

.

cNðLÞ

8><>:

9>=>;þ Aext sin oTtð Þ

c1ðxf Þ...

cNðxf Þ

8><>:

9>=>; (A.5)

Aext is the amplitude and oT the frequency of the external forcing term that produces the limit cycle.

After using the Ritz-Galerkin method to form the discrete beam model and to account for the nonlinear applied force of

the spring, the equations of motion were transformed back into physical coordinates using the relationship in (A.1). The

differential equations of motion can then be arranged in state space format.

_y

€y

� ¼

_y

� Mp

� ��1Cp

� �_yþ Kp

� �yþ Ff g �

( )

Mp

� � ¼ rAbL C½ ��T M½ � C½ ��1; Cp

� � ¼ C½ ��T C½ � C½ ��1;

Kp

� � ¼ EbI

L3C½ ��T K½ � C½ ��1; Ff g ¼ C½ ��T Qf g

(A.6)

The matrix C½ � has the numerical values of the mode vectors for specific position coordinates on the beam. Then, C½ � cancontain shape functions evaluated at the nodal degrees of freedom on the beam. In this study the number of mode shapes used

in the expansion and the number of degrees of measurement points (shown in Fig. 21.5) was N ¼ Nm ¼ 3. The nodes were

located at the center and tip of the beam as shown in Fig. 21.5.

References

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ASME/ASCE/AHS/ASC structures, structural dynamics and materials conference, Orlando, 12–16 Apr 2010

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184:35–58

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82:1413–1423

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6. Gibert C (2003) Fitting measured frequency response using non-linear modes. Mech Syst Signal Process 17:211–218

7. Sracic MW, Allen MA (2011) Numerical continuation of periodic orbits for harmonically forced nonlinear systems. Presented at the 29th

international modal analysis conference (IMAC XXIX), Jacksonville

8. Peeters M et al (2009) Nonlinear normal modes, part II: Towards a practical computation using numerical continuation techniques. Mech Syst

Signal Process 23:195–216


9. Carrella A, Ewins DJ (2011) Identifying and quantifying structural nonlinearities in engineering applications from measured frequency

response functions. Mech Syst Signal Process 25:1011–1027

10. Imregun M et al (1995) Finite element model updating using frequency response function data: I. Theory and initial investigation. Mech Syst

Signal Process 9:187–202

11. Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, vol 42. Springer, New York

12. Kerschen G, Peeters M, Golinval JC, Vakakis AF (2009) Nonlinear normal modes, part I: A useful framework for the structural dynamicist.


13. Sracic MW (2011) A new experimental method for nonlinear system identification based on linear time periodic approximations. Ph.D.

Engineering Mechanics, Department of Engineering Physics, University of Wisconsin-Madison, Madison

14. Sracic MW, Allen MS (2011) Identifying parameters of nonlinear structural dynamic systems using linear time-periodic approximations.

Presented at the 29th international modal analysis conference (IMAC XXIX), Jacksonville, 2011

15. Sracic MW, Allen MS. Identifying parameters of multi-degree-of-freedom nonlinear structural dynamic systems using linear time periodic

approximations. Mech Syst Signal Process, submitted July 2011.

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reverse path method. J Sound Vib 262:889–906

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20:505–592

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15:540–567

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21. Ginsberg JH (2001) Mechanical and structural vibrations theory and applications, 1st edn. Wiley, New York


Chapter 22

Identifying the Modal Properties of Nonlinear Structures

Using Measured Free Response Time Histories

from a Scanning Laser Doppler Vibrometer

Michael W. Sracic, Matthew S. Allen, and Hartono Sumali

Abstract This paper explores methods that can be used to characterize weakly nonlinear systems, whose natural

frequencies and damping ratios change with response amplitude. The focus is on high order systems that may have several

modes although each with a distinct natural frequency. Interactions between modes are not addressed. This type of analysis

may be appropriate, for example, for structural dynamic systems that exhibit damping that depends on the response

amplitude due to friction in bolted joints. This causes the free-response of the system to seem to have damping ratios

(and to a lesser extent natural frequencies) that change slowly with time. Several techniques have been proposed to

characterize such systems. This work compares a few available methods, focusing on their applicability to real

measurements from multi-degree-of-freedom systems. A beam with several small links connected by simple bolted joints

was used to evaluate the available methods. The system was excited by impulse and the velocity response was measured with

a scanning laser Doppler vibrometer. Several state of the art procedures were then used to process the nonlinear free

responses and their features were compared. First the Zeroed Early Time FFT technique was used to qualitatively evaluate

the responses. Then, the Empirical Mode Decomposition method and a simple approach based on band pass filtering were

both employed to obtain mono-component signals from the measured responses. Once mono-component signals had been

obtained, they were processed with the Hilbert transform approach, with several enhancements made to minimize the effects

of noise.

Keywords Iwan joint • System identification • Damping • Bolted joint • Nonlinear joint

22.1 Introduction

Many built-up systems consist of substructures that are assembled with bolted joints. Although some significant strides have

been made in recent years, it is still exceedingly difficult to predict the nonlinear damping behavior of bolted joints, caused

by micro- and macro-slip in the bolted joint interfaces. New experimental methods are needed to allow one to characterize

the nonlinear damping in real structures so better models can be created. In the recent literature, researchers have applied

several approaches in order to identify nonlinear damping from structures. The most common approach involves using some

form of time-frequency analysis [1]. For example, the Hilbert transform [2, 3] has been widely used to estimate the

instantaneous frequency and phase of a signal. This method is quite satisfactory for single frequency component signals

of single degree-of-freedom systems. Furthermore, the method is extended to multi-frequency component signals with the

Hilbert-Huang transform, which uses Empirical Mode Decomposition [4] to decompose the original response into several

single frequency component signals. In a more recent paper [5], the authors relate the Empirical Mode Decomposition

approach to the analytical slow flow analysis, and this approach provides a theoretical basis that promises to extend these

M.W. Sracic (*) • M.S. Allen

Department of Engineering Physics, University of Wisconsin-Madison, 535 Engineering Research Building,

1500 Engineering Drive, Madison, WI 53706, USA


H. Sumali

Component Science and Mechanics, Sandia National Laboratories, 5800, Albuquerque, NM 87185-1070, USA




269




concepts to multi-degree of freedom nonlinear systems. The wavelet transform is an alternative to the Hilbert-Huang

approach. In [6], the authors used the wavelet transform to analyze free-decay time responses of a built-up beam system,

but this type of analysis becomes challenging if the damping is not very light or if the nonlinearity is strong causing the

spectra to become difficult to interpret visually. Peeters et al. [7] have proposed an important extension to this approach

where the system is excited at a specific nonlinear normal mode and allowed to freely decay along that mode. A controlled

input (e.g. a sinusoidal input from a shaker) is typically required. However, attaching a shaker adds mass and damping to the

structure and inhibits its free response.

Several methods have been suggested to identify joint properties using measured frequency response functions,

for example [8–11]. However, these methods may be sensitive to measurement noise, may only provide valid models for

certain frequency ranges, or may require one to assume some information regarding the model for the joints a priori. In any

event, these approaches rely on linear theory, so they don’t seem to be able to predict the amplitude dependent damping that

is characteristic of many systems with bolted joints.

While several methods are available to identify nonlinear models of systems with bolted joints, all of the available

methods have limitations and none has proved to be the best method in all situations. Furthermore, few of the methods have

been applied to real measurements from high order systems. This work will compare several of the most promising methods

in order to evaluate their relative merits. In order to ground the comparison in a real, yet relatively simple system, a test

structure was created that consists of a free-free steel beam (i.e. suspended with elastic strings) with several steel links

attached. The links are bolted to the beam in various combinations using various torque values, and the beam is excited by an

impulsive force and allowed to freely vibrate while its velocity response is measured with a scanning laser Doppler

vibrometer (SLDV). The SLDV is non-contact, so the ring-down responses are not affected by the sensor. The time response

is recorded as the beam freely vibrates. Additionally, the linear frequency response function is also estimated using the

impulsive force, which was also measured. Both the time histories and linear frequency response functions are used to

characterize the damping of this high order nonlinear system. First, standard experimental modal analysis is performed and

the best-fit linear damping is extracted at different bolt torques in order to get a baseline linear approximation of the damping

trends for different torques. Then, the time histories are interrogated using both the Hilbert-Huang Transform with Empirical

Mode Decomposition and single-mode band-pass filtering, in order to isolate individual frequency component signals

(Intrinsic Mode Functions). In the end, a curve-fitting procedure that was presented in [12] is used to fit the nonlinear time

dependent properties of the Intrinsic Mode Function, and the nonlinear time dependent frequency and damping is extracted.

The rest of this paper will review the theory for the methods that will be used, introduce the linked-beam experiment,

show and discuss the results from when the proposed methods are applied to the responses from the experiment, and discuss

and present some conclusions based on the applied techniques.

22.2 Theory

The free response of a general nonlinear system can be represented by the following state space equation

yðtÞ ¼ f ðx; tÞ (22.1)

where xðtÞ is the time dependent state vector and f is a nonlinear function that describes how the state and input combine to

define the response. The function f is assumed to be sufficiently smooth so that all partial derivatives are well defined.

When this is the case, the system can be linearized about specific points in the state space or about entire trajectories (e.g.

periodic orbits [13]). In general the linearized modes of a nonlinear system interact and exchange energy, so one must

consider all of the linearized modes and their nonlinear couplings to construct the free response. Indeed, this characteristic

has even been exploited to create a very effective nonlinear vibration absorber [14]. On the other hand, systems with weak

nonlinearities are frequently observed to have insignificant modal interactions, in which case the free response can be

expressed as follows. Let the response y(t) define the free-decay velocity of the system such that yj(t) ¼ vj(t) is the jthmeasured velocity, which conforms to the property just described. From here forward, the response v(t) will be assumed to

be from the jth degree of freedom and the subscript will be dropped. Then, for a quasi-linear system (i.e. a nonlinear

system with smooth nonlinearities that vary slowly with time), the free decay velocity can be represented with the

following equation

yjðtÞ ¼ vðtÞ ¼Xmr¼1

ArðtÞ cos od;rtþ ’0;r

� �(22.2)


where m is the number of frequencies present in the response, ArðtÞ is the time varying amplitude for the rth frequency

component, od;r is the rth damped natural frequency, and ’0;ris the rth phase variable. If the response is linear, then the

frequency will be constant and the amplitude given by ArðtÞ ¼ A0e�zron;r t, where on;r is the natural frequency and is related

to the damped natural frequency by od;r ¼ on;r

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� z2

p. If the velocity response is nonlinear, then the damped natural

frequency and hence the argument of the cosine function may vary with time and the amplitude may not be a simple

exponential. The experimental methods of interest in this work seek to characterize the time dependent frequency and

damping in order to obtain the amplitude-frequency and amplitude-damping relationships for the system.

Although it will not be pursued in this work due to space limitations, the complexification approach discussed in [5]

can be used to compute the time varying amplitude and frequency of a nonlinear system from its equation of motion,

establishing a more solid theoretical foundation for (22.2).

22.2.1 Zeroed-Early Time Fast Fourier Transform

The first method considered is the Zeroed early-time fast Fourier transform (ZEFFT) that was presented recently by Allen

and Mayes [15]. This frequency domain technique was shown to be quite effective at detecting nonlinearity even in

relatively high order systems with quite severe nonlinearities. This method is briefly reviewed below.

For many systems, nonlinearities are only present when the system responds with large amplitude, so the response

becomes more linear as the response amplitude diminishes. At very low amplitude one may reach a point where the response

is linear so that the free response can be written as

vðtÞ ¼Xmr¼1

Rr expðlrtÞ þ R�r expðl�r tÞ (22.3)

where Rr and lr are the rth residue and eigenvalue, respectively, and the complex conjugate is denoted with ()*. If the

system has under-damped modes, then the complex conjugate eigenvalue pairs are defined by lr ¼ �zron;r þ iod;r where

the coefficient of critical damping, zr, has been introduced and i ¼ ffiffiffiffiffiffiffi�1p

. The Fourier transform can be used to compute

the frequency domain counterpart to (22.3) (assuming that the response is zero for t < 0).

VðoÞ ¼Xmr¼1

Rr

io� lrþ R�

r

io� l�r(22.4)

As described in [15], this is identical to the expression for the frequency response of a linear system in terms of its modes,

except that the residues in (22.4) have a different definition than the residues of a linear frequency response function. Even

then, the spectrum has the same shape as the frequency response of a linear mode near each natural frequency.

Now if the signal is artificially set to zero up to a certain time denoted tz (i.e. vz(t) ¼ 0 for t < tz and vz(t) ¼ v(t)otherwise), then the Fourier transform of vz(t) is

VzðoÞ ¼Xmr¼1

Rrelr tz

io� lrþ ðRre

lr tzÞ�io� l�r

� �eiotz (22.5)

and the residues change to reflect the initial value of the response at time tz but otherwise the spectrum has approximately the

same shape, especially near the peaks. If the spectra are compared for various zero times (various amounts of the initial

response erased) one would see peaks that have essentially the same shape but with decreasing amplitude.

On the other hand, if the system has the nonlinearity that was described previously then the influence of the nonlinearity

will diminish as tz increases, and when one compares VðoÞ and VzðoÞ the spectrum will show that the effective frequency

(and perhaps damping) of the system has changed. Plots of the spectra versus the truncation time, tz, give a qualitative

description of the nonlinearity in the system. The method can be extended to give quantitative measures of nonlinearity

(i.e. using the backwards extrapolation for nonlinearity detection (BEND) technique described in [15]), although in this work

it will be used primarily to detect nonlinearity and evaluate it qualitatively.

22 Identifying the Modal Properties of Nonlinear Structures Using Measured. . . 271

22.2.2 Hilbert-Transform

Slow-flow analysis is often realized through the Hilbert transform, which can be applied as follows. First, the discrete-time

analytic signal is formed by augmenting the real response signal v(t) with its Hilbert transform ~vðtÞ as follows.

VðtÞ ¼ vðtÞ þ i~vðtÞ (22.6)

Note that in this case, the signal v(t) is assumed to be mono-component (contain only one frequency component), so the

‘r’ subscript is dropped from these equations. The magnitude of the analytic signal, VðtÞj j, is the envelope of the response.If the signal is mono-component then the envelope can be readily related to the damping in the system.

AðtÞ ¼ VðtÞj j (22.7)

Assuming that the damping and frequency are slowly varying functions of time, the amplitude or decay envelope can be

estimated as the following

AðtÞ ¼ A0 exp �zðtÞonðtÞð Þ (22.8)

where A0 is the initial amplitude, and the natural frequency onðtÞ and coefficient of critical damping zðtÞ are both functions

of time for a general nonlinear system. The phase can be obtained from the analytic form of the measured signal (i.e. (22.6))

using the following equation.

’ðtÞ ¼ tan�1 ~vðtÞ=vðtÞð Þ (22.9)

In order to obtain the damped natural frequency, some authors have time-differentiated the phase signal [2]. However,

most measured signals contain a certain amount of noise which can corrupt the time-differentiated signal. Following [12], an

alternative approach is to use the measured response at time instants t ¼ t0; t1; . . . ; tN�1, (N ¼ number of data points)

and fit the phase signal ’ðtÞ with a polynomial of degree, p.

’ðt0Þ’ðt1Þ...

’ðtN�1Þ

8>>><>>>:

9>>>=>>>;

¼t0p � � � t0 1

t1p � � � t1 1

..

. � � � ...

1

tN�1p � � � tN�1 1

26664

37775

bp

..

.

b1b0

8>>><>>>:

9>>>=>>>;

(22.10)

The polynomial coefficients b0; b1; . . . ; bp can be obtained by a least squares solution of the above system of equations.

Then, since the instantaneous frequency is the time derivative of the phase, the time-varying damped oscillation frequency

can be estimated as the time-derivative of the previous equation.

odðtÞ ¼ d’ðtÞdt

¼pt0

p�1 � � � 1 0

pt1p�1 � � � 1 0

..

. � � � ... ..

.

ptN�1p�1 � � � 1 0

26664

37775

bp

..

.

b1b0

8>>><>>>:

9>>>=>>>;

(22.11)

The next step is to estimate the decay envelope. Again, assuming that the signal is nonlinear, the decay envelope will be

time varying and can be well approximated with a polynomial. The coefficients of the polynomial can be calculated from the

following equation (assuming a third order polynomial).

ln V t0ð Þj jln V t1ð Þj j

..

.

ln V tN�1ð Þj j

8>>><>>>:

9>>>=>>>;

¼t03 t0

2 t0 1

t13 t1

2 t1 1

..

. ... ..

. ...

tN�13 tN�1

2 tN�1 1

26664

37775

c3c2c1

ln A0ð Þ

8>><>>:

9>>=>>;

(22.12)


Now the above cubic regression analysis gives a nonlinearly decaying envelope,

AðtÞ ¼ A0 exp �c1t� c2t2 � c3t

3� �

(22.13)

which implies that the following relationship holds.

zðtÞonðtÞ � CðtÞ ¼ c1 þ c2tþ c3t2

� �(22.14)

Then, using the relationship between the damped natural frequency, the natural frequency, and the coefficient of critical

damping, ðodðtÞÞ2 ¼ ðonðtÞÞ2 1� ðzðtÞÞ2� �

, the time-varying natural frequency onðtÞ can be computed.

onðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiodðtÞð Þ2 þ CðtÞð Þ2

q(22.15)

Finally, the time-varying damping ratio z(t) can be computed.

zðtÞ ¼ �CðtÞonðtÞ (22.16)

If the measured response is linear, then only the linear terms will be significant in the polynomial regressions in

(22.10)–(22.12), and the procedure identifies the constant linear natural frequency and coefficient of critical damping.

22.2.3 Empirical Mode Decomposition

In general the response of a nonlinear system is composed of oscillations of multiple different frequencies, so the method in

the previous section cannot be directly applied. One must first isolate individual oscillatory components (often called

intrinsic mode functions or IMFs) so that their time varying frequency and damping can be characterized. The Empirical

Mode Decomposition method was developed to isolate the intrinsic mode functions, which are constrained to obey two

properties. For each intrinsic mode function, the number of local extrema and the number of zero crossings must be equal or

differ by no more than one. Additionally, the envelopes defined by the local maxima and the local minima must have a mean

of zero. Using these two properties, the IMFs can be successively removed from the full, multi-component signal with

an iterative process, which is well described in a few references [1, 4]. To start the process, the measured response (i.e. vðtÞ)is interrogated for its local maxima and minima. A cubic spline is fit to the local maxima and then to the local minima to form

the upper and lower envelops, respectively. Then, the mean of these curves is calculated and designated as m1. The first

estimate for the IMF is formed by subtracting the mean signal from the original signal.

h1 ¼ vðtÞ � m1 (22.17)

The estimate may need to be refined in order to satisfy the intrinsic mode function criteria, so one iterates on the estimate

by repeating the previous process until the kth IMF estimate

h1k ¼ h1ðk�1Þ � m1k (22.18)

satisfies the criteria for an intrinsic mode function. Then, the first intrinsic mode function c1 ¼ h1k can be subtracted from the

original signal

r1 ¼ vðtÞ � c1 (22.19)

In order to start sifting for the remaining IMFs, the procedure is repeated now using the residual signals (i.e. r2 ¼ r1 � c2,. . ., rm ¼ rm�1 � cm). Once the mth residual rm is monotonic, or has only one local extremum [4], then the decomposition is

complete. The procedure is ad hoc, in general, but it often works quite well and, as was shown in [5], it can be directly linked

to the theoretical slow-flow analysis.


22.3 Experimental Application to Linked-Beam System

As mentioned previously, a structure was designed in order to evaluate these methods. The structure was designed to have

the following features:

• Numerous modes in the 0–2,000 Hz range.

• The modes are well separated in frequency.

• Modular attachments can be bolted to the main structure in various combinations to vary the nonlinear damping.

• The attachments do not cause modes to switch order (e.g., the third lowest mode in frequency should remain the third no

matter which links are attached). This feature would allow modal damping to be more easily related to the combination of

attachments.

• The structure can be assembled and disassembled with good repeatability.

• The joints can be readily modeled with sufficient detail using the finite element method (FEM).

• The structure can be modeled analytically with reasonable accuracy.

• The structure can be accurately measured with out-of-plane scanning LDV (so data from many measurement points could

be obtained without modifying the structure).

Figure 22.1 shows a schematic of the beam and three links that were bolted to the beam. The beam was 508 mm long,

50.8 mm wide, and 6.35 mm thick. Six through-holes were drilled on the beam’s midline for ¼-in. fine-thread bolts, which

were used to attach the three links. The bolt hole pattern started at 63.5 mm from one end, and the spacing between holes was

76.2 mm. The three links were 12.7 mm wide and 3.175 mm thick. Through-holes were also drilled near the ends of the links

for the ¼-in. bolts. The links were fastened to the beam with the bolts and a washer between each pair of facing surfaces. All

parts of the structure were made of AISI 304 stainless steel. All mating surfaces on the beam and on the links were polished

to a roughness of 0.1 mm or smoother.

The experiment was designed to minimize the effects of the boundary conditions on the measured damping. A clamped

boundary condition, for example, would cause significant damping at the clamp which could dominate the measured

damping and make it difficult to calculate the damping caused by the bolted joints. Thus, a free-free boundary condition

was chosen. The beam was excited by impacts in order to avoid using excitation hardware with surfaces that rub such as

those of attached force transducers and stingers. The cables required to attach conventional sensors (e.g. accelerometers) can

also introduce damping so the vibration responses were measured with a laser Doppler vibrometer.

The finite element method (FEM) was used to predict the natural frequencies and mode shapes of the structure. Because

FEM modal analysis cannot account for nonlinear rubbing interfaces, the interfaces were fused together in the model so that

the whole built-up structure was modelled as monolithic. The FEMmodel was used to iterate the design towards the features

mentioned above.

While the structure was designed to allow various combinations of link attachments, this paper discusses only a case

where all three links are bolted to the beam. To characterize the effects of bolt torques on modal damping, the torque on all

the six bolts was varied using 9.04 Nm (80 Lbf.in.), 10.2 Nm (90 Lbf.in.), and 12.4 Nm (110 Lbf.in.).

22.3.1 Experiment

The beam was suspended in a manner that emulated the free-free boundary condition as shown in Fig. 22.1a. Two strings

suspended the beam from overhead points. Four elastic strings kept the beam from swinging too much out-of-plane since that

would cause the laser spot to depart from the measurement point of interest. A small patch of retro-reflective tape was

Beam

Link C

Link B

Link A

Washers

View C-C: Nut

BeamLink

Bolt head

C-C

Fig. 22.1 Finite element

model of the linked-beam


adhered to the beam at each measurement point (see Fig. 22.3). These patches ensured that the LDV sensor received

adequate light even if the test article rotated relative to the laser.

The structure was excited with an impulsive force from a shaker-impactor as follows. An APS 400 long-stroke shaker

carried a force transducer at the end of its armature. A special waveform command was designed and applied to

the amplifier, which caused the shaker to push the force transducer into the beam then to retract quickly after impact. The

force transducer was then held in the retracted position so that the swinging of the beam did not cause a second impact. After

the rigid-body deflections of the structure dissipated, the shaker brought the force transducer close to the structure again and

the process was repeated.

The peak force of the impact was around 150 N. Upon each impact, the laser Doppler vibrometer (LDV) computed the

mobility (frequency response function) between the input and response. The sampling rate of 12,800 samples/s gave a

bandwidth of 5,000Hz. The scanning LDV recorded the mobility at 63 points shown in Fig. 22.3 using four averages at each

point. The impact location was behind point 28. Point 22 was on link B, and point 53 was on the main beam.

22.3.2 Nonlinearity Detection with Zeroed-Early Time Fast Fourier Transforms

The measurements were first interrogated using the Zeroed Early-time FFT (ZEFFT) method [15], in order to obtain a

qualitative understanding of the degree to which the structure is behaving nonlinearly. The measurements with the bolts at

a torque of 10.2 Nm were first investigated. There were 62 measurement points, so the ZEFFTs of the entire dataset were

computed and the average of the magnitude of the ZEFFTs was computed over all of the measurement points for each zero

time. Figure 22.4 shows the resulting average spectrum for various values of tz. The system is predominantly linear, so the

spectrum near each peak must be closely examined to see any sign of nonlinearity. Figures 22.5 and 22.6 show expanded

views near the first and third bending modes respectively. The former shows that the first natural frequency appears to be

constant, within the resolution of the measurement at least. The third natural frequency seems to show a very small shift from

Fig. 22.2 (a) Test structure

suspended by two vertical in-

plane strings and four out-of-

plane elastic strings and

(b) close up showing the

beam, one link with its bolted

joints, and the force transducer

on the shaker armature

Fig. 22.3 Laser Doppler

vibrometer scan points

on the structure


801.1 to 801.5 Hz (at tz ¼ 1.2 s. or later). All of the other peaks were similarly inspected and small, 0.5 Hz frequency shifts

were also noticeable in the modes at 1,192 and 2,871 Hz. All of the other peaks seemed to have constant frequencies.

While this analysis suggests that the frequencies of these modes remain essentially constant, visual inspection does not

reveal whether the damping is changing nonlinearly. The BEND procedure described by Allen and Mayes [15] could be used

to assess this to some extent, although this will not be pursued in this work. Since the nonlinearity seems to be quite weak,

it might be reasonable to approximate this system as linear. This is explored in the following section.

22.3.3 Linear Modal Analysis Using Low-Excitation Frequency Response Functions

Since the ZEFFT method showed weak evidence of nonlinearity, the standard approach using linear modal analysis was first

attempted to see whether a linear model might be adequate to characterize the system at each torque value. This section

discusses results from a separate modal test where the peak impact force was 50 N instead of 150 N. The purpose of the test

0 500 1000 1500 2000 2500 300010-2

10-1

100

101

102

103

Frequency - Hz

Mag

nit

ud

e

NLDetect: Composite of FFTs of Time Response

0.025977

0.32199

0.61801

0.91402

1.21

1.5061

1.8021

Fig. 22.4 Composite ZEFFT

of the response of beam at a

torque level of 10.2 Nm

135 136 137 138 139 140

101

102

103

Frequency - Hz

Mag

nit

ud

e


0.025977 0.32199

0.61801

0.91402

1.21

1.5061 1.8021

Fig. 22.5 Expanded view

of Fig. 22.4 near the natural

frequency of the first bending

mode


was to investigate the effect of bolt torques on linearized damping. The LDV software computes an ‘average spectrum’,

which is an average of the mobilities at all the scanned points. The resonant frequencies of the structure correspond to the

peaks of the average spectrum. In this paper, those peaks will be called ‘peaks of average mobility’ (PAM). The PAM gives a

good estimate of the natural frequencies of the structure. These estimates were useful in determining poles in the linear

experimental modal analysis (EMA). The LDV software also plotted the mobility values at all scanned points at a chosen

frequency. At each PAM, the spatial distribution of those mobility values (i.e. operating deflection shape) gives a good

estimate of the mode shape since none of the modes have close natural frequencies.

Linear curve fitting was performed using accelerances (computed from the mobilities) from the measurements at each

torque level. ATA Engineering’s AFPoly software was used to perform the curve fitting. The low end of the analysis

frequency band was set to 30 Hz to exclude all rigid-body modes caused by the soft free-free suspension cords. The mode

indicator function (MIF) and ‘stability’ diagram showed estimates of where the natural frequencies are likely to be. In almost

all cases, the peaks of the MIF were collocated with the trains of poles on the stability diagram. These frequencies were very

close to the PAM from the LDV software. Thus, for the most part the EMA was straightforward. Appendix A shows the MIF

and stability diagram, along with sketches of the mode shapes. Near 2,500 and 4,000 Hz there is strong indication of a mode.

However, observation of the shape from the PAM led to the conclusion that the modes around those two frequencies involve

primarily in-plane motion. Because the links were attached to only one side of the beam, the in-plane motion resulted in a

little out-of plane motion. Those modes are probably not well measured by the laser and hence will not be analyzed here.

EMA was difficult near certain frequencies because the three nominally identical links resonate at similar frequencies.

The combinations of one, two and three links resonating together created a high modal density; five modes were found

between 2,870 and 3,015 Hz. In that range, the PAM and stability diagram had to be used together to identify the modes.

Despite the high modal density, the five modes in that range were identified. For example, Fig. 22.7 shows good agreement

between accelerances from measurement (solid curves) and from the synthesis (dashed curves) of the identified modes for

the bolt torque case of 9.04 Nm. A careful comparison does show differences between the reconstruction and the

measurements that are on the order of 10% of the peak, but differences such as this can arise for many reasons and they

are not large enough that one would typically call the linear model into question.

Table 22.1 shows the modal damping ratios for 21 modes with three bolt torques: ‘low’ torque of 9.04 Nm, ‘medium

torque’ of 10.2 Nm, and ‘high’ torque of 12.4 Nm. The plots of most of the mode shapes are from a finite element analysis

(FEA). The rest of the plots (modes 4c, A1, etc.) came from the spatial distribution of mobility amplitudes at the PAM,

because the monolithic FEA did not predict those modes. (The modes in question seemed to depend strongly on

the characteristics of the interfaces in the joints.) The modes are shown in Table 22.1 from top to bottom in the order

of increasing natural frequencies. The bars in the chart show the identified linear modal damping of each mode as percentage

of critical damping. Three bars are shown for each mode: the bar on the top represents damping for the low bolt torque,

the middle bar for the medium torque, and the bottom bar for the high torque.

799 800 801 802 803 804 805 806 807

10-1

100

101

Frequency - Hz

Mag

nit

ud

e


0.025977

0.32199

0.61801

0.91402

1.21

1.5061

1.8021

Fig. 22.6 Expanded view of

Fig. 22.4 near the natural

frequency of the third bending

mode


Mode 4c had much higher damping than the rest of the modes, so its damping was multiplied by 0.1 prior to displaying it

on the plot. This mode is of questionable accuracy; its shape is very similar to mode 4, but it has a strong local motion of link

C. This mode could be an artifact caused by processing nonlinear measurements with linear frequency response estimation

and modal analysis procedures.

Using Table 22.1, one can assess the effect of bolt torques on modal damping. The damping for these 13 modes clearly

decreases as the bolt torques increase. In five other modes, the low torque still gives higher damping than the high torque, but

the medium torque gives the lowest or highest damping among the three torques. Three modes (1, 4c and 6) exhibit damping

that increases with higher torque, which is the opposite trend from other modes.

22.3.4 Nonlinear Modal Analysis Using Free Ring-Down History

The linear modal analysis in the preceding section gave an estimate of the modal damping of the structure. The following

section discusses three techniques to study the nonlinear damping. The free response velocity due to 150 N peak impacts was

measured at each of the points in Fig. 22.3 for 2.56 s at all three of the torque levels. Figure 22.8 shows the response at the

torque level of 10.2 Nm for point 28, which is at the same location as the impactor. The dashed blue curve is the measured

response. There is a considerable amount of low frequency vibration in the response (likely due to rigid body motion of the

beam on the suspension system). The response was high-pass filtered with an eighth order Butterworth filter and a cut off

frequency of 30 Hz. The filtered response is plotted with the red curve, which seems to remove the low frequency motion.

This signal will be used as a benchmark in this paper, although there were many other measurement points available. The

bolted joints in this beam provide very complicated nonlinear damping relationships, and before the spatial information

provided by the numerous measurement points can be taken into account we seek first to characterize a single response.

Figure 22.9 shows the FFT magnitude of the velocity response of point 28 (after filtering and drop out bridging). There are

many sharp peaks in the response out to 12 kHz, so the signal certainly has multiple frequency components.

The response of point 28 is characteristic of many of the responses measured on the beam. There are many modes

involved in each response, but most of the modes are well separated. The goal is to determine which modes contain

nonlinearity and to try to identify the time dependent properties of those nonlinearities. Since the free velocity response

contains multiple significant modes, it is necessary to first isolate mono-component signals which contain the time

dependent properties of a single mode.

22.3.4.1 Application of Empirical Mode Decomposition

In order to isolate mono-component signals, the Empirical Mode Decomposition (EMD) method was applied to the free

velocity response. For this paper, the EMD implementation from Ortigueira that is available on MATLAB Centrals File

exchange was used, and this algorithm was based on [16]. The EMD procedure extracted 12 intrinsic mode functions (IMFs)

2850 2900 2950 3000 305010-1

100

101

102

103

104

Frequency, Hz

|Acc

eler

atio

n/F

orce

|, m

/s2 /

N

22 Meas22 Synth53 Meas53 Synth

Fig. 22.7 Accelerances at

points 22 (higher magnitudes)

and 53 (lower magnitudes) in

the 2,850–3,050 Hz range for

bolt torque case 9.04 Nm;

solid curves are frommeasurements; dashed curvesare AFPoly synthesis


from the free velocity response. Figure 22.10 shows plots of the first three IMFs. The time domain signals are plotted

in Fig. 22.10a, and the frequency domain signals are plotted in Fig. 22.10b from 0 to 7,000 Hz. The spectra of the first IMF

(top plot of (b)) still contains several significant peaks near 4,000, 5,600, and 6,600 Hz as well as several less coherent peaks

at lower frequencies. The spectrum of the second IMF contains one dominant peak near 400 Hz, and the third IMF contains

two dominant peaks near 130 and 360 Hz.

The EMD algorithm sifted through the components of the free velocity response and extracted several signals that

contained significantly less frequency content. The second IMF, for example, seems to contain only one major component.

However, as seen in the first and third IMF, the signals still contain the effects of several modes. Moreover, the EMD process

adds a certain amount of broad band noise to the responses, which can be seen in the frequency spectra of the IMF signals.

Table 22.1 Beam with three links ABC, various bolt torques

# Shape 1

2

T1

3

4

4c *0.1 T2

5

6

T3

A1

T4

A+ B-C- B+ C- B1

7

T5

8

T6

7b

9

Bar chart on the right depicts the critical damping ratio for each mode for low torque ¼ 9.04 Nm (blue bar), medium torque ¼ 10.2 Nm

(green bar), and high torque ¼ 12.4 Nm (red bar)Mode 4c was much more heavily damped than the rest, so its damping ratios were multiplied by 0.1 in order to fit on the bar chart

The table on the left depicts the corresponding mode shapes


0 0.5 1 1.5 2 2.5-0.2

-0.1

0

0.1

0.2

0.3

time, s

Poi

nt 2

8 ve

loci

ty, m

/s

Drive Point (28) Velocity Ringdown

MeasuredHP Filt, 30Hz

Fig. 22.8 Free velocity response of point 28: measured (dashed blue), high-pass filtered with a 30 Hz cut off frequency (red)

10−1

10−2

10−3

10−4

10−5

IFF

T(V

eloc

ity)

I, m

/s

10−6

10−7

0 2000 4000 6000Frequency, Hz

FFT of Response 28

8000 10000 12000

Fig. 22.9 FFT magnitude of the free velocity response of point 28

0 0.5 1 1.5 2 2.5 3-0.05

0

0.05

Vel

ocity

, m/s

IMF 1

0 0.5 1 1.5 2 2.5 3-0.05

0

0.05

Vel

ocity

, m/s IMF 2

0 0.5 1 1.5 2 2.5 3-0.05

0

0.05

time, s

Vel

ocity

, m/s IMF 3

0 1000 2000 3000 4000 5000 6000 7000

10-5

|Vel

ocity

|, m

/s

IMF 1

0 1000 2000 3000 4000 5000 6000 7000

100

|Vel

ocity

|, m

/s IMF 2

0 1000 2000 3000 4000 5000 6000 7000

10-5

Frequency, Hz

|Vel

ocity

|, m

/s IMF 3

a b

Fig. 22.10 Application of Empirical Mode Decomposition where the first three intrinsic mode functions are shown: (a) time domain,

(b) frequency domain

Since there are numerous modes in the free velocity response, it is possible that the EMD procedure has

difficulty isolating all of the individual components. The algorithm may perform better if the response was first low-pass

or high-pass filtered to reduce the number of modes that need to be separated. The authors are exploring these ideas, but

the Hilbert transform method used in this paper performs best when the signal contains only one frequency component,

so the EMD signals were not processed further. Instead, a different sifting procedure was implemented as described in the

next section.

22.3.4.2 Band-Pass Filtering of Individual Modes

Next, the authors applied a single mode filtering procedure where different Butterworth filters were applied to the time

domain free velocity response in order to isolate individual frequency components in the signal. The ZEFFT analysis did not

show much frequency shift in any of the modes, so the stiffness of the system is predominantly linear. Hence, it is unlikely

that any of the modes contain harmonic peaks at higher frequencies, but the spectra can always be visually scanned to see

if this type of the nonlinear phenomenon is occurring. The free velocity response did not show signs of harmonics of any of

the modes occurring at higher frequencies, so it was assumed that each mode could be isolated with a single bandpass filter.

Figure 22.11 shows the results when the first two modes were isolated using this filtering procedure. First, the free velocity

response was filtered with a tenth order low-pass Butterworth filter with a cut-off frequency of 240 Hz. The frequency

spectra of the measured signal and the filtered signal are shown in Fig. 22.11a. The filtered time domain signal is shown

10−10

10−10

10−8

10−8

10−6

10−6

10−4

10−4

10−2

10−2

100

100

0.08

0.05

−0.05

0

0.06

0.04

0.02

−0.02

−0.04

−0.06

−0.08

0

0 500 1000

|Vel

ocity

|, m

/s|V

eloc

ity|,

m/s

Vel

ocity

, m

/sV

eloc

ity,

m/s

1500

Filtered Response

Filtered Response

Measured Response

Measured Response

2000 0 1 20.5 1.5 2.52500

0 500 1000 1500 2000 0 1 20.5 1.5 2.52500

Filtered Mode 1a

c d

b Filtered Mode 1

Filtered Mode 2 Filtered Mode 2

Frequency, Hz

Frequency, Hz

time, s

time, s

Fig. 22.11 Single mode filtering of the free velocity response of point 28: (a) tenth order low-pass Butterworth filter for Mode 1 with cutoff

frequency of 240 Hz, (b) filtered time domain response of Mode 1, (c) fifth order band-pass Butterworth filter for Mode 2 with cutoff frequencies

300 and 450 Hz, (d) filtered time domain response of Mode 2


in Fig. 22.11b. In order to isolate the second mode, the measured signal was filtered with a fifth order band-pass Butterworth

filter with cut-off frequencies of 300 and 450 Hz. The frequency spectra and the time domain response are shown

in Fig. 22.11b,c, respectively.

The single mode filter approach effectively isolates a single mode and produced a mono-component time domain signal

for both the two modes that are shown. This could be applied to most of the modes in the free velocity response, especially

since the modes are well separated. For a structure with close modes, a different approach may be needed. For example a

band of close modes could first be isolated using a band-pass filter similar to the one used for Mode 2 above. Then, one might

try Empirical Mode Decomposition on the resulting signal, and it may be more effective since the overall number of modes

in the band is likely to be much fewer than the measured signal. These ideas are being explored for future work.

22.3.4.3 Hilbert Transform with Curve Fitting

Once one has isolated mono-component signals from the measured response, the Hilbert transform curve fitting approach

described in Sect. 22.2.2 can be applied to extract the nonlinear time dependent properties of the system. First, the phase

signal from (22.6) is fit with a 5th degree polynomial. The time derivative of the polynomial fit is equal to the damped natural

frequency, and this can be formed with (22.11). Then, the amplitude of the Hilbert transformed signal can be fit with a cubic

polynomial in order to extract the time dependent damping. Figure 22.12 shows the Hilbert transformed velocity Amplitude

(a) and phase (b) as well as the curve fits of those signals. Figure 22.13 shows the time dependent natural frequency

and damping that were extracted from the curve fits. The natural frequency changes by less than 1% over the course of the

free response, however, this is likely spurious since the modes were found to be predominantly linear in frequency.

The damping changes a significant amount over time, and as shown in Fig. 22.14, the damping is clearly nonlinear since

it is a function of the amplitude of the response. The same procedure can be followed for the filtered signal of Mode 2.

The damping-amplitude relationship of Mode 2 is plotted in Fig. 22.15.

0 1 2 3-500

0

500

1000

1500

2000

2500

Ang

le (

rad)

time, s

Curve Fit of Phase

Hilbert-Phase5th Degree Polynomial

0 1 2 310-8

10-7

10-6

10-5

Mag

nitu

de

time, s

Curve Fit of Amplitude

Hilbert-AmplitudeCubic Polynomial

a bFig. 22.12 Curve fit of the

amplitude (a) and phase

(b) of the Hilbert transform

of the filtered Mode 1

0 1 2 3135.8

135.9

136

136.1

136.2

Nat

ural

Fre

quen

cy, H

z

Time, s

Natural Frequency vs Time

0 1 2 30

0.05

0.1

0.15

0.2

0.25

zeta

, % o

f Crit

ical

Time, s

Damping vs Timea bFig. 22.13 Time varying

natural frequency and

damping that were extracted

from the curve fits in

Fig. 22.12


22.3.4.4 Discussion

The Empirical Mode Decomposition method and the single mode filtering method were both used to isolate

mono-component signals from the measured response. The single mode filtering method was found to produce very clean

signals with a single, distinct frequency component. The identified nonlinear damping functions had clear amplitude

dependence, and those functions were different due to the differences in filtered modes. Since the beam has well spaced

modes, the single mode filtering approach could be used to identify the nonlinear damping functions on the remainder of the

modes. For other systems with closely space modes, a combination of band-pass filtering and Empirical Mode Decomposi-

tion may produce the mono-component signals needed to apply the Hilbert transform curve fitting method. In any event,

it seems that these tools can be readily applied to characterize the damping nonlinearity in this structure.

22.4 Conclusions

This paper reviewed a few methods for system identification of nonlinear systems with slowly time-varying nonlinear

properties, especially damping. In particular, the Empirical Mode Decomposition method was compared with a method

based on band-pass filtering around each resonance, in order to obtain single mode responses that could be processed using

the Hilbert transform. The Zeroed Early-Time FFT (ZEFFT) was also discussed and applied to the measurements, and was

found to allow one to quickly and quite robustly identify which modes had natural frequencies that were amplitude

dependent. That method was not explored further in this work, although it might be helpful in obtaining more quantitative

results in the future. The Hilbert transform method seemed to provide the most convenient approach for quantifying the

10-7 10-6 10-5-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Amplitude

zeta

, % o

f crit

ical

Damping vs. AmplitudeFig. 22.14 Nonlinear

damping plotted versus the

amplitude of the Hilbert

transformed signal of Mode 1

10-8 10-7 10-60

0.02

0.04

0.06

0.08

0.1

0.12

Amplitude

zeta

, % o

f crit

ical

Damping vs. AmplitudeFig. 22.15 Nonlinear

damping plotted versus the

amplitude of the Hilbert

transformed signal of Mode 2


dependence of the frequency and damping on time (and hence amplitude), however the signal of interest must be first

decomposed into mono-component signals. The Empirical Mode Decomposition method and a band-pass filtering method

were both used, but the former gave quite unsatisfactory results and was not pursued further. This may not be a fault of the

algorithm; the authors are certainly not experts and were at the mercy of a particular implementation of the EMDmethod. On

the other hand, simple band-pass filtering was quite effective for the system considered here, although it is difficult to be sure

that the filter has not distorted the signal of interest. In any event, once a mono-component response was available the Hilbert

transform method was quite effective.

The proposed methods were applied to free velocity response measurements of a linked-beam system that contains bolted

joints. The ZEFFTs of the measured responses showed little sign of variation in the natural frequencies of the modes, and this

seems to imply that the stiffness of the system is predominantly linear. Therefore, linear modal analysis was performed at

different torque values to assess any trends in damping versus torque. Some trends were clearly observed but it is difficult to

assess the accuracy of the damping measurement in each configuration and to determine whether the trends observed, where

damping both increased and decreased with increasing torque, were meaningful. The measurements were then processed

with Empirical Mode Decomposition and with band-pass filtering in order to isolate individual modes. The Empirical Mode

Decomposition method was not successful in extracting mono-component signals from the measurements, perhaps because

so many modes were excited in the free response. The band-pass filtering approach worked very well to isolate mono-

component signals, since this system has well spaced modes. Once the modes were isolated, the Hilbert transform curve fit

approach identified significantly nonlinear damping from the measurements, and the damping-amplitude relationships were

displayed. Despite the initial success of the band-pass filtering and Hilbert transform approach, all of the methods are being

investigated further to understand which methods work best in a variety of situations.

Acknowledgment Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned

subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-

AC04-94AL85000.


Appendix A

Linear Curve Fitting of the 9 Nm Data

with ATA Engineering’s AFPoly

Mode shapes and “Stability Diagram” are shown below.

# Shape Modal Indicator Function (curve) and pole orders (squares or triangles)

1

2

T1

3

4

4c

T2

5

6

T3

A1

T4

A+ B-C-

B+ C- B1

7

T5

8

T6

7b

9


References

1. Pai PF (2010) Instantaneous frequency of an arbitrary signal. Int J Mech Sci 52:1682–1693

2. Feldman M (1997) Non-linear free vibration identification via the Hilbert transform. J Sound Vib 208:475–489

3. Feldman M (2011) Hilbert transform in vibration analysis. Mech Syst Signal Process 25:735–802

4. Huang NE, Shen Z, Long SR et al (1998) The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time

series analysis. Proc R Soc Lond A 454:903–995

5. Kerschen G, Vakakis AF, Lee YS et al (2008) Toward a fundamental understanding of the Hilbert-Huang transform in nonlinear structural

dynamics. J Vib Control 14:77–105

6. Heller L, Foltete E, Piranda J (2009) Experimental identification of nonlinear dynamic properties of built-up structures. J Sound Vib

327:183–196

7. Peeters M, Kerschen G, Golinval JC (2010) Phase resonance testing of nonlinear vibrating structures. In Presented at the 28th international

modal analysis conference (IMAC XXVIII), Jacksonville

8. Celic D, Boltezar M (2008) Identification of dynamic properties of joints using frequency-response functions. J Sound Vib 317:158–174

9. KimW-J, Lee B-Y, Park Y-S (2004) Non-linear joint parameter identification using the frequency response function of the linear substructure.

J Mech Eng Sci 218:947–955

10. Ren Y, Beards CF (1995) Identification of joint properties of a structure using FRF data. J Sound Vib 186:567–587

11. Tsai J-S, Chou Y-F (1988) The identification of damping characteristics of a single bolt joint. J Sound Vib 125:487–502

12. Sumali H, Kellogg RA (2011) Calculating damping from ring-down using Hilbert transform and curve fitting. In: Presented at the 4th

international operational modal analysis conference (IOMAC), Istanbul

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Process 25:2705–2721

14. Tsakirtzis S, Lee YS, Vakakis AF et al (2010) Modelling of nonlinear modal interactions in the transient dynamics of an elastic rod with an

essentially nonlinear attachment. Commun Nonlinear Sci Numer Simulat 15:2617–2633

15. Allen MS, Mayes RL (2010) Estimating the degree of nonlinearity in transient responses with zeroed early-time fast Fourier transforms. Mech

Syst Signal Process 24:2049–2064

16. Rato RT, Ortigueira MD, Batista AG (2008) On the HHT, its problems and some solutions. Mech Syst Signal Process 22:1374–1394


Chapter 23

Nonlinear System Identification of the Dynamics

of a Vibro-Impact Beam

H. Chen, M. Kurt, Y.S. Lee, D.M. McFarland, L.A. Bergman, and A.F. Vakakis

Abstract We study the dynamics of a cantilever beam with two rigid stops of certain clearances by performing nonlinear

system identification (NSI) based on the correspondence between analytical and empirical slow-flow dynamics. First, we

perform empirical mode decomposition (EMD) on the acceleration responses measured at ten, almost evenly-spaced,

spanwise positions along the beam leading to sets of intrinsic modal oscillators governing the vibroimpact dynamics at

different time scales. In particular, the EMD analysis can separate any nonsmooth effects caused by vibro-impacts of

the beam and the rigid stops from the smooth (elastodynamic) response, so that nonlinear modal interactions caused by

vibro-impacts can be explored only with the remaining smooth components. Then, we establish nonlinear interaction models

(NIMs) for the respective intrinsic modal oscillators, where the NIMs invoke slowly-varying forcing amplitudes that can be

computed from empirical slow-flows. By comparing the spatio-temporal variations of the nonlinear modal interactions for

the vibro-impact beam and those of the underlying linear model (i.e., the beam with no rigid constraints), we demonstrate

that vibro-impacts significantly influence the lower frequency modes introducing spatial modal distortions, whereas the

higher frequency modes tend to retain their linear dynamics in between impacts.

Keywords Nonlinear system identification • Empirical mode decomposition • Vibro-impact beam • Intrinsic mode

oscillation • Nonlinear interaction model

23.1 Introduction

Experimental modal analysis based on Fourier transforms (FTs) has beenwell established based on the assumption of linearity

and stationarity of the measured signals (see, for example [1]). In many practical situations, however, the measured data is

likely to exhibit strong nonlinearity and nonstationarity, particularly when the tested systems involve nonlinearities due to

complexity caused by multi-physics nonlinear interactions [2]. In addition, FT-based methods are not able to properly isolate

and extract nonlinearity and nonstationarity from themeasured data, frequently leading to wrong conclusions (for example, to

misinterpretations of internal and combination resonances as natural frequencies). As a result, there has been the need for an

effective, straightforward, system identification and reduced-ordermodelingmethod for characterizing strongly nonlinear and

nonstationary, complex, multi-component systems in multi-physics applications. Reviews of nonlinear system identification

(NSI) and reduced-order modeling (ROM)methods are provided in [3, 4]. Typical nonparametric NSImethods include proper

orthogonal decomposition (POD, also known as Karhunen-Loeve decomposition [5–8]), smooth orthogonal decomposition

[9], Volterra theory [10, 11], Kalman filter [12], and so on. As for the methods of nonlinear parameter estimation, we mention

H. Chen • Y.S. Lee (*)

Department of Mechanical and Aerospace Engineering, New Mexico State University, 1040 S. Horseshoe St., Las Cruces, NM 88003, USA


M. Kurt • A.F. Vakakis

Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, 104 S. Wright St., Urbana, IL 61801, USA


D.M. McFarland • L.A. Bergman

Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 W. Green St., Urbana, IL 61801, USA




287







the restoring force surface method [13], NARMAX (Nonlinear Auto-Regressive Moving Average models with eXogenous

inputs) methods [14], harmonic balance method [15], methods based on Hilbert transform [16, 17], and others.

Use of POD has been rather popular in studying system identification and nonlinear normal modes of coupled beams [18]

and rods [19], and in structural damage detection [20]. For example, the method of POD has been utilized for studying

chaotic vibrations of a 10-degree-of-freedom (DOF) impact oscillator and a flexible-beam impact oscillator [21, 22]. In these

studies, the spatial structure of impacting responses under a harmonic excitation of the boundary was demonstrated to be

close to what can be obtained by averaging over many impulse-response tests on the linear system (even if the system is

strongly nonlinear). Moreover, POD was applied for model reduction of a vibro-impact (VI) rod [23], and also for extracting

dominant coherent structures of a VI beam from experimental time-series data [24] with the goal to eventually derive low-

dimensional ROMs through a Galerkin reconstruction process based on the extracted mode shape functions.

We note, however, that these techniques are only applicable to specific classes of dynamical systems; in addition, some

functional form is assumed for modeling the system nonlinearity. Recently, a nonlinear system identification (NSI) method

with the promise of broad applicability was proposed by Lee et al. [25]. This method was based on empirical mode

decomposition (EMD) [26], under the key assumption that the measured time series can be decomposed in terms of a finite

number of oscillating components. These are in the form of fast (nearly) monochromatic oscillations modulated by slowly

varying amplitudes. The empirical slow-flow model of the dynamics is obtained from EMD, and its correspondence with the

analytical slow-flow model has been established [27], paving the way for constructing physics-based local nonlinear

interaction models (NIMs) [28]. A NIM consists of a set of intrinsic modal oscillators (IMOs) that can reproduce the

measured time series over different time scales and account for (even strongly) nonlinear modal interactions across scales.

Hence, it represents a local model of the dynamics, identifying specific nonlinear transitions. By collecting energy-

dependent frequency behaviors from all identified IMOs, a frequency-energy plot can be constructed, which depicts global

features of the dynamical system. The method requires no a priori system information but only measured (or simulated) time

series; i.e., it is purely an output-based approach. Applications of the proposed NSI methodology have been provided with

studies of targeted energy transfers in a 2-DOF dynamical system [28], instability generation and suppression in a 2-DOF

rigid aeroelastic wing model [29], and the dynamics of a rod coupled to an essentially nonlinear end attachment [30].

In this paper, we explore the nonlinear dynamics of a VI beam (whose setup is similar to that used in [24]) by performing

the aforementioned NSI method [25] to reveal coherent structures (e.g., Dawes [31]) in terms of IMOs of strongly nonlinear

dynamics due to vibro-impacts. Study of such systems will provide essential dynamical features of structures with defects

with applications to structural health monitoring and damage detection (e.g. [32, 33]). For this purpose, this paper has the

following structure. Section 2 provides a discussion of the VI beam model including geometry, measurement locations,

method of excitation, (linearized) natural frequencies and mode shapes; in Sect. 3 the proposed NSI method is applied to the

numerically-obtained acceleration data of the VI beam and those of the underlying linearized beam for comparison

purposes; then, concluding remarks are provided in Sect. 4.

23.2 System Descriptions

We consider the uniform, homogeneous cantilever beam (made of steel with the density r ¼ 7850 kg/m3 and Young’s modulus

E ¼ 200 GPa) depicted in Fig. 23.1, with dimensions Lxhxt ¼ 1.311 � 0.0446 � 0.008 m so that the cross-sectional area and

the second moment of area with respect to the z axis are A ¼ 3.57 � 10�4 m2 and Izz ¼ 1.9 � 10�9 m4, respectively (we refer

to Fig. 23.1 for a definition of the system of axes). Table 23.1 summarizes the positions of the accelerometers x1–x10 along thebeam span, the position of the laser displacement sensor xLDS, and the placement of the two symmetric rigid stops xSTP causingvibro-impacts. The leading ten natural frequencies (theoretical and experimental) on in Hz are listed in Table 23.2, with

the corresponding normalized mode shape functions fn(x/L), n ¼ 1,. . ., 10, being presented in Fig. 23.2 [34, 35].

1x 2x 3x 4x 5x 6x 7x 8x 9x 10xLDSx

x

y

y

zh

t1x 2x

3x 4x

5x 6x 7x 8x9x 10x

LDSx

STPx( )p t

( , )v x t

L

z

x

Fig. 23.1 Experimental setup for the VI beam: x1–x10, xLDS, and xSTP respectively denote the spanwise locations of the accelerometers, of the laser

displacement sensors, and of the rigid stops

288 H. Chen et al.

Two clearance levels between the cantilever beam and the rigid stops are considered, namely, infinite clearance

corresponding to the case of the linear cantilever beam, and 4 mm clearance corresponding to the case of the strongly

nonlinear VI beam. Experimental procedures for measuring time series data involve (1) applying impulsive excitation p(t)by varying magnitude at position x3 by means of an impact hammer, selecting the excitation frequency band by using several

types of tips on the impact hammer (e.g., plastic, rubber and metal), and (2) measuring the resulting accelerations at x1�x10and the displacement at xLDS.

In thisworkwe utilize numerically generated acceleration signals froma reduced-ordermodel based on the assumed-modes

method, and such numerical solutions are updated and validated by the experimental measurements. That is, the beam

was excited at each node with an impact hammer, and averages of four measurements were taken at each node; from

the resulting 100 transfer functions, the leading ten mode shapes, modal damping factors, and natural frequencies were

obtained and used to update the assumed-modes model. In the assumed-modes method the analytical natural frequencies

were replaced with the measured ones, and numerically simulated time series were obtained by solving the reduced

system of differential equations. Details of this computation can be found in [34]. We remark that the 5th mode, whose

linearized natural frequency is equal to 209 Hz, has a node at x9, which is located very close to the point of vibro-

impacts xSTP. Furthermore, the impulsive excitation is applied at location x3, which is also close to another node of the

5th mode. As shown below, this will affect the results of EMD analysis used for reconstructing the 5th mode at those

particular points (i.e., there will arise issues of observability) in the sense that the flexible dynamics of the beam at

these locations is small and consequently is dominated by the vibro-impacts (non-smooth effects). Similar observations

apply for the 8th mode, which possesses a node near the excitation point (x3).

23.3 Nonlinear System Identification of the VI Beam

In this section we apply the NSI methodology to two typical cases: (1) The linear beam (i.e., a cantilever beam with infinite

clearances at the impact boundaries); and (2) the vibro-impact (VI) beam with 4 mm symmetric clearances. In particular, by

comparing the system identification results for the VI beam to those of the linear beam, we study the effects of the strongly

Table 23.1 Positions of the accelerometers, rigid stops and laser displacement sensors of the VI beam

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 xSTP xLDS

Positions (mm) 131 263 395 527 657 787 917 1,052 1,215 1,311 1,185 1,230

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

1

fn(x

/L

)

x/L

f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10

Fig. 23.2 Experimental setup for the VI beam: x1–x10, xLDS, and xSTP respectively denote the spanwise locations of the accelerometers, of the laser

displacement sensors, and of the rigid stops

Table 23.2 The leading ten linear natural frequencies in Hz for the beam in Fig. 23.1

o1 o2 o3 o4 o5 o6 o7 o8 o9 o10

Theoretical 3.8 23.8 66.6 130.5 215.7 322.2 450.0 599.1 769.5 961.2

Experimental 3.7 23.2 64.9 126.9 209.4 314.7 433.9 580.7 751.3 926.7

23 Nonlinear System Identification of the Dynamics of a Vibro-Impact Beam 289

nonlinear dynamics induced by the vibro-impacts. Typically, we will consider the acceleration signals at position x9 for thiscomparison because the VI effects are expected to be most significant there due to its proximity to the impact position. We

also illustrate overall spatio-temporal variations of the beam dynamics caused by vibro-impacts. The basic elements of the

NSI methodology are referred to [25].

23.3.1 Linear Beam

By linear beam we mean the cantilever beam in Fig. 23.1 without any rigid stops (or with impacting boundaries of infinite

clearances). Then, since the beam is homogeneous and uniform, we can assume that its transverse vibrations can be

approximately governed by the Bernoulli-Euler beam model with the following equation of motion,

rA€vðx; tÞ þ EIzzv0000ðx; tÞ ¼ pðtÞdðx� x3Þ (23.1)

where v x; tð Þ denotes the displacement of the beam in the transverse (y) direction at ðx; tÞ (cf. Fig. 23.1); pðtÞ ¼ P0dðtÞ is theimpulsive excitation at t ¼ 0, where dðtÞ and dðxÞ denote Dirac delta functions; and primes and dots are partial differentia-

tion with respect to x and t, respectively. Then, the general solution for (23.1) can be written as

vðx; tÞ ¼X1

m¼1AmfmðxÞe�zmomt cosðomdt� ymÞ (23.2)

where omd ¼ omð1� z2mÞ1=2; om is the natural frequency of the m-th linear bending mode; zm is the modal damping factor

(when a certain viscous damping is assumed in the system); and fmðxÞ is the normalized mode shape function for the m-th

mode (cf. Fig. 23.2). The corresponding acceleration can be written as

aðx; tÞ€vðx; tÞ ¼X1

m¼1�AmfmðxÞe�zmomt cosðomdt� �ymÞ (23.3)

where �Am ¼ Amo2m and �ym ¼ ym þ 2tan�1½ð1� z2mÞ1=2=zm�.

Consider now the acceleration response of the linear beam at position x9 depicted in Fig. 23.3. The wavelet and Fourier

transforms clearly depict the ten dominant fast frequencies identified from experimental modal analysis (see Table 23.2).

−1000

−500

0

500

1000

a(x

9,t)

Fre

quen

cy (

Hz)

Time (s)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

500

1000

0 100 200 300 400 500 600 700 800 900 1000

100

0 10 20 30

10−5

Frequency (Hz)

Frequency (Hz)Frequency (Hz)200 210 220

10−8

10−7

10−6

(A) (B)

Close−up (A)

Close−up (B)

Fig. 23.3 Wavelet and Fourier transforms of the acceleration for the linear beam at position x9

290 H. Chen et al.

As discussed in Sect. 2, the harmonic at 209 Hz appears to be negligible because the position x9 is close to one of the nodesfor the 5th mode. We write

a9ðtÞ � aðx9; tÞ �X10

m¼1�Amfmðx9Þe�zmomt cosðomt� �ymÞ (23.4)

for small damping zm. By means of EMD analysis, we wish to obtain the relation between the acceleration time series and the

intrinsic mode functions (IMFs) such that

a9ðtÞ �X10

m¼1cmða9; tÞ �

X10

m¼1�Amfmðx9Þe�zmomt cosðomt� �ymÞ (23.5)

where cmða9; tÞ denotes the m-th IMF of the acceleration at position x9 (and is usually associated with the m-th normal mode

vibration that can be observed at the same position of the beam). Figure 23.4 depicts the ten dominant IMFs from the

advanced EMD analysis algorithm introduced in [27], demonstrating that the relation (23.5) is valid except for the 5th mode

of the beam due to observability issues. We establish the reduced-order model (ROM) for the acceleration in Fig. 23.3 for the

linear beam dynamics at position x9 in terms of intrinsic modal oscillators (IMOs). That is, we write the IMO corresponding

to each IMF [27], and the instantaneous slowly-varying envelope and phase of the m-th IMF are computed respectively as

AmðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficmða9; tÞ2 þ H½cmða9; tÞ�2

q; ymðtÞ ¼ tan�1f H½cmða9; tÞ�=cmða9; tÞg � omt (23.6)

where m ¼ 1,. . .,10. Since the slowly-varying complex forcing amplitude LmðtÞ is computed from the time series (or IMF)

in an effort to match the solution xmðtÞ of the IMO with the corresponding IMF, we can write xmðtÞ � cmða9; tÞ. During this

validation process the damping factor 0<zm<1 is chosen such as to minimize the error between xmðtÞ and cmða9; tÞ. Then, theoriginal response can be reconstructed as the sum of all IMO solutions; that is, the following expression holds:

a9ðtÞ �X10

m¼1xmðtÞ �

X10

m¼1cmða9; tÞ (23.7)

Figure 23.5 compares the 10th and 5th IMFs with the corresponding IMO solutions, exhibiting good agreement. Finally,

we consider the physical meaning of the complex-valued forcing function LmðtÞ for the m-th IMO of the linear problem,

since such a term is known to be associated with nonlinear modal interactions in nonlinear dynamical systems [28, 29]. In

our linear beam problem, the slowly-varying envelope AmðtÞ and phase ymðtÞ can be identified from (23.5) as

AmðtÞ ¼ �Amfmðx9Þe�zmomt; ymðtÞ ¼ ��ym ¼ constant (23.8)

Then, the slow-flow variable can be expressed as

’mðtÞ � jom�Amfmðx9Þe�j�yme�zmomt ) _’mðtÞ � �jzmo

2m�Amfmðx9Þe�j�yme�zmomt (23.9)

If zm ¼ zm (i.e., the damping factor in the IMO is the same as the modal damping factor identified from experimental

modal analysis, and carries a direct physical meaning), then we can easily show thatLmðtÞ � 0. This idea may sound feasible

and reasonable, because the resulting reduced-order model will be the same as that obtained from the typical linear modal

analysis with the coordinates, xm;m ¼ 1; � � � ; 10; being the modal coordinates. Furthermore, the solution for the IMO will

appear as a free damped response, which may naturally satisfy the relation in (23.5). However, as is the case for many other

nonlinear system identification methods where it is of more interest to check whether the proposed parametric model is able

to reproduce the measured (or simulated) dynamics, the damping factor in the IMO is not necessarily the same as the

physical one (i.e., zm 6¼ zm, in general). In this case, the complex forcing amplitude LmðtÞ can be expressed as

LmðtÞ � jðzm � zmÞo2m�Amfmðx9Þe�j�yme�zmomt (23.10)

The absolute value of the complex number (23.10) is a monotonically and exponentially decaying function; such forcing

function will not generate any modal interactions (as is supposed to be the case for a linear system). Nonetheless, the solution

for the IMO, which is strongly driven by the forcing LmðtÞejomt because zm � zm, can approximately reproduce the IMF in

(23.5). Similar discussions can be made not only for the response at position x9, but also for those at all other positions alongthe linear beam.


−50

0

50

c 8(a

9,t

)F

requ

ency

(Hz)

Time (s)0 0.2 0.4 0.6 0.8 1

0

500

1000

≈ 580Hz

Time (s)

−500

0

500

c 9(a

9,t

)F

requ

ency

(Hz)

0 0.2 0.4 0.6 0.8 10

500

1000

≈ 750Hz

−500

0

500

c 10(a

9,t

)F

requ

ency

(Hz)

Time (s)0 0.2 0.4 0.6 0.8 1

0

500

1000 ≈ 926Hz

−100

0

100

c 7(a

9,t

)F

requ

ency

(Hz)

Time (s)0 0.2 0.4 0.6 0.8 1

0

500

1000

≈ 436Hz

−50

0

50

c 6(a

9,t

)F

requ

ency

(Hz)

Time (s)0 0.2 0.4 0.6 0.8 1

0

500

1000

≈ 314Hz

−5

0

5c 5

(a9,t

)F

requ

ency

(Hz)

Time (s)0 0.2 0.4 0.6 0.8 1

0

200

400

600

≈ 209Hz

−20

−10

0

10

20

c 4(a

9,t

)F

requ

ency

(Hz)

Time (s)0 0.2 0.4 0.6 0.8 1

0

50

100

150

200

250

≈ 125Hz

−50

0

50

c 3(a

9,t

)F

requ

ency

(Hz)

Time (s)0 0.2 0.4 0.6 0.8 1

0

50

100

150

200

250

≈ 64Hz

−10

0

10

c 2(a

9,t

)F

requ

ency

(Hz)

Time (s)0 0.2 0.4 0.6 0.8 1

0

50

100

≈ 23Hz

−4

−2

0

2

4

c 1(a

9,t

)F

requ

ency

(Hz)

Time (s)0 0.2 0.4 0.6 0.8 1

0

10

20

30

40

≈ 4Hz

a b

c d

e f

g h

i j

Fig. 23.4 The ten dominant IMFs extracted from the acceleration response in Fig. 23.3: (a) through (j) sequentially depict the tenth to first

IMFs, respectively

292 H. Chen et al.

23.3.2 Vibro-Impact Beam

We now consider the cantilever beam in Fig. 23.1 but with the two symmetric rigid stops of 4 mm clearances at position xSTP.If the displacement jvðxSTP; tÞj<4mm, then the dynamics of the beam is linear and can be described by (23.1). Whenever the

beam displacement jvðxSTP; tÞj ¼ 4mm, a vibro-impact occurs resulting in a new impact load �pðtÞdðx� xSTPÞ applied to the

beam as well as causing energy dissipation due to inelastic impact. Mathematically speaking, the nonsmoothness due to the

vibro-impacts means that the displacement response is of class C0 (i.e., continuous but not continuously differentiable). For

this strongly nonlinear nonsmooth dynamical system, there is no closed-form solution available, in general. Furthermore,

such a VI dynamical system may possess a very complicated topological structure of periodic orbits (e.g., see [36]). This is

mainly because nonsmooth dynamical systems may involve complicated dynamics such as grazing bifurcations [37] and

chaos [21]. We wish to model and understand the nonlinear dynamics of the VI beam by applying the proposed NSI method.

As for the case of the linear beam problem of the previous section, we consider the acceleration signal at position x9(depicted in Fig. 23.6), where the effects of vibro-impacts generate multiple broadband perturbations in the wavelet

transforms. In particular, comparing the Fourier transform of the linear beam response (dashed line) with that of the VI

beam, this broadband excitation of the beam due to vibro-impacts is significant. Figure 23.7a depicts the numerically

computed displacement and the corresponding impact load on the beam at xSTP in order to identify the instants of vibro-

impacts (i.e., the time instants when the beam displacement at xSTP reaches the thresholds4 mm). It was shown in [38] that

the nonlinear modal interactions due to vibro-impacts are purely due to the smooth parts of the VI dynamics, whereas the

nonsmooth parts tend to create frequency-energy relations involving numerical artifacts. Such numerical artifacts could lead

to wrong conclusions regarding the nonlinear resonances involved in the nonlinear modal interactions between the measured

IMFs. Furthermore, it was demonstrated that the smooth parts of the VI dynamics can be obtained by separating the

nonsmooth effects by means of EMD analysis [38]. Typically, the nonsmooth part is computed as the first IMF with the help

of masking and mirror-image signals [27]. The characteristics of the nonsmooth IMF were explored in previous works by

relating them to Fourier series expansions of saw-tooth wave signals [39], and also by a partial-differential-equation-based

sifting process [38] noting that EMD acts, in essence, as a dyadic filter bank.

Figure 23.7b depicts such a nonsmooth IMF for the acceleration signal in Fig. 23.6. Superposition of the impact instants

identified from Fig. 23.7a illustrates that the isolated nonsmoothness agrees reasonably well with the time instants of vibro-

impacts. We note that the numerical displacement was calculated from the reduced-order model through the assumed-modes

method, which means that some other modes higher than 10th may need to be included to get a better match between the

numerical simulations and experimental measurements. Some quantitative discrepancies prevail after 0.2 s with the current

reduced-order model.

Now EMD is applied to the remaining smooth part of the acceleration signal after subtracting the nonsmooth IMF in

Fig. 23.7b from the original acceleration in Fig. 23.6. The ten dominant IMFs are depicted in Fig. 23.8. By superimposing the

vertical dashed line at each impact instant identified from Fig. 23.7a, one can at least qualitatively observe the effects of

vibro-impacts on each IMF at position x9; for example, the vibro-impacts seem to directly influence higher IMFs (above the

5th). Indeed, considering these higher frequency IMFs we note linear dynamical behavior between consecutive vibro-

impacts, in the form of exponentially decaying damped responses. On the other hand, lower IMFs do not seem to exhibit

such straightforward patterns, implying that these IMFs may undergo more strongly nonlinear modal interactions and may

be more significantly influenced by the strong nonlinearities due the vibro-impacts.

0 0.2 0.4 0.6 0.8 1−600

−400

−200

0

200

400

600

Time (s)

c10(a9; t) IMOClose−up

Close−up

0 0.2 0.4 0.6 0.8 1−4

−2

0

2

4

Time (s)

c5(a9; t) IMO

Close−up

Close−up

a b

Fig. 23.5 Comparison of the IMFs in Fig. 23.4 with their corresponding IMO solutions: (a) tenth IMF; (b) fifth IMF


As in the linear beam, we can also establish a nonlinear interaction model (NIM) for the IMFs obtained in Fig. 23.8 in

terms of a set of IMOs. Computing the nonlinear modal interaction forcing LmðtÞ from each IMF by means of the slow-flow

correspondence, we solve the ten IMOs respectively. Figure 23.9 compares the IMFs with the corresponding IMO solutions

for 10th and 5th IMFs, which show good agreement. We sum all IMO solutions to reconstruct the original signal, and this

exhibits a perfect match as depicted in Fig. 23.10. That is, the NIM we established has been validated so that it can be used to

study the nonlinear dynamics of the VI beam (at position x9) as an alternative reduced-order model.

Now, the physical meaning of Lmðak; tÞ;m; k ¼ 1; � � � ; 10; in the nonlinear dynamics of the VI beam can be explored by

comparing it with that for the linear beam. We first note that the magnitude of Lmða9; tÞ for all IMOs of the linear beam

appears as almost a straight line on a logarithmic scale (cf. Fig. 23.11), which makes sense due to the form of (23.10).

−4000

−2000

0

2000

4000

a(x

9,t

)

0 500 1000 1500

100

Frequency (Hz)

Fre

quen

cy(H

z)

Time (s)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

500

1000

1500

Fig. 23.6 Wavelet and Fourier transforms of the acceleration for the VI beam at x9 (the Fourier transform in Fig. 23.3 is superimposed as a dashed

line to illustrate the effects of vibro-impacts in frequency domain)

−5

0

5

v(x

ST

P,t

) (m

m)

t1 t2,3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13

0 0.2 0.4 0.6 0.8 1−2

−1

0

1

2−2

Time (s)

p(t

) (k

N)

t1t2

t3t4

t5t6

t7t8

t9t10

t11t12

t13

−2000

−1000

0

1000

2000

c NS(a

9,t

)

t1 t2,3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13

Fre

quen

cy (

Hz)

Time (s)0 0.2 0.4 0.6 0.8 1

0

500

1000

1500

a b

Fig. 23.7 Depiction of the vibro-impacts: (a) the displacement response of the VI beam simulated at position xSTP and its corresponding impact

loads on the beam from the rigid stops; (b) the nonsmooth component of the acceleration in Fig. 23.6 is decomposed via EMD analysis (note that

the dashed lines at t ¼ tk; k ¼ 1; � � � ; 13 imply the impact instants identified from the impact force �pðtÞ)

294 H. Chen et al.

−2000

0

2000

c 10(

a9,t

)F

requ

ency

(Hz)

Time (s)0 0.2 0.4 0.6 0.8 1

0

500

1000

1500

≈ 926Hz

−1000

−500

0

500

1000

c 9(a

9,t

)F

requ

ency

(Hz)

Time (s)0 0.2 0.4 0.6 0.8 1

0

500

1000

1500

≈ 750Hz

−500

0

500

c 8(a

9,t

)F

requ

ency

(Hz)

Time (s)0 0.2 0.4 0.6 0.8 1

0

500

1000

1500

≈ 580Hz

−200

−100

0

100

200

c 7(a

9,t

)F

requ

ency

(Hz)

Time (s)0 0.2 0.4 0.6 0.8 1

0

500

1000

1500

≈ 436Hz

−100

−50

0

50

100

c 6(a

9,t

)F

requ

ency

(Hz)

Time (s)0 0.2 0.4 0.6 0.8 1

0

100

200

300

400

500

≈ 314Hz

−50

0

50c 5

(a9,t

)F

requ

ency

(Hz)

Time (s)0 0.2 0.4 0.6 0.8 1

0

100

200

300

400

500

≈ 209Hz

−100

0

100

c 4(a

9,t

)F

requ

ency

(Hz)

Time (s)0 0.2 0.4 0.6 0.8 1

0

100

200

300

400

500

≈ 125Hz

−100

0

100

c 3(a

9,t

)F

requ

ency

(Hz)

Time (s)0 0.2 0.4 0.6 0.8 1

0

50

100

150

200

≈ 64Hz

−50

0

50

c 2(a

9,t

)F

requ

ency

(Hz)

Time (s)0 0.2 0.4 0.6 0.8 1

0

50

100

≈ 23Hz

−2

0

2

c 1(a

9,t

)F

requ

ency

(Hz)

Time (s)0 0.2 0.4 0.6 0.8 1

0

10

20

30

40

50

≈ 3.5Hz

a b

c d

e f

g h

i j

Fig. 23.8 The ten dominant IMFs extracted from the acceleration response in Fig. 23.6: (a) through (j) sequentially depict the 10th to 1st IMFs,

respectively (note that the dashed lines imply the impact instants identified in Fig. 23.7)


Similarly, jLmða9; tÞj for the VI beam can also exhibit linearity with the same slope on average on a logarithmic scale as in

the case of the linear beam, but such a linear pattern appears only in between impact instants and, in particular, when m 6

(cf. Fig. 23.11a for the 10th IMO). The trajectory of L10ða9; tÞ in the complex plane for the linear beam appears as a single,

monotonic, decaying pattern (i.e., time-like behavior on a logarithmic scale), which implies no modal coupling or

interactions in the ROM. The trajectory of L10ða9; tÞ in the complex plane for the VI beam also exhibits such monotonic

behavior but only in between vibro-impacts (denoted by the intervals In; n ¼ 1; 2; � � � ); the role of the vibro-impacts is to

cause phase shifts of the slowly-varying forcing L10ða9; tÞ at the instants of vibro-impacts. On the other hand, the slowly-

varying complex forcing function for the 4th IMO of the VI beam does not exhibit any linear behavior but only a slowly-

varying wavy envelope regardless of vibro-impacts (cf. Fig. 23.11b). Such wavy patterns in the plot of jL4ða9; tÞj indicatethat certain modal interactions occur through nonlinear resonant conditions such as internal resonances or resonance

captures [29]. Also, nonlinear modal interactions are evidenced by the spiral (or non-time-like) patterns of the trajectory

in the complex plane.

From these two typical examples, we may conjecture the following: Whereas the higher IMOs (i.e., the IMOs associated

with higher frequency components) tend to maintain their linear dynamics in between impacts (although the overall

dynamics is strongly nonlinear), the lower IMOs exhibit strongly nonlinear modal interactions independent of vibro-impact

patterns. The role of vibro-impacts is just to exert broadband impulsive excitations on the linear beam causing instantaneous

phase shifts in the higher IMOs. To verify this conjecture we first compute Pearson’s linear correlation coefficient [40] for

the slowly-varying complex forcing amplitudes Lmðak; tÞ;m; k ¼ 1; � � � ; 10; of all IMOs for the linear and VI beams at all

the positions along the beam. This correlation coefficient is widely utilized in statistics as a measure of the linear dependence

between two variables, and a Matlab command, ‘corr.m’, was used in this work.

Figure 23.12 depicts the interpolated contour map of the absolute value of the linear correlation coefficient for each mode

number along the beam span. Note that by ‘mode number’ m in Fig. 23.12 we mean the IMO which is associated with the

0 0.2 0.4 0.6 0.8 1−3000

−2000

−1000

0

1000

2000

3000

Time (s)

c10(a9; t) IMO

0 0.2 0.4 0.6 0.8 1−50

0

50

Time (s)

c5(a9; t) IMO

a b

Fig. 23.9 Comparison of the IMFs in Fig. 23.8 with their corresponding IMO solutions: (a) tenth IMF; (b) fifth IMF

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−5000

0

5000

Time (s)

a(x

9;t

)

Original Reconstructed

Fig. 23.10 Comparison of the reconstructed acceleration from the ten IMO solutions plus the nonsmooth IMF with the original response

in Fig. 23.6

296 H. Chen et al.

m-th linear mode; hence, there is no such continuous distribution with respect to the vertical axis. From these simple

calculations, we find that the IMOs higher than the 4th possess strong linear dependence (linear correlation coefficient above

90%) between the linear and VI responses of the beam, regardless of the position along the beam. Again, it is noted that the

low correlation for the 3rd, 5th, 7th and 9th IMOs at the midspan of the beam is due to the fact that the position is very close

to one of the nodes for the respective linear modes. Similar explanations can be given for the 5th and 7th IMOs at position x9,and for the 8th IMO at position x8. Therefore, the aforementioned conjecture is confirmed by means of the linear correlation

coefficients between jLmðak; tÞj (and hence the corresponding IMO responses) for the linear and VI beams. That is, vibro-

impacts do not significantly alter the linear dynamics for the higher modes (typically, higher than 4th), but they significantly

affect the lower modes through strongly nonlinear modal interactions. This result agrees with Cusumano’s previous work

[22], where the topological characterization of the spatial structure of the VI beam vibrations was studied by means of the

two-point spatial correlation (i.e., correlation dimension) and POD. In particular, the estimate for the correlation dimension

of the VI dynamics obtained was lower than but near four, which dictates that a low-dimensional model can capture the

overall complicated, chaotic-like VI beam dynamics. Furthermore, if such complicated dynamics can be captured by a low-

dimensional model with several lower IMOs, then energy transfers (or cascades) from the higher to the lower modes through

certain nonlinear modal interactions, such as internal resonances, may be responsible [22].

While the linear correlation coefficient provides excellent physical insights into the VI beam dynamics, we note that it can

be regarded as a static global measure; that is, it does not contain any information regarding the temporal variations of the

vibro-impacts throughout the beam. A high linear correlation coefficient for certain IMOs at some position may indicate a

strong linear dependence between the VI beam and the underlying linear beam, and imply that the corresponding IMO of the

VI beam behaves linearly for that position. Nonetheless, this will not be apparent in the local dynamics (e.g., propagation

and/or localization of the effects of nonlinear modal interactions caused by vibro-impacts, such as the temporal localization

of the nonlinear dynamics for the 7th IMO). In particular, the linear correlation becomes a poor measure when the issue of

observability is involved (e.g., all the odd-number IMOs higher than 3rd at position x5 in Fig. 23.12).

b

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

106

107

108

Time; t (s)

|Λ4(a

9;t

)|

−1 −0.5 0 0.5 1

x 108

−4

−2

0

2

4

6

8

x 107

Re Λ4(a9;t )Im

Λ4(a

9;t

)

Linear Beam

VI BeamLinear Beam

VI Beam

I1 I2

I3I4

I1

I2

I3

I4

I5I5

a

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

6

107

108

109

1010

1011

Time; t (s)

|Λ10

(a9;t

)|

−2 −1 0 1 2 3 4

x 1010

−2

−1.5

−1

−0.5

0

0.5

1x 10

10

Re Λ10(a9;t )

ImΛ

10(a

9;t

)

Linear Beam

VI Beam

I1 I2

I3

I4

I5

I6

Linear Beam

I6

I5

I4

I1

I2

I3

VI Beam

Fig. 23.11 Comparison of the slowly-varying forcing functionsL10ða9; tÞ: (a)m ¼ 10 (10th IMO) and (b)m ¼ 4 (4th IMO) (Note that the dashedlines imply the impact instants identified in Fig. 23.7)


23.4 Conclusions

We presented the dynamics of a cantilever beam with two symmetric rigid stops with prescribed clearances by performing

nonlinear system identification (NSI) based on the correspondence between analytical and empirical slow-flow dynamics.

Performing empirical mode decomposition (EMD) analysis of the numerically-computed acceleration responses at ten,

almost evenly-spaced, spanwise positions along the beam, we constructed sets of intrinsic modal oscillators at different time

scales of the dynamics. In particular, the EMD analysis can separate nonsmooth effects due to vibro-impacts between the

beam and the rigid stops from the underlying smooth dynamics of the flexible beam, so that nonlinear modal interactions can

be explored only based on the remaining smooth components. Then, we established nonlinear interaction models (NIMs) for

the respective intrinsic mode oscillations, where the NIMs invoke slowly-varying forcing amplitudes (or nonlinear modal

interaction terms) that can be computed from empirical slow-flows and directly dictate nonlinear modal interactions between

different-scale dynamics. By comparing the spatio-temporal variations of the nonlinear modal interactions for the vibro-

impact beam and the corresponding linear beam model, we demonstrated that vibro-impacts significantly influence the lower

intrinsic mode functions through strongly nonlinear modal interactions, whereas the higher modes tend to retain their linear

dynamics between impacts. Also, computation of linear correlation coefficients as measures for linear dependence between

the dynamics of the linear and VI beams manifested the same results but only with spatial information about this correlation.

Acknowledgments This material is based upon work supported by the National Science Foundation under Grants Number CMMI-0927995 and

CMMI-0928062.

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Chapter 24

Modeling of Subsurface Damage in Sandwich Composites

Using Measured Localized Nonlinearities

Sara S. Underwood and Douglas E. Adams

Abstract Composite materials are being used more frequently in commercial and military aircraft structures. Many

nondestructive techniques have been developed to inspect composite materials for subsurface damage; however, many of

these existing inspection techniques aim to detect either linear changes in the material properties of the composite or

geometrical changes in the material to determine the presence of damage. Subsurface damage in a sandwich composite panel

is tested using a scanning laser vibrometer, and nonlinear vibration response properties are identified in the forced frequency

response of the composite panel. The nonlinear behavior identified in the composite panel is applied to a homogeneous,

isotropic beam model such that the forced frequency response of localized damage in the beam resembles the behavior

measured in the sandwich composite panel. Stiffness and damping nonlinearities induced locally in the analytical model are

used to show that multi-amplitude frequency response functions may be used as a means of detecting nonlinear behavior

attributed to composite damage in a composite material. The results of the analytical model show that the nonlinear behavior

due to damage displays a global behavior in an analysis of the frequency response of the system and is able to be identified

locally when consideration of the linear system dynamics are taken into account.

Keywords Composite damage • Multi-amplitude frequency response functions • Nonlinear behavior • Scanning laser

vibrometry

24.1 Introduction

Composite materials are being used more frequently in commercial and military aircraft structures. For example, honeycomb

core sandwich composites are being used in fixed wing structures, floor panels, rotor blades, and other parts of air vehicles in

which high strength and low weight material properties are needed. A drawback to using sandwich composite materials for

aircraft applications is that damage in composite materials often occurs beneath the surface, making it difficult to inspect these

aircraft using visual or line-of-sight techniques. Many existing methods for inspecting composite materials for subsurface

damage involve using linear or geometrical changes in the structural properties of the composite, typically measured between

undamaged and damaged states, as an indicator of damage. A nonlinear approach has the potential to remove the dependence

on historical baseline data or a reference specimen with manufactured defects to detect damage. Several researchers, for

example, [1–3], have applied nonlinear methods to detect damage in composite structures, which rely on the assumption that

damage displays localized nonlinear properties.

Nonlinearmethods for detecting damage in compositematerials have significant advantages over linearmethods in that they

are less susceptible to variations in environmental conditions and may be used to detect damage without the need for historical

baseline measurements or reference standards. Thesemethods address the nonlinear nature of damage; however, there is not an

agreement in the literature on the nature of the nonlinear behavior identified in the vicinity of subsurface composite damage.

An understanding of the type of nonlinear behavior seen in the vicinity of composite damage, such as one which can be applied

through measurements of forced harmonic excitation, may help guide inspection methods for composite materials.

S.S. Underwood (*) • D.E. Adams

Purdue University, School of Mechanical Engineering, Center for Systems Integrity, 1500 Kepner Drive, Lafayette, IN 47905, USA




301



In this paper, nonlinear behavior in the vicinity of core crack damage is investigated on a fiberglass sandwich panel. A

scanning laser vibrometer is used to collect frequency response measurements from the surface of the panel as it is excited at

multiple amplitudes of excitation. High and low amplitude frequency response functions measured with the laser vibrometer

are used to investigate and identify nonlinear behavior at the damage location. A single degree-of-freedom model is used to

confirm the nonlinear behavior identified in the frequency response function measurements and then the identified

nonlinearities are used to simulate damage in an analytical finite element model of a Bernoulli-Euler beam. The analytical

model is used to study the effects of the localized nonlinearities on the vibration response of the structure and the ability to

discern the damage location by considering the localized nature of composite damage identified through multi-amplitude

frequency response functions.

24.2 Experimental Investigation

In order to investigate nonlinear behavior due to subsurface damage in a composite material, a fiberglass, honeycomb core

sandwich panel, which resembles the material commonly used in rotor blade trailing edge structures, was obtained and

damaged. The panel was made of 0.060 in. fiberglass facesheets and a 0.50 in. polypropylene honeycomb core with a 10 mm

cell size. The panel measured 84 in. long by 6.5 in. wide and was clamped at the top and bottom to a rigid frame. Core crack

damage, such as that shown in Fig. 24.1a was introduced to the panel by placing a cut in the honeycomb core at the location

indicated by the small, solid rectangle in Fig. 24.1b.

A piezoelectric actuator (PCB model 712A02) with an attached 50 g mass was attached to an impedance head (PCB

model 288D01). The actuator stack was mounted to an aluminum block which was attached to the panel at a skewed angle in

order to excite the structure in multiple directions simultaneously. A three-dimensional scanning laser vibrometer (PCB

PSV-400-3D) was used to measure the surface velocity of the panel in three orthogonal directions as an excitation was

applied through the actuator. Frequency response functions relating the input force to the panel, measured through the

impedance head, to the output surface velocity of the panel in the transverse, lateral, and longitudinal directions were

obtained from the laser vibrometer measurements.

Three measurement points on the surface of the fiberglass panel were considered for analysis of the frequency response

behavior of the panel in the vicinity of composite damage. The locations of these points are indicated by the dark, medium,

and light colored circles in the panel diagram shown in Fig. 24.1b. The actuator was set to a sine sweep excitation from 100

to 5,000 Hz and laser vibrometer measurements were taken at high and low amplitudes of excitation. For each measurement

Fig. 24.1 Fiberglass panel (a)

core crack damage and (b)

diagram showing the damage

location and measurement

points (dark, medium, andlight colored circles)considered in the experimental

investigation

302 S.S. Underwood and D.E. Adams

point and excitation amplitude, five measurement averages were taken where the sample time for each measurement was

1.28 s and the frequency resolution was 781.25 mHz. The frequency response functions measured in the transverse direction

were considered for analysis of nonlinear behavior in the vicinity of damage.

24.3 Nonlinear Behavior of Damage

The frequency response functions obtained from high and low amplitude excitation levels were compared in order to

determine if nonlinear behavior is discernable in the vibration response of the panel in the vicinity of the core crack damage.

By using multi-amplitude frequency response functions, nonlinear behavior is identified by determining measurement points

where the measured frequency response functions change due to a change in excitation amplitude. The frequency response

function is an input–output relationship, and in a nonlinear dynamic system, the output response does not change

proportional to changes in input excitation as it does in a linear dynamic system. This non-proportional change in the

nonlinear case leads to a change in the measured frequency response function. Therefore, nonlinear behavior is able to be

identified when frequency response functions collected from multiple amplitudes of excitation are compared.

The high and low amplitude frequency response function measurements considered in the analysis of the frequency

response behavior of the panel in the vicinity of composite damagewere scaled and plotted simultaneously, alongwith a scaled

difference between the two values. This result is shown in Fig. 24.2 for the three measurement points depicted by the

corresponding colors in Fig. 24.1b. In Fig. 24.2, the bold line of each the dark, medium, and light colored lines representsthemeasured high amplitude response while the thin line represents the low amplitude response and the dashed line representsthe difference between the two responses. As seen in Fig. 24.2, nonlinear behavior, which is identified by large differences

between the high and low amplitude frequency response functions, occurs for each of the measurement points considered. In

addition, the point directly on the damage location, depicted by the dark colored lines, shows a significantly larger change infrequency response behavior than the points on the edge and away from the damage location, depicted by themedium and lightcolored lines, respectively.

The nonlinear behavior identified in the high and low amplitude frequency response comparison shown in Fig. 24.2

shows trends that resemble behavior which is expected in the presence of stiffness and damping nonlinearities. A single

degree-of-freedommodel was used to identify the form of the identified nonlinear stiffness and damping behavior seen in the

measured frequency response functions. A simple spring-mass-damper system, with mass M, stiffness K, and damping C,

was used for the single-degree-of freedom model. Newmark’s method was used to evaluate the response when an impulse

input was used to excite the mass. The parameters used in the model are shown in Table 24.1.

Frequency ranges from the frequency response comparison shown in Fig. 24.2, where the nonlinear behavior was most

apparent, were selected for further analysis. Figure 24.3a shows a frequency range where nonlinear stiffness behavior is

400 450 500 550 600 650 700 750 8000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Frequency (Hz)

Mag

nitu

de (

mm

/s/N

)

Low Amplitude Response

High Amplitude Response

Difference (Scaled)

Low Amplitude Response

High Amplitude Response

Difference (Scaled)

Fig. 24.2 Frequency response function comparison for three measurement points on the fiberglass panel

24 Modeling of Subsurface Damage in Sandwich Composites Using Measured Localized Nonlinearities 303

apparent. Here, the frequency response functions are diverging, and the largest difference between them is seen to occur

before the peak in the response. Similar behavior was simulated in the single-degree of freedom model using a hardening

cubic stiffness nonlinearity. This result is shown in Fig. 24.3b. The nonlinearity causes an upward shift in frequency of the

resonance frequency between the high and low amplitude simulations. The largest difference between the simulated

frequency response functions occurs prior to the resonance peaks as is seen in the measured frequency response data.

In addition to the nonlinear stiffness behavior identified in the frequency response comparison, nonlinear damping

behavior was also identified. Figure 24.4a shows an example of a frequency range where nonlinear damping behavior is

apparent. Here, the amplitude of the frequency response functions differs with the high amplitude response significantly

lower in magnitude than the low amplitude response. The largest difference between the responses is seen to occur over the

peak area in the response. Similar behavior was simulated in the single-degree of freedom model using a cubic damping

nonlinearity. This result is shown Fig. 24.4b. The nonlinearity causes a decrease in the magnitude of the response with no

affect on frequency, and the largest difference between the frequency response functions is seen at the resonance frequency.

This behavior is similar to what is seen in the measured frequency response data.

The analysis of the measured frequency response functions for the core cracking damage mechanism using a single

degree-of-freedom model for comparison showed that the data displays behavior similar to behavior seen for cubic stiffness

and cubic damping nonlinearities. With these nonlinearities, the internal forces in the structure can be described by:

f u; _uð Þ ¼ C _uþ Kuþ m1u3 and f u; _uð Þ ¼ C _uþ Kuþ m2 _u

3 (24.1)

for the cubic stiffness nonlinearity and the cubic damping nonlinearity, respectively. In (24.1), C and K represent the

damping and stiffness matrices of the structure, respectively, u and _u represent the displacement and velocity of the structure,

respectively, and m1 and m2 represent the respective coefficients for the nonlinearities.

a

400 410 420 430 440 4500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Frequency (Hz)

Mag

nitu

de

(m

m/s

/N)

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2x 10-3

Frequency Ratio, f/fn

Ma

gnitu

de

(m/s

/N)

Low AmplitudeHigh AmplitudeDifference (Scaled)

b

Fig. 24.3 (a) Frequency response comparison displaying a stiffness nonlinearity and (b) the single degree-of-freedom model result with a cubic

stiffness nonlinearity

Table 24.1 Parameters for the

single degree-of-freedom modelParameter Value

Mass, M (kg) 5

Stiffness, K (N/m) 1.0E + 07

Damping, C (N-s/m) 1840

g (Newmark parameter) 1/2

b (Newmark parameter) 1/6

Nonlinear stiffness constant, m1 (N/m3) 2.0E + 20

Nonlinear damping constant, m2 (N-(s/m)3) 4.0E + 10


24.4 Analytical Model

A finite element model was developed in Matlab in order to qualitatively understand the frequency response behavior of a

beam-like structural component with localized nonlinearities as the excitation amplitude is varied. An isotropic, homoge-

nous beam, such as that shown in Fig. 24.5, was used to develop the analytical model. The properties of the beam were

determined through micromechanical methods such that the first few bending modes of the beam matched the first few

bending modes of the panel used in the experimental investigations. The properties used in the model are given in

Table 24.2.

Subsurface damage

xa

d

n(u,u)• u(x,t)f(a,t)

Subsurface damageSubsurface damage

L

Fig. 24.5 Beam model with a localized nonlinearity n u; _uð Þ to represent damage

Table 24.2 Model properties

selected to simulate the fiberglass

sandwich panel

Property Value

Density, r (kg/m3) 460

Young’s modulus, E (Pa) 2.3e9

Beam length, L (m) 2.032

Beam width, b (m) 0.1

Beam height, h (m) 0.017

Cross-sectional area, A (m2) 0.0017

Area moment of inertia, I (m4) 4.1e-8

Mass proportional damping constant, a (s�1) 0.005

Stiffness proportional damping constant, b (s) 0.00005

a

560 570 580 590 600 610 6200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Frequency (Hz)

Mag

nitu

de (

mm

/s/N

)

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2x 10-3

Frequency Ratio, f/fn

Mag

nitu

de (

m/s

/N)

Low AmplitudeHigh AmplitudeDifference (Scaled)

b

Fig. 24.4 (a) Frequency response comparison displaying a damping nonlinearity and (b) the single degree-of-freedom model result with a cubic

damping nonlinearity


The beam was modeled using Ne Bernoulli-Euler beam elements with two nodes per element. This divided the beam into

Ne finite length segments of length Le ¼ L Ne= . For a beam response u x; tð Þ given as a function of the position x along the

beam and the time t, the discretized equation of motion for the finite element model of the beam is given by:

M½ � €uf g þ C½ � _uf g þ K½ � uf g ¼ ff g � n uf g; _uf gð Þf g (24.2)

In (24.2), the mass and stiffness matrices, [M] and [K], respectively, are determined by the Bernoulli-Euler beam

formulation, and a proportional damping matrix [C] is given by:

C½ � ¼ a M½ � þ b K½ � (24.3)

In (24.3), a and b are mass proportional and stiffness proportional damping constants, respectively. Since each

Bernoulli-Euler beam element has two nodes per element, there are four degrees of freedom per element, leading to a

total of 2Ne þ 2 degrees of freedom for the assembled finite element model. A fixed boundary condition was selected to

resemble the boundary condition used in the experimental investigation, giving 2Ne � 2 degrees of freedom of interest.

Two applied forces were used, including an applied forcing function represented by the vector ff g, and a nonlinear

forcing function represented by the vector n uf g; _uf gð Þf g. The applied forcing function models the excitation source as a

point excitation, fa(t), on the nodes of one or more elements, where the excitation is applied to the transverse degrees of

freedom at a location a along the beam. The nonlinear forcing function simulates damage and was applied across a single

element at a location d along the beam such that Newton’s third law of equal and opposite forces is enforced across the nodes

of the element on which the nonlinearities are applied. The cubic stiffness and damping nonlinearities identified using the

single degree-of-freedom model in the previous section were used for the nonlinear forces to represent composite damage.

The applied and nonlinear force vectors are given by:

ff g ¼

0

..

.

faðtÞ...

0

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;

2Ne�2x1

and n uf g; _uf gð Þf g ¼

0

..

.

�m1 uj � uk� �30

þm1 uj � uk� �3

..

.

0

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;

2Ne�2x1

or n uf g; _uf gð Þf g ¼

0

..

.

�m2 _uj � _uk� �30

þm2 _uj � _uk� �3

..

.

0

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;

2Ne�2x1

(24.4)

In (24.4), m1 and m2 are coefficients for the stiffness or damping nonlinearity, respectively, and the subscripts j and krepresent the nodes which the damaged element connects.

The number of elements selected for the finite element model simulation was Ne ¼ 13. For the fixed boundary condition,

this gave a total of 24 degrees of freedom. Two excitation types were studied, including an impulse input and a step sine

input. In both cases, the excitation was applied at a single node to a transverse degree of freedom. In each simulation, a single

element was damaged, and the excitation force was applied at a transverse degree of freedom away from the applied

nonlinear force simulating the damage. Newmark’s method, using Newmark parameters g ¼ ½ and b ¼ 1/6, was used to

evaluate the time response of the beam, and frequency response functions relating the displacement at each degree of

freedom to the applied excitation force were determined for high and low amplitudes of excitation.

24.5 Model Results

High and low amplitude excitations were applied to the analytical model, and frequency response functions were determined

from the simulated response. Figure 24.6a shows the high and low amplitude frequency response functions obtained at a

single node for the cubic stiffness nonlinearity case simulated with an impulse excitation. The linear and nonlinear restoring

force curves for this case are depicted in Fig. 24.6b. Comparison of the high and low amplitude frequency response functions


for this case shows that the cubic stiffness nonlinearity applied locally to a single element, had a global effect on the system.

Each transverse degree of freedom experienced a change in frequency response between the high and low amplitude

simulations.

A damage index computed by taking difference between the high and low frequency response functions across a selected

frequency range did not clearly indicate the damage location. Further investigations led to the hypothesis that the linear

response of the beam may mask the nonlinear response, making the localized nonlinearities difficult to detect. In the

frequency domain, the simulated output, {X}, in the model can be represented by:

Xf g ¼ H½ � Ff g þ FNf g½ � (24.5)

In (24.5), [H] is the frequency response function relating the simulated output to the simulated input, {F} is the frequencydomain spectrum of the linear applied input force, and {FN} is the spectrum of the nonlinear applied force used to

simulate damage. For an input location q, an output location p, and a nonlinear location n, the relationship used to compare

high and low amplitude frequency response functions, Hpqhigh and Hpqlow, respectively, is given by:

Hpqhigh � Hpqlow ¼ Xphigh

Fqhigh� Xplow

Fqlow¼ HpqFqhigh

Fqhigh� HpqFqlow

Fqlow

� �linear

þ HpnFnhigh

Fqhigh� HpnFnlow

Fqlow

� �nonlinear

¼ HpnFnhigh

Fqhigh� Fnlow

Fqlow

� �nonlinear

¼ 0 for all p except damaged p (24.6)

In (24.6), Hpq represents the linear frequency response function of the system, and Hpn represents the nonlinear frequency

response function of the system. The resulting relationship from the frequency response comparison evaluated in (24.6)

suggests that the global dynamics of the system affect the ability to discern the location of the applied nonlinearities.

However, if the frequency response function of the linear system is known, the output of the system can be pre-filtered by the

frequency response function, resulting in an ability to separate the effects of the linear and nonlinear applied forces by

considering a pre-filtered output variable Xpf:

Xf g ¼ H½ � Ff g þ FNf g½ �H½ ��1 Xf g ¼ Ff g þ FNf g

Xpf

� � ¼ Ff g þ FNf g (24.7)

-3 -2 -1 0 1 2 3

x 10-9

-1

-0.5

0

0.5

1

x 10-3

Deflection (m)

For

ce (

N)

Linear Restoring ForceNonlinear Restoring ForceLinear Restoring ForceNonlinear Restoring Force

0 100 200 300 400 500 600

10-13

10-12

10-11

10-10

Frequency (Hz)

Mag

nitu

de (

m/N

)

Low AmplitudeHigh Amplitude

a b

Fig. 24.6 For the cubic stiffness nonlinearity case with an impulse excitation: (a) high and low amplitude frequency response functions and

(b) restoring force curve for the linear and nonlinear restoring forces


In this case, a similar comparison between high and low amplitude outputs yields a result in which the nonlinear force is

able to be distinguished separate from the linear dynamics of the system:

Xpfhigh

Fqhigh� Xpflow

Fqlow¼ Fqhigh

Fqhigh� Fqlow

Fqlow

� �linear

þ Fnhigh

Fqhigh� Fnlow

Fqlow

� �nonlinear

¼ Fnhigh

Fqhigh� Fnlow

Fqlow

� �nonlinear

¼ 0 for all p except damaged p (24.8)

The result in (24.8) was applied to the finite element simulation for both cubic stiffness and cubic damping nonlinearity

cases when a step sine excitation at selected frequencies was used for the applied force. Figure 24.7a shows damage index

results for the cubic stiffness case where the damage was applied between the seventh and eighth transverse degrees-of-

freedom. Similarly, Fig. 24.7b shows damage index results for the cubic damping case where the damage was applied

between the third and fourth transverse degrees-of-freedom. In both cases, the damage index was computed using the

pre-filtered displacement output.

24.6 Discussion

The result of frequency response comparison for the finite element model of the beam using a pre-filtered output variable

showed that the locally applied nonlinearities were able to be detected by comparing the high and low amplitude responses.

Prior to filtering the output by the frequency response function of the linear system, the location of the nonlinearities was not

able to be distinguished due to being masked by the linear dynamics of the system. By pre-filtering the output by the linear

frequency response function of the system, the damage location in each case was able to be determined.

Prior to pre-filtering the response by the frequency response function of the system, the high and low amplitude frequency

response function comparison created a damage index which indicated that the regions with the largest nonlinear response

occurred near antinodes of the mode shape of the system which was most easily excited by the applied excitation. After

removing the influence of the linear system dynamics, the largest nonlinear response was located at the nodes where the

localized nonlinearities were applied. The step sine excitation provided a clearer indication of the damage location than the

impulse excitation, due to more energy being put into the system at frequencies where the nonlinearities are excited. In

addition, the step sine excitation results showed that the nonlinear behavior due to the simulated damage becomes more

pronounced at higher frequencies.

60

70

1234567891011120

0.5

1

0

10

20

30

40

50

60

70

1234567891011120

0.5

1

0

10

20

30

40

50

60

70

1234567891011120

0.5

1

0

10

20

30

40

50

60

70

1234567891011120

0.5

1

a bX

pfhi

gh

Transverse degree of freedom

Freq

uenc

y (H

z)

Fqh

igh

Xpf

low

Fql

ow

Xpf

high

Transverse degree of freedom

Freq

uenc

y (H

z)

Fqh

igh

Xpf

low

Fql

ow

Fig. 24.7 Damage index results obtained from a step sine excitation for (a) a cubic stiffness nonlinearity at transverse degrees-of-freedom seven

and eight and (b) a cubic damping nonlinearity at transverse degrees-of-freedom 3 and 4


24.7 Conclusion

The nonlinear nature of subsurface damage in composite materials was investigated through experimental measurements

and an analytical model of an isotropic, homogenous beam. A fiberglass sandwich panel was obtained and damaged in

order to investigate the nonlinear behavior measured using a scanning laser vibrometer. The panel was excited at multiple

amplitudes of excitation to allow for nonlinear behavior to be identified in a comparison of the high and low amplitude

frequency response measurements. Stiffness and damping nonlinearities were identified in the high and low

amplitude frequency response comparison, and a single degree-of-freedom model was used to show that the nonlinear

behavior identified resembles cubic stiffness and cubic damping behavior. The cubic stiffness and damping nonlinearities

were then investigated in an analytical model to study effects of the localized nonlinearities on the vibration response of the

system. It was found that, in order to locate the locally applied nonlinearities, it was necessary to pre-filter the simulation

output by the frequency response function of the linear system prior to taking into account the applied excitation force. This

pre-filtering was necessary to separate the linear system behavior from the nonlinear behavior of the damaged element. In

application to subsurface damage detection in composite materials, comparison of multi-amplitude frequency response

functions provides a method for identifying the presence of nonlinear behavior in the system. In order to locate the damage,

it is important to consider the linear dynamics of the system, where the challenge arises in separating the linear system

dynamics from the overall response of a structure.

References

1. Wong LA, Chen JC (2000) Damage identification of nonlinear structural systems. AIAA J 38:1444–1452

2. Zwink B, Koester D, Evans R, Adams DE (2008) Damage identification in composite sandwich helicopter blades using point laser velocity

measurements. In: proceedings of the sensor, signal and information processing workshop, Sedona, in print

3. Vanlanduit S, Guillaume P, Schoukens T, Parloo E (2000) Linear and nonlinear damage detection using a scanning laser vibrometer. Proc SPIE

4072:453–466


Chapter 25

Parametric Identification of Nonlinearity from Incomplete

FRF Data Using Describing Function Inversion

Murat Aykan and H. Nevzat Ozg€uven

Abstract Most engineering structures include nonlinearity to some degree. Depending on the dynamic conditions and level

of external forcing, sometimes a linear structure assumption may be justified. However, design requirements of sophisticated

structures such as satellites require even the smallest nonlinear behavior to be considered for better performance. Therefore,

it is very important to successfully detect, localize and parametrically identify nonlinearity in such cases. In engineering

applications, the location of nonlinearity and its type may not be always known in advance. Furthermore, in most of the

cases, test data will be incomplete. These handicaps make most of the methods given in the literature difficult to apply to

engineering structures. The aim of this study is to improve a previously developed method considering these practical

limitations. The approach proposed can be used for detection, localization, characterization and parametric identification of

nonlinear elements by using incomplete FRF data. In order to reduce the effort and avoid the limitations in using footprint

graphs for identification of nonlinearity, describing function inversion is used. Thus, it is made possible to identify the

restoring force of more than one type of nonlinearity which may co-exist at the same location. The verification of the method

is demonstrated with case studies.

Keywords Nonlinear identification • Nonlinear model testing • Experimental verification • Nonlinear parametric

identification

25.1 Introduction

System identification in structural dynamics has been thoroughly investigated over 30 years [1]. However, most of the

studies were limited to the linear identification theories. This short literature review does not cover linear identification

theories which are well documented [2, 3].

In the last decade, with the increasing need to understand nonlinear characteristics of complicated structures, there were

several studies published on nonlinear system identification [4–16]. Nonlinearities can be localized at joints or boundaries or

else the structure itself can be nonlinear. There are various types of nonlinearities, such as hardening stiffness, clearance,

coulomb friction, etc. [5].

Nonlinear system identification methods can be divided into two groups, either as time and frequency domain methods [4],

or as discrete and continuous time methods [6]. Most of the methods available require some foreknown data for the system.

Some methods require all or part of mass, stiffness and damping values [8–10] whereas some methods [4, 11–16] require

linear frequency response function (FRF) of the analyzed structure. In these methods nonlinearity type is usually determined

by inspecting the describing function footprints (DFF) visually. However, although the user interpretationmay be possible for

a single type of nonlinearity, it may not be so easy when there is more than one type of nonlinearity present [5]. Furthermore,

obtaining the linear FRF, which is usually presumed to be an easy task, may be difficult when nonlinearity is dominant at low

M. Aykan

Department of Mechanical Engineering, Middle East Technical University, Ankara 06800, Turkey

Defense Systems Technologies Division, ASELSAN Inc., Ankara 06172, Turkey

H.N. Ozg€uven (*)

Department of Mechanical Engineering, Middle East Technical University, Ankara 06800, Turkey




311


level excitations. There are also methods using neural networks and optimization for system identification [6]. Application of

optimization methods in nonlinear system identification is rather a new and promising approach. The major disadvantage of

these methods is generally the computational time required.

Nonlinearity identification method presented in this study consists of four main stages. Firstly, existence of nonlinearity

in the system is detected by performing step sine tests with different loads. Secondly, the location of the nonlinearity is

determined by using incomplete FRF data. The next step is the determination of the type of nonlinearity which is achieved by

investigating the restoring force function. Finally, in the parametric identification stage the coefficients of the nonlinear

elements are obtained by curve fitting techniques. The method proposed in this study is mainly an improved version of the

method developed earlier by Ozer et al. [12]. The improvement includes using incomplete FRF data which makes the

method applicable to large systems, and employing describing function inversion in order to reduce the effort in identifica-

tion of nonlinearity. Furthermore, using describing function inversion rather than footprint graphs makes it possible to

identify the total restoring force of more than one type of nonlinearity that co-exist at the same location.

25.2 Theory

Representation of nonlinear forces in matrix multiplication form using describing functions has been employed in identifi-

cation of structural nonlinearities by Ozer et al. [12]. They developed a method starting from the formulation given in their

earlier work [4] to detect, localize and parametrically identify nonlinearity in structures. As the basic theory of the method is

given in detail in reference [12], here it is briefly reviewed just for the completeness.

The equation of motion for a nonlinear MDOF system under harmonic excitation can be written as

½M�f€xg þ ½C�f _xg þ ½K�fxg þ j½D�fxg þ fNðx; _xÞg ¼ ffg (25.1)

where [M], [C], [K] and [D] stand for the mass, viscous damping, stiffness and structural damping matrices of the system,

respectively. The response of the system and the external force applied on it are shown by vectors {x} and {f}, respectively.{N} represents the nonlinear internal force in the system, and it is a function of the displacement and/or velocity response of

the system, depending on the type of nonlinearity present in the system. When there is a harmonic excitation on the system in

the form of

ff g ¼ Ff gejot (25.2)

the nonlinear internal force can be expressed as [17]

N x; _xð Þf g ¼ D x; _xð Þ½ � Xf gejot (25.3)

where [D(x, _x)] is the response dependent “nonlinearity matrix” and its elements are given in terms of describing functions

v as follows:

Dpp ¼ vpp þXnq¼1

q 6¼p

vpq p ¼ 1; 2; :::; n(25.4)

Dpq ¼ �vpq p 6¼ q p ¼ 1; 2; :::; n (25.5)

From the above equations it is possible to write the pseudo-receptance matrix for the nonlinear system, [HNL], as

HNL� � ¼ �o2 M½ � þ jo C½ � þ j D½ � þ K½ � þ D½ �� 1

(25.6)

The receptance matrix of the linear counterpart of the nonlinear system can also be written as

H½ � ¼ �o2 M½ � þ jo C½ � þ j D½ � þ K½ �� 1(25.7)

312 M. Aykan and H.N. Ozg€uven

From (25.6) and (25.7), the nonlinearity matrix can be obtained as

D½ � ¼ HNL� ��1 � H½ ��1

(25.8)

Post multiplying both sides of (25.8) by [HNL] gives

D½ � HNL� � ¼ I½ � � Z½ � HNL

� �(25.9)

where [Z] is the dynamic stiffness matrix of the linear part:

Z½ � ¼ H½ ��1 ¼ �o2 M½ � þ jo C½ � þ j D½ � þ K½ �� (25.10)

In order to localize nonlinearity in a system, a parameter called “nonlinearity index” is used. The nonlinearity index (NLI)for an pth coordinate is defined by taking any ith column of [HNL] and the pth row of [D] from (25.9) as follows:

NLIp ¼ Dp1 � HNL1i þ Dp2 � HNL

2i þ . . .þ Dpn � HNLni (25.11)

Here, theoretically, i can be any coordinate; however, in practical applications it should be chosen as an appropriate

coordinate at which measurement can be made and also be close to suspected nonlinear element. Equation (25.11) shows that

any nonlinear element connected to the pth coordinate will yield a nonzero NLIp. On the other hand, NLIp can be

experimentally obtained by using the right hand side of (25.9), which requires the measurement of the receptances of the

system at high and low forcing levels, presuming that low level forcing will yield FRFs of the linear part:

NLIp ¼ dip � Zp1 � HNL1i � Zp2 � HNL

2i � . . .� Zpn � HNLni (25.12)

25.2.1 Nonlinearity Localization from Spatially Incomplete FRF Data

The main disadvantage of the method discussed in [12] is that in order to calculate the NLIp the whole linear FRF matrix may

be required (if instead of theoretically calculated dynamic stiffness matrix, inverse of experimentally measured receptance

matrix is used). When this is the case, it may not be feasible to apply the method. In this study it is proposed to use

theoretically predicted values for unmeasured receptances calculated from the measured ones, and it is shown with case

studies that this approach yields acceptable results.

In modal testing of complicated structures usually a shaker is attached to a specific location on the test structure and

measurements are made at several locations. Usually test engineer excites the structure from 1 or 2 locations and measures

the responses from many points using accelerometers. This yields 1 or 2 columns of the FRF matrix. The number of

unknown elements can be reduced if reciprocity is used, which is one of the main assumptions of linearity. However, there

will be still unknown terms in the FRF matrix, especially the ones related with rotational degrees of freedommay be missing.

Although there are various methods to obtain FRFs at rotational degrees of freedom [18], measuring FRFs for rotational

degrees of freedom is usually found very difficult and it is avoided.

Nonlinearity localization by using the right hand side of (25.9) requires either the system matrices (that can be obtained

from the FE model) or the complete receptance matrix of the linear part so that it can be inverted to find [Z]. In order to

obtain the missing elements of the experimentally obtained receptance matrix, the application of a well known method is

proposed. Theoretically, if the modal parameters (natural frequency, damping ratio, modal constant, lower and upper

residues) of a structure are obtained by linear modal identification then missing elements of the receptance matrix can be

synthesized. In this study, the linear modal identification is performed by using LMS Test Lab software.

Once the modal parameters are identified, the unmeasured elements of the receptance matrix are calculated by using [2]

HpqðoÞ ¼XNr¼1

1

j2Or

ffiffiffiffiffiffiffiffiffiffiffiffi1�ðzrÞ2

p fprfqr

Orzr þ jðo� Or

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðzrÞ2

qÞþ

ð 1

j2Or

ffiffiffiffiffiffiffiffiffiffiffiffi1�ðzrÞ2

p fprfqrÞ�

Orzr þ jðoþ Or

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðzrÞ2

qÞþ UApq � LApq

o2(25.13)

25 Parametric Identification of Nonlinearity from Incomplete FRF Data Using Describing Function Inversion 313

where,

Or: undamped natural frequency of mode r

zr: damping ratio of mode r

fpr;fqr: mass normalized eigenvectors for mode r

UApq: upper residual

LApq: lower residual

N: number of modes considered

25.2.2 Nonlinearity Type Determination

After determining the locations of nonlinear elements in a structural system from nonzero NLI values, (25.8) is used to

evaluate the numerical values of describing functions for each nonlinear element at various response levels. The value of the

describing function, when there is single nonlinearity present in the system can be obtained from experimental data at

different response amplitudes by using Sherman-Morrison formulation to avoid inversion (see [12] for details). However,

when there are multiple nonlinearities present in the system, Sherman-Morrison formulation cannot be employed. Yet,

simultaneous solution of all describing function values is possible as long as the number of nonlinear elements do not exceed

the total DOF of the system, which would be rather unusual in practical applications. Then, the value of each describing

function can be plotted at different response amplitudes for obtaining Describing Function Footprints (DFF) which can be

used for determining the type of nonlinearity, as well as for parametric identification of nonlinear element(s). Another

common approach used for the same purpose is to obtain Restoring Force (RF) plots. Figure 25.1 presents RF and DFF plots

for some common nonlinear elements. It is clear that RF plots contain more physical information compared to DFF plots.

In this study, DFF calculated as described above is inverted to obtain RF function, which is graphically investigated to

evaluate the type of nonlinearity.

Nonlinearities in a structural system are usually due to nonlinear stiffness (piecewise stiffness, hardening cubic stiffness,

etc.) and/or nonlinear damping (coulomb friction, quadratic damping, etc.). Describing function formulation makes it

possible to handle stiffness and damping nonlinearities separately [19]. The real part of the describing function corresponds

to stiffness nonlinearities whereas the imaginary part corresponds to damping nonlinearities.

The DFF inversion has to be performed using different approaches for stiffness and damping nonlinearities when using

experimental data with no knowledge on the type of the nonlinearity.

The inverse of the describing function can be obtained approximately or analytically. Gibson [20] derived inverses for

real, imaginary and mean parts of a describing function. However, in this formulation the inversion of the real part and the

mean of the describing function requires the information about the type of nonlinearity, but the inversion for the imaginary

part works for any describing function and it does not require information about the type of nonlinearity. The only limitation

for the imaginary part is that the damping nonlinearity, which yields the imaginary part of DF, should not be frequency

dependent. The inverse of the imaginary part of the describing function is given as follows:

NðXÞ � p2

d

dXX2vðXÞ� �

(25.14)

In order to obtain the describing function inversion for the real part, the approximate inversion equations suggested by

Gelb and Vander Velde [19] are used:

NðXÞ � 3XX1

i¼0�2ð Þiv 2iþ1X

� �for n Xð Þ increasing with X (25.15)

NðXÞ � 3X

2

X1i¼0

� 1

2

� �i

vX

2i

� �for n Xð Þ decreasing with X (25.16)

where {N} represents the nonlinear internal force in the system.


The major drawback of these formulations is that when the describing function is inversely proportional to X, for instance

due to Coulomb friction, the summation gives alternating series and a correct result cannot be obtained. However for

damping the imaginary part of the describing function is to be inverted and this is achieved analytically as explained above.

Consequently, in this study it is proposed to use (25.15) or (25.16) for the real part of DF, which is due to stiffness type of

nonlinearity, and to employ (25.14) for the imaginary part of DF, which is due to damping type of nonlinearity.

25.2.3 Parametric Identification of Nonlinearity

There are numerous ways to calculate parametric values for DFF and RF functions. Optimization and black box methods

such as neural networks provide promising results if they are well guided. More direct approaches like graphical methods

require the engineer to be experienced.

In this study the parametric values of the nonlinearity are obtained from RF plots by curve fitting. It is also possible to

obtain the coefficients from DFF when the type of nonlinearity is known. However, for most of the nonlinearity types, DF

representation is far more complicated than the corresponding RF function. It should be noted that when the RF representa-

tion of nonlinearity is already obtained, it is of little importance what the coefficients of RF function are. All the required

information about nonlinear element is stored in the RF function itself which can be further employed in dynamic analysis

for different inputs. Determining RF function, rather than DF may be more important when there is more than one type of

nonlinearity at the same location, in which case it will be very difficult if not impossible to make parametric identification for

each nonlinearity by using DFF.

3000Cubic stiffnesscs.x3

x

x

m2m1

x•

f

f

f

RF

Ff

−Ff

RF

RF

δ

Coulomb damping

Piecewise stiffness

2000

1000

15

4

3

2

1

0

10

5

00 0.1 0.20.150.05

0

� 104

0 0.1

DFF

DFF

DFF

Displacement Amplitude

Displacement Amplitude

0 0.1 0.20.150.05Displacement Amplitude

Del

ta V

alu

eD

elta

Val

ue

Del

ta V

alu

e

0.20.150.05

Fig. 25.1 RF and corresponding DFF plots


25.3 Case Study

The nonlinear identification approach proposed in this study is applied to a 4 DOFs discrete system with a nonlinear elastic

element represented by k1* (a linear stiffness of 100 N/m with a backlash of 0.005 m) between ground and coordinate 1, and

a nonlinear hardening cubic spring k4* (¼ 106*x2 N/m) between coordinates 3 and 4, as shown in Fig. 25.2.

The numerical values of the linear system elements are given as follows:

k1 ¼ k2 ¼ k3 ¼ k4 ¼ k5 ¼ 500N=m

c1 ¼ c2 ¼ c3 ¼ c4 ¼ c5 ¼ 5Ns=m

m1 ¼ 1kg;m2 ¼ 2kg;m3 ¼ 3kg;m4 ¼ 5kg

(25.17)

The time response of the system is first calculated with MATLAB by using the ordinary differential equation solver

ODE45. The simulation was run for 32 s at each frequency to ensure that transients die out. The frequency range used during

the simulations is between 0.0625 and 16 Hz with frequency increments of 0.0625 Hz. The linear FRFs are obtained by

applying a very low forcing (0.1 N) from first coordinate as presented in Fig. 25.2. The nonlinear FRFs are obtained

by applying high forcing (10 N) to the system from the first coordinate as shown in Fig. 25.2. Before using the calculated

FRFs as simulated experimental data, they are polluted by using the “rand” function of MATLAB with zero mean, normal

distribution and standard deviation of 5% of the maximum amplitude of the FRF value. A sample comparison for the

nonlinear and linear FRFs (H11) is given in Fig. 25.3.

It is assumed in this case study that we have only the first columns of the linear and nonlinear receptance matrices. Then,

firstly the missing elements of the linear FRF matrix are calculated by using the approach discussed in Sect. 2.1, and the NLIvalues are calculated for each coordinate by using (25.12). The calculated values are shown in Fig. 25.4a. From Fig. 25.4a it

can easily be concluded that there are nonlinear elements between ground and coordinate 1, and between coordinates 3 and

4. Furthermore, since the nonlinearity can be stiffness and/or damping type, it is possible to make this distinction at this stage

by investigating the real and imaginary parts of the describing function. The real and imaginary parts of the describing

function can be summed over the frequency range and compared with each other. Figure 25.4b reveals that system has

stiffness type of nonlinearity since DF has much higher real part compared to imaginary part.

Using the method proposed, the describing functions representing these nonlinear elements are calculated at different

response amplitudes and are plotted in Fig. 25.5. From the general pattern of the curves it may be possible to identify the

types of nonlinearity. Fitting a curve to the calculated values makes the parametric identification easier. Although

identification of backlash may not be so easy from DFF, it is quite straightforward to identify the type of cubic stiffness

from Fig. 25.5b.

Alternatively, the types of nonlinear elements can be identified more easily if DF inversion method proposed in this study

is used. The inversion of DF is calculated for this case study by using the formulation given in Sect. 2.2, and RF plots

obtained are presented in Fig. 25.6. Figure 25.6a gives the RF plot for the nonlinearity between the first coordinate and

ground, whereas Fig. 25.6b shows the RF plot for the nonlinearity between coordinates 3 and 4. By first fitting curves to the

calculated RF plots, parametric identification can easily be made. The parametric identification results for the nonlinear

elements are tabulated in Table 25.1. As can be seen from the table, the identified values do not deviate from the actual

values more than 12%.

Although the DF inversion formulations are based on polynomial type describing functions, it is shown in this case study

that they work, at an acceptable level, for discontinuous describing functions such as backlash as well.

Fig. 25.2 Four DOFs discrete system with two nonlinear elements


25.4 Experimental Study

The proposed approach is also tested on the experimental setup used in a recent study [21]. The experimental setup and FRF

plots obtained with constant amplitude harmonic forces are given in Figs. 25.7 and 25.8, respectively.

The test rig consists of a linear cantilever beam with its free end held between two thin identical beams which generate

cubic spring effect. The cantilever beam and the thin nonlinear beams were manufactured from St37 steel. The beam can be

taken as a single DOF system with a nonlinear cubic stiffness located between the ground and the equivalent mass

representing the cantilever beam. This test rig is preferred for its simplicity in modeling the dynamic system since the thin

beams yield only hardening stiffness nonlinearity and the structure itself can bemodeled as a single degree of freedom system.

14a b

9000

Sum

of D

escr

ibin

g F

unct

ion

Val

ues

8000

7000

6000

5000

4000

3000

2000

1000

0

12

10

8

NLI

Val

ue

6

4

2

01 2 213

Coordinate Number

NLI Chart Sum of DF Values

Sum of Real PartsSum of Imaginary Parts

Coordinate 0-1 Coordinate 3-44

Fig. 25.4 (a) Nonlinearity index chart, (b) sums of real and imaginary parts of DF values at high forcing excitation

6�10−3

Linear

Nonlinear

Driving Point Linear and Nonlinear FRF Plots

5

4

3

Rec

epta

nce

(m/N

)

2

1

00 2 4 6 8

Frequency (Hz)10 12 14 16

Fig. 25.3 Driving point linear and nonlinear FRF plots


For a single degree of freedom system, the nonlinearitymatrix reduces to the describing function defining the nonlinearity [4]:

v ¼ H � HNL

HNLH(25.18)

150

100

50

00.005 0.01 0.015 0.02

Displacement (m)

Des

crib

ing

Fun

ctio

n V

alue

(N

/m)

Des

crib

ing

Fun

ctio

n V

alue

(N

/m)

Describing Function Values for Backlash Describing Function Values for Cubic Stiffness

Displacement (m)0.025 0.03 0.035 0.002

0

50

100

150

200

250

300

350

400

450

500a b

Calculated DFExact DF

Calculated DFExact DF

0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022

Fig. 25.5 Identified and exact DFs. (a) For nonlinear element between coordinate 1 and ground, (b) For nonlinear element between coordinates

3 and 4

00 0.005 0.01 0.015

Displacement (m) Displacement (m)

DF InvertedExact RF

DF InvertedExact RF

RF Plot for Backlash RF Plot for Cubic Stiffness

0.02 0.025

a b

0.03 0.035

12

10

8

6

4

2

00 0.005 0.01 0.015 0.02 0.025

0.5

1

1.5

RF

(N

)

RF

(N

)2

2.5

3

3.5

Fig. 25.6 Identified and exact RF plots. (a) For nonlinear element between coordinate 1 and ground, (b) for nonlinear element between

coordinates 3 and 4

Table 25.1 Parametric identification results for the nonlinear elements

Actual Identified Error %

Backlash (m) 0.0050 0.0044 12

Linear stiffness part of k1* (N/m) 100 95 5

k2* (cubic stiffness constant) N/m3 1,000,000 956,800 4


The describing function representation of the nonlinearity (n) can be graphically shown as a function of response amplitude,

which makes it possible to identify the type of nonlinearity and to make parametric identification by using curve fitting.

The nonlinear coefficient for the hardening cubic stiffness is first obtained by a static test. In the static test a load cell is

used to measure force and a linear variable differential transformer is used to measure displacement for stepped loadings

with 5 N increments. The force is applied at the point where the cantilever beam is attached to thin beams. The deflection is

also measured at the same point. The results of this test are presented as a force versus deflection curve in Fig. 25.9.

Then, by using the DFF and DF inversion approaches for nonlinear identification, both DF and RF plots are obtained for

the nonlinear element between the tip point of the cantilever beam and the ground (Figs. 25.10, 25.11). The cubic stiffness

constants identified by using DF and RF curves are 2.667 � 108 N/m3 and 2.656 � 108 N/m3, respectively. The cubic

stiffness constant obtained from static test, on the other hand is 2.437 � 108 N/m3. For visual comparison, force deflection

curves obtained from static test and DF inversion approaches are compared with the force deflection characteristics obtained

from DFF approach in Fig. 25.11. As can be seen, DFF and DF inversion approaches yield very close results.

Thus, it can be concluded that the accuracy in parametric identification of nonlinearity by DF inversion is comparable to

that of DFF method. However, the main advantage of DF inversion is that it gives better insight into the type of the

nonlinearity. Furthermore, when the RF function is obtained by DF inversion, it may be directly used in nonlinear model of

the system when time domain analysis is to be used. Then, it will be possible to identify the restoring force of more than one

type of nonlinearity which may co-exist at the same location.

Fig. 25.7 Setup used in the experimental study

Fig. 25.8 Constant force driving point FRF curves


Fig. 25.10 Measured describing function values and the curve fitted

Fig. 25.11 RF plots of nonlinearity for experimental study

Fig. 25.9 Static force–deflection curve for the cubic stiffness


25.5 Conclusions

It was recently shown [21] with an experimental case study that the method developed by Ozer et al. [12] for detecting,localizing and parametrically identifying nonlinearity in MDOF systems is a promising method that can be used in industrial

applications. In the study presented here some improvements are suggested to eliminate some of the practical limitations of

the previously developed method. The verification of the approach proposed is demonstrated with two case studies. The

main improvements are using incomplete FRF data which makes the method applicable to large systems, and employing

describing function inversion which makes the identification of nonlinearity easier.

The method requires dynamic stiffness matrix of the linear part of the system which can be obtained by constructing a

numerical model for the system and updating it using experimental measurements. In this study, however, it is proposed to

make linear modal identification by using one column of the receptance matrix of the system experimentally measured at

low forcing level, and then to calculate the missing elements of the complete FRF matrix so that the dynamic stiffness matrix

required for the identification can be obtained. Note that low forcing testing will not give the linear receptances if

nonlinearity is due to dry friction, since its effect will be dominant at low level vibrations. For this type of nonlinearity

high forcing testing will yield the linear receptace values. The approach suggested is first applied to a lumped parameter

system and it is shown that detection, localization and identification of nonlinear elements can successfully be achieved by

using only one column of the linear FRF matrix.

Secondly, it is proposed in this study to use RF plots obtained from DF inversion for parametric identification, instead of

DFF plots, in order to avoid the limitations in using footprint graphs. It is found easier to determine the type of nonlinearity

by using RF plots, rather than DFF plots, especially for discontinuous nonlinear functions such as backlash.

The application of the approach proposed is also demonstrated on a real structural test system, and it is concluded that the

accuracy in parametric determination of nonlinearity by DF inversion is comparable to that of DFF method, and since RF

plots give better insight into the type of nonlinearity this approach may be preferred in several applications to identify the

type of nonlinearity. Furthermore, when the RF function is obtained, it may be directly used in nonlinear model of the system

if time domain analysis is to be made. Using describing function inversion rather than footprint graphs makes it possible to

identify total restoring force of more than one type of nonlinearity that may co-exist at the same location. Thus, DF inversion

yields an equivalent RF function that can be used in further calculations without any need to identify each nonlinearity

separately. Consequently, it can be said that the approach proposed in this study is very promising to be used in practical

systems, especially when there are multiple nonlinear elements at the same location.

References



2. Maia NMM, Silva JMM (1997) Theoretical and experimental modal analysis. Research Studies Press, Baldock, Hertfordshire, England

3. Ewins DJ (1995) Modal testing: theory and practice. Research Studies Press, Baldock, Hertfordshire, England

4. Ozer MB, Ozg€uven HN (2002) A new method for localization and identification of nonlinearities in structures. In: proceedings of ESDA2002:

6th biennial conference on engineering systems design and analysis, Istanbul

5. G€oge D, Sinapius M, F€ullekrug U, Link M (2005) Detection and description of non-linear phenomena in experimental modal analysis via

linearity plots. Int J Non-linear Mech 40:27–48

6. Worden K, Tomlinson GR (2001) Nonlinearity in structural dynamics. Institute of Physics Publishing, Bristol

7. Siller HRE (2004) Non-linear modal analysis methods for engineering structures. PhD thesis in mechanical engineering. Imperial College

London/University of London

8. Narayanan S, Sekar P (1998) A frequency domain based numeric–analytical method for non-linear dynamical systems. J Sound Vib

211:409–424

9. Muravyov AA, Rizzi SA (2003) Determination of nonlinear stiffness with application to random vibration of geometrically nonlinear

structures. Comp Struct 81:1513–1523

10. Masri SF, Caughey TK (1979) A nonparametric identification technique for nonlinear dynamic problems. Trans ASME J Appl Mech

46:433–445

11. Elizalde H, Imregun M (2006) An explicit frequency response function formulation for multi-degree-of-freedom non-linear systems. Mech


12. Ozer MB, Ozg€uven HN, Royston TJ (2009) Identification of structural non-linearities using describing functions and the Sherman–Morrison

method. Mech Syst Signal Process 23:30–44

13. Thothadri M, Casas RA, Moon FC, D’andrea R, Johnson CR Jr (2003) Nonlinear system identification of multi-degree-of-freedom systems.

Nonlinear Dyn 32:307–322


14. Cermelj P, Boltezar M (2006) Modeling localized nonlinearities using the harmonic nonlinear super model. J Sound Vib 298:1099–1112

15. Nuij PWJM, Bosgra OH, Steinbuch M (2006) Higher-order sinusoidal input describing functions for the analysis of non-linear systems with

harmonic responses. Mech Syst Signal Process 20:1883–1904

16. Adams DE, Allemang RJ (1999) A New derivation of the frequency response function matrix for vibrating non-linear systems. J Sound Vib

227:1083–1108

17. Tanrikulu O, Kuran B, Ozg€uven HN, Imregun M (1993) Forced harmonic response analysis of non-linear structures. AIAA J 31:1313–1320

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Imperial College London

19. Gelb A, Vander Velde WE (1968) Multiple-input describing functions and nonlinear system design. McGraw Hill, New York

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Chapter 26

Finding Local Non-linearities Using Error Localization

from Model Updating Theory

Andreas Linderholt and Thomas Abrahamsson

Abstract Within the aerospace industry, linear finite element models are traditionally used to describe the global structural

dynamics of an aircraft. Ground vibration test data serve to facilitate the validation of models which are then used to

characterize the aeroelastic behavior of the aircraft and to predict the responses due to dynamic loads. Thus, it is vital that the

models contain the essential dynamics of the aircraft. Observed nonlinearities are judged to be local in nature whereas the

main part of the structure behaves linearly under normal loading. In this work we focus on the identification of nonlinear

effects and do that based on model updating theory. That includes methods for error localization with proper selection of

candidate error parameters. The nonlinearities are treated as local modeling errors not considered in the linear system model.

The error localization behavior is studied using synthetic test data from a simple system, known as the ECL Benchmark, with

known nonlinear properties.

Keywords Non linear •Model updating • Error localization • Parameter identifiability • Data informativeness • Optimization

26.1 Introduction

The coupling between an overall linear structure and local nonlinearities is a classical problem in industry. That is also the

case in the aircraft industry where the major part of a structure may be satisfactorily described by a linear model although

local nonlinearities do exist. Typically, joints between stores and the aircraft introduce nonlinear characteristics due to e.g.

gap and dry friction. Traditionally, linear finite element models are used to describe the global behavior of an aircraft since

they are computationally inexpensive and at the same time good from a global perspective. However, linear models may not

be capable of reproducing all the characteristics found in test data. In aircraft industry flight test data and ground vibration

test (GVT) data complement each other in characterizing the structural dynamics. The loading condition in the two situations

are different and during flight the effects of some types of nonlinearities may be suppressed while others are engaged/

increased compared to the loading conditions during a GVT. However, test data stemming from ground vibration tests form

the primary source for detailed model validation and updating. Thus, such data are concerned here.

The observed deviations between test data and the corresponding analytical data stemming from a linear model are here

thought of as being caused by errors within the nominal, linear, model. The task then becomes a model updating problem.

The parameters needed to substantially increase the model’s capability of representing the real structure are most likely not

included in the initial model. Therefore, a set of candidate parameters controlling nonlinear effects, opposite to what is used

within the vast majority of model updating exercises, have to be added. The candidate parameters have to be chosen using

engineering insight into the structure at hand.

A. Linderholt (*)

Department of Mechanical Engineering, Linnaeus University, SE-35195 V€axj€o, Swedene-mail: [email protected]

T. Abrahamsson

Department of Applied Mechanics, Chalmers University of Technology, SE-41296 G€oteborg, Swedene-mail: [email protected]



323



Before any updating exercises take place, the selection of the data to use in combination with the selection of model

parameters to be involved should be examined. Test data to be used for computational model updating have to fulfil at least

two requirements. Firstly, they should be informative with respect to the parameters used for updating meaning that the

test data should be sensitive to changes of the parameter values. Secondly, they have to change differently for changes

of different parameters or group of parameters which is to say that the parameters should be identifiable. Should these

two requirements not be fulfilled, it is simply impossible to get a reliable parameter value estimation made using that test

data [1].

The Cramer-Rao lower bound quantifies a limit to the accuracy of parameter estimates from the information in the test

data. The inverse of the Cramer-Rao lower bound is known as the Fisher information. The Cramer-Rao lower bound and the

FIM are useful quantities to assess test data informativeness and parameter identifiability and they can be estimated a priori

using a calculation model. The test data should be chosen such that the expected variances of the estimated parameters are

small. For multi-parameter problems, the FIM and its inverse are matrices. Hence, the quantification of an, in some aspect,

large FIM, or a small inverse of it, is not clear [2]. This is described further in Sect. 2.3.

The FIM and its inverse, and thereby the data informativeness and the parameter identifiability, varies with the response

data used. It is therefore important to include all available data that carries information that differentiate parameter settings.

This is partly controlled by the selection of excitation and response measurement during test; that is the actuator and

sensor placement together with the excitation time history. Another part is the choice of the perspective on which the test

data are looked upon. Sometimes, processing the data may hide or destroy information that are in the original data. Some

test data simply destroys or decreases the goodness of the overall test data; such data should be excluded before the

updating is made.

Another issue regarding measurements on nonlinear systems is which sampling frequency to use [3]. The well known

Nyquist’s sampling theorem, or Shannon’s criterion, states that the sampling frequency should be at least twice the highest

frequency of interest which for a linear system is equal to twice the largest excitation frequency. Ljung also shows that the

sampling frequency can be high which leads to increase of the variance of estimated parameters [4]. This phenomena is not

further considered here.

Here, the focus is on the selection of data to use for the optimization process, known as model updating, and the parameter

selection; these are coupled. The paper consists of theoretical part and a numerical example illustrating the theory. For this, a

model setting out from the ECL (Ecole Centrale de Lyon) nonlinear benchmark is used [5]. To the ECL-structure nonlinear

springs; one linear, one quadratic and one cubic, are added. Although, Georg Duffing treated only a few nonlinear systems,

the accepted de notion for any differential equation in which a cubic nonlinearity is included is nowadays a Duffing equation

[6]. Hence, the numerical example consists mathematically of the Duffing equation type.

26.2 Theoretical Background

In general, the forced vibration response of a nonlinear deterministic system can be expressed by the state-space system of

ordinary differential equations as

_x ¼ f ðx; uÞ; y ¼ gðx; uÞ (26.1a,b)

Here u is the excitation vector, x the state vector and y the response vector. For steady-state periodic excitation it is well-known that the linear system reacts to such stimulus by a periodic steady-state response with same periodicity after initial

transients have settle. However, for nonlinear systems it is also well-known that this is not generally the case. The steady state-

response of a nonlinear systemmight be periodicwith stimulus periodicity orwith other periodicity, or therewill be no periodic

response at all [7]. For stationary harmonic excitation with excitation frequency O, corresponding to period T ¼ 2p/O,the non-linear system may respond in a periodic steady-state with the periodicity nT; n 2 Zþ which we will refer to as being

n-cycle periodic.We confine our study to n-cycle periodic steady-state conditions and thereby rejects response solutions thatare chaotic, non-periodic or periodic with any other periodicity.

For a linear system with periodic excitation

uðtÞ ¼ < u1 exp ðiO tÞ þ u2 exp ð2iO tÞ þ . . .þ un exp ðn i O tÞ þ . . .þ u1=2 exp ðiO t=2Þ þ . . .þ u1=n exp ðiO t=nÞ� �(26.2)

324 A. Linderholt and T. Abrahamsson

where the harmonic order is v 2 Zþ, the response may be written as

yðtÞ ¼ < y1 exp ðiO tÞ þ y2 exp ð2iO tÞ þ yn exp ðn i O tÞ þ . . .þ y1=2 exp ðiO t=2Þ þ . . .þ y1=n exp ðiO t=nÞ�(26.3)

or

yðtÞ ¼ < HðOÞu1 exp ðiO tÞ þ H ð2OÞu2 exp ð2iO tÞ þ . . .þ H ðO=nÞ u1=n exp ðiO t=nÞ� �(26.4)

where HðoÞ is the system’s complex-valued frequency dependent transfer function at frequency o. For the mono-frequency

excitation

uðtÞ ¼ < u1 exp ðiO tÞð Þ (26.5)

the linear system’s stationary response is thus

yðtÞ ¼ < HðOÞ u1 exp ðiO tÞð Þ (26.6)

However, for the nonlinear system in n-cycle periodic steady-state, exposed to a mono-frequency loading, the response is

generally not mono-frequency and may be written as

yðtÞ ¼ < �y1 exp ðiO tÞ þXn max

n¼2

�yn exp ðiO tÞ þ �yn exp ðiO t=nÞð Þ( )

(26.7)

where �yn and

y�n are the superharmonic and subharmonic distortion amplitudes of order v respectively. The magnitude of

these generally depend on the stimulus magnitude u1. It is convenient for the sequel to denote the stimulus and response

magnitudes by load, order and frequency indices such that ulvk denotes the excitation level l of harmonic order v at the kthdiscrete frequency Ok and ylvk is then the vth order response due to that excitation.

26.2.1 d-Level Multi-Harmonics Frequency Response Functions

In lab testing, the mono-frequency excitation is an anomality and harmonic distortion is always present due to imperfections

in the test setup. Instead of trying to achieve a pure sinusoidal excitation in lab, which requires some closed loop control [8],

one might have a better chance to enforce a multi-sine stimulus with a distortion level that overshadows the intrinsic test

setup distortion. Let the d-level multi-harmonic excitation of load level l at the fundamental frequency Ok be defined as

uklðtÞ ¼ < ul exp ðiOk tÞ þ dulXn max

n¼2

exp ði n Ok tÞ þ exp ðiOk t=nÞð( )

(26.8)

We note that by letting d � 1 we create a harmonic function with small distortion to the fundamental harmonics atOk. We

may use that as the stimulus signal in lab testing or in simulation. After initial transients have settle, under n-periodic steady-stateconditions, the system response will eventually become periodic and before the full periodic condition arise we have

yklðtÞ ¼ < �yl1 exp ðiO tÞ þXn max

n¼2

�yln exp ði nOktÞ þ y�ln exp ðiOk t=nÞð( )

þ residual (26.9)

In testing or numerical simulation, we can find the complex-valued amplitudes of �y and y� by regression. We define the

d-level multi-harmonic frequency response function that relates the input at the ith dof to the response at the jth dof to be

the complex-valued quantities

�HlnijðOkÞ ¼ �ylniðOkÞ=ulj and HlnijðOkÞ ¼ y�lniðOkÞ=ulj (26.10a,b)

26 Finding Local Non-linearities Using Error Localization from Model Updating Theory 325

for the zeroth order and higher order harmonics and subharmonic orders respectively. We note in particular that, by

construction, for a linear system the following relations hold

�HlnijðOkÞ ¼ d �Hl1ijðnOkÞ and HlnijðOkÞ ¼ d �Hl1ijðOk=nÞn 6¼ 1 (26.11a,b)

i.e. the superharmonic and subharmonic frequency response function can by obtained from the frequency response function

of the fundamental harmonics by a scaling with the level parameter d.For a nonlinear system, relations as those of (26.11a,b) do not hold and the d-level frequency response functions

will depend on the excitation level l. A study of a quantity that relates to the difference between these functions of the

nonlinear system and the corresponding functions of the linearized systems might the give an insight into the nonlinear

behavior of the system. Before we proceed, we make a slight adjustment of our notation. By the d-level frequency responsefunction Hlvkij we mean the function value that pertain to the excitation level of index l (l ¼ 1,2,. . .,lmax), of indexed order

n (n ¼ 1,2,. . .,nmax) that includes both superharmonic and subharmonic orders, and at indexed fundamental frequency

Ok (k ¼ 1,2,. . .,kmax). The frequency response function value Hlvkij relates the output at dof i with the input at dof j. We also

use the Matlab colon notation in the respect we define the vector H:vkij as the vector with elements as function values Hlvkij

for indices l of all available loads as

H:nkij ¼ 8lvect

ðHlnkijÞ (26.12)

with fixed order index, frequency index and dof index. Similarly we mean by the vectors Hl: kij and Hlv: ij the vectors

Hl:kij ¼ 8nvect

ðHlnkijÞ and Hln:ij ¼ 8kvect

ðHlnkijÞ (26.13)

Also we define the vector operator for defined subsets of loads ls as

Hlsnkij ¼ l 2 lsvect

ðHlnkijÞ (26.14)

with similar notation for subsets of order indices and frequency indices. We also generalize this vectorization concept,

such that e.g.

H::kij ¼ 8ðl; nÞvect

ðHlnkijÞ (26.15)

is the vectors of data for all available load levels and orders and

H:nskij ¼ 8l; n 2 nsvect

ðHlnkijÞ (26.16)

and is the vector of data for all available load levels and for orders from the subset vs. The use of any other combination

of indices should now be obvious.

26.2.2 A Deviation from Linearity Criterion Function

For the linear system, the frequency response function are independent of the load level l and we denote it with Hlinvkij. In the

sequel we focus on the difference between the nonlinear system’s frequency response behavior and the linearized systems

counterpart. We define the deviation from linearity as

elnkij ¼ Hlnkij � Hlinnkij (26.17)


and use the same vectorizing notation for this quantity as before for H. In short we call the vector of elements elnkijthe vector e, with proper indexing when needed for clarity. The difference from linearity is a complex-valued

number and consequently e is a complex-valued vector. We define the real-valued scalar deviation from linearity criterion

function E as

E ¼ �eT e (26.18)

where overbar denotes the complex conjugate. This function is obviously zero for the linear system. In the following,

for identifiability purpose, we study the gradient, rE, of this function with respect to model parameters, pm, m ¼ 1,2,. . ..

rE ¼ @E=@p1 @E=@p2 . . .½ � (26.19)

The m:th component can be expressed as

@E=@pm ¼ 2< �eT @e=@pm� � ¼

Xk

< 2�e::k:: @e::k::=@pmf g ¼defXk

ek; m (26.20)

These gradient component functions ek;m are functions of discrete frequency Ok. We use these to evaluate the parameter

identifiability.

26.2.3 Ranking Identifiability of Parameters

The requirements for a test design is that the resulting test data should be informative with respect to the parameters and that

the parameters should be identifiable. The first means that a change of the value of any parameter should change the observed

response in a noticeable way. The latter is to say that test data should differentiate changes of different parameter values. In

computational model updating, these requirements are coupled to the usefulness of the test data in estimating model

parameter values accurately. The two requirements can jointly be stated as; regardless of which parameters that are

estimated, precise estimates require a small parameter covariance matrix. This should, if possible, be examined prior to

the test using a model.

The Fisher information, which is a scalar, is a measure of the information an observable variable, i.e. test data, Xcarries with respect to a single unknown parameter that the probability of X depends on. The probability function of X is

also the likelihood function for X given a certain parameter value p. That function is denoted ƒ(X: p). The partial

derivative of the natural logarithm of f with respect to p is denoted the score. An unbiased estimator is an estimator

having a mean that converges to the correct parameter setting as the number of test data tends towards infinity. That

means that the first moment of the score vanish. Further, the second moment of the score is known as the Fisher

information. The corresponding measure for a multi parameter problem is the Fisher information matrix (FIM) having

size p x p. When it comes to estimating parameter values, pest, the Fisher information matrix, here denoted JðpÞ, plays animportant role, see Walter and Pronzato [9] and Spall [10]. The reason for this is the Cramer-Rao theoretical lower bound

which establishes a limit on the expectation of the covariance matrix of the estimate of the parameter values. This limit is

coupled to the FIM according to;

X½pe s t � p�½pe s t � p�T � J�1 (26.21)

in which X denotes the expectation. The Cramer-Rao bound implies that, irrespective of the method used to quantify the

parameters from the data, there is a bound on the estimation precision that can not be overcome. Similar to the single

parameter situation, the Fisher information matrix is determined by the joint probability density function, f;

JðpÞ ¼ X@

@plog f

� �T @

@plog f

� �(26.22)


Here, the measurement noise is assumed to be of the simplest kind, from an analytical point of view. It consist of

independent Gaussian sequences having zero mean values. Further, it is assumed that the noise is uncorrelated and

statistically equivalent for all measured quantities; the noise variance equals s2. Then, the FIM expression simplifies into:

JðpÞ ¼ 1

s2XNk¼1

rET rE (26.23)

The gradient rE is a 1� m vector. Each of its columns is formed as a summation, see (26.20). Let all, N, possiblecomponents contributing in the summation be stacked on top of each other to form an N � mmatrix that is here denotedr ~E.The FIM can now be calculated as

JðpÞ ¼ 1

s2XNk¼1

r ~ET r ~E (26.24)

with unchanged result. However, the formation of r ~E is useful in the study of data informativeness and parameter

identifiability which are complementary. When the value of a model parameter is changed, the measurement data should

change noticeable in order to the data to be informative with respect to the parameters involved in the model updating.

Complementary, changing the values of two parameters should affect the measurement data differently. A combination of a

parameter selection and a test design rendering in that all pairs of columns of r ~E are orthogonal is the ultimate. The angle

between two columns is easily calculated and using the definition of the modal assurance criterion gives a good, normalized,

measure of the linear dependencies. When two columns associated with parameters pi and pj are orthogonal, the element of

the ith row and the jth column of the Fisher information matrix is zero.

If the ideal situation for which columns are orthogonal is not fulfilled, the linear relationship between groups of columns has

to be examined since changing one parameter (or a group of parameters) may change the dynamics of themodel in the sameway

as a change of another parameter (or another group of parameters). When this happens, the parameters are not identifiable from

test data, which means that the updating is unreliable. Proper measures have to be taken, possibly involving re-parameterization

or a change of the test design, see Linderholt and Abrahamsson [11]. The conclusion is that it is not enough to examine only

relations between pairs of parameters and this is where the FIM and its inverse is a more useful measure.

A number of criteria ranking the goodness of the FIM or its inverse have been proposed. The A-optimality focus on

minimizing the trace of the inverse of the Fisher information matrix. The D-optimality aims at maximizing the determinant

of the FIM. E-optimality, maximizes the minimum eigenvalue of the FIM. Further, the T-optimality maximizes the trace of

the FIM. Finally, the G-optimality seeks to minimize the maximum expected variance estimated parameter values. These are

just examples of frequently used matrix criteria. In this study, yet another criterion is chosen; that is the condition number of

the FIM. A low condition number indicated a well conditioned matrix which in turn is necessary for small parameter

estimate variances given by the inverse of the FIM.

Since each column in r ~E is partitioned into contributions from different sources such as different sensors or different

excitation frequencies etc., also the FIM is built up by a summation of FIM:s stemming from each combination of sensor,

load level, excitation frequency etc. Such a combination is here denoted with the index s.

JðpÞ ¼XSs¼1

sJðpÞ (26.25)

It is not known a priori how many and which of the contributing data that should be taken into account to get a low

condition number of the FIM. Examining all possible combinations of reductions is time consuming since the number of

combinations grows rapidly as a function of the number of possible sensors etc. Using a sub-set selection technique is an

inexpensive choice to achieve a good, although not guarantied to be optimal, solution, see Miller [12].

26.3 Numerical Example

We illustrate the identifiability properties of the parameters with respect to the deviation from linearity criterion function by

considering a simulation model of the ECL Benchmark setup [5]. The ECL Benchmark was designed with the parameter

identification of nonlinear systems in mind. It consists of a cantilever beam of simple cross section, supported in the free


end by a thin structural element that is subjected to tension and mainly adds membrane stiffness to the system. Since the

membrane is short it gives a restoring force to the beam end that is rather strongly cubic in the beam tip deflection. The beam

is loaded by a concentrated transversal force not far from its fixed support end. The system can be seen in Fig. 26.1.

In our simulation model we use three planar linear Euler-Bernoulli finite elements of equal length to model the beam.

That introduces three nodal translational and three nodal rotational degrees-of-freedom. The membrane member is modelled

as a short taut string which introduces nonlinear effects because of its nonlinear kinematics. The elasticity of the supporting

structure is represented by the elasticity of a fictitious rod element. The taut force of the membrane in its neutral position and

the total stiffness of the membrane and support rod are two introduced free parameters of the identifiability problem. To

increase the identifiability complexity, three more free parameters are introduces as the linear, quadratic and cubic stiffness

coefficients of a discrete spring that supports the free end of the beam. The discrete spring is not part of the ECL Benchmark.

Numerical data is given by the caption of Fig. 26.1.

In the simulation we apply a periodic force u(t) with three different magnitudes (ul ¼ 2; 5 and 10N for l ¼ 1,2,3) and

evaluate the response during each full period. We do a stationarity check by comparing the response from period to period

and assume that stationarity is achieved when the responses between two consecutive periods are the same, i.e. when

the norm of the difference of responses between periods is small. All cases here studied settled to periodic solutions.

We evaluate the multi-level FRF:s of the last simulated period for the given condition.

Figure 26.2 shows the d-level multi-harmonics frequency response functions with d ¼ 0.01. It can be noted that

the frequency response functions deviates from the linearized system’s frequency response functions mainly in the vicinity

of structural resonances where the deviation is significant.

Figures 26.3 and 26.4 show the gradient components ek;m, see (26.20), for two load levels ul ¼ 1 ¼ 2N and ul ¼ 2 ¼ 5N.

The gradients were evaluated by finite differentiation of the parameters pi with Dpi ¼ 0:01. We use a vectorizing operation

such that

e; m ¼ 8kvect

ek; m (26.26)

and study the correlation of these vector functions for the five parameters, m ¼ 1,2,. . .,5. We use as correlation index Cmn

the cosine-square of the angle between the vectors (similar to the well-known MAC index for eigenvectors) as

Cmn ¼ cos2 ffðe; m; e; nÞ� �

(26.27)

Graphical illustrations of these indices are shown as inserts in Figs. 26.3 and 26.4.

EI,m,Lb

q2q5q3

q1 q7q6q4

(1+p2)EA

1 32

u(t)(1+p2)EA

kNL(1+p1)R

kNL= (1+p3)k0(1+p4 / +p5( )2)q5 q52 q5 q53/

(1+p1)R Lm Lr

2´N(q5,q7)

N(q5,q7)N=EA(Lm(q5,q7)-Lm0)/Lm0

Nsin φ

φ

Fig. 26.1 Seven-degree-of-freedom model of ECL Benchmark setup. Photo of testbed from [Thouverez]. Beam 1 bending stiffness, mass per unit

length and total length are EI, m and Lb respectively. Taut membrane 2 has tensional stiffness EA and free length Lm0. Elastic supporting rod 3 hastensional stiffness EA and free length Lr. System pretension is R and the periodic loading is u(t). The loading of displaced membrane 20 is shown ininsert. The restoring force from the membraneNsinf acting at the beam end is a nonlinear function of the nodal displacements q5 and q7 because themembrane rotationf is not small. Spring stiffness kNL is nonlinear in q5. The freemodel parameters of the study are p1, p2, . . ., p5which in the nominal

setting are all zero. Numerical data: EI ¼ 672 Nm2, m ¼ 3.36 kg/m, EA ¼ 3150kN, Lb ¼ 593 mm, Lr ¼ 57 mm, R ¼ 10 N, k0 ¼ 175 N/m,

q52 ¼ 30 mm and q53 ¼ 33.5 mm


The seven degrees of freedom together with the three load levels (2, 5 and 10N) and the five orders (1, 2, 3, 1/2 and 1/3),form 105combinations each contributing to the FIM. By using a sub-set selection technique, one contribution at the time is

removed. The selection is made such that the condition number of the remaining FIM is kept as low as possible within each

step. The result is shown in Fig. 26.5. The result shows that by reducing the set of combinations by five, the condition number

of the FIM is lowered considerably.

Fig. 26.2 d-Level multi-

harmonics frequency response

functions at load amplitudes

Fl ¼ 2; 5 and 10 N for

l ¼ 1; 2; 3


Fig. 26.3 Functions ek;m evaluated for m1 ¼ 2N together with correlation indices Cmn or corresponding gradient vectors. We note that the vectors

are highly correlated for harmonic orders 1 and 3 at this load level


Fig. 26.4 Functions ek;mevaluated for load levels ml ¼2; 5; and 10 N together with

correlation indices Cmn for

corresponding gradient

vectors of harmonic order 1/3.

We note that the vectors are

highly uncorrelated for load

level 2N


The data stemming from degree-of-freedom no.6, a load level equal to 10N and order 1 is the first to be removed

according to the reduction criterion. The Auto MAC of the part of r ~E gradient stemming from that combination is shown

in Table 26.1. It is obvious that the column vectors are highly correlated. Thus, it is understandable that excluding such

data strengthens the information which results in a decrease of the condition number of the Fisher information matrix,

see Fig. 26.5.

26.4 Conclusions

It is shown that test data should be looked upon from different perspectives; processing data may hide or destroy

information. Here, information from sub and superharmonic component found from steady state oscillation of nonlinear

systems assist the information within the fundamental harmonic response. Instead of trying to get a pure mono-harmonic

excitation, which is hard in practise, it is here proposed to include a few sub and superharmonics having amplitudes that are

low compared to the amplitude of the fundamental harmonic but still large enough to dominate over the excitation noise.

Furthermore, the condition number of the Fisher information matrix, associated with the physical parameters selected, is

chosen as the optimization objective. Finally it is shown that disregarding data that have low information value can actually

increase the parameter identifiability and data informativity.

Acknowledgement We gratefully acknowledge the Swedish National Aviation Engineering Programme (NFFP) for their kind support of

this work.

References

1. Linderholt A (2003) Test design for finite element model updating, identifiable parameters and informative test data. Department of Applied

Mechanics, Chalmers University of Technology, G€oteborg2. Udwadia FE (1997) A finite element model updating formulation using frequency responses and eigenfrequencies. NAFEMS international

conference on structural dynamics modelling, pp 293–305

3. Worden K, Tomlinson GR (2001) Nonlinearity in structural dynamics: detection, identification and modelling. Taylor and Francis,

London, GB

90 75 60 45 30 15 0106

107

108

Five data sets removed from the setbuilding up the FIM

No. of remainingcombinations

Condition no.

Fig. 26.5 The condition number of the Fisher’s information matrix as a function of the remaining of contributing data sets i.e. combinations of

degree’s of freedom, load levels and orders

Table 26.1 The AutoMAC matrix associated with the data that should be excluded first among the 105 candidates.

The data stem from degree of freedom no. 6, loadlevel equal to 10N and order 1

1.00 0.85 1.00 0.98 0.89

0.85 1.00 0.85 0.84 0.76

1.00 0.85 1.00 0.98 0.89

0.98 0.84 0.98 1.00 0.96

0.89 0.76 0.89 0.96 1.00


4. Ljung L (1987) System identification: theory for the user. Prentice-Hall, Englewood Cliffs

5. Thouverez F (2003) Presentation of the ECL benchmark. Mech Syst Signal Process 17(1):195–202

6. Kovacic I, Brennan MJ (2011) The Duffing equation, nonlinear oscillations and their behaviour. Wiley, Chichester, West Sussex

7. Nayfeh AH, Mook DT (1979) Nonlinear oscillations. Wiley, New York

8. Josefsson A, Magnevall M, Ahlin K (2006) Control algorithm for sine excitation on nonlinear systems. International modal analysis conference

(IMAC) XXIV, St. Louis

9. Walter E, Pronzato L (1997) Identification of parametric models from experimental data. Springer, Great Britain

10. Spall C (1999) The information matrix in control, computation and some applications. Thirty-eighth conference on decision and control,

Arizona, pp 2367–2372

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17(3):579–588

12. Miller AJ (1990) Subset selection in regression. Chapman and Hall, London


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