Nonlinear design of tuned inertia damper: From analytical calculation to chaotic behavior prediction

In this study, a tuning method based on a bi-objective optimization approach—focusing on displacement and kinetic energy—is proposed for the design of Tuned Inertia Dampers (TIDs) for coupled linear and nonlinear primary systems. For the nonlinear primary system, the design criteria for the TID parameters are established through an analytical method, and the analytical expression for the steady-state frequency response of the nonlinear system is derived using the harmonic balance method (HB). A dimensional reduction analysis of the two-degree-of-freedom system is conducted, and the chaotic threshold is calculated by applying Melnikov function theory. The suppression effect of the TID parameters on the system's chaotic behavior is further verified through numerical simulations of the safety basin. Building upon this, a tuning strategy is developed with the goal of optimizing the balance between the peak displacement and kinetic energy responses of the primary system. The dynamic correlation between the optimal stiffness of the TID, the nonlinear stiffness coefficient of the primary system, and the inertia element is also explored. This method offers a design framework that combines both analytical accuracy and engineering applicability for nonlinear vibration control, particularly for structural systems with viscoelastic materials or large deformation characteristics.