Received 19 May 2016; accepted 22 July 2016; published 25 July 2016
People became interested in vibration when the first musical instruments, probably whistles or drums were discovered. Since then people have applied ingenuity and critical investigation to study the phenomenon of vibration. Many studies in existing period have been aggravated by the engineering applications of vibration, such as design of machines, foundations, structures, engines and turbine systems. Most major movers have vibrational problems because of the inbuilt unbalance in the engines. In spite of its detrimental effects, vibration can be utilized profitably in several industrial and consumer applications.
In many engineering applications different types of plates such as rectangular, parallelogram, circular etc. act as an integral part of the system. In contrast of uniform thickness of plate, the suitable variation in thickness in plate has a significant effect on its vibration. Thus, the choice of material depends on suitable properties of materials. On the whole, non-homogeneity is a significant constituent of any design which occurs as a result of variation in density. Literature shows that the vibration analysis has inspired many researchers to do work in this direction. Out of them few are given under. Kumar and Lal  worked on the vibrations of non-homogeneous orthotropic rectangular plates with bilinear thickness variation resting on Winkler foundation. Kumar and Tomar  had studied the free transverse vibrations of monoclinic rectangular plates with continuously varying thickness and density. Johri and Johri  had worked on the exponential thermal effect on vibration of non-homo- geneous orthotropic rectangular plate having bi-directional linear variation in thickness. Gupta et al.  did the vibration analysis of non-homogeneous circular plate of non-linear thickness variation by differential quadrature method. Li and Zhou  discussed the shooting method for non-linear vibration and thermal buckling of heated orthotropic circular plates. Chakraverty et al.  studied the effect of non-homogeneity on natural frequencies of vibration of elliptic plates. Gupta et al.  discussed the vibration of visco-elastic orthotropic parallelogram plate with linear thickness variation in both directions. Chen et al.  worked on the free vibration of non-ho- mogeneous transversely isotropic magneto-electro-elastic plates. Gurses et al.  analyzed the shear deformable laminated composite trapezoidal plates. Kitipornchai et al.  presented a global approach for vibration of thick trapezoidal plates. Sayad and Ghazy  studied the rayleigh-ritz method for free vibration of midline trapezoidal plates. Leung et al.  had studied the free vibration of laminated composite plates subjected to in-plane stresses using trapezoidal p-element. McGee and Butalia  presented the natural vibrations of shear deformable cantilevered skewed trapezoidal and triangular thick plates. Qatu  studied the vibrations of laminated composite completely free triangular and trapezoidal plates. Grigorenko et al.  used spline functions to solve boundary-value problems for laminated orthotropic trapezoidal plates of variable thickness. Feng and Min  worked on the vibrations of axially moving visco-elastic plate with parabolically varying thickness. Gupta and Sharma  evaluated the forced axisymmetric response of an annular plate of parabolically varying thickness. Liew and Lim  studied the transverse vibration of trapezoidal plates of variable thickness: symmetric trapezoids. Maruyama et al.  presented an experimental study of the free vibration of clamped trapezoidal plates. Karami et al.  used a differential quadrature method for skewed and trapezoidal laminated plates. Huang et al.  carried out experimental and numerical investigations for the free vibration of cantilever trapezoidal plates. Gupta and Sharma  studied the effect of thermal gradient on transverse vibration of non-homogeneous orthotropic trapezoidal plate of parabolically varying thickness. Gupta and Sharma  observed the effect of linear thermal gradient on vibrations of trapezoidal plates whose thickness varies parabolically. Gupta and Sharma  study the thermally induced vibration of non-homogeneous trapezoidal plate with varying thickness and density.
The existing work is an attempt to investigate the thermal effect on vibration of non-homogeneous trapezoidal plate of bi-parabolically varying thickness with linear density variation. To attain the natural frequencies for the first two modes of vibration Rayleigh-Ritz’s method has been applied. The deflection function has been taken to satisfy the C-S-C-S boundary condition. All the obtained results have been presented in tabular and graphical form.
2. Mathematical Formulation
2.1. Geometry of the Plate
For the study of transverse vibration a thin, symmetric, non-homogeneous trapezoidal plate with varying thickness and density has been taken. The geometry of the plate is shown in Figure 1.
2.2. Thickness and Density
The thickness of the plate which varies parabolically in both directions can be expressed as
Figure 1. Geometry of the trapezoidal plate.
The non-homogeneity occurs in the bodies because of imperfection of materials and it is assumed to arise due to the linear variation in density along the length of the plate. So, it can be stated as
It is assumed that the temperature of the non-homogeneous trapezoidal plate varies linearly along x-axis and is of the form
where represent the excess above the reference temperature at a distance and denotes the temperature excess above the reference temperature at the end.
The temperature dependence of the modulus of elasticity for most of the engineering materials is specified as 
where denotes the value of Young’s modulus at reference temperature and is the slope of variation of E with.
Using Equation (3) into Equation (4), one obtain
where known as thermal gradient.
3. Equation of Motion
The governing differential equation for kinetic energy T and strain energy V for a non-homogeneous trapezoidal plate with bi-parabolically varying thickness can be expressed as
where is the Poisson ratio, is the angular frequency of vibration and A is the area of the plate.
Flexural rigidity of the plate is given by
where, are non-dimensional variables. Here,
On using Equation (9) and (5), Equation (8) gives the value of flexural rigidity as follows
Now after putting Equations (1), (2) into Equation (6) and (10) into Equation (7), kinetic energy and strain energy become
Two terms deflection function for a C-S-C-S trapezoidal plate can be defined as,
where and are unknowns to be calculated.
In this manner, for vibrational analysis a trapezoidal plate whose two sides are clamped and two are simply- supported has been considered. The deflection function which is already discussed by Equation (13) satisfies the boundary conditions and presents an excellent evaluation to the frequency. Thus, the boundaries are given by four straight lines as follows:
4. Method of Solution
In addition the frequency is calculated through Rayleigh-Ritz technique which is based on the principle of conservation of energy i.e. the maximum strain energy must be equal to the maximum kinetic energy. Therefore, the resulting equation can be described by
Using boundary condition (14) into Equation (11) and (12), one gets
Now Equations (16) and (17) consists the values of T and V so, put these values into Equation (15), we obtain
is a frequency parameter.
Equation (18) includes two unknowns and which occurs as a result of using the deflection function. These two unknown can be evaluated from Equation (18) as follow:
On simplifying (22), we get
where, (m = 1, 2) involves parametric constants and the frequency parameter. For a non-zero solution, the determinant of co-efficient of Equation (23) must vanish. Therefore, for a (C-S-C-S) trapezoidal plate the frequency equation can be obtained as
The quadratic equation in is obtained through the Equation (24) which presents the two values of known as first and second modes of vibration respectively.
5. Results and Discussion
The existing work deals with the vibration behavior of a non-homogeneous trapezoidal plate whose thickness varies bi-parabolically and density varies linearly in one direction. All results acquired by Equation (24) provide the values of frequency parameter for different values of taper constants, thermal gradient, aspect ratios and non-homogeneity constant. The natural frequencies are estimated for first two modes of vibration. The value of Poisson’s ratio is considered as 0.33. With the help of tables and graphs all the results have been displayed.
Table1 includes the values of frequency parameter for a non-homogeneous trapezoidal plate where taper constant varies from 0.0 to 1.0, taper constant, thermal gradient, non-homogeneity constant and aspect ratios. It is evident from the table that as taper constant increases the values of frequency parameter also increases for both the modes of vibration. In addition when the value of non-homogeneity constant increases the frequency parameter decreases.
Table 2 contains the values of frequency parameter in which taper constant varies from 0.0 to 1.0, taper constant, thermal gradient, non-homogeneity constant and aspect ratios,. It is clear from the table that as taper constant increases the values of frequency parameter also increases for both the modes of vibration. Moreover when the value of non-homogeneity constant increases the frequency parameter decreases.
Table 3 depicts the values of frequency parameter for different values of thermal gradient from0.0 to 1.0, taper constants &, non-homogeneity constant and aspect ratios
Table 1. Values of frequency parameter (l) for different values of taper constant (b1) and constant aspect ratios (a/b = 1.0, c/b = 0.5).
Table 2. Values of frequency parameter (l) for different values of taper constant (b2) and constant aspect ratios (a/b = 1.0, c/b = 0.5).
Table 3. Values of frequency parameter (l) for different values of thermal gradient (a) and constant aspect ratios (a/b = 1.0, c/b = 0.5).
,. It is obvious from the table that as thermal gradient increases, the values of frequency parameter decreases for both the modes of vibration. It is also noted that on increasing the value of non- homogeneity constant, the frequency parameter decreases.
The value of non-homogeneity constant, the values of aspect ratio are 0.75 and 1.0 and the values of aspect ratio are (0.25, 0.50, 0.75, 1.0).
It is obvious from the above discussed Table 4 and Table 5 that the values of frequency parameter decrease by increasing the aspect ratio for both the modes of vibration. Moreover as taper constant increases frequency parameter also increases. It has been observed from the comparison of Table 4 and Table 5 that as one increases the aspect ratio from 0.75 to 1.0 the frequency parameter also increases.
Table 6 contains the values of frequency parameter for a non-homogeneous trapezoidal plate for which non-homogeneity constant varies from 0.0 to 1.0, the values of taper constants & , thermal gradient and aspect ratios,. From this table one can observe that as non-homogeneity constant increases, the frequency parameter decreases for both the modes of vibration.
Table 4. Values of frequency parameter (l) for different combinations of thermal gradient (a), taper constants (b1 & b2) fixed value of non-homogeneity constant (b = 0.4) and aspect ratio (a/b = 0.75).
Table 5. Values of frequency parameter (l) for different combinations of thermal gradient (a), taper constants (b1 & b2) fixed value of non-homogeneity constant (b = 0.4) and aspect ratio (a/b = 1.0).
Table 6. Values of frequency parameter (l) for different values of non-homogeneity constant (b) and constant aspect ratios (a/b = 1.0, c/b = 0.5).
Figure 2 depicts the behaviour of frequency parameter with taper constant. For first two modes of vibration the values of various plate parameters are taken as follows:
It is evident from the Figure 2 that frequency for both the modes of vibration increases as taper constant increases.
Figure 2. (a) & (b) Frequency parameter l vs. taper constant b1.
Figure 3 represents the variation of frequency parameter with taper constant. For first two modes of vibration the values of various plate parameters are taken as follows:
This figure explicates that as taper constant increases, the frequency parameter also increases for both the modes of vibration.
Figure 4 shows the variation of frequency parameter with thermal gradient. For first two modes of vibration the values of various plate parameters are taken as follows:
From the discussed Figure 4 the behaviour of the frequency parameter can be examined. It is found that as thermal gradient increases the values of frequency parameter decreases for both the modes of vibration.
Figure 5 displays the effect of aspect ratio c/b varies from 0.25 to 1.0, on the frequency parameter for different combinations of taper constants & and thermal gradient as follows:
Figure 3. (a) & (b) Frequency parameter l vs. taper constant b2.
In this case the value of non-homogeneity constant and two values of aspect ratio a/b = 0.75, 1.0 have been considered.
Now it can be easily observed from Figure 5 that as aspect ratio c/b increases the frequency parameter decreases for both the modes of vibration. It is also noticed that frequency parameter also increases as taper constant increases. Furthermore when the value of aspect ratio a/b is increased from 0.75 to 1.0 then frequency parameter increases for both the modes of vibration.
Figure 6 demonstrates the effect of non-homogeneity constant which varies from 0.0 to 1.0, on the frequency parameter for different combinations of taper constants & and thermal gradient as follows:
The behaviour of the frequency parameter is noticed and found that as non-homogeneity constant increases the frequency parameter decreases for both the modes of vibration. In addition frequency parameter in-
Figure 4. (a) & (b) Frequency parameter l vs. thermal gradient a.
Figure 5. (a) & (b) Frequency parameter l vs. aspect ratio c/b.
Figure 6. (a) & (b) Frequency parameter l vs. non-homogeneity constant b.
creases for both the modes of vibration as taper constant increases.
Study of vibration of plates is an important area owing to its extensive range of engineering applications such as in aeronautical, civil and mechanical engineering. Rayleigh-Ritz method gives a perfect and computationally proficient scheme for finding the vibration characteristics of transverse vibration of trapezoidal plate. Thus the natural frequencies for a symmetric, non-homogeneous trapezoidal plate have been acquired by varying values of taper constants, thermal gradient, aspect ratio and non-homogeneity constant. Tables and graphs state that the frequency increases with the increase of taper constants and decreases with the increase of thermal gradient, aspect ratio and non-homogeneity constant. A design engineer can directly observe the presented plots of figures to have the knowledge about particular mode to finalize the design of the structure. The material should be selected such that the total cost should be minimum and within definite confines.
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