Generalized multi-symplectic method for dynamic responses of continuous beam under moving load

Based on the multi-symplectic idea, a generalized multi-symplectic integrator method is presented to analyze the dynamic response of multi-span continuous beams with small damping coefficient. Focusing on the local conservation properties, the generalized multi-symplectic formulations are introduced and a fifteen-point implicit structure-preserving scheme is constructed to solve the first-order partial differential equations derived from the dynamic equation governing the dynamic behavior of continuous beams under moving load. From the results of the numerical experiments, it can be concluded that, for the cases considered in this paper, the structure-preserving scheme is generalized multi-symplectic if the viscous damping c ≤ 0.3751 when the continuous beam is under a constant-speed moving load and the structure-preserving scheme is generalized multi-symplectic if the viscous damping c ≤ 0.3095 when the continuous beam is under a variable-speed moving load with fixed step lengths Δt = 0.05 and Δx = 0.025. Similar to a multi-symplectic scheme, the generalized multi-symplectic scheme also has two remarkable advantages: the excellent long-time numerical behavior and the good conservation property.