The mini-oscillator technique: a finite element method for the modelling of linear viscoelastic structures

The Mini-Oscillator Technique:

A Finite Element Method

for the Modeling

of

Linear Viscoelastic Structures

Donaid

J.

McTavish

..

.

The Mini-Oscillator Technique:

A Finite Element Method for the

Modelling of Linear Viscoelastic Structures

by

Donaid

J.

McTavish

Subrnitted October 1987

Abstract

The use of finite elements to model complex structures has been traditionally effective with regard to ma ss properties and equilibrium elastic stiffness properties. This thesis presents a practical formulation for the analysis of structures whose constituent materials may be classed as linear viscoelastic.

Construction of viscoelastic finite element matrices fuHy compatible with the usual second-order equations of motion is demonstrated, given a knowledge of ma-terial properties. The viscoelastic mama-terial properties are represented in the Laplace-domain by the GHM (Golla-Hughes-McTavish) Analytic Model.

The Mini-Oscillator Technique is introduced and developed through the consid-eration of a single modulus finite element for which the mass and elastic stiffness matrices are known. The simplest case of a single degree of freedom finite element is examined in detail to expose a mechanical analogy from which the mini-oscillator technique derives its name. The procedure is then extended to the case of a general structural model with many elements. Each finite element may be of a different material with stiffness properties ranging between simple elastic and multi-modulus viscoelastic. Each viscoelastic modulus may be approximated to an arbitrary degree of accuracy using the GHM representation.

Energy dissipation characteristics of the structure are present in the finite ele-ment model proposed here, realized by a damping matrix in the equations of motion. For the modal analysis of lightly damped structures, convenient formulae are pro-vided which allow the solution of a reduced order system.

Comprehensive numerical examples illustrating the use of the mini-oscillator technique are included. An unconstrained flexible satellite in various configurations is considered in detail. A second example demonstrates the fitting of GHM model parameters to given material property data.

•

Acknowledgements

I gratefully acknowledge the advice and guidance of my thesis supervisor, Dr. P. C. Hughes. He and his former student, Dr. D. F. Golla, are the source of the original theory from which this work proceeds. The members of the Spacecraft Dynamics and Control and Space Robotics groups at UTIAS provided on numerous occasions a responsive sounding board for my ideas.

f

Contents

Abstract

Acknowledgments 1 Introduction

2 Viscoelastic Finite Element Matrices

2.1 Representation of Material Properties . . . .

2.2 Time-Domain Realization of a Viscoelastic Finite Element. 2.3 A Mechanical Analogy for the Mini-Oscillator . . . .

3 Modelling of Material Relaxation Properties

3.1 Time-Domain Realizations of

G(t) ..

.

. . . .

3.1.1 ~

> 1, An Overdamped Mini-Oscillator

.

3.1.2 ~ = 1, A Critically Damped Mini-Oscillator 3.1.3 ~

< 1, An Underdamped Mini-Oscillator

3.2 Energy Dissipation . . . .

3.3 Quality of GHM Model Approximations . . . .

4 DetaUed Form of the Finite Element Viscoelastic System

4.1 Preliminary Definitions 4.2 Element Matrices . .

4.3 The G lob al System . . .

5 Results for Light Damping

5.1 Perturbative Solution for Light Damping . . . .

5.2 Validity of the Perturbative Solution and Special Cases 5.3 Light Damping Ex'ample . . . .

5.3.1 Performance of Perturbation Formulae . . . .

ii iii 1 3 3 4 8 15 15 16

17

17

18 23 24 24 24 27 30 30 32 34 36

6 Numerical Example One 40 6.1 FLATSAT - A Two-Dimensional Flexible Viscoelastic Structure . 40 6.2 The Basic Finite Element Model of FLATSAT 41 6.3 Elastic Analysis of the Basic FLATSAT . . . . . . . . . . 6.4 Viscoelastic Analysis of the Basic FLATSAT . . . .

6.4.1 The Viscoelastic Beam Element - A Two Modulus Element 6.4.2 Analysis...

6.5 FLATSAT With A Stiffened Array . .

6.5.1 Analysis . . . . 6.6 FLATSAT With Viscoelastic Dampers

6.6.1 The Rod Damper Element. 6.6.2 Analysis... 43 44 44 50 50 51 54 54 55

7 Numerical Example Two 58

7.1 A Restrained Elastic Rod Supported by Elastomeric Pads 58 7.2 MOT Parameter Fitting . . . . . . . . 60 7.3 Viscoelastic Analysis . . . . . . . . . . 61

7.4 A Higher Order GHM Approximation 65

8 Concluding Remarks 67

A Linear Hereditary Constitutive Relations 69

A.1 A Simple Derivation . . . . . . . . . . . . 69 A.2 Various Forms of the Constitutive Relation 71 A.2.1 Separation of an Equilibrium Modulus 71 A.2.2 Strain Histories Discontinuous at t = 0 . 71 A.2.3 Commutativity of G and e in the Constitutive Relation 72 A.2.4 Laplace-Transformed Constitutive Relation 73 A.2.5 Complex Modulus and Loss Factor . . . . . . 73 A.2.6 Creep Representation of Material Properties 76 A.2.7 Fourier Transformed Constitutive Relation 78

A.3 Physical Restrictions on G(t) 79

A.3.1 Fading Memory. . . . . . . . 79 A.3.2 Energy Dissipation . . . . . . . . . 79 A.4 Three-Dimensional Linear Viscoelasticity 82

B Eigenproblem Perturbation Formulae 84

B.1 Solution for a General First-Order System . . . . . . . . . . . 84 B.2 Perturbative Solution for the Viscoelastic Finite Element System 86

.

'

B.2.1 Solution of the Equilibrium Elastic System 87

B.2.2 Solution of the Dissipation System . . . . . 88

B.2.3 The Viscoelastic System in Modal Form . . 89

B.2.4 Perturbations in the System Matrices Due to Light Damping 90

B.2.5 Solution . . . · · · · 91 C Shnple Finite Elements for Two-Dimensional Analyses

C.1 Rigid Body . . .

C.2 Beam Elements . . . . . . .

C.3 Rod Elements . . . . C.3.1 Simple Rod Element

C.3.2 Elements for Multi-Piece Rods

CA Coordinate Projections . . . .

C.4.1 Rotation from Local to Global Coordinates C.4.2 Remote Nodes

C.4.3 Examples D N omeclature

D.1 Lowercase Roman Symbols D.2 Uppercase Roman Symbols D.3 Greek Symbols . . . .

DA Matrix Conventions . . . .

D.4.1 Matrix Assembly Operators.

D.5 Other Symbols . . . .

D

.

6 ,B-Notation for Mini-Oscillator Parameters .

97 98 98 101 101 102 103 104 105 105 110 110 111 112 113 113 115 116

'

.

Chapter

1

Introduction

As the size and fiexibllity of current space structures increase, the need to charac-terize energy dissipation in an accurate and convenient manner also increases. The somewhat arbitrary "modal damping factors" too of ten tacked onto the results of sophisticated finite element (elastic) analyses of complex structures represent an unacceptable state of affairs. ,

In this new approach to finite element structural analysis with damping, based on Golla and Hughes [3], auxiliary "dissipation coordinates" are employed to augment the standard finite element nodal displacements. The familiar linear matrix-second-order (LMSO) form is retained along with the symmetry and definiteness properties of the constituent matrices. Thus, the full arsenal of analysis techniques employed by the structural dynamicist remains applicable.

The reason for the presence of auxiliary coordinates in the LMSO equation of motion lies the Laplace-domain representation of linear viscoelastic material prop-erties: the time-domain phenomenon of stress relaxation is realized by a modulus

function in the Laplace-domain. This material modulus function is represented

by the GHM (Golla-Hughes-McTavish) Analytic Model involving a sum of

ratio-nal functions of second degree, each of which will be referred to in this thesis as a

mini-oscillator term due to a straightforward, though incidental, mechanical

anal-ogy. The algebraic form of these mini-oscillator terms allows a LMSO realization in the time-domain, albeit with auxiliary coordinates. The mini-oscillator technique for the analysis of viscoelastic structures assumes isothermal conditions, viz. the heat energy dissipated from the structure causes negligible change in temperature and material properties.

It is true that the use of auxiliary coordinates to model the material modulus function leads to a LMSO system of order larger than the corresponding purely

el ast ic system. The degree to which it becomes larger depends upon the number of material moduli, the desired accuracy, and the frequency range over which the material damping properties are to be modelled. However, the strength of this new approach lies in the reteption of the LMSO form, allowing the creation of viscoelastic structural finite elements fully compatible with existing techniques used in dynamics analyses.

When energy dissipation is relatively low (light damping), a corollary to the

anal-ysis is presented which provides a path to the calculation of modal damping factors directly from a knowledge of the (undamped) elastic frequencies and the GHM mini-oscillator parameters which characterize the material dissipative viscoelastic properties. Thus, far lightly damped structures, an explicit expansion of the system order to include the auxiliary coordinates associated with the GHM representation is not necessary.

'

.

Chapter 2

Viscoelastic Finite Element

Matrices

2.1

Representation of Material Properties

Using the theory oflinear viscoelasticity [2,4] (also see Appendix A), the constitutive relation for a simple one-dimensional stress-strain system can be written:

(2.1)

where e is restricted to be zero for tE (-00,0). G(t) is identified as the material

relaxation function: the stress response to a unit-step strain input. Transforming the constitutive relation to the Laplace-domain yields:

a(s)

=

sG(s)e(s)

(2.2)

wherein 'sG(s)' will be referred to as the material modulus function.

The use of the mini-oscillator technique requires the representation of material properties by the GHM Analytic Model involving a series of mini-oscillator terms (MOTs). In terms of the material modulus function, such a representation would be:

(2.3)

ê

O is the GHM approximation to GO, the equilibrium elasticity constant pertaining

to the relaxation function G(t).

GO = lim G(t)

Three independent MOT parameters

(a,

w,?h

are associated with each term of the series.

The complex modulus provides a practical vehicle for the presentation of vis-coelastic material property data [8] (see also Appendix A, Section A.2.5). The Laplace transformed relaxation function may be used to de fine the complex modu-lus denoted

G*(w):

where

G*(w)

=

jwGUw)

=

G'(w)

+

jG"(w)

G'(w)

=

Re{ jwG(jw)}

G"(w)

= Im{

jwG(jw)}

Further, the materialloss factor, fJ, is defined thus:

G"(w)

fJ(w)

=

G'(w)

(2.5)

(2.6)

(2.7)

For a GHMrepresentation ofmaterial properties, the components ofthe complex modulus are given below.

[

( 2 A 2)

(2

A A ) 2

1

' A

a

2

w - Wk

+

ÇkWk

G (w) = G 1

+w

E

_{k w - Wk}

(2 A2)2

+

_w

2(2A A )2

_Çkwk

(2.8)

(2.9)

Figure 2.1 shows complex modulus data plotted for a single mini-oscillator term

GHM material modulus function with:

Aa

G = 1. al

=

1.

In the application of the mini-oscillator technique, the analyst is required to fit MOT parameters to actual material data. Knowing the frequency range over which the structural information is required and the level of accuracy desired in the results dictates the number of MOTs to be considered.

2.2

Time-Domain Realization of a Viscoelastic Finite

Element

Extending the linear hereditary stress-strain law of (2.1), the equation ofmotion for a single finite element may be written (assuming for now a single material modulus): (2.10)

o log G"(CO) -1 -2 "' .. ·1

_,

.. .. I , .. , , , · .. 1_, ···· ' .... , ... '" ..

,

.... -.... ~ . .. ... ····"f ..

!

..

!

.. .

.: ... _ .~ ...

,

! . I

,

.. .

,

,

,

,

,

,

. " , . ., ... , , ,

,

, I : •

,

,

____ ~ ______ .: ______ ~ _ J ______ .:. ______ ~ __ _ __: __ L _____ ~ ______ ..; ___ ,.; __ ':'_.J ______ ..:.. _____ _ · . . . _.. I , _.. . . . I . . . . I . . '"I "t'· ... ; ... : ... : ··1···

,

,.

,

.. ,

,

1. •• ,

,

_,

,

· . . .

.

,

.

. . . -: . . " : · : . : , . : : : : , : _____ ~ ______ ~ ___ ~ __ ~_J ______ ~_ · . . .

.

I

,

_.... • , ""' , : I I •.•.•....• : •.• 1. .1 .. "

,

,

,

:

,

:

,

-:'"1"

,

, ,

,

_ ___ ~ ___ :.. __ : __ L _____ ..: ______ ..: ___ ..: __ ..:.. _.l _ ____ _ . . : I . . . . I '" ... ~ .. '". .. ;.. ·1· ... , ... ; ... ~ .. -I .. I : I .

,

1. ..:. ...• " , I : I I : I I : I ... ~ . "" ... -... : .... -:-.. " , . I : ' I

,

, , , , ___ ..; ___ .,; __ ..:.._.J ______ ..:.. ______ :.. ___ :.. __ ~_L _____ ~ ______ ..; ___ ..; _ _ ..:.._.l ______ ..:.. ______ :.. ___ :.. : . : I . . . . I . . . . I . . . . . ':-. I ... . '" . . . . -•. _... .-, . ···1· .. ... ',·1· .... ;. .. j ... . I I : t . . . : . . . · . . 1

,

... .

,

1. ... : .... 1

,

.

,

,

,

,

,

, I I : I .. ,

,

.. .,_, . '7 " '1

,

. , ,

,

,

,

,

_3~--~~~~WW~----L-~~~~U---~--~~~wu~--~~~~~~ -3 -2 -1 o log CO I : I .1 , ... ,

:

~

:

···1

,

··· ... , ,

:

:

,

, .1... .1.. , . I I , ,

,

, , , , , , , , ,

.

....

...

-

!

.

·f· :c:....,;~,i---.;...---.;...~ log G' (CO)

,

,

,

, , ,

:

: :

,

,

,

,

,

, I . . : . : : , . : . : o 1---"----'--"'-....;...-+, ---'---~" "'-"'--~c:':_:-::_,--t---~---"---" --~ --t -- - - --~ ---_C - -_c __ ,_ . _I I . . ' ... ~ _• .. .l. _I .... _. ~ .. : _I .. , ,

,

, , , · .. ·, 1.. . . .•. '1, ' · .. ··1, ··

,

,

,

,

, , I , : I I... .... "I. . ... : .... 1 ..

,

, , ,

,

,

,

, , ,

,

,

,

,

,

,

,

,

,

,

, , , , : . . _, 1. ..,I

,

. . .. . . , ... _ .. 1.

,

. ,

,

,

, , , ,

,

,

, , , , , , , ,

,

_lL---~~~~~~--~~~~~~--~~~~~U---~~~~~u -3 -2 -1 o 1 log CO

Figure 2.1: Complex Modulus Data for a Single Mini-Oscillator Term GHM Model.

l

where q(t) is restricted to be zero for t <

o.

f(t) is the elemental no dal force vector corresponding to the coordinate vector q of dimension nq. The stiffness matrix

K

e is the same as that for an equilibrium elastic analysisj that is,

(2.11) N ote that an overbar is used to indicate the equilibrium elastic stiffness matrix with the mate rial modulus factored out:

(2.12) Transforming (2.10) into the Laplace-domain, the element al equation of mot ion becomes:

(2.13) To illustrate this procedure a single mini-oscillator term GHM model will now

be used to represent the material modulus function

8é(8):

(2.14) The following second-order time-domain system has the same Laplace-domain representation as (2.10), given (2.14):

ab

K ' ] [

~

1

+

[~

a;K' ][

H··

-:~:

1 [

i

1

= [

~

1

(2.15) For system equivalence, the initial conditions on

z

are prescribed to be zero.

The viscoelastic element matrices are refined somewhat through the spectral decomposition of the equilibrium stiffness matrix Ke to realize the desired defi-niteness properties. The equilibrium modulus GO is positive and

K

e is symmetrie and non-negative definite

(KeT

=

K

e

f.

0), possessing nq non-negative eigenvalues. We eonsider only the positive (non-zero) eigenvalues, À~, and the eorresponding

- e

orthonormal eigenvectors, r~, of K

to write:

(2.16)

t

wh ere

R

= row{ r~} (2.18)

Zero eigenvalues of

Re,

if any, represent rigid displacements of the element, which cannot lead to energy dissipation.

The desired form for the viscoelastic element matrices of (2.15) is achieved from the above factorization of Ke , th en letting

AT

z

=

R

z

(2.19)

Applying this substitution and multiplying the bottom row by

R

T, one obtains with the definition

R

=

RA

(2.20)

the following result:

(2.21) where DV - [ 0 0

1

- 0 a~A

-aR

1

aA

Note that the viscoelastic element matrices have the properties:

MvT =Mv

>

0

DvT =Dv ~ 0

K vT

=

KV

1.

0

(2.22)

Clearly, this formulation is LMSO compatible. The dissipation coordinates rep-resented by

z

may be considered internal coordinates with respect to the finite element. The positive definiteness of the diagonal matrix A allows the viscoelastic mass matrix M V _{to be positive definite just as the corresponding elastic mass matrix}

M

e_is.1 _{The viscoelastic element stiffness matrix KV shares the same nullity as its}

elastic counterpart Ke•

2.3

A Mechanical Analogy for the Mini-Oscillator

This simplest structure to consider is one having a single finite element possessing just one spatial degree of freedom with single mini-oscillator term GHM

represen-tation of material properties. The equilibrium elastic system corresponding to this

single DOF system has matrices:

q=

[q]

f=

[I]

Hence

mq+kq =

I

(2

.

23)

or, in the Laplace-domain

(2.24) This is the familiar linear elastic mass-spring system. If the spring stiffness is de-scribabie by a relaxation function with Ik' being the equilibrium modulus and a single term G HM model is applied then

2 - k [ (8 2

+

2~W8)

]-

1-

.

s mq

+

1

+

(2

A A A

2)

q =

+ l.c.

S

+

2ÇW8

+

W (2.25) For the simple equilibrium stiffness matrix Ke =

[kj,

spectral decomposition yields

ft

=

[1] and Ä

=

k[I]. Thus a single dissipation coordinate ( z

=

[z] )

is required to augment the elastic system. Therefore, the viscoelastic system takes the form:

[~

"b

k

][!]

+

[~ "~k]

[: ]

+ [

k

~,,~k -:~][:] [~

]

(2.26) The mechanical analogy which realizes this system is shown in Figure 2.2 The use of a mini-oscillator term to represent the viscoelastic stiffness is tantamount to a fictitious spring-mass-dashpot unit which is coupled to the spatial displacement of the system through the auxiliary coordinate 'z'. Holding the spatial coordinate

Iq' fixed, the mini-oscillator is described by the familiar second-order characteristic equation: 82

+

2~ws

+

C} = 0; hence the name "mini-oscillator".

In order to study the modal behaviour of th is simplest structure, it is convenient to non-dimensionalize (2.26) to obtain the following eigenproblem:

special cases this may be relaxed. For example, a lightweight spring connecting more massive components of a structure together may be assumed to be a mas8le8s element, in which case

M'

=

o.

For this situation, the viscoelastic element mass matrix will have the property MvT =

MV ~

O. Nevertheless, the assembied matrix system for a realstructure generally has M > 0 for

the global mass matrix of both an elastic and viscoelastic model.

t

f

where with - À À= -w and

w=~

w

w=

-w

being the natural frequency of the corresponding equilibrium elastic system. (2.27)

For the primary oscillatory mode, the plots in Figures 2.3-2.5 provide the follow-ing information for the mechanical analogy for variations in the parameters

(a,w,ç).

where and

such that

The angle cf> is the phase angle between the displacements q and z.

The last plot, Figure 2.6, shows the behaviour of the fictitious dissipation modes for variation in

ç

only.

When the dissipation modes are distinct, i.e. overdamped (two real eigenvalues), the data shown is:

where and

such that

When the dissipation modes are oscillatory the data shown is: where

and

such that

The baseline parameter values for the plots are:

(a,

w,

ç)

=

(I, I, 2)

Phase Angle o <I> -90 (deg) 1 o log -1 -2 -3 -4 -5 : , :

,

'~'T :

,

:

,

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,

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Figure 2.3: Variation of Eigen-Data with a.

2 3

log a

Phase Angle o Cl> -90 (deg) -180 , , ,

,

, .... _,

,

,

.

,

. ~ ... -- -:--:-~r---:---:--: : : I . . .... .: ... ; .. :.J ... : : : I · . : I · : : I ····":···";····1 __ ~_..;_:..1. __ ~ ___ :..._ · '.~.: .... ~. . : : " . . · : 'I : : ... .: ... .: .. :.J ... : .... : .. · ."' : . . . I . . ---:--:-~,.---:---:--_.. ~ .. ~"~'T""-~ ... . ,

,

. ~. , I , .• . • , ,

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:

.

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. .;.

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···i·· " i ··i·

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i

./

l.

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.

.

L .... : .. ;

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:.

....•... .:. j ..

1

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... .. i ...... L .. L

~

.

:.

.

'.L ... .; L ...

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...

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... L.:

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~~~~t~~

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}

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~~

.+~~t~

j

~~~~~~~~~~~~~~~~

t~~T

Tli~~~~~~~~~~

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~t~;,~~~~~r.~~i~~

~:

.,. : :-H··' ··'· . ...

V-

H·· ·i·· .;.; ·;1··· :·· ··'·1· .. ; .. i .. i··;I···· .. \ .. :

.;

+

... :

...

:t-

ii,

l:ë

1

'ë

':T,j

ë:

TA-;1l--

.:tji

~-

T:1T·

I-:

~

::

_·

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m-:·

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.. , .

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·

·

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r·

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-4 -5 : I . : I I : • . : I : • :. \ _6L-~~~~~UW~~~~~~UW~~~~~~~~~u.~~~~w -4 -3 -2 -1 o 1

Figure 2.4: Variation of Eigen-Data with

w.

2 3

log ro

Phase

Angle o _{I I : I : : I : : : I I : I}

i

····

·

····

·

·

·

.

.

..

..

!

... ':

.

.

'

H

'"

"'

1

"

d

·+··

·

·

·t

·

·

1··

d

.

!

..

.

...

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__

:~~_~~~:1~'~' ____ ~~':_"~"_"_"_:'_"_";~"_"_:''~:'+' __ ~ __ ~ __ ~:'_'~ __ ~~':~"~" __ ~~'~'~ 1 -,-... ' . ..; . I . . .. ~ • . . ., .• . .• ~... . . I I : I I I I . . : I : I .I .:.1 ... :.. . . .1.. I . " . I . I : I : r---:--- .,--- -1'

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... .

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•... : ... I . . I 1

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+

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... ~ :

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ç

Phase Angle <l> (deg) log 90 : I . .\ : I : I " : I : I . . : I · I . I . ~: I . J "1 . . .. .. :'" ':T ':'1" • . . . : I . . .. : " : I : : 1 ~ I : I . I . I ', : : I · I '\ '\ . I ~ . ~. i ~ . . ! . I . . , ~. i' . "' . '~'I ... ,. '~'I" ... ~'i" .... ! .. ~ .. ~. ---~r--~---~-;71---7--~~---~---- -~ï--7--~-~--~---~--71---~--~~---~--7-7-~ . ' : I . . : : I . . : : I . : I . . . : I . . : : I . . . , I ' . • : .:.\ ,: • .1. .:.:1... :.1 . · I . ' :1 " '\ . : .. :, -:- : ... : .. ;. ~.: .~. ~'1 '\ : I : I · I '\ I : I : I : : : I : I " : I I " 0 ';"1" . ~'1 '" ;'1 '.' r" ... -:" ...•. .~ ..• " . t-....;. __ .:...:.~"*L"'_.:;-.:"_----~ -~ ~ I . : I · I ~ : I : , . I . . ,',.~. I ,-, ... .••.. ..~.~. i . •• ~ • I • . . -: .• ~. I" .. •.• , '1 . • ~ ... • ., . " : I . : I : I : I ., : I " : : I : : I : I : I '1 : I ,:,J. .:.:1.. ,; .. :J .. .:.1. ':.:.J :1. :.J .... '\ : I . : I :, ' :1 . I ., I . . , : I . . . I . I . . '1 . r : . . r . . . . ·:::-:::-:-:~-:-::-:::f.:-:-:~:::::-:::-:::~1:-::-::-:~:-:-:::-;~~1:::-:-:;::::::;-: '\ '\ . : : I -~r--~--~-~-~r_{'fT' .. : ...} --~---,---~--~-ê~1---ê ---T'-~

·;·

r

··· ... :···:·

~

:

··· .. ·:

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···~

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~··~·r·

..

·~···

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,

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,

-902

~~~~~TTnn~~ln~~~~~~

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~~

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~~~~~~~~~~

1 o -1 -2 -3 -4 -5 -6 -2 -1 o 2 ... _, , _, .:.' .. "_"' ' "' ... ~ .... ~ ... ~ .. ,. . . ... ' 3 4 5 log 6 A

S

Chapter 3

Modelling of Material

Relaxation Properties

3.1

Tillle-Domain Realizations of

G(t)

A relaxation function representing a material modulus ean be written as the sum of its equilibrium value and a time-dependent part:

G(t)

= GO

[1

+

h(t)

1

(3.1)

where

CO = lim

G(t)

t-+oo

(3.2)

is the equilibrium el ast ie modulus.

The eonstitutive relation between stress and st ra in for a one-dimensional medium ean be written in the Laplaee-domain as:

o-(s)

sG(s)e(s)

(3.3)

CO [ 1

+

sh(s)

1

e(s)

The function

'sh(s)'

is introdueed as the material dissipation junction. The GHM

representation of

sh(s)

is given by the series:

(3.4)

assumed for the moment.

For illustration we will consider only a single term GHM model of

h(8)j

that is,

a single mini-oscillator:

(3.5)

or, equivalently: h(8)=a 8+(b1_+b2) 82

+

(bI

+

b2)8

+

b1b2

(3.6)

h,b

2

=

W[Ç=r=VÇ2-1J

wherein

(3.7)

3.1.1

?

>

1,

An Overdamped Mini-Oscillator

In this case, the poles bI and b2 are real, distinct, and positive, with

(3.8)

Then we may write

(3.9)

(3.10)

or

h(t)

(3.11)

or

h(t)

=

ae-?Glt

[COSh(WVÇ2

-1

)t+

(k)

sinh(wV

ç

2 -1

)t]

(3.12)

or

-?Glt

.

r:;;;---:

h(t)

=

a ~sinh[(wv

ç2 -

l)t

+

4>J

ç -

1

(3.13)

where

Vf2=1

tanh4> = A

<

1

ç

(3.14)

3.1.2

?

= 1,

A Critically Damped Mini-Oscillator

When the mini-oscillator is critically damped, there is a double real pole in the

bI

=

b2

=

W

(3.15) dissipation function. ~ then

h _

_(s

+

_2w)

(3.16)

(s) -

a

(s2

+

2ws

+

w2)

or

-

[1

w ]

(3.17)

hes)

=

a

(s

+

w)

+

(s

+

w)2

which has the time-domain realization

(3.18)

or

(3.19)

3.1.3

?

<

1,

An Underdamped Mini-Oscillator

In this case, the poles

bI

and

b

2 are a complex conjugate pair:

(3.20)

Then we may write

(3.21)

as before, with the time-domain realization

(3.22)

or

het)

or

or

where

h(t) = ae-?wt

[COS(WVI -

ç2)t

+

(h)

sin(wVI -

ç2

)t] -?wt

h(

t)

=

a

Jr=-psin[(

wV

1 -

ç2

)t

+

cfo] 1-~

~

tancfo = A ~ (3.24) (3.25) (3.26)

Figures 3.1-3.3 show time-domain realizations for the above three classes of

dissipation functions.

3.2

Energy Dissipation

In Appendix A, a number of conditions are presented which as a set describe

char-acteristics of a function

G(t),

such that

G(t)

is an acceptable material relaxation

function. These conditions ensure that a material described by

G(

t)

will dissipate

energy for an arbitrary history of deformation and otherwise behave in a manner consistent with ohserved material characteristics.

The conditions from Appendix A will be divided into two groups of criteria to be applied to mini-oscillator representations of material properties. The first group

results from nonnegative work con di ti ons alone:

I-a. lim

G(t)

==

GO =

sG(s)

I

~

0

t-+oo 8=0

i-ho

G(O)

= lim

sG(s)

~ 0

8--+00

I-C.

IG(t)1

~

G(O)

i-d. Im{

jwG(jw)}

==

G"(w)

~ 0, w>O

The second group of criteria emhodies stricter conditions for energy dissipation

than just nonnegative work. Thermodynamic considerations in concert with the

observed fading memory behaviour of stress relax at ion in real materials yield:

a.

1\

~=2

h(oot)

10 oot

Figure 3.1:

het)

for an Overdamped Mini-Oscillator.

a.

1\

~=O

h(oot)

10 oot

Figure 3.2:

het)

for an Critically Damped Mini-Oscillator.

a.

1\

~= 1/2

h(oot)

10 oot

G(l)

Figure 3.4: An Acceptable Relaxation Function.

u-a.

G(t)

~ 0

ii-b.

~G(t)

<

0

dt

-d

2

u-co

_dt

₂

_G(t)

_{~ 0}

Figure 3.4 shows the general characteristics of an acceptable relax at ion function. Consider now the GHM model of a material relaxation function given as a ma-teriai modulus function in the Laplace-domain,

(3.27)

and the implication of the above criteria.

Firstly, the nonnegative work conditions place the less stringent requirements on

I-a. -+

i-b. -+

I-C. -+

i-d. -+

Criterion (i-a.) requires, not surprisingly, that the equilibrium value of the material modulus is nonnegative. For the MOT representation it is dear that the constant

CO

be nonzero as weIl else the modulus is identically zero, thus:

(;0> 0

In light of (i-c.), criterion (i-b.) becomes extraneous. As a particular case of (i-c.) consider

G(t)

=

GO

in which case it is obtained that:

Since criterion (i-d.) is applicable for w

> 0

it is required:

While the above inequalities involving series of mini-oscillator parameters constitute necessary conditions for dissipativity it is useful to extract more practical sufficient conditions by applying the criteria to the individual terms of the MOT series. In particular, from the special case of (i-c.) we require

ak

> 0,

and if it is agreed that

Wk

>

0, then (i-d.) provides

ç

>

0. To consider more generally criterion (i-c.), the time-domain realizations ofindividual mini-oscillator terms given in Section 3.1 show that in fact this condition is satisfied given the aforementioned sufficient constraints

on

(Go,ak,wk,çk)'

To consider the criteria of group

(ii)

it is useful to refer to the time-domain realizations of Section 3.1. Using the material dissipation function

h(

t)

rather than

G(t)

and accepting the results of criteria (i), criteria (ii) become: lla. iib. llC.

-h(t)

~ -1 d

dth(t)

~ 0

Again the MOT series for

h(t)

will be considered on a term by term basis to derive sufficient conditions to satisfy criteria (ii). Recall the time-domain realizations for a single term

h(

t).

ç>

1

ç

= 1

ç

<

1

-?Git

.

r;:;;;--;

h(t)

= a

~sinh[(wy?2

-

1)t

+

4>] , 4>

<

0

ç

-1

h(t)

= ae-

wt (1

+

wt)

-?

Git

r;--;;;

11"

h(t)

= a

n

sin[(wy1-?2)t+4>] ,

4>E(0'-2) 1-ç

Criterion (ii-a.) is always satisfied for ? ~ 1. However in the case of an under-damped mini-oscillator, ?

<

1, this criterion may be violated if a is large. Secondly, criterion (ii-b.) is never satisfied for ?

<

1, satisfied for ç

=

1 ex cept at t

=

0, but always satisfied ç

>

1. Finally, criterion (ii-c.) can never be satisfied for all t, but holds true for ç ~ 1 when t is greater than some value. The larger ç is the smaller the range of t for which (ii-c.) is violated. In particular criterion (ii-c.) is unsatisfied over the following ranges.

? = 1

wt

<

1

?» 1

wt

<

2In2ç/2? In summary, the conditions

(;0

_>

0

ak

>

0

(3.28)

Wk

>

0

..

are sufficient to ensure that the MOT representation of a material modulus function will be dissipative in the sense of nonnegative work. For the special case of a single term MOT series, these are necessary conditions.

The additional condition:

(3.29) provides for the most realistic modelling of observed material relaxation behaviour particularly for a single term MOT series.

3.3

Quality of

GHM

Model Approximations

For a discretized single modulus system the eigenvalue problem which determines the modes and natural frequencies has the general form:

(3.30)

A The true material modulus function

ÀG('x)

is approximated by the MOT function

'xG('x).

For some particular generalized frequency ,X ( ,X = (T

+

jw ) it is of interest

to know the error in the GHM representation of the modulus function. We let

E('x)

=

'xG('x) - 'xG('x)

(3.31 )

It is anticipated that in many cases the fitting of a GHM model of mini-oscillator terms to a known viscoelastic material property will been carried out in the frequency-domain. Specifically, some GHM function is fitted to the complex modulus of the material of interest. The data fit to G*

(w)

will be over a limited range of frequency to some specified tolerance.

By manipulating Laplace and inverse Laplace transforms (see Appendix A for a similar example) it is possible to write an expression for

E('x).

E(À)

= -

₁

1

[E'(O)

cos

Ot - E"(O)

sin

Ot]

dO e->.t

dt

00

1

00

'Ir 0 0 (3.32)

wherein E' and E" are the errors in fitting the real and imaginary parts of the complex modulus.

E'(w)

E"(w)

ê'(w) - G'(w)

ê"(w) - G"(w)

(3.33)

Chapter 4

Detailed

Form of

the Finite

Element Viscoelastic System

4.1

Preliminary Definitions

The viscoelastic structure is represented by a global matrix system consisting of the

following:

• 'ne' finite elements, subscripted 'i'.

Each element is considered to he a structurally independent unit composed of a

single material. The material of the i-th element is characterized hy: • 'mi' material moduli, suhscripted "j',

with each modulus modeled with:

• '4;'

mini-oscillator terms (MOTs), subscripted 'k'.

4.2

Element Matrices

In a purely elastic system based on equilibrium (relaxed) values of the various ma-terial moduli, the element mass and stiffness matrices and corresponding degrees of

freedom are identified by:

M~

_,

K~

I q~ with

M~

>

0

..

The number of DOF for the i-th element is denoted n~i. Further, the element stiffness matrix,

Ki,

is expressed as a linear sum of eontributions from its mi

moduli.

mi mi

Ki

=

L

G?jK~j

=

L

Kij

(4.2)

j=1 j=1

where

ê?j

is the

GHM

approximation to

G?j'

the equilibrium value of the j-th modulus, and

Kij

is the eorresponding stiffness eontribution. Note that

Kij

=

G?jK~j

.

If a value for

G?j

is available, then one may select

ê?j

=

G?j.

Eaeh

Kij

is neeessarily symmetrie and non-negative definite as

Ki

is, having the factorization:

( 4.3)

where Aij is a diagonal matrix formed from the 'ni;' positive eigenvalues of

Kij

(subseripted

Cp')

and Rij is a matrix with the eorresponding orthonormalized eigen-veetors:

1Îï

j = row{

rijp}

sueh that

Also we define the following:

( 4.4)

(4

.

5)

(4.6)

Any zero eigenvalues of the K~j represent rigid body motion with no energy dissi-pation.

The

GHM

representation for the j-th material modulus (of the i-th element) is:

(4.7)

Eaeh ofthe lïj terms involves a different set ofthe three MOT parameters: (a, Cl, Ç)ijk.

The partitioned viseoelastie element mass, damping, and stiffness matrices, de-noted:

Mi, Di, Ki,

respectively, may now be written down:

M~= [

~~q

~~z

1

D~

= [0

0

1

I I

0

D~z

K~

= [

K~q

+

~~~

-K;z

1

(4.8)

I - K I K~z zq

wherein the constituent element submatrices are defined:

M~q

=

M~ _I ( 4.9)

M~z diag{ (Xijk

A!

Äij} (4.10)

Wijk

D~z d' { _{lag (Xijk-A-Äij} 2Çijk } _(4.11) Wijk

K~q

K~ _I (4.12) mi lij Kai . qq LL(XijkKij (4.13) j=lk=l

K~z

row{ (XijkRij} ( 4.14) K~q col{ (XijkRij } T

=

K~z ·T ( 4.15) K~z

=

diag{ (XijkÄij} (4.16)

For emphasis we reiterate the properties of the viscoelastic element matrices.

M~T

_,

M~

I

>

0

D~T _I D~ _I

_>

0 ( 4.17)

K~T

I K~ I <f- 0

The element DOF vector consists now of the original spatial coordinates augmented by dissipation coordinates:

where Zi = col{ Zijk} and Zijk = col{ Zijkp}

Each 'ZijkP' is a single dissipation coordinate associated with a (>"ijp,rijp) pair, k-th MOT of the j-th modulus, th us each Zijk possesses nij coordinates, so that the dimension of the viscoelastic element matrices is:

where

mi

nzi =

L

lïjnij

j=1

4.3

The Global System

The global system is an assembly of element matrices. The partitioned global system is written ( 4.18) where M [ Mqq

o

_{M zz} 0

1

( 4.19) D ( 4.20) K ( 4.21) and

"

[ ; 1

( 4.22)

The dimension of the global system is (nq

+

n z ). Clearly, it is a LMSO system with all matrices symmetrie and

M>O

D

~

0

( 4.23)

In general, elements are developed in alocal coordinate system not coincident with the global coordinate system. A projection of coordinates is required.

q~ is the n~i X 1 element DOF column vector in local coordinates.

qi is the nqi X 1 element DOF column vector in global coordinates.

then q~ = Ciqi (4.24)

The n~i X nqi element projection matrix C i projects the global DOF onto the elemental DOF. The "projection" which Ci represents is in general a transformation

between independent coordinates. The matrix ei is of ten, but not necessarily, a rotation matrix.

Since only the q~ need to be made compatible with the global coordinate system,

the Zi remain untransformed. Thus, the element matrices expressed in global

coordinates are:

e,

~qe,

[ T . ( 4.25)

[: :.l

( 4.26) [

C?K~qei

+

eiTK~;ei

-K~qei ( 4.27) with DOF:

_{[ qz,'}

_.·

_]

( 4.28)

The global system may be written as a matrix sum of the element matrices using

the artifice of element assembly matrices, written as:

Si

= [

SJi

S~i

] ( 4.29)

dim{ Sqi} nqi X nq

dim{ Szi} nzi X n z

Each row of Si, associated with one of the elemental DOF, is all zeros with the exception of a '1' in the column corresponding to a DOF index location in the global

system. If an elemental DOF is to be restrained in the global structure, then no

corresponding global DOF exists and the associated row of Si is all zeros. There

is a one-to-one correspondence between the columns of each Si and the DOF of

the global system. To illustrate, consider a 2-DOF element to be assembIed into a

6-DOF system. Suppose the first and second elemental DOF are associated with the fifth and first global DOF respectively, then the assembly matrix would take the form:

S= [0 0 0 0 1 0]

Writing expressions for the global matrices, then: Re M =

Lsl~Si

(4.31)

i=1

R. D

_LslDiSi

( 4.32)

i=1

Re K =

LslKiSi

( 4.33)

i=1

or, in detail, using the partitioned form:

( 4.34) ( 4.35) ( 4.36) ( 4.37) R.

L

Sql ClK::CiSqi

( 4.38)

i=1

( 4.39) Re

Kzq

LSzlK~qClSqi =

KqZT

( 4.40)

i=1

( 4.41)

The dimension of the global square matrix system is

(n

q

+

n

z ) .

dim{

q}

( 4.42)

ne mi

nz

dim{

z}

=

L L !.;jnij

Chapter 5

Results for

Light Damping

5.1

Perturbative Solution for Light Damping

For problems of unforced motion, the viscoelastic system takes the form:

()'~M

+

).vD

+

K)v

=

0

(5.1)

The column matrix v contains the rIq spatial degrees of freedom q augmented by

the t1z dissipation coordinates z. It is possible that 1Iz be as large as or significantly greater than rIq. When damping is "light" , a perturbative solution is possible which avoids the formulation and solution of this potentially large numerical system.

The presumption of light damping infers that the eigenvalues and eigenvectors of the damped system differ little from those of a corresponding undamped system. For the light damping formulae presented here the undamped eigenproblem is the equilibrium elastic system defined by

().2M

qq

+

Kqq) q

=

0

(5.2)

having solutions for nl flexible (non-rigid body) modes thus:

).2 = ).2 = -w2 _{-J. 0}

a a r

q= e_a

a=l,11J

(5.3)

The eigenvectors are orthonormal with respect to Mqq such that

(5.4)

The Wa are the natural frequencies of the undamped (equilibrium elastic) structure.

Any rigid body modes, those with ).2 = 0, remain rigid body modes with identical

By use of the eigenproblem perturbation formulae of Appendix B, approximate eigenvalues and eigenvectors for the fiexible modes of the viscoelastic system may be generated.

The perturbation approximations to the i-th eigenvalue and eigenvector, assum-ing

À;

=

-w;

is a distinct eigenvalue of the undamped problem, are:

(5.5)

and

(5.6)

GA 0' e (eTST CTr

e_{) [}

1 "'/

~A

_{o::...,....::,.e}

Le:--TS-.!.T.c..".ST~!:..-e)

_[

eo~

l}

pAp i qp P P ei " pAp Cl qp P P ~

+ap 4w~ 0 - ~ ap (w~ _ w2 )

, C l = l ' Cl

Cl;ti

Also note the conjugate viscoelastic modes:

ÀtI.",!+i

=

ÀtI i V"'!+i

=

Vi

(5.7)

From the real and imaginary parts of Àtli the effective modal "natural frequencies"

and "damping factors" are easily obtainable.

W~i =

Re{

Àtli}2

+

Im{ Àtli}2

(5.8)

2

Re{

ÀtI;}2

Çi =

Re{

ÀtliP

+

Im{ ÀtliP

such that

5.2

Validity of the Perturbative Solution and Special

Cases

The perturbative solution for the viscoelastic system is valid as written when one of the two following criteria is met.

• General Light Damping

The inclusion of viscoelastic element matrices in the structural model repre-sents a small change from the corresponding equilibrium elastic system:

• Small Loss Factor Materials

The magnitudes of the material moduli change little with frequency.

I.e. all 0:(3

«

1

Additionally the mode to be approximated is subject to a couple of restrictions.

• Distinct Elastic Frequency

For mode

'i',

the eigenvalue obtained from the equilibrium elastic analysis

must be distinct. The other modes of the system need not be subject to this.

0: =1=

i

More complicated eigenproblem perturbation formulae may be derived for the case of nondistinct elastic frequencies.

• Comparable Frequency

The equilibrium elastic frequency for the mode under consideration should be comparable in magnitude to the mini-oscillator frequencies. This is not a strict requiremen t.

w;

~ all w~

When

w;

is much different in magnitude than the w~ some modification to the

perturbation formulae is warranted to maintain validity.

A few special cases may be identified wherein some simplification of the pertur-bation formulae is possible.

1. Similar Moduli

The af3 for all the moduli of the strueture are equal or very nearly equal.

let all o.f3 = a. then

K~q = o.Kqq

and the perturbation formulae beeome:

I n .. (2A A) A2GAO\e( TST cT e)2

--w.o."" Çf3Wf3 Wf3 f3/\f3ei qf3 f3rf3 2 ,~ _f3=1 w_' · [( 2 _W A2)2 2(2 A A )2] i - Wf3

+

Wi Çf3Wf3 . { 1

.

+

JWi 1

+

20.···

and

(5.10)

(5

.

11)

The change in the spatial part of the eigenvector is only a sealing: the mode shapes remain the same.

2. Common Modulus

The entire structure shares a single eommon modulus, so that the global sys-tem may be written in the Laplaee-domain using MOT parameters as: