 Open Access
 Total Downloads : 51
 Authors : Dr. Renu Chaudhary , Dr. Rajendra Kumar
 Paper ID : IJERTV8IS060741
 Volume & Issue : Volume 08, Issue 06 (June 2019)
 Published (First Online): 08072019
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Damped Vibrations of an Isotropic Circular Plate of Parabolically Varying Thickness Resting on Elastic Foundation
Dr. Renu Chaudhary
Assistant Professor, Quantum University, Roorkee
Haridwar
Abstract: Damped vibrations of a circular plate of parabolically varying thickness resting on elastic foundation
Also D =
Dr. Rajendra Kumar
Assosiate Professor,
J.V. Jain College, Saharanpur
E rp r
12 1 2 , hence introducing the non
have been studied on the basis of classical plate theory. The
h r W
fourth order differential equation of motion is solved by the method of frobenius. Using high speed digital computer, frequencies, deflection functions and moments corresponding
dimensional variables, H =
, R , w
a a a
E
to the first two modes of vibrations are computed for circular plate with clamped and simply supported edge conditions for
where 'a' is the radius of the Plate and E =
a
and
various values of taper constants, damping parameter and
elastic foundation. These results have been presented both in tabular and graphical forms.
= . Here the thickness of plate H is assumed to
a
vary in the form H = Ho (1 R 2 ), where
INTRODUCTION
In the research work the focus has been laid down the effect of taper constant, damping and elastic foundation on frequencies of an Isotropic circular plate of linearly varying thickness has been studied. The object of the work presented here is to study the damped vibration of a
H0 = HR=0, = taper constant.
In the light of these assumptions
equation (1) takes the form,
circular plate of parabolically varying thickness resting on elastic foundation.
(1
2 3 4 w
R 4
R 4
12
12
R
(1
R2 R
2 (1
R2 3
1 3w
R R3
Here the fourth order differential equation of motion is solved by the method of Frobenius. The
2 2
2 2 2 2 3 2
transverse displacement is expressed as an infinite series in
6 1 R
24
1 R R 1 R R
terms of radial coordinates. The frequencies, deflection
2 2
2 2 2 w
2 2 1
functions and moment parameters corresponding to the first two modes of vibrations are computed for the circular plate
12 1 R
6 1 R
R 2
6 1 R R
with clamped and simply supported edge conditions for various values
of taper constant, damping parameter, and elastic
24 2 1 R 2 R 6 1 R 2 2 R 1 1 R 2 3 R 3
w 12 1 R 2 1 2 a 2 2 w 12 1 2
foundation.
Equation of Transverse Motion
0
0
0
0
R
EH 2
t 2
EH 3
K f w
4 W
D
D 3 W 2 D
2 D D 2 W
12 1 2 w
D r4
2 r
r r3 r
r r2 r2 r2

K
0
0
E H 3 t
0
(2)
+
2 D 1 D
D W 2 W W
Solution
r . r 2 r 2
r r 3 r h t 2

Kd
t K f W 0
For damped harmonic vibrations, the solution is
given by
(1) w (R, t) =
W Ret cos pt
(3)
Substituting (3) in (2) and solving we get,
3 W
3 W
1 R 2
12 1 R 2 R 1 R 2
R 1
the following indicial roots are obtained c= 0, 0, 2,
the following indicial roots are obtained c= 0, 0, 2,
2 further, equating to zero the coefficient of the next
4 4 W
4 4 W
3
3
4
4
R 4
R3
subsequent power of R, one finds that a =0 and a is
1
2
1
2
6 1 R 2 3 24 2 1 R 2 2 R 2 1 R 2 4 R 2
1
2
1
2
indeterminate for c=0 hence a2 can be written as an
2 3
2 3 2 W
2 3 1
arbitrary constants along with a0. Similarly equating to zero the coefficients of next higher power of R the constant a3 is
12 1 R
6 1 R
R2 6 1 R R
obtained in terms of a0, and a2 and a ( = 4, 5, 6, –) can be written in terms of a0 and a2.
24 2 1 R 2 2 R 6 1 R 2 3 R 1 1 R 2 4 R 3
Hence assuming ,
W E
C * 1 R2 D2 I *2 2 I * 1 R2 2 W =0 (4)
a Aa0
B a2 0,1,2,3,
(7)
R F k
The following solution, corresponding to c=0 is obtained,
12 K 1 v 2
W a A R a R 2 B R
E
E
where ,
F
F E
, C * 1
H
H
3
3
0
0 1
4
2
4
(8)
31 v2 K 2
Dk
E
, I * 1
H
H
2
2
0
It is evident that no new solution will arise corresponding to other values of c, i.e. for c=2, it is already contained in the solution (8) with arbitrary constants a0 and
12 1 v 2 a 2 p 2
2 ,
E
where p= circular frequency,
= Frequency parameter ,
a2.
Convergence of the Solution
Lamb's technique is applied to test the convergence of the solution (8). Rewriting recurrence relation
D = damping parameter ,
k
k
a 8 a 6
F2 6 a 4 . F3 4 a 2 . F4 2
a a
F1 8
a F1 8
a F1 8
EF = Elastic foundation parameter
A series solution for w is assumed in the form,
F5 0 F1 8
w R a RC
0
, a0 0 , (5)
F2 6
. F3 4
. F4 2
where c is exponent of singularity
F1 8
F1 8
F1 8
8 6
8 6
4
4
2
2
Substituting (5) in equation (4) one
obtains
F5 0 F1 8
a F RC 4 a
F RC 2 a
F RC a
a
a
1 2 3
where = limit
1 where
0 0 0
,
a F
RC 2 a F
RC 4 0
4
0
5
0
(6)
Hence the infinite series is uniformly convergent when  < 1. Hence the solution is convergent.
For the series expression (5) to be the solution the coefficient of different powers of R in the equation (6) must be identically equal to zero. Thus equating to zero the coefficient of lowest power of R, one gets the identical
equation , a0 F1 0 0 Since a0 0 i.e. F1 (0) =
0
Boundary Conditions and Frequency Equations
The frequency equations for clamped and simply supported cirular plates have been obtained by employing the appropriate boundary conditions.
Clamped Plate: For a circular plate clamped at edges r=a, the deflection w and slope of the plate element at edges should be zero.
F1 0 N1 1b0 3 N1 2b0 2 N1 3b0 1 N1 4b0 0 0
1c c 1c 2c 3 2 c c 1c 2 c c 1 c 0
c = 0, 0, 2, 2
W (r, t)r=a =
M M x 1 R2 3 1A R 2
W r,t 0
or W R=1 =
D
D
0 3
r r a
1 A
3
3
1 R2 3 21
1B R 2
W
1 B
3
0
R R1
3
EH 3
(11)
Using above equation and applying the boundary
0
0
conditions one obtains the frequency equation for (clamped
Where
D0 121 2
V1 V2
plate) as
0
4
4
V3 V
(9)
The values of for both edge conditions have been taken from equation (9) and (10).
Result and Discussion : Numerical results for an isotropic
circular plate of parabolically varying thickness resting on
where
V1 1 A
4
V2 1 B
4
elastic foundation have been computed by using computer technology. In all the cases considered the Poissons ratio
V3
4
A
V4 2
4
B
has been assumed to remain constant and it has been taken to be 0.3. Terms of series up to an accuracy of 108 in their absolute values have been retained. Frequency
Simply Supported Plate: For a circular plate simply supported at the edge r=a, the deflection W and the moments Mr at the edge should be zero.
parameter corresponding to first two modes of vibration of a clamped and simply supported isotropic circular plate has been computed for different values of taper constant,
W r,t
r a
Mr
r,tr a 0
or,
damping parameter and foundation effect have been
computed. All the results are tabulated in tables and graphically shown in figures (1.1) to (1.8). The results up
4
2 W
v W
to accuracy of 10 have been given in the tables.
WR1 R2
0
R R R1
Verification of work is obtained by allowing damping parameter and elastic foundation parameter to be zero, the problem reduce to well known problem of a
Applying these boundary conditions on the equation , one gets the frequency equation for simply supported plate as,
homogenous circular plate of parabolically varying thickness. The results so good agreements with the already published work of Gupta .
Figure (1.1) and (1.2) shows the effect of variation
V V
of a taper constant on the frequency parameter for a
2
2
1
1
0
(10)
circular plate of parabolically varying thickness resting on
V5 V6
elastic foundation, (i.e., for DK .01, EF .01, h =.1
and DK
.02, EF
.02, h =.1) with simply supported
where
V5 1 A
4
(SS) and clamped edge edge conditions. From figure it is observed that the first mode remain near about constant and
V6 2 1 1B
4
Deflection Functions and Moments
Again enforcing the boundary condition W=0 at X=1 and adopting the same value of a0 and a2 the non
the second mode will be decreases in frequency parameter with the increasing of taper constant on the both mode of vibration for simply supported and clamped edge plates.
Figure (1.3) and (1.4) shows the effect of variation of damping constant on the frequency parameter for a circular plate of parabolically varying thickness resting on elastic foundation (i.e., for .01, EF .01, h =.1 and
dimensional parameter is obtained in the form
.02, EF .02, h =.1) with simply supported and
clamped edge conditions. From figure it is observed that there is a decreasing in the frequency parameter with the increasing of damping parameter on the both mode of vibration but this decreasing is some greater for the first mode than the second mode for the simply supported and clamped edge plates.
Figure (1.5) and (1.6) shows the effect of variation of foundation parameter for a circular plate of parabolically varying thickness resting on elastic foundation (i.e., for
circular plate of parabolically varying thickness for different values of taper constant.
FIGURE 1.3 (H=0.1, =0.3)
.01 DK .01, h =.1 and .02, DK .02, h =.1)
with simply supported and clamped edge conditions. From figure it is observed that there is a increasing in the frequency parameter with the increasing of foundation effect but this increasing is some greater for the first mode than the second mode on the both mode of vibration for the simply supported and clamped edge plates.
Figure (1.7) and (1.8) shows the variation of deflection and moment parameter with respect to the different points on the plate surface from axis of symmetry.
FIGURE 1.1 (H=0.1, =0.3)
Graph = (Dk=E F =0.01) Graph ——— = (Dk=E F =0.02)
Graph = (Dk=E F =0.01) Graph ——— = (Dk=E F =0.02)
Frequency Parameter ()
Frequency Parameter ()
(SS) 

1.2 

1 

0.8 

0.6 

0.4 

0.2 

0 

0 0.01 0.02 
0.03 

Damping (DK) 

Mode 1 Mode 2 Mode 1 
Mode 2 
(SS) 

1.2 

1 

0.8 

0.6 

0.4 

0.2 

0 

0 0.01 0.02 
0.03 

Damping (DK) 

Mode 1 Mode 2 Mode 1 
Mode 2 
Variation of for the vibration of a damped simply supported circular plate of parabolically varying thickness for different values of damping parameter.
Frequency Parameter ()
Frequency Parameter ()
(SS) 

1.4 

1.2 

1 

0.8 

0.6 

0.4 

0.2 

0 

0 
0.01 0.02 0.03 

Taper constant () 

Mode 1 Mode 2 Mode 1 Mode 2 
(SS) 

1.4 

1.2 

1 

0.8 

0.6 

0.4 

0.2 

0 

0 
0.01 0.02 0.03 

Taper constant () 

Mode 1 Mode 2 Mode 1 Mode 2 
FIGURE 1.4 (H=0.1, =0.3)
Graph = ( =E F
=0.01)
0.9
0.8 0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.01
0.9
0.8 0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.01
Frequency Parameter ()
Frequency Parameter ()
Graph ——— = ( =E F =0.02)
(CC)
(CC)
Variation of for the vibration of a damped simply supported circular plate of parabolically varying thickness for different values of taper constant.
FIGURE 1.2 (H=0.1, =0.3)
Graph = (Dk=E F =0.01) Graph ——— = (Dk=E F =0.02)
0.02
0.03
0.02
0.03
Damping (DK)
Damping (DK)
Mode 1 Mode 2 Mode 1 Mode 2
Mode 1 Mode 2 Mode 1 Mode 2
Variation of for the vibration of a damped clamped circular plate of parabolically varying thickness for different values of damping parameter.
(CC) 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

0 
0.01 0.02 0.03 0.04 

Taper constant () 

Mode 1 Mode 2 Mode 1 Mode 2 
(CC) 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

0 
0.01 0.02 0.03 0.04 

Taper constant () 

Mode 1 Mode 2 Mode 1 Mode 2 
FIGURE 1.5 (H=0.1, =0.3)
Frequency Parameter ()
Frequency Parameter ()
Graph = ( =D K
= 0.01)
variation of for the vibration of a damped clamped
Graph ——— = ( =D K = 0.02)
(SS)
FIGURE = 1.8
( H=0.1, =0.3) 0.0, D =0.01, E =0.01
Frequency Parameter ()
Frequency Parameter ()
1.2
1
0.8
0.6
0.4
0.2
0
0.01 0.015 0.020 0.025
Elastic Foundation (EF)
Mode 1 Mode 2 Mode 1 Mode 2
Variation of for the vibration of a damped simply supported circular plate of parabolically varying thickness for different values of foundation parameter.
FIGURE 1.6 (H=0.1, =0.3)
Frequency Parameter ()
Frequency Parameter ()
Graph = ( =D K =0.01) Graph ——— = ( =D K =0.02)
(CC) 

1.2 

1 

0.8 

0.6 

0.4 

0.2 

0 

0.01 
0.015 
0.020 
0.025 0.030 

Elastic Foundation (EF) 

Mode 1 
Mode 2 
Mode 1 
Mode 2 
Variation of for the vibration of a damped clamped circular plate of parabolically varying thickness for different values of foundation parameter.
FIGURE = 1.7
k f
Graph =SS Graph———=CC
10
8
6
Moment (M)
Moment (M)
4
2
0
0
0
0.05
0.05
0.10
0.10
0.15
0.15
0.20
0.20
0.25
0.25
0.30
0.30
0.35
0.35
0.40
0.40
0.45
0.45
0.50
0.50
0.55
0.55
0.60
0.60
0.65
0.65
0.70
0.70
0.75
0.75
0.80
0.80
0.85
0.85
0.90
0.90
0.95
0.95
1
1
2
4
6
8
Mode 1 Mode 2 Mode 1 Mode 2
Moment parameter (M) for a circular plate of parabolically varying thickness
REFRENCES

Airey, J., The Vibration of Circular Plates and their Relation to Bessel Functions. Proc. Phys. Soc.(London), Vol. 23, (1911), pp. 225232.

Carrington, H., The Frequencies of Vibration of Flat Circular Plates Fixed at the Circumference., Phil. Mag., Vol. 50, No.6, (1925), pp.12611264.

Reid. W.P., Free Vibrations of a Circular Plate. J. Soc. Ind. Appl. Math., Vol. 10, No.4, Dec. (1962), pp.668674.

Ungar, E.E., Maximum Stresses in Beams and Plates Vibrating at Resonance. Trans. A.S.M.E., J. Eng. Ind., Vol.84B, (1962), pp.149155.

Bodine, R.Y., The Fundamental Frequencies of a Thin Flat Circular Plate SimplySupported along a Circle of Arbitrary Radius. A.S.M.E. Paper No.APMW10, J. Appl. Mech., Vol.26, (1959), pp. 666668.

Colwell, R.C. and Harday, H.C., The Frequencies and Nodal. Systems of Circular Plates. Phil. Mag., Ser.7, Vol.24, No. 165 , (1937), pp.10411055.

Waller, M.D., Vibrations of Free Circular Plates. Proc. Phys. Soc. (London), Vol. 50, (1938), pp. 7076.

Tomar, J.S, Gupta, D.C. and Jain, N.C., Free Vibrations of an Isotropic ElasticNonHomogeneous Circular Plate of Linearly Varying Thickness, J. Sound & Vib., Vol. 85, No.3, (1982),
(H=0.1, =0.3) 0.0,
Graph =SS Graph——–=CC
1.2
1
Deflection (W)
Deflection (W)
0.8
Dk =0.01,
Ef =0.01
pp.365370.

Tomar, J.S., Gupta. D.C. and Jain, N.C., Free Vibrations of an Isotropic NonHomogeneous Infinite Plate of Linearly Varying Thickness, J. Italian Ass Th. App. Mech., AIMETA. Vol. 18 (1983), pp. 3033.

Harris, G.Z., The Normal Modes of a Circular Plate of Variable Thickness, Quar. J. Mech. Appl.Maths., Vol.21, (1968), pp.321 327.

Jain, R.K., Axisymmetric Vibrations of Circular Plates of Linearly Varying Thickness, J. Appl. Maths. Phys. [ZAMOP],Vol.23, (1972).
0.6
0.4
0.2
0
0
1
1
0
0.05
0.05
0.10
0.10
0.15
0.15
0.20
0.20
0.25
0.25
0.30
0.30
0.35
0.35
0.40
0.40
0.45
0.45
0.50
0.50
0.55
0.55
0.60
0.60
0.65
0.65
0.70
0.70
0.75
0.75
0.80
0.80
0.85
0.85
0.90
0.90
0.95
0.95
0.2
Mode 1 Mode 2 Mode 1 Mode 2
Transverse Deflection (W) for a circular plate of parabolically varying thickness.