Benchmark free vibration solutions of spherical shells by the state space method incorporating precise matrix exponential computation. (June 2022) - Zhou, Chao; Zheng, Xinran; Wang, Zixuan; Xiong, Sijun; Shi, Yueqing; Li, Rui
Abstract: In this study, new benchmark free vibration solutions of spherical shells, including annular spherical shells and spherical caps, are explored by the state space method incorporating precise matrix exponential computation. A strategy of piecewise solution along the meridional direction is developed to tackle the governing partial differential equations (PDEs) with variable coefficients, and no assumption of modal displacement functions is imposed, which is different from the conventional state space solution framework. A precise computational method is employed to determine the state transfer matrix characterized by the matrix exponential, which eliminates potential numerical instabilities in calculation. To validate the obtained solutions and to provide useful benchmark results, comprehensive frequency and mode shape results for the spherical shells under typical boundary conditions are presented, which show good agreement with those by the FEM and the Rayleigh–Ritz method in the literature. In view of the virtues on solving the PDEs with variable coefficients, the developed approach is expected to facilitate the acquisition of more benchmark solutions of complex plate and shell problems. Highlights: State space method with precise matrix exponential computation is explored. Partial differential equations involving variable coefficients are solved. New benchmark free vibration solutions of spherical shells are obtained.