Finite Elements in Analysis and Design 7 (1991) 291-297 291 Elsevier

Vibration response of constrained viscoelastically damped plates: Analysis and experiments Y.S. Sh in a n d G.J . M a u r e r

Department of Mechanical Engineering Naoal Postgraduate School, Monterey, CA 93943, USA

Revised June 1990

Abstract. Direct frequency response analyses of the constrained viscoelastically damped flat plates were performed with modifications to account for the frequency-dependent properties of viscoelastic materials (i.e., loss factors and shear modulus). The modal strain energy method was also applied to each plate to compute the same type of responses. Comparisons of the results based on the direct frequency response analysis to the modal strain energy method were made for each plate. Experiments were also performed on the plates to establish the reference data for the comparisons.

I n t roduc t i on

A distributed thin viscoelastic material with an incorporated constraining layer has gained widespread acceptance as a means of passive vibration control of resonant response. A few prediction methods have been developed to predict the frequency response of constrained layered viscoelastically damped structural components and/or systems. However, accurate prediction of frequency responses of such systems is tedious and costly due to the frequency and temperature dependence of viscoelastic material properties such as shear modulus and material loss factor.

The response of the simple structures can be solved by analytical methods, but the finite element approach remains the method of choice when complex physical systems must be analyzed. One standard method of computing the frequency response of such damped struc- tures is through the repeated applications of the frequency response analysis over the incremen- tal steps in frequency that the proper viscoelastic material properties can be specified. A second approach is an approximation method using modal strain energy (MSE) [1]. Johnson and Kienholz [1] developed and applied the MSE method to three simple example problems for which other solutions were available: sandwich beams, sandwich rings and sandwich plates. The natural frequencies and modal loss factors were the quantities compared.

This paper investigates the frequency responses based on the two prediction methods described above as applied to the constrained layered viscoelastically damped flat plate. The first is called the direct method and the second the M S E method. Two different types of flat plates were used: free-free and clamped-clamped boundaries. The numerical results were compared with those of experiments.

Matrix equations of motion

T h e m a t r i x e q u a t i o n s o f m o t i o n fo r the d i s c r e t i z ed c o n t i n u o u s s y s t e m c a n b e e x p r e s s e d in t he f o l l o w i n g f o r m :

[ M ] { ~ ( t ) } + [ C ] { ~ ( t ) ) + [ K ] ( x ( t ) } -- { F ( t ) } , (1)

0168-874X/91/$03.50 © 1991 - Elsevier Science Publishers B.V.

292 Y.S. Shin, G.J. Maurer / Vibration response of oiscoelastically damped plates

where [M] is the mass matrix, [C] is the damping matrix, [K] is the stiffness matrix, (x(t)}, (~(t)}, (3/(t)} are the displacement, velocity and acceleration vectors respectively, and (F(t)} is the applied force vector.

Introducing the linear transformation from the physical coordinate system, {x(t)}, to the modal coordinate system, (q(t)}, into eqn. (1),

(/](t)} + [l~itoi] ( ~(t)} + [to2] { q(/)} = ( f ( / ) } ,

{ x ( / ) } = [ ~ ] ( q ( / ) } ,

( f ( t ) } = [ff~]T{F(t)},

(2)

(3)

(4)

where {//(t)}, {~(t)}, {q(t)} are the modal acceleration, velocity and displacement vectors respectively, [~ito~] is the diagonal modal damping matrix, [to2] is the diagonal modal frequency matrix, 7/i is the ith modal loss factor, to~ is the ith modal frequency, and [~] is the mode shape matrix. Equation (2) is the decoupled equation of motion in modal space.

For the MSE method, the ith modal loss factor, ~/~ in eqn. (2), can be approximated by the following expression [1]:

~i = ~v:i ( Vv:i/ VT:i ), ( 5 )

where Vv: ~ is the elastic strain energy in the viscoelastic layer at the ith modal frequency, VT: i is the total strain energy at the ith modal frequency, and 7/v:~ is the viscoelastic material loss factor at the ith modal frequency. Then the equation of motion (2) can be solved for the given forcing function.

Frequency response analysis

If the applied force is a harmonic function in time,

(F ( t )} = ( i f} e j" , (6)

and the corresponding response is also a harmonic function,

( x ( t ) } = ( x ) e j°', (7)

where { ff } is the applied force amplitude vector, (X } is the displacement response amplitude vector and to is the forcing frequency, then the coupled matrix equations of motion for the steady-state response in the frequency domain can be expressed by,

( - [ U l t o 2 +j[C(to)] + [ r ( t o ) ] ){ X(to)} = (F( to)}, (8)

where the frequency-dependent damping and stiffness matrices can be expressed as a summa- tion of each individual component,

N N

[c(to)l = E [cT(to)] = E (9) i=1 i=1

and N

[ r ( to)] = E [ r : ( to ) l , (a0) i=1

where [C7] is the/ th element damping matrix, [KT] is the ith element stiffness matrix, and 71~ is the i th element loss factor.

Then multiple numbers of direct frequency response analysis were performed using equation (8) to capture the effect of frequency-dependent viscoelastic material properties.

Y.S. Shin, G.J. Maurer / Vibration response of viscoelastically damped plates 293

For the MSE method, the decoupled modal equation of motion was used to calculate the steady-state response in the frequency domain and is expressed by:

(_to2[ I ] + jto[~,to,] + [ J ] ) ( Q ( t o ) ) = (f ' ( to) ), (11)

where

(q ( / ) } = (Q( to) ) e j'~t (12)

and

( ;(.)} =[+]T ( ;(.)};

again, *b is obtained from eqn. (5).

(13)

E x p e r i m e n t s

Two different plate type structures were used in this study. The constrained layered viscoelastically damped plate consists of a base plate, a viscoelastic thin layer and a constrain- ing layer. Two different types of base plate were used: a simple rectangular plate of uniform thickness and a rectangular milled plate machined at the center portion (Fig. 1). 3M ISD-112 was chosen for its high loss factor and reasonable shear modulus in the temperature range of 0 to 100 °F (Fig. 2). All base plates were machined from 6061 T6 aluminum block. The damping material applied to all models was 3M ISD-112 viscoelastic compound with a thickness of 1/16 inches (1.5875 mm). The constraining layer is 1/14 inch (6.35 mm) thick 6061 T6 aluminum plate. Each layer was bonded firmly by applying uniform pressure without entrapped air.

The input force was provided by a Wilcoxon Research F3 vibration generator mounted on the base plate. An HP-3562A Dynamic Signal Analyzer provided the control signal to the Wilcoxon power amplifier. A force transducer was mounted at the base of the vibration generator and was used to measure the input excitation force. This force input signal was amplified by a charge amplifier and fed to channel one of the HP-3562A. An Endevco miniature accelerometer was mounted at the base plate and coupled with signal conditioner, and fed to channel two of the HP-3562A. All measurements were performed with the

0.762 m (30 inches)

5

(a) Simple Damped Plate

; I 0 .0254m(1")

ilk

0.762 m (30 inches) 'r

I ~ 6.35mm(1/4") thick constraining layer 1.5875mm(1/16") thick viscoelastic layer 9.525mm(3/8") thick base plate

(b) Milled Damped Plate

Fig. 1. Constrained layered viscoclastically damped plate models.

294 Y.S. Shin, G.J. Maurer / Vibration response of viscoelastically damped plates

4 10

I 03

102

n- O I,- <0 101 LL

r,/) 0 • --= 10 ~

i0 -I

I 0 "2 10 "1 101

TEMPERATURE (DEG. F) 281 131 56 31 6 -19 -44 s

10

"103 Z~u.I / / ° 0

t0 ~ ~

" ~ 10

, n / / / . n ~ m , I ,I ~ Hm ,I Ill I~IJIIi 3 5 9 1011 13 10 10 10 7 10 10

REDUCED FREQUENCY (HZ)

Fig. 2. Reduced frequency nomogram for 3M ISD-112.

HP-3562A using a swept sine control signal, with ten averages at each data point. The measurement points for input excitation and output response are shown in Figs. 3 and 4. The average temperature across the plate was approximately 60 ° F.

§ I

excitation point

1 /

response measurement point

48 QUAD4 elements for constraining layer

I I ~ I I I I I I iii,ili,iiiiiii,iiiil,,ii,iiSi ~,iiiiiiii,~i~i~ili,ili,,tiii],iiiiiiiiiiii ~]iiiii ~iiiii,iiii,iiiiiii,,iiiiii,i,ii,iii

- - 48 QUAD4 elements for base plate

Fig. 3. Finite element model of simple damped plate.

\

\

exaltation point

7 /

/

Z response measurement point

48 QUAD4 elements for constraining layer

~ I I I I I I

~ - - 48 H E ~ elements for viscoelastic layer

80 QUAD4 elements for base plate

Fig. 4. Finite clement model of ndlled damped plate.

Y.S. Shin, <7..1. Maurer / Vibration response of viscoelastically damped plates 295

Finite element analysis

The finite element analyses were performed using MSC/NASa'gAr~ [2]. The base plate and the constraining layer were modelled using the four-node quadrilateral plate element QUAD4. For the viscoelastic layer, the eight-node solid element I-mXA was used to accurately represent the strain energy due to shearing of the layer. The finite element models are shown in Figs. 3 and 4.

The direct frequency response analysis using MSC/r~ASTRAN requires the viscoelastic material properties to be input as constants throughout the frequency range analyzed. This restriction limits the analysis to frequency spans over which the viscoelastic material properties have little change. As an approximation, a frequency span no greater than 100 Hz was used in the analysis. In order to give the best accuracy at the peak response values, the natural frequencies of each plate were estimated by modal analysis using the mid-range viscoelastic material properties. The shear modulus and loss factor of the viscoelastic layer were then obtained for each of these natural frequencies. Then the frequency response analysis was performed over 100 Hz spans centered at each natural frequency. The output responses were compiled and plotted as composite curves.

For the MSE method, modal analysis was per fo rmed to calculate the fraction of elastic strain energy in the viscoelastic material at each mode and to calculate the modal loss factor defined in eqn. (5). The viscoelastic material properties at constant mid-range frequency were used in the analysis. Once the modal loss factor of the constrained layered plate was calculated for each mode, modal frequency response analysis was performed once for each type structure.

R ~

A series of frequency responses was plotted for each plate type as shown in Figs. 5 through 8. Figures 5 and 6 show the experimental results for each plate type before and after the application of the viscoelastic and constraining layers on the base plate. A significant reduction of amplitudes at each resonance is observed.

=:: i==i=:~: =~=i=: :i~=:7:1==! ~ :!:1=::!==~==i:=i=~=1 • i..Jl..--.i .i-, ....... i ......... i ............. +-+-

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l o • . . . . . . ..... ' . ......,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . o.. . . . . . . . . : . . . . . . , . .~ . . ..... . . . . . . . . . . . . . . . . . . . . . . . + . - - . - ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . ~ . . . . . . + ' - oo " : o ' " " r "o"

• r ,; ............. .................................... ; ................... t .............. r'""v"r",'"'r'""r -':"""rr'l l . . . . . . . . . . . . . . . . . . . . . . " . . . . . . . . . . . . . . ; . . . . . . . . . . . . "v+~::l • ~ , o .............. + . . : : : : ~ : ~ . I ~ i : : : : .

I=::i ..................................... ~ ! ! i!!!!l . . . . . . . . . . . . . . :, . . . . . . . . . . . ~ . . . . . . . . " . . . . . . . ! . . . . . . i . . - . . ' [ . . . . . .

2 3

10 10 F R E O U E N C Y (HZ)

Fig. 5. Experimentally measured frequency response of simple plate before and after viscoelastic damping treatment.

2 9 6 Y.S. Shin, G.J. Maurer / Vibration response of viscoelastically damped plates

uJ a

_J

<

Fig. 6. Experimentally measured frequency response of milled plate before and after viscoelastic damping treatment.

The frequency response results were compared using (i) the direct method, (ii) the MSE method, and (iii) experiments for each plate as shown in Figs. 7 and 8. The experimental results show higher response amplitudes at each resonance than the direct and MSE methods. The results based on the direct and MSE methods compared fairly well.

The significant differences in amplitudes between experiment and analysis may be caused by real system damping being lower than predicted. Vibration energy absorbed by the viscoelastic layer is converted to heat and possibly increases the damping material temperature, resulting in a decrease of the loss factor. Another contributing factor may be the significant variation of measured viscoelastic material properties from one batch to another.

LU r',,

1--

10 !

<

10-i 5 " 1 0

. . . . . . . ; . . . . . . , . . . . . . ~ . . . . . ~ . . . . . , • • . = . . . . . . . . ÷ . . . . . . . ~

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i ~ / i i ~ . - i~ i i i i i i l ........... ! g ............... ! ~ " , ~ = ~ . ........ ~ ................................ ~ ................ : ................

~ , N i : ; ~ t • ........ ~ ~ - ~ : ......... ! ........ i l l . . . . .

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~ : " f " l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . "N ' t " " l " " r " l " : ~ , " " "~ . ' . " ' ~ " ' ~ "'1~ ~ " " ~ " ' I ' " ' T " '; ! " ~ ~ " ~ ' " " ~ " ~

L E G E N D : i " ~ ; : • '

D I R E C T M E T H O D

M S E M E T H O D

E X P E R I M E N T

. . . . . . . . . . . . . . . . . . . . . . . . . . . ÷ . . . . . . . . . . . . . . : . . . . . . . . . . . i . . . . . . . ~ - - . . - .~ . - . - - . : . - ...'~r •

I! " ' " ' ~ . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . . ": . . . . . . . . . . T ~ " ' 9 ~ ' " . . . . . . . ~ . . . . . . . . . . . . . . . . . . . . 6 . . . . . . . . . . . . . . " . . . . . . . . . . . ~ . . . . . . . . ÷ - - . - - - - ~ - - . . . - " - - . . " - . -

...... i .................... i .............. ........... ! ......... i! 10 2 10 3

FREQUENCY (HZ)

Fig. 7. Frequencyresponseofsimpledampedplateby ~ed~cc t method, the MSE method and by experiment.

Y.S. Shin, G.J. Maurer / Vibration response of oiscoelastically damped plates 2 9 7

......... ~-------~------:-----.*----i .................................... ~ ................... - .............. ~ ~ ........ ": ....... ~ -- 4 ~ ' • ' " : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . -." . . . . . . . . . . . . . . i . i . . . . . . . . -.' . . . . . . - i - - . . - - i . - :

i . . . . . . . . . . . ~. j . . . . . . . . . 1

........ -'-----.-;------,-.---~ ................ I "~---'- ......................................... ÷ ......................... i ....... - " ' " ~ ' ' " . ~ ......... "1 .......... ........L..-.i.....~.----I ........ .jt.....':~. .................. - ................... 4 .............. ~ ................... -." ............. ~---"-'-"1

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o ~ ~ I _~,.,,~ ! r,i~ ! i i i ~,, = , I /Z , ' \ : ~ 1 0 / :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ..................... " . . . . . . . . " . . . . . . . . . . . . . . . . . . . . . . : ............. Z'~ ................. : .................. • •

a_ ................ ~'~_~---I"; ............. ::.%.~,-~,..--, ............ -. -..--.~---~- ........ • .......... ~ ........ .~-...-'-"-----~--'--r---'l ........ - - - - .~ - - - . . ~ ....................... :~:~.-'--"~------;-r'~'~ ........ : ........... : ........ -.. ............. :""r ' ' l

10 -~ LEGEND ....... ; .................... ".: ............ : ; : . . : : : : : . . , : : : : : : : . ; . ~~ " : . . . . . . . , . .2

,.......~ ...................................... ~ . . . . . ~ : ~ , . ~ . ; ~ .

O,REOT ME~.OD t:-~...-.:..-.~:..:..:--..~::.-:-:.--.-.-..:---.=i=~~ MSE METHOD i "i" 'i" 'i ......... i ....... i ' ' " ' " ~ ' ~ " " .~ . . . . . . . . . - . . . . . . . ~ . . . . .

i i i ! i .i.....~ ~" EX~ER'MENT ' i i i i i i '

2 1113 5"10 10 FREQUENCY (HZ)

F i g . 8. F r e q u e n c y r e s p o n s e o f m i l l e d d a m p e d p l a t e b y t h e d i r e c t m e t h o d , t h e M S E m e t h o d a n d b y e x p e r i m e n t .

Cons ider ing the first resonance, the r e sonan t f requency based on the M S E m e t h o d is h igher than that of the d i rec t method . This is caused b y the fact tha t the viscoelas t ic ma te r i a l p roper t ies used in the m o d a l analysis for the M S E m e t h o d were the values in the mid - r ange frequency, resul t ing in higher stiffness and higher na tu r a l f requency in the lower f requency

range.

Conclusions

The m o d a l f requency response analysis resul t based on the M S E m e t h o d c o m p a r e d well wi th that of d i rec t f requency response analysis for the cons t r a ined v iscoelas t ica l ly d a m p e d p la te type structure, reassur ing tha t the M S E me thod will p rov ide a r easonab le solut ion. The s impl ic i ty of app l i ca t ion coup led with the low compu te r t ime requ i red makes the M S E m e t h o d the mos t a t t rac t ive choice for engdneering analysis .

The compar i sons of results be tween exper imen t and analys is ind ica te tha t numer ica l ly p red ic ted d a m p i n g m a y be higher than the actual , resul t ing in a p r ed i c t ed r e sonan t response which is s ignif icant ly lower than the actual . Since one of the con t r ibu t ing fac tors m a y be the s ignif icant va r ia t ion of measured viscoelast ic ma te r i a l p rope r t i e s f rom one b a t c h to another , s ta t is t ical analysis on viscoelast ic mate r ia l p rope r t i e s is needed to de t e rmine a r easonab le b o u n d for design values.

References

[ 1 ] J O H N S O N , C . D . a n d D . A . K I E N H O L Z , " F i n i t e e l e m e n t p r e d i c t i o n o f d a m p i n g i n s t r u c t u r e s w i t h c o n s t r a i n e d

v i s c o e l a s t i c l a y e r s " , AIAA J. 20 (9) , p p . 1 2 8 4 - 1 2 9 0 , S e p t e m b e r 1 9 8 2 .

[2] JOSEPH, J . A . ( ed . ) , MSC/NASTI~N Users Manual, V o l s . 1 & 2, M a c N e a l - S c h w e n d l e r C o r p . , L o s A n g e l e s , C A .