## 3 Analysis for Forced Vibration Excited by Piezoelectric Sheets
### Forced Vibration Analysis of a Touch Beam
Using the Equations (16), (18), and (19), the \(i^{th}\) order modal force of the cantilever touch beam excited by the piezoelectric sheets can be written as  
\[ \begin{array}{c}F^{(i)}(t)=2\sum\limits_{n=1}^{3}\int_{n-1}^{n}\frac{Wh_{21}}{ \lambda_{n}}V_{n}g^{i\omega t}\delta(x-x_{0})\phi_{n1}^{(i)_{n}}dx\\ =\frac{-2 Wh_{21}V_{n}g_{21}^{\omega t}}{h_{y}}\\ \left[(A_{11}^{(i)}\beta_{11}^{(i)2}\sin\beta_{11}^{(i)}x+B_{11}^{(i)}\beta_{1 1}^{(i)2}\cos\beta_{11}^{(i)}x+C_{11}^{(i)}\beta_{11}^{(i)2}\sinh\beta_{11}^{( i)}x+D_{11}^{(i)}\beta_{11}^{(i)2}\cosh\beta_{11}^{(i)}x)\right.\\ \left.+(A_{21}^{(i)}\beta_{21}^{(i)2}\sin\beta_{21}^{(i)}x_{0}+B_{21}^{(i)} \beta_{21}^{(i)2}\cos\beta_{21}^{(i)2}x_{0}+C_{21}^{(i)}\beta_{21}^{(i)2}\sinh \beta_{21}^{(i)2}x_{0}+D_{21}^{(i)}\beta_{21}^{(i)2}\cosh\beta_{21}^{(i)}x_{0} \right)\\ \left.-(A_{11}^{(i)}\beta_{11}^{(i)2}\sin\beta_{11}^{(i)}x_{0}+B_{11}^{(i)} \beta_{11}^{(i)2}\cos\beta_{11}^{(i)}x_{0}+C_{11}^{(i)}\beta_{11}^{(i)2}\sinh \beta_{11}^{(i)}x_{0}+D_{11}^{(i)}\beta_{11}^{(i)2}\cosh\beta_{11}^{(i)}x_{0} \right)\\ \left.+(A_{31}^{(i)}\beta_{31}^{(i)2}\sin\beta_{31}^{(i)}x_{0}+B_{31}^{(i)} \beta_{31}^{(i)2}\cos\beta_{31}^{(i)}x_{0}+C_{31}^{(i)}\beta_{31}^{(i)2}\sinh \beta_{31}^{(i)}x_{0}+D_{31}^{(i)}\beta_{31}^{(i)2}\cosh\beta_{31}^{(i)2}x_{0} \right)\\ \left.-(A_{21}^{(i)}\beta_{21}^{(i)2}\sin\beta_{21}^{(i)2}x_{0}+B_{21}^{(i)} \beta_{21}^{(i)2}\cos\beta_{21}^{(i)2}x_{0}+C_{21}^{(i)}\beta_{21}^{(i)2}\sinh \beta_{21}^{(i)}x_{0}+D_{21}^{(i)}\beta_{21}^{(i)2}\cosh\beta_{21}^{(i)}x_{0} \right]=O^{(i)}e^{i\omega t}\end{array}  \tag{20}\]  
where \(A_{n1}\sim D_{n1}\) is the mode coefficient of the \(n\)th segment of the touch beam. Therefore, the steady-state displacement forced response of the cantilevered touch beam under piezoelectric bimorph excitation is  
\[ \begin{array}{c}w(x,t)=\sum\limits_{i=1}^{\infty}\frac{O^{(i)}\phi_{n1}^{(i )}(x)e^{i(\omega-\varphi^{(i)})}}{{{K^{(i)}}\sqrt{{(1-r^{2})}^{2}+{(2{\zeta^{( i)}}r)}^{2}}}}\\ =\sum\limits_{i=1}^{\infty}A^{(i)}B^{(i)}e^{i(\omega t-\varphi^{(i)})}\end{array}  \tag{21}\]  
where \(A^{(i)}=\frac{O^{(i)}\phi_{n1}^{(i)}(x)}{{{K^{(i)From Figure 6, the following points are worth noting. * As the order of vibration increases, the resonant frequency increases and the total amplitude decreases. The 6 th order amplitude of the cantilever touch beam is \(1.44\times 10^{-6}\) m. The peak number of the vibration touch beam increases with increasing excitation frequency. The wavelength of one cycle is shortened, so the vibration number of the ciliary-like body keeping the same direction in one cycle is changing less. The subject will feel the touch beam friction change is more when the finger touches and moves along the beam in one direction. * The amplitude of the vibration of the ciliary-like body touch beam at different positions during vibration is also different, and it is distributed in a sinusoidal manner. The feeling of the changing friction of the touch beam is more obvious when the vibration amplitude is larger, and the friction changes of the touch beam are weaker in the place where the vibration amplitude is smaller. The four key electromechanical parameters of the touch beam were selected to draw the relevant forced response curve with the excitation frequency close to the natural frequency of the 6 th order, which is suitable as a working frequency. The forced response changes with three different values corresponding to the four parameters that can be obtained as shown in Figure 7. The four parameters investigated are excitation voltage \(V_{3}\), piezoelectric sheet thickness \(h_{\text{p}}\), touch beam elastic modulus \(E\), and piezoelectric coefficient \(e_{31}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Length \(L\)** & **Thickness \(H\)** & **Width \(W\)** & **Elastic Modulus \(E\)** & **Density \(\rho\)** \\ \hline
0.1 m & 5 mm & 5 mm & 105 GPa & 8500 kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the touch beam.