Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

47, 2, pp. 457-471, Warsaw 2009

FREE VIBRATION OF CLAMPED VISCO-ELASTIC

RECTANGULAR PLATE HAVING BI-DIRECTION

EXPONENTIALLY THICKNESS VARIATIONS

Arun K. Gupta

Maharaj Singh College, Mathematics Department, Saharanpur, India

e-mail: gupta arunnitin@yahoo.co.in

Anupam Khanna

Shobhit Institute of Engineering Technology – Gangoh, Department. of Applied Science,

Saharanpur, India; e-mail: anupam rajie@yahoo.co.in

Dharam V. Gupta

College of Engineering, Departments of Mathematics, Roorkee, India

e-mail: dvgupta 1@yahoo.co.in

Freevibrationofaclampedvisco-elasticrectangularplatehavingbi-direction
exponentially varying thickness has been analysed on the basis of classical
plate theory. For visco-elastic materials, basic elastic and viscous elements
are combined.We have assumed the Kelvinmodel for visco-elasticity, which
is a combination of elastic and viscous elements connected in parallel. Here,
the elastic element is constituted by a spring and the viscous one is a da-
shpot. An approximate but quite convenient frequency equation is derived
by using the Rayleigh-Ritz technique. Logarithmic decrement, time period
and deflection (at two different instant of time) for the first two modes of
vibration and for various values of the taper constants and aspect ratio are
calculated. Comparison studies have been carried out with bi-linearly thick-
ness variation to establish the accuracy and versatility of the method.

Key words: visco-elasticity, clamped rectangular plate, variable thickness

List of symbols

a – length of rectangular plate
b – width of rectangular plate
x,y – co-ordinates in plane of the plate



458 A.K. Gupta et al.

h – thickness of the plate at point (x,y)
E – Young’s modulus
G – shear modulus
ν – Poisson’s ratio

D̃ – visco-elastic operator
D1 – flexural rigidity, D1 =Eh

3/[12(1−ν2)]
ρ – mass density per unit volume of plate material
t – time
η – visco-elastic constants
w(x,y,t) – deflection of the plate
W(x,y) – deflection function
T(t) – time function
β1,β2 – taper constants in X- and Y -directions, respectively
Λ – logarithmic decrement
K – time period

1. Introduction

Plates of uniform and non-uniform thickness are widely used as structural
components in various engineering fields such as aerospace industry, missile
technology, naval ship design, telephone industry, etc. An extensive review on
linear vibration of plates has been given by Leissa (1987) in his monograph
and a series of review articles (Leissa, 1969). Several authors (Tomar and
Gupta, 1985; Laura et al., 1979) studied the effect of taper constants in two
directions on elastic plates, but none of them on visco-elastic plates. Sobotka
(1978) considered freevibrationsof visco-elastic orthotropic rectangular plates.
Gupta and Khanna (2007) studied the effect of linearly varying thickness in
both directions on vibration of a visco-elastic rectangular plate.

Young (1950) solved the problem of a rectangular plate by the Ritz me-
thod.Free vibrations of rectangular plateswhose thickness varies parabolically
were studied by Jain and Soni (1973). Bhatnagar and Gupta (1988) studied
the effect of thermal gradient on vibration of a visco-elastic circular plate of
variable thickness. Kumar (2003) discussed the effect of thermal gradient on
some vibration problems of orthotropic visco-elastic plates of variable thick-
ness. Gupta et al. (2007a) solved the problem of thermal effect on vibration
of a non-homogeneous orthotropic rectangular plate having bi-directional pa-
rabolically varying thickness. Gupta et al. (2007b) examined vibration of a
visco-elastic orthotropic parallelogram plate with linear variation of the thick-
ness.



Free vibration of clamped visco-elastic... 459

Visco-elasticity, as its name implies, is a generalisation of elasticity and
viscosity. The ideal linear elastic element is the spring.When a tensile force is
applied to it, the increase in distance between its two ends is proportional to
the force. The ideal linear viscous element is the dashpot.

The main objective of the present investigation is to study the effect of
taper constants on vibration of a clamped visco-elastic rectangular plate with
bi-direction exponentially thickness variations. It is assumed that the plate
is clamped on all four edges. To determine the frequency equation, Rayleigh-
Ritz’s technique has been applied. It is considered that the visco-elastic pro-
perties of the plate are of the Kelvin type.

Allmaterial constants, which are used in numerical calculations, have been
taken for the alloy DURALIUM, which is commonly used in modern techno-
logy.

Logarithmic decrement, timeperiodanddeflection (at twodifferent instant
of time) for the first two modes of vibration for various values of the aspect
ratio a/b and taper constants β1 and β2 are calculated. All the results are
illustrated with graphs.

2. Equation of motion and its analysis

The equations of motion of a visco-elastic rectangular plate of variable thick-
ness are (Gupta and Khanna, 2007)

[D1(W,xxxx+2W,xxyy+W,yyyy )+2D1,x (W,xxx+W,xyy )+

+2D1,y (W,yyy+W,yxx )+D1,xx (W,xx+νW,yy )+

+D1,yy (W,yy+νW,xx )+2(1−ν)D1,xyW,xy ]−ρhp
2W =0

(2.1)

T̈ +p2D̃T =0

where (2.1) aredifferential equationsofmotion for an isotropic plate of variable
thicknessmade of a visco-elastic material describing lateral deflection and free
vibration, respectively.

Here p2 is a constant.

The expressions for kinetic energy T1 and strain energy V1 are (Leissa,
1969)



460 A.K. Gupta et al.

T1 =
1

2
ρp2

a∫

0

b∫

0

hW2 dydx

(2.2)

V1 =
1

2

a∫

0

b∫

0

D1[(W,xx )
2+(W,yy )

2+2νW,xxW,yy+2(1−ν)(W,xy )
2] dydx

Assuming the thickness variation of the plate in both directions as

h=h0e
β1
x
aeβ2

y

b (2.3)

where β1 and β2 are the taper constants in the x- and y-directions, respec-
tively, and h0 =h at x= y=0.
The flexural rigidity of the plate can now be written as (assuming the

Poisson’s ratio ν is constant)

D1 =
Eh30

12(1−ν2)

(
eβ1

x
a

)3(
eβ2

y

b

)3
(2.4)

3. Solutions and frequency equation

To find a solution, we use the Rayleigh-Ritz technique. This method requires
that themaximumstrain energymustbeequal to themaximumkinetic energy.
So, it is necessary for the problem under consideration that

δ(V1−T1)= 0 (3.1)

for arbitrary variations of W satisfying relevant geometrical boundary condi-
tions.
For a rectangular plate clamped (c) along all the four edges, the boundary

conditions are

W =

{
W,x=0 at x=0 ∧ x= a

W,y=0 at y=0 ∧ y= b
(3.2)

and the corresponding two-term deflection function is taken as (Gupta and
Khanna, 2007)

W =
[x
a

y

b

(
1−
x

a

)(
1−
y

b

)]2[
A1+A2

x

a

y

b

(
1−
x

a

)(
1−
y

b

)]
(3.3)

which satisfies equations (3.2).



Free vibration of clamped visco-elastic... 461

Assuming non-dimensional variables as

X =
x

a
Y =
y

a
W =

W

a
h=
h

a
(3.4)

and using equations (2.4) and (3.4) in equations (2.2), one obtains

T1 =
1

2
ρp2h0a

5

1∫

0

b/a∫

0

eβ1Xeβ2Y
a
bW
2
dYdX

(3.5)

V1 =Q

1∫

0

b/a∫

0

(
eβ1Xeβ2Y

a
b

)3
·

·[(W,XX )
2+(W,YY )

2+2νW,XXW,YY +2(1−ν)(W,XY )
2] dYdX

where

Q=
Eh
3

0a
3

24(1−ν2)

Substituting the expressions for T1 and V1 from (3.5) into equation (3.1), one
obtains

V2−λ
2p2T2 =0 (3.6)

where

V2 =

1∫

0

b/a∫

0

(
eβ1Xeβ2Y

a
b

)3
·

·[(W,XX )
2+(W,YY )

2+2νW,XXW,YY +2(1−ν)(W,XY )
2] dYdX

T2 =

1∫

0

b/a∫

0

eβ1Xeβ2Y
a
bW
2
dYdX (3.7)

λ2 =
12ρ(1−ν2)a2

Eh
2

0

and λ is a frequency parameter.
For better accuracy of the results, the exponents are taken up to the fifth

degree of X and Y .
Equation (3.6) has the unknowns A1 and A2 due to substitution of W

from equation (3.3). These two constants are to be determined from equation
(3.6) as

∂(V2−λ
2p2T2)

∂An
=0 n=1,2 (3.8)



462 A.K. Gupta et al.

After simplifying equation (3.8), one gets

bn1A1+ bn2A2 =0 n=1,2 (3.9)

where bn1, bn2 (n = 1,2) involve the parametric constant and frequency
parameter.

For the non-trivial solution, the determinant of coefficients of equation
(3.9) must be zero. So one gets the frequency equation as

∣∣∣∣∣
b11 b12
b21 b22

∣∣∣∣∣=0 (3.10)

From equation (3.10), one can obtain a quadratic equation in p2 from which
two values of p2 can be found. After determining A1 and A2 from (3.9), one
can obtain the deflection function W in form

W =
[
XY
a

b
(1−X)

(
1−Y

a

b

)]2[
1+
−b11
b12
XY
a

b
(1−X)

(
1−Y

a

b

)]
(3.11)

if one chooses A1 =1.

4. Time functions of visco-elastic plates

Equation (2.1)2 is defined as a general differential equation of time functions
for free vibrations of visco-elastic plates. It depends on the visco-elastic ope-
rator D̃, which is

D̃≡ 1+
η

G

d

dt
(4.1)

for Kelvin’s model (Gupta andKhanna, 2007).

After substituting equation (4.1) into (2.1)2, one obtains

T̈ +
p2η

G
Ṫ +p2T =0 (4.2)

Expression (4.2) is a differential equation of the second order for the time
function T .

Solution to equation (4.2) will be

T(t)= ea1t(C1cosb1t+C2 sinb1t) (4.3)



Free vibration of clamped visco-elastic... 463

where

a1 =−
p2η

2G
b1 = p

√

1−
(pη
2G

)2
(4.4)

and C1, C2 are constants which can be determined easily from the initial
conditions of the plate. Assuming the initial conditions as

T =1 ∧ Ṫ =0 at t=0 (4.5)

and using equation (4.5) in (4.3), one obtains

C1 =1 C2 =−
a1
b1

(4.6)

Finally, one gets

T(t)= ea1t
(
cosb1t−

a1
b1
sinb1t

)
(4.7)

after substituting (4.6) into (4.3).

Thus, deflection of the vibrating mode w(x,y,t), which is equal to
W(x,y)T(t), may be expressed as

w=
[
XY
a

b
(1−X)

(
1−Y

a

b

)]2[
1−
b11
b12
XY
a

b
(1−X)

(
1−Y

a

b

)]
·

(4.8)

·ea1t
(
cosb1t−

a1
b1
sinb1t

)

bymaking use of equations (4.7) and (3.11).

The vibration period of the plate is

K =
2π

p
(4.9)

where p is the frequency given by equation (3.10).

The logarithmic decrement of vibrations, defined by the standard formula,
is

Λ= ln
w2
w1

(4.10)

where w1 is thedeflectionatanypointon theplate at the timeperiod K =K1,
and w2 is the deflection at the same point and the time period succeeding K1.



464 A.K. Gupta et al.

5. Numerical evaluations

The values of logarithmic decrement Λ, time period K and deflection w (at
two differenting instants) for a clamped visco-elastic rectangular plate for dif-
ferent values of taper constants β1, β2 and aspect ratio a/b at different points
for the first two modes of vibrations are calculated.

The following material parameters are used: E = 7.08 · 1010N/m2,
G=2.632·1010N/m2, η=14.612·105Ns/m2, ρ=2.80·103kg/m3, ν=0.345.
The data corresponds tuDURALIUM reported inGupta andKhanna (2007).

The thickness of the plate at the centre is h0 =0.01m.

6. Results and discussion

Numerical results for a visco-elastic isotropic clamped rectangular plate of
exponentially varying thickness in both directions have been accurately com-
putedby using the latest computer technology. Computations have beenmade
for the logarithmic decrement Λ, time period K and deflection w (for two
time instants) for different values of the taper constants β1, β2 and aspect
ratio a/b for the first two modes of vibrations. All results are presented in
Fig.1 to Fig.7. The comparison is made with the author’s paper (Gupta and
Khanna, 2007) on a plate with linearly variable thickness.

Fig. 1. Logarithmic decrement Λ versus taper constant β1



Free vibration of clamped visco-elastic... 465

In Fig.1, it can be easily seen that for a fixed value of the aspect ratio
a/b = 1.5 as the taper constant β1 increases, the logarithmic decrement Λ
decreases continuously for bothmodes of vibration for two values of β2.

Figure 2 shows a steady decrease in the time period K with an increase of
the taper constant β1 for a fixed aspect ratio a/b=1.5 and two values of β2.
It is simply seen that the time period K decreases as the taper constants
increase for both modes of vibration.

Fig. 2. Time period K versus taper constant β1

Fig. 3. Time period K versus aspect ratio a/b



466 A.K. Gupta et al.

Figure 3 shows the time period K for different values of the aspect ratio
a/b for both modes of vibration for uniform and non-uniform thickness with
the following constants:

(i) β1 =β2 =0.0

(ii) β1 =β2 =0.6

In both cases, one can note that the time period K decreases as the aspect
ratio a/b increases for both modes of vibration.

Figures 4, 5, 6 and 7, respectively, depict numerical values of the deflec-
tion w for a fixed aspect ratio a/b=1.5 for the first two modes of vibration
and different values of X and Y for the following cases:

(i) Fig.4 – β1 =β2 =0 and time is 0K

(ii) Fig.5 – β1 =β2 =0 and time is 5K

(iii) Fig.6 – β1 =β2 =0.6 and time is 0K

(iv) Fig.7 – β1 =β2 =0.6 and time is 5K

Fig. 4. Deflection w versus X; T =0K, β1 =β2 =0, a/b=1.5

Separate figures are given for the first and second mode of vibration
(Figs.5-7). One can conclude fromall the four figures that the deflection w for
the first mode of vibration initially increases and then decreases as X grows
for different values of Y ,.



Free vibration of clamped visco-elastic... 467

Fig. 5. Deflection w versus X; T =5K, β1 =β2 =0, a/b=1.5

Fig. 6. Deflection w versus X; T =0K, β1 =β2 =0.6, a/b=1.5

Also, one can see that the deflection w for the second mode of vibration
for Y =0.3 and Y =0.6 first decreases and then increases, while for Y =0.9,
the deflection increases first and then decreases as X grows.



468 A.K. Gupta et al.

Fig. 7. Deflection w versus X; T =5K, β1 =β2 =0.6, a/b=1.5

One can get results for higher modes of vibration by introducing more
terms to equation (3.3).

In the figures, Mode 1 and Mode 2 means the first and second mode of
vibration, respectively.

7. Conclusion

The results for a uniform isotropic clamped visco-elastic rectangular plate
are compared with the results published by the authors (Gupta andKhanna,
2007) and found to be in close agreement. The results of the present paper,
shown in Fig.1 to Fig.3, are given in Table 1 to Table 3 together with results
obtained in Gupta and Khanna (2007), which are placed in brackets in these
tables.

After comparing, the authors conclude that as the taper constant increases
for exponentially varying thickness, the timeperiodand logarithmicdecrement
decrease in comparison to the increasing taper constant for linearly varying
thickness. Therefore, engineers are provided with a method to develop plates
in a manner so that they can fulfill the requirements.



Free vibration of clamped visco-elastic... 469

Table 1. Logarithmic decrement for various parameters β1 and β2

β2 =0.2 β2 =0.6
β1 First Second First Second

mode mode mode mode

0.0 −0.183075 −0.729066 −0.238421 −0.959711
(−0.181522) (−0.722762) (−0.220184) (−0.881917)

0.2 −0.203070 −0.809745 −0.264408 −1.066887
(−0.200110) (−0.798014) (−0.242727) (−0.974714)

0.4 −0.226834 −0.905293 −0.295142 −1.192726
(−0.219396) (−0.876352) (−0.266093) (−1.071635)

0.6 −0.254990 −1.018582 −0.331403 −1.341444
(−0.239209) (−0.957267) (−0.290085) (−1.172157)

0.8 −0.288020 −1.152337 −0.373808 −1.517993
(−0.259428) (−1.040427) (−0.314561) (−1.275984)

1.0 −0.326156 −1.308829 −0.422684 −1.727861
(−0.279965) (−1.125632) (−0.339418) (−1.382989)

Table 2. Time period for various parameters β1 and β2

β2 =0.2 β2 =0.6
β1 First Second First Second

mode mode mode mode

0.0 599.1 151.5 460.1 115.8
(604.2) (152.8) (498.2) (125.8)

0.2 540.1 136.7 415.0 104.6
(548.1) (138.7) (452.0) (114.1)

0.4 483.6 122.6 371.9 94.0
(500.0) (126.5) (412.4) (104.1)

0.6 430.3 109.4 331.3 84.1
(458.6) (116.1) (378.3) (95.6)

0.8 381.0 97.1 293.8 75.0
(422.9) (107.1) (349.0) (88.2)

1.0 336.6 86.1 260.0 66.7
(392.0) (99.3) (323.5) (81.8)



470 A.K. Gupta et al.

Table 3.Time period for various parameters a/b, β1 and β2

β1 =β2 =0.0 β1β2 =0.6
a/b First Second First Second

mode mode mode mode

0.5 1650.2 412.5 818.8 204.9
(1650.1) (412.6) (934.9) (233.2)

1.0 1129.0 288.5 559.7 143.8
(1129.0) (288.5) (639.4) (163.3)

1.5 667.9 169.0 331.3 84.1
(667.9) (169.0) (378.3) (95.6)

2.0 412.6 103.3 204.7 51.3
(412.5) (103.2) (233.7) (58.3)

2.5 274.5 68.0 136.2 33.8
(274.4) (68.1) (155.5) (38.5)

References

1. Bhatanagar N.S., Gupta A.K., 1988, Vibration analysis of visco-elastic
circularplate subjected to thermalgradient,Modelling, Simulation andControl,
B, AMSE, 15, 17-31

2. GuptaA.K., JohriT.,VatsR.P., 2007a,Thermal effect onvibrationofnon-
homogeneous orthotropic rectangular plate having bi-directional parabolically
varying thickness, International Conference inWorld Congress on Engineering
and Computer Science 2007 (WCECS 2007), San Francisco, USA, 784-787

3. Gupta A.K., Khanna A., 2007, Vibration of visco-elastic rectangular plate
with linearly thickness variations in both directions, J. Sound and Vibration,
301, 450-457

4. Gupta A.K., Kumar A., Gupta D.V., 2007b, Vibration of visco-elastic or-
thotropic parallelogram plate with linearly thickness variation, International
Conference in World Congress on Engineering and Computer Science 2007
(WCECS 2007), San Francisco, USA, 800-803

5. Jain R.K., Soni S.R., 1973, Free vibrations of rectangular plates of parabo-
lically varying thickness, Indian J. Pure App. Math., 4, 267-277

6. Kumar S., 2003, Effect of thermal gradient on some vibration pro-
blems of orthotropic visco-elastic plates of variable thickness, Ph.D.Thesis,
C.C.S.University, Meerut, India

7. Laura P.A.A., Grossi R.O., Carneiro G.I., 1979, Transverse vibrations
of rectangular plates with thickness varying in two directions and with edges
elastically restrained against rotation, J. Sound and Vibration, 63, 499-505



Free vibration of clamped visco-elastic... 471

8. Leissa A.W., 1969, Vibration of plate,NASA SP-160

9. Leissa A.W., 1987, Recent studies in plate vibration 1981-1985.Part II, com-
plicating effects,The Shock and Vibration Dig., 19, 10-24

10. SobotkaZ., 1978,Freevibrationof visco-elastic orthotropic rectangularplate,
Acta Techanica CSAV, 6, 678-705

11. Tomar J.S., Gupta A.K., 1985, Effect of thermal gradient on frequencies
of an orthotropic rectangular plate whose thickness varies in two directions, J.
Sound and Vibration, 98, 257-262

12. Young D., 1950, Vibration of rectangular plates by the Ritz method, J. App.
Mech., Trans. ASME, 17, 448-453

Drgania swobodne utwierdzonej lepkosprężystej prostokątnej płyty

o dwukierunkowo wykładniczo zmiennej grubości

Streszczenie

Wpracy rozważonoproblemswobodnychdrgańutwierdzonej lepkosprężystej pro-
stokątnej płyty o dwukierunkowo wykładniczo zmiennej grubości na podstawie kla-
sycznej teorii płyt. Uwzględniono lepkosprężystewłaściwościmateriałupłyty, bazując
na podstawowych elementach reologicznych. Przyjęto model Kelvina, tj. równoległą
kombinację elementu sprężystego i wiskotycznego. Równanie ruchu płyty rozwiąza-
no metodą Rayleigha-Ritza, otrzymując przybliżoną, ale wygodną do analizy postać
wyrażeniaw dziedzinie częstości. Następnie wyznaczonowartość logarytmicznego de-
krementu tłumienia, okresu drgań i ugięcia płyty dla dwóch pierwszych funkcji wła-
snych dla różnych parametrów opisujących zmienną grubość i wymiary zewnętrzne
płyty. Wyniki obliczeń przy uwzględnieniu zmiany grubości płyty porównano z do-
tychczasowymi rezultatami badań w celu potwierdzenia dokładności i uniwersalności
metody.

Manuscript received May 18, 2008; accepted for print November 12, 2008