Optimum thickness distribution of unconstrained viscoelastic damping layer treatments for plates

Journal ofSound and VibroGon (1985) 103(2), 183-199

OPTIMUM THICKNESS DISTRIBUTION OF UNCONSTRAINED VISCOELASTIC DAMPING LAYER TREATMENTS FOR PLATES? A.

YILDIZ

Faculty of Naval Architecture, Technical University of Istanbul, Istanbul, Turkey AND

K. STEVENS Department of Mechanical Engineering, Florida Atlantic University, Boca Raton, Florida, U.S.A. (Received 2 September 1982, and in revised form 24 October 1984)

This paper concerns the optimum thickness distribution of unconstrained viscoelastic damping layer treatments for plates. The system loss factor is expressed in terms of the mechanical properties of the plate and damping layer and the layer/plate thickness ratio. Optimum distributions of the thickness ratio that maximize the system loss factor are obtained through sequential unconstrained minimization techniques. Results are presented for both simply-supported and edge-fixed rectangular plates with aspect ratios of 1.0 to 4.0. These results indicate that the system loss factor can be increased by as much as lOO%, or more, by optimizing the thickness distribution of the damping treatment. Also revealed are the regions of the plate where added damping treatments are most effective.

1. INTRODUCTION Many structures include plates and panels which are subject to flexural vibration. Attenuation of this vibration is important in reducing material fatigue, vibrational discomfort and noise problems, and it is frequently accomplished by the use of viscoelastic damping treatments. Viscoelastic damping materials usually are used in one of two basic configurations, constrained layer and unconstrained layer. Unconstrained layer treatments, also referred to herein as viscoelastic coatings for brevity, consist of a layer of viscoelastic damping material bonded to the surface of the structure. In constrained layer treatments, the damping layer is overlaid with, and bonded to, a stiffer constraining layer. Energy, in both configurations, is dissipated in the viscoelastic material as a result of the cyclic deformations induced by the flexural vibration of the underlying structure. Both types of damping treatments have their respective advantages and disadvantages, which are well established and have been discussed in the extensive literature on the subject. Simply put, unconstrained layer treatments (coatings) are effective for thin plates and panels, while constrained layer treatments are effective for thicker members. Only viscoelastic coatings are considered in this paper. A comprehensive review of the literature on viscoelastic damping treatments has been given in references [l, 21, and a good discussion of the current state of the art may be found in reference [3]. t This work was completed while the authors were associated with the Department Florida Atlantic University. Their support is gratefully acknowledged.

of Ocean

Engineering,

183 0022-460X/85/220183

+ 09 $03.00/O

0

1985 Academic

Press Inc. (London)

Limited

184

A. YILDIZ

AND

K. STEVENS

Effective design of damping layer treatments for plates and panels often requires that material costs and added weight be held to a minimum. This requirement can be met only by judicious trade-offs between the amount of damping material used and the amount of damping achieved. One way to proceed is to use a partial treatment. It is intuitively obvious, and has long been recognized, that the damping achieved by a complete treatment is not significantly greater than that of a partial treatment extending over some lesser fraction of the surface area. This result has been confirmed, both analytically and experimentally [4-61. Another way to proceed is to vary the thickness of the coating in such a way as to maximize the damping achieved with a fixed amount of damping material. This procedure leads to a nonlinear constrained optimization problem, and is the subject of this paper. While the optimum distribution of damping material along a beam has been studied by several investigators [7-111, this is believed to be the first application of optimization methods to the design of damping layer treatments for plates. The general approach used is described, and results are presented for three flexural modes of simply supported and edge-fixed rectangular plates with a typical damping treatment. 2.THEORY 2.1.

BASIC

APPROACH

The damping of a mechanical system can be expressed in terms of the system loss factor, 7, which, for steady state vibration, is the ratio of the energy dissipated per radian and the maximum stored strain energy of the system. However, rather than use this definition directly to compute the loss factor, it is simpler to use a different, but equivalent approach. By making use of the correspondence principle of Bland [12] for steady state sinusoidal oscillations of viscoelastic systems, it is possible to compute the loss factor (and natural frequency) of the damped system from the Rayleigh quotient for the corresponding elastic system [ 131. This is done by using, in the strain energy expressions, the complex moduli of the viscoelastic components in place of the elastic moduli of these same components in the corresponding elastic system. This approach has been confirmed and verified experimentally for plates with viscoelastic coatings [4-61. Consider the problem of an elastic plate with a viscoelastic coating on one side, as illustrated in Figure 1. In this figure, the plate is depicted as rectangular and the coating thickness as varying in a piecewise fashion over the surface. However, at this point in the analysis, the plate and coating need not be restricted to any particular geometries. For the corresponding elastic system, the Rayleigh quotient can be written as w2= U/T,

Figure 1. Plate with damping layer of varying thickness.

(1)

OPTIMUM

UNCONSTRAINED

DAMPING

185

LAYER

where U is the maximum strain energy, wZT is the maximum kinetic energy and o is the frequency of oscillation (a list of notation is given in the Appendix). For a purely elastic system, o is real-valued. For a system with viscoelastic components, it will be a complex-valued quantity which can be expressed in one of the following forms: w2= wz(l +in), or o’=Re(~~)+iIrn(w~). 0 2=wr+lwi, or (2) It is shown in reference [14] that o, is the natural frequency of the damped system. The loss factor is given by the expression I These relationships

= Im (02)/Re (6~‘).

(3)

can also be obtained by inspection from equations (2).

2.2. KINETIC AND STRAIN ENERGIES Consider a purely elastic system corresponding to a plate with a viscoelastic coating, as shown in Figure 1. The surface of the plate is assumed to be divided into m elements of area, Uj, with the thickness of the coating constant over each element, but variable from element to element. Co-ordinates are taken as shown, with z = 0 corresponding to the neutral surface. Subscripts p and c denote the plate and coating, respectively. The following assumptions are made: (i) transverse shear and rotatory and in-plane inertia effects in both the plate and the coating are negligible for the lower modes of vibration; (ii) there are no applied in-plane loads; (iii) displacements are small and changes in thickness are negligible; (iv) the viscoelastic coating is applied to only one side of the plate; (v) the plate and the coating are homogeneous and isotropic and subjected to a state of plane stress; (vi) displacements are continuous across the interface between the plate and the coating; (vii) Poisson’s ratio of the coating is a real constant and the coating is incompressible (v, = 0.5). Under these assumptions, classical plate theory can be used to determine the system strain energy, which can be expressed as the sum of the bending strain energies of the plate, U, and of the coating, U,: u=

up+ u,

(4)

The bending strain energy in a plate is [ 151 2u,=

(m,k + myky + m,k,)

dx dy,

where m, and m,,are the bending moments and mX,,is the twisting moment; k, &,,and kx, are the curvatures of the deflected neutral surface. Note that the assumptions made imply that the plate and coating undergo the same lateral deflections. Expressed in terms of the lateral deflection, W,the curvatures are 5,=-%.X,

h = -wyv,

kxy=-WXy,

(6)

where the subscripts on w denote partial derivatives. Addition of the damping layer causes the neutral surface of the system to shift toward the coating and away from the mid-plane of the plate. If the unknown distance from the neutral surface to the interface between the plate and the coating is denoted by H, the bending and twisting moments for the jth element of the plate are [15] H, m,=

H w,zdz,

my=

’

"1 u,,z dz,

mXY=

r,yz dz, (7) I -(fp-“,) I -_($.-H,) I-_(f,,-H,, where a, a,, and rV are the stress components and rp is the thickness of the plate. The condition that the resultant normal force on any transverse cross section must be zero in

186

A. YILDIZ

AND

K. STEVENS

the absence of applied in-plane loadings leads to the expression “,

H,+t crxp

I -($-“,)

dz+

I “,

uXx,dz = 0,

(8)

’

from which Hj can be determined. Here, 6 is the thickness of the jth element of coating. The stresses are [15] ffX= {F/(1 - v2)](KX+ v&y),

ry = {F/(1 - vZ)&+

%), (9)

rxy = {F/2(1 + v)]r,,,

where E is the modulus of elasticity, v is the Poisson’s ratio, and the strains E,, ey, and rXy are E, = -zw,,, Combination

Ey= -zwvv,

YXY = -zw,y

(10)

of equations (8)-( 10) yields, after simplification, Hj=(K,W,t,-K,W,t,)I2(K,W,+K,W,), Kp = Fpt,J(l-

Kc = E&l (1 - v:),

v;),

(11) wp = WXX + vpwyv,

w, = w,, + vcwyv

(12)

At this point, and only in equation (ll), it is asumed that the Poisson’s ratios for the plate and for the coating are approximately equal ( vp = v,). With this assumption, equation (11) reduces to Hj = tp( l-

nh:)/2( 1 + nhj),

n=EJE

P

(13714)

being the modulus ratio of the coating and plate and hj= tj/tp

(15)

the thickness ratio for the jth element. For typical plates and coatings, ncc 1, and the shift in the neutral surface is slight. Accordingly, the assumption of equal Poisson’s ratios in the determination of Hj introduces little error in the strain energies. Combination of equations (4), (6), (7), (9), (10) and (13) yields, for the strain energy in the plate, 2up=: j=l

II

D,[(V*w)*-2(1-

v,)G(w)] dx dy,

(16)

4

G(w) = k&,, - k’, = wxxw,,,, - w&

(17)

being the Gaussian curvature and DP=EP(t;-3t;Hj+3tpH;)/3(1-v;)

(18)

the flexural rigidity, modified to account for the shift in the neutral surface. Expressed in terms of the flexural rigidity of the bare plate, Do, Dp=40~[1_3Hj+3ti~], where D,, = Epti/ 12( 1 - vi)

and

(19) Hj= Hj/tp

(20921)

The strain energy in the coating is obtained in the same way as the plate strain energy. However, integrals through the thickness now range from z = Hj to z = Hj + tj and, of course, the properties of the plate are replaced by the properties of the coating. The strain

OPTIMUM

UNCONSTRAINED

DAMPING

187

LAYER

energy in the coating is 2u,=

Q[(V2w)‘-2(1 f j=l II 9

- v,)G(w)] dx dy,

0, = E,[ri’+3H,tf+3H,t,]/3(1-

(22)

v:,

(23)

being the modified flexural rigidity of the coating. Expressed in terms of the bare plate flexural rigidity, D,,, 0, =4D,n[hj+3hftij+3ZrjH;](1

- z$)/(l-

yf).

(24)

The kinetic energy factor, T, for the system is 2T=

p&w2 dx dy,

pPfPw2dx dy + : ;=1

(25)

where p is the mass density. 2.3.

EDGE-FIXED

AND

SIMPLY

SUPPORTED

PLATES

Attention is now restricted to simply supported or edge-fixed plates, for which the integral over the plate area of the Gaussian curvature terms in the strain energy expressions is zero. Using this result and substituting equations (16), (22) and (25) into equation (l), one obtains, for the Rayleigh quotient,

co2= Do

2 [A,Z; +BjZ," +C,Z,x+DjZF]

P&J,+

j=l

B, = -2( 1

Aj=4(1-3Gj+3Z?;),

(V2w)* dx dy,

Z,x”= I “I

(PJj)Z,""

1,

(26)

vp)Aj,

Dj = -2( 1 - VJ Cj,

cj=4n(hj+3hfHj+3hjB:)(l_v~)/(l_vf), Zj”=

-

f

j=l

Z,“”=

G(w) dx dy, w2 dx dy.

w2 dx dy,

(27)

The coefficients Aj-Dj are material and thickness parameters, while the integral parameters Zj and Z,, are functions of the plate mode shapes. Damping treatments encountered in practice almost always have a damping layer whose modulus is much less than that of the underlying structure (n = EC/ EP <<1). The parameter n enters into the analysis through the coefficients Aj-Dj in equation (27). Expanding these coefficients into a power series in n, and assuming that the coating is not significantly thicker than the plate so that nh also is small, one obtains from equation (26), with v, = 0.5,

1

flJ2=CO;1+ n f ajLj j=l

I/[

1+p:ll,L, j=l

1 .

(28)

In this equation, WE is the natural radian frequency of the undamped plate, Lj and Lj are integral parameters, cr is a thickness parameter, and p is the density ratio:

d = Wp,tp

4 = zj”/zp,

L,=(z,“-z,yz,

aj=(4/3)(4hj+6hf+3hj)(l-

vg),

P = Pcl Pp.

(29)

188 2.4.

DETERMINATION

OF

THE

A. YILDIZ

AND

K. STEVENS

NATURAL

FREQUENCY

AND

LOSS

FACTOR

Up to this point, the coating has been assumed to be elastic. It is now allowed to be viscoelastic, and the coating modulus, EC,is replaced by the complex modulus Er of the damping material. One has ET=E,(l+in,), (30) where EC and 7, are the storage modulus and loss factor of the viscoelastic material, respectively. Alternatively, one can replace the modulus ratio n with n*=

n(l+i7jC),

(31)

where n now is the ratio of the storage modulus of the damping material and the plate modulus. The damping of the bare plate also can be included by replacing the elastic modulus, EP of the plate by E,*=EP(l+iTP),

(32)

where nP is the plate loss factor. However, Q typically is quite small, and the total damping can be obtained with sufficient accuracy simply by adding nP to the value of 7 resulting from the damping treatment [ 141. In this spirit, it is assumed that qP = 0 in the analysis. Substituting equation (31) into equation (28) and making use of equations (2), one obtains for the system natural frequency and loss factor

_IL =n 77,

f j=l

ffjLj

/(

l+n

f j=l

ajLj . )

(34)

The numerator in equation (33) represents the modification to the bare plate natural frequency of the added stiffness of the damping treatment, and the denominator represents the effect of the added mass. Equation (34) indicates clearly that the system loss factor depends not only upon the modulus ratio and thickness of the coating, but upon the distribution of coating thickness, as well. Even though the analysis yields the system natural frequency, in this paper only the loss factor is considered. 2.5. COMPUTATION OF THE LOSS FACTOR In order to compute the loss factor, it is necessary to evaluate the integral parameters Lj in equation (34). To do this, the plate area is divided into sixteen elements of equal size, as shown in Figure 2. The coating thickness is taken to be constant over each element, but variable from element to element. More elements could be used, and might yield improved values of damping. However, there is a practical limit to the number of different pieces of damping material that can be accommodated without incurring excessive labor costs for installation. Because of the symmetry of the boundary conditions, values of Lj are symmetric with respect to axes through the center of the plate and parallel to its sides. The thickness distribution of the coating is also taken to be symmetric about these same axes. Accordingly, corresponding elements of coating on opposite sides of the axes of symmetry will provide equal contributions to the overall system damping (assuming the same damping material throughout). Because of the assumed symmetry of the thickness distribution, there are only four different coating thicknesses, t,-z~, to be considered. These are numbered as shown in Figure 2.

OPTIMUM

UNCONSTRAINED

DAMPING

LAYER

189

(a) v

0.520

0.240

Figure 2. Assumed damping layer thickness distributions for (a) edge-fixed plates and (b) simply-supported plates.

It has been shown [ 161 that addition of a typical damping treatment has little effect upon the flexural mode shapes. Hence, the lateral deflection amplitude, w, in the integral parameters Lj is taken to be the mode shape of the undamped plate. The accuracy of this procedure has been demonstrated analytically and experimentally [4-61. For edge-fixed plates, the flexural mode shapes can be represented with good accuracy by a single product of the mode shapes for beams with fixed ends [ 171. Thus one can take w = F,(x)G,o)),

(35)

F,,,(x) = (cash /3,x - cos p,,,x) - q,, (sinh /3,x - sin &,,x), G,(y) = [cash /?.y -cos p,,y) - cu,(sinh P,,y -sin Pny).

(36)

Values of the parameters LY,,CY,,/3,,,and /In are given in reference [ 181, and expressions for the various integrals of F,(x) and G,(y) involved in the computation of the parameter Lj may be found in reference [19]. The integers m and n correspond to the various flexural modes. For simply supported plates, the exact mode shapes are sine functions of the form w=sin(m7rx/a)sin(~y/b),

(37)

where m and n are integers which define the various modes and a and b are the plate dimensions. 3. OPTIMIZATION 3.1. STATEMENT OF THE PROBLEM As mentioned previously, the objective of this study was to determine the thickness distribution of a given amount of damping material which maximizes the system loss

190

A. YILDIZ

AND

K. STEVENS

factor. For the symmetric thickness distribution shown in Figure 2, with a given value 01 the modulus ratio n, the loss factor is a function only of the coating-plate thickness ratios, hP Thus, the problem is to determine the values of the four thickness ratios, hi, hz, h, and hq, such that (38) is a maximum. By dividing the numerator and denominator in equation (38) by cj ajL+ it can be seen that the objective function can be rewritten as max F( hj)+max The optimization

C CujLp

(39)

is carried out subject to the following constraints: i) f( hj) = Constant; iii) hj s h,

j=l,4;

ii) hj>O,

j=l,4;

h is a parametric constant.

(40)

The first constraint fixes the amount of damping material. Here it is assumed that it is a volume of material equal to that of a complete damping treatment with uniform thickness the same as that of the plate. By referring to Figure 2, this constraint can be expressed as 0*2704h, + 0*2496h2 + 0.2496h3 + 0.230431, = 1

and 0*2500h, + 0.2500h2 +0*2500h,

+O-2500h,

= 1,

(41)

for the edge-fixed and simply supported cases, respectively. The second constraint is introduced to ensure non-negative coating thicknesses, while the third constraint ensures finite coating thicknesses and fixes the maximum admissible thickness ratio, h. In this study, h was varied from l-0 to 2.0, in increments of 0.1. The problem at hand is a non-linear, constrained optimization problem. In order to achieve a solution, a new function was constructed by adding penalty terms and Lagrange multipliers to the original objective function in such a way that the new function has an unconstrained maximum at the same point as the maximum of the given constrained problem. By gradually removing the effect of the constraints on the new objective function by controlled parameters, a sequence of unconstrained problems that have solutions converging to a solution of the original constrained problem was generated. The algorithm used was similar to those presented in references [20-231, and is described in detail in reference [24]. 3.2. OPTIMIZATION RESULTS Optimum thickness distributions were determined for three flexural modes of edge-fixed and simply-supported plates with aspect ratios ranging from 1.0 to 4.0, in increments of O-5. The modes considered were (m = 1, n = l), (m = 1, n = 2), and (m = 2, n = 2). For square plates, these correspond to the first three flexural modes. For other cases, the particular modes to which these values of M and n correspond depend upon the plate aspect ratio [25]. The maximum admissible thickness ratio was varied from 1.0 to 2.0, in increments of 0.1. It was assumed that vP = 0.3 and n = O-01, which is representative of a commercially available damping material on an aluminum plate. The percentage increase in the loss factor ratio, n/n0 obtained with a coating of optimum thickness distribution over that obtained with the same amount of material

II 0 0t1 Of1 011 E6 LL E9

EP

zs

fZ.0

cz.0

69 es fC SI 0

25 IZ.0 81.0 91.0 VI.0 OLI 091 OPI OZI 001 Z8 99 IS ZE 51 0 EC-0 ZE-0 62-0 LZ.0 sz.0 EZ.0 1z.o IZ.0 81-O 91-o PI-0

fP SZ

8E.O LE.0 PC-0 ZE.0 62-O 9z.o EZ.0 IZ.0 81.0 91-O PI-O OPI Of1 011 L6 18 L9 LS OS IE PI 0

I=M)

-

00.0 00.0

L

00.0 so.0 9z.o 9t.o L9.0 88-O 60-I Of.1 oz.1 01.1 00.1

60.1 OE.1 oz.1 01.1 00-I

P8-I 06.1 08-I OL.1 09.1 OS.1 OP.1

P8.1 06.1 08.1 OL.1 09-I OS.1 OP.1 05.1 oz.1 01.1 00-I

09.1 OE.1 oz.1 01.1 00.1

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OP.1 OE.1 oz.1 01.1 00.1

0.z 6-I 8-I L.1 9.1 s.1 P.1 E.1 z.1 I.1 0.1

P.1 E.1 z-1 I.1 0.1

s.1 =bl

o.z=x

Of.1 oz.1

WeId

0*1=x

alq!ss!wpe

0.z .6-I 8-I L.1 9.1 s.1 P-1 E-1 z.1 I.1 0.1

(I=U=uJ JOJ) ssauqcq~ urtuqdg

i

0O.Z 06.1 08.1 OL.1 09-I OS-1 OVI 05.1 oz.1 01.1 00.1

EE.0 L9.0 00.1

ZL PS PE 91 0

LE.0 9E.O EC.0 OE.0 8Z.O sz-0 EZ-0 IZ-0 81.0 91.0 PI.0

(I=U

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IZ-0 81.0 91.0 PI.0

IZ.0 61.0 LI.0 s1.0 PI.0 081 OLI OS1 OEI 011 06 IL PS PE 91 0

Jh/h

00.0 00.0 00.0 00-O 00-O 00-o 00-O 00.0 EC.0 L9.0 00.1

zc.0 IE.0 6Z.0 LZ-0 PZ.0 zz.0 IZ.0 oz.0 LI.0 SI-0 PI.0 OLI 091 OPI OZI 001 P8 L9 zs ZE SI 0

I=uJ)

9E.O 9E.O EE-0 OS-0 LZ.0 sz.0 EZ.0 IZ.0 81.0 91.0 PI.0

95 II 0 ZE.0 oc-0 82.0 9z.o tz-0 ZZ-0 oz.0 oz.0 LI-0 SI*0 @I-0

(z=u

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01.1 00.1

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00.0 SO-0 9z.0 W.0 L9.0 88-O 60.1 OE.1 oz.1 01.1 00-I

OEI ozr 011 88 EL 09 OS w 95 II 0

z=w)

soyw

00.0 00-O 00.0 00-o 00.0 00.0 00.0 00.0 EE.0 L9-0 00.1

(z=u

wa %

sapory

1.0 1.1 1.2 1.3

R=3.0 1.00 1.10 1.20 1.30

1.00 1.10 1.20 1.30 1.40 1.50 l-60 1.70 1.80 1.90 1.85 1.00 l-10 1.20 l-30

1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 l-90 2.00

1.50 1.60 1.70 1.80 1.90 1.84

1.50 1.60 1.70 1.80 1.90 2.00

1.5 1.6 1.7 1.8 1.9 2.0

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

h,

I h,

R =2.5

Plate aspect ratio

Maximum admissible thickness ratio

1.00 0.70 0.40 0.10

1.00 0.70 0.40 0.10 0.00 o*OO 0.00 0.00 0.00 0.00 0.00

0.88 0.67 046 0.26 0.05 0.00

h,

h,

,

1.00 1.10 1.20 1.30

1.00 1.10 1.20 1.30 1.18 0.95 0.73 0.50 0.28 0.05 0.00

o-00 0.00 0.00 0.00 o-00 040

Optimum thickness ratios (for m=n=l)

TABLE 1 (cont.)

0.14 0.16 0.18 0.21

0 16 34 54

0 16 34 53 72 91 110 130 150 170 180

88 110 130 150 170 180

0.26 0.28 0.31 0.34 0.37 0.38 0.14 0.16 0.18 0.21 0.23 0.26 0.29 0.32 0.35 0.37 O-38

n=l) % Diff.

(nl=l llll)C

0.14 0.16 0.18 o-21

0.14 0.16 0.18 0.21 0.24 0.26 0.29 0.32 o-35 0.38 0.39

0.26 0.29 0.32 0.35 0.38 0.39

(m=l 17/S -

0 16 35 55

0 16 35 54 73 93 110 130 160 180 180

92 110 130 160 180 180

n=2) % Diff.

Modes

0.14 0.15 0.17 0.19

0.14 0.15 0.17 0.19 0.21 0.22 0.24 0.27 0.29 0.32 0.33

0.22 0.24 0.27 0.29 0.32 0.33

(m=2 VlrlC

0 10 25 42

25 43 52 64 79 96 110 130 140

0 10

95 78 110 130 140

64

n=2) % Diff.

Y $ z w

;k:

z

N

1.00 1.10 1.20 l-30 1.40 1.50 l-60 1.70 1.80 l-90 1.85

1.0 1-l 1.2 1.3 l-4 1.5 l-6 1.7 l-8 l-9 2.0

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2-o

R=3-5

R = 4.0 l-00 1.10 1.20 1.30 140 1.50 1.60 l-70 I.80 1.90 1.85

1.40 1.50 1.60 1.70 l-80 1.90 1.85

1.4 1.5 1.6 1.7 1.8 l-9 2.0

1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 l-90 2.00

l-00 1.10 1.20 l-30 1.40 1.50 1.60 1.70 I.80 1.90 2.00

1.40 1.50 1.60 1.70 1.80 1.90 2-00

1.00 o-70 o-40 0.10 O-00 0.00 OX@ 0.00 o*OO 0.00 o-00

l-00 0.70 o-40 o-10 O-00 om 0.00 o*OO 0.00 O-00 o-00

040 040 O-00 040 040 040

040

l-00 1.10 1.20 1.30 1.18 0.95 0.73 0.50 0.28 0.05 0.00

l-00 1.10 1.20 1.30 1.18 o-95 0.73 0.50 0.28 0.05 0.00

1.18 0.95 0.73 o-50 0.28 0.05 0.00

0.14 0.16 0.18 0.21 0.24 0.26 0.29 0.32 0.35 0.38 0.39

o-14 O-16 0.18 0.21 0.24 0.26 0.29 0.32 o-35 0.38 0.39

0.24 0.26 0.29 0.32 o-35 O-38 0.39

0 16 35 55 74 93 110 130 160 180 190

0 16 35 55 74 93 110 130 160 180 190

73 92 110 130 150 180 180

o-14 O-16 0.18 0.21 0.24 O-26 0.29 0.32 o-35 0.38 0.39

0.14 0.16 0.18 o-21 O-24 O-26 O-29 O-32 0.35 O-38 0.39

O-24 0.26 0.29 0.32 o-35 O-38 0.39

0 16 35 55 73 93 110 140 160 180 190

0 16 35 55 73 93 110 140 160 180 190

73 93 110 130 160 180 190

0.14 0.15 0.17 o-19 0.21 o-22 O-24 0.27 O-29 O-32 o-33

0.14 o-15 0.17 0.19 0.21 0.22 0.24 0.27 O-29 0.32 o-33

0.21 0.22 O-24 O-27 0.29 0.32 o-33

0 10 25 42 52 64 79 96 110 130 140

0 10 25 42 52 64 79 96 110 130 140

52 64 79 96 110 130 140

1.0 1.1 1.2 l-3 l-4 l-5 1.6 1.7 l-8 1.9 2-o

1-o 1.1 1.2 1.3 l-4 1.5 l-6 1.7 l-8 1.9 2.0

1-o 1.1 1.2 l-3 l-4 l-5 1.6

R=l.O

R=1*5

R=2.0

Plate aspect ratio

Maximum admissible thickness ratio r

damping

-

l-00 1.10 1.20 1.30 1.40 1.50 1.60

1.00 1.10 l-20 l-30 l-40 1.50 l-60 1.70 1.80 l-90 2.00

1.00 1.10 l-20 l-30 140 1.50 l-60 l-70 1.80 1.90 2-00

h,

l-00 1.10 1.20 1.30 l-40 1.50 1m

l+O l-10 1.20 1.30 1.20 1.00 O-80 0.60 0.40 0.20 O-00

la0 1.10 1.20 l-30 1.20 1X)0 0.80 0.60 040 0.20 0.00

1.00 1.10 1.20 1.30 l-20 1.00 WRO

l-00 o-70 o-40 0.10 o*OO O-00 0.0

l-00 l-10 1.20 1.30 1.40 1.50 l-60 1.70 1.80 1.90 2*00

la0 0.70 040 0.10 O-00 O-00 o*OO OXlO 0.00 om OaO

,

la0 1.10 l-20 1.30 1.40 1.50 l-60 l-70 1.80 1.90 2aO

ratios

0.14 0.15 o-17 o-19 o-21 0.23 0.25

0.14 0.14 0.17 o-19 o-21 0.23 0.25 O-27 0.29 0.32 o-35

o-14 o-15 o-17 o-19 o-21 O-23 0.25 O-27 O-29 O-32 o-35

(m=l

ratios for S-S-S-S

2

l-00 0.70 040 o-10 O-00 o*OO O-00 O-00 om OXlO OaO

Optimum thickness (for m=n=l)

Optimum

TABLE

0 11 25 42 55 68 81

0 10 23 40 53 66 81 97 120 130 150

0 10 23 40 53 66 81 97 120 130 150

n=l)

plates

0.14 0.15 o-17 o-20 0.22 O-24 0.26

o-14 0.15 o-17 0.20 0.22 O-24 0.26 O-29 0.31 0.34 O-36

o-14 0.15 0.17 0.19 o-21 O-23 0.25 0.27 0.30 0.32 o-35

Cm=1

0 12 28 46 61 76 94

0 11 26 44 59 74 90 110 130 150 170

10 24 41 55 68 83 100 120 140 160

0

n=2)

Modes

o-14 0.14 0.15 0.17 0.18 0.19 O-20

o-14 0.14 o-15 0.17 0.18 0.19 0.20 0.21 0.23 0.25 0.27

0.14 0.14 o-15 o-17 O-18 o-19 0.20 0.21 0.23 0.25 0.27

Cm=2

0 3 13 27 33 38 45

0 3 13 27 33 38 45 56 69 84 100

3 13 27 33 38 45 56 69 84 100

0

n=2)

la0 1.10 l-20 1.30 l-40 1.50 1.60 1.70 1.80 l-90 2*00

1.0 1.1 l-2 1.3 1.4 1.5 1.6 l-7 l-8 1.9 2.0

1.0 1-l l-2 l-3 1.4 1.5 1.6 1.7 l-8 1.9 2-o

R=3.0

R =3.5 l-00 1.10 1.20 1.30 1.40 l-50 1.60 1.70 l-80 1.90 2.00

l-00 l-10 l-20 1.30 1.40 l-50 1.60 1.70 1.80 l-90 2.00

l-00 l-10 l-20 1.30 l-40 1.50 l-60 1.70 1.80 l-90 2.00

1.0 1-l l-2 1.3 l-4 l-5 1.6 1.7 1.8 l-9 2-o

R=2-5

l-00 l-10 l-20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00

l-00 1.10 1.20 l-30 l-40 1.50 1.60 1.70 1.80 1.90 2.00

1.70 1.80 1.90 2aO

1.70 1.80 1.90 2.00

1.7 1.8 1.9 2.0

l-00 l-10 l-20 l-30 1.20 1.00 0.80 0.60 0.40 0.20 0~00

l-00 1.10 1.20 1.30 1.20 la0 0.80 0.60 0.40 o-20 om

1.00 1.10 l-20 1.30 1.20 1.00 0.80 0.60 o-40 o-20 0.00

O-60 0.40 0.20 0.00

la0 o-70 o-40 0.10 OaO OaO 0.00 O+Kl 0.00 0.00 oal

1.00 0.70 0.40 0.10 o*OO oal 0.00 0.00 o*OO O-00 OaO

1.00 0.70 o-40 o-10 O-00 o-00 Od.KI o-00 o-00 o*OO om

0.00 om 0.00 O-00

0.14 0.16 O-18 0.20 0.22 O-24 O-26 0.29 0.31 0.34 0.37

o-14 0.15 0.18 0.20 0.22 0.24 0.26 0.28 0.31 0.34 O-36

0.14 o-15 0.17 o-20 o-22 O-24 O-26 0.28 0.30 o-33 0.36

0.27 o-30 0.32 o-35

0 14 31 50 64 78 94 110 130 150 170

0 13 29 48 62 76 91 110 130 150 170

0 12 28 46 59 73 88 110 120 140 160

100 120 140 160

o-14 o-15 O-18 0.20 0.22 0.24 0.27 O-29 0.32 0.35 0.38

0 13 28 47 63 79 97 120 140 160 180

0 13 29 47 63 79 96 120 140 160 180

0 12 28 46 62 78 96 110 130 150 170

0.14 0.15 0.17 0.20 0.22 0.24 0.27 O-29 O-32 o-35 0.37 o-14 o-15 0.18 o-20 o-22 0.24 O-27 O-29 0.32 o-35 O-38

110 130 150 170

O-29 0.32 0.34 0.37

0.14 0.14 0.15 0.17 O-18 0.19 0.20 0.21 0.23 0.25 0.27

0.14 0.14 0.15 0.17 0.18 0.19 0.20 0.21 0.23 0.25 0.27

0.14 0.14 0.15 0.17 0.18 0.19 0.20 0.21 0.23 0.25 0.27

0.21 0.23 0.25 0.27

0 3 13 27 33 38 45 56 69 84 100

0 3 13 27 33 38 45 56 69 84 100

0 3 13 27 33 38 45 56 69 84 100

56 69 84 100

R=4.0

Plate aspect ratio

1-o 1.1 1.2 1.3 l-4 1.5 1.6 1.7 1.8 1.9 2.0

Maximum admissible thickness ratio r h, 1.00 l-10 1.20 1.30 1.40 1.50 l-60 1.70 1.80 1.90 2.00

h, 1.00 l-10 l-20 l-30 1.40 1.50 1.60 1.70 1.80 1.90 2.00

1.00 1.10 1.20 1.30 1.20 1.00 0.80 0.60 040 0.20 0.00

h,

h,

\

1.00 0.70 o-40 o-10 O-00 O-00 0.00 0.00 o-00 O-00 0.00

Optimum thickness ratios (for m=n=l)

TABLE 2 (cont.)

C

0.14 0.16 0.18 o-21 O-23 0.24 0.27 0.29 0.32 o-34 0.37

(Xl 0 15 32 51 65 79 95 110 130 150 170

n=l) % Diff. Oil4 0.15 0.18 0.20 o-22 0.24 0.27 0.30 0.32 0.35 0.38

(m=l VlrlC

13 29 47 63 80 98 120 140 160 180

0

n=2) % Diff.

Modes

o-14 o-14 0.15 0.17 0.18 0.19 0.20 0.21 0.23 O-25 0.27

(m=2 77lv7,

0 3 13 27 33 38 45 56 69 84 100

n=2) % Diff.

OPTIMUM

UNCONSTRAINED

DAMPING

197

LAYER

applied to a uniform thickness over the entire plate was computed by using the expression % difference = loo x n/qC(optimum thickness) - r)/n,(uniform ‘I/ TJuniform thickness)

thickness)

.

(42)

Results of the optimization are summarized in Tables 1 and 2. These results indicate that optimal distribution of the damping material can increase the system loss factor by as much as 100%) or more. In all the cases considered, the greater the maximum admissible thickness ratio, the greater the amount of damping that can be achieved. The trends are similar for both the edge-fixed and simply supported cases, and for the various modes and plate aspect ratios considered. In Tables 1 and 2, values of the optimum thickness ratios h,-h4 for the various elements of the plate area define the optimum thickness distribution of the damping material for the mode m = n = 1. Values of the optimum thickness ratios for the other modes considered are not listed because of space limitations. The optimum thickness distribution depends upon the boundary conditions, plate aspect ratio, mode of vibration and maximum admissible thickness ratio, but follows the general trends shown in Figure 3. As can be

Figure 3. Optimum coating thickness distribution (typical case). (a) S-S-S-S plate; (b) C-C-C-C

plate.

seen from this figure, or from Tables 1 and 2, the central part of the plate is the most important insofar as damping is concerned. This is true for both the edge-fixed and simply supported cases. For edge-fixed plates, the side elements (elements 2 and 3) usually are more important than the comer elements (element 4), while the reverse is true for simply supported plates. These results indicate clearly the regions of the plate where damping material can be most profitably applied, and should serve as useful guidelines in the design of partial damping treatments. It is important to note that no symmetry conditions were included among the constraint equations. Thus, the thickness ratios for a square plate (R = 1) with m = n = 1 or m = n = 2 do not. reflect the result h2 = hg, as might be expected. That is, the thickness ratios are not symmetric about the plate diagonal. However, symmetry is preserved in the sense that the values of h2 and h, can be interchanged without altering the damping values.

198

A.

YILDIZ

AND K. STEVENS

This can be seen most easily from the objective functions. For the example of a clamped square plate with the parameter values vP =0.3 and n =O.Ol, this function is F(hj)=2~1157h;+1~0772(h:+h:)+05845h:+3~1735h~ + 1~6157(h~+h~)+0~8767h:+1~5867h,+0~8079(h,+h,)+0~4383h,.

(43)

4. CONCLUDING REMARKS

The results presented reveal that the loss factor of plates with unconstrained layer viscoelastic damping treatments can be increased significantly (as much as lOO%, or more) by optimizing the thickness distribution of the damping material. Also revealed are the regions of the plate where added damping treatments are most effective. While the results obtained are for a particular modulus ratio (n = O*Ol), they are believed to be representative of those for plates with other modulus ratios, so long as n <<1.

REFERENCES

5.

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

B. C. NAKRA 1976 The Shock and Vibration Digest 8, 3-12. Vibration control with viscoelastic materials. B. C. NAKRA 1981 The Shock and Vibration Digest 13, 17-20. Vibration control with viscoelastic material-II. L. ROGERS (editor) 1978 Air Force Fhght Dynamics Laboratory, Report No. AFFDL-TM-78-78FBA. Conference on aerospace polytechnic viscoelastic damping technology for 1980s. K. K. STEVENS, C. H. KUNG and S. E. DUNN 1981 Proceedings Noise Con. 81, Raleigh, North Carolina, June, 1981, pp. 445-448. Damping of plates by partial viscoelastic coatings-Part I-Analysis. S. E. DUNN, C. H. KUNG, V. R. JAISING and K. K. STEVENS 1981 Proceedings Noise-Con. 81, Raleigh, North Carolina, June, 1981, pp. 449-452. Damping of plates by partial viscoelastic coatings-Part II-Experimental. K. K. STEVENS, C. H. KUNG and S. E. DUNN 1981 Proceedings Inter-Noise 81, Amsterdam, October, 1981, pp. 363-366. Partial damping layer treatments for plates. D. I. G. JONES 1973 Journal ofSound and Vibration 29, 423-434. Parametric study of multiplelayer damping treatments on beams. D. K. RAO 1978 Acustica 39, 264-269. Vibration damping of tapered unconstrained beams. R. LUNDEN 1979 Journal ofSound and Vibration 66, 25-37. Optimum distribution of additive damping for vibrating beams. R. LUNDEN 1980 Journal ofSound and Vibration 72,391-402. Optimum distribution of additive damping for vibrating frames. R. PLUNKETT and C. T. LEE 1970 Journal of the Acoustical Society of America 48, 150-161. Length optimization for constrained viscoelastic layer damping. D. R. BLAND 1960 Theory of Linear Viscoelasticity. London: Pergamon Press. M. E. MCINTYRE and J. WOODHOUSE 1978 Acustica 39,209-224. The influence of geometry on linear damping. CHUN-HUA KUNG 1981 M.S. Thesis, Florida Atlantic Llniwrsity. A method of analysis for evaluating the effectiveness of partial damping layer treatments for square plates. S. TIMOSHENKO and S. WOINOWSKY-KRIEGER 1959 Theory ofPlates and Shells. New York: McGraw-Hill, second edition. V. R. JAISING 1981 MS. Thesis Florida Atlantic University. Analysis of damping layer treatments for plates using experimentally determined mode shapes. A. W. LEISSA 1973 Journal of Sound and Vibration 31,257-293. The free vibration of rectangular plates. D. YOUNG and R. P. FELGAR 1949 Bureau of Engineering Research, University of Texas, Publ. No. 4913. Tables of characteristic functions representing normal modes of vibration of a beam. R. P. FELGAR 1950 The University of Texas, Circular No. 14. Formulas for integrals containing characteristic functions of a vibrating beam. E. J. HANG and J. S. ARORA 1979 Applied Optimal Design. New York: Wiley-Intersceince.

OPTIMUM

UNCONSTRAINED

DAMPING

LAYER

199

21. A. V. FIACCO and G. P. MCCORMICK 1979 Nonlinear Programming, Sequential Unconstrained Minimization Techniques. New York: John Wiley. 22. R. FLETCHER and M. J. D. POWELL 1963 Computer Journal 6, 163-168. A rapidly convergent descent method for minimization. 23. L. C. W. DIXON et al. (Editors) 1980 Nonlinear Optimization, Theory and Algorithms. Boston: Birkhausen. 24. A. YILDIZ 1982 Department of Ocean Engineering, Florida Atlantic University, Report No. FAU-CAV-82-2. Optimum coating thickness distribution for simply-supported or clampedclamped plates. 25. A. W. LEISSA 1969 NASA SP-160. Vibration of plates.

APPENDIX: a

length of plate in x-co-ordinate direction surface area of jth element of coating plate surface area constants defined in equation (27) length of plate in y-co-ordinate direction plate aspect ratio plate flexural rigidity modulus of elasticity (storage modulus for viscoelastic materials) beam mode shapes Gaussian curvature $1 tp distance from plate-coating interface to neutral surface

a1 aP

A, 4, C,,D, b alb D0 E F,, G G(w) h, HJ 4 I;, I:“, z;,,, kx, k_,.,k, Lj, L,

m, n m,, m,,,m;, n T u, U, W

Y a1 a,, Pm e,, ag, Yxy 77 v

X9

P u,, ov, r,, w 00

NOTATION

z ’

Hjlrp integrals defined in equation (27) curvatures of deflected neutral surface integral parameters defined in equation (29) integers in mode shapes expressions bending and twisting moments E,lEp kinetic energy factor plate strain energy coating strain energy plate lateral deflection co-ordinates in plane of plate thickness parameter defined in equation (29) mode shape parameters strain components in rectangular coordinate system loss factor Poisson’s ratio density stress components in rectangular co-ordinate system radian frequency natural radian frequency of bare plate

Subscripts P C j X Y

denotes denotes denotes denotes denotes

plate coating jth element of coating partial derivative with respect partial derivative with respect

to x to y