Exact Solutions to H∞ and H2 Optimizations of Passive Resonant Shunt Circuit for Electromagnetic or Piezoelectric Shunt Damper

Abstract

In this work, exact closed-form solutions are derived for optimizing the resonant shunt circuits of electromagnetic shunt dampers (EMSDs), which use an electromagnetic transducer, and piezoelectric shunt dampers (PZSDs), which use a piezoelectric element, shunted with an electric circuit. Modeling of the EMSD and PZSD is unified by nondimensional parameters. The optimization criteria selected for the EMSD and PZSD are H∞-norm minimization, H2-norm minimization, and exponential time-decay rate maximization. The aim of this study is to derive for the first time the exact solutions that have not previously been investigated, including cases that consider the inherent damping of the primary system. This paper comprehensively summarizes the exact solutions based on the optimization criteria together with approximated solutions obtained by the fixed-point method, which is commonly used to optimize the dynamic vibration absorber (DVA).

Multimodal vibration damping using a simplified current blocking shunt circuit

“…As shown by Ikegame et al (2019), this problem is equivalent to a piezoelectric transducer shunted with a parallel RL shunt. The optimal formulae are then obtained by first computing the series of coefficients and finally and the optimal inductance L and resistance R are given by and respectively; see Tang et al (2016); Ikegame et al (2019) for details.…”

“…The analytical H ∞ solution was found by Tang et al (2016) for an electromagnetic transducer shunted with a series RC circuit. As shown by Ikegame et al (2019), this problem is equivalent to a piezoelectric transducer shunted with a parallel RL shunt. The optimal formulae are then obtained by first computing the series of coefficients…”

“…The exact H ∞ -optimal solution corresponding to the minimization of the maximum amplitude of the frequency response function (FRF) of a single degree-of-freedom (SDOF) oscillator was found by Soltani et al (2014) for the series RL shunt circuit. It was derived for parallel RL shunt circuits by Ikegame et al (2019), by noting the analogy of this problem with series RC shunts for electromagnetic transducers (Tang et al, 2016).…”

A digital nonlinear piezoelectric tuned vibration absorber

“…The shunt circuit parameters can be tuned in a number of ways according to the objectives at hand, see, e.g., the recent paper by Ikegame et al [4]. Because this study considers harmonic forcing, the H ∞ -optimum tuning given by Soltani et al [3] is adopted.…”

“…This strategy results in the so-called equal-peak design, because the frequency response function of the controlled structure exhibit two resonance peaks of equal amplitude. A recent paper by Ikegame et al [4] gathers tuning formulas for usual configurations and objectives at hand. Gripp and Rade [5] reviewed a number of works and applications of piezoelectric shunt damping.…”