Analysis of Free Vibrations of Axisymmetric Functionally Graded Generalized Viscothermoelastic Cylinder Using Series Solution

Abstract

Purpose

The analysis of free vibrations of axisymmetric functionally graded isotropic viscothermoelastic hollow cylinder has been investigated in the radial direction. The material of viscothermoelastic cylinder has been assumed to be graded in the radial direction due to simple exponent law. The governing partial differential equations are transformed into ordinary differential equations with th e help of time harmonic vibrations.

Methods

The extended power series solution of matrix Fröbenius method has been applied to derive the analytical solutions for stresses, displacement and temperature change. The frequency equations of various types of modes of vibrations have been derived in a compact form. To investigate the numerical features of vibrations, the analytical results of frequency equations have been further solved with the help of fixed-point numerical iteration technique using MATLAB software tools. The polymethyl methacrylate material has been used for numerical computations.

Results and Conclusions

The numerical results have been presented for frequency shift, natural frequencies, thermoelastic damping, stresses, displacement and variation of temperature. With the increase in the values of grading index, variation of vibrations in field functions go on decreasing. The numerical results show alternate variations in homogenous materials in contrast to inhomogeneous materials. The behaviour of deformation, temperature change, frequency shift and thermoelastic damping have been monitored (increase or decrease) with the help of grading index (in-homogeneity parameter).

Appendix

$$g_{1} (s_{j} + 1) = \left( {\begin{array}{*{20}c} {(s_{j} + 1)^{2} - \eta^{2} } & 0 \\ 0 & {\left( {(s_{j} + 1)^{2} - \frac{{\gamma^{2} }}{4}} \right)} \\ \end{array} } \right)\,$$

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$$g_{2} (s_{j} ) = \left( {\begin{array}{*{20}c} 0 & {A^{*} \left( {s_{j} + 1 + \frac{\gamma }{2}} \right)} \\ {B^{*} \left( {s_{j} + 1 + \frac{2 - \gamma }{2}} \right)} & 0 \\ \end{array} } \right)$$

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$$\begin{aligned} e_{11}^{2} (s_{j} ) = g_{12}^{0} (s_{j} )\,e_{21}^{1} (s_{j} ) - g_{11}^{0} (s_{j} )\,\,\, \hfill \\ \,e_{22}^{2} (s_{j} ) = g_{21}^{0} (s_{j} )\,e_{12}^{1} (s_{j} ) - g_{22}^{0} (s_{j} )\, \hfill \\ \end{aligned}$$

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$$\begin{aligned} e_{12}^{3} (s_{j} ) = g_{12}^{1} (s_{j} )\,e_{22}^{2} (s_{j} ) - g_{11}^{1} (s_{j} )\,e_{12}^{1} \,(s_{j} )\,\,\,\, \hfill \\ \,e_{21}^{3} (s_{j} ) = g_{21}^{1} (s_{j} )\,e_{11}^{2} (s_{j} ) - g_{22}^{1} (s_{j} )\,e_{21}^{1} \,(s_{j} ) \hfill \\ \end{aligned}$$

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$$\begin{aligned} e_{11}^{4} (s_{j} ) = g_{12}^{2} (s_{j} )\,e_{21}^{3} (s_{j} ) - g_{11}^{2} (s_{j} )\,e_{11}^{2} \,(s_{j} )\,\,\,\, \hfill \\ \,e_{22}^{4} (s_{j} ) = g_{21}^{2} (s_{j} )\,e_{12}^{3} (s_{j} ) - g_{22}^{2} (s_{j} )\,e_{21}^{2} \,(s_{j} ) \hfill \\ \end{aligned}$$

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$$\begin{aligned} e_{12}^{5} (s_{j} ) = g_{12}^{3} (s_{j} )\,e_{22}^{4} (s_{j} ) - g_{11}^{3} (s_{j} )\,e_{12}^{3} \,(s_{j} )\,\,\,\, \hfill \\ \,e_{21}^{5} (s_{j} ) = g_{21}^{3} (s_{j} )\,e_{11}^{4} (s_{j} ) - g_{22}^{3} (s_{j} )\,e_{21}^{3} \,(s_{j} ) \hfill \\ \end{aligned}$$

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$$e_{11}^{2k} (s_{j} ) = g_{12}^{2k - 2} (s_{j} )\,e_{21}^{2k - 1} (s_{j} ) - g_{11}^{2k - 2} (s_{j} )\,e_{11}^{2k - 2} \,(s_{j} )\,\,\,\,$$

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$$e_{22}^{2k} (s_{j} ) = g_{21}^{2k - 2} (s_{j} )\,e_{12}^{2k - 1} (s_{j} ) - g_{22}^{2k - 1} (s_{j} )\,e_{21}^{2k - 1} \,(s_{j} )$$

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$$e_{12}^{2k + 1} (s_{j} ) = g_{12}^{2k - 1} (s_{j} )\,e_{22}^{2k} (s_{j} ) - g_{11}^{2k - 1} (s_{j} )\,e_{12}^{2k - 1} \,(s_{j} )\,\,\,\,$$

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$$e_{21}^{2k + 1} (s_{j} ) = g_{21}^{2k - 1} (s_{j} )\,e_{11}^{2k} (s_{j} ) - g_{22}^{2k - 1} (s_{j} )\,e_{21}^{2k - 1} \,(s_{j} )$$

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