## 1 Introduction  
Piezoelectric materials, as an intelligent material, have been widely used in the manufacture of many intelligent devices, such as sensors, actuators, and transducers [1; 2; 3]. The working principle of piezoelectric sensors and piezoelectric actuators depends mainly on the conversion between force and electricity in the vibration process of piezoelectric materials [4; 5]. Thus, an understanding of the vibration problems of the piezoelectric structures is the premise for designing piezoelectric sensors and piezoelectric actuators. However, the vibration problems of the piezoelectric structures are often difficult to solve analytically, because of the force-electric coupling characteristics of piezoelectric materials. So, it is necessary and meaningful to find an efficient analytical method for solving the vibration problems of piezoelectric structures and to give their analytical solutions. In the existing works, many researchers have studied the vibration problems of piezoelectric structures, and some achievements have been made. [PERSON] et al. [6] studied the free vibration of a rotating circular nanoplate made of two directional functionally graded piezo materials (two directional FGPM) based on the first shear deformation theory (FSDT). [PERSON] and [PERSON] [7] presented theoretical and experimental investigations into the nonlinear flexural vibrations of a structure composed of a host beam with piezoelectric ceramic actuators symmetrically bonded to its top and bottom surfaces. [PERSON] et al. [8] studied the dynamic analytical solution of a piezoelectric stack utilizedin an actuator and a generator based on the linear piezo-elasticity theory. [PERSON] et al. [9] studied the nonlinear shear-induced flexural vibrations of piezoceramic actuators. [PERSON] and [PERSON] [10] demonstrated the effect of large deformations on piezoelectric materials and structures under time varying loads. [PERSON] et al. [11] studied the natural vibration and transient response of a functionally graded piezoelectric material (FGPM) curved beam with a numerical method. [PERSON] et al. [12] discussed the influence that piezoelectric materials exert on the vibration behavior of a stepped cantilever beam with surface bonded or embedded piezoelectric materials. [PERSON] et al. [13] studied the free vibration of statically thermal post-buckled functionally graded material beams with surface-bonded piezoelectric layers subjected to both temperature rise and voltage. [PERSON] et al. [14] investigated the post buckling and vibration characteristics of a piezolaminated composite plate subjected to thermo-piezoelectric loads. [PERSON] and [PERSON] [15] studied the free vibration of the piezoelectric-elastic laminated beams by combining the state-space method and differential quadrature method. [PERSON] et al. [16] derived the free vibration frequency of the state space solution and the solution for a simple support piezoelectric laminated beam. [PERSON] et al. [17] presented a new method of a state-space-based differential quadrature for free vibration of generally laminated beams. [PERSON] and [PERSON] [18] presented a one-dimensional beam finite element with electric degrees of freedom for the dynamic analysis of hybrid piezoelectric beams, using the coupled efficient layer wise theory. The summation of the results of the existing works shows that there are only a few related vibration experiments of piezoelectric structures, which means that the reliability of the analytical solution cannot be guaranteed sufficiently. Moreover, there has been no unified and effective method for solving the vibration problems of piezoelectric structures. The parameter perturbation method is a general analysis method for solving approximate solutions of nonlinear mechanical problems, which was first used to calculate the influence of the small celestial body on the motion of the large object, and has been widely used in the theoretical research of physics and mechanics.