Thickness-shear vibration analysis of rectangular quartz plates by a numerical extended Kantorovich method B. Liu a,⇑ , Y.F. Xing a , M. Eisenberger b , A.J.M. Ferreira c,d a The Solid Mechanics Research Centre, Beihang University (BUAA), Beijing 100191, China b Faculty of Civil and Environmental Engineering, Technion Israel Institute of Technology, Rabin Building, Technion City, Haifa 32000, Israel c Faculdade de Engenharia da Universidade do Porto, Porto, Portugal d Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia article info Article history: Available online 27 August 2013 Keywords: Thickness-shear vibration Quartz plate Kantorovich method Differential quadrature Finite element method abstract Thickness-shear vibration analysis of rectangular quartz plates was carried out by a numerical extended Kantorovich method (NEKM) proposed in this paper. Thickness-shear vibration is characterized by high frequency vibration that needs huge computational cost. The NEKM reduced the two-dimensional elas- ticity problem into two one-dimensional ones and then solved them iteratively as the Kantorovich method. But all the processes of reduction and solvation in NEKM were carried out by using a differential quadrature finite element method (DQFEM). Therefore, the NEKM is much simpler than the Kantorovich method that does all the processes analytically with great complexity. Because the two-dimensional problem was reduced to one-dimensional ones and the DQFEM was very accurate, the NEKM was very efficient and accurate. The correctness and high accuracy of the NEKM were validated through numerical comparison with the results of the Kantorovich method in literature. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction Piezoelectric resonators are vibrating crystals as key elements of electric circuits called oscillators [1]. Piezoelectric crystals are widely used to make acoustic wave resonators as a frequency standard for time-keeping, frequency generation and operation, telecommunica- tion, and sensing [2]. Quartz is the most widely used crystal for resona- tors. A large portion of quartz crystal resonators, usually in circular or rectangular platform with all edges free [3], are operated with the so- called thickness-shear modes of a plate [4–6]. Considerable efforts have been spent in analysis of piezoelectric resonator with approximate and numerical methods since Mindlin [7,8]. Although accurate and reliable results have been obtained through the efforts of many researchers for several decades [3,9], with emerging problems and practical needs, the sizes of quartz crystal resonators are shrinking fast and their frequency are increasing rapidly. Therefore, precise analysis of thickness-shear vibrations of crystal plates is even more important now. Quartz plates of AT-cut have orthotropic property and at least the first-order shear deformation theory that includes transverse shear deformation effects should be used for thickness-shear vibration analysis [10,11]. Pure thickness-shear modes only exist in unbounded plates without edge effects. Thickness-shear modes in a finite plate are coupled to flexural modes. It is easy to find relatively simple solutions for unbounded plates with free edges. However, exact solutions could be obtained only for rectangular crystal plates with a pair of opposite edges simply supported [10]. Solutions for the case of rectangular plates with all four edges free are not expressible in terms of a finite number of elementary func- tions [12]. Exact solutions for completely free rectangular Mindlin plates can only be obtained with some restrictions on length/width and Poisson’s ratios [13]. The additional complications of anisotropy and high frequency make general solutions even more intractable. Therefore, most literatures on thickness-shear vibration of rectan- gular plates solve the problem by reducing the two- or three-dimen- sional differential equations to one-dimensional ones [12,14,15] or by only satisfying the major boundary conditions of the dominant thickness-shear mode [3]. The computational cost is very expensive if a numerical procedure is used [3]. Gorman [16] has used the super- position method to obtain analytical solutions for completely free Mindlin plates. However, the superposition method has only been used in low frequency vibrations. Recently, Wang et al. [8] used the Kantorovich method [17] for thickness-shear vibration of rectan- gular Mindlin plates. The method is computationally very efficient, however, is also very complex as most analytical methods are [18]. In this paper, the differential quadrature finite element method (DQFEM) [19,20] was incorporated into the Kantorovich technique, which leaded to a new method named as a numerical extended Kant- orovich method (NEKM). The NEKM was applied to thickness-shear vibration analysis of rectangular quartz plates. The NEKM reduced the two-dimensional elasticity problem into two one-dimensional ones and then solved them iteratively like the Kantorovich method. 0263-8223/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2013.08.021 ⇑ Corresponding author. Tel.: +86 13681162181. E-mail address: liubo68@buaa.edu.cn (B. Liu). Composite Structures 107 (2014) 429–435 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct