Closed-form solutions for the optimal parameters of three inerter-enhanced dampers (IEDs) equipped on a ground acceleration-excited structure

Abstract Replacing the viscous damper of tuned mass damper (TMD) with the proposed inerter-enhanced dampers (IEDs), novel vibration mitigation methods, namely the IED-TMDs, are proposed. Unlike the TMD, which brings only one additional freedom into the system, the proposed IED-TMDs introduce more freedoms into the considered dynamic system. As a result, the traditional fixed-point theory cannot be used. To address this issue, this paper develops an extended fixed-point theory. Firstly, the inerter and the springs of the IED-TMDs are optimized considering that all four fixed points are of the same height. The closed-form solutions for the optimal inerter and springs of the IED-TMDs are obtained. Secondly, to obtain the optimal damping ratio for the IED-TMDs with multi-fixed points, a new optimization criterion is introduced. Different from the traditional fixed-point theory which controls the slope of the transfer function at the fixed points, the new optimization criterion assumes that the local peaks of the transfer function in between the four fixed points have the same height as the fixed points. And, a flat plateau is achieved in the transfer function. Further, the closed-form solutions for the optimal damping ratio are simplified in consideration of actual applications. Finally, the vibration mitigation performance of the IED-TMDs is evaluated. Results show that the vibration mitigation performance of IED-TMDs is superior to that of the conventional TMD. This superior vibration mitigation performance is more significant for the IED-TMDs with a smaller mass ratio.