Forced Vibration Analysis in Axisymmetric Functionally Graded Viscothermoelastic Hollow Cylinder Under Dynamic Pressure

Abstract

The forced vibrations of axisymmetric inhomogenous isotropic viscothermoelastic hollow cylinder under periodic dynamic pressure have been studied. The material was taken inhomogenous due to easy exponent law in radial direction. By applying time harmonics variation technique, the partial differential equations were converted into ordinary differential equations. These ordinary differential equations were solved by applying series solution of matrix Fröbenius method analytically to represent displacement, temperature and stresses. Numerically simulated outcomes were presented graphically to express the effect of functionally graded material disk for different values of grading parameter. With the increase in value of grading index, variation of field functions go on decreasing. The variation of field functions show high variation in homogenous materials in contrast to low variation in inhomogeneous materials.

Appendix

$$ g_{1} (s + 1) = \left( {\begin{array}{*{20}c} {(s + 1)^{2} - \eta^{2} } & 0 \\ 0 & {\left( {(s + 1)^{2} - \frac{{\gamma^{2} }}{4}} \right)} \\ \end{array} } \right); g_{2} (s + 1) = \left( {\begin{array}{*{20}c} 0 & {A^{*} \left( {s + 1 + \frac{\gamma }{2}} \right)} \\ {B^{*} \left( {s + 1 + \frac{2 - \gamma }{2}} \right)} & 0 \\ \end{array} } \right); $$

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$$ e_{12}^{1} (s_{j} ) = A^{*} \frac{{\left( {s_{j} + \frac{2 + \gamma }{2}} \right)}}{{(s + 1)^{2} - \eta^{2} }},\quad e_{21}^{1} (s_{j} ) = B^{*} \frac{{\left( {s_{j} + \frac{4 - \gamma }{2}} \right)}}{{(s + 1)^{2} - \frac{{\gamma^{2} }}{4}}};\quad j = 1,2,3,4.; $$

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$$ g_{11}^{k} (s_{j} ) = \frac{i\varOmega }{{\tilde{\delta }_{0} \left( {(s_{j} + k + 2)^{2} - \eta^{2} } \right)}}, g_{22}^{k} (s_{j} ) = \frac{{\varOmega^{*} \varOmega^{2} \tilde{\tau }_{0} }}{{\left( {(s_{j} + k + 2)^{2} - \frac{{\gamma^{2} }}{4}} \right)}}; $$

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$$ g_{12}^{k} (s_{j} ) = \frac{{A^{*} \left( {s_{j} + k + \frac{2 + \gamma }{2}} \right)}}{{\left( {(s_{j} + k + 2)^{2} - \eta^{2} } \right)}}, g_{21}^{k} (s_{j} ) = \frac{{B^{*} \left( {s_{j} + k + \frac{4 - \gamma }{2}} \right)}}{{\left( {(s_{j} + k + 2)^{2} - \frac{{\gamma^{2} }}{4}} \right)}}; $$

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

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

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

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

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

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

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

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