Bending waves of a rectangular piezoelectric laminated beam
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RESEARCH PAPER
Bending waves of a rectangular piezoelectric laminated beam C. P. Wei1 · C. X. Xue1 Received: 11 March 2020 / Revised: 20 May 2020 / Accepted: 24 June 2020 © The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract A simple nonlinear model is proposed in this paper to study the bending wave in a rectangular piezoelectric laminated beam of infinite length. Based on the constitutive relations for transversely isotropic piezoelectric materials and isotropic elastic materials, combined with some electric conditions, we derive the bending wave equation in a long rectangular piezoelectric laminated beam by using energy method. The nonlinearity considered is geometrically associated with the nonlinear normal strain in the longitudinal beam direction. The shock-wave solution, solitary-wave solution and other exact solutions of the bending wave equation are obtained by the extended F-expansion method. And by using the reductive perturbation method we derive the nonlinear Schrodinger (NLS) equation, further more, the bright and dark solitons are obtained. For those soliton solutions, and some parameters derived by the process of solving soliton solutions, some conclusions are drawn by numerical analysis with some fixed conditions.
Keywords Bending wave · Rectangular piezoelectric laminated beam · Extended F-expansion method · Nonlinear Schrodinger equation
1 Introduction Soliton as a solution with the properties of particle and structure to a kind of nonlinear equations was first found in the water wave of canal by the Scottish engineer Russell in 1834. In these equations, a balance between the nonlinear effect, which makes wave form steep, and the dispersion effect, which makes wave form disperse, leads to the formation of solitons. For those nonlinear equations with soliton solutions such as the Korteweg-de Vries equation, the nonlinear Schrodinger (NLS) equation, etc., there are many results were found. In 1967, a method for solving the Korteweg-de Vries equation was proposed by Gardner et al. [1]. Abundant exact solutions of the NLS equation were obtained by using the homogeneous balance principle and F-expansion method [2]. Wang and Zhang [3] proposed a improved extended F-expansion method to solve the nonlinear partial differential equations. An effective method to construct some new types of localized nonlinear wave solutions of the (2+1)-
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C. X. Xue [email protected] Department of Mechanics, School of Science, North University of China, Taiyuan 030051, China
dimensional generalized Hirota-Satsuma-Ito equation was presented [4]. The long-time asymptotics of the focusing Kundu–Eckhaus equation with nonzero boundary conditions at infinity was investigated [5]. A new two-component nonlinear wave system was studied [6]. For the Whitham– Broer–Kaup equation, all the traveling waves of it were obtained [7]. The (2+1)-dimensional variable-coefficient nonlinear Schrodinger equation with partial nonlocality was studied [8]. Dar
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