Closed-Form Solution for Forced Vibration of Hybrid Cable-Damper Systems via Complex Mode Superposition

The analysis and control of stay cable vibrations under environmental excitation hinge on the effective solution of forced vibration problems. This study addresses cable systems equipped with viscous dampers by modeling them as tensioned strings with concentrated damping, and introduces a closed-form solution approach for forced vibration based on an analytical complex mode superposition method. First, the weighted orthogonality of complex modes is utilized to derive the analytical expression for free vibration response under arbitrary initial conditions. By transforming the unit impulse excitation problem into an equivalent initial-value problem, the system’s impulse response function is obtained. On this basis, a unified closed-form analytical framework for the system response under arbitrary spatiotemporal loads is established through Duhamel’s integral and the linear superposition principle. Results demonstrate that, despite the hybrid damped nature of the system, its displacement response can still be expressed as a superposition of analytical complex mode functions, with both the mode shapes and the response expressions given in closed form. Numerical examples show that, compared to the conventional real mode superposition method and the finite difference method, the proposed approach requires only a small number of complex modes to achieve comparable accuracy, significantly improving computational efficiency. This method provides a novel analytical tool for precise dynamic analysis and efficient vibration control of hybrid systems such as stay cables.