Effect of negative stiffness nonlinearity on the vibration control effectiveness of tuned negative stiffness inerter damper

Recently, tuned negative stiffness inerter damper (TNSID) integrating the advantages of both negative stiffness and inerter elements has been proposed and verified to be effective for structural vibration control. In previous studies, the stiffness of the negative stiffness element was normally regarded as a constant for simplicity. However, it is well-known that negative stiffness is generally accompanied by nonlinearity, which may lead to undesirable nonlinear vibrations and unstable responses. Therefore, in this paper, the influence of the nonlinearity of negative stiffness, including weakening and strengthening nonlinear negative stiffness, on the performances of two representative TNSIDs with different configurations is investigated. The working mechanisms and optimal parameters of the TNSIDs with simplified linear negative stiffness are introduced firstly. Following that, based on the harmonic balance method, the dynamic responses of the two TNSIDs with different nonlinear negative stiffness are attained. The unstable problems and performance degradation caused by the nonlinearity of negative stiffness are analysed. Parametric studies are further carried out to evaluate the effects of the nonlinear coefficient, excitation amplitude, internal frequency ratio, and damping ratio on the performances of the two nonlinear TNSIDs. Moreover, to eliminate the impact of the negative stiffness nonlinearity, a new structural parameter optimization is conducted by minimizing the maximum displacement amplitude-frequency response of the TNSIDs. The results indicate that the nonlinearity of negative stiffness could lead to the bend and jump of the displacement amplitude-frequency response curves of TNSIDs, as well as superharmonic resonances, especially when the performance of TNSIDs is sensitive to the variation of negative stiffness. Besides, the unstable areas and the maximum displacement amplitude-frequency responses of TNSIDs are dependent on the excitation amplitude and nonlinear coefficient. To avoid the unstable vibration phenomenon and maintain effective vibration control performance, TNSID-P2 is recommended to be equipped with a weakening negative stiffness under both moderate and strong excitations, while TNSID-P1 should be designed with a weakening negative stiffness and work under a moderate excitation due to its performance being highly sensitive to the variation of negative stiffness.