This paper considers the multi-symplectic formulations of the generalized fifth-order KdV equation in Hamiltonian space. Recurring to the midpoint rule, it presents an implicit multi-symplectic scheme with discrete multi-symplectic conservation law to solve the partial differential equations which are derived from the generalized fifth-order KdV equation numerically. The results of the numerical experiments show that this multi-symplectic algorithm is good in accuracy and its long-time numerical behaviour is also perfect.
Based on Hamilton＇s principle, a new kind of fully coupled nonlinear dynamic model for a rotating rigid-flexible smart structure with a tip mass is proposed. The geometrically nonlinear effects of the axial, transverse displacement and rotation angle are considered by means of the first-order approximation coupling （FOAC） model theory, in which large deformations and the centrifugal stiffening effects are considered. Three kinds of systems are established respectively, which are a structure without piezoelectric layer, with piezoelectric layer in open circuit and closed circuit. Several simulations based on simplified models are presented to show the differences in characteristics between structures with and without the tip mass, between smart beams in closed and open circuit, and between the centrifugal effects in high speed rotating state or not. The last simulation calculates the dynamic response of the structure subjected to external electrical loading.
Based on the new explicit Magnus expansion developed for nonlinear equations defined on a matrix Lie group, an efficient numerical method is proposed for nonlinear dynamical systems. To improve computational efficiency, the integration step size can be adaptively controlled. Validity and effectiveness of the method are shown by application to several nonlinear dynamical systems including the Duffing system, the van der Pol system with strong stiffness, and the nonlinear Hamiltonian pendulum system.
