Analytical Solutions to H₂ and H_∞ Optimizations of Resonant Shunted Electromagnetic Tuned Mass Damper and Vibration Energy Harvester

When optimized, tuned mass dampers (TMDs) can effectively mitigate the vibration of the primary structure, because additional resonance and damping are introduced by the auxiliary mass-spring-damper system. Similar effect can be realized without auxiliary mass when an electromagnetic transducer shunt with the R-L-C resonant circuit is placed between the primary structure and the base. This paper is to analytically optimize the parameters of the R-L-C circuits for vibration mitigation. Both $H2$ and $H∞$ optimization criteria are investigated, which are to minimize the root-mean-square (RMS) vibration under random excitation and the peak magnitude in the frequency domain, respectively. The concise closed-form solutions of the optimal parameters are then summarized together with the ones obtained the using fixed-point method, for practical implementation convenience. The $H2$ and $H∞$ optimizations of energy harvesting are also discussed in this paper. Furthermore, we also investigate the sensitivity of system performances to the tuning parameter changes of the electromagnetic shunt circuit.