## 2 The Basic Equations and Biparametric Perturbation Solution  
### The Mechanical Model and Basic Equations  
A piezoelectric strip with isotropy in plane \(x-y\) and anisotropy in plane \(x-z\) is clamped at the left end in order to construct a cantilever beam whose length is \(l\), width is \(b\), thickness is \(h\), and uniformly-distributed mass is \(\overline{m}\). Then, an initial displacement \(w_{0}\) is given at the right end of the piezoelectric cantilever beam, and the piezoelectric cantilever beam will generate a free damping vibration under the drive of the initial displacement, as shown in Figure 1.  
From the literature [26], we can know that the free damping vibration equation of a general cantilever beam with equal cross section is as follows:  
\[El_{y}\frac{\partial^{4}w(x,t)}{\partial x^{4}}+\overline{m}\frac{\partial^{2 }w(x,t)}{\partial t^{2}}+c\frac{\partial w(x,t)}{\partial t}=0, \tag{1}\]  
in which \(El_{y}\) is the bending stiffness, \(c\) is the damping parameter, and \(c=2\xi\overline{m}\omega\), in which, \(\xi\) is the damping ratio and \(\omega\) is the vibration frequency. The boundary conditions are as follows:  
\[\left\{ \begin{array}{l}w(x,t)=0\\ \frac{\partial w(x,t)}{\partial x}=0\end{array} \right.,\text{ at }x=0 \tag{2}\]  
and  
\[\left\{ \begin{array}{l}El_{y}\frac{\partial^{2}w(x,t)}{\partial x^{2}}=0\\ El_{y}\frac{\partial^{3}w(x,t)}{\partial x^{3}}=0\end{array} \right.,\text{ at }x=l \tag{3}\]  
and the initial conditions are as follows:  
\[\left\{ \begin{array}{l}w(x,t)=w_{0}\\ \frac{\partial w(x,t)}{\partial t}=0\end{array} \right.,\text{ at }x=l,t=0. \tag{4}\]  
It should be noted that the beam considered here is a piezoelectric beam, thus \(E\) should be the equivalent elastic modulus under the coupling of the elastic and piezoelectric properties in a  
Figure 1: Scheme of the mechanical model of the piezoelectric cantilever beam. one-dimensional pure bending problem. The constitutive relations of the piezoelectric materials are, in a two-dimensional case, as follows:  
\[\left\{ \begin{array}{l}\varepsilon_{x}=s_{11}\sigma_{x}+s_{13}\sigma_{z}+d_{31} E_{z}\\ \varepsilon_{z}=s_{13}\sigma_{x}+s_{33}\sigma_{z}+d_{33}E_{z}\\ \gamma_{zx}=s_{44}\tau_{zx}+d_{15}E_{x}\end{array} \right. \tag{5}\]  
and  
\[\left\{ \begin{array}{l}D_{x}=d_{15}\tau_{zx}+\lambda_{11}E_{x}\\ D_{z}=d_{31}\sigma_{x}+d_{33}\sigma_{z}+\lambda_{33}E_{z}\end{array} \right., \tag{6}\]  
in which, \(\sigma_{x}\), \(\sigma_{z}\), and \(\tau_{zx}\) are the stress components; \(D_{x}\) and \(D_{z}\) are the electric displacement components; \(\varepsilon_{x}\), \(\varepsilon_{z}\), and \(\gamma_{zx}\) are the strain components; \(E_{x}\) and \(E_{z}\) are the electric field intensity components; \(s_{11}\), \(s_{13}\), and \(s_{33}\) are the flexibility coefficients; \(d_{31}\), \(d_{33}\), and \(d_{15}\) are the piezoelectric coefficients; and \(\lambda_{11}\) and \(\lambda_{33}\) are the dielectric coefficients. For a one-dimensional pure bending problem, there is no stresses \(\sigma_{z}\) and \(\tau_{zx}\), as well as the corresponding strain \(\varepsilon_{z}\) and \(\gamma_{zx}\), only existing \(\sigma_{x}\) and \(\varepsilon_{x}\), from the point of view of deformation. Thus, the constitutive relations can be simplified as follows:  
\[\left\{ \begin{array}{l}\varepsilon_{x}=s_{11}\sigma_{x}+d_{31}E_{z}\\ \varepsilon_{z}=0\\ \gamma_{zx}=0\end{array} \right. \tag{7}\]  
and  
\[\left\{ \begin{array}{l}D_{x}=\lambda_{11}E_{x}\\ D_{z}=d_{31}\sigma_{x}+\lambda_{33}E_{z}\end{array} \right.. \tag{8}\]  
Generally, \(D_{x}\gg D_{z}\) and \(D_{z}\) are very small a in one-dimensional problem. Thus, we can suppose \(D_{z}\approx 0\).