## 3 Analysis for Forced Vibration Excited by Piezoelectric Sheets
### Forced Vibration Analysis of a Touch Beam
 \tag{6}\]  
Figure 4: Dynamics model of the cantilever touch beam. where \(k_{ij}^{p}=k_{ji}^{p}=\sum\limits_{n=1}^{3}\left[\int_{n-1}^{l_{n}}\frac{\partial{ \it{N}}_{n}}{4}\dot{C}_{1}^{(i)}\phi_{n1}^{(j)_{n}}(x)\phi_{n1}^{(j)_{n}}(x){\rm d }x\right]\). The strain energy of a touch beam that does not include a piezoelectric sheet is  
\[ \begin{array}{c}V=\frac{1}{2}\sum\limits_{n=1}^{3}\left\{\int_{l_{n-1}}^{l_{ n}}\left(EI\right)_{n}\left[\frac{\partial^{2}w(x\prime t)}{\partial x^{2}} \right]^{2}{\rm d}x\right\}\\ =\frac{1}{2}\sum\limits_{n=1}^{3}\left\{\int_{l_{n-1}}^{l_{n}}\left(EI\right)_ {n}\left[\sum\limits_{i=1}^{\infty}\phi_{n}^{(i)_{n}}(x)q_{n}^{(i)}(t)\right] \left[\sum\limits_{j=1}^{\infty}\phi_{n}^{(j)_{n}}(x)q_{n}^{(j)}(t)\right]{\rm d }x\right\}\end{array}  \tag{7}\]  
where \(k_{ij}=k_{ji}=\sum\limits_{n=1}^{3}\int_{l_{n-1}}^{l_{n}}\left(EI\right)_{n} \phi_{n}^{(i)_{n}}(x)\phi_{n}^{(j)_{n}}(x){\rm d}x\), \(E\) is the elastic modulus of the touch beam, and \(I\) is the polar moment of inertia of the touch amount. Therefore, the total strain energy of the cantilever touch beam when vibrating is  
\[ \begin{array}{c}V_{n}=2V_{p}+V\end{array}  \tag{8}\]  
The kinetic energy of the cantilevered touch beam coupled to the piezoelectric sheet in vibration is  
\[\begin{array}{c}T_{n}=\frac{1}{2}\sum\limits_{n=1}^{3}\left\{\int_{n-1}^{n} \left(\rho S\right)_{n}\left[\frac{\partial{\it{\it{\it{\it{\it{\it{\it{\it{\it{ \it{\it{\it{\it{\it{\it{\it{\itit{\it{\it{\itit{{\ititit{ itit{{ ititit  
where \(M^{(i)}\), \(C^{(i)}\), and \(K^{(i)}\) are the \(i\) modal mass, modal damping, and modal stiffness of the cantilever touch beam, respectively, and the expressions of modal stiffness and modal mass are  
\[K^{(i)}=\sum\limits_{n=1}^{3}\int_{l_{n-1}}^{l_{n}}{(ES)_{n}\Bigg{(}\frac{\mathrm{ d}\phi_{n1}^{(i)}}{\mathrm{d}x}\Bigg{)}^{2}\mathrm{d}x} \tag{14}\]  
\[M^{(i)}=\sum\limits_{n=1}^{3}\int_{l_{n-1}}^{n}{(\rho S)_{n}\phi_{n1}^{(i)}} \mathrm{d}x \tag{15}\]  
Modal force is  
\[F^{(i)}(t)=\frac{Wh_{1}}{4}\sum\limits_{n=1}^{3}\int_{l_{n-1}}^{l_{n}}{e_{31}E _{3}\phi_{n1}^{(i)}}^{(i)}(x)\mathrm{d}x=O^{(i)}e^{i\omega t} \tag{16}\]  
where \(O^{(i)}=\frac{Whe_{11}}{4 lp}\sum\limits_{n=1}^{3}\int_{l_{n-1}}^{l_{n}}{V_{3} (x)\phi_{n1}^{(i)}}^{(i)}(x)\mathrm{d}x\). According to [PERSON]'s integral, the steady-state forced vibration response of the cantilever touch beam under piezoelectric bimorph excitation is  
\[w_{2}(x,t)=\sum\limits_{i=1}^{\infty}\phi_{n1}^{(i)}(x)q_{n1}^{(i)}(t)=\sum \limits_{i=1}^{\infty}\frac{O^{(i)}\phi_{n1}^{(i)}(x)e^{i(\omega t+\varphi^{( i)})}}{K^{(i)}\sqrt{\left(1-r^{2}\right)^{2}+\left(2\zeta^{(i)}r\right)^{2}}} \tag{17}\]  
where \(A^{(i)}=\frac{O^{(i)}\phi_{n1}^{(i)}(x)}{K^{(i)}}\), \(B^{(i)}=\left[\sqrt{\left(1-r^{2}\right)^{2}+\left(2\zeta^{(i)}r\right)^{2}} \right]^{-1}\), \(r\) is the frequency ratio, \(r=\omega/\omega_{n}\), \(\omega\) is the excitation frequency, \(\omega_{n}\) is the natural frequency of the touch beam, \(\zeta\) is the relative damping coefficient of the touch beam, \(\zeta=C/(2M\omega_{n})\), and \(\varphi\) is the initial state constant: \(\varphi=\arctan(2\zeta r/(1-r^{2}))\), \(0\leq\varphi\leq\pi\). As shown in Figure 5, the cantilever touch beam is subjected to the excitation force \(f(t)=Fe^{i\omega t}\) of the piezoelectric sheet at \(x=x_{0}\), and the excitation force of the piezoelectric sheet is regarded as the concentrated excitation force at \(x=x_{0}\). At this time, the concentrated excitation force can be expressed as a distributed force using a unit pulse function whose independent variable is \((x-x_{0})\). The distributed force generated by the piezoelectric sheet excitation can then be expressed as  
\[f(x,t)=f(t)\delta(x-x_{0}) \tag{18}\]  
The nature of the unit impulse function is  
\[\int_{-\infty}^{\infty}\delta(x-x_{0})G(x)=G(x_{0}) \tag{19}\]  
where \(\delta\) is Dirac delta function. Figure 5: Excitation of a cantilever touching beam by a piezoelectric bimorph.