Shock and Vibration

Volume 2016 (2016), Article ID 4015363, 12 pages

http://dx.doi.org/10.1155/2016/4015363

## An Enhanced Plane Wave Expansion Method to Solve Piezoelectric Phononic Crystal with Resonant Shunting Circuits

^{1}Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China^{2}Hubei Key Laboratory of Engineering Structural Analysis and Safety Assessment, Huazhong University of Science and Technology, Wuhan 430074, China^{3}State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China

Received 28 October 2015; Revised 25 January 2016; Accepted 15 February 2016

Academic Editor: Sergio De Rosa

Copyright © 2016 Ziyang Lian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An enhanced plane wave expansion (PWE) method is proposed to solve piezoelectric phononic crystal (PPC) connected with resonant shunting circuits (PPC-C), which is named as PWE-PPC-C. The resonant shunting circuits can not only bring about the locally resonant (LR) band gap for the PPC-C but also conveniently tune frequency and bandwidth of band gaps through adjusting circuit parameters. However, thus far, more than one-dimensional PPC-C has been studied just by Finite Element method. Compared with other methods, the PWE has great advantages in solving more than one-dimensional PC as well as various lattice types. Nevertheless, the conventional PWE cannot accurately solve coupling between the structure and resonant shunting circuits of the PPC-C since only taking one-way coupling from displacements to electrical parameters into consideration. A two-dimensional PPC-C model of orthorhombic lattice is established to demonstrate the whole solving process of PWE-PPC-C. The PWE-PPC-C method is validated by Transfer Matrix method as well as Finite Element method. The dependence of band gaps on circuit parameters has been investigated in detail by PWE-PPC-C. Its advantage in solving various lattice types is further illustrated by calculating the PPC-C of triangular and hexagonal lattices, respectively.

#### 1. Introduction

The propagation of elastic or acoustic waves in phononic crystals (PCs) has attracted growing attention during the past two decades [1, 2], both because of its amazing physical properties and because of its potential applications. One of the most attractive properties of PCs is the band gaps, in which elastic or acoustic waves are attenuated significantly. According to the generation mechanisms of Bragg scattering and locally resonant (LR), the band gaps can be divided into Bragg band gap and LR band gap [3].

The PC has the potential to be used as a vibration isolator because of the band gap, but one of main difficulties of isolating mechanical vibration lies in how to attenuate low-frequency vibration. Compared to the Bragg band gap, the LR band gap has a lower frequency and can be generated by a smaller PC, which enables the PC to isolate low-frequency vibration for the high-precision machine [3–5]. Yet a unit cell of conventional LR PCs commonly consists of a heavy core and a soft coat, as a mass-spring oscillator, to obtain a low resonance frequency. Therefore, a lower frequency LR band gap implies a heavier PC. By replacing the heavy mass-spring oscillator, a light inductor-capacitor oscillator can be constituted by the equivalent capacitance of piezoelectric phononic crystal (PPC) and the inductance of the resonant shunting circuit. The beam-like piezoelectric phononic crystal connected with resonant shunting circuits (PPC-C) was studied to obtain LR band gaps for controlling the propagation of vibration [6–10]. Furthermore, the PPC-C has another significant advantage that its band gaps can be tuned just by circuit parameters, but reconfiguring the structure is not needed [11–16]. This property will further enhance the application flexibility of the PPC-C.

The PPCs have been researched by Transfer Matrix method [10], plane wave expansion (PWE) method [17, 18], Finite Difference Time Domain method [19], and Finite Element method [20, 21]. Nevertheless, among these methods, only Transfer Matrix method [6–9, 11–13] and Finite Element methods [15, 16, 22] have been used to study the PPC-C. Transfer Matrix method is just applicable to one-dimensional PPC-C, such as beam [7] and circle plate [23]. Up to now, two-dimensional plate-like PPC-C have been researched merely by Finite Element method [15, 16, 22]. Although the utilization of Finite Element method becomes popular nowadays, it is useful to have another method to handle this kind of problem.

Two-dimensional and three-dimensional PPCs were investigated by the methods of Finite Difference Time Domain [19] and conventional PWE [17, 18]. However, without resonant shunting circuits are connected with piezoelectric patches of the PPC. Under this circumstance, we only need to consider two kinds of electric boundary conditions for the piezoelectric patches, that is, short and open circuits. When electrodes of one piezoelectric patch are shorted, electric field is equal to zero in a certain direction, but when these electrodes are open, normal component of electric displacement is equal to zero. Electric field or electric displacement can be expressed directly by displacements. Hence the band structure can be obtained by solving governing equation in terms of displacements. Yet for the PPC with circuits (PPC-C), electrical parameters, such as electric potential and electric displacement, are not contained in the equations of motion. Only one-way coupling, that is, from displacements to electrical parameters, can be taken into account. Firstly, displacements are always obtained from the equations of motion and then get electrical parameters according to circuit equation. Hence the influence of the circuit on structure movement is not taken into consideration. Thus the PPC-C cannot be solved accurately by conventional PWE.

The PWE method is one of the most extensively used methods to calculate band structure because of its convenience [24]. Kushwaha et al. [25] firstly obtain the band structure of a PC by PWE. PWE method includes three steps as follows. Firstly, all parameters such as modulus, density, and Poisson’s ratio are expanded as the Fourier series in the reciprocal space. Secondly, equations of motion are transformed into the standard eigenvalue problem based on the Bloch theorem. Finally, band structure of the dispersion relations between the frequency and wave vector is obtained by solving the eigenvalue equation. It is convenient that wave vector just needs to traverse along the boundary of the irreducible region of the first Brillouin zone. Compared with Finite Element method, the PWE shows great advantages in dealing with scatterers of different geometries and different arrangements, because it does not require meshing and reconstructing the Finite Element matrix. But the conventional PWE has a disadvantage of the slow convergence rate, especially for systems of either very high or very low filling ratios or of large elastic mismatch. The fictitious band gaps always appear as redundant lines in the band structure. Nevertheless, an improved PWE has a good convergence and can provide much more accurate numerical results [26, 27].

The main aim of the paper is to propose an enhanced PWE method to solve the PPC-C, which is abbreviated as PWE-PPC-C. A two-dimensional PPC-C model is established to demonstrate solving process of the PWE-PPC-C. The model is composed of two portions: elastic plate and elastic-piezoelectric composite plate. In general, the conventional PWE does not directly utilize the continuity conditions of forces and bending moment between two portions of the PPC. Moreover, the voltage appears in the expression of bending moment of elastic-piezoelectric composite plate but does not exist in that of elastic plate. To deal with this voltage mismatch, the voltage is included in equations of motion of these two portions of the PPC to ensure the continuity of bending moment at the interface between these two portions. And the voltage is introduced by multiplying an infinitely small quantity for the elastic plat. Then band structure is obtained by solving the equations of motion and circuit equation. The paper is organized as follows: Section 2 presents governing equations of a PPC-C. The equations of motion are expanded by PWE-PPC-C and Bloch theorem in Section 3. Section 4 describes two approaches to solve the PPC with two kinds of common shunting circuits, respectively. The validity and correctness of the PWE-PPC-C are verified by comparisons with Transfer Matrix method and Finite Element method in Section 5. Finally, some conclusions are given in Section 6.

#### 2. Governing Equations of a Two-Dimensional PPC-C

In order to demonstrate solving process of the PWE-PPC-C, we propose a two-dimensional PPC-C which consists of an isotropic elastic plate and rectangular piezoelectric patches. These piezoelectric patches are bonded on the upper and lower surfaces of the plate and are arrayed as orthorhombic system. Its lattice and arrangement are shown in Figures 1 and 2(a). With The same polarization direction, each pair of parallel piezoelectric patches is connected by a resonant shunting circuit, which can be simplified by a complex impedance . Each unit cell of the PPC can be divided into two portions: the elastic plate as a matrix and the elastic-piezoelectric composite plate as a scatterer. Subscripts () of physical quantities correspond to coordinate system ().