Jean-François Deü

A two-dimensional closed-form solution for the free-vibrations analysis of piezoelectric sandwich plates

This work presents a two-dimensional (2D) closed-form solution for the free-vibrations analysis of simply-supported piezoelectric sandwich plates. It has the originality to consider all components of the electric field and displacement, thus satisfying exactly the electric equilibrium equation. Besides, the formulation considers full layerwise first-order shear deformation theory and through-thickness quadratic electric potential. Its independent mechanical and electric variables are decomposed using Fourier series expansions, then substituted in the derived mechanical and electric 2D equations of motion. The resulting eigenvalue system is then condensed so that only nine mechanical unknowns are retained. After its validation on single- and three-layer piezoelectric, and hybrid sandwich plates, the present approach was then used to analyze thickness modes of a square sandwich plate with piezoceramic faces and elastic cross-ply composite core. It was found that only the first three thickness modes are global, thus can be modeled by the mixed equivalent single-layer/layerwise approach, often retained in the literature; the remaining higher thickness modes being characteristic of sandwich behavior; i.e., dominated by the deformations of either the core or the faces. These results, together with presented through-thickness variations of the mechanical and electric variables clearly recommend full layerwise modeling. Several numerical results are provided for future reference for validation of 2D approximate analytical or numerical approaches; in particular, of 2D piezoelectric adaptive finite elements.

Piezoelectric Transverse Shear Actuation and Sensing of Plates, Part 1: A Three-Dimensional Mixed State Space Formulation:

This paper, in two parts, presents in this first part an exact three-dimensional solution of transverse shear actuation and sensing of simply-supported piezoelectric laminated plates. For this, the piezoelectric laminae are considered initially polarized along their first materialaxis (x1) and subjected to through-thickness electric field. Hence, transverse shear strains can be induced or measured under electric or mechanical loading, respectively. The formulation is based on a new mixed state space approach that retains the standard state mechanical displacement and transverse stress variables augmented with the electric potential and transverse displacement. The latter are condensed analytically at the piezoelectric lamina transfer matrix level. Thus, the solution of the assembled system is reduced to that of only a third-order one. Both open- and closed-circuit electric boundary conditions are considered for sensing and actuation problems, respectively. Corresponding computational procedures are also detailed. In part two of the paper, the present approach is applied to transverse actuation and sensing of sandwich plates. Parametric studies are also conducted for physical understanding of these mechanisms. Presented analytical results are useful for future reference and comparison with other approximate laminate electromechanics.

Structural Vibration Reduction by Switch Shunting of Piezoelectric Elements: Modeling and Optimization

This work deals with the reduction of structural vibrations by means of synchronized switch damping techniques on piezoelectric elements. Piezoelectric patches are attached to the vibrating structure and connected to an electrical circuit that includes a switch. The latter allows to continuously switch the piezoelectric elements from an open-circuit state to a specific electric impedance, synchronously with the mechanical oscillations. The present study focuses on two goals: (i) the quantification of the added damping, (ii) the optimization of the electric circuit parameters, carried out on a one degree of freedom model. The free and forced responses of one mode of the mechanical structure are studied in detail. The precise time response of the system is obtained with semi-analytical models for the two cases where the electrical impedance is a simple resistance (synchronized switch damping on short circuit) or a resistance in series with an inductance (synchronized switch damping on inductor). The damping added by the device is estimated. In all cases, the main result of the study is that the piezoelectric coupling factor is the only parameter to optimize and has to be maximized in order to maximize the added damping. An optimal value of the electric circuit quality factor is obtained when using an inductance, for free and forced response.

Structural-Acoustic Vibration Reduction Using Switched Shunt Piezoelectric Patches: A Finite Element Analysis

In this paper, we present a finite element formulation for vibration reduction in structural-acoustic systems using passive or semipassive shunt techniques. The coupled system consists of an elastic structure (with surface-mounted piezoelectric patches) filled with an inviscid linear acoustic fluid. An appropriate finite element formulation is derived. Numerical results for an elastic plate coupled to a parallelipedic air-filled interior acoustic cavity are presented, showing the performances of both the inductive shunt and the synchronized switch shunt techniques.

Multimodal vibration damping of a beam with a periodic array of piezoelectric patches connected to a passive electrical network

A multimodal damping strategy is implemented by coupling a beam to its analogue electrical network. This network comes from the direct electromechanical analogy applied to a transverse lattice of point masses that represents the discrete model of a beam. The mechanical and electrical structures are connected together through an array of piezoelectric patches. A discrete and a semi-continuous model are proposed to describe the piezoelectric coupling. Both are based on the transfer matrix formulation and consider a finite number of patches. It is shown that a simple coupling condition gives a network that approximates the modal properties of the beam. A multimodal tuned mass effect is then obtained and a wide-band damping is introduced by choosing a suitable positioning for resistors in the network. The strategy and the models are experimentally validated by coupling a free-free beam to a completely passive network. A multimodal vibration reduction is observed, which proves the efficiency of the control solution and its potential in term of practical implementation.

Piezoelectric Transverse Shear Actuation and Sensing of Plates, Part 2: Application and Analysis

An exact three-dimensional mixed state space formulation has been presented in Part 1 of this paper for the piezoelectric transverse shear actuation and sensing of plates. In this second part, its application to three-layer adaptive plates with piezoceramic core is studied theoretically and numerically for validation purpose. Then, a detailed parametric analysis is conducted to investigate the influence of several parameters such as, the plate thickness ratio, the piezoceramic core thickness, the mechanical and electric loading amplitudes, and the piezoceramic sensor/actuator position through a multilayer cross-ply hybrid laminate. It was found that, to represent both sensing and actuation behaviors, mechanical in-plane displacements and transverse deflection should be taken layerwise linear (in “zig-zag”) and constant through the plate thickness, respectively. The electric potential was found always linear inside the piezoceramic layer. Variation of the piezoceramic core thickness did not affect much the distributions patterns. However, the induced plate voltage and center deflection under varied mechanical and electric loading amplitudes were found to be proportional to the applied mechanical load and actuation voltage. The optimum position of the piezoceramic transverse shear sensor/actuator through-the-thickness of a multilayer cross-ply hybrid plate was found to be at its mid-plane. These results are useful for future reference and comparison with other approximate analytic and numerical solutions. They can also be used to construct such two-dimensional models.

Wafer-scale fabrication of self-actuated piezoelectric nanoelectromechanical resonators based on lead zirconate titanate (PZT)

In this paper we report an unprecedented level of integration of self-actuated nanoelectromechanical system (NEMS) resonators based on a 150 nm thick lead zirconate titanate (PZT) thin film at the wafer-scale. A top-down approach combining ultraviolet (UV) lithography with other standard planar processing technologies allows us to achieve high-throughput manufacturing. Multilayer stack cantilevers with different geometries have been implemented with measured fundamental resonant frequencies in the megahertz range and Q-factor values ranging from ~130 in air up to ~900 in a vacuum at room temperature. A refined finite element model taking into account the exact configuration of the piezoelectric stack is proposed and demonstrates the importance of considering the dependence of the beam’s cross-section upon the axial coordinate. We extensively investigate both experimentally and theoretically the transduction efficiency of the implemented piezoelectric layer and report for the first time at this integration level a piezoelectric constant of d31 = 15 fm.V−1. Finally, we discuss the current limitations to achieve piezoelectric detection.

Piezoelectric Shunt Vibration Damping of Structural-Acoustic Systems: Finite Element Formulation and Reduced-Order Model

For noise and vibration attenuation, various approaches can be employed depending on the frequency range to attenuate. Generally, active or passive piezoelectric techniques are effective in the low-frequency range, while dissipative materials, such as viscoelastic or porous treatments, are efficient for higher-frequency domain. In this work, a reducedorder model is developed for the approximation of a fully coupled electromechanicalacoustic system using modal projection techniques. The problem consists of an elastic structure with surface-mounted piezoelectric patches coupled with a compressible inviscid fluid. The piezoelectric elements, connected with resonant shunt circuits, are used for the vibration damping of the coupled system. Numerical examples are presented in order to illustrate the accuracy and the versatility of the proposed reduced-order model, especially in terms of prediction of attenuation. [DOI: 10.1115/1.4027133]

Multimodal vibration damping of a plate by piezoelectric coupling to its analogous electrical network

Multimodal damping can be achieved by coupling a mechanical structure to an electrical network exhibiting similar modal properties. Focusing on a plate, a new topology for such an electrical analogue is found from a finite difference approximation of the Kirchhoff-Love theory and the use of the direct electromechanical analogy. Discrete models based on element dynamic stiffness matrices are proposed to simulate square plate unit cells coupled to their electrical analogues through two-dimensional piezoelectric transducers. A setup made of a clamped plate covered with an array of piezoelectric patches is built in order to validate the control strategy and the numerical models. The analogous electrical network is implemented with passive components as inductors, transformers and the inherent capacitance of the piezoelectric patches. The effect of the piezoelectric coupling on the dynamics of the clamped plate is significant as it creates the equivalent of a multimodal tuned mass damping. An adequate tuning of the network then yields a broadband vibration reduction. In the end, the use of an analogous electrical network appears as an efficient solution for the multimodal control of a plate.

Topology optimization of shunted piezoelectric elements for structural vibration reduction

Passive structural vibration reduction by means of shunted piezoelectric patches is addressed in this article. The concept of topology optimization, based on the solid isotropic material with penalization method, is employed in this work to optimize, in terms of damping efficiency, the geometry of piezoelectric patches, as well as their placement on the host elastic structure. The proposed optimization procedure consists of distributing the piezoelectric material in such a way as to maximize the modal electromechanical coupling factor of the mechanical vibration mode to which the shunt is tuned. An original finite element formulation, suitable to any elastic structures with surface-mounted piezoelectric patches, is proposed to solve the electromechanical problem. Numerical examples validate and demonstrate the potential of the proposed approach for the design of piezoelectric shunt devices.