# New Optimal Design of Multimode Shunt-Damping Circuits for Enhanced Vibration Control
## 1 Introduction  
For many years, there has been growing scientific interest in the use of piezoelectric materials for active or passive vibration control. Passive vibration control of structures can be achieved by using shunted piezoelectric elements connected to passive electric circuits. The fundamental idea behind piezoelectric shunt damping is the transformation of mechanical energy into electrical energy by piezoelectric material, which is then dissipated by heating through a resistor circuit. To efficiently suppress the vibration of one or more vibration modes, several shunt circuit layouts have been developed (see, e.g., [1, 2]). [PERSON] was the first to introduce the concept of piezoelectric shunt damping in his work [3], experimentally demonstrating the feasibility of employing external electrical circuits to damp mechanical vibrations in optical systems. However, the pioneering work of [PERSON] and [PERSON] [4] provided the theoretical foundation of the resistive shunt (R shunt) and resonant shunt (RL shunt) techniques. In the latter technique, the shunting circuit is tuned to the resonance frequency of the mode to be damped in a fashion similar to the technique used to tune a mechanical vibration absorber. Since then, several studies have been performed on the passive vibration control of a single structural mode [5, 6, 7]. However, in practical applications, single-mode damping is insufficient for flexible mechanical structures containing an infinite number of structural modes (or resonant frequencies). A method for multimode shunt damping is to use as many piezoelectric transducers as the number of vibration modes to be damped. Recently, a new technique for multimode vibration control using multiple transducers was described in [8]. It is a design of an electrical network which is connected to a structure through an array of piezoelectric transducers. The characteristics of this network are specified in terms of modal properties. Additionally, a new approach for multi-resonant, flexural, vibration-control produced by broadband stochastic excitations was provided in [9]. The study used multiple elementary shunts connected to piezoelectric patches bound to thin plate structures. A design optimization procedure was proposed, which focuses on finding shunt networks, which can be built using small-size RLC components and, thus, embedded directly on the piezoelectric patch transducer. However, in many circumstances, this may not be a viable solution because, for damping many modes, a large number of transducers is required. This has urged researchers to develop multimode shunt-damping circuits that require only one piezoelectric transducer. First, [PERSON] [10] provided a multimode shunt-damping circuit using a single piezoelectric element by extended single-mode shunt damping. A shunt circuit consists of parallel \(RLC\) shunt branches, with the very first branch being an RL circuit. The proposed multimode dampener requires as many parallel branches as modes to be damped. No closed-form tuning solution was proposed for this technique, and the author proposed a numerical optimization to determine optimal component values. This multimode shunt circuit was applied to a cantilevered beam and was demonstrated experimentally as a two-mode dampener. [PERSON] [11] proposed another piezoelectric shunt circuit for multimode damping which employs a _blocking_ circuit in series with a parallel resistor-inductor shunt circuit for each mode to be damped. Closed-form tuning solutions were provided for the inductive and capacitive components of the proposed circuit. Optimal resistance values for this circuit were provided in [12] by a systematic \(H_{2}\) optimization approach. However, the complexity and order of current-blocking circuits limit their application to a maximum of three modes even in their simplest version. An alternative method for multimode piezoelectric shunt damping using a single piezoelectric element was provided by [PERSON] and coauthors [13, 14, 15].