Analytical H_∞ and H₂ optimization for negative-stiffness inerter-based systems

Two novel dynamic vibration absorbers containing inerter and negative stiffness (NI-DVA) are presented to reduce the vibration of a primary system which is subjected to the acceleration excitation from the base. The NI-DVAs which are called N-TID and N-TVMD, are composed of a parallel inerter and negative stiffness spring that connected with a spring and a dashpot in different layouts. The analytical expressions of optimal stiffness ratio, damping ratio and negative stiffness ratio for both H_∞ and H₂ optimization design are derived based on the fixed-points theory and extreme value theory under the Routh-Hurwitz stability condition. These analytically optimal design parameters of NI-DVAs are expressed as functions of the inertance mass ratio. The effectiveness of the analytical solutions of optimal parameters for two NI-DVAs is verified by comparing them with the results obtained from the numerical searching optimization method. The vibration reduction performance of the negative stiffness element in the N-TID and N-TVMD is evaluated by comparing it with TID and TVMD harmonic excitation, white noise, and non-stationary earthquake waves. The numerical studies suggest that the presence of the negative stiffness element in NI-DVAs possesses superior vibration reduction performance compared with their original counterparts. Though higher improvement in vibration reduction performance is observed from N-TID due to the existence of a negative stiffness element, N-TVMD outperforms N-TID in terms of vibration reduction under both harmonic and random acceleration excitation.