viscoelastic vibration toolbox

8/11/2019 Viscoelastic Vibration Toolbox

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Viscoelastic Vibration Toolbox

For Use with MATLAB r

Users Guide Etienne Balmes

Version 1.0 Jean-Michel Leclere


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Contents

1 Modeling viscoelastic materials 51.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Representing complex modulus . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Non parametric (tabular) representations . . . . . . . . . . . 71.2.2 Simple rheological models . . . . . . . . . . . . . . . . . . . . 91.2.3 High order rational models . . . . . . . . . . . . . . . . . . . 121.2.4 Fractional derivative models . . . . . . . . . . . . . . . . . . . 13

1.3 Environmental factors . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.1 Inuence of temperature . . . . . . . . . . . . . . . . . . . . . 131.3.2 Other environment factors . . . . . . . . . . . . . . . . . . . . 18

1.4 Determining the complex modulus . . . . . . . . . . . . . . . . . . . 181.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Viscoelastic FEM models 21

2.1 Viscous and structural damping . . . . . . . . . . . . . . . . . . . . . 222.1.1 Properties of the damped 1 DOF oscillator . . . . . . . . . . 232.1.2 Real modes and modal damping . . . . . . . . . . . . . . . . 252.1.3 Selection of modal damping coe ffi cients . . . . . . . . . . . . 27

2.2 Viscoelastic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2.1 Frequency domain representation with variable coe ffi cients . 292.2.2 State-space representations . . . . . . . . . . . . . . . . . . . 30

2.2.3 Second order models with internal states . . . . . . . . . . . . 312.2.4 Fractional derivatives . . . . . . . . . . . . . . . . . . . . . . 332.3 Spectral decomposition and reduced models . . . . . . . . . . . . . . 33

2.3.1 Complex modes of analytical models . . . . . . . . . . . . . . 342.3.2 Complex mode eigenvalue problems with constant matrices . 352.3.3 Model reduction methods . . . . . . . . . . . . . . . . . . . . 372.3.4 Equivalent viscous damping . . . . . . . . . . . . . . . . . . 382.3.5 Case of viscoelastic models . . . . . . . . . . . . . . . . . . . 40

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CONTENTS

2.4 Meshing of sandwich models . . . . . . . . . . . . . . . . . . . . . . . 402.4.1 Mesh convergence and non conformity . . . . . . . . . . . . . 44

3 Toolbox tutorial 473.1 Viscoelastic modeling tools . . . . . . . . . . . . . . . . . . . . . . . 493.2 Representing viscoelastic materials . . . . . . . . . . . . . . . . . . . 49

3.2.1 Introducing your own nomograms . . . . . . . . . . . . . . . 493.2.2 Selecting a material for your application . . . . . . . . . . . . 50

3.3 Viscoelastic device meshing tools . . . . . . . . . . . . . . . . . . . . 513.3.1 Generation of sandwich models . . . . . . . . . . . . . . . . . 51

3.3.2 Meshing foam llings . . . . . . . . . . . . . . . . . . . . . . . 523.3.3 Exporting submeshes to NASTRAN . . . . . . . . . . . . . . 523.4 Data structure reference . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4.1 Dynamic sti ff ness coeffi cients zCoefFcn . . . . . . . . . . . . 533.4.2 Format reference . . . . . . . . . . . . . . . . . . . . . . . . . 563.4.3 Input denitions . . . . . . . . . . . . . . . . . . . . . . . . . 563.4.4 Sensor denitions . . . . . . . . . . . . . . . . . . . . . . . . . 573.4.5 Response post-processing options . . . . . . . . . . . . . . . . 57

3.5 Sample parametric analysis step by step . . . . . . . . . . . . . . . . 583.5.1 Model parameterization . . . . . . . . . . . . . . . . . . . . . 583.5.2 Parametric model generated within NASTRAN (fo by set DMAP) 603.5.3 Parametric model from NASTRAN element matrices . . . . . 633.5.4 Full and reduced order solution within SDT . . . . . . . . . . 63

3.6 Fluid/structure coupling . . . . . . . . . . . . . . . . . . . . . . . . . 643.6.1 Summary of theory . . . . . . . . . . . . . . . . . . . . . . . . 64

3.7 NASTRAN DMAP routines . . . . . . . . . . . . . . . . . . . . . . . 673.8 Advanced connection models . . . . . . . . . . . . . . . . . . . . . . 68

3.8.1 Screw models . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.8.2 Physical point with rotations . . . . . . . . . . . . . . . . . . 68

3.9 Validation examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.9.1 Validity of free and constrained layer models . . . . . . . . . 68

4 Toolbox Reference 714.1 fe2xf . . . . . . . . . . . . . . . . . 724.2 fevisco . . . . . . . . . . . . . . . . . 774.3 m visco, mvisco 3m . . . . . . . . . . . . . 864.4 nasread . . . . . . . . . . . . . . . . 914.5 naswrite . . . . . . . . . . . . . . . . 95

Bibliography 101

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Index 104

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CONTENTS

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1

Modeling viscoelasticmaterials

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 61.2 Representing complex modulus . . . . . . . . . . . 6

1.2.1 Non parametric (tabular) representations . . . . . 71.2.2 Simple rheological models . . . . . . . . . . . . . . 91.2.3 High order rational models . . . . . . . . . . . . . 12

1.2.4 Fractional derivative models . . . . . . . . . . . . . 131.3 Environmental factors . . . . . . . . . . . . . . . . . 13

1.3.1 Inuence of temperature . . . . . . . . . . . . . . . 131.3.2 Other environment factors . . . . . . . . . . . . . . 18

1.4 Determining the complex modulus . . . . . . . . 181.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . 19


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1 Modeling viscoelastic materials

1.1 Introduction

Linear viscoelasticity assumes [1] that stress is a function of strain history. Thistranslates into the existence of a relaxation function given by

(t) = 0

(t )h( )d (1.1)Using Laplace transform, one sees that this hypothesis is equivalent to the existenceof a complex modulus such that

(s) = (s)(s) = ( (s) + i (s))(s) (1.2)From a practical point of view, one can solve viscoelasticity problems as elasticityproblmes with a complex modulus that depends on frequency. This property isknown as the elastic/viscoelatic equivalence principle [1].

For a strain tensor, the number of idependent coe ffi cients in is identical to that inHookes law for an elastic material (for the same reasons of material invariance). Forhomogeneous and isotropic materials, one thus considers a Youngs modulus and aPoisson coeffi cient that are complex and frequency dependent.

The separate measurement of E () et () is however a signicant experimentalchallenge [2] that has no well established solution. Practice is thus to measure acompression E () or a shear modulus G() and to assume a constant Poissonsratio, although this is known to be approximate.

Section 1.2 analyzes the main representations of complex moduli. Section 1.3 listsrepresentations used to account for the e ff ect of environmental factors (temperature,pre-stress, ...). Domains of applicability for di ff erent models is the given at the end

of the chapter.For more details, Ref. [1] a detailed account of viscoelasticity theory. Refs [ 2, 3]present most practical representations of viscoelastic behaviour.

1.2 Representing complex modulus

Even though, viscoelastic constitutive laws are fully characterized by the measure-ment of a complex modulus. As will be detailed in the next chapter, numericalsolvers are often associated with specic analytical representations of moduli. Thissection illustrates the main classes of modulus representations.

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1.2.1 Non parametric (tabular) representations

At each frequency, the complex modulus describes an elliptical stress/strain relation.

For = cos t , one has = Re( E s (1 + i)eit ) = E cost k sin t as shown ingure 1.1. On calls, storage modulus the real part of the complex modulus E s =Re(E ) and loss factor the ratio of the imaginary and real parts = Im( E )/ Re(E ).The name loss factor derives from the fact that is equal to the ratio of energydissipated for one cycle E d =

T 0 dt by 2 the maximum potentional energy E p =

1/ 2E . This denition is used to compute a loss factor for systems with small nonlinearities (see section 2.1.3).

1

!

"

E s

1

E s #

-E s #

-1

Figure 1.1: Elliptical stress/strain cycle

The raw measure of viscoelastic characteristics gives a complex modulus which typ-ically has the characteristics shown in gure 1.2.

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Figure 1.2: Sample constitutive law for a viscoelastic material

The solution of frequency domain problems (frequency response functions, com-plex mode extraction) requires the interpolation of measurements between frequencypoints (shown as a solid line) and the extrapolation in unmeasured low and highfrequency areas (shown in dotted lines in the gure).

For interpolation, a moving average, wheighted for 2 or 3 experimental points, iseasily implemented and fast to compute. For extrapolation at low frequencies, onewill assume a real assymptote E (0) (i.e. (0) = 0) from basic principles : sincethe relaxation function is real, its Fourier transform is even and thus real at 0. For

high frequencies, one uses a complex asymptote E , . There are mathematicallimitations, on possible values for this assymptote but in practice, there are alsoother physical dissipation mechanisms that are not represented by the viscoleasticlaw so that the real objective is to avoid numerical pathologies when pushing themodel outside its range of validity.

The main advantage of non parametric representations of constitutive laws is to al-low fully general accounting of behaviour that are strongly dependent on frequency,

temperature, pre-stress, ... Moreover, the direct use of experimental data avoidsstandard steps of selecting a representation and identifying the associated parame-ters. Since good software to perform those steps is not widespread, avoiding themis really useful.

Disadvantages are very few and really mostly linked to the extraction of complexmodes and as a consequence time response simulations using modal bases. And eventhese diffi culties can be circumvented.

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1.2.2 Simple rheological models

The classical approach in rheology is to represent the stress/strain relation as a series

of springs and viscous dashpots. Figure 1.3 shows the simplest models [ 3].

E C

(a)

E

(b)

E(1+j ! )

(c)

E0

E1 1

(d)

C C

Figure 1.3: Material damping models with 2 or 3 parameters : (a) Maxwells model,(b) vicous damping (Kelvin-Voigts model), (c) structural damping, (d) standardviscoelastic solid

Maxwells model is composed of a spring and a dashpot in series. Each elementcarries the same load while strains are added. The stress / strain relation is, in thefrequency domain, written as

=1E

+ 1Cs

= Es

s + E/C

1 (1.3)

where

= C E

(1.4)

is a settling time that is caracteristic of the model and is associated to a pole at = E/C = 1/ .The viscous model, called Kelvin Voigt Model , is composed of a spring and a dashpotin parallel. Each element undergoes the same elongation but strains add. Dampingis then represented by the addition of strains proportionnal to deformation anddeformation velocity. In the frequency domain the modulus has the form

E (s) = E 0 (1 + s) (1.5)

Structural damping, also called hysteretic damping, is introduced by consideringa complex sti ff ness that is frequency independent. The complex modulus is thus

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characterized by constant storage moduli and loss factorsE (s) = E 0 (1 + j ) (1.6)

The standard viscoelastic solid has a constitutive law given by

(s) = E 0 + 1E 1

+ 1C 1s

1 =

E 0E 1 + ( E 0 + E 1)C 1sE 1 + C 1s

= E 01 + s/z1 + s/p

(1.7)

leading to a time domain representation

E 1(t) + C

1 (t) = E

0 + E

1C

1 (1.8)

Figure 1.4 shows the frequency domain properties of these models. Maxwells modelis only valid in the high frequency range since its static sti ff ness is zero. The viscousdamping model is unrealistic because the loss factor goes to innity at high frequency(there deformation locking).

101

102

103

104

Frequency

R e ( E )

101

102

103

104

!

Frequency

abcd

Figure 1.4: Complex modulus associated to standard models: (a) Maxwells model,(b) vicous damping (Kelvin-Voigts model), (c) structural damping, (d) standardviscoelastic solid

The structural damping model approximate the modulus by a complex constant.Although it does not represent accurately any viscoleatic material, such as the oneshown in gure 1.2, it leads to a good approximation when the global behavior of the structure is not very sensitive to the evolutions of the complex modulus withfrequency. This model is generally adapted for materials with load damping (metals,concrete, ...).

The standard viscoelastic model has the main characteristics of real materials : lowand high frequency assymptotes, maximum dissipation at the point of highest slope

of the real part of the modulus. This model is characterized a nominal level E 0 and10


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two pairs of related quantities: a pole

p = E 1/C 1 = + 1 + 2 , (1.9)where the modulus attens out, and a zero

z = ( E 0E 1)/ ((E 0 + E 1)C 1) =

+ 1 + 2 (1.10)

giving the inexion point where the modulus starts to augment; or the maximumloss factor

= ( p z)

2 pz =

1

2

E 1

E 0(E 0 + E 1)(1.11)

and the frequency where is maximum is reached (between the pole and the zero)

= pz = E 1C 1 E 0E 0 + E 1 (1.12)

The maximum loss factor is higher when the pole and zero are well separated andthe low and high frequency moduly di ff er. This is consistent with the fact thatmaterials that dissipate well also have storage moduli that are strongly frequencydependent.

Figure 1.5 illustrates problems encountered with three parameter models. The pa-rameters of the standard viscoelastic model where chosen to match the low and highfrequency assymptote and the frequency of the maximum in the loss factor. Oneclearly sees that the slope is not accurately represented and more importantly thatthe loss factor maximum and evolution in frequency are incorrect. These limitations

have motivated the introduction of more complex models detailed in the followingsections.

100

101

102

103

104

105

106

100

150

200

Frequency

S t o r a g e m o

d u

l u s

( d B )

100

101

102

103

104

105

1060

5

L o s s

f a c t o r

Figure 1.5: E amplitude () and loss factor (- - -) of ISD112 (+) overlaid with a 3parameter model

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1.2.3 High order rational models

The rst generalization is the introduction of higher order models in the form of

rationnal fractions. All classical representations of rational fractions have been con-sidered in the litterature. Thus, fractions of high order polynomials, decompositionsin products of rst order polynomials giving poles and zeros, sums of low orderfractions

E (s) = E 0

1 +n

i=1 i (s)i

1 +n

i=1 i (s)i

= E

n

i=1(s zi )

n

i=1(s i )

= E 0 +

n

i=1

E i ss + E i /C i (1.13)

are just a few of various equivalent representations.

(a)

(b)

Figure 1.6: Examples of generalized damping models (a) Kelvins chain, (b) gener-alized Maxwell model

Figure 1.6 shows two classical representations. Maxwells generalized model intro-duces a pole ( j = E j /C j , associated to settling time j = C j /E j ) and high fre-quency stiff ness E j for each spring/dashpot branch.

The physical meaning of each representation is identical. Hower, each representationmay have advantages when posing equations for implementation in a numericalsolver. For example, section 2.2.3 will models using particular representations of themodulus of order 2 (so called GHM models for Golla, Hughes, McTavish [4, 5]) andorder 1 (so called ADF models for Anelastic Displacement Field [ 6, 7, 8]). Thissection will also show how these models correspond to the formalism of materials

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Rational fractions have a number of well known properties. In particular, the modu-lus slope at a given frequency s = i is directly caracterized by the position of polesand zeros in the complex plane (see Bode diagram building rules in any course in

controls [9]). The modulus slopes found in true materials are generally such that itis necessary to introduce many poles to accurately approximate experimental mea-surements. This motivated the use of fractional derivatives described in the nextsection.

1.2.4 Fractional derivative models

The use of non integer fractions of s allows a frequency domain representation withan arbitrary slope. A four parameter model is thus proposed in [ 10]

E (s) = E max + E min E max

1 + ( s/ ) (1.14)

where the high E max and low E min frequency moduli are readily determined andthe and coeffi cients are used to match the frequency of the maximum loss factorand its value. Properties of this model are detailed in [ 2].

One can of course use higher order fractional derivative models. To formulateconstant matrix models one is however bound to use rational exponents (voir sec-tion 2.2.4).

The main interest of fractional derivative models is to allow fairly good approxi-mations of realistic material behavior with a low number of parameters. The maindiffi culty is that their time representation involves a convolution product. One will

nd in Ref. [11] a recent analysis of the thermomechanical properties of fractionalderivative models as well as a fairly complete bibliography.

1.3 Environmental factors

For usual damping materials, the complex modulus depends not only of frequencybut also of other environmental factors such as temperature, prestress, etc. Thefollowing sections discuss these factors and the representation in contitutive laws.

1.3.1 Inuence of temperature

Temperature is the environmental factor that has the most inuence on viscoelasticmaterial characteristics. [2]. At various temperatures, these materials typically

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have four diff erent regions shown in gure 1.7: glassy, transition, rubberlike anduid. Depending on the considered material, the operating temperature can be inany of the four regions. For polymer blends, each polymer can be in a di ff erent

region.

Temprature

M o

d u

l e d e s

t o c

k a g e

E e

t f a

c t e u r

d e p e r t e !

(a) (b) (c) (d)

E

!

Figure 1.7: Evolution of complex modulus with temperature at a xed frequency.Regions : (a) vitrous, (b) transition, (c) rubberlike, (d) uid.

In the rst region, associated with low temperatures, the material in its vitrousstate is characterized by a storage modulus that reaches its maximum value and haslow variations with temperature. The loss factor is very small and diminishes withtemperature. Material deformations are then small.

The transition region is characterized by a modulus decreasing with temperature anda loss factor peaking in the midle of the region. Typically the maximum correspondsto the point of maximum slope for the storage modulus. The associated temperature

is called transition temperature. Note that this can be confusingsince the transitiontemperature depends on the frequency used to generate gure 1.7.

In the rubberlike region storage modulus and loss factor are both characterized byrelatively small values and low temperature dependence. The fourth region corre-sponds to a uid state. This region is rarely considered because of its inherentinstability.

For damping applications, one typically uses viscoelastic materials in the transition

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region. This choice is motivated by the fact that the loss factors presents a maximumin this area, thus allowing an e ffi cient use of the material damping properties.

100 10

2 104 10

6

020

4060

1

2

3

x 107

Frequency HzTemperature C

R e a l ( E )

100 10

2 104 10

6

020

4060

0

0.5

1

Frequency HzTemperature C

!

Figure 1.8: Variations of ISD 112 modulus in the frequency / temperature domain

In the frequency / temperature domain, gure 1.8 illustrates the existing inverserelation of the e ff ects of temperature and frequency. Experimentally, one often ndsthat by shifting isotherm curves along the frequency axis by a given factor T , oneoften has good supperposition.

xxx gure Gerald xxx.

This property motivates the introduction of the reduced frequency (T ) and adescription of the complex modulus under the form

E (, T ) = E ( (T )) (1.15)

The validity of this representation is called the frequency temperature superposition principle [2, 12] and the curve E is called a master curve.

Various authors have given thermodynamic justications to the frequency temper-ature superposition principle . These are limited to unique polymers. For polymerblends, which have signicant advantages, it is not justied.

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O

102

100

102

104

106

108

106

107

108

109

10 10

Frquence rduite !" T

M o

d u

l e d e s t o c k a g e

E ( P a

) [ f a c

t e u r

d e p e r t e #

]

[102 ]

[101 ]

[10 0]

E

#

T2

T1

T0

T1

T2

Px

1O

2O

3

102

100

102

104

106

108

101

100

101

102

10 3

F r q u e n c e

( H z

)

O3

Figure 1.9: Reduced frequency nomogram

The superposition principle is used to build a standard representation called nomo-gram which simplies the analysis of properties as a function of temperature T andfrequency . The product T corresponds to an addition on a logarithmic scale.One thus denes true frequencies on the right vertical axis, and isotherm lines allow-ing to read the reduced frequency graphically on the horizontal axis. For a frequency j and a temperature T k , one reads the nomogram in three step shown in gure 1.9

(1) one seeks the intersection P of the horizontal j line and the of sloped T kline,

(2) the abscissa of point P gives the reduced frequency j T (T k)), (3) the intersection of the vertical line at that reduced frequency with the

storage modulus E and loss factor master curves gives their respective valuesat j , T k .

Various parametric expression have been proposed to model the temperature shift

factor T . The empirical equation of Williams-Landel-Ferry [ 13], called WLF equa-16


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ence between T and a given reference temperature T 0). One clearly sees that thesecurves mostly di ff er in their low temperature behavior. In practical applications, onecan usually adjust parameters of any law to be appropriate. The solution preferred

here is to simply interpolate between points of an T table.

1.3.2 Other environment factors

Between the other environmental factors inuencing the behavior, one essentiallydistinguishes non linear e ff ects (static and dynamic) and history e ff ects (expositionto oil, high temperatures, vacuum, ...).

Non linear dynamic e ff ects are very hard to characterize since high amplitude varia-tions of the induced strain are typically correlated with signicant energy dissipationand thus temperature changes. The e ff ects of level and temperature changes are thencoupled for materials of interest which are typically in thei transition region. Ex-perimentally, such non linear studies are thus limited to the rubber like region. Theeff ects are similar to those of temperature although of smaller magnitude.

Non linear static e ff ects, that is e ff ects linked to a static prestress assumed to beconstant, are signicant and easier to characterize. It is known to be essentialwhen considering machinery suspensions or constrained viscoelatic sandwiches wherepress-forming induced signicant pre-stress [14].

History e ff ects are generally associated with extreme sollicitations that one seeks toavoid but whos probability of occurrence is non zero.

As for temperature, one generally represent the e ff ect of other environmental fac-tors as shift factors [2], although the supperposition hypothesis may not be as wellveried [14].

1.4 Determining the complex modulus

A number of methods and a few standards [ 15], exist for the experimental determi-nation of the complex modulus. Only the main test categories will be listed here

traction/compression tests under sinusoidal excitation are used to measure theproperties of materials that are su ffi ciently sti ff to allow testing without com-bination of the material sample with metallic components. With signicantexperimental precautions, this technique has also been applied to lms. De-pending on the experimental setup, one will determine the complex modulus

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directly on isofrequencies or isotherms.

Oberst [ 16] and sandwich beams are used to determine the properties of aviscoelastic layer glued onto a metalic beam. Sti ff materials work in tractioncompression in a free layer, while soft ones work in shear in a metal/visco/metalsandwich. Vibration analysis of the beam gives resonance frequencies and asso-ciated damping ratio. By inverse analysis of analytical [17] or numerical solu-tions, one determinse the complex modulus at the resonances under the modaldamping assumption (see section 2.1.3). Using changes in the beam/plate di-menstions and temperatures, one can obtain a large number of points on thenomogram.

shear tests allow the direct determination of the complex modulus of lmsbetween two rigid surfaces. This test is more representative of material sollic-itation for sandwich structures. It is the only one applicable to test the e ff ectof pre-stress in sandwich structures.

Building a test rig with no perturbing modes being quite di ffi cult, modulus carac-terization is always performed on fairly narrow frequency bands. The frequency-temperature supperposition hypothesis is thus made to create master curves over awide frequency band. The next possible step is the determination of the coe ffi cientsof an analytical representation. Identication tools developped in control theory(ARMA models, ...) are suited for rational fraction models. It is however di ffi cult

to enforce a good reproduction of quantities that are typically judged as important :low and high frequency moduli, maximum loss factor, ... In other case a non linearoptimization is readily implemented using optimization tools available in MATLAB.

1.5 Conclusion

For a given complex modulus, it is diffi

cult to validate the fact that general ther-modynamic principles are veried. Ref. [11] gives an analysis based on the factthat each component of a generalized maxwell model must verify the second prin-ciple of thermodynamics and thus be associated with positive stifness and dampingcoeffi cients.

In practice, the criteria that are used to judge the quality of a constitutive model arethe accurate reproduction of experimental complex modulus measurements within

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the tested frequency/temperature range, and the likelihood of extrapoations. Fig-ure 1.11 illustrates the fact that with rational fractions one can fairly easily satisfythe rst objective (reproduce the modulus in a narrow band in bold), but extrapo-

lations can be di ffi cult (thin lines are here of poor quality).

100

101

102

103

104

105

10 7

108

109

Frequence (Hz)

M o

d u

l e d e

S t o c

k a g e

( P a

)

100

101

102

103

104

105

0

0.5

1

1.5

F a c

t e u r

d e p e r t e

Figure 1.11: Complex modulus of the TA viscoelastic () and estimation with a 3pole model (- - -) whose frequencies are indicated as vertical lines

On the other hand, one should note that the determination of the complex modulusis often diffi cult and that the frequency / temperature supperposition principle isan hypothesis. It just happens to be reasonably valid in many cases.

In conclusion, for the material stand-point, accurate representations are either tab-ulated (our point of view is that this is actually the most practical representation),

rational fractions with a suffi

cient degree or fractional derivative models. The fol-lowing chapters will address di ffi culties in using these representations for modelingdamped structures.

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2

Viscoelastic FEM models

2.1 Viscous and structural damping . . . . . . . . . . . 222.1.1 Properties of the damped 1 DOF oscillator . . . . 232.1.2 Real modes and modal damping . . . . . . . . . . 252.1.3 Selection of modal damping coe ffi cients . . . . . . 27

2.2 Viscoelastic models . . . . . . . . . . . . . . . . . . 292.2.1 Frequency domain representation with variable co-

effi cients . . . . . . . . . . . . . . . . . . . . . . . 29

2.2.2 State-space representations . . . . . . . . . . . . . 302.2.3 Second order models with internal states . . . . . . 312.2.4 Fractional derivatives . . . . . . . . . . . . . . . . 33

2.3 Spectral decomposition and reduced models . . . 332.3.1 Complex modes of analytical models . . . . . . . . 342.3.2 Complex mode eigenvalue problems with constant

matrices . . . . . . . . . . . . . . . . . . . . . . . . 352.3.3 Model reduction methods . . . . . . . . . . . . . . 372.3.4 Equivalent viscous damping . . . . . . . . . . . . 382.3.5 Case of viscoelastic models . . . . . . . . . . . . . 40

2.4 Meshing of sandwich models . . . . . . . . . . . . . 402.4.1 Mesh convergence and non conformity . . . . . . . 44

2 Viscoelastic FEM models


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The design of viscoelastic treatments is typically composed of a series of steps whichare outlined in this section

denition of dynamic objectives

potential area localization by combination of technological constraints on place-ment and sensitivity analysis on treatment potential

This chapter analyzes properties of models used to represent damped structures. Inthe case of linear viscoelasticity these models have the general form

[Z (s, T )] {q (s)} = [b]N NA {u(s)}NA 1{y(s)}NS 1 = [c]NS N {q (s)}N 1

(2.1)

where physical loads [b]{u(s)} are decomposed into vectors b, characterizing spatiallocalization, and u(s), characterizing time/frequency responses and vector {y} rep-resents outputs (physical quantities to be predicted, assumed to depend linearly inDOFs q .

Section 2.1 details the case of models with viscous and structural damping. Althoughone saw in the last chapter that these models did not allow a correct representationof material behavior over a wide frequency band, they are more easily accessible andcan be used to illustrate the di ff erence in needs when modeling vibration dampingin a structure and dissipation in a material.

Section 2.2 details strategies that can be used to create models with realistic repre-

sentation of viscoelastic materials.Section 2.3 deals with issues of spectral decompositions and model reduction whichare central to the analysis of the vibratory behavior of damped structures.

The objective of this chapter is to introduce equations used to solve viscoelasticproblems. The next chapter will focus on numerical techniques used for the resolu-tion.

2.1 Viscous and structural damping

In this section one analyses the properties of classical models with viscous andstructural damping of the form

22


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Ms 2 + Cs + K + iD N N {q (s)}N 1 = [Z (s)] {q (s)} = [b]N NA {u(s)}NA 1{y(s)}NS 1 = [c]NS N {q (s)}N 1

(2.2)where matrices are assumed to be constant.

This type of models does not allow a correct representation of the local behavior of damping treatments (constitutive laws detailed in the preceding chapter). At thelevel of a complete structure, it is however often possible to represent the e ff ect of various damping mechanisms by a viscous or structural model. One then uses aglobal behavior model. It does not necessarily have a local mechanical meaning, but

this does not lower its usefulness.

The main results introduced in this section are

properties of the damped oscillator;

the conditions of validity of the modal damping assumption which leads tomodels where the response is decomposed in a sum of independent oscillators;

techniques used to estimate equivalent viscous damping models.

2.1.1 Properties of the damped 1 DOF oscillator

This section illustrates the properties of the single degree of freedom oscillator witha viscoelastic sti ff ness shown in gure 2.1.

K(s)

Figure 2.1: Oscillator with a viscoelastic sti ff ness

For a viscous damping K (s) = k + cs, the load to displacement transfer is given by

H visc (s) = 1

ms 2 + cs + k =

1/m(s

)(s

)

(2.3)

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whose poles (root of the transfer denominator) are = n id , d = n 1 2

n = k/m , = c

2 km (2.4)For structural damping K (s) = k(1 + i), the load to displacement transfer is givenby

H (s) = 1

ms 2 + k(1 + i) = H visc

ik cs(ms 2 + cs + k)(ms 2 + k(1 + i))

(2.5)

its pole with a positive imaginary part is identical to that of the viscous model for = 2 (2.6)which leads to a di ff erence H H visc that is zero at resonance n . The pole with anegative imaginary part is unstable (positive real part) which is a classical limitationof the structural damping model.

For a damper following the 3 parameter law of a standard viscoelastic solid ( 1.7),the system has a pair of complex poles , and one real pole

H (s) = 1

ms 2 + k 1+ s/z1+ s/p=

1 + s 1+2 / m(s2 + 2 s + 2)(1 + s )

(2.7)

and the model caracteristics depend on those poles as follows

p =

1 + 2 z =

1 + 2 k = m

2

1 + 2 (2.8)

For low damping of the conjugate pair of poles (that is

1) and and in

the same frequency range, p and z are close which leads to a small maximum lossfactor and a response that is very similar to that of the oscillator with viscousdamping ( 2.3).

Figure 2.2 shows that for viscous, structural and viscoelastic dynamic sti ff ness mod-els for the oscillator (c), the dynamic exibilities (a-b) are almost exactly overlaid.This is linked to the fact that the real parts of the dynamic sti ff ness coincide natu-

rally since they are they not (viscous or structural) or little (viscoelastic) inuencedby the damping model and the imaginary parts (d) are equal at resonance.

This equivalence principle is the basis for the Modal Strain Energy (MSE) methodthat will be detailed in section 2.1.3.

24


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!"!

!"#

!"$

!"%

&'()*(+,-

. / 0 1

2 3 1 5 6 7

3

.

! # ! ! " ! #8 !"

! %9(.5123

: ; . 7 1 2

3

/

K(s)

,

!"#

!"$

&'()*(+,-

: ; . 7

1 ! < =

2 3

>

Figure 2.2: Low sensitivity of the dynamic exibility to the damping model ( viscous, o structural, + standard viscoelastic)

2.1.2 Real modes and modal damping

For an elastic model, normal modes are solution of the eigenvalue problem

[M ]{ j } 2 j + [K ]N N { j }N 1 = {0}N 1 (2.9)associated with elastic properties (sometimes called the underlying conservativeproblem). They verify two orthogonality conditions in mass

{ j }T [M ] [k] = \ j jk \ (2.10)and sti ff ness

{ j }T

[M ] [k] =\ j

2 j jk \ (2.11)

There are di ff erent standard scaling for normal modes, and one will assumed thatthey are scale so as to obtain j = 1 which greatly simplies equation writing. Theother standard scaling, often used in experimental modal analysis, sets one DOF(node, direction) of j to unity, j is then called the generalized mass at this DOF.

The basis of normal modes is classically used to build reduced model by congruenttransformation ( 2.54) with {q } = [T ]{q r } = [1...NM ]{q r }. In the resulting coor-

25



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dinates, called principal coordinates, the mass and sti ff ness matrices are diagonal((2.10)-(2.11) conditions). But this is not the case for the viscous and hystereticdamping matrices T T CT and T T DT .

The modal damping assumption (also called Basiles hypothesis in French ter-minology) consists in an approximation of the response where the o ff -diagonal termsof the damping matrices in principal coordinates are neglected. In practice, oneeven further restricts the model to viscous damping in principal coordinates sincethe result can then be integrated in the time domain. On thus has

s2 [I ] + s \ 2 j j \ +\ 2 j \ { p} (s) = [ ]

T {F (s)} + {F d} (2.12)where the approximation is linked to the fact of neglecting coupling terms describedby

{F d} = s \ 2 j j \ T [Cs + D] [] { p} (s) (2.13)

Damping is exactly modal ( F d = 0) if the matrices C and/or D are linear combina-tions of products of M and K [18]

[C ] = kl [M ]k [K ]l (2.14)

Rayleigh damping [19], typically called proportional damping , where[C ] = [M ] + [K ] , (2.15)

is thus often used. This is a behavior model, that can be easily adjust to predictcorrect damping levels for two modes since

[ j ] = 12 j+ j

2 (2.16)

Over a wide frequency band, it is however very unrealistic since 12j gives highdamping ratio for low frequencies j and

j2 for high frequencies. Rayleigh damping

is thus really inappropriate to accurate damping studies.

When one starts from a local description of dissipation in the materials or interfaces,modal damping is an approximation whose validity needs to be understood. To do so,one should analyze whether the coupling force ( 2.13) can be neglected. To validatethe domain of validity of this hypothesis, one considers a two DOF sytem [20, 21]

1 00 1 s

2 + 11 12 21 22s +

21 0

0 22 p1 p2

= b1b2{u(s)} (2.17)

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from which one determines an expression of the response of mode 1 given by

p1 = (1 + e1) 1 b1u

s2 + 11s + 21 + e2 (2.18)

withe1 =

12 21s2

(s2 + 11s + 21)(s2 + 22s + 22) (2.19)

and

e2 = 12sb2u

(s2 + 11s + 21)(s2 + 22s + 22) (2.20)

The matrix being positive denite, one has 12 21 / ( 11 22) < 1. Thus, the e1term can be close to 1 and coupling be signicant, if and only if both factors in thedenominator are small simultaneously, that is

min( 11, 22)/ |1 2 | 1 (2.21)Similarly, the e2 ter, is only signicant in special cases of closely spaced modes orsuch that b1 b2.

In practice, the validity of the modal damping assumption is thus linked to thefrequency separation criterion ( 2.21) which is more easily veried for small dampinglevels.

In cases where modal damping is not a good approximation (that is criterion ( 2.21)is not veried), a generalization of the modal damping model is the use of a viscousdamping matrix that is non diagonal in principal coordinates. One the generallytalks of non proportional damping since condition ( 2.14) is not veried. The use of non proportionally damped models will be discussed in section 2.3.4.

For groups of modes that do no verify condition ( 2.21), one can still use an as-sumption of modal damping by block [22] where the off diagonal terms of j C kare considered for j and k in the same group of modes. Between modes of di ff er-ent groups, the reasoning developed above remains valid and the error induced byneglecting damping coupling terms is small.

2.1.3 Selection of modal damping coe ffi cients

Modal Strain Energy method

The Modal Strain Energy method (MSE [23]) is a classical approximation base onthe choice of an equivalent viscous damping coe ffi cient chosen by evaluating the lossfactor for a cycle of forced response along a particular real mode shape. As we saw

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Experimental and design damping ratio

The other classical approach is to use modal damping ratio determined experimen-

tally. Identication techniques of experimental modal analysis [25, 26] give methodsto determine these ratios.

For correlated modes (when a one to one match between test and analysis is estab-lished), one thus uses a damping ratio jT est while typically preserving the analysisfrequency.

For uncorrelated modes, one uses values determined as design criteria. = 10 3 for apure metallic component, 10 2 for an assembled metallic structure, a few percent inthe medium frequency range or a civil engineering structure. Each industry typicallyhas rules for how to set these values (for example [27] for nuclear plants).

2.2 Viscoelastic models

This section details models of structures used to account more precisely for theconstitutive behavior of various viscoelastic materials.

For frequency response computations, section 2.2.1 shows how the complex dynamicstiff ness is built as a weighted sum of constant matrices associated with the variousmaterials.

For eigenvalue computations or time responses on the full model, the introductionof state space models (section 2.2.2) or second order models with internal states

(section 2.2.3) allow constant matrix computations. Such formulations could beused for frequency domain solutions but they are higher order and the increasein DOF count limits their usefulness. For the case of fractional derivative models(section 2.2.4), only modeshape computations are accessible.

2.2.1 Frequency domain representation with variable coe ffi cients

For a structure composed of elastic and viscoelastic materials frequency domaincomputations only require the knowledge of the complex modulus E i (s,T, 0) foreach material. By using the fact that sti ff ness matrices depend linearly on theconstitutive law coe ffi cients, one can represent the dynamic sti ff ness of a viscoelasticmodel as a linear combination of constant matrices (for independent complex moduliin the same material their may be more than one matrix associated to a givenmaterial)

29



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[Z (E i , s )] = Ms 2 + K e + iK ei + sC +i

E i (s,T, 0)K vi (E 0)

E 0 (2.28)

This representation is the basis for the development of solvers adapted for structureswith viscoelastic materials.

For frequency domain computations, it is rather ine ffi cient to reassemble Z at eachoperating point E i , s . Two solutions can be implemented easily. On can store thevarious M, K e, K ei ,C ,K vi matrices and evaluate the weighted sum ( 2.28) at eachoperating point, or store element matrices and reassemble with a weighing coe ffi cientassociated with the material property of each element.

2.2.2 State-space representations

One discusses here state space representations associated with analytical represen-tations of the complex modulus discussed in section 1.2.3.

State space models are rst order di ff erential equations, assumed here with constantcoeffi cients, with the standard form

{x (t)} = [A]{x (t)} + [B ]{u (t)}{y (t)} = [C ]{x (t)} + [D]{u (t)} (2.29)

The matrices are called : A transfer, B input, C observation and D direct feed-trough. The rst equation is the evolution equation while the second is called theobservation equation.

The usual technique for time integration of mechanical models is to dene a statevector combining displacements and velocities, leading to a model of the form

q q =

0 I

M 1K M

1C q q +

0M 1b {u(t)}

{y(t)} = [c 0] q

q

(2.30)

This model is rarely used as such because M is often singular or, for a consistentmass, leads to matrices M 1K and M 1C that are full. When using generalizedcoordinates, one can however impose a unit mass matrix and this easily use themodel form. Some authors [ 28] also prefer this form to dene eigenvalue problemsbut never build the matrices explicitely so that the full matrices are not built.

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which can be easily integrated (this approach is used in ABAQUS [31] for example).This formalism corresponds to the separate treatment of bloc rows in ( 2.33) or (2.34)which is simple in the time domain but leads to a non-linear problem in the frequency

domain, unless operators are dened implicitely as in Ref [ 28].

2.2.4 Fractional derivatives

The internal state formalism, can also be used to represent fractional derivativeconstitutive laws if one uses non integer but rational derivatives. For a commondenominator p, one will use a modulus of the form

E (s) = E max p

k=1

E ksk/p + k

(2.38)

and build a state space model of the form{x(s)} s1/p = [A]{x(s)} + [B ]{u(s)} (2.39)

where the state vector will combine fractional derivatives of the displacement sk/p q where k = 1 : 2 p 1 and of the internal state q vk = E k(sk/p + k ) q (see [32] forexample).

In practice, the number of blocs in the state vector being proportional to p, constantmatrix representations are thus limited to small values of p. This limits practicaluses of fractional derivative models to frequency domain response and non-lineareigenvalue computations (see section ?? ).

2.3 Spectral decomposition and reduced models

For an input [ b]{u(s)} characterized by the frequency domain characteristics of u(s)and spatial content of b, component mode synthesis and substructuring methodsprovide approximations of the solution of problems ( 2.2), (2.33), or (2.34).

For damped problems, one should distinguish

exact spectral decompositions using complex modes as treated in ections 2.3.1and 2.3.2;

model reduction methods which only seek to approximate the transfer spec-trum by projecting the model on bases built using solutions of problems thatare simpler to solve than the complex eigenvalue problem. These approachesare detailed in sections 2.3.3, 2.3.4 and 2.3.5.

33


l d f l l d l


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2.3.1 Complex modes of analytical models

All the problems that where introduced earlier can, in the frequency domain, berepresented as frequency response computations of the form

{y(s)} = [H (s)] {u(s)} = [c]{q (s)} = [c] [Z (s)] 1 [b]{u(s)} (2.40)which involve the inverse of the dynamic sti ff ness Z (s).

In the very general case where complex moduli are supposed to be analytic functionsin the complex plane (locally regular), the dynamic sti ff ness Z (s) is also an analyticfunction. Observation and input matrices c and b being constant, the poles (i.e.singularities) of H (s) correspond to non zero solutions of

[Z ( j )] { jD } = {0} and { jG }T

[Z ( j )] = {0} (2.41)which denes the generalized non linear eigenvalue problem associated with a visce-lastic model.

Near a given pole, analytic functions have a unique Laurents development

f (z) =+

k=

ak(s )k

avec ak = 12 i f (s)(s )k+1 ds (2.42)

where is a abitrary closed direct contour of the singularity .As a result, near an isolated pole j one has

[Z (s)] 1 = { jD } { jG }T

j (s j ) + O(1) (2.43)

where the normalization coe ffi cient j depends on the choice of a norm when solv-ing (2.41) and is determined by

j = { jG }T [Z (s)]

s{ jD } (2.44)

To simplify writting, it is desirable to use j = 1 which si the usual scaling forconstant matrix eigenvalue problems described in the next section.

Having determined the set of poles in a given frequency band, having normalizedthe associated modes so that j = 1, one obtains a rst order development in s

H (s) = [c] [Z (s)] 1 [b] = j

s j

{c jD } T jG bs j

+ [c] [Z (0)] 1 [b] (2.45)

where the [c] [Z (0)] 1 [b] terms correspond to the exact static response to loadsassociated with the input shape matrix b. This static correction term is well knownin component mode synthesis applications and is analyzed in section 2.3.3.

34

I i f f l id i i id l ibili


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It is often useful to consider a representation using residual exibility

H (s) = [ c] [Z (s)] 1 [b] =2NM

j =1

{c jD } T jG bs j

+ [c] [Z (0)] 1 [b]2NM

j =1

c jD T jG b

j(2.46)

2.3.2 Complex mode eigenvalue problems with constant matrices

The solution of the non linear eigenvalue problem ( 2.41) is diffi cult (see section ?? ).Solution algorithms are thus greatly simplied by restating the problem as a classicalrst order eigenvalue problem with constant matrices.

For models with viscous and structural damping ( 2.2) or viscoelastic models of form (2.34), on thus generally solves the eigenvalue problem associated with ( 2.31),that is

C M M 0 j +

K 00 M

{ j } = {0} (2.47)

Because of the block form of this problem, one can show that

{ j } = j j j (2.48)

and one thus gives the name complex mode both to j and j .

The existence of 2N eigenvectors that diagonalize the matrices of ( 2.47) is equivalentto the verication of two orthonormality conditions

[]T C M M 0 [] = T C + T M + T M = \ I \ 2N

[]T K 00 M [] = T K T M = \ \ 2N

(2.49)

For a model represented in is state space form, such as ( 2.33), one solves the leftand right eigenvalue problems

([E ] j + [ A]) { jD } = {0} et { jG }T ([E ] j + [A]) = {0} (2.50)and uses standard orthonormality conditions

{ jG } [E ]{kD } = jk et { jG } [A]{kD } = j jk (2.51)In the complex plane, one should distinguish complex poles associated with vibrationmodes and real poles which can be related with a damping ratio above 1 (super-critical damping that is not common) or to material relaxation (be linked to polesof the complex modulus). To understand this distinction one considers the vis-coelastic oscillator ( 2.8). The viscoleastic behavior leads to one real pole , whereas

35


supercritical damping corresponds to > 1 and leads to two real poles


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supercritical damping corresponds to > 1 and leads to two real poles = 2 1 (2.52)

In a viscoelastic computation where the constitutive law contains real poles the rootlocus of the solution is similar to that shown in gure 2.3. One must thus distinguishthe classical spectrum of vibration modes and the real poles associated with materialrelaxation

2 1.5 1 0.5 0 0.5 1 1.5 2

x 104

10000

8000 6000

4000

2000

0

Im( ! )

R e ( ! )

Relaxation

Vibration

Figure 2.3: Poles of a viscoelastic beam in traction

It is important to note that the number of real poles is directly associated withthe number of DOFs in the internal states q v . The number of these poles will theincrease with mesh renement. In practice, one thus cannot expect to compute thereal poles and associated modes in the area of relaxation poles. This selection of convergence area is the an aspect that needs to be accounted for in the developmentof partial eigenvalue solvers for damped problems.

A second consequence of the increase in the number of real poles is the potentialimpossibility to compute modes with supercritical damping. The authors experienceis that this limitation is mostly theoretical since in practice modes with supercriticaldamping are rare.

The complex modulus, being the Fourier transform of a real valued relaxation func-tion, should be symmetric in frequency ( E () = E ()). For a modulus represen-tation that does not verify this hypothesis, as is the case of structural damping, onlypoles with positive imaginary parts have a meaning. For response synthesis, one willthus take the conjugates j , j of modes computed with positive imaginary parts.

The computation of complex modes can be used to approach transfer functionsusing the developments ( 2.43) and ( 2.45). Orthogonality conditions given abovecorrespond to the j = 1 normalization.

36

2 3 3 Model reduction methods


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2.3.3 Model reduction methods

To simulate the dynamic response it is not useful and rarely possible from a nu-merical cost standpoint to use the full model ( 2.2) for direct time simulations (seesection ?? ). Model reduction methods (modal analysis, substructuring, componentmode synthesis, ...) seek an approximate solution within a restricted subspace. Onethus assumes

{q }N 1 = [T ]N NR {q R }NR 1 (2.53)and seek solution of ( 2.1) whos projection on the one the dual subspace T T is zero(this congruent transformation corresponds to a Ritz-Galerkin analysis). Transferfunctions are the approximated by

[H (s)] = [c] (Z (s)) 1 [b] [cT ] T T Z (s)T 1

T T b (2.54)One can note that for a non-singular transformation T (when {q } = [T ]{q R } isbijective) the input u / output y relation is preserved. One says that the transferfunctions are objective quantities (they are physical quantities that are uniquelydened) while DOFs q are generally not objective.

Classical bases used for model reduction combine modes and static responses tocharacteristic loads [ 33]. On distinguishes

bases containing free modes and static responses to applied loads {b}

[T ] = [1 . . . NM ] [K ]

1 [b]NR

j =1

[c]{ j

} { j

}T [b]2 j

(2.55)

For component mode synthesis (component model reduction to prior to a cou-pled system prediction), free modes have been used by [ 34], MacNeal [35], andmany others.

bases containing static displacements associated with displacement enforced

on an interface and xed interface modes[T ] = 01:NM,c I K 1cc K ci (2.56)

For CMS, the use of static terms only is called Guyan condensation [ 36].Adding xed interface modes leads to the Craig Bampton method [ 37].

37


damped modes can be considered as elastic models with an external damping


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damped modes can be considered as elastic models with an external dampingload. Static correction for the e ff ects of damping loads can then be incorpo-rated. The rst order correction proposed in [38]

[T ] = [T 0] [K ] 10 [K vi [1:NM ]] (2.57)

2.3.4 Equivalent viscous damping

Section 2.1.2 showed that the modal damping assumption was justied under thehypothesis of modal separation ( 2.21). When dealing with a real basis, it is of-ten possible to compute an equivalent viscous damping model approximating thedamped response with good accuracy.

To build this equivalence, one distinguishes in reduction bases ( 2.55), ( 2.56), or other,blocs T m associated with modes whose resonance is within the frequency band of interest and T r associated with residual exibility terms. As shown in gure 2.4,one is interested in the modal contribution of the rst while for the others only the

asymptotic contribution counts.

Figure 2.4: Transfer function decomposition into modal contributions and residualterms

For a real basis reduction, one is thus interested in an approximation of the form

[H (s)] [cT ] T T m Z (s)T m T T m Z (s)T rT T r Z (s)T m T T r Z (s)T r

1

T T b (2.58)

The objective in distinguishing T m and T r is to guarantee that T T r Z (s)T r does notpresent singularities within the band of interest (it represents residual exibility notresonances).

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z

Volume

Shell

Shell

Rigid links

Sandwich shell FEM Model

Figure 2.5: Shell/volume/shell model forsandwiches

! " #

! $ #

! % #

! "&

!$

&

! %&

Figure 2.6: Problems with thickness def-initions in shells with signicant curva-ture

map used as a meshing support.

For sti ff layers, shells are preferred over volumes, because volume element formula-tions are sensitive to shear locking when considering high aspect ratio (dimensionsof the element large compared to thickness).

For soft layers, the use of a volume element both necessary, because shell elementswill typically not correctly represent high shear through the thickness, and accept-able, because almost all their energy is associated with shear so that they will notlock in bending [38]. Note that shear corrections used in some FEM codes to allowbending representation with volumes may have to be turned o ff to obtain appro-priate results. Finally there are doubts on how to properly model the through thelayer compression sti ff ness of a very thin viscoelastic layer (this can have signicanteff ects on curved layers).

The demo basic sandwich generates curves for the validation of shell/volume/shellmodel used to represent constrained layer treatements as rst discussed in [38]. Theidea is to vary the properties of a central volume layer between a very soft modulusand the skin modulus.

For a very soft value, gure 2.7 shows convergence to the assymptotic value of asingle skin plate. For a modulus of the viscoelastic core equal to that of the skins,

one should converge to the frequency of a plate model with thickness equal to thesum of skins + viscoelastic core. Figure 2.7 shows that the high modulus assymptoteis slightly higher for the shell / volume / shell model. This is due to shear lockingin the very thin volume layer (well known and documented problem that low ordervolumes cannot represent bending properly). This di ffi culty can be limited by usingvolume element with shear locking protection. The gure also shows that dampingis optimal sowhere between the low and high modulus values.

41


2.5


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0 50 100 150 200 250 3001

1.5

2

x

x

Frequency [Hz]

D a m

p i n g

[ % ]

10 5 10 6 10 7 10 8 10 9 10 1020

25

30

35

40

45

50

G (Mpa)

F r e q

( H z )

Figure 2.7: Constrained layer model validity.

Figure 3.2 illustrates the validity of a shell/volume model as compared to a singleshell based on composite shell theory. Figure ?? illustrates that the results are nearlyidentical, provided that volume elements with proper shear locking protection areused. For a stanard isoparametric volume, a shell/volume model tends to be be to

stiff

(shear locking associated with bending).

42

4

4.5


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0 20 40 60 80 100 120 140 160 180 2001

1.5

22.5

3

3.5

4

Frequency [Hz]

D a m

p i n g

[ % ]

10 4 10 5 10 6 10 7 10 8 10 9 10 1010

15

20

25

30

G (Mpa)

F r e q

( H z

)

Shell+volumeLaminated plate

Figure 2.8: Free layer model validity.

The element degree does not seem critical to obtain accurate predictions of theresponse. The use of multiple element through the viscoelastic layers has also beenconsidered by some authors but the motivation for doing so is not understood.

For press formed sandwiches, there are further unknowns in how the forming processaff ects the core thickness and material properties. In particular, most materials usedfor their high damping properties are also very sensitive to static pre-stress. For asimple folded plate, gure 2.9 illustrates how the modal frequencies and energydistribution in the viscoelastic layer are modied if the shear modulus is multipliedby 10 in the fold. Such behavior was found in tests and motivated the study inRef. [14], where the eff ect of static pre-stress is measured experimentally. Overall,predicting the e ff ects press forming or folding sandwiches is still a very open issue.

43



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Figure 2.9: Energy density in the viscoelastic layer of a simple folded sandwich plate.(Top) High sti ff ness viscoelastic in the fold. (Bottom) equal sti ff ness in the fold andelsewhere.

A nal diffi culty is to deal properly with boundary conditions of the skin layers.Since diff erential motion of the skins plays a major role in the e ff ectiveness of thecore, the boundary conditions of each layer has to be considered separately. This

is easily illustrated by the generation of cuts in constraining layers [ 42, 41] (andcut optim demo).

2.4.1 Mesh convergence and non conformity

As illustrated in gure ?? the dissipation if often localized on a fairly small sub-partof the structure. It is thus quite important to validate the accuracy of predictions

obtained with various mesh renements. Figure 2.10 illustrates a convergence studywhere a constrained layer damping treatment is rened and one compares the strainenergy density maps for two levels of renement. The strain energy maps, clearlyindicate edge eff ects, which are typical of constrained layer treatments. In suchstudies the authors have usually found, that the distribution of constraints is wellpredicted and energy fractions (strain energy in the viscoelastic compared to totalstrain energy in the model) predicted with the ne and coarse meshes do not show

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signicant di ff erences.


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Figure 2.10: Zoom on the rened mesh of a constrained layer damping treatmentplaced on a volume model. Comparisons of strain energy maps for two levels of renement.

When considering free placement of damping devices (see section ?? ), one is rapidly

faced with the problem of incompatible meshes. For discrete connections, whereloads are transmitted at isolated points with at most one point on a given elementof the supporting structure, the problem is very much related to that of the repre-sentation of weld spots and strategies that use the underlying shape functions aremost eff ective (see SDT feutilb MpcFromMatch command).

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Toolbox tutorial

3.1 Viscoelastic modeling tools . . . . . . . . . . . . . . 493.2 Representing viscoelastic materials . . . . . . . . . 49

3.2.1 Introducing your own nomograms . . . . . . . . . 493.2.2 Selecting a material for your application . . . . . . 50

3.3 Viscoelastic device meshing tools . . . . . . . . . . 513.3.1 Generation of sandwich models . . . . . . . . . . . 513.3.2 Meshing foam llings . . . . . . . . . . . . . . . . . 52

3.3.3 Exporting submeshes to NASTRAN . . . . . . . . 523.4 Data structure reference . . . . . . . . . . . . . . . 53

3.4.1 Dynamic sti ff ness coeffi cients zCoefFcn . . . . . . 533.4.2 Format reference . . . . . . . . . . . . . . . . . . . 563.4.3 Input denitions . . . . . . . . . . . . . . . . . . . 563.4.4 Sensor denitions . . . . . . . . . . . . . . . . . . . 573.4.5 Response post-processing options . . . . . . . . . . 57

3.5 Sample parametric analysis step by step . . . . . 583.5.1 Model parameterization . . . . . . . . . . . . . . . 583.5.2 Parametric model generated within NASTRAN (fo by set

DMAP) . . . . . . . . . . . . . . . . . . . . . . . . 603.5.3 Parametric model from NASTRAN element matrices 633.5.4 Full and reduced order solution within SDT . . . . 63

3.6 Fluid/structure coupling . . . . . . . . . . . . . . . 643.6.1 Summary of theory . . . . . . . . . . . . . . . . . . 643.7 NASTRAN DMAP routines . . . . . . . . . . . . . 673.8 Advanced connection models . . . . . . . . . . . . 68

3.8.1 Screw models . . . . . . . . . . . . . . . . . . . . . 683.8.2 Physical point with rotations . . . . . . . . . . . . 68

3.9 Validation examples . . . . . . . . . . . . . . . . . . 68

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3.9.1 Validity of free and constrained layer models . . . 68


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3.1 Viscoelastic modeling tools


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The viscoelastic modeling tools provide packaged solutions to address the following

problems

Generation of sandwich structures

Handling of tabulated and analytical representations of viscoelastic constitu-tive laws.

Parameterization of FEM models to allow multiple viscoelastic materials.

Approximate solutions for the viscoelastic response

Vibroacoustic response generation

These tools are not distributed with SDT . Please contact SDTools for licensinginformation.

3.2 Representing viscoelastic materials

3.2.1 Introducing your own nomograms

You can introduce your own nomograms in the m visco database. By simply deningan mvisco *.m le (mvisco 3m.m serves as a prototype. The data structure denesa reference elastic material in mat.pl , complex modulus and shift factor tables, annally additional properties stored in mat.nomo (which will be better documentedlater).

mat.pl=[1 fe_mat(m_elastic,SI,1) 1e6*2*1.49 .49 1500 1e6]; mat.name=ISD112 (1993); mat.type=m_visco; mat.unit=SI; mat.T0=[0];

mat.G=[ % Freq, Re(G) Im(G)1 1.72688e+004 3.51806e+003

10 2.33865e+004 5.35067e+003

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100 3.49390e+004 8.25596e+0031000 5.76323e+004 1.67974e+004


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10000 1.03151e+005 5.72383e+0041e+005 2.10295e+005 1.79910e+0051e+006 6.59947e+005 6.57567e+0051e+007 2.06023e+006 1.95406e+0061e+008 5.83327e+006 3.57017e+0061e+009 1.48629e+007 5.60247e+0061e+010 3.25633e+007 7.33290e+0061e+011 6.16925e+007 5.40189e+0061e+012 1.01069e+008 2.48077e+006

];

mat.at=[ % T, at-10 1.32885e+007

0 9.16273e+00510 1.14678e+00520 2.45660e+004

30 9.00720e+00340 2.99114e+00350 1.27940e+00360 7.10070e+00270 2.88513e+00280 1.96644e+00290 1.37261e+002

100 1.03674e+002110 6.84906e+001120 4.66815e+001

]; mat.nomo={w,[-1 0 12],Eeta,[4 9 2],unit,SI, ...

www,www.3m.com, ...file,ISD_112_93.png,Rect,[145 35 538 419], ...type,G};

3.2.2 Selecting a material for your applicationcf=feplot; m_visco(database,cf); % select all materialsfecom(curtabMat);

m_visco(deffreq,cf) % set frequencies vector for all the material

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m_visco(defT,cf) % set temperature vector for all the material


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m_visco(nomo,cf) % list with all nomograms

Freq=stack_get(cf.mdl,info,Freq,getdata);T=stack_get(cf.mdl,info,Range,getdata);Mat=stack_get(cf.mdl,mat);

m_visco(nomo,cf)

3.3 Viscoelastic device meshing tools

3.3.1 Generation of sandwich models

Starting from an undamped structure without treatment, you often want to gen-erate models for viscoelastic patches applied to the structure. This is done usingfevisco MakeSandwich commands. Modeling issues associated with this meshingare discussed in section 2.4.

For example, the generation of a three layer sandwich with the original layer 0.01thick (leading to a 0.005 o ff set), a volume of thickness 0.002, and a second 0.01 thickshell looks like

model=femesh(testquad4 divide 10 12);

model.Elt=feutil(orient 1 n 0 0 1,model);sandCom=[makesandwich shell 0 0 .005 ...

volume 101 .002 shell 102 -.005 .005];treated=withnode{x>.5 & y>.5}; model=fevisco(sandCom,model,treated);cf=feplot;cf.model=model;fecom(colordatamat);

You can also specify normals using a map.

model=femesh(testquad4 divide 3 4); model.Elt=feutil(orient 1 n 0 0 1,model);sandCom=[makesandwich shell 0 0 .005 volume 101 .2 ];% use a normal map that specify the direction of extrusionMAP=feutil(getnormal map node,model); MAP.normal(:,1)=2; model=fevisco(sandCom,model,withnode{x>.5 & y>.5},MAP);cf=feplot;cf.model=model;fecom(;colordatamat;view 1);

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The second strategy is more adapted when testing multiple viscoelastic congura-tions since it is more robust at preserving all options of the original le. The feviscoWriteInclude command is meant for that purpose It lets you select newly meshed


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WriteInclude command is meant for that purpose. It lets you select newly meshedviscoelastic parts using any selection (their MatId for example) and generates theNASTRAN bulk containing the associated nodes, elements, material and elementproperty cards, RBE2 entries connected to the selected elements.

3.4 Data structure reference

3.4.1 Dynamic sti ff ness coe ffi cients zCoefFcn

The normal mode of operation is to display your full model in an feplot gure. Therst step is to dene a set of parameters, which is used to decompose the full modelmatrix in a linear combination. The elements are grouped in non overlapping sets,indexed m, and using the fact that element sti ff ness depend linearly on the con-

sidered moduli, one can represent the dynamic sti ff ness matrix of the parametrizedstructure as a linear combination of constant matrices

[Z (Gm , s )] = s2 [M ] + [K e] +m

pm [K vm ] (3.1)

By convention, K e represents the sti ff ness of all elements not in any other set. Whilethe architecture is fully compatible there is no simplied mechanism to parametrizethe mass.

The full contant matrices M, K e , K vm can be assembled using fevisco MakeModelor are generated within NASTRAN. Reduced models can be generated with fe2xfdirect commands, or with a NASTRAN DMAP step12 , matrices are stored in themodel stack as an SE,MVR entry and information on how to build the coe ffi cientsof the linear combination in info,zCoef viewed in the gure below.

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The columns of the cf.Stack {info,zCoef } cell array, which can be modied

with the feplot interface, give

the matrix labels Klab

the values of coeffi cients in ( 3.1) for the nominal mass (typically mCoef=[1 00 ... ] )

the real valued coe ffi cients zCoef0 in (3.1) for the nominal sti ff ness K 0

the values or strings .zCoefFcn to be evaluated to obtain the coe ffi cients.

The problems handled by fe2xf and computations at multiple frequencies and de-sign points. Frequencies are either stored in the model using a call of the form model=stack set(model,info,Freq,w hertz colum) or given explicitly as anargument (the unit is then rad/s).

Design points (temperatures, optimization points, ...) are stored as rows of theinfo,Range entry, see fevisco Range for generation.

When computing a response, fe2xf zCoef starts by putting frequencies in a localvariable w (which by convention is always in rd/s), and the current design point(row of info,Range entry or row of its .val eld if it exists) in a local variablepar . zCoef2:end,4 is then evaluated to generate weighting coe ffi cients zCoef giving

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the weighting needed to assemble the dynamic sti ff ness matrix ( 3.1). For examplein a parametric analysis, where the coe ffi cient par(1) stored in the rst column of Range . One denes the ratio of current sti ff ness to nominal Kvcurrent = par (1)


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Range . One denes the ratio of current sti ness to nominal Kvcurrent par (1)Kv (nominal ) as follows

% external to fexfzCoef={Klab,mCoef,zCoef0,zCoefFcn;

M 1 0 -w.^2;Ke 0 1 1+i*fe_def(DefEta,[]);Kv 0 1 par(1)};

model=struct(K,{cell(1,3)});

model=stack_set(model,info,zCoef,zCoef); model=stack_set(model,info,Range,[1;2;3]);

%Within fe2xfw=[1:10]*2*pi; % frequencies in rad/s model.Range=stack_get(model,info,Range,getdata);for jPar=1:size(model.Range,1)

zCoef=fe2xf(zcoef,model,w,model.Range(jPar,:));disp(zCoef)

% some work gets done here ...end

To use a viscoelastic material, you can simply declare it using an AddMat commandand use a visc entry in the zCoefFcn column (the stack name for the materialand the zCoef matrix name must match).

cf=fevisco(testplateLoadMV feplot);% define material with a unit conversion mat=m_visco(convert INSI,m_visco(database Soundcoat-DYAD609));% MatId for original, name of parametercf.mdl = fevisco(addmat 101,cf.mdl,Constrained 101,mat);cf.Stack{zCoef}(4,4)={_visc};% Third coefficient will use material with namecf.Stack{zCoef}{4,1}fe2xf(zcoef,cf,500:10:4000,5);

At time of computation, the matrix coe ffi cient in ( 3.1) is found as Gm (s,T, 0 )Gm 0 wherethe reference modulus is found in the cf.Stack {Constraint 101 }.pl entry. Forsuch materials one assumes Poissons ratio to be real and the considered viscoelasticmaterial to have its sti ff ness dominated by either shear or compression. Based on this

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assumption, one considers the frequency dependence of either the shear or Youngsmodulus. The error associated to neglecting the true variation of other moduli isassumed to be negligible (methodologies to treat cases where this is not true are not


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g g ( gaddressed here).

3.4.2 Format reference

A viscoelastic model (see section 3.5.3 for the typical generation procedure), storesthe information needed to compute ( 3.1) in a data structure (generic type 1 su-perelement handled by fe super ) with the following elds

.Opt Options characterizing the model. In particular the second row de-scribes the type of each matrix in model.K (1 for stiff ness, 2 for mass,... )

.K a cell array of matrices giving the constant matrices in ( 3.1). Onenormally uses M, K e , K vm ,... . These matrices should be real and onlythe combination coe ffi cients should be complex.

.Klab a cell array giving a labels for matrices in the

.K eld..kd optional static preconditioner (nominally equal to ofact(k0) ) use to

compute static response based on a known residue.DOF DOF denition vector associated with matrices in .K.Mt a cell array of data structures describing viscoelastic materials.

This cell array is generated automatically by fevisco MakeModelbut can also be built with direct calls to m visco . For example model.Mt= { m visco(database ISD112) }.

.Stack standard model stack where one denes viscoelastic materials (seefevisco AddMat), frequencies, zCoef , reduced model, range (seefevisco Range ) ... as well as the case that contains descriptions forloads, sensors, parameters, ...

Additional elds used than may be used by reduced models are

.br input shape matrix for reduced model

.cr input shape matrix for reduced model

3.4.3 Input denitions

Inputs at DOFs are declared as fe case entries.

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If you have stored full basis vectors when building a reduced model ( MVR.TR eld),you may often be interested in redening your inputs. To do so, you can use fe casecommands to dene your loads, and MVR=fe sens(br&lab,MVR,model) , to rede-


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yne the reduced inputs accordingly.

3.4.4 Sensor denitions

MV=fevisco(testplate);MV=fe_case(MV,SensDof,BasicAtDof,[1;246]+.03);% Surface velocity

xxx% 4 strain sensors along a linepos=linspace(.5,1,4);pos(:,2)=.75; pos(:,3)=.006;data=struct(Node,pos,dir,ones(size(pos,1),1)*[1 0 0]);MV=fe_case(MV,SensStrain,ShearInLayer,data)Sens=fe_case(MV,sens)

If you have stored full basis vectors when building a reduced model ( MVR.TR eld),you may often be interested in redening your sensors. To do so, you can use thefe case commands to set your sensors, and fe sens(cr&lab,cf) to redene thereduced sensors accordingly (this calls Sens=fe case(sens,model) to generatethe data structure combining the observation equations of all your sensors).

[MV,cf]=fevisco(TestPlateLoadMVR);cf.Stack{MVR}% redefine sensors

i1=feutil(findnode x==0 & z==0 epsl1e-3,MV);MV=fe_case(MV,SensDof,Sensors,i1+.03);% reset reduced sensor representationfe_sens(cr&lab,cf);cf.Stack{MVR}

3.4.5 Response post-processing options

For a given reduced (or full) model you may want to post-process the computedfrequency responses before saving them. This is in particular important to analyzeresponses on large sensor sets (panel velocities, stresses, ...) which would require alot of storage space if saved at all frequencies.

The list MifDes xxx

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3.5 Sample parametric analysis step by step


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3.5.1 Model parameterization

fevisco handles parametric models with variable sti ff ness expressed as linear com-binations of constant matrices ( 3.1). The initial step of all parametric studies is todene the parameter sets.

For all models this can be done using fe case par commands (or upcom ParStackcommands for an upcom superelement).

model=demosdt(demoubeam); model=fe_case(model,par,Top,withnode {z>1});

If the parameters correspond to viscoelastic materials, one needs to declare which,of the initially elastic materials, are really viscoelastic. This is done using feviscoAddMat calls which associate particular MatId values with viscoelastic materials se-

lected in the m visco

database.Up=fevisco(testplate upreset);Up=stack_rm(Up,mat);Up = fevisco(addmat 101,Up,First area,ISD112 (1993));Up = fevisco(addmat 103,Up,Second area,ISD112 (1993));

Viscoelastic materials are then considered as parameters by fevisco .

END OF SECTION IS NOT FINALIZEDFor example, a selection based on ProId .

model=fe_case(model,par,FrontStruts,-k ProId 168;par,BackStruts ,-k ProId 304};

The PREDIT mat model XXX

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Figure 3.1: The PREDIT mat model

For example, in the PREDIT mat model, one can dene each of the 4 squares of viscoelastic materials on the pane as a parameter using ProId

cf=feplot(model); MV=cf.mdl; %XXX

One can then visualize each parameter using feplot model properties windows,within the Case tab

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3.5.2 Parametric model generated within NASTRAN (fo by setDMAP)

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The following gives a simple example of a beam separated in two parts. The step12calls run NASTRAN with the fo by set DMAP where an eigenvalue computationis used to generate a modal basis that is then enriched (so called step 1) beforegenerating a parametric reduced model (step 2)


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generating a parametric reduced model (step 2).

The steps of the procedure are the following

load the initial model into Matlab using a nasread command.

Generate through naswrite commands, or manual editing, a RootName bulk.bdfle that contains bulk data information, EIGRL and and possibly PARAMcards.

NOTE elements that will be parameterized should have the lossfactor of their material set to zero .

Dene element groups associated with parameters (see section 3.5.1).

These can be easily checked using feplot (open the Edit:Model propertiesmenu and go to the Case tab). cf.sel= selection {1,2 } commands. Oncethe job is written, elements sets will be written in a parameter sets.bdf le(using nas2up WriteSetC commands).

Sensors using SensDof case entries, see section 3.4.4.

Generate the RootName step12.dat le and run the job using

cf=feplot; % the model should be displayed in feplotcf.mdl=nas2up(JobOpt,cf.mdl); % Init NasJobOpt entry to its defafevisco(writeStep12 -run,RootName,cf)fecom(cf,ProModelSave,FileName); % save your model for reload

where the write command edits the nominal job les found in getpref(FEMLink,

If you need to edit the bulk le, for job specic aspects of the Case ControlSection (denition of MPC and SPC for example), omit the -run , do yourmanual edits, then run the job (for example nas2up(joball memory=8GB,Root

You can also pre-specify a series of EditBulk entries so that your job can runautomatically. For example

edits={insert, SOL 103 ,, GEOMCHECK NONE;replace, SPC = 10,, SPC = 1};

model=stack_set(model,info,NastranJobEdit,edits);

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which are here proportional to G()/G i,nom = 1 + iglob + T ()i,nom hence, giventhe complex modulus, the table is given by


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T = 1i,nom Re(G(

))Gi,nom 1 + i Im (G(

))Gi,nom glob (3.3)Generation of variable coe ffi cients to be used in the toolbox from NASTRAN inputis obtained with MVR.zCoefFcn=fevisco(nastranzCoefv1,model) .

3.5.3 Parametric model from NASTRAN element matrices

An alternative to the step12 procedure Viscoelastic computations are performed onan upcom model typically imported from NASTRAN after a SOL103 (real eigenvalue)run with PARAM,POST,-4 to export the element matrices. Thus for a NASTRAN runUpModel.dat which generated UpModel.op2 and where the upcom superelement issaved in UpModel.mat a typical script begins with Up=nasread(UpModel.dat,BuildO

In some cases, one may want to use reduced basis vectors that are generated by anexternal code (typically NASTRAN). The problem is then to generate the reducedmodel MVR without asking fevisco to generate the appropriate basis. You cansimply call the DirectReduced command

Up=fevisco(testplate upreset);Up=stack_rm(Up,mat);Up = fevisco(addmat 101,Up,First area,ISD112 (1993));Up = fevisco(addmat 103,Up,Second

viscoelastic vibration toolbox

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