## 2 The Basic Equations and Biparametric Perturbation Solution
### The Biparametric Perturbation Solution
\tag{30}\]  
The solution of Equation (29) is as follows  
\[Y_{0}^{0}(t)=B_{1}\sin(\omega_{1}t)+B_{2}\cos(\omega_{1}t), \tag{31}\]  
in which \(\omega_{1}=\sqrt{D^{4}I_{y}/\overline{m}s_{11}}\), \(\omega_{1}\) is the vibration frequency of the cantilever beam without a piezoelectric property, and \(B_{1}\) and \(B_{2}\) are the undetermined constants that can be determined by Equation (30). Substituting Equation (31) into Equation (30), one has the following:  
\[B_{1}=0,B_{2}=\frac{w_{0}}{A_{1}[\sin Dl-\sinh Dl+\frac{\sin Dl+\sinh Dl}{ \cos Dl+\cosh Dl}]\left(\cosh Dl-\cos Dl\right)]}. \tag{32}\]  
Comparing the coefficients of \(\left(d_{31}\right)^{2}\) and \(\left(c\right)^{1}\), we may obtain the first-order perturbation equations as follows,  
For \(\left(d_{31}\right)^{2}\):  
\[\overline{m}\lambda_{33}s_{11}\frac{d^{2}Y_{1}^{1}}{dt^{2}}-\overline{m}\frac {d^{2}Y_{0}^{0}}{dt^{2}}+D^{4}\lambda_{33}I_{y}Y_{1}^{1}=0. \tag{33}\]  
The corresponding initial conditions are as follows:  
\[\left\{ \begin{array}{l}v(x)Y_{1}^{1}=0\\ v(x)\frac{dY_{1}^{1}}{dt}=0\end{array} \right.,\text{ at }x=l,t=0. \tag{34}\]  
Solving Equation (33), we may obtain the following:  
\[Y_{1}^{1}(t)=B_{3}\sin(\omega_{1}t)+B_{4}\cos(\omega_{1}t)-\frac{\omega_{1}}{ 2\lambda_{33}s_{11}}B_{2}t\sin(\omega_{1}t), \tag{35}\]  
in which \(B_{3}\) and \(B_{4}\) are the undetermined constants that can be determined by Equation (34). Substituting Equation (35) into Equation (34), one has the following  
\[B_{3}=0,B_{4}=0. \tag{36}\]  
For \(\left(c\right)^{1}\):  
\[\frac{d^{2}Y_{2}^{1}}{dt^{2}}+\frac{1}{\overline{m}}\frac{dY_{0}^{0}}{dt}+ \frac{D^{4}I_{y}}{\overline{m}s_{11}}Y_{2}^{1}=0. \tag{37}\]The corresponding initial conditions are as follows  
\[\left\{ \begin{array}{l}v(x)Y_{2}^{\rm I}=0\\ v(x)\frac{dY_{2}^{\rm I}}{dt}=0\end{array} \right.,\text{ at }x=l,t=0. \tag{38}\]  
Solving Equation (37), gives the following  
\[Y_{2}^{\rm I}(t)=B_{5}\sin(\omega_{1}t)+B_{6}\cos(\omega_{1}t)-\frac{1}{2\overline {m}}B_{2}t\cos(\omega_{1}t), \tag{39}\]  
in which \(B_{5}\) and \(B_{6}\) are undetermined constants, which can be determined by Equation (38). From Equations (38) and (39), we can obtain the following:  
\[B_{5}=\frac{1}{2\overline{m}\omega_{1}}B_{2},B_{6}=0. \tag{40}\]  
Comparing the coefficients of \((d_{31})^{4}\), \((c)^{2}\), and \(c(d_{31})^{2}\), we may obtain the second-order perturbation equations as follows,  
For \((d_{31})^{4}\):  
\[\overline{m}\lambda_{33}s_{11}\frac{d^{2}Y_{1}^{\rm II}}{dt^{2}}-\overline{m} \frac{d^{2}Y_{1}^{\rm I}}{dt^{2}}+D^{4}\lambda_{33}I_{y}Y_{1}^{\rm II}=0. \tag{41}\]  
The corresponding initial conditions are as follows  
\[\left\{ \begin{array}{l}v(x)Y_{1}^{\rm II}=0\\ v(x)\frac{dY_{1}^{\rm II}}{dt}=0\end{array} \right.,\text{ at }x=l,t=0. \tag{42}\]  
From Equation (41), it can be obtained that  
\[Y_{1}^{\rm II}=B_{7}\sin(\omega_{1}t)+B_{8}\cos(\omega_{1}t)-\frac{3\omega_{1 }}{8(\lambda_{33}s_{11})^{2}}B_{2}t\sin(\omega_{1}t)-\frac{(\omega_{1})^{2}}{ 8(\lambda_{33}s_{11})^{2}}B_{2}t^{2}\cos(\omega_{1}t), \tag{43}\]  
in which \(B_{7}\) and \(B_{8}\) are undetermined constants, which can be determined by Equation (42). Substituting Equation (43) into Equation (42), one has the following  
\[B_{7}=0,B_{8}=0. \tag{44}\]  
For \((c)^{2}\):  
\[\frac{d^{2}Y_{2}^{\rm II}}{dt^{2}}+\frac{D^{4}I_{y}}{\overline{m}s_{11}}Y_{2} ^{\rm II}=-\frac{1}{\overline{m}}\frac{dY_{2}^{\rm I}}{dt}. \tag{45}\]  
The corresponding initial conditions are as follows:  
\[\left\{ \begin{array}{l}v(x)Y_{2}^{\rm II}=0\\ v(x)\frac{dY_{2}^{\rm II}}{dt}=0\end{array} \right.,\text{ at }x=l,t=0.