A Symplectic Precise Integration Perturbation Series Method for Linear Structural Dynamic Response Problems

The symplectic algorithm for Hamiltonian systems is renowned for its high accuracy and efficiency in long-term computations. However, its direct application to non-conservative problems in engineering dynamics is limited by the presence of dissipative effects and external load terms. To overcome these challenges, this paper presents a novel symplectic precise integration method based on the perturbation series approach, specifically tailored for structural dynamic response problems. By introducing a state vector, the general structural dynamic response equation is reformulated into a generalized Hamiltonian framework. Subsequently, perturbation theory is applied to transform this generalized Hamiltonian equation into a sequence of linear Hamiltonian equations, effectively converting the problem of solving non-conservative equations into one of solving multiple conservative equations. The precise integration method is then used to derive a step mapping matrix that significantly improves accuracy without substantially increasing computational cost. Numerical examples are provided to highlight the efficacy and practical applicability of the proposed method in addressing complex structural dynamic response problems.