Extended-variable Birkhoffian variational integrators for forced and damped structural systems

For numerical structural dynamics under nonconservative conditions such as external forcing and damping, this paper proposes an extended-variable Birkhoffian symplectic variational integrator. By augmenting the state, external loads and damping are incorporated uniformly into the Birkhoffian representation, and a second-order central-difference scheme is derived from a discrete variational principle that preserves the symplectic structure at the discrete level, thereby mitigating long-time energy and phase drift. A modal decoupling strategy is employed to address high-dimensional engineering models. The method is straightforward to implement, applies uniformly to linear and nonlinear systems, and is applicable to large-scale problems. Numerical studies spanning single-degree-of-freedom linear and nonlinear systems and a high-dimensional rudder structure demonstrate stable accuracy, together with more consistent energy behavior and phase fidelity for stiff systems and long-horizon simulations. These results indicate that the proposed approach provides a practical and portable route for engineering-grade structural dynamics computations under nonconservative conditions.