## 3 Analysis for Forced Vibration Excited by Piezoelectric Sheets
### The Active Average Acceleration of the Ciliary-Like Body Structure  
According to the introduction of the principle, the modulation of the friction coefficient is determined by the inertial force when the ciliary-like body vibrates, and then the strength of inertial force is determined by the vibration acceleration, though the real-time acceleration of the ciliary-like body's vibration acts on the skin intermittently. Therefore, we need to establish the average acceleration acting on the skin. As shown in Figure 10, the touch beam moves from the dotted line to the solid line, and the ciliary-like body acts on the finger in half a cycle. As the vibration frequency of the touch beam is large and the period is extremely short, the variable acceleration motion of the ciliary-like body can be approximated as a uniform acceleration motion, so each ciliary-like body takes an average acceleration of half a vibration period. At the \(i\) order vibration frequency, the average ciliary-like body acceleration with position coordinate \(x_{j}\) can be expressed as  
\[a_{c}=\frac{2}{T}\int\limits_{0}^{\frac{T}{2}}\big{|}w^{(i)_{\pi}}(x_{j},t) \big{|}\mathrm{d}t \tag{23}\]  
where \(x_{j}\) is the position coordinate of the \(j\) ciliary-like body.  
Figure 10: Schematic diagram of the acceleration of the ciliary-like body.  
Figure 9: The effects of parameter changes on the sixth frequency response of the touch beam. (**a**) Change of the piezoelectric constant; (**b**) Change of the excitation voltage; (**c**) Change of modal stiffness; (**d**) Change of the damping ratio.  
In order to further study the effect of the ciliary-like body on the equivalent friction coefficient, it is necessary to analyze the functional relationship between the average acceleration of the ciliary-like body and the system parameters. The parameters are the same as those in Tables 1 and 2. In the case of the structure of the touch beam, the relationship between the vibration acceleration \(a_{c}\) of the ciliary-like body and the excitation frequency \(\omega\), the excitation voltage amplitude \(V_{3}\), the piezoelectric coefficient \(e_{31}\), and the position coordinate \(x_{j}\) is shown in Figure 11.  
From Figure 11 the following can be noted.  
* As the excitation frequency increases, the vibration acceleration of the cantilever touch beam ciliary-like body gradually increases, and the slope of the curve gradually increases. When the excitation frequency is increased from 10,000 Hz to 20,000 Hz, the vibration acceleration of the ciliary-like body is increased by about 45,000 m/s\({}^{2}\), and the excitation frequency is the 6-order natural frequency of 24,221 Hz. The maximum vibration acceleration of the ciliary-like body is 69,324 m/s\({}^{2}\).
* The vibration acceleration of the ciliary-like body is linearly positively correlated with the amplitude of the excitation voltage. When the excitation voltage is 100 V, the vibration acceleration of the ciliary-like body is 69,324 m/s\({}^{2}\). When the excitation voltage is 200 V, the vibration acceleration doubles to 13,8648 m/s\({}^{2}\).
* The vibration acceleration of the ciliary-like body is linearly positively correlated with the piezoelectric constant of the piezoelectric piece. When the piezoelectric constant is 3.0 c/m\({}^{2}\), the vibration acceleration is 10,864 m/s\({}^{2}\).
* The vibration acceleration of the ciliary-like body changes in different positions of the touch beam. As the position coordinates increase, the vibration acceleration exhibits a pulsating cycle change. The second peak at the left is 69,324 m/s\({}^{2}\), and the acceleration at some points is zero.  
Figure 11: Image of the vibration acceleration of the ciliary-like body and its parameters. (**a**) \(a_{c}\)-\(\omega\) image of ciliary-like body; (**b**) \(a_{c}\)-\(V_{3}\) image of ciliary-like body; (**c**) \(a_{c}\)-\(e_{31}\) image of ciliary-like body; (**d**) \(a_{c}\)-\(x_{j}\) image of ciliary-like body.