Arun Kumar Gupta^{1}, Pragati Sharma^{2}
^{1}Department of Mathematics, M.S. College, Saharanpur, U.P., India
^{2}Department of Mathematics, H.C.T.M., Kaithal, Haryana, India
Correspondence to: Arun Kumar Gupta, Department of Mathematics, M.S. College, Saharanpur, U.P., India.
Email:  
Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved.
Abstract
The goal of present investigation is to study the effect of thermal gradient on the vibrations of nonhomogeneous trapezoidal plate whose thickness varies parabolically and density varies linearly. Effect of other plate parameters such as nonhomogeneity constant, taper constant and aspect ratios have also been studied. CSCS boundary condition with two term deflection is taken into consideration. RayleighRitz method is used to find the solution of the problem. Results are calculated with great accuracy and presented in tabular form. Comparison of the results with published data has also shown.
Keywords:
Vibration, Trapezoidal Plate, Thermal Gradient, NonHomogeneity, Parabolically Varying Thickness
Cite this paper:
Arun Kumar Gupta, Pragati Sharma, "Study of Thermally Induced Vibration of NonHomogeneous Trapezoidal Plate with Varying Thickness and Density", American Journal of Computational and Applied Mathematics, Vol. 2 No. 6, 2012, pp. 265275. doi: 10.5923/j.ajcam.20120206.05.
1. Introduction
Thermal effect on vibration of nonhomogenous plates are of great interest in the field of engineering such as for better designing of gas turbines, jet engine, space craft and nuclear power projects. Such structures are exposed to high intensity heat fluxes and thus material properties undergo significant changes, in particular the thermal effect on the modules of elasticity of the material can not be taken as negligible. Plates of variable thickness are frequently used in order to economize the plate material or to lighten the plates, especially when it is used in the wings of highspeed and high performance aircrafts. The study of vibration of plates has acquired great importance in the field of research, engineering and space technology. In the engineering we cannot move without considering the effect of vibration because almost machines and engineering structuresexperiences vibrations. Structures of plates have wide applications in ships, bridges etc. In the aeronautical field, analysis of thermally induced vibrations in nonhomogeneous plates of variable thickness has a great interest due to their utility in aircraft wings.Many analyses show that plate vibrations are based on nonhomogeneity of materials. Nonhomogeneity can be natural or artificial. Nonhomogeneous materials such as plywood, delta wood, fibrereinforced plastic etc, are used in engineering design and technology to strengthen the construction. Study of the effect of vibration cannot be restricted only in the field of science but, our day to day life is also affected by it. Vibration of plates of various shapes, homogeneous or nonhomogeneous, orthotropic or isotropic, with or without variation in thickness, have been studied by various authors, with or without considering the effect of temperature.A large number of researchers have reported about the vibration analysis of different types of plates. Some of them are mentioned here.Sharma et al.[1] studied the free transverse vibrations of nonhomogeneous circular plates of linearly varying thickness. Gupta et al.[2] worked on the vibrations of nonhomogeneous rectangular plate of variable thickness in both directions with thermal gradient effect. Gutierrez and Laura[3] calculated the fundamental frequency of vibrating rectangular, nonhomogeneous plates. Gupta and Kumar[4] studied the effect of thermal gradient on free vibration of nonhomogeneous viscoelastic rectangular plate of parabolically varying thickness.Gupta et al.[5] observed the transverse vibration of nonhomogeneous orthotropic viscoelastic circular plate of varying parabolic thickness. Lal et al [6] studied the transverse vibrations of nonhomogeneous rectangular plates of uniform thickness using boundary characteristic orthogonal polynomials. Gupta et al.[7] observed the effect of nonhomogeneity on the free vibrations of orthotropic viscoelastic rectangular plate of parabolic varying thickness. Gupta and Kumar[8] studied the thermal effect on vibration of nonhomogenous viscoelastic rectangular plate of linear varying thickness. Gupta et al.[9] worked on the vibration of viscoelastic orthotropic parallelogram plate with parabolically thickness variation. ElSayad and Ghazy[10] studied the RayleighRitz method for free vibration of midline trapezoidal plates. Gupta and Kumar[11] did the free vibration analysis of nonhomogeneous viscoelastic circular plate with varying thickness subjected to thermal gradient. Lal and Dhanpati[12] worked on the transverse vibrations of nonhomogeneous orthotropic rectangular plates of variable thickness. Gupta et al.[13] did the vibration analysis of nonhomogeneous circular plate of nonlinear thickness variation by differential quadrature method. Chakraverty et al.[14] noticed the effect of nonhomogeneity on natural frequencies of vibration of elliptic plates. Chen et al[15] studied on free vibration of nonhomogeneous transversely isotropic magnetoelectro elastic plates. Gupta et al  Figure 1. Use of plates 
[16] observed the effect of nonhomogeneity on vibration of orthotropic viscoelastic rectangular plate of linearly varying thickness. Gupta and Sharma[17] observed the effect of thermal gradient on transverse vibration of nonhomogeneous orthotropic trapezoidal plate of parabolically varying thickness. Chen et al[18] studied the free vibration of cantilevered symmetrically laminated thick trapezoidal plates. Bambill et al.[19] studied the transverse vibrations of rectangular, trapezoidal and triangular orthotropic, cantilever plates. Gupta and Sharma[20] observed the thermally induced vibration of orthotropic trapezoidal plate of linearly varying thickness. Gurses et al.[21] analysed the shear deformable laminated composite trapezoidal plates. Kitipornchai et al.[22] discussed a global approach for vibration of thick trapezoidal plates. Gupta and Sharma[23] studied the thermal gradient effect on frequencies of a trapezoidal plate of linearly varying thickness. Gupta and Sharma[24] studied the thermal effect on vibration of nonhomogeneous trapezoidal plate of linearly varying thickness. The authors have so far not come across any paper dealing with parabolically varying thickness and linearly varying density. In the present work the effect of nonhomogeneity, taper constant, thermal gradient and aspect ratios has been studied. The frequencies for the first two modes of vibration are obtained for CSCS nonhomogeneous trapezoidal plate by RayleighRitz method. The authenticity and accuracy of numerical results of the present work has been verified with the authors published paper[25]. Results are presented in tabular form. Comparison of results has also been presented.
2. Theoretical Formulation
2.1. Geometry of the Plate
The plate under consideration is shown in figure 2 which is thin, symmetric and nonhomogeneous trapezoidal plate. Here h_{0 }is the maximum plate thickness occurring at the left edge and is the minimum plate thickness occurring at the right edge.
2.2. Thickness and Density
The thickness of the plate is parabolic in x direction and is of the form  (1) 
where h_{0} is thickness along the edge = x/a= 1/2. Nonhomogeneity of plate arises due to variation in density which is linear in x direction and is of the form  (2) 
It is assumed that the plate considered here is subjected to a steady onedimensional temperature distribution along the length i.e. in the x direction,  Figure 2. Geometry of trapezoidal plate with variable thickness 
 (3) 
where denotes the temperature excess above the reference temperature at a distance and denotes the temperature excess above the reference temperature at the end . For most of the elastic materials, modulus of elasticity is described as  (4) 
where E_{0} is the value of Young’s modulus along the reference temperature i.e. at = 0 and is the slope of the variation of E with. On substituting value of from equation (3) into (4)  (5) 
where .
2.3. Equation of Motion
The governing differential equation for kinetic energy T and strain energy V is given by  (6) 
and  (7) 
in which is the flexural rigidity of the plate, which is given by  (8) 
Also  (9) 
is the flexural rigidity of the plate at the side , A is the area of the plate and is the mass density per unit area of the plate.Using equation (9) and (5) in (8) flexural rigidity is given by  (10) 
Using equation (10) in (7) & (1), (2) in (6)  (11) 
 (12) 
2.4. Deflection Function and Boundary Condition
A two term deflection function is taken as  (13) 
where A_{1} and A_{2} are constants to be evaluated. Eq. (13) satisfy boundary conditions and provide a good estimation to the frequency. ClampedSimply supportedClampedSimply supported plate is taken into consideration as shown in figure.  Figure 3. CSCS boundary condition of a trapezoidal plate 
Also for the plate considered here boundaries are defined by four straight lines  (14) 
3. Method of Solution
RayleighRitz technique is used to find the solution of the problem. According to it maximum kinetic energy must be equal to maximum strain energy, so it is necessary for the problem under consideration that  (15) 
Using equation (14) in (11) and (12)  (16) 
 (17) 
Now (15) becomes  (18) 
where  (19) 
 (20) 
and is a frequency parameter.Equation (18) involves the unknown A_{1} and A_{2} arising due to the substitution of from eq (13). These two constants are to be determined from eq. (18), as follows  (21) 
On simplifying (21), One gets  (22) 
where involve parametric constant and the frequency parameter.For a nonzero solution, it is desired that coefficient of eq. (22) must be zero. So one gets the frequency equation as  (23) 
From eq. (23), one can obtain a quadratic equation in from which the two values can found which constitutes first and second mode of vibration.
4. Results and Discussion
Frequencies for the first two modes of vibrations are computed for nonhomogeneous trapezoidal plate whose thickness varies parabolically and density varies linearly. Different values of nonhomogeneity constant (), taper constant (), thermal gradient () and aspect ratios (a/b, c/b) has been considered. All results are presented in tabular form.Table 1 and 2: These tables shows the effect of taper constant (0.0 to 1.0) on the frequency parameter for = 0.0, 0.4, = 0.4, 1.0, a/b = 1.0 and c/b = 0.5, 1.0.Table 3 and 4: These tables shows the behaviour of the frequency parameter with thermal gradient ( 0.0 to 1.0), for = 0.0, 0.4, = 0.4, 1.0, a/b = 1.0 and c/b = 0.5, 1.0.Table 5, 6, 7 and 8: These tables show the effects of the frequency parameter with aspect ratio c/b for different combinations of and as follows.(i) = 0.0, = 0.0(ii) = 0.0, = 0.4(iii) = 0.4, = 0.0(iv) = 0.4, = 0.4.Nonhomogeneity constant varies from 0.0 to 0.4, aspect ratio a/b varies from 0 .75 to 1.0 and aspect ratio c/b varies from 0.25 to 1.0.Table 9 and 10: In these tables effect of nonhomogeneity constant (0.0 to 1.0) on frequency parameter has been shown. Four combinations of and (as in table 5 to 8), two values of aspect ratio c/b (0.5, 1.0) and one value of aspect ratio a/b (1.0) has been taken. In table 1 and 2 it is observed that the frequency parameter increases with increasing values of taper constant for both the modes of vibration. The rate of increase as well as the value of frequency parameter is higher in second mode in comparison to the first mode. Further on comparing the results of table 1 and 2 it is found that on increasing the value of aspect ratio c/b from 0.5 to 1.0, frequency parameter decreases for both the modes of vibration.In table 3 and 4 it is seen that frequency parameter decreases with increasing values of thermal gradient whatever be the values of other plate parameters. The rate of decrease as well as the value of frequency parameter is higher in second mode in comparison to the first mode. Also on increasing aspect ratio c/b from 0.5 to 1.0, frequency parameter decreases for both the modes of vibration.It is noticed from table 5, 6, 7 and 8 that the frequency parameter decreases with increasing values of aspect ratio c/b. Further if we compare table 5 and 6 (a/b = 0.75) it is found that the frequency parameter decreases with increase in nonhomogeneity constant i.e. from 0.0 to 0.4. Similar pattern is observed on comparing table 7 and 8 (a/b = 1.0).Now if we compare table 5 and 7 in which aspect ratio a/b increased from 0.75 to 1.0 and nonhomogeneity constant is 0.0. It is observed that frequency parameter increases for both the mode of vibration. Similar pattern is observed on comparing table 6 and 8 in which nonhomogeneity constant is 0.4. Table 9 and 10 clearly shows that frequency parameter decreases with increasing value of nonhomogeneity constant. Also when aspect ratio c/b increased from 0.5 to 1.0, frequency parameter decreases for both the mode of vibrationTable 1. Frequency parameter ( ) for a trapezoidal plate for different values of taper constant ( ), thermal gradient ( ), nonhomogeneity constant ( = 0.4, 1.0) and aspect ratios (a/b = 1.0), (c/b = 0.5) 
 

Table 2. Frequency parameter ( ) for a trapezoidal plate for different values of taper constant ( ), thermal gradient ( ), nonhomogeneity constant ( = 0.4, 1.0) and aspect ratios (a/b = 1.0), (c/b = 1.0) 
 

Table 3. Frequency parameter ( ) for a trapezoidal plate for different values of thermal gradient , taper constant ( ), nonhomogeneity constant ( = 0.4, 1.0) and aspect ratios (a/b = 1.0), (c/b = 0.5) 
 

Table 4. Frequency parameter ( ) for a trapezoidal plate for different values of thermal gradient , taper constant ( ), nonhomogeneity constant ( = 0.4, 1.0) and aspect ratios (a/b = 1.0), (c/b = 1.0) 
 

Table 5. Frequency parameter ( ) for a trapezoidal plate for different combinations of thermal gradient ( ), taper constant ( ) and fixed value of nonhomogeneity constant ( = 0.0) & aspect ratio (a/b = 0.75) 
 

Table 6. Frequency parameter ( ) for a trapezoidal plate for different combinations of thermal gradient ( ), taper constant ( ) and fixed value of nonhomogeneity constant ( = 0.4) and aspect ratio (a/b = 0.75) 
 

Table 7. Frequency parameter ( ) for a trapezoidal plate for different combinations of thermal gradient ( ), taper constant ( ) and fixed value of nonhomogeneity constant ( = 0.0) and aspect ratio (a/b = 1.0) 
 

Table 8. Frequency parameter ( ) for a trapezoidal plate for different combinations of thermal gradient ( ), taper constant ( ) and fixed value of nonhomogeneity constant ( = 0.4) and aspect ratio (a/b = 1.0) 
 

Table 9. Frequency parameter ( ) for a trapezoidal plate for different value of nonhomogeneity constant with different combinations of thermal gradient ( ) and taper constant ( ) and aspect ratios (a/b = 1.0), (c/b = 0.5) 
 

Table 10. Frequency parameter ( ) for a trapezoidal plate for different value of nonhomogeneity constant with different combinations of thermal gradient ( ) and taper constant ( ) and aspect ratios (a/b = 1.0), (c/b = 1.0) 
 

5. Confirmation of Results
The accuracy of the present computations is compared with the published results[25] for CSCSnonhomogeneous trapezoidal plate with nonhomogeneity constant = 1.0, aspect ratio a/b = 1.0, c/b = 0.5, 1.0 and thermal gradient = 0.0 to 1.0 and taper constant = 0.0 to 1.0. Table 1, 2, 3 and 4 shows a comparison of the values of frequency parameter obtained in the present problem and published paper of the authors[25]. A very close agreement is seen between the present results and of the published paper in which homogeneous plate with parabolically varying thickness (density is constant) has been considered.
6. Conclusions
The main purpose of the present work is to study the effect of thermal gradient on the frequencies of CSCS nonhomogeneous trapezoidal plate with other plate parameters as taper constant, nonhomogeneity constant and aspect ratios. Thickness of the plate varies parabolically and density varies linearly. RayleighRitz technique is used to find frequencies for first two modes of vibration. Study shows that frequency parameters increases with increasing value of taper constant and aspect ratio a/b whereas it decreases with increasing value of nonhomogeneity constant, thermal gradient and aspect ratio c/b.
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