## 1 Introduction
Parameter selection is an important problem while using the parameter perturbation method. In order to solve the difficulty of parameter selection, [PERSON] and [PERSON] [19] put forward the concept of the free parameter perturbation method, that is, there is no need to point out the physical meaning of the perturbation parameters during perturbation, which provides a new idea for solving the parameter selection problem of the parameter perturbation method. [PERSON] et al. [20] solved the Hencky membrane problem without a small-rotation-angle assumption by using a single-parameter perturbation method. There are many perturbation methods based on a single parameter, which are not described in detail here. For the difficulty in the selection of the perturbation parameters, another idea is to select multiple parameters, that is, the so-called \"multi-parameter perturbation method\". For the multi-parameter perturbation method, [PERSON] and [PERSON] [21] solved the cylindrical orthotropic circular plate problem under a uniform load by using the biparametric perturbation method. The application of the multi-parameter perturbation method in beam problem was proposed by Professor [PERSON] [22] in 2002, and the classical Euler-Bernoulli beam equation was solved by using the load and height difference of the beam as the perturbation parameters. Later, [PERSON] and [PERSON] [23] simplified the bending moment by using the quasi linear analysis method, so that the parameter perturbation process was directly aimed at the algebra equation rather than the integral equation, which greatly simplified the perturbation process. Recently, [PERSON] et al. [24; 25] comprehensively analyzed the large deflection problem of beams with a height difference under various boundary conditions, and put forward the so-called \"biparametric perturbation method\" definitely, and successfully applied this method to the solution of the bimodular von-Karman thin plate equation. So far, the application of the biparametric perturbation method in the vibration problems of piezoelectric structures has not been reported. In this study, we will derive the theoretical solution of the free damping vibration problem of the piezoelectric cantilever beams by the biparametric perturbation method and perform the related vibration experiments to verify the validity of the theoretical solution presented here. The whole paper is organized as follows. In Section 2, the mechanical model and the basic equations of the free damping vibration problem of piezoelectric cantilever beams will be established, the piezoelectric parameter and damping coefficient will be selected as the perturbed parameters, and the theoretical solution of the problem studied here will be given. A comparison of the theoretical solution and the existing solution will be presented, and the influence of the piezoelectric properties will be analyzed and discussed in the Section 3. Next, in Section 4, we will show the related vibration experiments of the piezoelectric cantilever beam and compare the experimental results with the theoretical results. According to the results mentioned above, some main conclusions will be drawn in Section 5.