Dynamics of piezoelectric structures with geometric nonlinearities: A non-intrusive reduced order modelling strategy
Introduction
The dynamics of piezoelectric systems has been the focus of a large and still increasing amount of studies in the last few decades. Indeed, predicting the dynamics of piezoelectric systems is relevant for numerous applications, such as energy harvesting [1], [2], Micro-Electro-Mechanical Systems (MEMS) [3] and vibration reduction using shunt damping [4], [5] or active control [6]. More recently, a particular focus has been made on the nonlinear effects to improve the efficiency of electromechanical systems [7], [8], [9], [10], [11].
Two main sources of nonlinearities can influence the dynamical behaviour of thin piezoelectric structures. Firstly, the geometric nonlinearities arise in the case of large transverse displacements, creating a significant stretching in the plane of the structure known as the bending-membrane coupling phenomenon [12], [13], [14], [15], [16]. More complex nonlinearities, that stem from the large rotations of the structure cross-sections, can also be observed [17], [18]. Secondly, material nonlinearities can occur when piezoelectric patches are subjected to high electrical drives, an effect that was investigated on several occasions (see for instance [19], [20], [21], [22], [23]). The present paper is devoted to geometrical nonlinearities only, identified as the predominant ones in some experimental investigations on MEMS with piezoelectric transduction [24], [25], [26].
Piezoelectric structures in real-life applications can present complex geometries, often laminated with several layers of different materials [27], [28], [29]. Thus, their modelling often relies on the use of a finite element code, which requires a specific electromechanically coupled variational formulation. Since the pioneering proposition in the 70’s [30], extensive studies were devoted to the development of piezoelectric finite elements (see among others [31], [32], or the reviews [33], [34]). Indeed, a large variety of laminated thin-walled piezoelectric elements (beams, plates, shells) with specific kinematics have been introduced (see [35], [36], [37] among others). However, these coupled electromechanical formulations are rarely included in commercial finite element codes, in particular when geometrically nonlinear problems are addressed. This justifies the 3D formulation developed in this paper, in order to take a broad-scope approach without restraining to specific structural finite elements. Moreover, such an approach can be addressed with standard commercial finite element codes including both piezoelectric coupling effects and geometrically nonlinear capabilities [38], [39].
In order to connect the piezoelectric structure to an electronic circuit, it is convenient to consider global electric variables on the piezoelectric patches (voltage and current/charge), instead of the standard local electric variables used in most finite element formulations. It can be performed thanks to a condensation of the electric degrees of freedom internal to the piezoelectric patches, and by exploiting the equipotentiality properties of the electrodes [40]. It is also possible to express explicitly the global electric variables if the piezoelectric patches are assumed to be thin [41]. Such procedures were initially proposed for linear structures and are here extended to the case of geometrical nonlinearities.
By using direct time integration, simulating the dynamics of a full finite element model presents a high computational cost, even in the linear case [42]. To reduce the number of variables without losing accuracy, reduced-order modelling based on a modal approach was proposed at several occasions. In this case, the linear modal piezoelectric coupling is expressed with only one coefficient per mode, called the modal Electro-Mechanical Coupling Factor (EMCF), identified as the key parameter of the reduced-order modelling procedure [41], [43], [44], [45]. To estimate this parameter, the more common method relies on the estimation of an effective EMCF obtained thanks to the natural frequencies with the piezoelectric patches in short and in open circuits [46]. However, this method is based on a modal truncation valid if natural frequencies are widely spaced. Another method to compute the EMCF without restrictive assumption has been recently proposed, by computing the global electric charge with short-circuited electrodes in reaction to the modal displacement [47].
In the geometrically nonlinear case, reduced-order models (ROMs) have also been the focus of intensive studies for elastic structures in the past two decades [14]. A natural strategy is a simple expansion on a truncated linear modal basis, with the ROM coefficients efficiently estimated by an enforced displacement method (the so-called STiffness Evaluation Procedure (STEP) [12]). However, the truncation of the modal basis is an issue since a large number of eigenmodes is often required in the basis to obtain converged dynamic solutions. In particular, the coupling between low frequency bending modes and high frequency modes associated with in-plane [48], [15] or thickness [49] motions must be taken into account. Many strategies were proposed to overcome this issue, among which: static condensation methods [50], [49], modal derivatives and quadratic manifolds [51], [52], dual modes [53], spectral submanifolds [54] or invariant manifolds [55], [56], [57]. A series of recent studies compare those methods and promote the use of normal form reductions [58], [59], [56], [57], [60]. Most of those methods are non-intrusive since they are based on the direct use of output results of a commercial finite element code equipped with geometrically nonlinear capabilities, as opposed to direct methods which require computations implemented at the level of the element [61].
The focus of the present study is to address both geometric nonlinearities and piezoelectricity, for ROMs obtained with a non-intrusive approach. This has been addressed only at three occasions in the literature, to the knowledge of the authors. A first study was based on a home-made finite element code with beam elements [48]. The present paper is a natural extension of this investigation, additionally allowing for the predictive modelling of arbitrary geometries. The second one proposes a method relying on the use of a commercial finite element code, by founding on a thermoelastic analogy [62]. This has the advantage of relying on a convenient non-intrusive approach. However, this analogy only partially accounts for the three-dimensional and the reversibility nature of the piezoelectric constitutive law. Finally, a recent contribution [63] proposes simulations of the resonant nonlinear behaviour of micromirrors under piezoelectric actuation, including nonlinear geometrical effects. A home made finite element code is used with the harmonic balance method, without model order reduction. The combined effects of both piezoelectric and geometrically nonlinear effects results potentially in parametric resonances. Thus, their accurate modelling can lead to efficient predictive simulations of nonlinear vibrations with parametric excitation. This phenomenon was addressed in a large number of studies in the MEMS community (see [25] and references therein) for increasing the quality factor of thin piezoelectric MEMS for which strong nonlinear responses were obtained.
In this overall context, for an elastic structure with piezoelectric elements, the present paper focuses on four main aspects. The first one is concerned with a generic formulation of a 3D finite element (FE) discretized model, that includes both geometric nonlinearities and electromechanical coupling, with global electric variables (voltage and electrode charges). We address secondly the associated modal model, which is reduced around a given resonance by static condensation of the higher order slave modes. Then, the modified STEP (M-STEP) method, introduced in [49] for elastic structures, is extended to a piezoelectric structure, to efficiently compute the electromechanical coefficients of the ROM, in the case of a symmetric structure in the transverse direction. Finally, the ROM is validated by comparison to analytical solutions and full 3D FE computations, thanks to dynamic responses in the frequency domain. One of the main originality is the emergence of special terms in the formulation, due to the conjugate effect of piezoelectricity and geometrical nonlinearities, and physically responsible for parametric excitation and instabilities of thin symmetric structures. Another originality is the systematic consideration of both converse (actuation) and direct (detection) piezoelectric effects.
Section snippets
Notations and assumptions
A piezoelectric structure is considered in its reference configuration as defined in Fig. 1. It occupies a domain , of boundary , of the three-dimensional Euclidean space. It is subjected to a prescribed displacements on a part of its boundary , whereas a surface force density is applied on the complementary part . Note that subscripts denote the three-dimensional vector and tensor components and repeated subscripts imply summation, according to standard indicial
Identification of the modal model
This section focuses on the computation of the coefficients of Eqs. (12). The linear parameters and are directly computed by the modal analysis of Eq. (11), available in any finite element code. We focus here on the computation of the coefficients and and the capacitances . Since the three main corresponding effects are additively written, we propose here to compute these different coefficients separately. Indeed, if short-circuited piezoelectric patches are
The case of flat structures
The procedure of the previous section can be applied to compute the coefficients related to any modal basis, for any geometry of the electromechanical structure. However, a large number N of eigenmodes can be required in the basis to obtain converged dynamic solutions, especially when 3D finite elements are used, because of the coupling between low frequency bending modes and high frequency modes associated with in-plane [48], [15] or thickness [49] motions. As reported in the introduction,
Validation test cases
In this section, two test cases are presented. They consist in piezoelectric symmetric beams, with geometries chosen such that they can be investigated with the analytical model of E. In the first example, a hinged-hinged beam with very thin piezoelectric patches is discretized with a refined mesh to compare the ROM coefficients computed with the method of Sections 3 Identification of the modal model, 4.2 The modified STEP (M-STEP) to analytical values. The advantage of the hinged-hinged
Dynamic response computation of a realistic structure
The previous section aimed at computing the dynamic response of two ideal test cases, in order to provide a numerical validation of the methodology. In this section, we apply the same approach to a more realistic structure: a clamped–clamped bimorph, shown in Fig. 12. It is composed of an elastic central layer of length , width and thickness and two piezoelectric patches of length , width and thickness , glued at an axial location defined by . The width of the patches is chosen
Conclusion
In this work, we presented a complete and efficient strategy to compute the nonlinear free and forced frequency responses of thin structures with piezoelectric patches, subjected to geometrical nonlinearities, using a reduced order model (ROM) and a continuation method to obtain the periodic responses and their stability. The ROM includes an efficient coupling to any electrical circuit, since the electrical degrees of freedom are global (the voltages between the electrodes of the piezoelectric
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.
Acknowledgements
The authors acknowledge the Arts et Métiers High Performance Computing Center Cassiopee with which the Abaqus time integrations were performed. The French Ministry of Research is thanked for the financial support of this study, through the Ph.D. Grant of the first author.
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