The reduction of operational amplifier electrical outputs to improve piezoelectric shunts with negative capacitance

https://doi.org/10.1016/j.jsv.2021.116163Get rights and content

Highlights

  • Analysis of OP-AMP outputs in negative capacitance circuit for piezoelectric shunt.

  • Methods for decreasing OP-AMP output without deteriorating attenuation.

  • Comparison between different negative capacitance layouts.

  • Guidelines for designing and building negative capacitances for piezoelectric shunt.

Abstract

One way to enhance the performance of vibration control with piezoelectric shunt is to use a negative capacitance in the shunt circuit. This component is very effective and provides good results in terms of attenuation improvement without significantly increasing the complexity of the shunt network. However, negative capacitances are built using operational amplifiers and, in some applications, the risk of saturation of the outputs of the operational amplifier exists. This constitutes a non-negligible aspect since it leads to a non-proper functioning of the control system which significantly deteriorates the control performance or even triggers instability phenomena. In light of this limitation, this paper proposes strategies to decrease the outputs of the operational amplifier in order to reduce the risk of saturation acting just on the values of the circuit components, without worsening the attenuation performance. However, when the achievable reduction is not sufficient, it is also possible to act on other components accepting a deterioration of the attenuation performance. Guidelines are provided for properly choosing the best shunt circuit configuration accounting for both the extent of the operational amplifier outputs and the control performance. The paper also evidences that the mechanical part of the system cannot be neglected in the analysis when assessing the operational amplifier outputs. Furthermore, two different circuit types used to build the negative capacitance are compared in terms of output requirements. This analysis shows that there is no circuit always less demanding than the other and that the choice of the circuit is not always straightforward. Therefore, a multi-degree of freedom model is presented, which is essential to understand which configuration of the negative capacitance has to be used in a given engineering application. All the presented outcomes are validated through an experimental campaign.

Introduction

The use of piezoelectric shunt for attenuating vibrations in light structures is a well-known technique where piezoelectric transducers act at the same time as both sensors and actuators, and which has been extensively studied and employed in engineering applications (e.g. hard-disk vibrations [1], vibrations of turbine blades [2,3]). Piezoelectric shunt usually consists in the electrical connection between a piezoelectric transducer bonded to the vibrating structure and a properly designed passive electrical network [4], [5], [6], [7], [8]. According to the layout of the electrical network, it is possible to perform mono-modal control (e.g. [9], [10], [11], [12], [13], [14]) or multi-mode control with either single (e.g. [15], [16], [17], [18], [19]) or multiple piezoelectric transducers (e.g. [20,21]). Hence, in the classical approach, the piezoelectric shunt damping is a passive technique and does not imply the use of either expensive electronic devices or real-time controllers. On the one hand, this represents an advantage making the control system inexpensive, stable and easy to be implemented. On the other hand, the power involved is limited and the control performances are lower if compared to the traditional active control strategies [22].

An effective approach for improving the control performance of the piezoelectric shunt is the addition of synthetic circuits in the shunt impedance (e.g. [23], [24], [25], [26], [27], [28], [29]), even in a context of periodic structures (e.g. [30,31]). This is a technique often employed also in other smart control approaches like, as an example, electro-magnetic shunt (e.g. [32], [33], [34]). In all these cases, the use of synthetic circuits showed to provide significant improvement in terms of vibration attenuation performance (e.g. [35]). However, in these cases, due to the presence in the circuit of components which have to be supplied, the approach becomes semi-active.

Among the approaches which rely on additional synthetic circuits in piezoelectric shunt, the use of negative capacitances (NC) has been shown to be reliable and effective. The addition of NCs proved to be able to artificially increase the modal electro-mechanical coupling factor for all the modes of a system. This coupling factor is one of the main parameters affecting the control performance since it is representative of the efficiency of the conversion between mechanical and electrical energy (i.e. the increase of the absolute value of the modal electro-mechanical coupling factor related to the ith mode, ki, increases the achievable attenuation level on the corresponding mode) [6,36]. For this reason, NCs can be successfully applied to improve the effectiveness of any type of passive shunt circuit. In this scenario, their use becomes attractive especially when they are coupled to simple shunt impedances, allowing for both good attenuation levels and easy-to-implement systems at the same time. Even if the NCs can be coupled to any passive shunt impedance, many works in the literature have focused on the coupling of NCs to resistive (e.g. [22,[37], [38], [39], [40], [41]]) and resonant shunt impedances (e.g. [22,[42], [43], [44]]). A resistive impedance is made from a single resistance, while a resonant impedance is made from either the parallel or the series of an inductance and a resistance. When no NCs are used, the tuned resonant shunt offers attenuation levels much higher than the resistive shunt, but, when NCs are added in the shunt circuit, the attenuation provided by the resistive shunt becomes closer and closer to that of the resonant shunt [45]. Former studies already proved that resonant shunt shows low robustness to system changes and uncertainties (e.g. [12,[46], [47], [48]]). Conversely, the resistive shunt coupled to NCs shows to be highly robust [45] and able to damp more than one mode at the same time [38,45,49].

Considering the aforementioned reasons, the piezoelectric shunt based on resistive impedances coupled to NCs is a very effective, reliable, inexpensive and easy-to-implement approach to damp vibrations. The main issue of this approach is related to possible dynamic instabilities of the electro-mechanical system (composed by the vibrating structure, the piezoelectric actuator and the shunt impedance) due to the active nature of the NCs. Indeed, since NCs do not exist in nature, they are implemented by using operational amplifiers (OP-AMP) [50] and this makes the control approach semi-active, as mentioned. This problem has been widely studied in the literature and different works provide the limits on the NC value to avoid instability (e.g. [22,42,49]). Nevertheless, there is also another important problem: the OP-AMPs can undergo to saturation of their outputs, thus leading to a non-proper functioning of the control system, and sometimes triggering instability phenomena [49,51]. This problem is not deepened in the literature but it is very important because it limits the applicability and the reliability of the piezoelectric shunt based on NCs.

The most intuitive approach to overcome the aforementioned problem is to use high power amplifiers, but this increases significantly the cost of the control system and, in case of an increase of the disturbance forces acting on the mechanical system, saturation can still occur. Another possibility is to use more complex circuits simulating NCs (e.g. [26]); even if the approach is effective, the complexity of the shunt circuit increases significantly. Therefore, different researchers have studied whether it is possible to decrease the output of the OP-AMP without changing the traditional layout of the NC circuit. Beck et al. [51] described how the different electric elements (i.e. resistances and capacitances) present in the NC circuit are able to change (i.e. increase/decrease) the OP-AMP voltage and power output as a function of the frequency in the case of an NC connected in series to the shunt resistance and the piezoelectric actuator. Qureshi et al. [52] proposed a similar study, but considering a resonant shunt coupled to an NC. Other related works are those of Václavíc et al., who studied the energy flow in applications related to vibration isolation using NCs connected in series to the shunt resistance and the piezoelectric actuator [53] and proposed the use of switching amplifers to decrease the electric power consumption of the shunt control [54]. Even if the referenced works provided a significant insight of the problem, there are still open points to be addressed.

Under a general point of view, the main target of most of the referenced works is to study the dependence of the OP-AMP outputs on the values of the components of the NC circuit (i.e. resistances and capacitances) and the shunt impedance. On the one hand, this approach shows whether it is possible to act on the circuit parameters to reduce the OP-AMP outputs but, on the other hand, it loses sight of the main target of the piezoelectric shunt damping, that is the vibration attenuation. These studies, indeed, do not evidence either how a reduction of the OP-AMP output, achieved by changing the values of the circuit parameters, affects the control effectiveness or whether it is possible to achieve the reduction of the OP-AMP output mantaining a given control performance. Furthermore, the main focus of the analyses in the literature is on the effect of the electrical part of the electro-mechanical system on the OP-AMP outputs, while the effect of the dynamics of the mechanical system is not deepened.

In this scenario, this paper aims to fill some of these gaps in order to provide a more general overview of the problem. Particularly: (i) the electro-mechanical system will be considered not just under the electrical point of view but also the mechanical part will be taken into account in the analysis, showing how the dynamics of a multi-degree-of-freedom (MDOF) mechanical system is able to influence the OP-AMP output; (ii) an analysis aiming at investigating whether and how it is possible to decrease the demand on the OP-AMP outputs given a certain level of vibration reduction will be presented. It will be shown that some electric parameters can be tuned in order to decrease the OP-AMP outputs without changing the attenuation performance provided by the shunt. Furthermore, this analysis will be shown to be general and not dependent on the type of passive shunt impedance coupled to the NC. This analysis will also evidence that it is possible to act separately on the voltage and current outputs of the OP-AMP, with the consequent possibility to lower the power consumption of the NC circuit; (iii) without the constraint of maintaining constant the level of vibration attenuation, a further analysis will be performed to evidence the effect of the circuit parameters on both the OP-AMP outputs and the attenuation performance; (iv) it is well-known that there are different ways to connect the piezoelectric actuator to the NC and the resistance (the classical series and parallel connections will be taken into account here) and there are also different circuits to build NCs. The paper will analyse all these different cases, providing a detailed description of the OP-AMP outputs in the different configurations. This will allow for direct comparisons among the different solutions and will provide useful information about which configuration must be preferred in given practical cases; (v) based on the above points, the paper provides general guidelines on how to build an NC according to given targets in terms of both OP-AMP outputs and attenuation performance.

The paper is structured as follows: Section 2 will present the model of the electro-mechanical system used in this paper and the circuits adopted for building the NCs. Section 3 will explain how the OP-AMP outputs can be changed without changing the attenuation performance, and will discuss the different possible layouts of the shunt impedance. Section 4 will show how the mechanical behaviour of the electro-mechanical system influences the OP-AMP outputs. The results of this section will also allow to show the effects of the parameters able to change at the same time the OP-AMP outputs and the attenuation performance. Section 5 proposes an MDOF model of the electro-mechanical system, that is needed for assessing which NC layout should be preferred in a given specific engineering application. Section 6 provides guidelines on how to build the shunt impedance according to given targets in terms of both OP-AMP outputs and attenuation performance. Finally, Section 7 will present the experiments carried out to validate the previous theoretical results.

Section snippets

The model of the electro-mechanical system

The model used in this paper was originally developed by Thomas et al. [6,55] and Ducarne et al. [56], and then refined by Berardengo et al. [49]. Here, the model is briefly recalled for the sake of clarity in order to provide the basics needed for the comprehension of this work. The interested reader can find more details in the referenced articles.

A generic structure is excited by an external forcing F and a piezoelectric actuator, bonded to the structure, is shunted with an electric

The change of the OP-AMP outputs without changing the attenuation performance

According to Berardengo et al. [49], the attenuation performance is fixed as soon as the value of the NC (i.e. C1 or C2) and the value of the shunt resistance R are set. Therefore, in order to see whether it is possible to change the OP-AMP outputs without changing the performance of the control, the trend of the Mvo and Mio functions must be analysed keeping fixed the values of C1, C2 and R, which are thus treated as constants in this analysis.

In this case, a change of the OP-AMP outputs can

The effect of the NC and R values on the OP-AMP outputs

In section 3, attention was focused on how to change the OP-AMP outputs, given a certain attenuation performance. However, it is worth analysing also what occurs if this constraint is removed and, thus, studying the effect of the values of the NC and R on the OP-AMP voltage and current outputs. Indeed, this will allow to understand also how the vibration level of the electro-mechanical system affects the OP-AMP outputs.

As an example, suppose to change the value of R. On the one hand, this

The MDOF model

This subsection explains how to derive the mentioned MDOF model. The frequency range of the first n modes (with n<N) is taken into account. The n modes considered are those in the frequency range where the power of the disturbance F is significant. According to Section 2.1, the equations of motion are described by limiting Eq. (1) to the modes of interest:q¨i+2ξiωiq˙i+ωi2qiχiV=Fii{1,,n}Considering the electrical equation, it is not sufficient to limit the summation in Eq. (2) to the modes

Guidelines

This section provides some guidelines about how to build a shunt circuit based on an NC coupled to a resistance taking into consideration both the vibration attenuation and the OP-AMP outputs. As mentioned, the OP-AMP outputs are functions of many different parameters such as, as examples, the frequency band of the external disturbance, the value of τ used that depends on the type of control strategy, the values of the eigenvector components for the modes in the frequency range of the

Experimental tests

This section aims at validating most of the theoretical results discussed in the previous sections of the paper. Section 7.1 describes the set-up used for the tests, while Sections 7.2 and 7.3 discuss the experimental results for the series and parallel NC connections, respectively.

Conclusion

This paper has addressed the piezoelectric shunt damping improved by the use of NCs from the OP-AMP output point of view. Indeed, NCs are built employing OP-AMPs and the paper has shown how it is possible to decrease their outputs by keeping the same level of vibration attenuation. Since it is possible to act separately on the OP-AMP voltage and current outputs, it is also possible to lower the power consumption of the NC circuit by properly tuning the values of the circuit elements.

Author credit statement

M. Berardengo: Study conception and design, Data acquisition, Analysis and interpretation of data, Drafting of manuscript, Critical revision. S. Manzoni: Study conception and design, Data acquisition, Analysis and interpretation of data, Drafting of manuscript, Critical revision. O. Thomas: Study conception and design, Critical revision. C. Giraud-Audine: Study conception and design, Critical revision. L. Drago: Data acquisition, Analysis and interpretation of data. S. Marelli: Data

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This research has financially been supported at University of Parma by the Programme "FIL-Quota Incentivante" of University of Parma and co-sponsored by Fondazione Cariparma. Furthermore, the Italian Ministry of Education, University and Research is acknowledged by the staff of Politecnico di Milano for the support provided through the Project "Department of Excellence LIS4.0 - Lightweight and Smart Structures for Industry 4.0".

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