Numerical modeling of sound and vibration reduction using viscoelastic materials and shunted piezoelectric patches

doi:10.1016/j.compstruc.2017.07.024

Computers & Structures

Volume 232, May 2020, 105822

https://doi.org/10.1016/j.compstruc.2017.07.024 Get rights and content

Highlights

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Noise and vibration control in low and high frequency ranges.
•
Complex multiphysics visco-electro-mechanical-acoustic formulation and simulation.
•
Variational coupling of BEM and FEM for fluid-structure problems.
•
Reduced order-model for composite structures with piezo and viscoelastic layers.
•
Original coupled modal base for piezoelectric shunt.

Abstract

In this paper, a reduced order coupled finite element/boundary element method (FEM/BEM) for control of noise radiation and sound transmission of vibrating structure by viscoelastic and passive piezoelectric treatments is presented. The system consists of a sandwich structure composed of elastic faces and a viscoelastic core (with surface mounted piezoelectric patches) and coupled to external/internal acoustic domains. The passive shunt damping strategy is employed for vibration attenuation in the low-frequency range while dissipative viscoelastic materials are used for the higher-frequency domain. The originality of the present paper lies (i) in evaluating the classically used FEM/BEM methods for structural-acoustics problems when taking account smart systems at the fluid-structure interfaces and viscoelastic interlayers and (ii) in development of a reduced-order model, based on a normal mode expansion, able to capture the main characteristics of the system dynamical behavior.

Introduction

Sandwich structures with a viscoelastic layers are commonly used in many applications for vibration damping and noise control. In such structures, the main energy loss mechanism is due to the transverse shear of the viscoelastic core. However, accurate modeling of structures with viscoelastic materials is difficult because the measured dynamic properties of viscoelastic material are frequency and temperature dependent. This motivated several authors to develop accurate numerical methods of modeling the effects of viscoelastic damping mechanisms which introduce frequency dependence [16], [28], [2], [3]. A review of these methods can be found in [34]. Concerning the application of these structures in noise and vibration attenuation, we can cite for example [13], [1]. In [13], the measured sound transmission loss of multilayered structures is compared with transfer matrix method results (assuming infinite layers) and a wave based model (taking into account finite dimensions) to show the importance of the finite dimensions in a broad frequency range. The effects of viscothermal fluid in laminated double glazing are investigated in [1] using a finite element approach. However, at low frequency, the acoustic performance of this type of systems is greatly deteriorated and the viscoelastic layer is not effective in treating the fall of sound transmission loss and the vibration reduction. The aim of this reported research work is to reduce the vibration and the sound transmission at low resonance frequencies by a passive piezoelectric shunt technique through a fully coupled finite element/boundary element modeling of the problem. In this technology, the structure is equipped with piezoelectric patches that are connected to a passive electrical circuit, called a shunt. The piezoelectric patches transform mechanical energy of the vibrating structure into electrical energy, which is then dissipated by Joule heat in the shunt circuits. Several shunt circuits can be considered: the classical R- and RL-shunts, proposed by Hagood and Von Flotow [21] and improvements of those techniques, using several piezoelectric elements [11], [10], [8], active fiber composites [6] or adaptive shunts [29], and recently semi-passive techniques, commonly known as switch techniques [20], [14], [23]. As those techniques are passive (or semi-passive if some electronic components have to be powered), a critical issue is that their performances, in terms of damping efficiency, directly depend on the electromechanical coupling between the host structure and the piezoelectric elements, which has to be maximized and necessitates the development of predictive models.

The present work concerns the numerical modeling of noise and vibration reduction of laminated structures with viscoelastic interlayers by using shunted piezoelectric elements. The frequency domains of interest are the low and medium frequency ranges. The aim is to propose an efficient reduced order coupled finite element/boundary element model able to predict the shunt damping around the resonance frequencies of the system. In the first part of this paper, a finite element formulation (FEM) of sandwich structures with viscoelastic core and equipped with shunted piezoelectric patches is presented. This formulation involves structural displacement in the structure (sandwich structure with piezoelectric elements), acoustic pressure in the fluid cavity and the electric charge and voltage between the electrodes in the piezoelectric patches. The charge/voltage variables are intrinsically adapted to include any external electrical circuit into the electromechanical problem and to simulate the effect of shunt damping techniques. Moreover, since the elasticity modulus of the viscoelastic core is complex and frequency dependent, this formulation is complex and nonlinear in terms of frequency. The direct solution of this problem can only be considered with models which do not imply a prohibitive number of degrees of freedom. This has severe limitations in attaining adequate accuracy and wider frequency ranges of interest. A reduced order-model is then proposed to solve the problem at a lower cost in the second part of this paper. The proposed methodology, based on a normal mode expansion, requires the computation of the uncoupled structural and acoustic modes. The uncoupled structural modes are the real and undamped modes of the sandwich structure without fluid pressure loading at fluid-structure interface and with short-circuited and open-circuited patches, whereas the uncoupled acoustic modes are the cavity modes with rigid wall boundary conditions at the fluid-structure interface. It is shown that the projection of the full-order coupled finite element model on the uncoupled bases, leads to a reduced order model in which the main parameters are the classical fluid-structure coefficient, the residual stiffness complex coupling factors and the electromechanical coupling factors. Because of its reduced size, this model is proved to be very efficient for simulations of steady-state and frequency analyses of the fully coupled visco-electro-mechanical-acoustic system and the computational effort is significantly reduced. Note that the computing of eigenmodes in the medium frequency range presents no difficulty in this work. Indeed, the new numerical computing tools allow easy access to these modes which has not been the case before. As a next step, the direct boundary element method (BEM) is used for modeling the scattering/radiation of sound by the structure coupled to an external acoustic domain. The BEM is derived from a boundary integral equation involving the surface pressure and normal acoustic velocity at the boundary of the acoustic domain. The coupled FEM-BEM model is obtained by using a compatible mesh at the fluid-structure interface. In the last part of the paper, a numerical example is presented in order to validate and analyze results computed from the proposed formulations.

Section snippets

Numercial modeling of the structure

Consider a thin sandwich structure, denoted $Ω_{E}$ , made of elastic faces and viscoelastic core (Fig. 1). A prescribed force density $F^{d}$ is applied to the external boundary $Γ_{σ}$ of $Ω_{E}$ and a prescribed displacement $u^{d}$ is applied on a part $Γ_{u}$ of $Ω_{E}$ . The structure is coupled to internal and external fluids denoted $Ω_{Fi}$ and $Ω_{Fe}$ , respectively. These fluids are considered homogeneous, inviscid and compressible and the gravity effects are neglected. We denote by $Σ_{i}$ and $Σ_{e}$ the internal and external

Numerical modeling of fluid

This section deals with the numerical modeling of the fluid. The internal fluid is modeled using the finite element method. The external fluid is modeled using the boundary element method.

Reduced FE/BE formulation for the fluid-structure coupled problem with piezoelectric shunt systems and viscoelastic materials

By combining Eq. (21) with Eqs. (28), (40), we find the following coupled FE/BE matrix equation $[\begin{matrix} Φ_{s}^{T} (K_{u}^{*} (ω) + C_{uV} K_{V}^{- 1} C_{uV}^{T}) Φ_{s} & Φ_{s}^{T} C_{uV} K_{V}^{- 1} & - Φ_{s}^{T} C_{up} Φ_{f} & - Φ_{s}^{T} C_{up} \\ K_{V}^{- 1} C_{uV}^{T} Φ_{s} & K_{V}^{- 1} & 0 & 0 \\ 0 & 0 & Φ_{f}^{T} K_{p} Φ_{f} & 0 \\ 0 & 0 & 0 & H \end{matrix}] [\begin{matrix} q_{s} \\ Q \\ q_{f} \\ P_{Σ e} \end{matrix}] + i ω [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & R & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] [\begin{matrix} U \\ Q \\ q_{f} \\ P_{Σ e} \end{matrix}] - ω^{2} [\begin{matrix} Φ_{s}^{T} M_{u} Φ_{s} & 0 & 0 & 0 \\ 0 & L & 0 & 0 \\ Φ_{f}^{T} C_{up}^{T} Φ_{s} & 0 & Φ_{f}^{T} M_{p} Φ_{f} & 0 \\ ρ_{F} GT & 0 & 0 & 0 \end{matrix}] [\begin{matrix} q_{s} \\ Q \\ q_{f} \\ P_{Σ e} \end{matrix}] = [\begin{matrix} Φ_{s}^{T} F \\ 0 \\ 0 \\ 0 \end{matrix}]$

This matrix equation represents the reduced order model of the fully coupled visco-electro-mechanical-acoustic system.

The matrices $H$ and $G$ , associated to the BE formulation for the external fluid, are complex-valued, nonsymmetric,

Numerical example

In this last section, numerical results obtained with a Matlab program developed by the author are proposed in order to validate and analyze the results calculated from the proposed formulation.

We consider a flexible sandwich plate (in-plane ( $x, y$ ) dimensions are A = 1.5 m, B = 1.25 m) composed of two identical glass plates bonded together by a PVB interlayer. The thickness of the outer and inner glass ply is 3 mm and those of the PVB interlayer is 1.14 mm (Fig. 3). The inner face of the plate is

Conclusions

In this work, a coupled finite element/boundary element formulation (FEM/BEM) used to model the reduction of noise and vibrations by viscoelastic materials and piezoelectric techniques is presented. The multiphysics studied system is constituted of a sandwich structure with a viscoelastic core, equipped with shunted piezoelectric patches and coupled to interior and exterior acoustic domains. A reduced-order FE model, based on normal mode expansion, is developed for modeling the internal

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Numerical modeling of sound and vibration reduction using viscoelastic materials and shunted piezoelectric patches

Highlights

Abstract

Introduction

Section snippets

Numercial modeling of the structure

Numerical modeling of fluid

Reduced FE/BE formulation for the fluid-structure coupled problem with piezoelectric shunt systems and viscoelastic materials

Numerical example

Conclusions

Appl Acoust

Compos Struct

Comp Meth Appl Mech Eng

Compos Part B: Eng

J Sound Vib

Compos Struct

Compos Part B: Eng

Finite element model for hybrid active-passive damping analysis of anisotropic laminated sandwich structures

J Sand Struct Mater

Finite element and boundary methods in structural acoustics and vibration

Fractional calculus - a different approach to analysis of viscoelastically damped structures

AIAA J

Structural vibration control via RL-shunted active fiber composites

J Intell Mater Syst Struct

Finite element analysis of the dynamic behavior of a laminated windscreen with frequency dependent viscoelastic core

J Acoust Soc Am

Broad band vibration control through periodic arrays of resonant shunts: experimental investigation on plates

Smart Mater Struct

Theory of viscoelasticity: an introduction

Wave motion optimization in periodically distributed shunted piezocomposite beam structures

J Intell Mater Syst Struct

Passive damping of beam vibrations through distributed electric networks and piezoelectric transducers: prototype design and experimental validation

Smart Mater Struct

Sound transmission through finite lightweight multilayered structures with thin air layers

J Acoust Soc Am