Damped Vibrations of an Isotropic Circular Plate of Parabolically Varying Thickness Resting on Elastic Foundation

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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 = H|R=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

(S-S) 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 ()

(S-S)

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

(S-S)

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 ()

(S-S)

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

(S-S)

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)

(C-C)

(C-C)

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.

(C-C)

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

(C-C)

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)

(S-S)

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)

(C-C)

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 =S-S Graph———-=C-C

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

  1. Airey, J., The Vibration of Circular Plates and their Relation to Bessel Functions. Proc. Phys. Soc.(London), Vol. 23, (1911), pp. 225-232.

  2. Carrington, H., The Frequencies of Vibration of Flat Circular Plates Fixed at the Circumference., Phil. Mag., Vol. 50, No.6, (1925), pp.1261-1264.

  3. Reid. W.P., Free Vibrations of a Circular Plate. J. Soc. Ind. Appl. Math., Vol. 10, No.4, Dec. (1962), pp.668-674.

  4. Ungar, E.E., Maximum Stresses in Beams and Plates Vibrating at Resonance. Trans. A.S.M.E., J. Eng. Ind., Vol.84B, (1962), pp.149-155.

  5. Bodine, R.Y., The Fundamental Frequencies of a Thin Flat Circular Plate Simply-Supported along a Circle of Arbitrary Radius. A.S.M.E. Paper No.APMW-10, J. Appl. Mech., Vol.26, (1959), pp. 666-668.

  6. 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.1041-1055.

  7. Waller, M.D., Vibrations of Free Circular Plates. Proc. Phys. Soc. (London), Vol. 50, (1938), pp. 70-76.

  8. Tomar, J.S, Gupta, D.C. and Jain, N.C., Free Vibrations of an Isotropic ElasticNon-Homogeneous Circular Plate of Linearly Varying Thickness, J. Sound & Vib., Vol. 85, No.3, (1982),

    (H=0.1, =0.3) 0.0,

    Graph =S-S Graph——–=C-C

    1.2

    1

    Deflection (W)

    Deflection (W)

    0.8

    Dk =0.01,

    Ef =0.01

    pp.365-370.

  9. Tomar, J.S., Gupta. D.C. and Jain, N.C., Free Vibrations of an Isotropic Non-Homogeneous Infinite Plate of Linearly Varying Thickness, J. Italian Ass Th. App. Mech., AIMETA. Vol. 18 (1983), pp. 30-33.

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

  11. 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.

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