Mechanical Vibration of Orthotropic Square Plate with Clamped Boundary Conditions
Ashish Kumar Sharma1,Vandna2, Vijyeta Verma3
1Dept. of Mathematics, IEC University. Baddi, H.P, India.
2,3Research Scholar, Dept. of Mathematics, IEC University, Baddi, H.P, India
*Corresponding Author E-mail: ashishk482@gmail.com, vandnamehta9091@gmail.com, v.verma286@gmail.com
ABSTRACT:
A mathematical model is constructed to support the engineers in designing numerous mechanical structures mostly used in satellite and aeronautical engineering. The present work analyses the vibration behavior of non-homogeneous orthotropic visco-elastic square plate of parabolically varying thickness on the idea of classical plate theory while the all edges are clamped and are subjected to parabolically thermal variation. For non-homogeneity of the plate material is far assumed that the density of the plate material varies parabolically along the x-direction and y-direction. For visco-elastic substances, primary elastic and viscous factors are mixed. The Kelvin version for visco-elasticity is taken into consideration right here that is a mixture of elastic and viscous elements related in parallel. Rayleigh-Ritz approach is used to assess the essential frequencies. Each modes of the frequency are calculated with the aid of the trendy computational approach i.e. MATLAB, for the various values of taper parameters and temperature gradient. All the outcomes are concluded in the graphs.
Keyword: visco-elastic, square plate, vibration, thermal gradient, taper constant.
1. INTRODUCTION:
The thermal effect of non-homogenous visco-elastic plates on vibration is of remarkable significance in the discipline of engineering with correspondence such as improved designing of gasoline generators, jet engines, space craft and nuclear power projects in which metals and their alloys showcase visco-elastic behavior. Consequently, for those reasons such structures are exposed to high-intensity warmness fluxes and for that reason the materialundergo substantial adjustments. Mainly, the thermal effect of elasticity of the material cannot be taken as negligible. Space generation is growing very unexpectedly in the present generation, and the significance of analyzing the vibration of plates of sure factor ratios with some simple restraints at the barriers has increased. The vehicles of rockets and aircraft in bloodless areas are advanced with using tender filaments in aerospace shape supported with elastic or visco-elastic media. While finalizing a design, a creation engineer has to apprehend the primary few modes of vibration, as they may be significant.
Plates of variable thickness had been appreciably used in civil, electronic, mechanical, aerospace and marine engineering society. The realistic significance of such plates has made vibration evaluation important, especially because the vibratory response desires to be correctly determined in the design method which will avoid resonance excited by means of inner or external forces. The plate kind’s structural components in aircraft and rockets must perform below elevated temperatures that motive non-homogeneity inside the plate fabric, i.e. elastic constants of the substances become features of space variables. An updated survey of the research in this area shows that authors have stumble upon various fashions to account for non-homogeneity of plate substances and plenty of researchers have proposed handling vibration.
A collection of research papers on vibration of plates with different shapes and boundary conditions is given by Leissa [1] in his monograph. Leissa [2] discussed different models on free vibration of rectangular plates.Anukul De andD. Debnath [3] studied about vibration of orthotropic circular plate with thermal effect in exponential thickness and quadratic temperature Distribution. Khanna, A., Kaur, N., and Sharma, A. K. [4] have discussed effect of varying poisson ratio on thermally induced vibrations of non-homogeneous rectangular plate. Sharma, S. K., and Sharma, A. K. [5] have discussed the mechanical vibration of orthotropic rectangular plate with 2d linearly varying thickness and thermal Effect. Khanna, A., and Sharma, A. K. [6] have solved the problem on vibration analysis of visco-elastic square plate of variable thickness with thermal gradient. Kumar Sharma, A., and Sharma, S. K.[7] have discussed the vibration computational of visco-elastic plate with sinusoidal thickness variation and linearly thermal effect in 2d. Khanna, A., Kumar, A., and Bhatia, M. [8] has investigated the computational prediction on two dimensional thermal effects on vibration of visco-elastic square plate of variable thickness. Khanna, A., and Sharma, A. K. [9] studied natural vibration of visco-elastic plate of varying thickness with thermal effect. Kumar Sharma, A., and Sharma, S. K. [10] discussed free vibration analysis of visco-elastic orthotropic rectangular plate with bi-parabolic thermal effect and bi-linear thickness variation. Sharma, S. K. and Sharma, A. K [11] discussed effect of bi-parabolic thermal and thickness variation on vibration of visco-elastic orthotropic rectangular plate. Khanna, A., and Sharma, A. K. [12] analyzed a computational prediction on vibration of square plate by varying thickness with bi-dimensional thermal effect. Khanna, A., and Sharma, A. K. [13] discussed effect of thermal gradient on vibration of visco-elastic plate with thickness variation. Khanna, A., and Sharma, A. K [14] have studies the mechanical vibration of visco-elastic plate with thickness variation. Khanna, A., Kaur, N., and Sharma, A. K. [15] discussed about effect of varying poisson ratio on thermally induced vibrations of non-homogeneous rectangular plate.
The analysis present is to study the effect of parabolic non-homogeneity on thermally-prompted vibration of an orthotropic visco-elastic square plate of parabolically varying thickness.It’s far clamped on all four edges. The assumption of small deflection and linear orthotropic visco-elastic residences are made. It’s far further assumed that the visco-elastic residences of the plate are of the Kelvin kind. The first two modes of the frequency are calculated for the various values of thermal gradient,non homogeneity and taper constants.
2. Equation of Motion and Analysis:
The governing differential equation of transverse motion of a visco-elastic square plate of variable thickness.
(1)
Where
are bending
moments Expression for the bending moments are given below
(2)
On
substituting the value of from equation
(2) in (1) we get
Now equation (3)is the equation of motion for orthotropic square plate of variable thickness.
Here
D = 
, 
,
is constant,
are flexural
rigidity in x- axis and y-axis as:
(4)
is the
rheological operator 
are modulus of
elasticity in X-and Y- direction respectively.
Are poison
ratios and 
be the shear
modulus.
Assume that the temperature varying parabolicaly in both direction.
(5)
Where
be the
temperature at any point on the plate and 
denotes the
temperature at any point on the boundary of plate and “b” be the length of a
square plate.
In
a most of engineering materials temperature dependents upon the modulus of
elasticity and it can be expressed as 
(6)
Here
are the values
of the young modulus respectively, along X- axis and Y-axis at the reference
temperature 
β is the
slope of the variation of the modulus of elasticity with .
Thus modulus variation becomes:
(7)
Where
(0 ≤
is thermal
gradient.
Now the expression for Kinetic energy U and Strain energy R are written as:
and
Assume
ρ is mass density and thickness 
of the plate
varying parabolicaly in x- directionsand y- directions as:
{(
) (
)}
(10)
and
{(
)( 
)} (11)
where
,
is taper
constant and 
is the non
homogeneity constant
3. Solution and Frequency:
Here we can use Rayleigh –Ritz technique to obtain the frequency equation, maximum strain energy must be equal to the maximum kinetic energy is required to use this technique. It is compulsory for the following equation must be satisfied:
`(12)
For the arbitrary variation of Z satisfying relevant geometrical boundary conditions which are:
(13)
And
two term deflection function is taken as where 
(x, y) is
denotes the 1st two modes of frequency with two constants
satisfy the
boundary conditions.
(14)
Now, the non-dimensional variables are
By using (7), (10),(11) and (15) in equation (8), (9) we get
and
Where
(18)
On substituting the values of U and R from equation (15) and (16) in equation (12), we get
(19)
where
and
Now equation (19) involves two un-known 
, arising due to
the substitution of Z (x, y) Now these two constants are determined from
equation (19), as follows:
On simplifying equation (23) we get:
(24)
Where n=1, 2, and 
involve parametric
constants and the frequency parameter P. For a non-trivial solution, the
determinant of the coefficient of equation (24) must be zero. So, we get the
frequency as follows:
With the help of equation of (25) we get
the quadratic
equation in
, which gives
two values of can found these
two values represent first two modes of vibration of frequency i.e. 
( first mode)and
(second mode)
for different value of taper constant and thermal gradient for a clamped plate.
RESULT AND DISCUSSION:
Calculations has been total for rate of visco-elastic orthotropic regionpiece for choice values of taper constants η1 andη2 thermal gradient Ψ and non- homogenous constant β1 and β2for different points for first two modes of vibrations have been calculated numerically.
Figure I:
Show
that, increasing the value of frequency as compared with previous figure for
both of the modes of vibration is shown for increasing value of 
constant from 0.0 to 1.0for different
cases:
Case
I:
,
Case
II:
,
Case
III:
,
Case
IV:
,
Case
V:
,
Case
VI:
Fig. I Frequency vs. Taper constant η1
FigureII: -
In
this fig., value of the frequency increasing in both the modes of vibration
when the value of 
constant increasing
from 0.0 to 1.0 for
different cases:
Case
I:
,
Case
II:
,
Case
III:
,
Case
IV:
Case
V:
,
Case
VI:
Fig. II Frequency vs. Taper constant η2
Figure
III: -It
is clearly seen that value of frequency decreases as value of non homogeneity
increases from
0.0 to 1.0 for, both modes of vibrations for different cases:
Case
I:
,
Case
II:
Case
III:
,
Case
IV:
,
Case
V:
,
Case
VI:
Fig. III Frequency vs.non homogeneity β1
FigureIV:- Clearly seen
that value of frequency decreases as value of non homogeneity
increases from
0.0 to 1.0 for, both modes of vibrations for different cases:
Case
I:
,
Case
II:
,
Case
III:
,
Case
IV:
,
Case
V:
,
Case
VI:
Fig. IV Frequency vs.non homogeneity β2
FigureV
:-Sharply,
decreasing the value of frequency as compared with previous figure for both of
the modes of vibration on increasing value of 
from 0.0 to 1.0
for
different cases:
Case
I:
,
Case
II:
,
Case
III:
,
Case
IV:
,
Case
V:
,
Case
VI:
Fig. V Frequency vs. Thermal Effect Ψ
CONCLUSION:
The effect of non-homogeneity, which is supposed to arise due to variation in Young’s moduli and density on natural frequencies of square orthotropic plate of parabolically varying thickness in X-axis and Y-axis, has been studied on the basis of classical plate theory. In present paper the value of frequencies increase for both the modes of vibrations for the corresponding values of taper parameter increases and frequencies decrease when thermal gradient increases. By choosing appropriate values of varying parameters, desired or required values of frequencies can be obtained. So, predominant purpose for our studies is to expand a theoretical mathematical version for scientists and layout engineers in order to make a use of it with a sensible method, for the welfare of the human beings as well as for the advancement of technology.
REFERENCES:
1 A. W. Leissa, “Vibration of plates”,Tech. Rep. SP-160, NASA, 1969.
2 [A. W. Leissa, “The free vibration of rectangular plates”,Journal of Sound and Vibration, vol. 31, no. 3, pp. 257–293, 1973.
3 Sharma, S. K., and Sharma, A. K. “Mechanical Vibration of Orthotropic Rectangular Plate with 2D Linearly Varying Thickness and Thermal Effect” International Journal of Research in Advent Technology, 2(6), 2014, 184-190.
4 Kumar Sharma, A., and Sharma, S. K. “Vibration Computational of Visco-Elastic Plate with Sinusoidal Thickness Variation and Linearly Thermal effect in 2D” Journal of Advanced Research in Applied Mechanics and Computational Fluid Dynamics, 1(1), 2014, 46-54.
5 Kumar Sharma, A., and Sharma, S. K. “Free Vibration Analysis of Visco-elastic Orthotropic Rectangular plate with Bi-Parabolic Thermal Effect and Bi-Linear Thickness Variation” Journal of Advanced Research in Applied Mechanics and Computational Fluid Dynamics, 1(1), 2014, 10-23.
6 Sharma, S. K., and Sharma, A. K. “Effect of Bi-Parabolic Thermal and Thickness Variation on Vibration of Visco-Elastic Orthotropic Rectangular Plate” Journal of Advanced Research in Manufacturing, Material Science and Metallurgical Engineering,1(2), 2014, 26-38.
7 Ashish Kumar Sharma, Manoj Kumar Dhiman, Vijyeta Verma, Vandana “Vibration of Clamped Non-homogeneous Square Plate with non-uniform varying Thickness and Thermal Effect”, Research Journal of Science and Technology, 2017.
A Quintic Spline Technique: Effect on Frequency of C-S-C-S and S-S-S-S Rectangular Plate with Varying Thickness and Temperature EffectRomanian Journal of Acoustics and Vibration; Bucharest
9 Ashish Kumar SHARMA Amit SHARMA, “Mathematical Modeling of Vibration on Parallelogram Plate with Non Homogeneity Effect”, Romanian Journal of Acoustics and Vibration, 2016.
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Received on 24.08.2017 Modified on 18.10.2017 Accepted on 20.01.2018 ©A&V Publications All right reserved Research J. Science and Tech. 2018; 10(4):237-243. DOI:10.5958/2349-2988.2018.00034.7 |
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