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

Reeta Bhardwaj,Sudeshna Ghosh,Amit Sharma,Kamal Kumar,Naveen Mani,Alessandro Lo Schiavo

doi:10.1155/2021/4723759

Reeta Bhardwaj, Sudeshna Ghosh + Show 4 more

Open Access

https://doi.org/10.1155/2021/4723759

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Abstract

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. The 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. This 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.

Highlights

Vibration, sometimes, is described as a kind of waste energy
It can be 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 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 supported circular plate is higher when compared with the time period K of the clamped circular plate
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 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

Summary

Introduction

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. E effect of exponential Young’s modulus and density on the asymmetric vibrations of nonhomogeneous, clamped, 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]. 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 supported circular plates, and the result is presented in the form of tables and graphs. The main conclusions of the studies have been discussed

Analysis

R zΦ zR

Numerical Results

Discussion and Conclusions

E: Young’s modulus φ: Transverse deflection ω