Static and forced vibration analysis of layered piezoelectric functionally graded structures based on element differential method

Highlights

• A strong-form method is developed for structure integrated with piezoelectric layers;

• A unified manner for dynamic responses of the piezoelectricity is established;

• The homogeneous materials and functionally graded materials are both considered;

• The impact of boundary conditions on deflection and shape control is investigated.

Abstract

A novel strong-form numerical algorithm, piezoelectric vibration element differential method (PVEDM), is proposed for simulating the static deflection and forced vibration of the structure integrated with piezoelectric layers, with the host structure being homogeneous or functionally graded materials. A unified manner for the steady-state and dynamic responses of piezoelectric structures is set up by the proposed method, which draws on the merits of the finite element method and collocation method. In the whole process of assembling the system of equations, variational principle and integration are not required. Furthermore, the influence of boundary conditions on static deflection, and static shape control are investigated. Three examples of static and dynamic responses from one-layer structure, bimorph structure to the structure bonded with piezoelectric layers are given in turn. By comparing with analytical solution or ABAQUS, precise results are achieved, which verifies the the accuracy of the method.

Introduction

With the great development of aerospace technology and some other advanced industries in recent years, the demand for advanced composites with excellent properties such as lightweight [1] and self-adjusting [2], [3], [4] is also increasing year by year. For the sake of safety, the above-mentioned structures usually spend a lot of time eliminating the vibration caused by external factors, which will lead to structural fatigue [5], [6], [7], instability, or other problems affecting the work of the structure. Due to the existence of the piezoelectric effect, by using the characteristics of piezoelectric materials, the vibration of the core structure can be effectively controlled by integrating the piezoelectric layers into the core structure to improve the performance of the structure. Bailey and Hubbard Jr [8] proposed the distributed parameter control law and distributed-parameter actuators, which can improve the accuracy because of all modes being considered in the analysis. He et al. [9] realized the shape control and vibration suppression of the laminated structure with piezoelectric layers by using a constant velocity feedback control algorithm based on the classical laminated plate theory. Selim et al. [10] developed an element-free improved moving least-squares Ritz method to study the active vibration control of multilayer graphene nanoplatelets plates bonded with piezoelectric layers. Besides, they discussed some factors affecting the natural frequency, including thickness-width ratio, volume fraction, and so forth. Therefore, according to the contents mentioned above, it is important to understand the steady-state and vibration characteristics of the structure integrated with piezoelectric layers.

Essentially, the second-order partial differential equations (PDEs) can well describe the mechanical and electrical responses of piezoelectric structures. Based on the Stroh formalism, the formulation of the analytical solution for the vibration of the laminated plates integrated with piezoelectric patches under arbitrary boundary conditions at the edges was derived by Vel et al. [11]. The analytical solution about free vibration for the moderately thick plates with circular geometries was performed by Liu et al. [12] with the aid of Mindlins plate theory. Under the condition of simply supported, free and forced vibration of piezoelectric shell structures with hybrid angle-ply were carried out by Kapuria et al. [13] in two-dimensional space. The behavior of free vibration of thin-circular FGM plates integrated with piezoelectric layers under the clamped condition was analyzed by Ebrahimi and Rastgo [14]. However, owing to the complexity of the partial differential equations, the analytical results is only limited to handle problems with special geometries or special boundary conditions.

In order to solve thorny practical issue with complicated geometry or external load, many efficient and accurate numerical methods have been proposed and employed widely in engineering and science. Denpending on the ways of forming the system of equations, there are two categories of numerical methods: weak-form methods and strong-form methods. Among the weak-form methods, the finite element method (FEM) and some improved methods based on FEM occupy the dominant position. Lam et al. [15] derived the formulation for the vibration control of piezoelectric composite laminates with FEM based on the laminated plate theory. Wang [16] investigated the steady-state and dynamic response of the piezoelectric bimorph structure with FEM, which improved the calculation accuracy of displacement and electric potential. Phuc and Khue [17] combined the third-order shear deformation theory with FEM to analyze the free and forced oscillation of the piezoelectric functionally graded plates in the thermal environment, which pointed out the direction for the design and optimization of the structures. Jiang et al. [18] utilized the zonal free element method, which combines the advantages of FEM and mesh-free method (MFM), to solve the steady-state problems of the piezoelectric model with complex geometry. Nguyen et al. [19] examined the influences of materials properties and external conditions on natural frequencies of functionally graded piezoelectric material porous plates with the isogeometric Bezier finite element method, which absorbs the merits of the FEM and isogeometric analysis (IGA). Wang et al. [20] developed the scaled boundary finite element method (SBFEM) with the precise integration technique (PIT) to obtain accurate results of bending of the piezoelectric laminated plates with complex geometries.

Compared with the weak-form methods, the strong-form methods had also made a lot of contributions to the vibration problem of laminated plates. By using the differential quadrature method [21], Tornabene et al. [22], [23] developed the local generalized differential quadrature method to calculate the natural frequencies of the functionally graded sandwich shells structures and studied the free vibrations of doubly-curved shells by combining the moving least squares and non-uniform rational B-splines. Besides, Fantuzzi et al. [24] performed the free vibration of FGM plates of arbitrary shapes with cracks. Moreover, they proposed the improved strong-form FEM(SFEM) [25], [26] for the analysis of free vibration, static and dynamic responses of laminated plates, then the stability and accuracy of their method were deeply discussed.

Certainly, there are many other methods to study the steady-state or transient response of piezoelectricity. The boundary element method (BEM), which meshes the model on the boundary, had been adopted to deal with steady-state [27] and transient analyses [28] of piezoelectricity. Momeni and Fallah used the meshfree finite volume method (MFV) [29] to obtain reliable results about active control of the piezoelectric laminated composite plates, whose material properties vary with temperature. The proper generalized decomposition method (PDG) with the variable separation technique was employed by Infantes et al. [30] not only got the precise results for forced vibration of composite plates with piezoelectric layers but also reduced the computational cost. A Petrov-Galerkin finite element interface method was modified by Wang et al. [31] for the band structure compitation of piezoelectric phononic crystals. The characterization method, which was based on particle swarm optimization algorithm, was proposed by Sun et al. [32] for piezoelectric material.

According to the basic principles of various methods mentioned above, the strong-form method, which does not require complex integration operations, is more suitable for multi-fields coupling analysis than the weak-form methods. Recently, the element differential method (EDM) was proposed by Gao et al. to deal with the heat conduction problems [33], solid mechanics problems [34], the thermal-mechanical problems [35] and the steady-state problem of piezoelectricity [36]. The method draws on the merits of FEM and collocation method, which makes the method have two bright spots. One is that owing to the isoparametric elements borrowed from the finite element method, EDM can easily characterize the physical variables and deal with complex geometric models, so stable results can be obtained. The other is that attributing to analytical expressions of the shape functions with respect to global coordinates, EDM can directly discrete the governing equation to generate the final system of equations, like most collocation methods.

In this paper, based on the element differential method, the piezoelectric vibration element differential method (PVEDM) is proposed to solve the steady-state and transient responses of the structure integrated with piezoelectric layers. Besides, the homogeneous materials and functionally graded materials are all considered in the examples given in this paper. And the problem of making the structure vibrate by applying actuator voltage is also studied, in which the frequency of alternating voltage is the same as the first natural frequency of the structure. Moreover, the influence of three kinds of boundary conditions on maximum deflection is investigated. And then the steady-state shape control of the structure integrated with piezoelectricity by applying actuator voltage to the upper and lower surfaces of the piezoelectric layer is performed.

The outline of the article is as follows. At first, the unified equations for piezoelectric vibration are presented in Section 2. Then, the basic principle and calculation process of the proposed PVEDM for steady-state and dynamic responses are given in Section 3. Besides, a brief introduction of the Newmark time integral method is also given in this section. In Section 4, three examples about the static and dynamic responses of single-layer, double-layer, and three-layer structures are analyzed to verify the accuracy and effectiveness of the proposed method. Finally, in Section 5, there are several conclusions.