of forcing frequencies and temperatures on free-layer viscoelastic damping treatment of plates ... moduli as a function of temperature, since polymer moduli are.
Sung Yi Lecturer. School of Mechanical and Production Engineering, Nanyang Technological University, Singapore, 2263
Dynamic Responses of Plates With Viscoelastic Free Layer Damping Treatment
M. Fouad Ahmad Research Scientist.
H. H. Hilton Professor Emeritus.
Dynamic transient responses of plates with viscoelastic free damping layers are studied in order to evaluate free layer damping treatment performances. The effects of forcing frequencies and temperatures on free-layer viscoelastic damping treatment of plates are investigated analytically. Young's modulus ratio of structures to viscoelastic damping materials and the damping layer thickness effects on the damping ability are also explored.
Aeronautical and Astronauticai Engineering Department and National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana, iL 61801
Introduction The ability to control vibrations and noise is of great importance in automobiles, flight vehicle structures and electronic devices. Viscoelastic materials have been and are being used as shock absorbers, vibration and flutter dampers, acoustical barrier, etc., because of their ability to reduce structural fatigue and vibration and attenuate structure-borne noise. When deformation energy is lost as heat through viscous action then this mechanism is known as viscoelastic damping, which is solely due to material properties. Structural damping on the other hand refers to energy dissipation at joints, fasteners, and interfaces. Structural damping is due to dry solid friction and its constitutive relations are independent of frequency, displacements, velocities and accelerations. For this case, elastic stiffnesses can be replaced by frequency independent complex ones (Hilton, 1991). Viscoelastic materials, however, obey differential or integral stresses-strain relationships, which are associated with stresses, strains and their time derivatives. Lazan (1968) has called structural damping rate-independent damping since material properties such as Young's modulus and specific damping coefficients are all independent of strain and stress time derivatives, frequency and temperature, while viscoelastic damping corresponds to rate-dependent damping where constitutive relations are functions of those variables. Viscoelastic and structural damping have been analyzed by Hilton (1991) using generalized Kelvin models. Hilton and Yi (1992) and Yi (1992) formulated expressions for stored and dissipated energies in anisotropic viscoelastic bodies using generalized Maxwell models, and the relationships between shapes of master relaxation modulus curves and dissipation energy and the latter's influence on passive structural motion control have been investigated. Recently, Yi (1992) also developed more computationally efficient and accurate algorithms for analyzing transient responses of viscoelastically damped composite structures in the real time domain than the previous studies of Bagley and Torvik (1983), Holzlohner (1974), Golla and Hughes (1985), Johnson and Kienholz (1982), and Yamada et al. (1970). Recursion formulas also were obtained in order to reduce computer storage, and only two previous time solutions are required to compute the next time solution. Contributed by the Technical Committee on Vibration and Sound for publication in the JOURNAL OF VIBRATION AND ACOUSTICS . Manuscript received July 1993; revised April 1994. Associate Technical Editor: S. C. Sinha.
Free layer damping treatment is the simplest procedure for introducing light weight damping into structures (see Fig. 1). In the forced vibration analysis, overshoot and settling time indicate the behavior of the transient response and these are important parameters in designing damped structures. Therefore, in order to effectively design damping treatment, investigations of such parameters by means of transient analysis are essential. In this paper, forced oscillation of a plate with viscoelastic free-layer damping treatment is studied using the finite element method. The loading frequencies and temperature effects of the damping treatment on plates are examined and the damping layer thickness effects on the damping performances are also investigated.
Geometric Relations and Viscoelastic Damping Mechanics Based on the Mindlin plate theory which accounts for transverse shear deformations, the time-dependent displacement fields for plates in a plane stress state are taken as u = -z^^(x, t),
V = -zOy(x, t),
w = ^ ( x , t)
(1)
where w is the transverse displacement, and 9^ and 9^ are the small angles of rotations of lines which are normal to the undeformed mid-surface in the x and y planes, respectively. The geometric relationships can then be expressed as exi'ii,
t)-- = -z
(d9, t) = —z \dy
89, dx
e,(x ,
dx ) •
?«(> :, 0
=
dw 'dy
0 = -z ae. dy '
r«(x> t) -e,.
dw dx (2)
Master relaxation modulus curves for viscoelastic materials can be determined by using the time temperature superposition principle or vibrating beam tests (Hilton, 1964). One way to characterize viscoelastic material properties is to measure their moduli as a function of temperature, since polymer moduli are functions of time as well as being highly sensitive to temperature. By using time-temperature superposition, moduU master curves and shift factors are generated. The superposition principle states that the viscoelastic modulus at one temperature can
362 / Vol. 118, JULY 1996
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where i = v — 1. The complex moduli can be separated into the storage moduli Cjj(x, uj) and loss moduli C"j(x, u>) Cij(x, ui) = Cyix, a>) -I- iC'ljix, )-^t]
UMo)
}j
II
11
ii
\ -
Rmrpitp)
...
(17)
1
•ji
il
M
\ \
-fi
j.j ii •
•ij
11
:|| •i'iii
•ij •ij
1
li
..
i!ii
-0.2
'
li
....... Ti
eXp[-C,Atp)/Kp]-
il
... •fi
0.2
-
11
-
U„{tp^x)
- S%(Atp)--^-[U,(tp^t)
ii
w
+ U„itp-x)
ti
it \ •n
-- ... ll1
III!
1!
fi
If
•it
il
^ •if
tv
•fi-
•«ii
\ \ iifl
•il
.li •ii
lit
Ijjs N i
li
*Ni
It
••
•fj'
ii-ii
i
•J
where 10"
5Ur,.) =
= Utj) -
10'
(sees)
Fig. 3 Master curve for viscoelastic damping material Young's modulus at 18°C
Utj-i)r
R'nrpito) = 0,
Discussion of Results
R':„rpitp) = exp[-AC.(/p)/X,J ^ \Knmrp' \^rpitp-\)'
10"
10
Time
S^iAtj) = j ' j Slpis')ds'ds, ^Utj)
10
J'exp[-(;,-C;)/M ds.
f/„(f;,-l) " S rpitp-2) ' U „itp-2)
- S^^piAtp^,)- t / r ( ? , - , ) } + /?f„,,(?^_,)].
(18)
Use of these recursion formulas makes it possible for the solution to be marched forward in the time domain until the desirable time is reached. Memory storage depends on the number of relaxation moduli, the expanded Prony series terms, and degrees of freedom, etc., but does not depend on the number of time step. Additional details for the transient viscoelastic finite element formulation can be found in Yi (1992).
A numerical example is presented to verify the accuracy and convergence of the analysis of transient responses of viscoelastically damped structures. For the sake of simplicity, consider the one degree of freedom system which is excited by a unit step load. The time-dependent stiffness of the system is described by the following Prony series Kit) = Ko-ll + e x p ( - 1 0 0 ]
(19)
where Ko is 100 Pa and the mass is 100 kg. Then the exact longitudinal response of a single degree of freedom system becomes Uit) = 10^^ - 1.02 X 10-*-exp(-9.90 - exp(-0.050
2.0 10"'
X [10^^-cos iait) + 5.0817142 X lO^'^-sin ia^t)]
1.510"'
•
present numerical solution closed form solution steady state response
0.0 1 0"
I
-5.010'' 0
10
20
I
I I I I
30
I l l
40
50
Time (sees) Fig. 2 Step response of a single degree of freedom system
364 / Vol. 118, JULY 1996
60
(20)
where ai = 1.00379305. The dynamic response calculated by the present numerical procedure is compared with the closed form solution in Fig. 2. Two solutions are in excellent agreement for a time step Af = 4 X 10"^ sees. Consider plates simply supported along all edges. MindlinReissner type theories which account for first order transverse shear deformations are employed in the present study. Twenty five 9-node isoparametric plate elements with three degrees of freedom for each node are used. The material behavior of the plate is linearly elastic with properties for the isotropic plate E, = 68.95 GPa (10 Msi), v,, = 0.3 and y,, = 9134.4 kg/m' (0.33 lb/in'). The plate dimensions are 0.508 m (length) X 0.508 m (width) (20 in. X 20 in.) and the thickness h, is 0.0127 m (0.5 in.). Isothermal conditions are assumed throughout the loading history. The viscoelastic damping layer is attached to the top of the plate as shown in Fig. 1. Young's modulus of the viscoelastic damping material is time and temperature dependent and the master relaxation curve for the viscoelastic damping material at 18°C is plotted in Fig. 3. Poisson ratio is taken to be constant with respect to time. At t = 0 sees, Young's modulus E^iO) for the damping material is 13.79 GPa (2 X 10^ psi), and Poisson ratio i/^ = 0.3 with material density y^ = 6920 kg/m^ Transactions of the ASME
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RI
18 "C
ni
a.
n. .X E
2.5 1 0 '
-40°C
2.5 10"^ 11
• • ^
o OL
0)
E
g
1.5 10"'
;ii :'* ;: M ;; ^;
"-
^irt i'! i,i M n M
c
2.010"'^
1.510"^
i:i i'-i i/i lA i'^
a> o (0
a m U (U V) 0)
1.010"^
5.0 10"''
tfi
'
1 ,' ? (
c
10
LJ_I
0.0 10"
0.4 0.2
0.4
0.6
0.8
1.2
1.6
0.8
Time (sees) Time
(sees)
Fig. 4 Vibration response at center of plate witfi free-layer damping treatment (EJE, = 5, T = 40°C)
(0.25 lb/in'). Time step size is taken as A? = 10"' sees and shift factor is 10 at 40°C. First, the effects of free damping layer thickness are studied. The plate is subjected to a uniformly distributed unit step load P„ = 271.46 Pa at a temperature of 40°C. Plates with various damping layer thicknesses (h,,) are considered and the corresponding transverse responses of square plates are calculated. Dynamic transverse deflections at the center of the damped plates with five different damping layer thicknesses are plotted in Fig. 4. The results show that vibration amplitudes significantly decrease with increasing damping layer thickness. The vibration is damped out after 1. sees for h^/h, = 0.6 and 0.6 sees for hjh^ = 1 respectively. Increasing the damping layer thickness also results in decreasing vibration amplitudes at the initial time and vibration frequencies. For thicker damping layers, small creep deformations occur because of large amounts of damping material relative to the plate. Another numerical example is undertaken in order to show the effects of temperature on the performance of viscoelastic free-layer damping treatment. The plate is subjected to a uniformly distributed unit step load at 18°C (shift factor = 1) and 40°C (shift factor = 10). It can be readily seen from Fig. 5 that changes in temperature affect both amplitude and frequency responses. This is due to the smaller damping material capabilities with decreasing temperatures. Fast material degradation induced by higher temperature results in smaller amplitudes and larger vibration frequencies. Similar results were also observed in the previous study (Hilton and Yi, 1992). As shown in Fig. 5, after 2 sees, the vibration amplitudes for plates with damping treatments at 40°C are reduced by 95 percent, while those at 18°C decrease only by 56.7 percent. During the same time period, the relaxation modulus of the damping material at 40°C degrade by 99 percent while that at 18°C reduced by 95 percent. Young's modulus ratio (EJE,,) effects on damping are also investigated. Three models for EJE,, = 5, 10, 100 are used. As seen in Fig. 6, the results indicate that the damping capacity decreases as the Young's modulus ratio increases. At EJEj = 100, the vibration amplitudes are reduced only by 9 percent in a 2 sees period. The damping material with larger initial stiffness degrades more. More energy is dissipated for the larger initial stiffness (t = 0) of the damping material. However, for larger Journal of Vibration and Acoustics
Fig. 5 Vibration response at center of plate witti free-layer damping treatment [EJE^ = 5, h^/he = 0.2)
initial moduli, the vibration frequency increases since the stiffness of the structure increases. Next, the time dependent responses of viscoelastically damped plates subjected to uniformly distributed sinusoidal loading, P{t) = P„ sin (ujt), are studied. The Young's modulus ratio is 5 and a time step Af = 2 X 10"' sees is used. Two different forcing frequencies u> = 10^ and 10' rad/sec are employed in this study. At a; = 10^ rad/sec, dynamic responses for plates without damping treatment are evaluated and plotted in Fig. 7. Also the transverse defections of the damped plates are calculated at 40°C are illustrated in Fig. 7. Vibration amplitudes of the plate decrease rapidly and reach a steady state after about 0.9 sees. Those are also significantly reduced by applying viscoelastic damping materials, i.e., after 2 sees, they decrease
4.010' E/E .= 5 s c s d - E/E.=100 s a
(II Q.
5,
C
