Tuning methods for tuned inerter dampers coupled to nonlinear primary systems

This study develops displacement- and kinetic energy-based tuning methods for the design of the tuned inerter dampers (TIDs) coupled to both linear and nonlinear primary systems. For the linear primary system, the design of the TID is obtained analytically. The steady-state frequency–response relationship of the nonlinear primary system with a softening or hardening stiffness nonlinearity is obtained using the harmonic balance (HB) method. Analytical and numerical tuning approaches based on HB results are proposed for optimal designs of the TID to achieve equal peaks in the response curves of the displacement and the kinetic energy of the primary system. Via the developed approaches, the optimal stiffness of the TID can be obtained according to the stiffness nonlinearity of the primary system and the inertance of the absorber. Unlike the linear primary oscillator case, for a nonlinear primary oscillator the shape of its resonant peaks is mainly affected by the damping ratio of the TID, while the peak values depend more on the stiffness ratio. The proposed designs are shown to be effective in a wide range of stiffness nonlinearities and inertances. This study demonstrates the benefits of using inerters in vibration suppression devices, and the adopted methods are directly applicable for nonlinear systems with different types of nonlinearities.