## 3 Analysis for Forced Vibration Excited by Piezoelectric Sheets  
### Forced Vibration Analysis of a Touch Beam  
The forced vibration dynamic equations of a touch beam have to be established based on free vibration solution results. In [19], the establishment process and analysis results of the free vibration equation for the ciliary-like body touch beam were released. It was also proved that the ciliary-like body contributes little effect on the dynamic response characteristics by simulation and theoretical methods. Therefore, the ciliary-like body beam can be simplified to behave as a plain beam as shown in Figure 4. Due to the condition of the touch beams where one of the ends is fixed, we consider the boundary condition to be that of the cantilever. The dynamic model of the cantilever touch beam is established, which is divided into three segments: \(l_{1}\), \(l_{2}\), and \(l_{3}\). The piezoelectric piece is attached to \(x_{0}\). The piezoelectric sheets are connected with the driving signals. As the piezoelectric touch beam performs the haptic execution function after the vibration tends to stabilize, the focus here is to analyze the steady-state response of the touch beam. Figure 3: Working principle. Figure 2: The dimensions of the touch beam. We suppose the voltage of the excitation signal is a sinusoidal signal, and the excitation voltage supplied to the two piezoelectric sheets can be expressed as \(V_{e}=V_{3}(x_{0})e^{i\omega}\). Therefore, the average electric field can be considered as  
\[E_{3}(t)=\frac{V_{3}(x_{0})e^{i\omega t}}{h_{p}} \tag{1}\]  
where \(h_{p}\) is the thickness of piezoelectric sheets, \(V_{3}\) is the excitation voltage amplitude, and \(w\) is the excitation frequency. The bending vibration displacement of the touch beam can be assumed as \(w\left(x,t\right)\). The general or total solution of the nth segment of the touch beam can be expressed using the method of separation of variables as  
\[w(x,t)=\sum\limits_{i=1}^{\infty}\phi_{n}^{(i)}(x)q_{n}^{(i)}(t) \tag{2}\]  
where \(\phi_{n}^{(i)}\left(x\right)\) is the intrinsic mode function of the \(i\)-th order of the nth segment of the touch beam, and \(q\)_(t)_ is the generalized coordinate of the bending vibration of the \(i\)-th order of the nth segment of the touch beam. The solution method and process of \(\phi_{n}^{(i)}\) is revealed in [19], so there are no more details listed here. For the piezoelectric sheet's strain, two piezoelectric sheets are attached to the upper and lower surfaces of the touch beam separately at the abscissa \(x_{0}\), so the strain of the piezoelectric sheet is obtained  
\[\varepsilon_{1}=\frac{h}{2}\frac{\partial^{2}w(x,t)}{\partial x^{2}} \tag{3}\]  
where \(h\) is the thickness of the touch beam. From the second piezoelectric equation, the unit stress can be obtained. \[\sigma_{1}(x,t)=-e_{31}E_{3}+\frac{h}{2}c_{11}^{E}\frac{\partial^{2}w(x,t)}{ \partial x^{2}} \tag{4}\]  
where \(e_{31}\) is piezoelectric constant and \(c_{11}^{E}\) is the elasticity coefficient of piezoelectric material. The distributed strain energy of the piezoelectric sheet can be written as  
\[V_{p}=\sum\limits_{n=1}^{3}\left[\frac{W}{2}\int_{l_{n-1}}^{l_{n}}\sigma_{1}(x,t)\varepsilon_{1}\mathrm{d}x\right] \tag{5}\]  
where \(W\) is the width of the touch beam. Then, substituting Equations (2)-(4) into Equation (5), the strain energy of the piezoelectric sheet becomes  
\[ \begin{split} V_{p}=\sum\limits_{n=1}^{3}\left\{\frac{W_{W}}{4} \int_{l_{n-1}}^{l_{n}}(-e_{31}E_{3}\sum\limits_{i=1}^{\infty}\phi_{n}^{(i)}( x)q_{n}^{(i)}(t))\mathrm{d}x+\frac{W_{W}}{8}\int_{l_{n-1}}^{l_{n}}\left\{c_{11} ^{E}\left[\sum\limits_{i=1}^{\infty}\phi_{n}^{(i)}q_{n}^{(i)}(t)\right]\cdot \left[\sum\limits_{i=1}^{\infty}\phi_{n}^{(i)}q_{n}^{(i)}(t)\right]\right\} \mathrm{d}x\right\}\\ =\sum\limits_{n=1}^{3}\left\{\frac{W_{W}}{4}\sum\limits_{i=1}^{ \infty}q_{n}^{(i)}(t)\int_{l_{n-1}}^{l_{n}}\left[-e_{31}E_{3}\phi_{n}^{(i)}(x) \right]\mathrm{d}x+\frac{1}{2}\sum\limits_{i=1}^{\infty}k_{ij}^{\prime}q_{n}^ {(i)}(t)q_{n}^{(i)}(t)\right\}\end{split}