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
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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.
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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
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coatings-Part
II-Experimental.
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D. R. BLAND 1960 Theory of Linear Viscoelasticity. London: Pergamon Press.
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on linear damping.
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V. R. JAISING 1981 MS. Thesis Florida Atlantic University. Analysis of damping layer treatments
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A. W. LEISSA 1973 Journal of Sound and Vibration 31,257-293. The free vibration of rectangular
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R. P. FELGAR 1950 The University of Texas, Circular No. 14. Formulas for integrals containing
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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
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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