Vibration analysis of circular cylindrical shells made of metal foams under various boundary conditions

Abstract

This study investigates the free vibration of metal foam circular cylindrical shells under various boundary conditions. The elasticity modulus and mass density of the shells vary gradually and continually in the thickness direction. Two types of porosity distribution are taken into account including symmetrical and unsymmetrical distributions. Love’s shell theory is employed to formulate the governing equations and then the Rayleigh–Ritz method is utilized to solve natural frequencies of the system. The results show that the porosity coefficient has important effect on the natural frequencies of metal foam shells. Its effect also relates to the boundary conditions of the shells. Moreover, different porosity distributions make the metal foam shells possess different vibration characteristics, which is quite obvious at large porosity coefficient. As the circumferential wave number increases, the natural frequencies of the metal foam shells tend to the same under various boundary conditions. Additionally, the present results are verified by the comparison with the published ones in the literature.

Appendix

The coefficients C_ij (i, j = 1, 2, 3) in Eq. (28) are defined as follows:

$$\begin{aligned} C_{11} & = \frac{{A_{11} m^{2} \pi^{3} R}}{2L} + \frac{{A_{66} \pi n^{2} L}}{2R} - \frac{\pi RL}{2}\rho_{m} \omega^{2} , \\ C_{12} & = - \frac{{mn\pi^{2} }}{2}\left( {A_{12} + A_{66} + \frac{{B_{12} }}{R} + \frac{{2B_{66} }}{R}} \right), \\ C_{13} & = - \frac{{m\pi^{2} }}{2}\left( {A_{12} + \frac{{B_{12} n^{2} }}{R} + \frac{{2B_{66} n^{2} }}{R} + \frac{{B_{11} m^{2} \pi^{2} R}}{{L^{2} }}} \right), \\ C_{22} & = \frac{{m^{2} \pi^{3} }}{L}\left( {\frac{{A_{66} R}}{2} + 2B_{66} + \frac{{2D_{66} }}{R}} \right) + \frac{{n^{2} L\pi }}{R}\left( {\frac{{A_{22} }}{2} + \frac{{B_{22} }}{R} + \frac{{D_{22} }}{{2R^{2} }}} \right) - \frac{\pi RL}{2}\rho_{m} \omega^{2} , \\ C_{23} & = \frac{{m^{2} n\pi^{3} }}{L}\left( {\frac{{B_{12} }}{2} + B_{66} + \frac{{D_{12} }}{2R} + \frac{{D_{66} }}{R}} \right) + \frac{n\pi L}{2R}\left( {A_{22} + \frac{{B_{22} }}{R}} \right) + \frac{{n^{3} \pi L}}{{2R^{2} }}\left( {B_{22} + \frac{{D_{22} }}{R}} \right), \\ C_{33} & = \frac{{m^{2} n^{2} \pi^{3} }}{LR}\left( {D_{12} + 2D_{66} } \right) + \frac{{n^{2} \pi L}}{{R^{2} }}\left( {B_{22} + \frac{{D_{22} n^{2} }}{2R}} \right) + \frac{{m^{2} \pi^{3} }}{L}\left( {B_{12} + \frac{{D_{11} m^{2} \pi^{2} R}}{{2L^{2} }}} \right) + \frac{{A_{22} \pi L}}{2R} - \frac{\pi RL}{2}\rho_{m} \omega^{2} . \\ \end{aligned}$$