Analysis of Damping and Non-homogeneity through spline ...

112

International Journal for Environmental Rehabilitation and Conservation

ISSN: 0975 — 6272

IX (2): 112— 124

www.essence-journal.com

Original Research Article

Analysis of Damping and Non-homogeneity through spline

interpolation technique of an isotropic Rectangular plate of

Parabollically varying thickness resting on Elastic foundation

Kumar, Ajendra1; Gupta, Manu2 and Kumar, Ankit1

1Department of Mathematics and Statistics, Gurukula Kangri Vishwavidyalaya, Haridwar, India 2Department of Mathematics, J.V. Jain College, Saharanpur, India

Corresponding Author: [email protected]

A R T I C L E I N F O

Received: 11 July 2018 | Accepted: 29 October 2018 | Published Online: 31 December 2018

DOI: 10.31786/09756272.18.9.2.216

EOI: 10.11208/essence.18.9.2.216

Article is an Open Access Publication.

This work is licensed under Attribution-Non Commercial 4.0 International

(https://creativecommons.org/licenses/by/4.0/)

©The Authors (2018). Publishing Rights @ MANU—ICMANU & ESSENCE—IJERC.

A B S T R A C T In this research paper we analyze the effect of damping and non-homogeneity of an isotropic

rectangular plates of parabollically varying thickness which rests on a winkler-type elastic foundation

on the basis of classical plate theory (CPT).The governing equation of motion/ mathematical model

of plate equation is solved by quintic spline interpolation method together with boundary conditions

for clamped-clamped (C-C) and clamped simply supported (C-SS) edges. Three modes of vibration

have been computed using MATLAB software and comparison of the calculated results with already

published work have also been presented.

K E Y W O R D S

Vibration | Non-homogeneity | Elastic foundation | Damping | spline technique

C I T A T I O N

Kumar, Ajendra; Gupta, Manu and Kumar, Ankit (2018): Analysis of Damping and Non-homogeneity through

spline interpolation technique of an isotropic Rectangular plate of Parabollically varying thickness resting on

Elastic foundation. ESSENCE Int. J. Env. Rehab. Conserv. IX (1): 112—124

https://doi.org/10.31786/09756272.18.9.2.216 https://eoi.citefactor.org/10.11208/essence.18.9.2.216

ESSENCE—IJERC | Kumar et al. (2018) | IX (1): 112—124

113

Introduction

The study of vibration is a special case of

mechanical oscillations. There are several

theories that have been developed to describe

the vibration of plates. But the most

commonly used are the Kirchhoff love theory

and the Mindlin Reissner theory. Due these

theories and rapidly growing technology; the

importance of study of vibration is increasing

day by day (Civalek and Acar, 2007). Elastic

plates of non- homogeneous type (fibre

reinforced material) with varying thickness

have great importance as structural

components in different engineering and

industrial fields like telephone industry,

missile technology, naval ship design and

aerospace industry etc. since they provide

great strength, light weight ,resistance to

corrosion and improved performance at

higher temperature. The variation in thickness

provides advantage in reduction of weight and

size which in turns helps in providing plates

efficiency for bending and buckling and also

helps in reduction of cost of material. The

elastic foundation factor have application are

in pressure technology which are used in

petrochemical industry and various other

industry. Structural damping is much needed

in study of vibration because it enhances the

performance of plate design by increasing

stiffness and thermal stability. Numerous

studies have been completed for study of free

and damped vibration of isotropic and

orthotropic plates. In 1973 Leissa studied the

free vibration of rectangular plates (Leissa,

1973) and Jain and Soni “Free vibration of

rectangular plates of parabollically varying

thickness” (Jain and Soni (1973). Gupta and

Lal (1978) worked on Transverse Vibration of

Non-uniform rectangular plate on elastic

Foundation (Gupta and Lal, 1978). Gupta and

Lal have also studied the transverse vibrations

of rectangular plate of exponentially varying

thickness resting on an elastic foundation by

using quintic spline method (Gupta and Lal,

1978). Later on, Lal R., assumed the

transverse vibration of non-homogeneous

rectangular plate of uniform thickness using

boundary characteristic orthogonal

polynomials (Lal et al., 1997). Recently Rana

and Robin have studied the damped vibration

of rectangular plates of variable thickness

(linear variation) resting on elastic foundation

considering spline technique (Robin and

Rana, 2016). In continuation Gupta M. have

discussed Damped vibrations of rectangular

plates of parabollically varying thickness

resting on elastic foundation considering the

effect of thermally induced non –

homogeneity (Gupta, 2016). Bahmyari and

Rahbar-Ranji (2012) studied plates free

vibration using element free Galerkin method

of orthotropic plates with variable thickness

resting on non-uniform elastic foundation.

Chakraverty and Pradhan (2004) too

presented results on free vibration of

exponential functionally graded rectangular

plates in thermal environment with general

boundary conditions. Also Sharma et al.

(2012) conducted Vibration analysis of non-

homogeneous orthotropic rectangular plates

of variable thickness resting winkler

foundation. Thus various approaches for

solution of mathematical model for plate

equation are available in literature however

quintic spline method provides us highly

accurate results with less computational

efforts for given boundary conditions so we

have used this method for obtaining the first


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three mode of vibration for two different

boundary conditions(c-c and c-ss).

Mathematical formulation

In the present paper we consider plate which

is an isotropic non-homogeneous rectangular

plate of length ‘a’, width ‘b’ and thickness

' ( , ) 'h x y .Plate density is ' ' and it rests on a

winkler-type elastic foundation occupying the

domain 0 ,0x a y b in x-y plane.

The mathematical equation which governs the

vibration of damped plate is given by

2 2 2 2 2 2 22 2

2 2 2 2 2( ) (1 ) 2 0,f

D w D w D w w wD w h k k w

y x x y x y x y t t

(I)

Where 3 2( , ) /12(1 )D Eh x y is ‘flexural

rigidity’ of plate at any point in the middle

plane of the plate, k is the damping constant,

fk is elastic foundation parameter and

( , , )w x y t is the transverse deflection in plate.

Let the two opposite edges y=0 and y=b of the

plate to be simply supported and thickness

h = h(x, y) varies parabollically in the

direction of x-axis. Thus, ‘h’ is independent

of y . .i e h=h(x). For a harmonic solution, it is

assumed that the deflection function w is of

the form

( , , ) ( ) sin costm yw x y t W x e pt

b at y=0 and y=b, (II)

Where p denotes the ‘circular frequency’ of vibration and m is is a positive integer.

Thus Eq. (1) on substituting Eq. (2) becomes

4 33 3 2

4 3

2 22 2 23 2 2 3

2 2 2

23 2

4 4 2 2 23 3 2

4 2 2

cos 2 6 cos

6 6 3 2 cos

2 6 cos

6

W E h WEh pt h Eh pt

x xx x

E E h h h m Wh h Eh Eh Eh pt

x x x bx x x

E m Wh Eh pt

x b x

m m EEh h h

b b x

2 22

2

2 2 2

6 3 cos

12 1 cos 2 sin 0,f

h E h hEh Eh W pt

x x x x

h p k k W pt hp kp W pt

(III)

We now introduce the following non-dimensional variables,

h

Ha

, x

Xa

, E

Ea

, W

Wa

, a

, 2

2 2 am

b

And putting physical quantities of interest to our study which are variation in thickness (taper

constant) and non-homogeneity, given by:


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20 1H H X , 2

0 1E E X , 20 1 X , where 0 0X

H H

, 00X

E E

,

00X

where α is a taper constant and is non-homogeneity parameter .

And performing suitable mathematical calculations the following equation is obtained:

4 3 2

0 1 2 3 44 3 20

W W W WA A A A A W

XX X X

(IV)

where

42 20

4 32 2 21

4 3 22 2 2 2 2 2 22

3 42 2 2 2

4 32 2 23

4 32 2 2

42 2 2 2 2 24

1 1 ,

4 1 1 2 1 1 ,

2 1 2 4 1 2 4 1 1

6 1 1 2 1 1 ,

4 1 1 2 1 1 ,

2 1 2 4 1

1 1 2 4 1 1

A X X

A X X X X X

A X X X X X X

X X X X

A X X X X X

X X X

A X X X X

22

32 2

22 * 2 2 * 2 * 2

6 1 1

1 1 ,k f

X

X X

d I E X C I X

Here , .f kE d are foundation parameter, damping parameter and frequency parameter respectively

given as 2 2 2 2 2 2

02 2

20 0 00

3 1 12 1 12 1, ,

f

k f

k a p kd E

a E E E

.

In order to determine solution of equation

(IV) together with boundary conditions

considered at the edge X=0 and X=1,we use

the quintic spline technique . For suppose

W(x) be the function with continuous

derivatives in [0, 1] and interval [0, 1] be

divided into ‘n’ sub intervals by means of

points iX such that

0 1 20 ... 1nX X X X ,

Where 1, 0,1, 2,..., .iX X i X i n

n

The approximating function W X for W(x)

be a quintic spline which has the following

properties:

(i) W X is quintic polynomial in each

interval , 1 .r rX X

(ii) , 0,1,2,..., .rW X W X r n

(iii) 2 3 4

2 3 4, ,

W W W Wand

X X X X

are

continuous.

In view of above axioms, the quintic spline

technique give


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4 1 5

0 00 0

ni

i j ji j

W X a a X X b X X

(V)

where 0,

,

j

j

J j

if X XX X

X X if X X

and ' , 'i ja s b s are constants.

Thus for the satisfaction at the thn knot, eq. (IV) reduced to

2

4 04 0 4 0 3 1 2

3 0 2

3 2

4 0 3 0 2 0 1 3

4 3 2

4 0 3 0 2 0 1 0 0 4

5 4 3

4 3 2

2

1 0

2 2

3 6 6

4 1 2 2 4 2 4

5 2 0

6 0 1 2 0

mm

m

m m m

m m m m

m j m j m j

j

m j m j

A X XA a A X X A a a

A X X A

A X X A X X A X X A a

A X X A X X A X X A X X A a

A X X A X X A X Xb

A X X A X X

1

0

0 .n

j

(VI)

For 0 1 ,m n above system contains (n+1)

homogeneous equation with (n+5) unknowns,

, 0 1 4ia i and. The above system of

equations can be represented in matrix form

as

0 ,A B (VII)

Where A denotes a matrix of order

1 5 ,n n while B and 0 are column

matrices of order (n+5).

Boundary conditions and frequency

equation

The following two cases of boundary

conditions have been considered:

(i) (C-C): clamped at both the edge

X=0 and X=1.

(ii) (C-SS): clamped at X=0 and

simply supported at X=1.

The relation that should be satisfied at

clamped and simply supported respectively

are

2

20, 0

dW d WW W

dX dX (VIII)

Apply the boundary conditions C-C to the

displacement function (eq. V) one obtains a

set of four homogeneous equation in terms of

(n+5) unknown constants which can be

written as

0ccB B (IX)

Where ccB is a matrix of order 4× (n+5)

Therefore the (VII) together with ( IX) gives

a complete set of (n+5) homogeneous

equations having (n+5) unknowns which can

be written as

0cc

AB

B

(X)

For a non-trivial solution of (eq. X), the

characteristic determinant must vanish, i.e.

0cc

A

B (XI)

Similarly for (C-SS) Plate the frequency

determinant can be obtained as


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0.ss

A

B (XII)

Where ssB is a matrix of order 4 ( 5).n

Numerical Results and Discussion

The frequency equation XI and XII provides

the values of frequency parameter forvarious

values of plate parameters.in present problem

first three modes of vibration have been

computed for above mentioned two boundary

conditions for different values of (i) taper

constant =0.0(0.1)0.4, (ii) foundation

parameter pf =0.0(0.005) 0.02 and (iii )non-

homogeneity parameter =0.0(0.05) 0.2

considering kd =0.0 and kd =0.075

respectively.

A comparison of the results with those

available in the literature obtained by other

methods with in permissible range of plate

parameters has been presented in table

1which shows a comparison of results for

homogeneous 0.0 isotropic plates of

uniform thickness 0.0 taking as

0.03h with exact solution [2] , with those

obtained by chebyshev collection technique

[6], Frobenious method [3], differential

quadrature method [11]for m=1, two value of

aspect ratio a/b=0.5, 1.0 .

Table 2 (a) and 2 (b) show the numerical value

of frequency parameter Ω with the increasing

value of taper parameter for homogeneous

0 and non-homogeneous 0.4

respectively, including both boundary

conditions C-C and C-SS. These results are

also shown in fig.1 (a), 1(b) and 1(c) for the

fixed value of foundation parameter pf and

non-homogeneity parameter for first three

modes of vibration of C-C and C-SS plate.

Fig. 1(a) shows the behavior of frequency

parameter Ω decreases with the increasing

value of taper parameter αfor two different

values of foundation parameter 0.0, 0.01pf

non-homogeneity parameter 0.0, 0.4 and

without damping parameter 0.0, 0.0075

for both plates. It has been seen that the rate

of decrease of Ω with taper parameter α for C-

C is higher than that for C-S plate keeping all

other parameter fixed. A similar inference can

be seen from fig. 1(b) and 1(c) when the plate

vibrating in the second as well as in the third

mode of vibration except that the rate of

decrease of Ω with αis lesser as compared to

the first mode.

Table 3(a) and 3(b) gives the inference of

foundation parameter pf on frequency

parameter Ω for two values of damping

parameter 0.0, 0.0075kd respectively, for

fixed value of taper parameter 0.0, 0.4

and non-homogeneity parameter

0.0, 0.4. it is observed that the frequency

parameter Ω increases continuously with the

increasing value of foundation parameter for

clamped and simply supported plates,

however be the value of other plate

parameters. It has been seen that the rate of

increase of Ω for C-SS plate is higher than C-

C plate for three modes of vibration. Further

fig. 2(a) gives the conclusion of foundation

parameter pf on for the first mode of

vibration. The rate increases with the increase

in the value of foundation parameter pf ,

further it decrease when increase the number

of modes, as clear from 2(a) and 2(b). Frome

fig. 2(b) the effect of foundation parameter is


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found to increase the frequency parameter,

however the rate of increase get reduced to

more than half of the first mode for both C-C

and C-SS. In case of third mode, this rate of

increase further decrease and becomes nearly

half of the second mode as is evident of fig.

2(c). Observation of the results shows that

presence of an elastic foundation increase the

frequency parameter in all the cases.

Table 4(a) 4(b) show the effect of non-

homogeneity parameter on frequency

parameter Ω for 0.0, 0.0075kd

respectively, for the fixed value foundation

0.0, 0.01pf and 0.0, 0.4. fig. 3(a)

gives the graph between frequency parameter

Ω and non-homogeneity parameter for the

first mode of vibration. It is observed that, for

both the value of damping parameter

. . 0.0, 0.0075k kd i e for d the frequency

parameter Ω increases continuously with

increasing value of non-homogeneity

parameter for both the boundary

conditions, whatever be the value of other

plate parameters. When the plate is vibrating

in the second mode (fig. 3(b)), the frequency

parameter is found to increase less as

compared to first mode with increasing the

value of for both the boundary conditions

in all the cases. In the same way when plate is

vibrating in the third mode (fig. 3(c)), we

observed it is the same as for the second mode

with the difference that the rate of increase of

with is much higher as compared to the

first two modes.


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120


121


122


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Conclusion

The calculations and results presented here

are computed using MATLAB software

within the permissible range of parameters

and up to accuracy 810 as desired in the

problem, which shows the real nature of

vibrational problems. From the graph and its

table we observe that variation in various

parameters like thickness, non-homogeneity;

elastic foundation and damping parameter

which are of great interest because the main

causes of plate failures in civil or in industrial

machines or high cyclic fatigue in plates is

from effect of these parameters. Due to this it

is necessary to determine the vibration

frequencies for the assessment of failure life

of plates used in the structures build. Thus the

present study shall be helpful in designing of

plate which requires determination of their

natural frequencies and mode shape.

References

Bahmyari, E. and Rahbar-Ranji, A. (2012):

Free vibration analysis of orthotropic

plates with variable thickness resting on

non-uniform elastic foundation by

element free Galerkin method. Journal

of Mechanical Science and Technology,

vol. 26: 2685-2694.

Chakraverty, S. and Pradhan, K. K. (2004):

Free vibration of exponential

functionally graded rectangular plates

in thermal environment with general

boundary conditions. Aerosp sci

Technol, vol. 36, pg. 132-56, (2004).

Civalek, O. and Acar, M. H. (2007): Discrete

Singular Method for the analysis of


124

Mindlin plates on elastic foundation,

International Journal of Press Vessel

Piping. 84: 527-535.

Gupta, M. (2016): Damped vibrations of

rectangular plates of parabollically

varying thickness resting on elastic

foundation considering the effect of

thermally induced non –homogeneity,

Aryabhatta Journal of Mathematics

and Informatics, 8: 2394-9309.

Gupta, U.S. and Lal, R. (1978): Transverse

Vibration of Non-uniform rectangular

plate on elastic Foundation, Journal of

sound and vibration, 61(1): 127-133.

Jain, R. K. and Soni, S. R. (1973): Free

vibration of rectangular plates of

parabollically varying thickness, Indian

Journal of Pure and Applied

Mathematics, 4(3): 267-277.

Lal, R., Gupta, U. S., Reena (1997): Quintic

spline in the study of transverse

vibrations of non-uniform orthotropic

rectangular plates. Journal of sound and

vibration 207: 1-13.

Lal, R. (1979): Vibration of elastic plates of

variable thickness, PhD Thesis,

University of Roorkee.

Leissa, A. W. (1973): The free vibration of

rectangular plates, J sound and

vibrations. 31(3): 257-293.

Robin and Rana, U.S. (2016): Study of

damped vibration of non- homogeneous

rectangular plate of variable thickness,

International Journal of Engineering

and applied science. 3(1).

Sharma S., Gupta U. S., Singhal P. (2012):

Vibration analysis of non-

homogeneous orthotropic rectangular

plates of variable thickness resting

winkler foundation, Journal of applied

science and engineering. 15(3): 291-

300.

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