Generalized Multi-Symplectic Integrator for Vibration of a Damping String with the Driving Force

Based on multi-symplectic theory, a generalized multi-symplectic formulation is used to simulate the vibration of a string with damping and driving forces. First, by introducing the canonical momenta, the generalized multi-symplectic form for the governing equation of a damping string is derived. Then, two multi-symplectic Box schemes are developed to obtain the discretization schemes for the generalized multi-symplectic form. Finally, numerical examples are carried out to verify the effectiveness of proposed schemes. Numerical results show that the proposed schemes can directly preserve the dynamic characteristics of the system, especially the non-conservative properties caused by damping and driving forces. The research can give a different approach when dealing with the non-conservative dynamics problems.