Instability in the transformation between extensional and flexural modes in thin-walled cylindrical shells

Abstract

Abstract:With the inclusion of all terms up to the third-order in the membrane strain to consider the geometric nonlinearity in deformation, a third-order implicit model and a fourth-order explicit model for the vibration transformation between extensional and flexural modes in thin-walled cylindrical shells are established and solved numerically. Numerical instability is observed in numerical solutions based on the explicit model. It is found that such numerical instability does not result from the accumulated numerical errors in the numerical integration process, but from the neglecting of higher order terms in the formulation of the problem. With the inclusion of all terms up to the fourth-order in the strain energy, the explicit model can predict a stable vibration history with periodic mode transformations between the two modes. The same 2:1 internal resonance of the vibration mode transformation is predicted by both models. As flexural stress growth is concerned, the two models matches very well during the first group of peaks but lag in phase gradually appears in later group of stress peaks based on the implicit model. This is understood to be resulting from the limited number of terms included in the series expansions based on the explicit model, which allows the most likely excited flexural mode get a larger share of energy transferred from the principle mode and as a result flexural stress arrives at peaks earlier based on the explicit model.

Non-linear vibrations of shells: A literature review from 2003 to 2013

“…[144] by solving the problem via a new technique based on the Richardson extrapolation method in [145], studying the dynamics of compound cylindrical shells with spherical caps in [146], and investigating the dynamic behavior of cylindrical shells in an elastic medium in [147]. Moreover, the stability of vibration mode transformation between extensional and flexural modes, and two to one internal resonance in thin walled circular cylindrical shells subjected to large initial radial velocity or impulse was investigated by Shi et al [148].…”