On Free Vibration Analysis of FGPM Cylindrical Shell Excited Under d₁₅ Effect

Abstract

Piezoelectric vibrating shell is the major component of the various equipments related to mechanical, nuclear, aeronautical and aerospace engineering. Shells especially exhibit certain effects that are not present in beams or even in plates. Analysis of vibrational characteristics of functionally graded piezoelectric circular cylindrical shell is quit complex than beams and plates. This is because the coupling of vibration of shell between the three directions can no longer be neglected. Dynamic analysis of FGPM continuum is quite complex, due to the synchronization of electric and mechanical terms. Closed form solution for complex FGPM model by analytical method is quite a tedious task. Approximation method provides an alternate path for solving such kind of problems. Paper contains free vibration analysis of a full three dimensional FGPM cylindrical shell model excited under d₁₅ effect. Shear piezoelectric coupling coefficient d₁₅ is much higher than the other piezoelectric coefficients d₃₁ or d₃₃. Rayleigh–Ritz, an approximation method, is used to obtain the eigenvectors and eigenfrequencies for FGPM cylindrical shell. Orthogonal polynomial functions are used with Rayleigh–Ritz method to formulate the linear eigenvalue problem. Higher order polynomials are generated by Gram–Schmidt method. MATLAB 2015a Symbolic Toolbox is used for calculation and obtained solution is validated using commercial finite element software COMSOL Multiphysics 2014.

Appendix 1

Expressions for the matrices $\Gamma _{piezo}$, $\Gamma _{dielectric}$ and $\Gamma _{mech}$ are given below:

$$\begin{aligned}\Gamma _{piezo} & = \left[ {\begin{array}{*{20}c} {\left[ {e_{31} \left( r \right)} \right]\left\{ {\Psi _{1,r} } \right\}\left\{ {\Psi _{4,z} } \right\}^{T} + \frac{1}{r}\left[ {e_{31} \left( r \right)} \right]\left\{ {\Psi _{1} } \right\}\left\{ {\Psi _{4,z} } \right\}^{T} + \left[ {e_{15} \left( r \right)} \right]\left\{ {\Psi _{1,z} } \right\}\left\{ {\Psi _{4,r} } \right\}^{T} } \\ {\frac{1}{r}\left[ {e_{31} \left( r \right)} \right]\left\{ {\Psi _{2,\theta } } \right\}\left\{ {\Psi _{4,z} } \right\}^{T} + \frac{1}{r}\left[ {e_{15} \left( r \right)} \right]\left\{ {\Psi _{2,z} } \right\}\left\{ {\Psi _{4,\theta } } \right\}^{T} } \\ {\left[ {e_{33} \left( r \right)} \right]\left\{ {\Psi _{3,z} } \right\}\left\{ {\Psi _{4,z} } \right\}^{T} + + \left[ {e_{15} \left( r \right)} \right]\left\{ {\Psi _{3,r} } \right\}\left\{ {\Psi _{4,r} } \right\}^{T} + \frac{1}{{r^{2} }}\left[ {e_{15} \left( r \right)} \right]\left\{ {\Psi _{3,\theta } } \right\}\left\{ {\Psi _{4,\theta } } \right\}^{T} } \\ \end{array} } \right] \\\Gamma _{dielectric} & = \left[ { - \left[ {\varepsilon_{11}^{S} \left( r \right)} \right]\left\{ {\Psi _{4,r} } \right\}\left\{ {\Psi _{4,r} } \right\}^{T} - \frac{1}{{r^{2} }}\left[ {\varepsilon_{11}^{S} \left( r \right)} \right]\left\{ {\Psi _{4,\theta } } \right\}\left\{ {\Psi _{4,\theta } } \right\}^{T} - \left[ {\varepsilon_{33}^{S} \left( r \right)} \right]\left\{ {\Psi _{4,z} } \right\}\left\{ {\Psi _{4,z} } \right\}^{T} } \right] \\ \end{aligned}$$

$$\Gamma _{mech} = \left[ {\begin{array}{*{20}l} {\frac{1}{{r^{2} }}\left[ {C_{11}^{E} \left( r \right)} \right]\left\{ {\Psi _{1} } \right\}\left\{ {\Psi _{1} } \right\}^{T} } \hfill & {} \hfill & {\frac{1}{{r^{2} }}\left[ {C_{11}^{E} \left( r \right)} \right]\left\{ {\Psi _{1} } \right\}\left\{ {\Psi _{2,\theta } } \right\}^{T} } \hfill & {} \hfill & {\frac{1}{r}\left[ {C_{13}^{E} \left( r \right)} \right]\left\{ {\Psi _{1} } \right\}\left\{ {\Psi _{3,z} } \right\}^{T} } \hfill \\ { + \frac{2}{r}\left[ {C_{12}^{E} \left( r \right)} \right]\left\{ {\Psi _{1} } \right\}\left\{ {\Psi _{1,r} } \right\}^{T} } \hfill & {} \hfill & { + \frac{1}{r}\left[ {C_{12}^{E} \left( r \right)} \right]\left\{ {\Psi _{1,r} } \right\}\left\{ {\Psi _{2,\theta } } \right\}^{T} } \hfill & {} \hfill & { + \left[ {C_{13}^{E} \left( r \right)} \right]\left\{ {\Psi _{1,r} } \right\}\left\{ {\Psi _{3,z} } \right\}^{T} } \hfill \\ { + \left[ {C_{11}^{E} \left( r \right)} \right]\left\{ {\Psi _{1,r} } \right\}\left\{ {\Psi _{1,r} } \right\}^{T} } \hfill & {} \hfill & { - \frac{1}{{r^{2} }}\left[ {C_{66}^{E} \left( r \right)} \right]\left\{ {\Psi _{1,\theta } } \right\}\left\{ {\Psi _{2} } \right\}^{T} } \hfill & {} \hfill & { + \left[ {C_{44}^{E} \left( r \right)} \right]\left\{ {\Psi _{1,z} } \right\}\left\{ {\Psi _{3,r} } \right\}^{T} } \hfill \\ { + \left[ {C_{44}^{E} \left( r \right)} \right]\left\{ {\Psi _{1,z} } \right\}\left\{ {\Psi _{1,z} } \right\}^{T} } \hfill & {} \hfill & { + \frac{1}{r}\left[ {C_{66}^{E} \left( r \right)} \right]\left\{ {\Psi _{1,\theta } } \right\}\left\{ {\Psi _{2,r} } \right\}^{T} } \hfill & {} \hfill & {} \hfill \\ { + \frac{1}{{r^{2} }}\left[ {C_{66}^{E} \left( r \right)} \right]\left\{ {\Psi _{1,\theta } } \right\}\left\{ {\Psi _{1,\theta } } \right\}^{T} } \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {\frac{1}{{r^{2} }}\left[ {C_{11}^{E} \left( r \right)} \right]\left\{ {\Psi _{2,\theta } } \right\}\left\{ {\Psi _{1} } \right\}^{T} } \hfill & {} \hfill & {\frac{1}{{r^{2} }}\left[ {C_{11}^{E} \left( r \right)} \right]\left\{ {\Psi _{2,\theta } } \right\}\left\{ {\Psi _{2,\theta } } \right\}^{T} } \hfill & {} \hfill & {\frac{1}{r}\left[ {C_{13}^{E} \left( r \right)} \right]\left\{ {\Psi _{2,\theta } } \right\}\left\{ {\Psi _{3,z} } \right\}^{T} } \hfill \\ { + \frac{1}{r}\left[ {C_{12}^{E} \left( r \right)} \right]\left\{ {\Psi _{2,\theta } } \right\}\left\{ {\Psi _{1,r} } \right\}^{T} } \hfill & {} \hfill & { + \left[ {C_{44}^{E} \left( r \right)} \right]\left\{ {\Psi _{2,z} } \right\}\left\{ {\Psi _{2,z} } \right\}^{T} } \hfill & {} \hfill & { + \frac{1}{r}\left[ {C_{44}^{E} \left( r \right)} \right]\left\{ {\Psi _{2,z} } \right\}\left\{ {\Psi _{3,\theta } } \right\}^{T} } \hfill \\ { - \frac{1}{{r^{2} }}\left[ {C_{66}^{E} \left( r \right)} \right]\left\{ {\Psi _{2} } \right\}\left\{ {\Psi _{1,\theta } } \right\}^{T} } \hfill & {} \hfill & { + \frac{1}{{r^{2} }}\left[ {C_{66}^{E} \left( r \right)} \right]\left\{ {\Psi _{2} } \right\}\left\{ {\Psi _{2} } \right\}^{T} } \hfill & {} \hfill & {} \hfill \\ { + \frac{1}{r}\left[ {C_{66}^{E} \left( r \right)} \right]\left\{ {\Psi _{2,r} } \right\}\left\{ {\Psi _{1,\theta } } \right\}^{T} } \hfill & {} \hfill & { + \left[ {C_{66}^{E} \left( r \right)} \right]\left\{ {\Psi _{2,r} } \right\}\left\{ {\Psi _{2,r} } \right\}^{T} } \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & { - \frac{1}{r}\left[ {C_{66}^{E} \left( r \right)} \right]\left\{ {\Psi _{2} } \right\}\left\{ {\Psi _{2,r} } \right\}^{T} } \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {\frac{1}{r}\left[ {C_{13}^{E} \left( r \right)} \right]\left\{ {\Psi _{3,z} } \right\}\left\{ {\Psi _{1} } \right\}^{T} } \hfill & {} \hfill & {\frac{1}{r}\left[ {C_{13}^{E} \left( r \right)} \right]\left\{ {\Psi _{3,z} } \right\}\left\{ {\Psi _{2,\theta } } \right\}^{T} } \hfill & {} \hfill & {\left[ {C_{44}^{E} \left( r \right)} \right]\left\{ {\Psi _{3,r} } \right\}\left\{ {\Psi _{3,r} } \right\}^{T} } \hfill \\ { + \left[ {C_{13}^{E} \left( r \right)} \right]\left\{ {\Psi _{3,z} } \right\}\left\{ {\Psi _{1,r} } \right\}^{T} } \hfill & {} \hfill & { + \frac{1}{r}\left[ {C_{44}^{E} \left( r \right)} \right]\left\{ {\Psi _{3,\theta } } \right\}\left\{ {\Psi _{2,z} } \right\}^{T} } \hfill & {} \hfill & { + \left[ {C_{33}^{E} \left( r \right)} \right]\left\{ {\Psi _{3,z} } \right\}\left\{ {\Psi _{3,z} } \right\}^{T} } \hfill \\ { + \left[ {C_{44}^{E} \left( r \right)} \right]\left\{ {\Psi _{3,r} } \right\}\left\{ {\Psi _{1,z} } \right\}^{T} } \hfill & {} \hfill & {} \hfill & {} \hfill & { + \frac{1}{{r^{2} }}\left[ {C_{44}^{E} \left( r \right)} \right]\left\{ {\Psi _{3,\theta } } \right\}\left\{ {\Psi _{3,\theta } } \right\}^{T} } \hfill \\ \end{array} } \right]$$