This work addresses passive reduction of structural vibration by means of shunted piezoelectric patches. The two classical resistive and resonant shunt solutions are considered. The main goal of this paper is to give closed-form solutions to systematically estimate the damping performances of the shunts, in the two cases of free and forced vibrations, whatever the elastic host structure is. Then it is carefully demonstrated that the performance of the shunt, in terms of vibration reduction, depends on only one free parameter: the so-called modal electromechanical coupling factor (MEMCF) of the mechanical vibration mode to which the shunts are tuned. Experiments are proposed and an excellent agreement with the model is obtained, thus validating it.

This article is devoted to the numerical simulation of the vibrations of an elastic mechanical structure equipped with several piezoelectric patches, with applications for the control, sensing and reduction of vibrations. At first, a finite element formulation of the coupled electromechanical problem is introduced, whose originality is that provided a set of non‐restrictive assumptions, the system's electrical state is fully described by very few global discrete unknowns: only a couple of variables per piezoelectric patches, namely (1) the electric charge contained in the electrodes and (2) the voltage between the electrodes. The main advantages are (1) since the electrical state is fully discretized at the weak formulation step, any standard (elastic only) finite element formulation can be easily modified to include the piezoelectric patches (2) realistic electrical boundary conditions such that equipotentiality on the electrodes and prescribed global charges naturally appear (3) the global charge/voltage variables are intrinsically adapted to include any external electrical circuit into the electromechanical problem and to simulate shunted piezoelectric patches. The second part of the article is devoted to the introduction of a reduced‐order model (ROM) of the problem, by means of a modal expansion. This leads to show that the classical efficient electromechanical coupling factors (EEMCF) naturally appear as the main parameters that master the electromechanical coupling in the ROM. Finally, all the above results are applied in the case of a cantilever beam whose vibrations are reduced by means of a resonant shunt. A finite element formulation of this problem is described. It enables to compute the system EEMCF as well as its frequency response, which are compared with experimental results, showing an excellent agreement. Copyright © 2009 John Wiley & Sons, Ltd.

This paper presents a finite element formulation for transient dynamic analysis of sandwich beams with embedded viscoelastic material using fractional derivative constitutive equations. The sandwich configuration is composed of a viscoelastic core (based on Timoshenko theory) sandwiched between elastic faces (based on EulerBernoulli assumptions). The viscoelastic model used to describe the behavior of the core is a four-parameter fractional derivative model. Concerning the parameter identification, a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Curve-fitting aspects are focused, showing a good agreement with experimental data. In order to implement the viscoelastic model into the finite element formulation, the Grünwald definition of the fractional operator is employed. To solve the equation of motion, a direct time integration method based on the implicit Newmark scheme is used. One of the particularities of the proposed algorithm lies in the storage of displacement history only, reducing considerably the numerical efforts related to the non-locality of fractional operators. After validations, numerical applications are presented in order to analyze truncation effects (fading memory phenomena) and solution convergence aspects. IntroductionMany investigations have demonstrated the potential of viscoelastic materials to improve the dynamics of lightly damped structures. There are numerous techniques to incorporate these materials into structures. The constrained layer passive damping treatment is already largely used to reduce structural vibrations, especially in conjunction with active control [2, 13]. One of the crucial questions is how to quantify such a material damping if the viscoelastic solid has a weak frequency dependence on its dynamic properties over a broad frequency range. Classical linear viscoelastic models, using integer derivative operators, convolution integral or internal variables, become cumbersome due to the high quantity of parameters needed to describe the material behavior. In order to overcome these difficulties, fractional derivative operators acting on both, strain and stress can be employed.Until the beginning of the 80s, the concept of fractional derivatives associated to viscoelasticity was regarded as a sort of curve-fitting method. Later, Bagley and Torvik [1] gave a physical justification of this concept in a thermodynamic framework. Their fractional model has become a reference in literature. Special interest is today dedicated to the implementation of fractional constitutive equations into FE formulations. In this context, the numerical methods in the time domain are generally associated with the Grünwald formalism for the fractional order derivative of the stress-strain relation in conjunction with a time discretization scheme. Padovan [7] derived several implicit, explicit and predictor-corrector type algorithms. Escobedo-Torres and Ricles [6] analyzed a numerical procedure based on the central difference method and its ...

No abstract

No abstract

a b s t r a c tThis article presents a finite element reduced order model for the nonlinear vibrations of piezoelectric layered beams with application to NEMS. In this model, the geometrical nonlinearities are taken into account through a von Ká rmá n nonlinear strain-displacement relationship. The originality of the finite element electromechanical formulation is that the system electrical state is fully described by only a couple of variables per piezoelectric patches, namely the electric charge contained in the electrodes and the voltage between the electrodes. Due to the geometrical nonlinearity, the piezoelectric actuation introduces an original parametric excitation term in the equilibrium equation. The reduced-order formulation of the discretized problem is obtained by expanding the mechanical displacement unknown vector onto the short-circuit eigenmode basis. A particular attention is paid to the computation of the unknown nonlinear stiffness coefficients of the reduced-order model. Due to the particular form of the von Ká rmá n nonlinearities, these coefficients are computed exactly, once for a given geometry, by prescribing relevant nodal displacements in nonlinear static solutions settings. Finally, the low-order model is computed with an original purely harmonic-based continuation method. Our numerical tool is then validated by computing the nonlinear vibrations of a mechanically excited homogeneous beam supported at both ends referenced in the literature. The more difficult case of the nonlinear oscillations of a layered nanobridge piezoelectrically actuated is also studied. Interesting vibratory phenomena such as parametric amplification or patch length dependence of the frequency output response are highlighted in order to help in the design of these nanodevices.

No abstract

Abstract. 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.