Time Period of the Vibration of the Circular Plate with Circular Variation Both in Thickness and Density

An analysis was carried out to investigate the time period of the thermally induced vibration of clamped and simply supported circular plates with circular variation both in thickness and density. Prior to this study, the variations considered were either linear, quadratic, parabolic, or exponential in nature. To study thermal effect, one-dimensional linear temperature variation on the plates is taken into consideration. Rayleigh–Ritz method is applied to compute the time period of the first three modes of vibration for both plates by varying tapering parameter, thermal gradient, and density. Convergence study of frequency modes for both plates conducted suggests that the convergence rate in case of circular variation is faster than the other studies done. A comparison of time period with the available published results is done. ,e comparison done concludes that time period obtained in the present study by varying thermal gradient and tapering parameter is found to be less than the other studies done for the same set of parameters. ,is study helped to establish the fact that, by using circular variation in plate parameters, we can get less time period of frequency modes in comparison to other variations considered till date.


Introduction
Vibration, sometimes, is described as a kind of waste energy. In modern days, vibrational study is involved in a wide range of industrial applications and research. Study of natural vibration of nonuniform plates is very essential because vibration is used in many engineering and science applications. Different shapes of plates are the choices of many engineering applications, depending on the requirement of the systems. Also, different shapes of plates with different types of variations in plate parameters are used according to the choices of researchers in order to make good structural designs. e study of the vibration of plates such as circular, elliptical, square, rectangular, and skew plates with linear and nonlinear variations in thickness, density, and temperature has been carried out by many researchers and is well documented in the literature. To the best of our knowledge, vibration of circular plates with circular variations in both thickness and density has not been considered yet. e problem of bending and vibration of a simply supported rectangular plate with linearly varying thickness is discussed in [1]. e differential quadrature method was implemented to study the natural vibrations of clamped, simply supported, and free circular plates (nonhomogeneous and isotropic) of nonlinear thickness variation and computed first three frequencies [2]. An exact solution of inhomogeneous circular plates has been discussed by using the novel analytical method, and the effect of Poisson's ratio, rotatory inertial, and shear deformation is examined [3]. e Rayleigh-Ritz method along with characteristic orthogonal polynomials has been applied to study the fundamental mode of vibrations of circular or elliptic plates [4]. An analysis of the vibration of circular annular disks on the clamped, free, and flexible boundary at the inner and outer edges has been studied, and the studies focused on frequency parameters for various plate parameters [5].
e Ritz method along with Chebyshev polynomials has been used to study the effect of different material compositions, Poisson's ratio, and the plate geometry on the frequencies and mode shapes of FGM plates [6]. e vibration of FG isotropic circular plates with exponential thickness has been studied using the displacement method on two edge conditions [7]. e effects of material heterogeneity and multifield coupling on the static behavior of the FG circular plate made of MEE materials under tension and bending have been discussed [8]. Levy approach and quintic spline method have been used to examine the effect of damping, elastic foundation, and taperness on damped vibrations of the homogeneous rectangular plate of linearly varying thickness resting on an elastic foundation [9]. Vibrations of FG annular plates on the ring support have been analyzed and presented numerically on the basis of classical plate theory [10]. e effect of exponential Young's modulus and density on the asymmetric vibrations of nonhomogeneous, clamped, simply supported, and free circular plates with parabolic thickness has been studied using the Ritz method, and the first three natural frequencies have been presented [11]. e vibration of the nonuniform skew plate with both circular variation in density and Poisson's ratio and the natural vibration of the nonuniform skew plate with both circular variation in thickness and Poisson's ratio have been analyzed using the Rayleigh-Ritz method under the temperature field, and frequency modes comprising the effect of various plate parameters have been computed [12,13]. e effect of circular variation in Poisson's ratio on frequencies of the nonuniform rectangle plate and effect of circular variation in density on frequencies of the nonuniform square plate have been presented using the Rayleigh-Ritz method under the temperature field [14,15]. Rayleigh-Ritz method is used to analyze the vibration of the circular plate with linearly varying thickness and temperature; and deflection, time period, and logarithmic decrement have been computed for different values of plate parameters [16]. Generalized differential quadrature method is applied to study free thermal vibrations of FGM; clamped and simply supported circular plates and first three modes of vibration have been computed [17].
In this study, time period of frequency modes (first three) comprises the effect of various plate parameters (circular variation in both density and thickness and linear variation in thermal gradient) for clamped and simply supported circular plates, and the result is presented in the form of tables and graphs. As far the knowledge of the authors, prior to this work, the effect of circular variation in plate parameters on the circular plate was not examined. In order to authenticate the findings of the present study, a comparative study of time period is conducted with the available published results for the clamped circular plate. is paper has been organized in the following manner. First, mathematical formulation of the problem has been discussed, and then the numerical results have been reported. In the end, the main conclusions of the studies have been discussed.

Analysis
Consider a circular plate of radius r 1 having circular thickness l(r), circular density ρ(r), and Poisson's ratio ] referred to cylindrical polar coordinates (r, θ, z) (refer to Figure 1) e maximum strain energy V s and kinetic energy T s of the plate are given by the expression derived in [18]: where D � El 3 /12(1 − ] 2 ) is the flexural rigidity, E is Young's modulus, and ϕ is known as the transverse deflection.
where ω is the circular frequency.
Applying the Rayleigh-Ritz method, the variable J is represented in the following manner: Substituting equations (1) and (2) in equation (3), we get Introducing nondimensional variable Φ � ϕ/r 1 and R � r/r 1 along with circular variation in both density ρ and thickness l, we get where ρ 0 and l 0 are nonhomogeneity and tapering parameters of the plate, respectively.
In [16], the temperature was assumed to be linear in the radial direction. In this study, we also assumed the same. Under this assumption, the temperature of the plate takes the form where η and η 0 are the temperature on the plate and at the center of the plate, respectively. e temperature-dependent modulus of elasticity is taken as in [19] and can be expressed in the following manner: where ϱ is the slope of variation and E 0 is Young's modulus at η � 0. Substituting equation (6) in equation (7), where κ � ϱη 0 is the thermal gradient. Using nondimensional variables Φ and R and substituting equations (5) and (8) in equation (4), we get where D 0 � E 0 l 3 0 /12(1 − ] 2 ) and λ 2 � ρ 0 a 4 ω 2 l 0 /D 0 . Now, assuming the mode shape function as in [20], we have As the functional in equation (9) contains the negative power of R, take Φ 1 (R) � RΦ 1 (R) as in [16], and using this relation and substituting equation (10) in equation (9), we get We choose deflection function, Φ 1 , in the radial direction which satisfies the geometric boundary condition as where p can be 0, 1, and 2 in accordance to free, simply supported, and clamped boundary conditions, respectively. C i , i � 0, 1, 2, . . . , N, are arbitrary constants. In order to minimize equation (11), we require the following condition: From equation (13), we have a system of homogeneous equations in C i whose nonzero solution gives the frequency equation as where P � [p ij ] N+1 and Q � [q ij ] N+1 are square matrices and i, j � 0, 1, 2, . . . , N. e time period K is calculated as where λ is the frequency obtained from equation (14).

Numerical Results
In this section, we will report the results obtained from numerical simulations.

Time Period Analysis.
e time period, K, comprising the effect of circular variation in both thickness and density is calculated for clamped and simply supported circular plates under the linear temperature variation effect, and results are presented with the help of tables. e value of Poisson's ratio ] � 0.345 throughout the calculation. Table 1 shows the time period K for clamped and simply supported circular plates corresponding to tapering parameter ϖ, for the variable value of thermal gradient κ and Mathematical Problems in Engineering nonhomogeneity φ, i.e., κ � 0.0, φ � 0.2 and κ � 0.4, φ � 0.6. It can be easily seen that time period K is increasing for the increasing value of tapering parameter ϖ for all the aforementioned values of thermal gradient κ and nonhomogeneity φ. e time period K is also increasing when thermal gradient κ as well as nonhomogeneity φ increases from 0.0 to 0.4 and 0.2 to 0.6, respectively. e time period K of the simply supported circular plate is higher when compared with the time period K of the clamped circular plate. Also, the rate of increment in time period K for the simply supported circular plate is much higher in comparison to the rate of increments in time period K for the clamped circular plate.
Time period, K, for clamped and simply supported circular plates corresponding to thermal gradient κ, for the variable value of tapering parameter ϖ and nonhomogeneity φ, i.e., ϖ � 0.0, φ � 0.2 and ϖ � 0.4, φ � 0.6, is presented in Table 2. One can conclude that increasing value of thermal gradient κ causes the increase in time period K for all the aforementioned values of tapering parameter ϖ and nonhomogeneity φ. e time period K is also increasing when thermal gradient κ as well as nonhomogeneity φ increases from 0.0 to 0.4 and 0.2 to 0.6, respectively. e behavior of time period K of the simply supported circular plate is the same as the behavior of time period K reported in Table 1, but for the clamped plate, the time period K of the first mode is slightly less for ϖ � 0.0, φ � 0.2 than the time period K of the first mode on ϖ � 0.4, φ � 0.6 at κ � 0.0. Here, the rate of increment in time period K for the clamped circular plate is much lesser in comparison to the rate of increments in time period K for the simply supported circular plate. Table 3 displays the time period K for clamped and simply supported circular plates corresponding to nonhomogeneity φ, for the variable value of thermal gradient κ and tapering parameter ϖ, i.e., κ � 0.0, ϖ � 0.2 and κ � 0.4, ϖ � 0.6. Table 3 concludes that increasing value of nonhomogeneity φ causes the decrease in time period K for the aforementioned variable value of thermal gradient κ and tapering parameter ϖ, but the time period K for both clamped and simply supported circular plates increases when the value of thermal gradient κ and tapering parameter ϖ varies from 0.0 to 0.2 and 0.2 to 0.6, respectively. e rate of decrement in time period K for the clamped circular plate is much lesser when compared to the rate of decrement in time period K for the simply supported circular plate.
In view to understand the obtained results as discussed in this section, a graphical illustration of results is given (refer to Figures 2-7).

Convergence Studies.
In this section, we will focus on the convergence studies done for clamped and simply supported circular plates computed for plate parameters in the range specified. e results are shown in Figure 8 for φ � ϖ � κ � 0.0 and ] � 0.345. e convergence study is done for two modes λ 1 and λ 2 of vibrations for both clamped and simply supported circular plates. It was observed that, for approximately N � 7 onwards, the value of the modes was constant up to five decimal places. e aforementioned results show that the modes converge.

Results' Comparison.
In order to validate the findings of the present study, a graphical comparison of time period K of the clamped circular plate is given with the available published results corresponding to tapering parameter ϖ and nonhomogeneity φ. Figure 9 presents the comparison of time period K of the clamped circular plate with the time period K obtained in [16] corresponding to tapering parameter ϖ for fixed values of thermal gradient κ � 0.0 and nonhomogeneity φ � 0.0. From Figure 9, one can easily conclude that time period K as well as variation in time period (rate of increment) K of the present study is less when compared with the time period K obtained in [16] for both fixed values of thermal gradient κ and nonhomogeneity φ.
A comparison of time period K of the clamped circular plate with the time period K obtained in [16] corresponding to nonhomogeneity φ for fixed values of thermal gradient κ � 0.0 and tapering ϖ � 0.0 is presented in Figure 10. Here, the time period K of the present study is higher than the time period K obtained in [16] for fixed values of thermal gradient κ and tapering parameter ϖ, but the variation in time period K (rate of decrement) of the present study is much lesser when compared with the variation of time period K obtained in [16].         [16] κ = φ = 0.0 Figure 9: Comparison of the time period of the clamped circular plate with [16] corresponding to tapering ϖ.  [16] κ= ϖ = 0.0 Figure 10: Comparison of the time period of the clamped circular plate with [16] corresponding to nonhomogeneity φ.

Discussion and Conclusions
In this section, we will summarize our major findings. e time period for the first three modes of vibration corresponding to the plate parameters, nonhomogeneity, tapering, and thermal gradient was discussed for both clamped and simply supported plates. e study done concluded that, for the case of circular variation in plate parameters, namely, nonhomogeneity and tapering, the time period calculated is smaller in comparison to other variations studied in the literature. Not only this, the change in time period is less rapid as compared to other studies done. For these facts, the mode convergence for circular variation is for a smaller value of N in comparison to other studies where the value of N is higher than our study. is further leads to the conclusion that, for circular variations, the required computational time is less in contrast to other studies.
Based on numerical illustration and comparison, the authors would like to conclude the following facts: (i) e time period K in case of circular thickness (present study) is less when compared with the time period K in case of linear thickness [16]. e rate of increment in time period K in case of circular thickness (present study) is also less when compared with the rate of increment in time period K in case of linear thickness [16]. e time period K of the present study and that obtained in [16] coincide at ϖ � 0.0 ( Figure 9). (ii) e time period K in case of circular density (present study) is higher than the time period K in case of linear density [16], but the rate of decrement in time period K reported in case of circular density (present study) is much less when compared with the rate of decrement in time period K reported in case of linear density [16]. Here also, the time period K of the present study and that obtained in [16] coincide at φ � 0.0 ( Figure 10).
(iii) Time period K of clamped and simply supported circular plates increases with increasing value to tapering ϖ and thermal gradient κ (Tables 1 and 2), but the time period K of clamped and simply supported circular plates decreases with the increasing value of nonhomogeneity φ (Table 3). (iv) Time period K of the simply supported circular plate is higher than the time period K of the clamped circular plate, but the variation in time period K (rate of increment/decrement) of the clamped circular plate is less when compared with the variation in time period K of the simply supported circular plate (Tables 1-3). (v) Points (i) and (ii) emphasize that time period K as well as variation in time period K can be controlled by choosing an appropriate variation in plate parameters, while points (iii) and (iv) focus on how the time period K is affected by plate parameters.

Symbols
l: ickness of the plate r 1 : Radius of the plate ρ: Mass density per unit volume of plate material ]: Poisson's ratio V s : Strain energy T s : Kinetic energy D: Flexural rigidity E: Young's modulus ϕ: Transverse deflection ω: Circular frequency Φ: Mode shape function φ: Nonhomogeneity constant κ: ermal gradient ϖ: Tapering parameter η: Temperature on the plate Φ 1 : Deflection function λ: Frequency K: Time period N: Number of terms K i , i � 1, 2, 3: Time period for the first three modes of vibration.

Data Availability
e research data used to support the findings of this study are currently under embargo, while the research findings are commercialized. Requests for data, 6 months after the publication of this article, will be considered by the corresponding authors.

Conflicts of Interest
e authors declare that they have no conflicts of interest.