Shock and Vibration

Volume 2016 (2016), Article ID 9739217, 18 pages

http://dx.doi.org/10.1155/2016/9739217

## The Behaviour of Mistuned Piezoelectric Shunt Systems and Its Estimation

^{1}Department of Industrial Engineering, Università degli Studi di Parma, Parco Area delle Scienze 181/A, 43124 Parma, Italy^{2}Department of Mechanical Engineering, Politecnico di Milano, Via La Masa 34, 20156 Milan, Italy

Received 20 May 2016; Revised 18 July 2016; Accepted 26 July 2016

Academic Editor: Nicola Caterino

Copyright © 2016 M. Berardengo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper addresses monoharmonic vibration attenuation using piezoelectric transducers shunted with electric impedances consisting of a resistance and an inductance in series. This type of vibration attenuation has several advantages but suffers from problems related to possible mistuning. In fact, when either the mechanical system to be controlled or the shunt electric impedance undergoes a change in their dynamical features, the attenuation performance decreases significantly. This paper describes the influence of biases in the electric impedance parameters on the attenuation provided by the shunt and proposes an approximated model for a rapid prediction of the vibration damping performance in mistuned situations. The analytical and numerical results achieved within the paper are validated using experimental tests on two different test structures.

#### 1. Introduction

Vibration attenuation in light structures is a widely studied topic and often takes advantage of the use of smart materials, which are characterised by useful properties. Indeed, these materials are inexpensive when compared to other control systems, and they are characterised by low weight. This last feature is a fundamental aspect because it avoids introducing high load effects on the controlled structure. Among smart materials, piezoelectric elements (particularly piezoelectric laminates, which are used in this paper) are among the best materials to attenuate vibrations in bidimensional (e.g., plates) and monodimensional (e.g., beams) structures [1–3]. There are several control techniques for light structures that rely on this type of actuator, and one of the most attractive is the shunt of the piezoelectric element. In this case, a properly designed electrical network is shunted to the piezoelectric bender bonded to the structure. The ability of the piezoelectric element to convert mechanical energy into electrical energy and vice versa [4, 5] is used, which allows a passive attenuation of the structure’s vibration. This method was initially proposed by Hagood and von Flotow [4]. This technique is extremely attractive because it is cheap, it does not introduce energy into the system, that is, it cannot lead to instability, and it does not require any feedback signal.

When a monoharmonic control is required, the most effective shunt electric impedance consists of a resistance and an inductance in series [2, 4, 6–8] (resonant shunt or* RL* shunt). These two elements, along with the capacitance of the piezoelectric actuator (i.e., the piezoelectric actuator is modelled here as a capacitance and a strain-induced voltage generator in series; see Section 2), constitute a resonant circuit, which is the electric equivalent of the mechanical tuned mass damper (TMD) [2]. Therefore, this circuit is able to damp the structural vibration corresponding to a given eigenmode as soon as its dynamic features are tuned to those of the vibrating structure.

There are several methods in the literature that explain how to select the values of and to optimise the vibration attenuation. Hagood and von Flotow [4] proposed two different tuning strategies based on considerations on the shape of the system transfer function and on the pole placement techniques for an undamped structure. Both these tuning methods are based on the classical TMD theory. Høgsberg and Krenk [9, 10] developed another calibration method based on the pole placement for* RL* circuits in series and parallel. The values of and are selected to guarantee equal modal damping of the two modes of the electromechanical structure and good separation of the complex poles. Thomas et al. [11] proposed two different methods, even for damped structures, that relied on the transfer function criteria and pole placement and provided closed formulas to estimate the attenuation performance.

Although all of the mentioned tuning strategies work extremely well, one significant issue of shunt damping using* RL* impedances is that this type of electrical circuit is not adaptive. This in turn means that it is not possible to follow possible changes in the dynamic behaviour of either the vibrating structure (e.g., a temperature shift can change the eigenfrequency of the mode to be controlled) or the impedance itself (e.g., a temperature shift can cause a significant change in the value [12]). Hence, this control technique often works in mistuned conditions, even when starting from a perfect tuning condition. This mistuning leads to severe worsening of the attenuation performance.

A few techniques based on adaptive circuits were proposed to overcome the limitations due to uncertainties in the mechanical and electrical quantities. Based on the single-mode control, Hollkamp and Starchville developed a self-tuning* RL* circuit that was able to follow any change in the frequency of the mode to be controlled [13]. This technique is based on a synthetic circuit (which provides both the resistance and the inductance) consisting of two operational amplifiers and a motorised potentiometer. Despite its effectiveness, this method only considers a mistuning due to a change in the eigenfrequency to be controlled and does not consider other types of changes or uncertainties, such as ones related to electrical parameters. Furthermore, this method is active, thus losing the advantage provided by the passive shunt technique. Other recent studies by Zhou et al. [14, 15] attempted to determine methods to limit the problem of mistuning by using nonlinear elements when the disturbance was harmonic and using more than one piezoelectric actuator bonded to the vibrating structure. Although these techniques can be effectively employed, their use implies the loss of the two primary features of the resonant piezoelectric shunt: linearity (and thus ease of use) and passivity. Therefore, the analysis of the performances of traditional* RL* shunts in mistuned conditions still has significant relevance.

Although the problems related to mistuning are evidenced in literature [16–18], there have been few analyses on shunt robustness. These analyses are of significant interest for numerous engineering applications where electrical power is often limited or even avoided, thus preventing the use of adaptation systems for the shunt impedance (e.g., space applications). In situations where passivity is requested, it is important to analyse the behaviour of the shunted system in the presence of mistuning because it worsens the attenuation performance. Recently, Berardengo et al. [19] studied the robustness of different optimisation methods for* RL* circuits and determined the most robust method. Based on the outcomes of [19], this paper aims to further investigate the robustness of* RL* shunt damping. The word robustness is intended here as the capability of the shunt impedance to attenuate the vibrations even when in mistuned conditions. Therefore, this paper analyses the behaviour of mistuned electromechanical systems, thus depicting the relationship between the attenuation and the system parameters (e.g., coupling coefficient, mechanical nondimensional damping ratio, and eigenfrequency) in tuned and mistuned conditions. Furthermore, this paper demonstrates that the loss of attenuation primarily depends on only one bias (i.e., either the bias on the damping or the eigenfrequency of the electric resonant circuit) if the electrical damping is overestimated, whereas the effects of the two bias types (on the electrical eigenfrequency and damping) combine with each other when the electrical damping is underestimated. Based on these results, an approximated analytical model is proposed to estimate the attenuation performance with different amounts of mistuning using a small number of numerical simulations.

To summarise, the goals of this paper are to investigate how mistuned systems (which are often encountered in real applications) behave and consequently propose an approximated model that is able to predict the behaviour of the mistuned system with the least amount of numerical simulations. To reach the above goal, the authors highlight the relationship between the attenuation and all of the problem parameters and demonstrate that some of these relations can be approximated linearly in a logarithmic scale. Moreover, the authors bring to evidence the cases where the loss of performance depends on just one mistuning type (i.e., either the bias on the damping or the eigenfrequency of the electric resonant circuit), even though mistuning occurs on both, as well as the cases where both the mistuning types have an influence. All of these observations allow for the development of the mentioned approximated model for mistuned systems, which enhances the knowledge of their behaviour. Moreover, using this new simplified model, the authors demonstrate that an initially overestimated value of is able to decrease the loss in performance due to mistuning and explain why this phenomenon occurs. Additionally, this allows for guidelines to be provided on how to tune the shunt parameters when a mistuning is expected.

This paper is structured as follows. Section 2 discusses the model of the electromechanical system used in this paper. Section 3 highlights the linear relationship between the attenuation and the system parameters, which will be employed in Section 4 to analyse the effects of mistuning and propose an approximated model to describe the attenuation performance in the presence of mistuning. Lastly, Section 5 validates the previous results using experiments.

#### 2. Model of the Electromechanical System

As mentioned in the previous section, the goal of this paper is to study the vibration attenuation of the controlled system in mistuned conditions. Thus, the most intuitive and used index to evaluate the attenuation performance is the ratio between the maximum of the dynamic amplification modulus in uncontrolled and controlled conditions [11, 19]. Therefore, for the performance analysis, it is necessary to derive the expression of the frequency response function of the electromechanical system and thus to introduce the model used to describe its electrodynamic behaviour.

The piezoelectric actuator is modelled here as a capacitance and a strain-induced voltage generator in series (Figure 1(a)). The induced voltage is , whereas the voltage between the electrodes of the piezoelectric bender is . is equal to when the piezoelectric actuator is open-circuited and null when the actuator is short-circuited. takes different values when an impedance is shunted to the electrodes of the actuator (Figure 1(b)) because a current flows in the circuit. Moheimani et al. [20, 21] proved that systems controlled by piezoelectric actuators shunted with electric impedances can be modelled as a double feedback loop (Figure 2(a)). The inner loop of Figure 2(a) can be observed as a controller , which can be expressed in the Laplace domain as follows:where is the Laplace variable.