Effect of Pasternak Foundation on Axisymmetric Vibration of Polar Orthotropic Non-Homogeneous...

IJCMNS, Vol. 5, No. 1, January-June 2012, pp. 151-163,

Effect of Pasternak Foundation on AxisymmetricVibration of Polar Orthotropic Non-Homogeneous

Circular Plate of Variable Thickness

Seema Sharma1, R. Lal2 and Shivani Srivastava3

ABSTRACT

Present paper analyses the axisymmetric vibration of polar orthotropic non-homogeneous circularplate of linearly varying thickness resting on Pasternak foundation. A linear type variation in

Young’s modulli and density has been assumed. Ritz method has been employed to obtainapproximate solution of the problem. The choice of this method has the advantage of high

accuracy and computational efficiency, which greatly depends upon the nature of admissiblefunctions. The first three natural frequencies have been obtained for different values of taperparameter, density parameter, non-homogeneity parameter, foundation stiffness parameters and

rigidity parameter. Normalized transverse displacements of the specified plates for first threemodes of vibration for clamped and simply-supported boundary condition have been presented.

Comparison studies have been carried out which establish the accuracy of the present method.

Keywords: Axisymmetric, Linear thickness, Non-homogeneous, Pasternak foundation.

1. INTRODUCTION

Non-homogeneous circular plates of composite material are being widely used in variousengineering application fields due to their high specific stiffness and light weightcharacteristics. Many structural components in aerospace, mechanical and civil engineeringare supported on elastic medium. The vibrational characteristics of plates resting on anelastic medium are different from those of the plates supported only on the boundary. A

number of papers [1-10] have appeared dealing with natural frequencies of homogeneous/non-homogeneous isotropic/polar orthotropic plates of non-uniform thickness to investigatethe effect of elastic foundation. Gupta et al., [1, 2] analysed the effect of Winkler foundationon buckling and vibrations of polar orthotropic circular plates of linear and parabolicthickness variations, respectively. Gupta et al., [3] investigated the effect of Winkler

1 Deptt. of Mathematics, Gurukul Kangri University, Hardwar, E-mail: [email protected]

2 Deptt. of Mathematics Indian Institute of Technology, Roorkee, E-mail: [email protected]

3 Deptt. of Mathematics Gurukul Kangri University, Hardwar, E-mail:[email protected]

© Serials PublicationsISSN: 0973-581X

152 Seema Sharma, R. Lal and Shivani Srivastava

foundation on vibrations of polar orthotropic Mindlin circular plate. Liew et al., [4] obtainedthe natural frequencies of Mindlin’s rectangular plates resting on Winkler foundation. Guptaand Bhardwaj [5] studied vibrations of polar orthotropic circular plates of quadraticallyvarying thickness resting on Winkler foundation. Buckling and asymmetric vibrations of

polar orthotropic circular plates resting on Winkler foundation have been investigated byGupta et al., [6]. Zhou et al., [7] made three dimensional vibration analysis of thick circularplates on Pasternak foundation. Lal and Dhanpati [8, 9] analysed the effect of Winkler andPasternak foundation, respectively, on vibrations of non-homogeneous orthotropicrectangular plates of varying thickness. The effect of Pasternak foundation on vibrations ofpolar orthotropic annular plates of variable thickness has been investigated by Sharma

et al., [10].

In this paper, the effect of Pasternak foundation on natural frequencies and mode shapesof non-homogeneous polar orthotropic circular plate of linear thickness variation has beenanalysed using Ritz method. The results are correct upto at least four significant digits andcan be used as benchmark for future researchers.

2. METHOD OF SOLUTION: RITZ METHOD

Consider a non-homogeneous orthotropic circular plate of radius a, thickness h (r), density

� (r) and resting on a Pasternak foundation with spring and shear stiffness parameters Kf

and Gf , respectively, elastically restrained against rotation by springs of stiffness k�, referredto cylindrical polar coordinate (r, �, z), where the axis of the plate is taken as the line r = 0and its middle surface as the plane z = 0. (Fig. 1)

Figure 1: Polar Orthotropic Circular Plate Resting on Pasternak Foundation

Effect of Pasternak Foundation on Axisymmetric Vibration of Polar Orthotropic... 153

The maximum potential energy and kinetic energy are given by

2 22 2 2

max 2 20 0

2 222

0

1 1 12

2

1 ( , ).

2

�

� �

�

�

� � ��

� � � � ��

� �

�

a

r

f f

W W W WU D v D

r r r r r r

W W aK W G rd rd ak

r r(1)

22 2

max

0 0

1

2

�

� �� a

T hW rd rd . (2)

where 30

12(1 )�� r

r

E hrD ,

30

12(1 )�

�� r

E hD and Er, E�, �r, �� are Young’s modulii and Poisson’ss

ratios in proper directions with Er �� = E��r.

Ritz method requires that the functional

22 2 2

max max 2 20 0

2 2

2

22 22 2

0 0 0

1 1( ) 2

2

1

1 ( , ) 1

2 2

�

�

�

� �

�

� � ��

� � � � ��

� ��

� �

� � �

a

r

f f

a

W W WJ W U T D v

r r r r

W WD K W G rd rd

r r r

W aak d hW rd rd

r(3)

be minimized.

Introducing the non-dimensional variables � WaW and � r

aR along with the relations

Er = E1(1 + �R), E� = E2(1 + �R), � = �0(1 + �R) and considering the thickness variation ash = h0(1 + �R) where h0 is the thickness of the plate at its centre, the functional J (W) givenby equation (3) becomes

0

0

2 222

22 13 2

20 0

2

22 2 12 2

0 0 0

1

( ) (1 )(1 )2 1

2

(1)(1 )(1 ) ,

2

�

�

� �

�

� ��

� � ��

� �

� � �

r

r

W Wp

D R R R WJ W R R KW G Rd Rd

rW W

R R r

D WK d R R W Rd Rd

R (4)


where,

0 0 0 0

4 43 4 22 21 0 0 0, , , , ,

12(1 )��

��

� ��

� � �f f

rr r r r r r

a K a G akE h E a hD p K G K

E D D D D

Now, approximate transverse deflection W—

in terms of a set of linearly independent

coordinate functions, which satisfy the boundary conditions of the problem. The deflectionfunction assumed here is based upon the static deflection for polar orthotropic circularplates. Assume the deflection function as

4 1 2

0 0

( ) (1 )�

� �

� � � � � �� m m

p ii i i i i

i i

W A F R A R R R , (5)

where, Ai are unknown coefficients and �i, �i are unknown constants. As each coordinatefunction has to satisfy the elastically restrained against rotation condition at the boundaryi.e. R = 1 (Leissa [11], pp. 14), we have

23

21

1(1 )� �

�

� ��

� �R

dW d W dWK

d R R dRdR, (6)

(1) 0�W . (7)

The unknown constants �i and �i are determined using these boundary conditions.

Substituting the value of W—

from equation (5) into (4), the functional J (W—

) becomes

0

0

2 2

2

0 03

1

0 00

2

0 0

22 12

00 01

1

(1 ) (1 )2

( )2

2

� �

�

� �

� �

�

��

� ��

��

� ��

� �

� �

� ��

� �

��

m m

i i i ii i

m mr

i i i ii i

m m

i i i ii i

mr

i ii R

A F p A FR

R RD

A F A FJ W Rd RdR

K A F G A F

DK A F d

2

0

(1 ) (1 ) .�

� ��

� ��

m

i ii

R R A F Rd Rd (8)

The minimization of the functional J (W—

) given by (8) requires

( )0, 0,1, ...,

��

� i

J Wi m

A. (9)


This leads to a system of homogeneous equations in Ai , i = 0, 1, 2, …, m, whosenon-trivial solution leads to the frequency equation

2 0� � �A B , (10)

where, A = [aij] and B = [bij] are square matrices of order (m + 1) given by

2

1 23

1

0

( )(1 )(1 ) ( )

( )� �

�

� ��

� ��

�i j i j

ij i j i j i j R

i j j i

pF F F F

Ra R R KF F GF F RdR K F F

F F F FR

, (11)

and

1

0

(1 )(1 )� � � � ��ij i jb R R F F Rd R (12)

for i = 0, 1, …, m; j = 0, 1, …, m.

3. RESULTS AND DISCUSSION

The frequency equation (10) provides the values of frequency parameter � of

non-homogeneous orthotropic circular plate of linearly varying thickness resting on a

Pasternak foundation. In order to investigate the effect of various parameters namely rigidity

ratio p2(= 0.75, 1.0, 2.0, 5.0); non-homogeneity parameter � (= – 0.5 (0.2) 1.0); density

parameter � (= – 0.5 (0.2) 1.0); taper parameter � (= – 0.5 (0.2) 0.5), on the natural frequencies

of clamped and simply supported plates resting on Pasternak foundation with stiffness

parameters K (= 0 (100) 500) and G (= 0 (5) 25) for �� = 0.3, numerical results have been

computed for the first three modes of vibration. The boundary conditions clamped and

simply supported have been established by assigning K� = 1020 and 0.0, respectively.

The natural frequencies obtained from Ritz method are upper bounds of the exact ones

and therefore, convergence should be monotonic from above as the number of terms of

admissible functions increases. Figures 2(a, b) show the convergence of normalized

frequency parameter �/�* for clamped and simply supported plates, respectively, for p2 = 5;

� = – 0.5; � = – 0.5; � = – 0.5; K = 500 and G = 25. It is found that 14 terms of admissible

functions give first three frequencies accurate up to at least four significant digits.


The numerical results for specified plate parameters have been presented in Tables (1-2)

and Figures (3-6). Tables (1-2) present the values of frequency parameter � for different

values of parameters namely rigidity ratio p2 (= 0.75, 1.0, 2.0, 5.0), non-homogeneity

parameter � (= – 0.5, 1.0), density parameter � (= – 0.5, 1.0), taper parameter � = – 0.3 and

0.3, spring stiffness parameter K (= 500) and shear stiffness parameter G (= 0, 25) for first

three modes of vibration for both clamped and simply supported plates, respectively. It has

been seen that the values of frequency parameter � are higher for clamped plate as compared

to those of simply supported plates. Also, the value of frequency parameter � increases by

increasing values of rigidity ratio p2, non-homogeneity parameter �, spring stiffness

parameter K and shear stiffness parameters G, while it decreases by increasing density

parameter �.

Figures 3(a, b, c) show the behavior of frequency parameter � with spring stiffness

parameter K for � = – 0.3, 0.3; � = – 0.5; � = – 0.5, G = 25 and p2 = 1, 2, for clamped and

simply supported plates vibrating in fundamental, second and third mode, respectively.

The frequency parameter � is found to increase with increasing value of K. The rate of

increase is more pronounced for the plates with thinner edge as compared to the plates with

thicker edge. This rate of increase is slightly higher for simply supported plates than clamped

plates. The rate of increase decreases when plate becomes tangentially stiff. The rate of

increase decreases with the increase in the number of modes.

Figure 2: Convergence of the Normalized Frequency Parameter / * for (a) Clamped and(b) Simply Supported Plate with Number of Terms m Used for the First Three Modes ofVibration for p2 = 5, = – 0.5; = – 0.5; = – 0.5; K = 500; G = 25. *-The Results Using20 Terms


Table 1Frequency Parameter for Clamped Plate

� – 0.3 0.3

� – 0.5 1.0 – 0.5 0.5

Mode �� – 0.5 1.0 – 0.5 1.0 – 0.5 1.0 – 0.5 1.0p2

K = 500, G = 0

0.75 26.9231 27.9118 20.9011 21.9178 25.8508 28.8914 19.9825 22.7810

1.00 27.0576 28.1463 20.9351 22.0276 26.0717 29.3546 20.0916 23.07212.00 27.4624 28.8860 21.0542 22.3985 26.7920 30.8861 20.4724 24.0590

5.00 28.2433 30.4317 21.3481 23.2628 28.3776 34.2824 21.3968 26.3306

I K = 500, G = 25

0.75 30.9348 31.8985 23.7365 24.7906 29.3402 32.3352 22.4556 25.2888

1.00 31.0213 32.0672 23.7656 24.8727 29.5031 32.7008 22.5387 25.51652.00 31.3098 32.6396 23.8722 25.1676 30.0702 33.9696 22.8446 26.32795.00 31.9501 33.9507 24.1453 25.9065 31.4223 36.9653 23.6349 28.3183

K = 500, G = 0

0.75 41.8526 52.5091 29.6199 38.1047 49.9349 67.6856 36.4065 50.35541.00 42.5915 53.7588 30.0676 38.9174 51.1447 69.5526 37.1907 51.6241

2.00 44.9149 57.7195 31.4771 41.4807 54.9833 75.5563 39.6698 55.67465.00 49.6335 65.7651 34.3660 46.6853 62.7893 87.8607 44.7152 63.9402

II K = 500, G = 25

0.75 54.5138 63.1444 39.0597 46.0736 58.7962 74.9401 42.9683 55.83111.00 55.1506 64.1969 39.4048 46.7134 59.7978 76.5674 43.5793 56.9001

2.00 57.2111 67.6291 40.5437 48.8166 63.0789 81.9276 45.5981 60.42175.00 61.5251 74.8687 43.0141 53.3316 70.0518 93.3081 49.9644 67.9406

K = 500, G = 0

0.75 77.1028 106.3029 53.9911 76.1083 102.3904 143.1573 73.3340 104.6574

1.00 78.5048 108.4463 54.9048 77.5540 104.4424 146.1149 74.7143 106.71682.00 82.9151 115.3183 57.7471 82.1228 111.0253 155.8692 79.0809 113.3866

5.00 91.6359 129.0552 63.3310 91.1668 124.1908 175.8816 87.7341 126.8388

III K = 500, G = 25

0.75 95.1780 119.7287 66.9715 99.5967 113.1396 151.4647 81.2210 110.9115

1.00 96.4387 121.6728 67.7399 91.2846 114.9900 154.2192 82.4212 112.78952.00 100.4755 127.9988 70.1976 87.2106 121.0182 163.3920 86.3019 118.95515.00 108.6088 140.9086 75.1977 99.5967 133.3584 182.5409 94.2488 131.6826


Table 2Frequency Parameter for Simply Supported Plate

� – 0.3 0.3

� – 0.5 1.0 – 0.5 0.5

Mode �� – 0.5 1.0 – 0.5 1.0 – 0.5 1.0 – 0.5 1.0p2

K = 500, G = 0

0.75 26.7954 27.1271 20.1941 20.4606 24.3249 24.7999 17.7672 18.34251.00 26.9070 27.2810 20.2075 20.5093 24.4444 25.0634 17.8160 18.49712.00 27.2331 27.7646 20.2559 20.6862 24.8509 26.0038 17.9990 19.0730

5.00 27.8286 28.7975 20.4322 21.2138 25.8639 28.4661 18.6066 20.7392

I K = 500, G = 25

0.75 30.5157 30.7376 22.9988 23.1712 27.4883 27.9168 19.4122 20.6210

1.00 30.5876 30.8580 23.0166 23.2184 27.5907 28.1448 20.1878 20.75702.00 30.8229 31.2627 23.0681 23.3801 27.9474 28.9722 20.3462 21.2609

5.00 31.7323 32.2621 23.3260 23.9428 28.9395 31.2574 20.9618 22.8154

K = 500, G = 0

0.75 37.6827 44.2944 26.0628 31.5472 40.8260 52.3543 29.5406 38.6700

1.00 38.2384 45.2872 26.3818 32.1726 41.7960 53.9247 30.1387 39.70342.00 39.9449 48.3719 27.3245 34.0625 44.7928 58.8764 31.9439 42.89505.00 44.0373 55.6646 29.9408 38.8823 52.0071 70.4954 36.6679 50.7580

II K = 500, G = 25

0.75 51.2707 56.1105 36.0701 40.1205 50.5950 60.3698 36.4261 44.44581.00 51.7912 56.9500 36.3265 40.6026 51.3808 61.6959 36.8779 45.2847

2.00 53.3352 59.5551 37.0268 42.0513 53.8265 65.9560 38.2426 47.92955.00 57.6680 66.4074 39.7189 26.4881 60.4546 76.7127 42.5335 55.1460

K = 500, G = 0

0.75 67.1742 90.7091 46.6437 64.4657 86.5383 119.0904 61.6650 86.67681.00 68.3876 92.6126 47.4212 65.7314 88.3631 121.7759 62.8731 88.51792.00 71.8963 98.3207 49.5655 69.3821 93.8166 130.1420 66.3488 94.0556

5.00 81.1251 112.6130 55.7475 79.1174 107.6111 150.6700 79.7198 108.2289

III K = 500, G = 25

0.75 87.6596 105.9197 60.9424 75.2657 98.4733 128.1622 70.1650 93.3324

1.00 88.7317 107.6031 61.5697 76.3321 100.0741 130.6154 71.1779 94.97182.00 91.7332 112.6284 63.2024 79.3793 104.8530 138.2938 74.0787 99.9279

5.00 100.8942 126.3197 69.2294 88.5842 117.9771 158.0023 82.8845 113.4288


Figure 3: Frequency Parameter for (a) Fundamental (b) Second and (c) Third Mode for G = 25, = – 0.5, = – 0.5. , : p2 = 2; , : p2 = 1; , : = 0.3; , : = – 0.3.

- - - - - - - - - - - - Simply Supported Plate; –––––––––– Clamped Plate.

Figure 4: Frequency Parameter for (a) Fundamental (b) Second and (c) Third Mode for K = 500, = – 0.5, = – 0.5. , : p2 = 2; , : p2 = 1; , : = 0.3; , : = – 0.3

- - - - - - - - - - - - Simply Supported Plate; –––––––––– Clamped Plate

Figures 4(a, b, c) depict variation of frequency parameter � with shear stiffness

parameter G for � = – 0.3, 0.3; � = – 0.5; � = – 0.5, K = 500 and p2 = 1, 2 for both clampedand simply supported plates vibrating in fundamental, second and third mode, respectively.It is observed that the frequency parameter � increases with increasing values of G. Therate of increase in the value of frequency parameter � with increase in G is higher for the


plates with thinner edge as compared to the plates with thicker edge. This rate of increasefor isotropic plate is slightly more than that for tangentially stiff plate. Also, this rate ofincrease, pronounces with the increase in the number of modes.

Figure 5(a) show the effect of taper parameter � on frequency parameter � for two sets

of non-homogeneity parameters namely (i) � = � = – 0.5 and (ii) � = � = 1.0 and K = 500,G = 25, p2 = 1, 2, for clamped and simply supported plates vibrating in fundamental mode.It is observed that the frequency parameter � decreases with increasing values of taperparameter � with few exceptions. In these cases, the frequency first decreases and thenincreases giving rise to local minima. For orthotropic non-homogeneous clamped plate,there is a local minima in the vicinity of � = – 0.3 for � = � = 1.0, which shifts to � = 0.3 for

� = � = – 0.5. For isotropic non-homogeneous simply supported plate, a local minima isobserved at � = – 0.1. From Figures 5(b, c), when plate is vibrating in second and thirdmode, respectively, it is found that the frequency parameter � increases with increasingvalues of taper parameter �, except for orthotropic clamped and simply supported plateand isotropic clamped plate vibrating in second mode. In these cases, the frequency firstdecreases and then increases with local minima. For � = � = – 0.5, a local minima is

Figure 5: Frequency Parameter for (a) Fundamental (b) Second and (c) Third Mode for K = 500,G = 25. , : p2 = 2; , : p2 = 1; , : = 1; = 1; , : = – 0.5; = – 0.5;- - - - - - - - - - - - Simply Supported Plate; –––––––––– Clamped Plate


Table 3Comparison of Frequency Parameter for Isotropic (p2 = 1) Homogeneous ( = 0.0, = 0.0)

Circular Plate of Uniform Thickness ( = 0.0)

Boundary

Mode Clamped Simply supported

I 10.2158 10.2158* 10.2159° 4.9352 4.9352* 4.9352°

II 39.7711 39.7711* 39.7766° 29.7200 29.7200* 29.7220°

III 89.1041 89.1041* 89.1708° 74.1961 74.1961* 74.1938°

* values taken from Leissa [11].o values taken from Pardoen [12].

observed in the vicinity of � = – 0.3 for clamped plate which shifts towards � = – 0.1 forsimply supported plate, which further shifts to � = 0.0 for isotropic clamped plate for� = � = 1.0.

Figures 6(a, b) show the normalized transverse displacements for non-homogeneous

orthotropic circular plates of linearly varying thickness resting on Pasternak foundationi.e., � = 1.0, � = 1.0, p2 = 2, � = – 0.5, 0.5, K = 500, G = 25 for clamped and simplysupported boundary, respectively. It is observed that the corresponding nodal radii for theplates with thinner edge are less than those for the plates with thicker edge.

Figure 6: Normalized Transverse Displacement for (a) Clamped and (b) Simply Supported Platefor K = 500, G = 25, p2 =2.0, =1.0, = 1.0. : = – 0.5; : = 0.5


Table 4Comparison of Frequency Parameter for Isotropic (p2 = 1) Non-Homogeneous

( = 1.0, = 1.0) Circular Plate of Linear Thickness Variation ( = – 0.5)

Boundary

Mode Clamped Simply supported

I 6.9743 6.9743* 3.6678 6.9743*

II 28.2756 28.2756* 21.5609 28.2756*

III 64.1358 64.1358* 53.8779 64.1358*

* values taken from Sharma [13].

Table 3 shows the comparison of frequency parameter Ù for isotropic homogeneouscircular plate of uniform thickness without foundation with those obtained by Leissa[11]and Pardoen[12]. Comparison of the results of this study for isotropic non-homogeneous

circular plates of linear thickness variation without foundation with those obtained bySharma[13] using differential quadrature method has been given in table 4. A close agreementof results shows the accuracy and versatility of the present method.

REFERENCES[1] Gupta U. S., Lal R., and Jain S. K., “Buckling and Vibrations of Polar Orthotropic Circular

Plates of Linearly Varying Thickness Resting on an Elastic Foundation”, Jl. Sound Vib., 147(3),

(1991), 423-434.

[2] Gupta U. S., Lal R., and Jain S. K., “Vibration and Buckling of Parabolically Tapered PolarOrthotropic Plates on Elastic Foundation”, Indian Jl. Pure Appl. Maths., 24, (1993), 607-631.

[3] Gupta U. S., Lal R., and Sagar R., “Effect of Elastic Foundation on Axisymmetric Vibrations of

Polar Orthotropic Mindlin Circular Plates”, Indian Jl. Pure Appl. Maths., 25(12), (1994),1317-1326.

[4] Liew K. M., Han J. B., Xiao Z. M., and Du H., “Differential Quadrature Method for Mindlin

Plates on Winkler Foundation”, Int. Jl. Mech. Sci., 38(4), (1996), 405-421.

[5] Gupta A. P., and Bhardwaj N., “Free Vibration of Polar Orthotropic Circular Plates of QuadraticallyVarying Thickness Resting on Elastic Foundation”, Appl. Mathematical Modelling, 29(2), (2005),

137-157.

[6] Gupta U. S., Ansari A. H., and Sharma S., “Buckling and Vibration of Polar Orthotropic CircularPlate Resting on Winkler Foundation”, Jl. Sound Vib., 297, (2006), 457-476.

[7] Zhou D., Lo S. H., Au F. T. K., and Cheung Y. K., “Three Dimensional Free Vibration of Thick

Circular Plates on Pasternak Foundation”, Jl. Sound Vib., 292, (2006), 726-741.


[8] Lal R., and Dhanpati, “Transverse Vibrations of Non-Homogeneous Orthotropic RectangularPlates of Variable Thickness: A Spline Technique”, Jl. Sound Vib., 306(1-2), (2007), 203-214.

[9] Lal R., and Dhanpati, “Effect of Non-Homogeneity on Vibrations of Orthotropic Rectangular

Plates of Varying Thickness Resting on Pasternak Foundation”, ASME Jl. Vib. Acoust., 131,(2009), 1-9.

[10] Sharma S., Gupta U. S., and Lal R., “Effect of Pasternak Foundation on Axisymmetric Vibration

of Polar Orthotropic Annular Plates of Varying Thickness”, ASME Jl. Vib. Acoust., 132(4), (2010),1-13.

[11] Leissa A. W., “Vibration of Plates”, NASA Report No. SP-160, (1969).

[12] Pardoen G. C., “Asymmetric Vibrations and Stability of Circular Plate”, Comp. and Struct., 9,

(1978), 89-95.

[13] Sharma S., “Free Vibration Studies on Non-Homogeneous Circular and Annular Plates”, Ph.D.

Thesis, I.I.T. Roorkee, Roorkee, (2006).

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