Coupling dynamic behaviors of spatial flexible beam with weak damping

As a typical high-dimensional nonlinear dynamic problem, the difficulties of the dynamic analysis on the spatial flexible damping beam mostly result from the coupling between the spatial motion and the transverse vibration. Considering the coupling effect and the weak structure damping, the dynamic behaviors of the spatial flexible beam are investigated by a complex structure-preserving method in this paper. Based on the variational principle, the dynamic model of the spatial flexible damping beam is established, which can be decoupled into two parts approximately, one controls the spatial motion and another mainly controls the transverse vibration. For the first part, the classic fourth-order Runge–Kutta method can be used to discrete it expediently. For the latter part, based on the generalized multi-symplectic idea, the approximate symmetric form as well as the generalized multi-symplectic conservation law is formulated, and a 15-point scheme equivalent to the Preissmann scheme is constructed. Numerical iterations are performed for six typical initial cases between the two parts to study the dynamic behaviors of the spatial flexible beam. In the numerical experiments, the effects of the damping factor and the initial conditions (including the initial radial velocity and the initial attitude angle) on the dynamic behaviors of the spatial flexible beam are investigated in detail. From the numerical results, it can be concluded that the damping effect on the long-time dynamic behaviors of the spatial flexible beam could not be neglected even if it is weak; the numerical method proposed in this paper owns the tiny numerical dissipation as well as the excellent long-time numerical stability.