analysis of damping and non-homogeneity through spline
TRANSCRIPT
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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114
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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115
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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117
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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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
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Free vibration analysis of orthotropic
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non-uniform elastic foundation by
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Chakraverty, S. and Pradhan, K. K. (2004):
Free vibration of exponential
functionally graded rectangular plates
in thermal environment with general
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Technol, vol. 36, pg. 132-56, (2004).
Civalek, O. and Acar, M. H. (2007): Discrete
Singular Method for the analysis of
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