detection of structural on - university of cincinnati€¦ · this investigation on a multiple...

DETECTION OF STRUCTURAL NON-LINEARITIES USING THE FREQUENCY

RESPONSE AND COHERENCE FUNCTIONS

A thesis submitted to the

Division of Research and Advanced Studies of the University of Cincinnati

in partial fulfillment of the

requirements for the degree of

MASTER OF SCIENCE

in the Department of Mechanical Engineering of the College of Engineering

2000

by

Thomas Roscher

Dipl.-Ing. (FH), Hochschule fuer Technik und Wirtschaft, Dresden, 1999

Committee Chair: Dr. Randall J. Allemang

ABSTRACT

It has been observed during modal tests of a structure, that high forcing level cause the Frequency

Response Function (FRF) estimates to show non-coherent behavior over certain frequency bands due to

non-linearities. This investigation on a multiple degree of freedom (MDOF), multi-connected simulation

model is intended to find corresponding non-linearities for the above-mentioned problems, whereas FRFs

and Coherence (COH) functions are the objects in focus. Special attention is paid to the typical testing

case, where broadband techniques are used and the presence of noise has to be assumed. In detail, these

testing cases include single input multiple output (SIMO) and multiple input multiple output (MIMO)

situations. The effects of cubic, softening, deadzone, hardening/softening, softening/hardening

displacement related non-linearities will be analyzed. Furthermore, velocity related non-linearities like

quadratic, non-symmetric and coulomb damping terms are applied to the simulation model. In addition, a

technique for lumped mass systems will be introduced which is capable of eliminating the effects of

non-linear motion between two DOFs, making it possible to separate structural non-linearity induced

distortions from measurement noise and digital signal processing errors.

ACKNOWLEDGEMENT

This work was done at the University of Cincinnati Structural Dynamics Research Laboratory (UC-SDRL).

I would like to thank all members of the UC-SDRL for their help and support throughout my work. I

appreciated all formal and informal discussions (because that is, what keeps progress alive); all suggestions

and honest criticism, which offered different points of view on the subject and by these means, shaped the

thesis in a crucial way.

I would also like to address my special thanks to my advisor and committee chair Dr. Randall Allemang.

His guidance and council were invaluable, even before I was taking on my thesis work and continued all

along the entire study. Without him, and that is true, this project would not have been possible. I also want

to express my gratitude to all members of my thesis committee, Dr. Edward Berger, Dr. Dave Brown and

Dr. Allyn Phillips. Their ideas, comments and suggestions provided a significant share to the final result.

Let me express my feelings in only one sentence: I consider myself lucky to have met all of you.

I

TABLE OF CONTENTS

1 MOTIVATION AND OVERVIEW .............................................................................................. 1

2 LINEAR AND NON-LINEAR VIBRATION............................................................................... 1 2.1 Linear Theory and Modal Analysis .............................................................................................. 1 2.2 Non-Linear Overview................................................................................................................... 1

3 SIMULATION CONSIDERATIONS ........................................................................................... 1 3.1 Simulation Model ......................................................................................................................... 1 3.2 Numerical Precision...................................................................................................................... 1

4 APPLICATION OF NON-LINEAR INTERACTIONS.............................................................. 1 4.1 Combined Coherence Function .................................................................................................... 1 4.2 Characteristic Effects of Non-Linearities ..................................................................................... 1

4.2.1 Hardening Stiffness.............................................................................................................. 1 4.2.2 Softening Stiffness ............................................................................................................... 1 4.2.3 Non-Symmetric Stiffness ..................................................................................................... 1 4.2.4 Deadzone/ Play..................................................................................................................... 1 4.2.5 Non-Linear Damping ........................................................................................................... 1

4.3 Non-Linear Effects in the Presence of Noise................................................................................ 1

5 SUMMARY/ FUTURE WORKS................................................................................................... 1

6 REFERENCE LIST........................................................................................................................ 1

7 APPENDIX...................................................................................................................................... 1 7.1 SIMULINK® Model ..................................................................................................................... 1 7.2 Definition of Physical Parameters, 4-DOF-Model, Linear ........................................................... 1 7.3 List of Cases in Section 4.2 .......................................................................................................... 1

II

LIST OF FIGURES Figure 2-1: Transfer Function ........................................................................................................................1 Figure 2-2: Superposition Principle ...............................................................................................................1 Figure 2-3: Types of Non-Linearities ............................................................................................................1 Figure 2-4: Reduction of Linear Stiffness Factors in Serial Connection .......................................................1 Figure 3-1: 4-DOF-Model, a) Physical Scheme, b) Connection Scheme ......................................................1 Figure 3-2: Frequency Response Function, Linear Model.............................................................................1 Figure 3-3: Frequency Response H32 and Error Estimate (|tol| = 1⋅10-5/1⋅10-8m) ..........................................1 Figure 3-4: Response x4(t) for different Tolerances and Error Estimation, One Ensemble ...........................1 Figure 3-5: Error Estimate RMS(dx4) ............................................................................................................1 Figure 4-1: a) MDOF System, b) Internal and External Forces on MDOF System.......................................1 Figure 4-2: a) SDOF System, b) Linearizing Concept for Cubic Stiffness....................................................1 Figure 4-3: FRFs and COH, Hardening Stiffness, Inp. 3, k13 .......................................................................1 Figure 4-4: Coherence and Combined Coherence, Hardening Stiffness, Inp. 3, k13.....................................1 Figure 4-5: FRFs and COH, Hardening Stiffness, Inp. 2, k13 .......................................................................1 Figure 4-6: Coherence and Combined Coherence, Hardening Stiffness, k13 ................................................1 Figure 4-7: FRFs and COH, Hardening Stiffness, Inp. 2, k13 .......................................................................1 Figure 4-8: Correlation between {H12} and {H32} .........................................................................................1 Figure 4-9: Coherence and Combined Coherence, Hardening, Inp. 1, k23 ...................................................1 Figure 4-10: FRFs and COH, Hardening Stiffness, Inp. 1, k23 .....................................................................1 Figure 4-11: FRFs and MCOH, Hardening Stiffness, Inp. 1+2, k23 .............................................................1 Figure 4-12: Multiple Coherence and Combined Coherence, Hardening Stiffness, k23 ...............................1 Figure 4-13: FRFs H21 and H31, MCOH, Hardening Stiffness, Inp. 1+2, k23 ...............................................1 Figure 4-14: a) SDOF System, b) Linearizing Concept for Softening Stiffness............................................1 Figure 4-15: Ordinary and Combined Coherence, Softening Stiffness, Inp. 1, k23 ......................................1 Figure 4-16: FRFs and COH, Softening Stiffness, Inp. 1, k23 ......................................................................1 Figure 4-17: FRFs and MCOH, Softening Stiffness, Inp. 1+2, k23...............................................................1 Figure 4-18: Multiple Coherence and Combined Coherence, Softening Stiffness, Inp. 1+2, k23 .................1 Figure 4-19: FRFs and Coherence, Softening Stiffness, Inp. 3, k3all............................................................1 Figure 4-20: Ordinary and Combined Coherence, Softening Stiffness, Inp. 3, k3all ....................................1 Figure 4-21: Linearizing Concept for Non-Symmetric Stiffness ...................................................................1

III

Figure 4-22: SDOF System with Hardening/Softening Stiffness ..................................................................1 Figure 4-23: FRFs and COH, Hardening/Softening Stiffness, Inp. 2, k13 ....................................................1 Figure 4-24: Ordinary and Combined Coherence, Hardening/Softening Stiffness, Inp. 2, k13.....................1 Figure 4-25: FRFs and MCOH, Hardening/Softening Stiffness, Inp. 1+2, k23.............................................1 Figure 4-26: Multiple and Combined Coherence, Hardening/Softening, Inp 1+2, k23 .................................1 Figure 4-27: FRFs and COH, Hardening/Softening Stiffness, Inp. 3, k3all ..................................................1 Figure 4-28: Ordinary and Combined Coherence, Hardening/Softening Stiffness, Inp. 3, k3all...................1 Figure 4-29: a) SDOF System, b) Linearizing Concept.................................................................................1 Figure 4-30: FRFs and COH, Deadzone, Inp. 2, k13 .....................................................................................1 Figure 4-31: Distortion Correlation for Deadzone, k13 .................................................................................1 Figure 4-32: Ordinary and Combined Coherence, Deadzone, Inp. 2, k13 .....................................................1 Figure 4-33: FRFs and COH, Deadzone, Inp. 3, k3all...................................................................................1 Figure 4-34: Ordinary and Combined Coherence, Deadzone, Inp. 3, k3all...................................................1 Figure 4-35: FRFs and COH, Quadratic Damping, Inp. 1, c13 .....................................................................1 Figure 4-36: FRFs and COH, Softening/Hardening Damping, Inp 1, c13.....................................................1 Figure 4-37: Ordinary and Combined Coherence, Coulomb Friction, Inp. 1, c13.........................................1 Figure 4-38: FRFs and COH, Coulomb Friction, Inp. 1, c13 ........................................................................1 Figure 4-39: FRF Estimation as Function of Spectral Averages....................................................................1 Figure 4-40: H21 and H31 as Function of Averages, Softening Stiffness, Inp. 1 ............................................1 Figure 4-41: Ordinary and Combined Coherence, Softening Stiffness, Inp. 1 ..............................................1

IV

NOMENCLATURE

a ......................................................... Non-linear Scaling Factor

C, c .......................................... Damping Matrix, Damping Element

f ......................................................... Force (time domain), Frequency

fd, fr ......................................................... Damping, Restoring Force

Δf ......................................................... Frequency Resolution

F ......................................................... Force (frequency domain)

GFF ......................................................... Power Spectrum, Input

GXF ......................................................... Cross Power Spectrum

GXX ......................................................... Power Spectrum, Output

h ......................................................... Stepsize

H ............................ Frequency Response between Input q and Output p

j ......................................................... Index, Imaginary Unit

K, k ......................................................... Stiffness Matrix, Stiffness Element

M, m ......................................................... Mass Matrix, Mass Element

p ......................................................... Output Location

q ......................................................... Input Location

r ......................................................... Correlation Coefficient

s ......................................................... Non-linear Scaling Factor

Δt ......................................................... Sample Time

T ......................................................... Time Period

Δv ......................................................... Relative Velocity

x ......................................................... Displacement

x& ......................................................... Velocity

x&& ......................................................... Acceleration

X ......................................................... Response (frequency domain)

Δx ......................................................... Relative Displacement

ε ......................................................... Non-linear Scaling Factor

ω ......................................................... circular Frequency

V

COH ......................................................... Ordinary Coherence Function

CCOH ......................................................... Combined Coherence Function

DOF ......................................................... Degree of Freedom

FRF ......................................................... Frequency Response Function

MCOH ......................................................... Multiple Coherence Function

MCCOH .......................................... Multiple Combined Coherence Function

MDOF ......................................................... Multi Degree of Freedom

MIMO ......................................................... Multiple Input, Multiple Output

SDOF ......................................................... Single Degree of Freedom

SIMO ......................................................... Single Input, Multiple Output

RMS ......................................................... Root Mean Square

Detection of Non-Linearities Motivation and Overview

1

1 MOTIVATION AND OVERVIEW

Experimental and analytical modal analysis, as known today, is based on several assumptions, which all

apply the powerful theory of linear algebra. One of the most important assumptions to be aware of is

linearity. Only when the structure exhibits linearity in the frequency range of excitation will the theory of

linear algebra be applicable and yield valid information. If, on the other hand, the structure does not behave

according to the linearity assumption, serious errors will result. Even though the frequency response

function may appear very smooth after many averages are taken, the corresponding coherence function will

show drops over a range of frequencies, which are believed to be caused by structural non-linearities. The

intention of this thesis is to investigate the effects of structural non-linearities on the frequency response

and coherence functions. This thesis will also investigate methods for distinguishing between errors

induced by digital signal processing errors (e.g. leakage) and the effects of structural non-linearities. A

study will be performed by means of a simulation model with several degrees of freedom, which provides

the advantage of knowing the parameters of the structure under consideration exactly and therefore

allowing an interpretation of results.

The next Chapter briefly reviews linear theory and the application of linear theory in modal analysis. In

addition, a short introduction to non-linear vibration and phenomena will be given. Chapter 3 focuses on

the set up of the simulation model and analyzes the issue of numerical precision. Chapter 4 introduces a

method to detect structural non-linearities by eliminating the effects of non-linear motion in the

measurements between degrees of freedom. Also, in Chapter 4 non-linear elements are applied to the

system in order to investigate their characteristic effects on the frequency response and coherence

functions, first in a noise-free environment and second when the presence of measurement noise has to be

Detection of Non-Linearities Motivation and Overview

2

assumed. Chapter 5 concludes and summarizes the work, giving recommendations for future work. A

reference list is provided in Chapter 6 and Chapter 7, as appendix, contains pertinent information to

complete the thesis.

Detection of Non-Linearities Linear and Non-Linear Vibration

3

2 LINEAR AND NON-LINEAR VIBRATION

2.1 Linear Theory and Modal Analysis

For mechanical systems, as for all other systems, the input-output relationship is usually of strong interest

and is foremost in determining system performance under given service conditions. In the field of

vibrations, the typical inputs applied are forces and moments, whereas the system response will be

measured in accelerations, velocities or displacements. A schematic picture of a dynamic system is shown

in Figure 2-1. Note, that no restriction in terms of system behavior is assumed.

Figure 2-1: Transfer Function

In most practical, real-world cases the analytical approach to determining the system properties, without

knowledge of outputs versus given inputs, will not lead to successfully identifying the system parameters.

Therefore, corresponding inputs and system responses will be used to estimate the structure’s behavior

indirectly. In order to proceed with that indirect technique, referred to as modal analysis, some restrictions

have to be applied. Specifically, the structure is assumed to obey linearity, reciprocity, observability and

time invariance.

[H] {F} {X}

Input System Output


4

If time invariance is assumed, the same test performed at a later time should provide the same system

properties as the first test. Typically, the critical conditions for a structure, with respect to time invariance,

are the property changing effects of different temperature, humidity and life cycle. Since it is known that

most structures are indeed dependent on the surrounding conditions, the tests to determine the unknown

system properties are usually performed under the anticipated operating conditions. Therefore time

invariance is a reasonable and often valid assumption.

Observability is satisfied if enough measurements are taken to describe the motion of the structure

completely. The level of observability necessary usually depends on the model chosen to describe the

structure and on the frequency range of interest. It requires knowledge of the information wanted,

sometimes a pretest and sufficient experience are necessary to choose the right number of measurements to

be acquired during the modal analysis to conform to the assumption of observability.

The important principal of reciprocity expresses the independence of path, stating that the relationship, or

transfer function, between input q and output p on a given structure will yield the exact same relationship,

or transfer function, as if point p is used as input and point q as output.

Furthermore, in order to apply the techniques of modal analysis, linearity has to hold true (besides the

assumptions of time invariance, observability, and reciprocity). Even if non-linear behavior of all real

structures is well accepted, the linearity assumption can be made for most of the testing conditions since

the error introduced by that assumption will be small and therefore negligible. In order to confirm linearity,

not only a reciprocity check has to be performed but also the principle of superposition must be true. If one

imagines exciting a system with a certain input F1 at point q which will result in a response X1 at point p,

furthermore exciting point q in a separate test by F2 resulting in X2 at point p, superposition then states, that


5

input (F1 + F2) at point q yields (X1 + X2) at point p for a given structure. Figure 2-2 shows a graphical

interpretation of the principle of superposition.

Figure 2-2: Superposition Principle

In order to model a system, if the assumption of linearity is valid, the equations of motion for a given

multi-degree of freedom (MDOF) structure can be written, in the time-domain, as:

[ ]{ } [ ]{ } [ ]{ } { })()()()( tftxKtxCtxM =++ &&& (2-1)

Taking the Fourier Transform of Equation (2-1) and therefore mapping a set of differential equations from

the time domain to an equivalent set of algebraic equations in the frequency domain will yield:

[ ]{ } [ ]{ } [ ]{ } { })()()()(2 ωωωωωω FXKXCjXM =++− (2-2)

It is important to realize, that response vector {X} and reference vector {F} in Equation (2-2) are frequency

and not time dependent. Exploiting the algebraic advantages one can simplify Equation (2-2) to:

[ ] [ ] [ ][ ]{ } { }[ ]{ } { })()()(

)()(2

ωωωωωωω

FXBFXKCjM

=

=++− (2-3)

+ =

F1

X1

F2

X2

(F1 + F2)

(X1 + X2)

q

p

q q

p p


6

Matrix [B] in Equation (2-3), the so called impedance matrix, contains all system information and relates

responses {X} to external forces {F}. Since one is usually more interested in relating applied external forces

to the corresponding system responses, the inverse of the impedance matrix has to be found:

)}()]{([)}({)}({)]([)}({ 1

ωωωωωω

FHXFBX

=

= −

(2-4)

As mentioned above, the system properties for a chosen model are generally unknown in practical

applications and an analytical solution for the components of matrix [H], the frequency response function

(FRF), in Equation (2-4) can not be found. Therefore, using an indirect approach, the FRF matrix [H] can

be estimated by exciting the system and measuring corresponding responses. In this sense, Equation (2-4)

will be rewritten so that measured responses {X} will be normalized by the input vector {F}, yielding the

frequency response [H]:

)}({)}({)]([

ωωω

FXH = (2-5)

It has to be emphasized, that the outlined theory is only valid under the assumption of linearity, since the

equations of motion in Eq. (2-1) used as starting point, a force equilibrium formulation, only express linear

relationships.

In real world experiments, a more practical approach is used to determine the frequency response function.

The next section, therefore, discusses modal analysis techniques and signal processing issues regarding the

estimation of the frequency response function (FRF) and coherence function (COH). Since these

techniques are well known and described, the review will be concise. For more information, the reader is


7

pointed to the appropriate literature, references [6], [9], [10], [12] and [24] should be considered as

examples.

The excitations used in modal analysis can be grouped primarily into pseudo harmonic, random, and

transient excitation. Pseudo harmonic signals, as for example slow swept sine, will be employed to

especially investigate non-linear behavior of structures, where the extended time needed for this excitation

is accepted. Impact testing is very popular because its easy application, whereas the analysis associated

with impacting proves very sophisticated. In this study, the focus will rely on random excitation, since

these signals are widely used to determine the frequency response function for a given structure and

random excitation is very likely to be applied to non-linear structures, assumed to obey linearity.

Therefore, evidence of non-linearity is very likely to appear in the FRFs computed using random

excitations.

When using random data for frequency response estimation it must be acknowledged that obtaining only

one (1) ensemble of measurements may not contain all the information needed and therefore several

spectral averages should be taken in order to yield reliable, confident results. Utilizing a number of spectral

averages to compute auto and cross power spectra, yields

( )[ ] ( ){ } ( ){ }∑=AvgN

H

AvgFF

NGFF

1

1 ωωω (2-6)

( )[ ] ( ){ } ( ){ }∑=AvgN

H

AvgXX

NGXX

1

1 ωωω (2-7)

( )[ ] ( ){ } ( ){ }∑=AvgN

H

AvgFX

NGXF

1

1 ωωω (2-8)


8

where {F} and {X} denote the Fourier Transform of inputs, and outputs and {F}H, and {X}H denote the

complex conjugate . The frequency response function [H] can be computed by:

( )[ ] ( )[ ] ( )[ ] 1−= ωωω GFFGXFH (2-9)

The formulation in Equation (2-9) is referred to as an H1 estimation and minimizes the noise on the

responses. Using an H2 algorithm will minimize the noise on the input and an HV estimation is designed to

minimize the noise on both input and output measurements in a least squares sense ([6],[24]).

After computing the frequency response, the coherence function (COH) is used to determine the level of

linearity between inputs and responses as function of frequency. The ordinary coherence for the single

reference case, between input q and response p can be formulated as:

( )( )

( ) ( )ωω

ωω

ppqq

pqpq GXXGFF

GXFCOH

⋅=

2

(2-10)

To understand the physical significance of the coherence function, one might imagine that the ratio

between output and input is evaluated for each spectral average and at each spectral line. Coherence values

of one (1) then correspond to a constant ratio for each spectral average and indicate that the output is linear

with respect to the input. Reasons for low ordinary coherence values are structural non-linearities, signal

processing errors (e.g. leakage) and multiple inputs. To account for multiple input cases, the multiple

coherence function (MCOH) is defined as:


9

( )( ) ( ) ( )

( )∑∑= =

∗⋅⋅=

i iN

q

N

t pp

ptqtpqp GXX

HGFFHMCOH

1 1 ω

ωωωω (2-11)

When applying the appropriate coherence function, only unmeasured inputs and signal or structural

non-linearities remain as source for low coherence values. Before turning to the effects of structural

non-linearities, a brief discussion of digital signal processing errors will be pursued.

As it is well known, by choosing a sample time (Δt) and a observation period (T) for a certain test, the

maximum frequency (fmax) to be observed (Shannon’s Sampling Theorem) and the frequency resolution

(Δf) possible (Rayleigh’s Criteria) are determined. If the measurements contain frequencies higher than fmax

an error called aliasing will result. Aliasing is induced in the measurements since the magnitude of

frequencies higher than fmax are “mistaken” by the signal processing for magnitudes of frequencies within

the legitimate frequency range defined by Δt. The only means of reducing the influence of aliasing, when

that error is expected, is the application of low-pass filters and/or drastic oversampling ([12],[24]).

Another digital signal processing error, introduced by violating the assumption of periodicity for the

Fourier Transform, is called leakage ([6],[24]). This error occurs when the time data has content whose

frequencies are not integer multiples of Δf. The magnitude of the frequencies that are not integer multiples

of Δf will “leak” into nearby frequencies which are periodic with respect to T and therefore cause a

distortion in the associated frequency spectrum. If leakage is expected, which is always the case for pure

random excitation, increasing the frequency resolution and applying a windowing concept are the common

approaches used to reduce the influence of leakage. As was already mentioned above, the effects of

leakage will be “perceived” as non-linearities by the coherence function estimate, since part of the

magnitude of the response, caused by leakage, at a certain spectral line can not be accounted for by the

corresponding input magnitude. It is also known, that in lightly damped systems, resonances and


10

anti-resonances are very sensitive to leakage which results in low coherence values in those frequency

regions.

2.2 Non-Linear Overview

Even though the assumption of linearity introduces convenient analyzing methods it has to be realized that

all real structures will sooner or later exhibit non-linear behavior. Depending on the forcing level, the

response of the system may be well approximated by linear motion and the error introduced by using linear

techniques will be insignificant. On the other hand, if the motion of the system is not well described by

linear approximation, linearizing techniques yield large and unacceptable errors. One familiar example of

the importance of response levels is the linearized pendulum equation, which approximates small angle

response reasonable but does not sufficiently describe large amplitude oscillation.

There are many ways of grouping the different types of non-linearities. One way is to recognize the

response variable that the non-linearity is acting upon. By formulating the general equation of motion with

constant mass,

( ) ( )( ) ( )( ) ( )tftxvtxutxm =++ &&& (2-12)

the velocity related function u and displacement related function v can be defined. Displacement

non-linearities can often be characterized as a hardening stiffness (frequently assumed to be cubic)

softening stiffness, hardening/softening stiffness, and softening/hardening terms (see Figure 2-3). Another

displacement related non-linearity is known as deadzone or backlash where no restoring force is present for

a certain range of displacement values. If one looks at velocity related non-linearities practical experience

shows that quadratic damping, softening/hardening damping (car-shocks), and coulomb friction are


11

possible relationships besides linear damping. In most cases, the assumed non-linearity will only be a

rough approximation of the actual relationship, but will better describe the true relationship than a linear

assumption.

Departing from linear vibration will give rise to a variety of new and often unexpected phenomena. One of

the more easy to accept, but nonetheless very important properties of non-linear systems, is the failure of

the principle of superposition. Twice the input force will not produce twice the response, but will deviate

from the contemplated linear relationship depending on the specific type of non-linearity. Also, the shifting

of natural frequencies for different excitation levels will indicate non-linear behavior and the direction of

shift can help identify the underlying non-linearity. The non-linear regime can also be represented by

effects not as easily explained and understood, such as the jump-phenomenon or the occurrence of sub- or

super-harmonics and secondary resonances.

As diverse and numerous the effects of non-linear relationships in dynamics are, there exists an even larger

number of techniques and methods to approach non-linear systems and extract the needed information.

Historically, the starting point was the formulation and analysis of SDOF systems and today the focus of

many techniques is still based on a SDOF system. This fact underlines the complexity of non-linear

motion, ranging from the question of stability to the advent of chaos. Many of these analysis techniques

require highly sophisticated and advanced mathematics and are very hard to apply to real world structures

in order to analyze their non-linear behavior. Still, there is a great demand on understanding non-linear

behavior of practical structures and experimental methods have been developed to investigate and evaluate

non-linear relations. Methods especially designed to study the effects of non-linearities are, for example,

techniques using the distribution of time histories (Bendat, [7],[8]), power spectral analysis at different

force levels, higher order frequency estimations, Hilbert transforms, Nyquist plots, and sinusoidal

excitation for each spectral line.


12

Figure 2-3: Types of Non-Linearities

It has to be understood that for practical MDOF systems the non-linear effects will not show up as clearly

as in non-linear theory and will often seem to be just measurement distortions to the examiner.

Furthermore, for MDOF systems, the non-linear evidence will be a function of relative response level

between certain DOFs and since this relative motion in turn is a function of frequency, the non-linear

distortion will also be a function of frequency. So, in order to apply analysis tools, the first step will be

determining whether the distortions seen in real measurements are in fact caused by structural

Displacement Related

F

Δx

Deadzone

Hardening/ Softening

Δx

F

Softening/ Hardening

Δx

F

Softening

Δx

F

Hardening

Δx

F

Δv

F

Quadratic with Sign

Δv

F

Soft/ Hardening

Δv

F

Coulomb

Velocity Related


13

non-linearities. Only after completing the detection step can analysis tools be applied to investigate the

non-linear relationship observed.

It has to be emphasized that this simulation study is mainly intended to detect and analyze the systematic

effects of non-linearities on the frequency response and coherence function and preferably find differences

in the non-linear effects as compared to system noise. Therefore, little attention will be paid to techniques

or excitations especially designed or intended to investigate non-linear effects on given structures. The aim

is to set up a modal analysis procedure, as in real experiments, but using the advantage of knowing the

model and relationships between certain degrees of freedom. Furthermore, since it is not reasonable (or

feasible) to cover all possible non-linear combinations applicable to the simulation model, the study

concentrates on cases of practical importance considering both the most likely non-linear effects and

typical testing conditions.

If non-linear system behavior is likely to occur, test setup in terms of types of excitation, excitation level,

location of excitation and points of response measurements have to be considered very carefully. For

example, it has already been shown (Adams, [20]), that it is essential to use spatial and temporal

information in order to successfully identify and analyze non-linear motions. Therefore, SIMO and MIMO

testing procedures should be preferred when expecting non-linear relations.

Also, the issue of observability, already a crucial subject in linear vibration analysis, becomes more

important when dealing with non-linear structures. Imagine two separate points on a complex structure,

where a serial connection of stiffness factors relates the motion of these two points (see Figure 2-4). If this

structure is excited at a linear level, the corresponding stiffness for serial connections will replace the

different stiffnesses between these two degrees of freedom, no distortion will result and measurements

acquired from these two points will reflect a linear system. If, on the other hand, the excitation level is


14

chosen to be higher and one of the serial stiffnesses between these two points now exhibits non-linear

behavior, the measurements will very likely show distortions caused by that non-linearity. In order to

locate or analyze this non-linear stiffness between these two points, more measurements have to be made.

As can be seen, this fact suggests that the level of observability of a structure varies according to the

system behavior and since the system behavior is determined by the excitation level, observability will vary

with changed forcing levels.

Figure 2-4: Reduction of Linear Stiffness Factors in Serial Connection

In the same sense, observability has to be considered more carefully in non-linear structures, the subject of

energy distribution becomes more important when investigating non-linearities. Using SIMO testing

procedures might induce non-linear behavior in the vicinity of the input location for high forcing levels and

furthermore may not excite the structure well at remote points. Therefore not only the excitation level but

also the number and location of inputs will determine if the structure responds in a non-linear regime.

DOF i

DOF j

k2

k3

k1

1

321

111−

⎟⎟⎠

⎞⎜⎜⎝

⎛++=

kkkktotal

Detection of Non-Linearities Simulation Considerations

15

3 SIMULATION CONSIDERATIONS

3.1 Simulation Model

This section will describe the setup procedure of the simulation models using a linear 4-DOF model as an

example. Further, even when models with different DOFs are used or a particular type of non-linearity is

applied to the physical model, the corresponding simulation model can still be developed in the same way

as this example. Consider the multi-connected model given in Figure 3-1. Spring connections are marked

by kij and the velocity proportional damping connections by cij, where indices i and j refer to the degrees of

freedom they are connecting (i < j by definition and for clarity).

Figure 3-1: 4-DOF-Model, a) Physical Scheme, b) Connection Scheme

1. DOF

2. DOF

3. DOF

4. DOF

k14c14

k01

k13

k34 k24

k12

k23

c34

c24

c12

c01

c23

c13

Ground

1. DOF 3. DOF

2. DOF 4. DOF

a) b)


16

As the first step towards a simulation-model, the equations of motion have to be found. By using Newton’s

Method for each degree of freedom and rearrange the equations, the equations of motion take the familiar

form of:

( )( ) 1414313212114131201

41431321211413120111

fxkxkxkxkkkkxcxcxcxccccxm=−−−++++

−−−++++

K

K&&&&&& ( 3-1.1)

( )( ) 24243232242312112

424323224231211222

fxkxkxkkkxkxcxcxcccxcxm=−−+++−

−−+++−

K

K&&&&&& ( 3-1.2)

( )

( ) 34343342313223113

434334231322311333

fxkxkkkxkxkxcxcccxcxcxm=−+++−−

−+++−−

K

K&&&&&& ( 3-1.3)

( )

( ) 44342414334224114

434241433422411444

fxkkkxkxkxkxcccxcxcxcxm=+++−−−

+++−−−

K

K&&&&&& ( 3-1.4)

It is worthwhile to note, that each equation of motion describes a state of dynamic equilibrium that

balances the internal and external forces acting on each degree of freedom. Internal forces are caused by

the existence of inertia, damping and stiffness and are a function of acceleration, relative velocity, and

relative displacement whereas external terms are caused by the external forces applied to the corresponding

degree of freedom. Equations (3-1) can now be rewritten in the form of Eq. (3-2), required by integration:

{ } { }( )kcxxfgm

x nn

n ,,,,1&&& = ( 3-2)

Therefore, the equations of motion for the 4-DOF-Model, in a slightly concise form, become:


17

{ } { }⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧−

⋅+

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧−

⋅+=

∑∑

14

13

12

1

4321

14

13

12

1

432111

11

kkk

k

xxxx

ccc

c

xxxxfm

x

ii

&&&&&& (3-3.a)

{ } { }⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧−

⋅+

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧−

⋅+= ∑∑

24

23

2

12

4321

24

23

2

12

432122

21

kk

kk

xxxx

cc

cc

xxxxfm

x ii&&&&&& (3-3.b)

{ } { }⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

−⋅+

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

−⋅+=

∑∑34

3

23

13

4321

34

3

23

13

432133

31

kk

kk

xxxx

cc

cc

xxxxfm

xii

&&&&&& (3-3.c)

{ } { }⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

−

⋅+

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

−

⋅+=

∑∑ 4

34

24

14

4321

4

34

24

14

432144

41

ii kkkk

xxxx

cccc

xxxxfm

x &&&&&& (3-3.d)

Equations (3-3) can now be used to build the simulation model in SIMULINK®, a MATLAB®

implemented simulation software. SIMULINK® is based on the idea of describing the governing equations

of motion by block diagrams. Each mathematical operation necessary for modeling the equations of

motion, such as summation, multiplication, or integration, can be expressed by appropriate blocks.

Appendix 7.1 shows the complete SIMULINK® model for the linear 4-DOF-System, a sub-block of a DOF

(DOF_1) and a sub-block (spring-damper-unit, SDU_1_2), modeling the stiffness-damping connection

between degrees of freedom 1 and 2.

It should be mentioned that the governing differential equations could also be integrated by using

MATLAB® alone, but employing SIMULINK® with its graphical description of the system makes the

formulation of differential equations easier, user-friendly and more resistant to errors introduced by

implementing the equations of motion in the conventional way. Furthermore, the application of changes to

an existing model can be done quickly and more efficiently.


18

After the simulation model is set up, a verification check is performed, validating the correct performance

of the SIMULINK® model. Using the linear parameters given in Appendix 7.2, the eigenvalues,

eigenvectors, and frequency response functions (FRFs) of the 4-DOF model are computed theoretically.

Applying the same parameters to the simulation model and comparing the FRFs estimated by the

simulation to the theoretical FRFs, confirmed the behavior of the SIMULINK® simulation model.

Figure 3-2 shows the normalized responses of all 4 DOFs for an input applied at DOF 2.

0 5 10 15 20 25 30-200

-100

0

100

200

Frequency in Hz

Phas

e in

[de

g]

0 5 10 15 20 25 3010

-7

10-6

10-5

10-4

10-3

10-2

1,2,3,4/2 FRF - Magnitude and Phase , Simulation (s) vs. Theory (d)

H in

[m

/N]

Figure 3-2: Frequency Response Function, Linear Model

3.2 Numerical Precision


19

When conducting simulation studies, one of the most essential issues is numerical precision. This part of

the thesis will discuss the considerations to be made regarding numerical precision, by first looking at the

ordinary differential equation solver employed and then using a particular example. Since often no

analytical solution for the non-linear equations of motion can be found, numerical integration has to be

employed. As a consequence of using numerical integration, an approximating technique, it must be

understood that the results will not be exact, but must be accurate enough to allow valid conclusions.

There exist numerous algorithms for the numerical integration of sets of ordinary differential equations

(ode), from the easy, straightforward methods, such as Eulers or Heun’s Method without error control, to

more advanced algorithms, such as Runge-Kutta Methods (RK) of different orders. Usually the more

sophisticated methods become more accurate but they are, at the same time, more computationally

demanding. SIMULINK® offers several implemented ode-solvers based on fixed step algorithms (e.g.

Euler [ode1], Heun [ode2], Bogacki-Shampine [ode3], Runge-Kutta [ode4]) and variable step algorithms

(e.g. Bogacki-Shampine [ode23], Runge-Kutta [ode45]). For this study, only the class of variable step

solvers is of interest due to their capability of providing error control during the integration process.

The ode45-solver, which implements a Runge-Kutta algorithm of fourth and fifth order, deserves special

attention. The RK 45 algorithm has been applied to a wide range of problems, realizing good accuracy and

reasonable computational time. For a detailed discussion of this and other integration methods, the

interested reader is pointed to references [15], [16] and [18]. Still, the properties of the ode45 solver

necessary to understanding the decisions on numerical precision are pertinent here and should therefore be

examined briefly. For any solver to be able to integrate the problem, the governing differential equations

have to be in a first order form. Since the equations of motion for a mechanical system are in general of

second order, the familiar technique of state space expansion has to be applied. After doing so, the

differential equations will have the form:


20

( )yxfy ,=& ( 3-4)

It should be mentioned that in Equation (3-4) quantities x and y do not have to be scalars but could be

vectors, each entry representing one degree of freedom. The initial condition task of numerical integration

now becomes, to determine a good estimate of y(n+1), let that estimate be called ( )1~ +ny , from given x(n)

and y(n):

( ) ( )( )( ) ?1~

,)(=+

=

nynynxfny&

( 3-5)

In Equation (3-5), n and (n+1) correspond to time t and (t+h) respectively, where h is usually referred to as

stepsize. It must be noted, that (t+h) does not necessarily coincide with the desired output times at

increments of Δt, since meeting the accuracy requirements often forces h to be much smaller than Δt. The

Runge-Kutta algorithms determine the estimate of the new state (n+1) by computing several auxiliary

slopes for values of y in between t and (t+h), weighting them differently and finally calculating ( )1~ +ny .

Depending on how many auxiliary slopes are used, the order of the RK method is defined. For example, if

four slopes are calculated the order of that particular RK algorithm is said to be four, likewise for any other

number of intermediate gradients.

As mentioned above and indicated in the last paragraph, by integrating the equations of motion

numerically, a truncation error will be admitted into the result. In order to reduce the integration error, the

MATLAB® employed RK-method ode45 uses the following technique to determine a reasonable local

error estimate. An RK algorithm of order five is used to calculate an estimate of y(n+1), called ( )1~5 +ny .

Since the fourth order coefficients are embedded in the fifth order algorithm, it is easy and time efficient to


21

retrieve an estimate of y(n+1) based on the fourth order method, called ( )1~4 +ny . An error estimation, not

the true error (!), can then be defined as:

( ) ( ) ( )1~1~1~5445 +−+=+ nynyne ( 3-6)

In order to control the error, tolerance settings must be used. Using a smaller stepsize h, if the error

becomes too large (violating a given tolerance setting) and increasing h when the state is changing slowly.

This way, the accuracy needed can be achieved but at the same time computational effort can be saved by

doing time effective integration. SIMULINK® offers error control by letting the user choose relative and

absolute tolerance values. For each integration step the error estimate from Equation (3-6) has to fulfill the

inequality shown in Equation (3-7):

( )abstolyreltole _,_max~ ⋅≤ ( 3-7)

Depending upon the magnitude of the function at a particular point in time, either the relative or the

absolute tolerance will be active. As can be seen from Equation (3-7), this formulation is set up from the

point of view of most effective integration, since only the rougher tolerance applicable is considered (max).

Therefore, choosing the relative and absolute tolerance values for integration should be made carefully in

order to yield valid results. This will be depicted in the next part of this section by a simulation example

discussing the issue of numerical precision by integrating the equations of motion several times with

different error tolerances.

For this demonstration, the vibration behavior of the linear 4-DOF model shown in Figure 3-1 will be

simulated by putting in a pure random force at DOF 2, recording the input and computing all responses.

The pure random force is set up to yield the maximum frequency to be observed with fmax = 102.4Hz and a

desired frequency resolution of Δf = 0.05Hz.


22

In general, the critical question involves the values of relative and absolute tolerance values that should be

chosen for the integration process. In order to do so, the equations of motion must be integrated first with

default settings for the tolerances to establish the order of magnitude of the vibration time histories. With

the structure’s given physical properties and an input RMS-force level of F = 100N, the RMS values of the

outputs become:

{ } mx rms

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0176.00173.00172.00136.0

As can be seen, all output motions are displaying the same order of magnitude which is 1⋅10-2 m. Now, for

example, allowing an absolute tolerance of tol_abs = 1⋅10-5m will result in four significant (or valid) digits

after the decimal point. In order to check if the absolute tolerance of tol_abs = 1⋅10-5m already produces

accurate results, the equations of motion are integrated again three more times with smaller absolute

tolerances, namely

mabstol 876 101/101/101_ −−− ⋅⋅⋅=

providing the same input force for each tolerance setting. The relative tolerance remains unchanged during

this test at a level of tol_rel = 1⋅10-6. By looking at Equation 3-7 it will be realized that the relative

tolerance hardly will be activated during the integration because of the small magnitude of the time

histories. This way the only changes in the time histories will be caused by the tighter absolute tolerance

settings.


23

Figure 3-3: Frequency Response H32 and Error Estimate (|tol| = 1⋅10-5/1⋅10-8m)

In Figure 3-3 H32 is shown, both, for the absolute tolerance values of tol_abs = 1⋅10-5m and

tol_abs = 1⋅10-8m. As can be seen, no significant difference is visible and H32(1⋅10-5m) and H32(1⋅10-8m)

seem identical. Furthermore, the magnitude of the estimate of the absolute and relative error between the

two FRFs is plotted, using the more accurate FRF estimation (tol_abs = 1⋅10-8m) as reference. In the

frequency range of interest (0 to 40Hz) the absolute error is roughly four to five decades below the

magnitude of vibration, with slightly higher values of error at the resonance frequencies. At higher

frequencies on the other hand (above 50Hz), the absolute error remains constant, resulting in an increasing

relative error due to the decreasing magnitude of vibration for that frequency range. Overall, it should be

noted that the accuracy gained by an absolute tolerance of tol_abs = 1⋅10-5m seems sufficient and does not

improve significantly by using a smaller tolerance.

If one is integrating a particularly long time history, one might suppose, that the error made from one

integration step to the next accumulates and therefore the end portion of the integration record might not be

as reliable as the beginning. Even though this will be especially true for diverging states, it is not


24

necessarily true for the vibration histories under investigation. In Figure 3-4, one ensemble of response

x4(t) is plotted for two absolute tolerance values (tol_abs = 1·10-5 m/1·10-8 m), where an error estimate is

also shown. Note, that the order of magnitude for the error estimation is 1·10-7 m, whereas the order of

vibration is 1·10-2 m. It can be furthermore noticed, that the error does not increase as integration time

progresses.

0 2 4 6 8 10 12 14 16 18 20-5

0

5x 10

-7

T ime in [s]

Error Estimate

0 2 4 6 8 10 12 14 16 18 20-0.04

-0.02

0

0.02

0.04

x 4 in [

m]

|tol| = 1e-08m

0 2 4 6 8 10 12 14 16 18 20-0.04

-0.02

0

0.02

0.04Time History x

4(t)

x 4 in [

m]

|tol| = 1e-05m

Figure 3-4: Response x4(t) for different Tolerances and Error Estimation, One Ensemble

In general, a large number of consecutive ensembles are integrated. In order to investigate the development

of error, the RMS values of the difference between corresponding time histories (error estimate) for each

ensemble and absolute tolerance value are computed. Figure 3-5 shows the error estimate RMS(dx) for x4(t)

as a function of ensembles, where different plots correspond to different absolute error tolerance settings.

Two important facts might be deduced by analyzing the information provided in Figure 3-5. First, that

indeed the error does not accumulate when integration time progresses and secondly, as might be

suspected, the reduction of absolute tolerance by one order reduces the absolute error by one order as well.


25

Figure 3-5: Error Estimate RMS(dx4)

To summarize the results from this section concerned with numerical precision, the following conclusions

can be drawn. The one-step ordinary differential equation solver ode45 produces accurate and valid results

for the system in focus. Although, in order to set appropriate, accuracy yielding tolerances for the

integration, the magnitude of vibration should be known for all degrees of freedom. If the order of

magnitude is the same for all time histories, only one absolute tolerance for all time histories will be

sufficient; if not, different absolute tolerances should be chosen. Furthermore, since in this example only

linear motion was present, a non-linear case, especially those with discontinuous derivatives, should be

treated more carefully. This can be done by checking the results (as in the linear case), by integrating the

time histories with a smaller tolerance and comparing the computed frequency response and/or coherence

functions with respect to their convergence.

Detection of Non-Linearities Application of Non-Linear Interactions

26

4 APPLICATION OF NON-LINEAR INTERACTIONS

4.1 Combined Coherence Function

One is often faced with the question, are low coherence values in a certain frequency range caused by

signal processing errors or are structural non-linearities the reason for the distortions? This part of the

thesis proposes a method, which can be used as check for whether non-linear motion is present in the

response histories and whether structural non-linearities are in fact the source of low coherence. Therefore,

a detection method for the presence of structural non-linearities will be provided.

Figure 4-1: a) MDOF System, b) Internal and External Forces on MDOF System

When building a model of a real structure, one often discretizes the object and thinks of it as a number of

degrees of freedom (lumped masses) connected by stiffness and damping terms. Figure 4-1 a) shows the

F F

a)

b )


27

general scheme of a MDOF system. It should be noted that the connections between the DOFs drawn

depict generic relations and are not subject to any linear assumption.

Formulating the dynamic force balance for any DOF of that structure, the sum of all inertia, stiffness,

damping related and external forces acting on that particular DOF vanishes. At this point, the assumption

of constant mass is made and all forces caused by stiffness and damping will be called internal forces. In

Figure 4-1 b) the structure is shown with all internal forces and one external force acting upon it. The force

balance in terms of acceleration, usually referred to as the equation of motion, for each DOF now becomes:

( )next

nn ff

mx ∑ ∑+⋅= int

1&& (4-1)

In a physical sense, Equation (4-1) states, that the acceleration of every DOF is the sum of all internal and

external forces acting on that degree of freedom scaled by the associated mass.

Deriving Equation (4-1) for the acceleration present at DOFs i and j will yield:

[ ])()()()()(1)1()1()1()1(11 niinjiijiiiiiiiiiii

ii xvxvxvxvxvf

mx Δ++Δ++Δ+Δ−−Δ−⋅= ++−− KKK&&

(4-2.a)

[ ])()()()()(1)1()1()1()1(11 njjnjjjjjjjjjiijjjj

jj xvxvxvxvxvf

mx Δ++Δ+Δ−−Δ−−Δ−⋅= ++−− KKK&&

(4-2.b)

In Equations (4-2.a) and (4-2.b) v(Δx) denotes force interactions between different DOFs as a function of

relative motion Δx. As can be realized, by applying Newton’s law, force vij(Δxji) acting on DOF i because

of relative motion Δxji is the same magnitude but opposite direction to the force acting on DOF j because of


28

the relative motion between DOFs i and j. It is important to note, that this is true, independent of the nature

of the relation between the two DOFs. Therefore, by adding Equation (4-2.a) to (4-2.b),

∑ += jiij xxx &&&&&& (4-3)

and under the condition that mi ≈ mj, the contribution of the interaction force between DOFs i and j to the

motion can be minimized or even eliminated. It has to be admitted, that by creating the virtual coordinate

ijx&& , a loss of information about connection ij will result. But if low coherence is caused by non-linear

behavior between DOFs i and j, adding motions ix&& and jx&& , and computing the coherence for the sum of

motions versus a given input will result in drastically improved coherence values (here defined as

combined coherence (CCOH)). In this way, a detection method for non-linear relationships is possible and

is indicated by higher combined coherence values.

This summation is limited to time domain histories (and their Fourier Transforms) but not to acceleration

only. Applying the summation to Fourier Transforms the only difference is that the summation has to be

done at each spectral line, and not at each time point. Since in a discrete sense displacement and velocity

are only scaled by the corresponding sample time (time domain) or frequency (frequency domain), above

arguments are still valid. Furthermore, leakage is not to be affected by this technique because of its digital

signal processing nature and therefore low coherence caused by leakage or aliasing should not improve.

As can be seen in the derivation, the crucial condition for that technique is the approximate equality of the

associated masses of DOFs i and j. If this does not hold true, this method will fail and not improve the

coherence function. On the other hand, if the condition mi ≈ mj is not valid but one has knowledge of the

relationship between mi and mj, it is possible to scale the motions according to the mass ratio and correct

the relationship. After scaling the motions, the CCOH can be applied and will indicate non-linear motion, if


29

present. This concludes the introduction of the CCOH function. Examples will be discussed in Sections 4.2

and 4.3 where this technique is actually applied.

4.2 Characteristic Effects of Non-Linearities

This section is intended to analyze the effects of non-linear connections upon the frequency response and

coherence function without the presence of measurement noise. In this way, it should be possible to find

the characteristic effects for particular non-linearities, which will allow a detection and determination of

the actual structural non-linearity by only recognizing the effects on FRFs and COH. Also, the proposed

method of combined coherence will be examined. The 4-DOF system developed in Section 3.1 will serve

as simulation example.

4.2.1 Hardening Stiffness

The hardening stiffness will be generated by a cubic offset to the linear term. Therefore the restoring force

can be formulated by:

( )3xxkf linr Δ⋅+Δ⋅= ε (4-4)

In Equation (4-4) ε is used to adjust the severity of the non-linear effect and note additionally, that for

small motion (small excitation) the relationship converges towards the linear case. As a first approach to

identification, the hardening stiffness is applied to a SDOF system in order to characterize its effects.

Figure 4-2 a) shows the FRF and COH for increasing level of excitation.


30

Figure 4-2: a) SDOF System, b) Linearizing Concept for Cubic Stiffness

When increasing the force level for the given SDOF system, it can be observed that the resonance

frequency shifts to higher values, the resonance amplitude decreases, and the distortion effects become

more apparent. The frequency shift can be explained by the “linearizing” concept of the FRF estimation,

depicted in Figure 4-2 b). By forcing at higher levels, resulting in higher motion, the “linearized” stiffness

k_lin2 is greater than k_lin1 and therefore, the resonance frequency will be shifted to higher values when

forced at higher levels.

I) Non-linearity between DOF 1 and 3, SIMO, Input at DOF 3

This case simulates a SIMO situation where the stiffness connection between DOF 1 and DOF 3 becomes

non-linear (ε = 50000). Compared to real world testing considerations this is a valid assumption, since

single input configurations might induce non-linear behavior in the vicinity of the input location. The

excitation chosen is pure random (RMS: F = 30N). Figure 4-3 shows the FRF estimation (H1 algorithm)

and ordinary coherence for all outputs versus input at DOF 3.

0 5 10 15 20 25 300

0.5 1

Frequency in Hz

0 5 10 15 20 25 30

10 -5

10 -4

10 -3 FRF - Magnitude and Coherence, Cubic Stiffness, 10N (s), 20N(d), 40N (dd)

F

ΔxF1

F2

k_lin1

k_lin2F = f( Δ x)

Hardening Nonlinearity

b ) a)


31

It can be seen from Figure 4-3 that, compared to the linear case, frequency responses H13 and H33 are most

affected by the non-linearity. This could be expected since the non-linear relationship is placed between

DOFs 1 and 3. Furthermore, one can notice a shift and a decrease in magnitude of the third resonance

frequency. Again, this behavior could be expected when applying this type of non-linearity and was

explained in the SDOF example above.

Looking at the coherence function for all 4 responses the leakage-caused coherence drop at the first, very

lightly damped resonance frequency can be noticed. The driving point measurement H33 also shows a

distinct coherence drop at the anti-resonance. As mentioned above, these low coherence values can be

attributed to leakage and are not caused by the applied structural non-linearity. On the other hand, the sharp

coherence drop at f ≈ 10Hz, noticeable in COH13 and COH33, is not associated with any resonance or

anti-resonance. It turns out that this drop in coherence occurs at exactly 3 times the first resonance and is

therefore evidence of a secondary resonance in the system, a typical non-linear phenomena. The low

coherence in the frequency range f ≈ 13 – 17Hz at COH13, COH23 and COH33 can be explained by the large

relative motion between DOF 1 and 3 in that frequency range. Besides the mentioned effects, there is also a

noise-like distortion at the high frequencies showing up in H13 and resulting in low values for COH13. In

summary it can be said, that the effects of the cubic non-linearity are dominantly present at the responses to

which the non-linearity is connected and have only minor influence on responses 2 and 4.


32

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1

F req u en c y in Hz

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3

1 0-2 1 /3 F R F - M ag n i t u d e an d C o h e r e n c e , F = 3 0 N

N o n -L in e arL in e ar

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



Figure 4-3: FRFs and COH, Hardening Stiffness, Inp. 3, k13


33

As a next step, the combined coherence (CCOH) will be computed for responses 1 and 3 in order to

eliminate the effects of the non-linear relation. In order to do so, responses x1 and x3 are added and the

coherence is calculated for this combined response. Figure 4-4 shows the CCOH(13)3 in comparison with

ordinary coherence COH13 and COH33.

Figure 4-4: Coherence and Combined Coherence, Hardening Stiffness, Inp. 3, k13

It can be seen, that the effects of the secondary resonance at f ≈ 10Hz is completely eliminated and the high

frequency range (f > 20Hz) also shows combined coherence values of one (1). In the frequency range of

f ≈ 14 – 17Hz the CCOH does not show improved values, supposedly due to the limitations of the

technique of combined coherence. First, the difference in mass and therefore the scaling difference has to

be recognized (m1 = 12kg, m3 = 9kg) and second since the MDOF system is a multi-connected structure,

the non-linear effects not only enter directly but will be still present even if the effects of the direct

0 5 10 15 20 25 30 35 40 0 0.5

1

Frequency in Hz

0 5 10 15 20 25 30 35 40 0 0.5

1

0 5 10 15 20 25 30 35 40 0 0.5

1 COH 1/3 , COH 3/3 and CCOH (1+3)/3, F = 30N


34

connection between DOF 1 and 3 are removed. The influence of leakage remains unaffected by the

summation of the responses, which was expected.

In summary, this case showed the following non-linear effects: shifting resonance frequencies according to

the hardening stiffness term; secondary resonance effects correlated with the first resonance; a distortion of

the frequency response in the range where the relative motion between DOF 1 and 3 is large and therefore

the non-linear effects are large too; and a noise-like distortion of the driving point frequency response at

high frequencies.

II) Non-linearity between DOF 1 and 3, SIMO, Input at DOF 2

This SIMO case is designed to simulate the situation where the non-linearity is somewhere in the system

and the input is applied to a DOF which is not associated with the non-linearity. As under Case I), the

excitation is a pure random signal with RMS value of F = 30N, and the severity of the non-linear offset is

held constant (ε = 50000). Frequency response and coherence function for all responses are shown in

Figure 4-5.


35

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3





36

As can be seen from Figure 4-5, the FRFs corresponding to a response connected to the non-linearity

(1 and 3) are affected mostly by the cubic stiffness. As in Case I) a secondary resonance shows clearly at a

frequency of f ≈ 10Hz. The frequency shift appears to be not as large as in Case I), due to the fact that the

excitation is not placed directly at the non-linear relation and therefore the relative motion between DOF 1

and 3 is not as large as in Case I). One can also notice a periodic repeat of low and high coherence parts in

COH12 and COH32. Focusing on the frequency range of these low coherence values (and FRF distortion), it

shows, that they are correlated with the first resonance frequency at f ≈ 3.3Hz by odd multiples of the first

resonance. In fact, distortions occur at f ≈ 5⋅3.3Hz = 15.9Hz, f ≈ 7⋅3.3Hz = 23.1Hz, f ≈ 9⋅3.3Hz = 29.7Hz

and so forth. As the secondary resonance at f ≈ 10Hz, these distortions appear to be secondary resonances

too, justified by the cubic nature of the non-linearity.

0 5 10 15 20 25 30 35 400

0.5

1

Frequency in Hz

0 5 10 15 20 25 30 35 400

0.5

1

0 5 10 15 20 25 30 35 400

0.5

1

COH 1/2 , COH 3/2 and CCOH (1+3)/2, F = 30N

Figure 4-6: Coherence and Combined Coherence, Hardening Stiffness, k13

The combined coherence CCOH of response 1 and 3 versus input 2 is computed and plotted in comparison

to COH12 and COH32, see Figure 4-6. Except for leakage, the distortion in the coherence functions is


37

almost totally removed and CCOH(13)2 indicates pure linear relation between (x1 + x3) and reference at

DOF 2. This allows the conclusion that the low coherence values in COH12 and COH32 are caused by

non-linear motion and not by measurement noise and signal processing errors.

It has also been noticed that the distortions caused by secondary resonance effects seem to be correlated.

Figure 4-7 shows H12 and H32 in a frequency range of f = 20 – 25Hz. The frequency distortions appear to

behave in an out of phase magnitude manner, which should not be expected when random noise in the data

is present.

20 20.5 21 21.5 22 22.5 23 23.5 24 24.5 250

0.5

1

Frequency in Hz

20 20.5 21 21.5 22 22.5 23 23.5 24 24.5 25

10-6

10-5 1/2 and 3/2 FRF - Magni tude and Coherence, F = 30N


Trying to quantify the correlation between H12 and H32, the correlation coefficient r has been computed for

log(|H12|) and log(|H32|), where the linear trend is removed from log(|H12|) and log(|H32|) before calculating

the correlation in order to emphasize the distortion and not the decaying effect of the system response. The

correlation coefficient for the frequency range f = 20 – 25Hz is determined to be r(H12,H32) = -0.76, in fact

indicating an out of phase correlation. For ease of writing, the notation {Hpq} will be used for log(|Hpq|),


38

where the linear trend is removed from log(|Hpq|). Figure 4-8 shows a plot of {H12} versus {H32}, visually

representing the correlation in the frequency range of distortion.

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

log |H12|

log

|H32

|

FRF Correlation, 1/2 vs. 3/2, f = 20 to 25 Hz

Corr. = -0.75895

Figure 4-8: Correlation between {H12} and {H32}

In summary, for this case, the presence of secondary resonances and the typical frequency shift has been

observed. The distortion because of large relative motion is decreased in comparison to Case I) since the

input is not directly applied at the non-linear location.

III) Non-linearity between DOF 2 and 3, SIMO, Input at DOF 1

In this case the location of the cubic stiffness in the 4-DOF model is moved to the connection between

DOF 2 and 3. All other connections remain unchanged and express linear relation. The scaling factor is

chosen to be ε = 50000 and a pure random signal (F = 100N) is used to excite the system at DOF 1.

Figure 4-10 contains the FRFs and coherence functions for this particular SIMO case.


39

0 5 10 15 20 25 30 35 400

0.5

1

Frequency in Hz

0 5 10 15 20 25 30 35 400

0.5

1

0 5 10 15 20 25 30 35 400

0.5

1


Figure 4-9: Coherence and Combined Coherence, Hardening, Inp. 1, k23

The non-linear stiffness at k23 only affects the responses to which the non-linearity is connected, namely

response 2 and 3. FRFs H11 and H41 do not show any significant distortion caused by the applied

non-linearity. By looking at H21 and H31, it can be noticed that the secondary resonances associated with

the first resonance frequency are not present in the data. Frequency distortion in the range f ≈ 15 – 20Hz

can be explained by the large relative motion between DOF 2 and DOF 3 in that frequency range and is not

caused by a secondary resonance effect. Furthermore, in this case the frequency shifting effect has changed

from affecting mainly the third resonance in Cases I) and II) to influence the fourth resonance.


40

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3

1 0-2 1 /1 F R F - M ag n i t u d e an d C o h e r e n c e , F = 1 0 0 N

N o n -L in earL in ear

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3





41

As for Cases I) and II) the combined coherence CCOH(23)1 is computed and plotted in Figure 4-9 versus the

ordinary coherence functions COH21 and COH31.

In summary, this case, as Cases I) and II), has shown that only the responses and therefore the frequency

response functions in fact connected with the non-linear behavior will be significantly effected by the

non-linearity. Furthermore, one has seen, even though secondary resonances are typical non-linear

phenomena, they do not necessarily have to show up.

IV) Non-linearity between DOF 2 and 3, MIMO, Input at DOF 1 and 2

This set up is chosen to investigate the effects of a MIMO testing situation and a more uniform energy

distribution in the system is realized. It should be expected that the non-linear behavior will be more

apparent because of the aforementioned reason. At each input (DOF 1 and 2), the excitation signal applied

has an RMS value of F = 50N, which is half the magnitude at each input compared to the SIMO Case III),

where an RMS value of F = 100N was used for the single input. The severity of the cubic stiffness, the

location of the non-linearity, and all other conditions remain unchanged with respect to Case III).

Figure 4-11 shows the FRFs and multiple coherence functions computed.


42

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3

1 0-2 1 /1 F R F - M ag n i t u d e an d M C o h e r e n c e , F = 5 0 N


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3


N o n -L in e arL in ear

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



Figure 4-11: FRFs and MCOH, Hardening Stiffness, Inp. 1+2, k23


43

It can be seen in Figure 4-11 that the FRFs associated with responses 2 and 3 are affected most by the

non-linearity. Since the cubic stiffness is at the same location as in Case III), the shift of resonance

frequency also influences the fourth resonance. Besides the frequency shift, a distortion is noticed, more or

less at all FRFs in the frequency range f ≈ 15 – 20Hz, which is caused by the fact that the relative motion

between DOF 2 and 3 in that frequency range is especially large. Even in H41 and H11, which do not have a

DOF in common with the non-linear connection, this influence is still apparent.

0 5 10 15 20 25 30 35 400

0.5

1

Frequency in Hz

0 5 10 15 20 25 30 35 400

0.5

1

0 5 10 15 20 25 30 35 400

0.5

1

MCOH 2 , MCOH 3 and MCCOH (2+3)

Figure 4-12: Multiple Coherence and Combined Coherence, Hardening Stiffness, k23

In Figure 4-12, the multiple combined coherence for (x2 + x3) MCCOH(23) is plotted along with MCOH2

and MCOH3. The fact that no complete clear up of the combined coherence can be accomplished, is

probably caused by the fact, that non-linear motion enters x2 and x3 not only through the non-linear relation

directly in between these two DOFs. Motions x1 and x4 exhibit non-linear vibration too, which will be

transmitted to x2 and x3. Therefore the total elimination of the cubic stiffness effect will not be possible.


44

Upon a closer look at the high frequency distortions of H21 and H31, a correlation can be recognized, as in

Case II). A zoom of the FRFs H21 and H31 to the range f ≈ 25 – 30Hz is shown in Figure 4-13. Also the

correlation coefficient between {H21} and {H31} has been computed to r({H21},{H31}) = -0.73 which

indicates a strong out of phase magnitude correlation.

25 25.5 26 26.5 27 27.5 28 28.5 29 29.5 300

0.5

1

Frequency in Hz

25 25.5 26 26.5 27 27.5 28 28.5 29 29.5 3010

-8

10-7

10-6

10-5 2/1 and 3/1 FRF - Magni tude and MCoherence, F = 50N

Figure 4-13: FRFs H21 and H31, MCOH, Hardening Stiffness, Inp. 1+2, k23

4.2.2 Softening Stiffness

The softening stiffness non-linearity will be formulated by the following equation:

( ) ( ) ( )[ ]aaxssakxsigf linr lnln −+Δ⋅⋅⋅⋅Δ= (4-5)

As can be seen from Equation (4-5), the derivative at Δx = 0 is the linear stiffness factor klin and therefore

the restoring force will converge towards the linear case for small excitation. Scaling factors a and s are

used to adjust the severity of the non-linear effect.


45

Before applying the softening stiffness to the 4-DOF system, the effects of this non-linearity are

investigated on a simple SDOF model. The frequency response and the coherence function for increasing

excitation levels are shown in Figure 4-14 a). For higher excitation forces, the resonance frequency

decreases and the distortion effects becoming more dominant. The frequency shift can be explained by

looking at Figure 4-14 b). When the system is forced at a higher level, the linearized stiffness k_lin2 is not

as big as the linearized stiffness k_lin1, which corresponds to the smaller forcing level F1.

Figure 4-14: a) SDOF System, b) Linearizing Concept for Softening Stiffness

0 5 10 15 20 25 300

0.5 1

Frequency in Hz

0 5 10 15 20 25 30

10 -5

10 -4

10 -3 FRF - Magnitude and Coherence, SoftStiff, 10N (s), 20N (d), 40N (d)

F

ΔxF1

F2

k_lin 1

k_lin2

F = f(Δx)

Softening Nonlinearity

a) b )


46

V) Non-linearity between DOF 2 and 3, SIMO, Input at DOF 1

This case uses Equation (4-5) to formulate the restoring force between DOF 2 and 3. The RMS value of

pure random excitation at DOF 1 is F = 100N, scaling values s and a are chosen to s = 2500 and a = 10,

which relates to an approximate 30% deviation from the linear restoring force at RMS(Δx23).

As learned from the hardening stiffness simulations, the responses directly connected to the non-linearity

are most affected (i.e. H21 and H31, see Figure 4-16). The frequency shift of the fourth resonance and the

distortion in the range of large relative motion (f ≈ 15 – 20Hz) are most noticeable in H21 and H31. It can

also be mentioned that the first anti-resonance appears to be very sensitive with respect to the softening

stiffness. COH21 and COH31 show a drop in the region of the first anti-resonance, which could not be

justified by the fact of leakage (no driving point measurement and not a very steep drop in FRFs).

Computing the combined coherence for responses x2 and x3 proves that the drop at the anti-resonance is not

caused by leakage but instead is a non-linear effect, see Figure 4-15.

0 5 10 15 20 25 30 35 400

0.5

1

Frequency in Hz

0 5 10 15 20 25 30 35 400

0.5

1

0 5 10 15 20 25 30 35 400

0.5

1


Figure 4-15: Ordinary and Combined Coherence, Softening Stiffness, Inp. 1, k23


47

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



Figure 4-16: FRFs and COH, Softening Stiffness, Inp. 1, k23


48

As in the hardening stiffness case, a correlation of the high frequency distortion can be recognized. For

example the correlation coefficient for {H21} and {H31} in the frequency range of f = 25 – 30Hz is

determined to be r({H21},{H31}) = -0.83, indicating a strong out of phase correlation.

VI) Non-linearity between DOF 2 and 3, MIMO, Inputs at DOF 1 and 2

This MIMO case is intended to excite the softening system more uniformly and therefore emphasize the

non-linear relation. Input locations are chosen to be at DOF 1 and 2, the RMS value of each pure random

excitation is F = 50N. All system parameters and the parameters of the non-linearity remain unchanged

with respect to Case V). Figure 4-17 shows all eight frequency response functions. The effect of the

non-linearity, again, is largest at the responses directly connected to the softening stiffness. It can be seen

that the fourth resonance frequency, in comparison with the linear case, has been shifted to a lower value.

As noted already, it appears that the anti-resonances are sensitive to the softening stiffness, noticeable in

H31 and H32. On the other hand, by using the multiple coherence approach in characterizing the linear

relationship between inputs and output, the driving point anti-resonances (H11 and H22) do not result in a

coherence drop.


49

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3

1 0-2 1 /1 F R F - M ag n i t u d e an d M C o h e r e n c e , F = 1 0 0 N


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



Figure 4-17: FRFs and MCOH, Softening Stiffness, Inp. 1+2, k23


50

The noise-like distortion in H21 and H31 in the frequency range f = 20 – 40Hz was checked for correlation

and in fact the correlation coefficient between {H21} and {H31} was determined to be

r({H21},{H31}) = -0.96 (f = 25 – 30Hz). Using {H22} and {H32} in the same frequency range, a correlation

coefficient r({H22},{H32}) = 0.7 was found. As can be seen, in the first case an out of phase and in the

second, an in phase correlation is present. Therefore it appears that no prediction beyond the correlation

between two frequency responses can be made. One more comment should be made at this point. If, for

this example, no noise was apparent and only linear motion was observed (hence: smooth frequency

responses), a correlation coefficient computed between two FRFs in a frequency range f = 25 – 30Hz

would of course show a strong correlation. But, if the FRFs are distorted in a noise-like manner, a strong

correlation between these two frequency responses does not point towards random errors but to systematic

effects. Therefore finding a correlation, random error can be excluded as possible source.

0 5 10 15 20 25 30 35 40 0 0.5

1

Frequency in Hz

0 5 10 15 20 25 30 35 40 0 0.5

1

0 5 10 15 20 25 30 35 40 0 0.5

1 MCOH 2 , MCOH 3 and MCCOH (2+3), F = 50N

Figure 4-18: Multiple Coherence and Combined Coherence, Softening Stiffness, Inp. 1+2, k23


51

Figure 4-18 shows the combined coherence for this example, where it appears hard to tell if an

improvement has been made in comparison with MCOH2 and MCOH3. It is questionable if the combined

coherence would be applied to MCOH2 and MCOH3 in the first place, because the MCOHs do not indicate

a non-linear relationship between DOF 2 and 3 (e.g. corresponding drops in coherence).

VII) Non-linearity between DOF 3 and 1/2/4, SIMO, Input at DOF 3

This simulation setup is intended to create the excitation at a softening element of the model and therefore

introduce non-linear relationships in the vicinity of the reference. The input is chosen to be applied at

DOF 3 and all connections from DOF 3 to the DOFs 1/2/4 will behave non-linear. Equation (4-5) is

employed to determine the softening restoring forces, with s13 = 2000, s23 = 2500, s34 = 5000 and a = 10 for

all three non-linear connections. The RMS excitation level for the pure random signal is F = 50N. From the

FRFs in Figure 4-19, it can be seen, that all resonance frequencies, except the rigid body mode, are

decreasing in frequency and that all the FRFs are now affected since multiple non-linearities are present.

Secondary resonance effects (H13), distortions in the frequency range of large relative motion between

DOFs (e.g. H43), high frequency noise-like distortions (H13, H43) and a sensitivity to anti-resonances can be

found (H33).

Furthermore, since multiple non-linearities are present, applying the combined coherence does not show

any significant improvement (e.g. for x2 and x3, see Figure 4-20). This is a clear indicator that the distortion

in motion is not only caused by one (1) dominant non-linear interaction.


52

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



Figure 4-19: FRFs and Coherence, Softening Stiffness, Inp. 3, k3all


53

0 5 10 15 20 25 30 35 400

0.5

1

Frequency in Hz

0 5 10 15 20 25 30 35 400

0.5

1

0 5 10 15 20 25 30 35 400

0.5

1


Figure 4-20: Ordinary and Combined Coherence, Softening Stiffness, Inp. 3, k3all

This concludes the discussion of softening stiffness. In summary, it could be found that the softening effect

is evident in decreasing resonance frequencies when a higher forcing level is applied. It also has been

found that the anti-resonances are sensitive to softening stiffness terms in the system and even though it

was not shown it should be mentioned that a correlation in distortions between frequency responses was

observed.

4.2.3 Non-Symmetric Stiffness

Non-symmetric stiffness should be defined as hardening/softening or softening/hardening restoring force,

the first resisting compression more than pull and the latter having a smaller reaction force when

compressed in comparison with pulled. Since causing similar effects in the frequency and coherence


54

function these two non-linear interactions can be discussed in one section. The restoring force for both

characteristics can be described in a functional form by:

( )11−⋅

±⋅= Δ⋅± xs

linr es

kf (4-6)

Scaling factor s in Equation (4-6) is used to adjust the severity of the non-linear behavior for given

displacements and the sign models the particular non-symmetric stiffness (+ = softening/hardening,

- = hardening/softening). Figure 4-21 shows both non-linearities sketching the linearizing concept used to

describe the effects on the frequency response.

Figure 4-21: Linearizing Concept for Non-Symmetric Stiffness

For the hardening/softening stiffness, forcing at a higher level the “linearized stiffness” k_lin2 “seen” by

the frequency response estimation has a slightly smaller value than k_lin1. Therefore, a larger excitation

will result in a slight frequency shift to smaller values. The opposite will happen when forcing a system

with softening/hardening elements. It should also be mentioned that for comparable severity of

non-linearities and similar displacements the frequency shift effect of a non-symmetric stiffness will not be

as significant as for a pure hardening or softening stiffness term. Figure 4-22 shows the frequency response

F

ΔxF1

F2

k_lin1

k_lin2

F = f(Δx)

Soft/ Hardening Nonlinearity

F

Δx F1

F2

k_lin1

k_lin2

F = f(Δx)

Hard/ Softening Nonlinearity


55

for a SDOF system with a hardening/softening stiffness element. It can be seen that the frequency shift

effects are not as significant when increasing the force level but a secondary resonance evidence is present

at frequency close to twice the linear resonance. Furthermore, clear distortions are apparent in the low

frequency range, which are caused by the non-symmetric behavior of the non-linearity.

0 5 10 15 20 25 300

0.5

1

Frequency in Hz

0 5 10 15 20 25 30

10-5

10-4

10-3 FRF - Magnitude and Coherence, Hard/SoftStiff, 10N (s), 20N (d), 40N (dd)

Figure 4-22: SDOF System with Hardening/Softening Stiffness

VIII) Non-linearity between DOF 1 and 3, SIMO, Input at DOF2

For this case, a hardening/softening stiffness is placed between DOF 1 and 3 in the 4-DOF simulation

model, choosing scaling factor to s13 = 100. The input is applied at DOF 2 and RMS excitation level for the

pure random signal is F = 50N. Figure 4-23 displays all four frequency responses.


56

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



Figure 4-23: FRFs and COH, Hardening/Softening Stiffness, Inp. 2, k13


57

Most affected by the non-linearity are frequency responses H12 and H32, since DOFs 1 and 3 are directly

connected to the hardening/softening stiffness. It is interesting to note, that no significant frequency shift

occurs, even though it is known (hardening stiffness at k13) that the stiffness at this location strongly

influences the third resonance. Besides small secondary resonances of order three, a secondary resonance

at twice the rigid body resonance of f1 = 3.3Hz at f ≈ 6.6Hz can be found. Distortions at f ≈ 13Hz (4),

f ≈ 20Hz (6) and f ≈ 33Hz (10) are also believed to be caused by secondary resonances corresponding to

even multiples (in brackets) of the first resonance frequency.

0 5 10 15 20 25 30 35 400

0.5

1

Frequency in Hz

0 5 10 15 20 25 30 35 400

0.5

1

0 5 10 15 20 25 30 35 400

0.5

1


Figure 4-24: Ordinary and Combined Coherence, Hardening/Softening Stiffness, Inp. 2, k13

As in the cases discussed before, the combined coherence will be determined. This technique will still

work for a non-symmetric stiffness, since the method is based on a force equality principle and not

dependent on symmetry of the displacement-force relation. Computing CCOH for x1 and x3, a drastic


58

improvement in comparison to COH12 and COH32 can be recognized, proving that the drops in ordinary

coherence are in fact caused by non-linear relationship between DOF 1 and 3 (see Figure 4-24).

Since the same effects on the FRFs and coherence functions have been observed when applying a

softening/hardening stiffness of equal severity in equivalent testing situation, the results are not shown

here.

IX) Non-linearity between DOF 2 and 3, MIMO, Inputs at DOF 1 and 2

This case simulates a MIMO situation, intended to excite the hardening/softening stiffness in between

DOFs 2 and 3 well and investigating the effects of this particular non-linearity in a multi-reference set up.

Scaling factor s is chosen as s23 = 300 and a pure random signal is applied to DOFs 1 and 2, with an RMS

value of F = 50N each input. All eight frequency responses are estimated and shown in Figure 4-25.

Responses of DOFs 1 and 4 seem not affected by the non-linearity and FRFs H11, H12, H41, and H42 appear

to be measurements of a linear system. Frequency responses H21, H22, H31, and H32 on the other hand

clearly show distortions caused by the hardening/softening stiffness term. As in Case VIII), it should be

remarked that no significant shift of resonance frequencies can be observed, a characteristic of that type of

non-linearity. Besides not perfect coherence values at low frequencies, caused by the non-symmetric nature

of the hardening/softening stiffness, a distorted peak can be noted in a frequency of f ≈ 33Hz. It is believed

that this effect is caused by a secondary resonance. A check of correlation between {H21} and {H31} in the

frequency range of that peak (f = 30 – 35Hz) does not reveal a systematic relation (r({H21},{H31}) = -0.07).

If, on the other hand, the correlation coefficient is computed for the frequency range f = 20 – 25Hz, it is

found that {H21} and {H31} seem to be dependent (r({H21},{H31}) = -0.85).


59

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



Figure 4-25: FRFs and MCOH, Hardening/Softening Stiffness, Inp. 1+2, k23


60

Looking at the significant improvement showing combined coherence, Figure 4-26, it can be stated, that

the low coherence values of MCOH2 and MCOH3 are in fact caused by non-linear relationship between

DOFs 2 and 3.

0 5 10 15 20 25 30 35 400

0.5

1

Frequency in Hz

0 5 10 15 20 25 30 35 400

0.5

1

0 5 10 15 20 25 30 35 400

0.5

1

MCOH 2 , MCOH 3 and MCCOH (2+3)

Figure 4-26: Multiple and Combined Coherence, Hardening/Softening, Inp 1+2, k23

For this testing set up, the same results and characteristics were obtained when applying a

softening/hardening stiffness instead of the hardening/softening stiffness used in this example.

X) Non-linearity between DOF 3 and 1/2/4, SIMO, Input at DOF 3

In this case, all stiffness elements connecting DOF 3 with the other three DOFs in the 4-DOF model

become non-linear, expressing a hardening/softening dependence upon the relative displacement.

Non-linear parameters in Equation (4-6) are chosen as s13 = 100, s23 = 300 and s34 = 200. When exciting at

DOF 3, this will be a valid simulation of a system with a local non-linearity induced by the SIMO situation.


61

The RMS level of the pure random excitation is F = 50N. The resulting frequency responses are shown in

Figure 4-27. As can be seen in Figure 4-27, all FRFs now have evidence of non-linear behavior, whereas

no significant frequency shift has been observed, as could be expected. In frequency responses H13 and

H33, the drop caused by a secondary resonance of order two with respect to the first resonance is clearly

identified. It should also be noted that in the region of resonances all coherences express relatively high

values (except the leakage-caused drop at first resonance). Furthermore, all FRFs, except the driving point

measurement H33, show poor coherence at higher frequencies. Also, a correlation between the distortions in

the FRFs, as done in previous cases, could not be established. This might be due to the fact that multiple

non-linearities are acting and a systematic separation might not be possible.

The combined coherences CCOH(13)3, CCOH(23)3, and CCOH(34)3 are computed, see Figure 4-28, where

some improvements are recognizable but a thorough interpretation appears infeasible because of the

multiple influences of the applied non-linearities. For example, CCOH(13)3 shows a clear increase in the

region of the secondary resonance (f ≈ 6.6Hz) in comparison to COH13 and COH33 and also in a frequency

range of about f ≈ 12Hz or f ≈ 17Hz but at the same time decreases in a frequency range of f ≈ 10Hz, where

COH13 and COH33 displayed mostly coherent behavior.

Concluding to the discussion of the effects of non-symmetric stiffness terms, it can be stated that for this

type of non-linearity frequency shifts are not significant when increasing the forcing level, secondary

resonances of even multiples with respect to actual resonances are possible and due to the nature of the

non-linearity the occurrence of non-coherent behavior at very low frequencies is likely. Since similar

effects occur in the frequency response and coherence functions, it will be hard to identify or separate

hardening/softening versus softening/hardening stiffness relations by only analyzing FRFs or coherence.


62

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



Figure 4-27: FRFs and COH, Hardening/Softening Stiffness, Inp. 3, k3all


63

0 5 10 15 20 25 30 35 400

0.5

1

Frequency in Hz

0 5 10 15 20 25 30 35 400

0.5

1

0 5 10 15 20 25 30 35 400

0.5

1

COH 1/3 , COH 3/3 and CCOH (1+3)/3

0 5 10 15 20 25 30 35 400

0.5

1

Frequency in Hz

0 5 10 15 20 25 30 35 400

0.5

1

0 5 10 15 20 25 30 35 400

0.5

1

COH 2/3 , COH 3/3 and CCOH (2+3)/3

0 5 10 15 20 25 30 35 400

0.5

1

Frequency in Hz

0 5 10 15 20 25 30 35 400

0.5

1

0 5 10 15 20 25 30 35 400

0.5

1

COH 3/3 , COH 4/3 and CCOH (3+4)/3

Figure 4-28: Ordinary and Combined Coherence, Hardening/Softening Stiffness, Inp. 3, k3all


64

4.2.4 Deadzone/ Play

If, for a certain relative motion, no restoring force is present, one speaks about play or deadzone. This

non-linearity can be modeled as in part linear and zero in the region of play. The following formulation can

be used:

( )gxkf linr +Δ⋅= Δx < -g

0=rf -g ≤ Δx ≤ g ( 4-7)

( )gxkf linr −Δ⋅= g < Δx

In Equation (4-7), g denotes the play, where 2g is the total region of deadzone and klin the linear stiffness

factor. There are two main reasons why this non-linearity represents an interesting case to analyze. First,

because the deadzone relation has a non-continuous derivative, challenging the integration algorithm and

second, this non-linearity will behave more linear when forced at increasing forcing levels, as opposed to

all non-linear cases dealt with up to now. Figure 4-29 a) shows the frequency response and coherence for a

SDOF system with a deadzone stiffness term and Figure 4-29 b) demonstrates the linearizing concept of

this particular relation, which helps to understand the effects of increasing force level on the frequency

response and the coherence function.

For small excitation and therefore small relative motion, a given deadzone will result in large distortions

where on the other hand large excitation and large motion let the given deadzone appear small in relation to

the experienced displacement, making the distortion effects smaller as a result. One should expect better

FRF estimation and improved coherence for that type of non-linearity, when forcing at a higher level.


65

Figure 4-29: a) SDOF System, b) Linearizing Concept

XI) Non-Linearity between DOF 1 and 3, SIMO, Input 2

In this case, a deadzone stiffness is placed between DOFs 1 and 3 of the 4-DOF simulation model.

Non-linear parameter g is chosen as g = 0.0015m, representing half of the actual play. Input location is

DOF 2, exciting the system with a pure random signal characterized by a RMS value of F = 50N.

Estimated frequency responses and coherences are shown in Figure 4-30.

F

ΔxF1

F2

k_lin1

k_lin2F = f( Δ x)

Deadzone Nonlinearity

0 5 10 15 20 25 300

0.5 1

Frequency in Hz

0 5 10 15 20 25 30

10 -5

10 -4

10 -3 FRF - Magnitude and Coherence, Deadzone, 10N (s), 20N (d), 40N (dd)

a) b )


66

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



Figure 4-30: FRFs and COH, Deadzone, Inp. 2, k13


67

As can be seen from the FRFs and as it could be expected, the influence is greatest in H12 and H32, because

the non-linearity is directly connected to DOFs 1 and 3. All other stiffnesses in the model are behaving

linear. Also, the leakage caused drop in coherence is visible at all FRFs and at the deep anti-resonance in

the driving point measurement H22. Looking at H12 and H32, the motion in the frequency region of the

resonance shows mostly coherent behavior, not so around f ≈ 9 – 12Hz, where a secondary resonance and a

small relative motion between DOFs 1 and 3 are believed to be the reason for low coherence values.

Distortions in the high frequency range of H12 and H32 seem also caused by secondary resonances, but of

higher orders.

-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2-0.15

-0.1

-0.05

0

0.05

0.1

log |H12|

log

|H32

|

FRF Correlation, 1/2 vs. 3/2, f = 25 to 30 Hz

Corr. = -0.90709

Figure 4-31: Distortion Correlation for Deadzone, k13

As mentioned before, in this case also, the high frequency distortions have been found to be correlated. In

fact the correlation factor between {H12} and {H32} has been computed to be r({H12},{H32}) = -0.61

(f = 20 - 25Hz), r({H12},{H32}) = -0.91 (f = 25 - 30Hz), r({H12},{H32}) = -0.94 (f = 30 - 35Hz), and


68

r({H12},{H32}) = -0.98 (f = 35- 40Hz). In Figure 4-31, {H32} is plotted versus {H12} for a frequency range

of f = 25 - 30Hz.

By computing the combined coherence function for x1 and x3, and therefore removing the direct interaction

between DOF 1 and 3, a drastic improvement is noticeable versus the ordinary coherence functions COH12

and COH32 (see Figure 4-32).

0 5 10 15 20 25 30 35 400

0.5

1

Frequency in Hz

0 5 10 15 20 25 30 35 400

0.5

1

0 5 10 15 20 25 30 35 400

0.5

1


Figure 4-32: Ordinary and Combined Coherence, Deadzone, Inp. 2, k13

XII) Non-Linearity between DOF 3 and 1/2/4, SIMO, Input at DOF 3

In this case, the connections from DOF 3 to all other DOFs exhibit a play of 2g = 0.00075m, input location

is at DOF 3 and the pure excitation signal has a RMS value of F = 50N. Compared to Case XI), the

severity of the deadzone non-linearity has increased, since multiple non-linearities instead of only one

non-linearity are present. In Figure 4-33, FRFs and coherence functions are plotted.


69

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3


N o n -L in ea rL in e ar

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3


N o n -L in ea rL in e ar

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



Figure 4-33: FRFs and COH, Deadzone, Inp. 3, k3all


70

From Figure 4-33 it can be seen, that all except the first resonance frequency are at significantly lower

frequencies compared to the linear case. The linear frequency responses would be the asymptotic limits

when increasing the forcing level. Every FRF is clearly affected by the non-linearities applied, which is

evident in the distortion of the FRFs and in low coherence values. Non-coherent behavior is dominant from

a frequencies f > 20Hz, where better coherence can be noticed at the driving point measurement H33, which

could be explained by the direct input at DOF 3. Furthermore, for this case a correlation check between

distortions in FRFs was performed but did not yield solid results. It appears that the a strong correlation

might only be apparent when one dominant non-linearity is present.

0 5 10 15 20 25 30 35 400

1

Frequency in Hz

0 5 10 15 20 25 30 35 400

1

0 5 10 15 20 25 30 35 400

1

0 5 10 15 20 25 30 35 400

1

0 5 10 15 20 25 30 35 400

1

COH i /3 and CCOH (1+2+3+4)

Figure 4-34: Ordinary and Combined Coherence, Deadzone, Inp. 3, k3all

As in the cases before, the combined coherence function was computed for (x3 + x1), (x3 + x2) and (x3 + x4)

but no significant improvement in CCOH versus the ordinary coherence was found. This can be explained

by noting that adding only two motions together will not eliminate all non-linear effects since there are

multiple non-linearities within the system. Therefore, the combined coherence for the sum of all responses


71

(x1 + x2 + x3 + x4) versus input at DOF 3 has been estimated, see Figure 4-34. The combined coherence for

all responses shows in fact higher values, especially for high frequencies.

This concludes the discussion of non-linear stiffness effects upon the frequency response and coherence

function. It has been shown, that non-linear stiffness terms within the system alter frequency response and

consequently influence the coherence function, depending upon the nature of non-linearity. Basically four

different effects could be found: Shifted resonance frequencies; distortions in the frequency range of large

relative motion between certain DOFs; appearance of secondary resonances; and random-like distortions

for high frequencies. Furthermore, depending upon the type of non-linearity, distinct effects are apparent,

making the identification of particular non-linear stiffness dependence possible.

4.2.5 Non-Linear Damping

In this section the effects of non-linear damping will be analyzed, where three different types of non-linear

damping are considered: quadratic, non-symmetric, and coulomb damping.

XIII) Quadratic Damping between DOF 1 and 3, SIMO, Input at DOF 1

This case simulates a quadratic offset to the linear relation between DOF 1 and 3 and which will be

formulated by the following equation:

( )xxxcf lind &&& Δ⋅Δ⋅+Δ⋅= ε ( 4-8)


72

It has been found, that using the lightly damped system as reference and developing the non-linear cases

from this baseline did not show significant effects on the frequency responses and coherence functions.

Therefore a higher damping value for the anticipated non-linear damping connection between DOF 1 and 3

was chosen, c13 = 200Ns/m as oppose to originally c13 = 8Ns/m. Determining the severity of the non-linear

effect, ε was fixed as ε = 10. The system was excited by a pure random signal with a RMS value of

F = 100N.

Figure 4-35 shows the estimated frequency response functions. As can be seen, the most affected

frequency responses are H11 and H31, since DOFs 1 and 3 are directly connected to the quadratic damping

connection. Since non-linear damping is being examined, no significant frequency shifts are visible and

should not be expected. On the other hand, a magnitude change between resonances can be observed, due

to the non-linear damping. As it turns out, the non-linear connection with c13 = 200Ns/m and ε = 10

modeled by Equation (4-8) behaves mainly as a linear system with a higher linear damping value

(c13 ≈ 400Ns/m). So, for a higher excitation level, the non-linear system would behave as an even larger

damped linear system. Furthermore, ordinary coherence does not necessarily reflect the non-linear

behavior of the system, only for COH31 is minor distortion in the high frequency range noticeable.


73

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



Figure 4-35: FRFs and COH, Quadratic Damping, Inp. 1, c13


74

XIV) Softening/Hardening Damping between DOF 1 and 3, SIMO, Input at DOF 1

This case simulates a softening/hardening damping relation between DOFs 1 and 3. In the real world, this

damping characteristic can, for example, be found in car shocks. To formulate this particular type of

non-linearity, the same relation as for the softening/hardening stiffness is used, utilizing relative velocity as

the independent variable, and not displacement:

( )11−⋅⋅= Δ⋅ xs

lind es

cf & ( 4-9)

It can be seen, that Equation (4-9) converges towards the linear case for small relative velocity. As in the

case for quadratic damping, clin is chosen as c13 = 200Ns/m and scaling parameter s to s = 5. The 4-DOF

system is excited at DOF 1 by a pure random signal, RMS value of F = 100N.

All frequency responses are shown in Figure 4-36 and it can be seen, that no significant distortion can be

found in the FRFs and compared to the linear case (c13 = 200Ns/m) no noticeable deviation is apparent.

Analyzing the ordinary coherence functions, the only remarkable evidence is found in high frequency

distortions of COH31, all other coherences express mainly linear behavior. This fact might be due to the

nature of the non-symmetric non-linear damping term, making the “linearized” damping about the same

value as the linear damping value. A detailed explanation was given when non-symmetric stiffness was

discussed, so refer to Section 4.2.3.


75

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



Figure 4-36: FRFs and COH, Softening/Hardening Damping, Inp 1, c13


76

XV) Coulomb Damping between DOF 1 and 3, SIMO, Input at DOF 1

The third case, modeling a non-linear damping, will simulate coulomb friction between DOFs 1 and 3,

which can be described by:

( ) 13cxsigf d ⋅Δ= & ( 4-10)

Equation (4-10) simply states, that depending on the direction of velocity, a constant damping force will

counteract the motion. This case can not be derived from the linear damping and it should be noted, that

this reaction force has a non-continuous derivative making it especially hard for the integration algorithm.

In order to make an integration possible in reasonable time, the error tolerance had to be lifted but, in order

to guarantee reliable results, the integration was repeated with smaller, but still feasible tolerances until

convergence was observed. For the actual simulation model, the damping factor was chosen to

c13 = 200Ns/m as in Cases XIII) and XIV), forcing the system with a pure random signal at DOF 1

characterized by a RMS value of F = 100N.

From the frequency responses, it can be seen (Figure 4-38), that especially for H11 and H31 the difference

between the linear (c13 = 200Ns/m) and the non-linear solution is significant. While the resonance

frequencies remain basically constant, the magnitude in between resonances and the location of

anti-resonances changes. As it shows, the actual system exhibits a much larger damping value between

DOFs 1 and 3 as the nominal damping value of c13 = 200Ns/m expresses. When compared to linear system

responses with a damping value of c13 = 6400Ns/m a much better match was reached. This fact might be

explained by the step-nature of the reaction force, inducing a much higher damping than the steady state

value of c13 = 200Ns/m suggests.


77

Looking at the combined coherence CCOH(13)1, it can be seen, that the distortions in x1 and x3 and therefore

in COH11 and COH31 are in fact caused by the non-linear relation between these two degrees of freedom

(see Figure 4-37).

0 5 10 15 20 25 30 35 400

0.5

1

Frequency in Hz

0 5 10 15 20 25 30 35 400

0.5

1

0 5 10 15 20 25 30 35 400

0.5

1


Figure 4-37: Ordinary and Combined Coherence, Coulomb Friction, Inp. 1, c13

This concludes the discussions about the influence of non-linear damping upon the frequency response and

coherence function. It has been shown, that a non-linear damping term basically alters the magnitude of the

FRF and corresponding anti-resonance but no significant shift in resonance frequency has been observed.

Furthermore, it becomes hard to specify the particular non-linear damping relation due to the similar effects

of different damping non-linearities.


78

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



Figure 4-38: FRFs and COH, Coulomb Friction, Inp. 1, c13


79

4.3 Non-Linear Effects in the Presence of Noise

When using pure random signals for modal testing it must be acknowledged, that obtaining only one (1) set

of measurements will not excite all frequencies equally well and therefore, for reliable results, several

ensembles must be taken. Furthermore, noise, which usually is assumed to be of random distribution, has

to be expected on the measured time histories. Figure 4-39 shows the same frequency response estimation

of a linear system for different numbers of spectral averages. As can be seen, the variance in system

response due to the variance in excitation will average out and the frequency response will converge to the

true value. It can also be noticed, that when the system response is covered by noise because of small

response magnitude (anti-resonance and in the high frequency range), low coherence values will result.

The effects of leakage at the first resonance and the anti-resonance are visible too.

0 5 10 15 20 25 30 35 400

0.5

1

Frequency in Hz

0 5 10 15 20 25 30 35 4010

-7

10-6

10-5

10-4

10-3

10-2 FRF - Magnitude and Coherence, Avg = 1

0 5 10 15 20 25 30 35 400

0.5

1

Frequency in Hz

0 5 10 15 20 25 30 35 4010

-8

10-7

10-6

10-5

10-4

10-3

10-2 FRF - Magnitude and Coherence, Avg = 30

Figure 4-39: FRF Estimation as Function of Spectral Averages

In summary, when employing a pure random excitation to a linear system, frequency response and/or

coherence distortions can have their source in leakage, insufficient number of spectral averages and

measurement noise. If there is non-linear motion within the structure one additional source of frequency


80

response and coherence distortion is possible. After looking at the characteristic effects of non-linearities

without the presence of noise in Section 4.2, this portion of the paper tries to analyze the influence of

measurement noise and the dependence on the number of averages.

For example, a softening stiffness non-linearity, discussed in Section 4.2.2, is chosen. The softening term,

formulated by Equation 4-5, is applied to the connection between DOFs 2 and 3 of the 4-DOF model,

where non-linear scaling factors s and a are set to s = 2500 and a = 10. After exciting the system by a pure

random signal with a RMS value of F = 150N at DOF 1 and simulating the response, random noise was

added, both to reference and response time histories (1% noise level on input and output).

Frequency responses H21 and H31 are plotted in comparison to the linear response and as function of

spectral averages (Avg = 1/10/20/30), see Figure 4-40. For only one (1) average the variance in system

response and the influence of measurement noise in a frequency range of f > 20Hz (small signal to noise

ratio) is clearly recognizable. With only one average taken, no statement regarding the linearity relation can

be made since no base line has been established. After ten (10) averages, the FRFs appear smoother and the

influence of variance at different spectral lines decreases. Also, low coherence values are present at high

frequencies (f > 20Hz) and since the corresponding FRFs show corresponding distortions about a constant

magnitude, it could be assumed that the signal goes down into the noise floor. Notice that even though the

frequency responses are clearing up, low coherence values are present in a region of f ≈ 12 - 18Hz. The

same can be said about the frequency estimates for 20 and 30 averages. The low coherence values in a

frequency range of f ≈ 12 - 18Hz can not be accounted for by leakage, measurement noise, or insufficient

averaging.


81

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3

1 0-2 2 /1 F R F - M ag n i t u d e an d C o h e r e n c e , A vg = 1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3

1 0-2 3 /1 F R F - M ag n i t u d e an d C o h e r e n c e , A vg = 1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3

1 0-2 2 /1 F R F - M ag n i t u d e an d C o h e r e n c e , A vg = 1 0


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00

0 .5

1


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0

-8

1 0-7

1 0-6

1 0-5

1 0-4

1 0-3



Figure 4-40: H21 and H31 as Function of Averages, Softening Stiffness, Inp. 1


82

0 5 10 15 20 25 30 35 400

0 .5

1

F requenc y in Hz

0 5 10 15 20 25 30 35 400

0 .5

1

0 5 10 15 20 25 30 35 400

0 .5

1

C O H 2 /1 , C O H 3 /1 and C C O H (2 + 3 )/1 , Avg = 1 0

0 5 10 15 20 25 30 35 400

0 .5

1

F requenc y in Hz

0 5 10 15 20 25 30 35 400

0 .5

1

0 5 10 15 20 25 30 35 400

0 .5

1


0 5 10 15 20 25 30 35 400

0 .5

1

F requenc y in Hz

0 5 10 15 20 25 30 35 400

0 .5

1

0 5 10 15 20 25 30 35 400

0 .5

1


Figure 4-41: Ordinary and Combined Coherence, Softening Stiffness, Inp. 1


83

Knowing from Section 4.2.2 that these effects are indeed caused by the non-linear stiffness between

DOFs 2 and 3, the combined coherence is computed for (x2 + x3) at 10, 20 and 30 averages (see

Figure 4-41). It can be seen that the combined coherence exhibits higher values in a frequency range of

f ≈ 12 - 18Hz, confirming the assumption that these coherence drops in COH21 and COH31 are caused by a

non-linear relation. Furthermore, when coherence values start to decrease because of measurement noise

(f > 20Hz), the combined coherence indicates slightly higher values compared to ordinary coherence. This

in turn points out, that the drop in coherence is, in part, also induced by the softening stiffness.

Since it has been shown in Section 4.2 that a correlated, non-linearity caused, noise-like distortion in a high

frequency region is possible, the correlation factor between {H21} and {H31} has been computed. But no

evidence of a correlation could be found, for example r({H21},{H31}) = 0.04 (f = 20 –25Hz, Avg. = 10),

r({H21},{H31}) = 0.03 (f = 20 –25Hz, Avg. = 20) or r({H21},{H31}) = 0.12 (f = 20 –25Hz, Avg. = 30).

Therefore, the high frequency distortions in H21 and H31 seem to be in fact mainly caused by random

measurement noise since the deterministic signal is hidden in the noise floor. It should be mentioned, that

as extension to the SIMO test described here, MIMO situations have been applied (for example Inputs at

DOFs 1 and 2) where the distributed energy level realizes a more uniform excitation. For these cases, the

non-linearity between DOFs 2 and 3 is also more engaged in the system response and high frequency,

noise-like distortions are correlated which appears to be evidence of a non-linear relation. This fact

emphasizes the known argument that in order to investigate non-linear effects, the non-linearity itself has

to be excited well enough.

Detection of Non-Linearities Summary/ Future Work

84

5 SUMMARY/ FUTURE WORK

In this thesis a simulation study was conducted in order to investigate the possibility of detecting the

presence of structural non-linearities by using the frequency response and coherence functions. A MDOF

SIMULINK® model was generated and structural non-linearities (stiffness and damping related) have been

applied at different locations and in different testing situations. It was intended to duplicate the typical

modal analysis procedure (using pure random excitation), which estimates the frequency response

functions by measuring inputs and outputs.

In summary, it can be said, that the effects of structural non-linearities are diverse and dependent on the

type of non-linearity applied. It has been found, that not only large excitation and with that, large relative

motion between degrees of freedom, cause large non-linear effects. The investigation of deadzone stiffness

related non-linearity showed, that especially small excitation levels force the system to exhibit large

non-linear distortions, whereas higher excitation levels let the system appear more linear. If the non-linear

relationship is stiffness related, shifts of resonance frequencies, distortion in the frequency region of

extensive non-linear motion (small or large motion) between degrees of freedom, the occurrence of

secondary resonances, and high frequency distortions can be noticed. Furthermore it can be mentioned, that

these high frequency distortions appear noise-like but are in fact correlated, which may be used to detect

the presence of structural non-linearities. If the non-linear relationship is damping related, it has been

found, that, in order to show non-linear damping effects at all, the severity of the non-linear damping

relation and the damping itself has to be extremely large. Since the very nature of damping, non-linear

damping mainly does not affect the frequencies of natural resonances but the magnitude of vibration in

between resonances and the anti-resonances correlated with the non-linear element.

Detection of Non-Linearities Summary/ Future Work

85

This thesis also proposes a method, called combined coherence, which is capable of detecting the effects of

non-linear motion within measurements. The combined coherence is based on the idea of removing the

contribution of direct interaction between two degrees of freedom by forming a new coordinate (sum of

motions). A new coherence function (combined coherence, CCOH) between inputs and the new, virtual

coordinates provides a method, whether distortions in the original coherence functions are in fact caused by

structural non-linearities. It has also been shown that distortions in frequency response and coherence

functions caused by measurement noise or digital signal processing errors (leakage and aliasing) will not

be affected by the technique of combined coherence. This way, not only a detection scheme for the effects

of non-linear motion is realized but also the effects of structural non-linearities can be separated from

measurement noise and digital signal processing errors.

While though the combined coherence is developed on lumped mass structures and some knowledge about

the mass distribution is required, future work (after testing the combined coherence function on a

simulation model) should include the application to continuous, real world structures. The model of a

lumped mass structure might still be a valid approximation of the real system and some knowledge about

the mass distribution might be available in these practical testing situations. Furthermore, it should be

investigated, if the combined coherence method, a linear spectra based technique, can be modified in order

to use it as a post-processing tool. In this way, the advantages of structural non-linearity detection, and

computational and memory efficient analysis could be combined.

Detection of Non-Linearities Reference List

86

6 REFERENCE LIST [1] “The Behavior of Nonlinear Vibrating Systems”, Volume I+II

W. Szemplinska-Stupnicka

Kluwer Academic Publishers, 1990

ISBN 0-7923-0368-7/ 0-7923-0369-5

[2] “Some Engineering Applications in Random Vibrations and Random Structures”

G. Maymon

American Institute of Aeronautics and Astronautics, Inc., 1998

ISBN 1-56347-258-9

[3] “Applications of Random Vibrations”

N.C. Nigam, S. Narayanan

Springer Verlag, Berlin, Heidelberg, New York, 1994

ISBN 3-540-19861-X

[4] “Normal Modes and Localization in Nonlinear Systems”

A.F. Vakakis et al.

John Wiley & Sons, Inc., 1996

ISBN 0-471-13319-1

[5] “Nonlinear Stochastic Dynamic Engineering Systems”

F. Ziegler, G.I. Schueller

Springer Verlag, Berlin, Heidelberg, New York, 1988

ISBN 3-540-18804-5

[6] “Model Testing: Theory and Practice”

D.J. Ewins

Research Studies Press, Ltd., 1984

ISBN 0-86380-017-3


87

[7] “Nonlinear System Analysis and Identification from Random Data”

J.S. Bendat


ISBN 0-471-60623-5

[8] “Nonlinear Systems Techniques and Applications”

J.S. Bendat


ISBN 0-471-16576-X

[9] “Theory of Vibration with Applications”, 5th Edition

W.T. Thomson, M.D. Dahlen

Prentice-Hall, Inc., 1998

ISBN 0-13-651068-X

[10] “Mechanical Vibrations”, 2nd Edition

A.H. Church

John Wiley and Sons, Inc., 1963

[11] “The Volterra and Wiener Theories of Nonlinear Systems”

M. Schetzen


ISBN 0-471-04455-5

[12] “Shock and Vibration Handbook” 4th Edition

C.M. Harris

The McGraw-Hill Companies, Inc., 1996

ISBN 0-07-026920-3

[13] “Numerical Solution of Ordinary Differential Equations”

L.F. Shampine

Chapman and Hall, Inc., 1994

ISBN 0-412-05151-6


88

[14] “Network Analysis with Applications”, 2nd Edition

W.D. Stanley

Prentice-Hall, Inc., 1997

ISBN 0-13-260910-X

[15] “Advanced Engineering Mathematics”, 7th Edition

E. Kreyszig


ISBN 0-471-55380-8

[16] “Applied Numerical Methods for Digital Computation”, 4th Edition

M.L. James, G.M. Smith, J.C. Wolford

HarperCollins College Publisher, 1993

ISBN 0-06-500494-9

[17] “Simulink, Dynamic System Simulation for Matlab”

The Math Works; 1997

[18] “Matlab, The Language of Technical Computing”

The Math Works, 1996

[19] “Taschenbuch Mathematischer Formeln und Moderner Verfahren”

H. Stoecker

Verlag Harri Deutsch, 1993

ISBN 3-8171-1256-4

[20] “A Spatial Approach to Nonlinear Vibration Analysis”

D.E. Adams

Ph.D. Dissertation, University of Cincinnati, 2000

[21] “Detection of Nonlinear Dynamic Behaviour of Mechanical Structures”

M. Mertens, et al.

IMAC Conference, p.712-719, 1986


89

[22] “Detection, Identification and Quantification of Nonlinearity in Modal Analysis – A Review”

G.R. Tomlinson

IMAC Conference, p.837-843, 1986

[23] “Detecting Nonlinearities Utilizing the Multiple Input Frequency Response Function Estimation

Theory”

G.W. Hopton

Master Thesis, University of Cincinnati, 1987

[24] “Vibrations: Experimental Modal Analysis”

R.J. Allemang

Course Notes, University of Cincinnati, 1995

Detection of Non-Linearities Appendix

90

7 APPENDIX

7.1 SIMULINK® Model

4 DOF Model, SIMULINK®


91

2

x_1

1

x_dot_1

s

1

velocitys

1

position

Sum

x_dot_1

x_dot_4

x_1

x_4

f_1_4

SDU_1_4

x_dot_1

x_dot_3

x_1

x_3

f_1_3

SDU_1_3

x_dot_1

x_dot_2

x_1

x_2

f_1_2

SDU_1_2

x_dot_0

x_dot_1

x_0

x_1

f_0_1

SDU_0_1

Display

1/m1

1/m_n

9

f_1

8

x_4

7

x_dot_4

6

x_3

5

x_dot_3

4

x_2

3

x_dot_2

2

x_0

1

x_dot_0

4 DOF Model, SIMULINK®, Subblock DOF_1


92

1

f_1_2

u*k_1_2

k_n_(n+1)

u*c_1_2

c_n_(n+1)

Sum2

Sum1

Sum

4

x_2

3

x_1

2

x_dot_2

1

x_dot_1

4-DOF-Model, SIMULINK®, DOF_1, SDU_1_2


93

7.2 Definition of Physical Parameters, 4-DOF-Model, Linear

Mass in [kg]

DOF 1 2 3 4

1 12 2 7 3 9 4 14

Damping in [Ns/m]

DOF Ground 1 2 3 4

Ground 6 1 6 8 8 7 2 8 9 4 3 8 9 5 4 7 4 5

Stiffness in [N/m]

DOF Ground 1 2 3 4

Ground 22000 1 22000 19000 19000 24000 2 19000 24000 20000 3 19000 24000 20000 4 24000 20000 20000


94

7.3 List of Cases in Section 4.2

I) Hardening Stiffness, k13, SIMO, Input at DOF 3 (page 1)

II) Hardening Stiffness, k13, SIMO, Input at DOF 2 (page 1)

III) Hardening Stiffness, k23, SIMO, Input at DOF 1 (page 1)

IV) Hardening Stiffness, k23, MIMO, Inputs at DOFs 1 and 2 (page 1)

V) Softening Stiffness, k23, SIMO, Input at DOF 1 (page 1)

VI) Softening Stiffness, k23, MIMO, Inputs at DOFs 1 and 2 (page 1)

VII) Softening Stiffness, k3all, SIMO, Input at DOF 3 (page 1)

VIII) Hardening/ Softening Stiffness, k13, SIMO, Input at DOF 2 (page 1)

IX) Hardening/ Softening Stiffness, k23, MIMO, Inputs at DOFs 1 and 2 (page 1)

X) Hardening/ Softening Stiffness, k3all, SIMO, Input at DOF 3 (page 1)

XI) Deadzone Stiffness, k13, SIMO, Input at DOF 2 (page 1)

XII) Deadzone Stiffness, k3all, SIMO, Input at DOF 3 (page 1)

XIII) Quadratic Damping, k13, SIMO, Input at DOF 1 (page 1)

XIV) Softening/ Hardening Damping, k13, SIMO, Input at DOF 1 (page 1)

XV) Coulomb Friction, k13, SIMO, Input at DOF 1 (page 1)

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