Investigating Vibration Acceleration of a Segmented Piezoelectric Ciliary-Like Body Beam for a Tactile Feedback Device

Abstract

A piezoelectric Ciliary-like body beam of a tactile feedback device can realize a touchpoint of different tactile sensations under simple control when the finger movement changes to the opposite direction. In a previous published study, the friction of touch sensation was shown to depend on the acceleration of forced vibration of the ciliary-like body beam. For investigating the system parameters’ effect on vibration accelerations, the dynamic model of forced vibration of the touch beam is established, and the steady-state response of the touch beam excited by piezoelectric sheets is deduced. The influence of instantaneous acceleration and average acceleration of the touch beam on skin was analyzed, and an experiment was conducted to prove the theoretical analysis. The study results show that larger excitation voltage, larger piezoelectric constants, smaller elasticity modulus, and smaller damping ratio would enhance the displacement and acceleration of the forced response of the touch beam. Through the experimental results, the working mode and frequency of the touch beam was obtained, and the correctness of the theoretical analysis was verified.

1. Introduction

Virtual reality technology can provide real-time stimulation and interaction from the sensation channels of vision, hearing, touch, etc. Stimulations focused on vision and hearing have been developed in many virtual applications [1]. With the development of virtual technology, people are aware of the importance of tactile feeling technology. Tactile reproduction technology not only further supplements the information transmission channel, but also assists people in having an enhanced physical engagement when using virtual reality devices [2,3]. Among the working principles of tactile reproduction technology, the use of piezoelectric materials as actuators that create vibration or ultrasonic waves and transmit haptic information has obvious advantages.

In 1995, T. Watanabe proposed a tactile method to control the surface roughness. The method created a smoother feeling on a surface by applying ultrasonic vibration. The sensation could thus be controlled without altering the actual surface profile. The smooth feeling caused by this method was assumed to be the “air film squeeze effect” between the finger and the device surface [4]. As a solution to the lack of compactness and simplicity often encountered in haptic interfaces, Gaston M’Boungui proposes a device based on the friction coefficient control principle. A passive tactile feedback device with two-dimensional degrees of freedom is designed with the principle of active lubrication, and the effectiveness of friction control is verified by simulation [5]. In 2013, Fabrice Casset developed a unique solution based on PZT thin-film actuated plates allowing haptic feedback effect using low bias voltage due to the squeeze-film effect [6]. In 2017, Gözde Sari designed a piezoelectric touch screen, which generates high-amplitude vibration on the surface of the touch screen through piezoelectric actuators on the surface of the touch screen. In this study, the author explores the effects of the voltage, frequency, position of the piezoelectric sheet on vibration. The vibration mode and amplitude can be changed by changing the applied frequency and voltage [7]. The above solution can only cause the tactile feedback device to generate simple bending vibrations, and realize a certain point of vibration or no vibration. The diversity of tactile effects is not good and cannot achieve different tactile feelings at the same position.

In 2006, Masaya Takasaki introduced a solution for remote touch using an active surface acoustic wave tactile display [8]. The tactile feedback device was composed of a position sensor, an interdigital transducer, a piezoelectric substrate, a slider, etc., and the Rayleigh wave generated by the excitation signal acts on the slider, thereby continuously producing a touch sensation. In 2012, Frédéric Giraud proposed a transparent piezoelectric tactile stimulator [9]. Multiple piezoelectric plates were placed on both sides of the glass plate to stimulate the bending vibration of the glass plate, and four force sensors were placed to infer the fingertip’s position to achieve a good touch effect. In 2013, Frédéric Giraud designed a piezoelectric haptic knob with a number of piezoelectric plates around the knob to adjust the amplitude by detecting the position and pressure of the finger, and then adjusted the tactile stimuli [10]. In 2015, Yen-Ming Chen and others designed a one-dimensional piezoelectric touch screen that can provide touch position and sliding detection functions, and developed a proposed sliding analysis method to realize touch position detection, based on the touch mechanism proposed in the research. A previous work established an experimental structure to prove the effectiveness of the proposed piezoelectric touch screen [11]. In 2018, Muhammad Khurram Saleem designed a tactile feedback device for friction changes caused by ultrasonic vibration of glass plates. In this device, two piezoelectric plates were driven by the power amplifier to control the surface vibration of the glass plate. The force sensor measured the normal force and the lateral force of the fingers. A piezoelectric patch was used as the sensor to obtain the vibration amplitude. Through these sensors, the vibration of the glass plate was adjusted, which caused the continuous change of the tactile effect [12]. Many tactile feedback devices utilize position sensors to detect the spatial position of fingers to change the excitation frequency, so as to realize the continuous change of touch sensation. In realizing variable tactile sensations, this type of haptic device requires a closed-loop control system that needs to be supported by extra circuits. Researchers usually neglect the importance of structural dynamic design to implement more diverse tactile sensations under simple control. On the other hand, some applications of haptic devices are fabricated by array actuators. For example, a piezoelectric tactile feedback device developed by McGill at the University in Canada had 64 piezoelectric exciters, 112 contact pins, and 36 grooves [13]. The tactile feedback actuator can realize the touch sensation of lines, surfaces, shapes, and geometric boundaries, but the structure of the device was very complex. In 2002, Yasushi designed a texture reproduction tactile feedback device by means of piezoelectric sheet and pin array. The device connected the needle array excited by the piezoelectric plate and tactile plane as the skin tactile excitation, and the stylus array was arranged according to 5 × 2. The device used array touch to stimulate the human fingers, which produced different effects of tactile feedback [14]. In 2008, Shuichi Ino designed a broadband vibrotactile display, where a pin matrix tactile display composed of 12 piezoelectric bimorph actuators was used to elicit tactile sensations [15]. The piezoelectric bimorph actuator of the display possessed flat frequency characteristics in the perceivable bandwidth of vibratory sensations. In 2018, Gi-Hun Yang developed a vibrotactile pedestal device, which was a tactile-based information transmission device using multiple vibration actuators. The developed device consisted of 12 discrete vibrating tactile areas, mainly by using a large number of actuators to create the detailed haptics [16]. In 2015, Takayuki Hoshi of the University of Tokyo released a spatial ultrasonic tactile display device at the World Touch Technology Conference. The device uses a piezoelectric array driver to emit ultrasonic waves on a tactile plane to stimulate human skin to produce spatial tactile sensations [17]. The haptic feedback devices using array actuators have problems such as complex structure, they are not easy to carry, and have high cost. In the above-mentioned research report on the haptic feedback devices, it can be seen that few people have focused on the realization of a rich and diverse tactile pattern under open-loop control using the dynamic structural design of the haptic devices.

To explore and solve the above-mentioned issues, Xing presented a tactile feedback device fabricated by piezoelectric ciliary-like body beams [18,19,20]. The tactile feedback device can create a different rough or smooth feeling by changing the finger movement direction when the subject touches a point on the ciliary-like beam. In previous studies, the authors have investigated the free vibration characteristics of the ciliary-like beam. In addition, the anisotropic vibration tactile model of the ciliary-like beam was established for predicting the equivalent friction coefficient when locally covered and fully covered by skin. In [18], it was shown that the acceleration of the beam vibration is vital for effecting the equivalent friction coefficient, but the method of obtaining the forced acceleration response of the ciliary-like beam was not presented in detail. This paper presents the detailed solution of the forced response of the ciliary-like beam under piezoelectric excitation, and the instantaneous acceleration and the average acceleration effect on the skin are obtained. Meanwhile, the effect of the system parameters is also investigated. The research results obtained can provide a theoretical basis for the design of a ciliary-like body touch beam structure.

2. Principle

The piezoelectric haptic feedback actuator is shown in Figure 1, which includes the touch beam, LCD1602 liquid crystal display, IIC communication conversion module, Arduino controller, HC-06 Bluetooth, AD9850, operational amplifier, PDU100B piezoelectric drive module, BBxx12N-3W boost module, and other parts. An Android phone can communicate with the Arduino controller through Bluetooth. The Arduino outputs a digital signal to the DAC and converts it into a sinusoidal analog signal, which is then transmitted to the piezoelectric sheets through the operational amplifier and the piezoelectric driving module to excite the touch beam to resonate.

The key parts of the tactile device are the touch beams. Array bars are arranged with the same spacing on the touch beams to create the piezoelectric tactile feedback. Here, they are called the ciliary-like body, which is the main factor that produces tactile differences. One of the touch beams is shown in Figure 2. The left side length of the touch beam with a ciliary-like body is longer than the right side. This is because the location of the area without a ciliary-like body is calculated for mounting the piezoelectric sheets for exciting the expected working mode.

When a sinusoidal signal with a frequency close to the natural frequency of the touch beam is supplied to the piezoelectric sheets, the touch beam resonates and produces a bending vibration. The bars in different positions of the touch beam will vibrate along the normal direction of the standing wave. As shown in Figure 3, the ciliary-like body 3 and 4 under the finger is distributed on the right and left side of the vibration peak separately. At this time, the ciliary-like body distributed on the right side of the peak will give the finger upward inertial pressure and the rightward thrust. The ciliary-like body distributed on the left side of the peak will give the finger upward inertial force and leftward thrust. Therefore, when one finger moves to the right, the ciliary-like body on the right side of the peak will make the subject feel smoother than the feeling made by the ciliary-like bodies on the left side of the peak. Conversely, the result is the opposite, when the finger moves to the left. The change in tactile sensation is due to the fact that the equivalent friction coefficient between the finger and the surface is modulated by the vibrating ciliary-like body on the beam. The equivalent friction depends on the ratio of the ciliary-like body in both directions covered by the finger. The details for predicting the equivalent friction have been presented in [18]. We also can change the frequency and amplitude of the excitation signal to modulate a certain point of roughness feeling.

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: l₁, l₂, and l₃. The piezoelectric piece is attached to x₀. 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.

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 = V₃(x₀)e^iω. Therefore, the average electric field can be considered as

$$E_3(t)=\frac{V_3(x_0)e^{i\omega t}}{h_p}$$

(1)

where h_p is the thickness of piezoelectric sheets, V₃ is the excitation voltage amplitude, and w is the excitation frequency. The bending vibration displacement of the touch beam can be assumed as w (x, t). The general or total solution of the nth segment of the touch beam can be expressed using the method of separation of variables as

$$w(x,t)={\displaystyle \sum _{i=1}^{\infty }{\varphi }_n^{(i)}(x)q_n^{(i)}(t)}$$

(2)

where ϕ_n⁽ⁱ⁾ (x) is the intrinsic mode function of the i-th order of the n^th segment of the touch beam, and q(t) is the generalized coordinate of the bending vibration of the i-th order of the n^th segment of the touch beam. The solution method and process of ϕ_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

$${\epsilon }_1=\frac{h}{2}\frac{{\partial }^{2}w(x,t)}{\partial x^2}$$

(3)

where h is the thickness of the touch beam. From the second piezoelectric equation, the unit stress can be obtained.

$${\sigma }_1(x,t)=-e_{31}{E}_3+\frac{h}{2}{c}_{11}^E\frac{{\partial }^{2}w(x,t)}{\partial x^2}$$

(4)

where e₃₁ 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

$$V_p={\displaystyle \sum _{n=1}^3\left[\frac{W}{2}{\displaystyle {\int }_{l_{n-1}}^{l_n}{\sigma }_1(x,t){\epsilon }_1\mathrm{d}x}\right]}$$

(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

$$\begin{gathered} V_p={\displaystyle \sum _{n=1}^3\left\{\frac{Wh}{4}{\displaystyle {\int }_{l_{n-1}}^{l_n}(-e_{31}{E}_3{\displaystyle \sum _{i=1}^{\infty }{\varphi }_n^{(i)}{}^{″}}(x)q_n^{(i)}(t))\mathrm{d}x}+\frac{Wh}{8}{\displaystyle {\int }_{l_{n-1}}^{l_n}\left\{c_{11}^E\left[{\displaystyle \sum _{i=1}^{\infty }{\varphi }_n^{(i)}{}^{″}{q}_n^{(i)}(t)}\right]\cdot \left[{\displaystyle \sum _{j=1}^{\infty }{\varphi }_n^{(j)}{}^{″}{q}_n^{(j)}(t)}\right]\right\}}\mathrm{d}x\right\}} \\ ={\displaystyle \sum _{n=1}^3\left\{\frac{Wh}{4}{\displaystyle \sum _{i=1}^{\infty }{q}_n^{(i)}(t){\displaystyle \underset{l_{n-1}}{\overset{l_n}{\int }}\left[-e_{31}{E}_3{\varphi }_n^{(i)}{}^{″}(x)\right]\mathrm{d}x+\frac{1}{2}{\displaystyle \sum _{i=1}^{\infty }{k}_{ij}^p{q}_n^{(i)}(t)q_n^{(j)}(t)}}}\right\}} \end{gathered}$$

(6)

where $k_{ij}^p=k_{ji}^p={\displaystyle \sum _{n=1}^3\left[{\displaystyle {\int }_{l_{n-1}}^{l_n}\frac{Wh}{4}{c}_{11}^E{\varphi }_{n1}^{(i)}{}^{″}(x){\varphi }_{n1}^{(j)}{}^{″}(x)\mathrm{d}x}\right]}$.

The strain energy of a touch beam that does not include a piezoelectric sheet is

$$\begin{gathered} V=\frac{1}{2}{\displaystyle \sum _{n=1}^3\left\{{{\displaystyle {\int }_{l_{n-1}}^{l_n}{(EI)}_n\left[\frac{{\partial }^{2}w(x\prime t)}{\partial x^2}\right]}}^2\mathrm{d}x\right\}} \\ =\frac{1}{2}{\displaystyle \sum _{n=1}^3\left\{{\displaystyle {\int }_{l_{n-1}}^{l_n}{(EI)}_n}\left[{\displaystyle \sum _{i=1}^{\infty }{\varphi }_n^{(i)}{}^{″}(x)q_n^{(i)}(t)}\right]\left[{\displaystyle \sum _{j=1}^{\infty }{\varphi }_n^{(j)}{}^{″}(x)q_n^{(j)}(t)}\right]\mathrm{d}x\right\}} \end{gathered}$$

(7)

where $k_{ij}=k_{ji}={\displaystyle \sum _{n=1}^3{\displaystyle {\int }_{l_{n-1}}^{l_n}{(EI)}_n{\varphi }_n^{(i)}{}^{″}(x)}{\varphi }_n^{(j)}{}^{″}(x)\mathrm{d}x}$, 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

$$V_n=2V_p+V$$

(8)

The kinetic energy of the cantilevered touch beam coupled to the piezoelectric sheet in vibration is

$$\begin{gathered} T_n=\frac{1}{2}{\displaystyle \sum _{n=1}^3\left\{{\displaystyle {\int }_{n-1}^n{(\rho S)}_n{\left[\frac{\partial w}{\partial t}(x,t)\right]}^2\mathrm{d}x}\right\}} \\ =\frac{1}{2}{\displaystyle \sum _{n=1}^3\frac{1}{2}{\displaystyle \sum _{n=1}^3{\displaystyle \sum _{i=1}^{\infty }{\displaystyle \sum _{j=1}^{\infty }{\dot{q}}_n^{(i)}(t)}}}}{\dot{q}}_n^{(j)}(t){\displaystyle {\int }_{l_{n-1}}^n{(\rho S)}_n{\varphi }_n^{(i)}}(x){\varphi }_n^{(j)}(x)\mathrm{d}x \\ =\frac{1}{2}{\displaystyle \sum _{n=1}^3{\displaystyle \sum _{i=1}^{\infty }{\displaystyle \sum _{j=1}^{\infty }{k}_{ij}}}}{q}_n^{(i)}(t)q_n^{(j)}(t) \end{gathered}$$

(9)

where $\rho$ is the density of the touch beam and $S$ is the cross-sectional area of the touch team.

$m_{ij}=m_{ji}={\displaystyle \sum _{n=1}^3{\displaystyle {\int }_{n-1}^n\rho S{\varphi }_{n1}^{(i)}}}(x){\varphi }_{n1}^{(j)}(x)\mathrm{d}x$

Considering the effect of damping, the dissipation function can be expressed as

$$D_n=\frac{1}{2}{\displaystyle \sum _{n=1}^3{\displaystyle \sum _{i=1}^{\infty }{\displaystyle \sum _{j=1}^{\infty }{c}_{ij}{\dot{q}}_{n1}^{(i)}(t)}}}{\dot{q}}_{n1}^{(j)}(t)$$

(10)

where $c_{ij}=c_{ji}={\displaystyle \sum _{n=1}^3{\displaystyle {\int }_{l_{n-1}}^{l_n}C{\varphi }_{n1}^{(i)}(x)}{\varphi }_{n1}^{(j)}(x)\mathrm{d}x}$.

Substituting q_n into the Lagrangian equation, we can obtain

$$\frac{d}{dt}\left(\frac{\partial T_n}{\partial {\dot{q}}_{n1}^{(i)}}\right)-\frac{\partial T_n}{\partial {\dot{q}}_{n1}^{(i)}}+\frac{\partial V_n}{\partial q_{n1}^{(i)}}+\frac{\partial D_n}{\partial {\dot{q}}_{n1}^{(i)}}=0,i=1,2,\dots ,\infty$$

(11)

Substituting Equations (8)–(10) into Equation (11), we can obtain the transverse vibration equation of the touch beam under the excitation of the piezoelectric sheet as

$$\mathbf{M}\ddot{q}(t)+\mathbf{C}\dot{q}(t)+\mathbf{K}q(t)=F(t)$$

(12)

where M = [m_ij] is the generalized mass matrix of the touch beam, C = [c_ij] is the generalized damping matrix of the touch beam, K = [k_ij] is the generalized stiffness of the touch beam, and $k_{ij}^n=k_{ij}^p+k_{ij}$; $F(t)=\frac{Wh}{4}{\displaystyle \sum _{n=1}^3\left[{\displaystyle {\int }_{l_{n-1}}^n{e}_{31}{E}_3{\varphi }_{n1}^{(i)}{}^{″}(x)\mathrm{d}x}\right]}$ is a generalized force array.

The following equation can be obtained by the condition of the displacement mode orthogonality,

$$M_{}^{(i)}\ddot{q}(t)+C_{}^{(i)}\dot{q}(t)+K^{(i)}q(t)=F^{(i)}(t)$$

(13)

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

$$K^{(i)}={\displaystyle \sum _{n=1}^3{{\displaystyle {\int }_{l_{n-1}}^{l_n}{(ES)}_n\left(\frac{\mathrm{d}{\varphi }_{n1}^{(i)}}{\mathrm{d}x}\right)}}^2\mathrm{d}x}$$

(14)

$$M^{(i)}={\displaystyle \sum _{n=1}^3{\displaystyle {\int }_{l_{n-1}}^n{(\rho S)}_n{\varphi }_{n1}^{(i)}{}^2}}\mathrm{d}x$$

(15)

Modal force is

$$F^{(i)}(t)=\frac{Wh}{4}{\displaystyle \sum _{n=1}^3{\displaystyle {\int }_{l_{n-1}}^{l_n}{e}_{31}{E}_3{\varphi }_{n1}^{(i)}{}^{″}}(x)\mathrm{d}x}=O^{(i)}{e}^{i\omega t}$$

(16)

where $O^{(i)}=\frac{Wh{e}_{31}}{4h_p}{\displaystyle \sum _{n=1}^3{\displaystyle {\int }_{l_{n-1}}^{l_n}{V}_3(x)}}{\varphi }_{n1}^{(i)}{}^{″}(x)\mathrm{d}x$.

According to Duhamel’s integral, the steady-state forced vibration response of the cantilever touch beam under piezoelectric bimorph excitation is

$$\begin{gathered} w_2(x,t)={\displaystyle \sum _{i=1}^{\infty }{\varphi }_{n1}^{(i)}(x)q_{n1}^{(i)}(t)}={\displaystyle \sum _{i=1}^{\infty }\frac{O^{(i)}{\varphi }_{n1}^{(i)}(x)e^{i(\omega t+{\phi }^{(i)})}}{K^{(i)}\sqrt{{(1-r^2)}^2+{(2{\zeta }^{(i)}r)}^2}}} \\ ={\displaystyle \sum _{i=1}^{\infty }{A}^{(i)}{B}^{(i)}{e}^{i(\omega t-{\phi }^{(i)})}} \end{gathered}$$

(17)

where $A^{(i)}=\frac{O^{(i)}{\varphi }_{n1}^{(i)}(x)}{K^{(i)}}$, $B^{(i)}={\left[\sqrt{{(1-r^2)}^2+{(2{\zeta }^{(i)}r)}^2}\right]}^{-1}$, r is the frequency ratio, r = ω/ω_n, ω is the excitation frequency, ω_n is the natural frequency of the touch beam, ζ is the relative damping coefficient of the touch beam, ζ = C/(2Mω_n), and φ is the initial state constant: φ = arctan(2ζr/(1 − r²)), 0 ≤ φ ≤ π.

As shown in Figure 5, the cantilever touch beam is subjected to the excitation force f(t) = Fe^iωt of the piezoelectric sheet at x = x₀, and the excitation force of the piezoelectric sheet is regarded as the concentrated excitation force at x = x₀. At this time, the concentrated excitation force can be expressed as a distributed force using a unit pulse function whose independent variable is (x − x₀). The distributed force generated by the piezoelectric sheet excitation can then be expressed as

$$f(x,t)=f(t)\delta (x-x_0)$$

(18)

The nature of the unit impulse function is

$${\displaystyle {\int }_{-\infty }^{\infty }\delta (x-x_0)G(x)=G(x_0)}$$

(19)

where $\delta$ is Dirac delta function.

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

$$\begin{gathered} F^{(i)}(t)=2{\displaystyle \sum _{n=1}^3{\displaystyle {\int }_{l_{n-1}}^{l_n}\frac{Wh{e}_{31}}{4h_p}{V}_3{e}^{i\omega t}\delta (x-x_0){\varphi }_{n1}^{(i)}{}^{″}\mathrm{d}x}} \\ =\frac{-2Wh{e}_{31}{V}_3{e}^{i\omega t}}{h_p} \\ [(A_{11}^{(i)}{\beta }_{11}^{(i)}{}^2\sin {\beta }_{11}^{(i)}x+B_{11}^{(i)}{\beta }_{11}^{(i)}{}^2\cos {\beta }_{11}^{(i)}x+C_{11}^{(i)}{\beta }_{11}^{(i)}{}^2\sinh {\beta }_{11}^{(i)}x+D_{11}^{(i)}{\beta }_{11}^{(i)}{}^2\cosh {\beta }_{11}^{(i)}x) \\ +(A_{21}^{(i)}{\beta }_{21}^{(i)}{}^2\sin {\beta }_{21}^{(i)}{x}_0+B_{21}^{(i)}{\beta }_{21}^{(i)}{}^2\cos {\beta }_{21}^{(i)}{x}_0+C_{21}^{(i)}{\beta }_{21}^{(i)}{}^2\sinh {\beta }_{21}^{(i)}{x}_0+D_{21}^{(i)}{\beta }_{21}^{(i)}{}^2\cosh {\beta }_{21}^{(i)}{x}_0) \\ -(A_{11}^{(i)}{\beta }_{11}^{(i)}{}^2\sin {\beta }_{11}^{(i)}{x}_0+B_{11}^{(i)}{\beta }_{11}^{(i)}{}^2\cos {\beta }_{11}^{(i)}{x}_0^{}+C_{11}^{(i)}{\beta }_{11}^{(i)}{}^2\sinh {\beta }_{11}^{(i)}{x}_0+D_{11}^{(i)}{\beta }_{11}^{(i)}{}^2\cosh {\beta }_{11}^{(i)}{x}_0) \\ +(A_{31}^{(i)}{\beta }_{31}^{(i)}{}^2\sin {\beta }_{31}^{(i)}{x}_0+B_{31}^{(i)}{\beta }_{31}^{(i)}{}^2\cos {\beta }_{31}^{(i)}{x}_0+C_{31}^{(i)}{\beta }_{31}^{(i)}{}^2\sinh {\beta }_{31}^{(i)}{x}_0+D_{31}^{(i)}{\beta }_{31}^{(i)}{}^2\cosh {\beta }_{31}^{(i)}{x}_0) \\ -(A_{21}^{(i)}{\beta }_{21}^{(i)}{}^2\sin {\beta }_{21}^{(i)}{x}_0+B_{21}^{(i)}{\beta }_{21}^{(i)}{}^2\cos {\beta }_{21}^{(i)}{x}_0+C_{21}^{(i)}{\beta }_{21}^{(i)}{}^2\sinh {\beta }_{21}^{(i)}{x}_0+D_{21}^{(i)}{\beta }_{21}^{(i)}{}^2\cosh {\beta }_{21}^{(i)}{x}_0)] \\ =O^{(i)}{e}^{i\omega t} \end{gathered}$$

(20)

where A_n1 ~ D_n1 is the mode coefficient of the nth segment of the touch beam.

Therefore, the steady-state displacement forced response of the cantilevered touch beam under piezoelectric bimorph excitation is

$$\begin{gathered} w_{}(x,t)={\displaystyle \sum _{i=1}^{\infty }\frac{O^{(i)}\cdot {\varphi }_{n1}^{(i)}(x)e^{i(\omega t-{\phi }^{(i)})}}{K^{(i)}\sqrt{{(1-r^2)}^2+{(2{\zeta }^{(i)}r)}^2}}} \\ ={\displaystyle \sum _{i=1}^{\infty }{A}^{(i)}{B}^{(i)}{e}^{i(\omega t-{\phi }_{}^{(i)})}} \end{gathered}$$

(21)

where $A^{(i)}=\frac{O^{(i)}{\varphi }_{n1}^{(i)}(x)}{K^{(i)}}$,$B^{(i)}={\left[\sqrt{{(1-r^2)}^2+{(2{\zeta }^{(i)}r)}^2}\right]}^{-1}$, using the second derivative, the acceleration response of the contact beam can be obtained as $a=\ddot{w}(x,t)$.

According to Laplace transformation, the response function of the touch beam in the frequency domain can be given by

$$\left|H(\omega )\right|=\frac{O^{(i)}\cdot {\varphi }_n^{(i)}(x_k)}{K^{(i)}\sqrt{{\left(\frac{{\omega }_n^2-{\omega }^2}{{\omega }_n^2}\right)}^2+{(2{\zeta }^{(i)}\frac{\omega }{{\omega }_n})}^2}}$$

(22)

where x_k is the maximum position of the vibration amplitude of the touch beam, and the first peak or trough from the right side is selected as the research object.

4. Results Analysis for the Dynamic Characteristic

4.1. Analysis for the Forced Response

Using the dynamic response equations of the touch beam, numerical simulation analysis can be conducted to reveal the influence of system parameters on its dynamic performance. Here, the excitation voltage is V₃ = 100 V, the damping ratio ζ = 0.025, and the position on the beam x = 0.02 m is chosen for the investigation, and the other touch beam and piezoelectric sheet parameters are as shown in

Table 1. Parameters of the touch beam.

Length L	Thickness H	Width W	Elastic Modulus E	Density $\mathit{\rho }$
0.1 m	5 mm	5 mm	105 GPa	8500 kg/m3

and

Table 2. Parameters of the piezoelectric sheets.

Length lp	Thickness Hp	Width Wp	Elastic Modulus Ep	Density ρp	Piezoelectric Constant e31
10 mm	0.3 mm	5 mm	76.5 GPa	7500 kg/m3	2.0 c/m2

. By substituting each parameter into the above solution formula, the forced response of each order of the touch beam can be obtained. As the natural modes above the 3rd order can form at least one complete harmonic wave, they can be considered to be suitable as working modes of the touch beam. Assuming that the excitation frequency is equal to the natural frequency, the 3rd~6th order forced response curves are solved as shown in Figure 6. Assuming that the excitation frequency is equal to the natural frequency, the 1st and 2nd order vibration modes do not show a good periodic change in the mode shapes, so the 3rd–6th order forced response curves are solved.

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 6th order amplitude of the cantilever touch beam is 1.44 × 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 6th 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 V₃, piezoelectric sheet thickness h_p, touch beam elastic modulus E, and piezoelectric coefficient e₃₁.

From Figure 7, the following points are worth noting.

• We took the peak values of excitation voltage of 100 V, 150 V, and 200 V as calculation examples. As the value of the excitation voltage increases, the amplitude of the forced response gradually increases, and the larger the excitation frequency, the smaller the amplitude increase. It can be seen from the curves that when the excitation frequency is at the sixth-order natural frequency, the excitation voltage is increased from 100 V to 200 V, and the vibration amplitude is changed from 1.44 × 10⁻⁶ m to 2.62 × 10⁻⁶ m. More importantly, it can be speculated that increasing the excitation voltage can make the tactile change of the cantilever touch beam easier to perceive.
• The effect of choosing piezoelectric sheets of different thicknesses on the forced response is verified by bringing in piezoelectric sheets with thicknesses of 0.3 mm, 0.4 mm, and 0.5 mm. As the thickness h_p decreases, the vibration amplitude of the cantilever touch beam gradually increases. The larger the excitation frequency, the smaller the amplitude increases. For increasing the displacement of forced vibration to improve the sensitivity of touch sensation, we can use the thinner piezoelectric sheets as an excitation element.
• Investigating the influence of the material properties of the touch beam substrate on the vibration amplitude of the forced response, especially the elastic modulus E, can provide a basis for selecting materials during design. We chose the elastic modulus of the touch beam as a research subject and analyze the effect of the three materials of brass, phosphor bronze, and spring steel, respectively, on the forced response. As the elastic modulus increases, the forced response of the cantilever touch beam gradually decreases. Therefore, the material of the touch beam substrate with the higher elastic modulus would reduce the forced response displacement to a greater extent.
• In addition, the piezoelectric constant that measures the quality of the piezoelectric material is also an index that affects the forced response performance. The piezoelectric constant e₃₁ was changed so that e₃₁ takes values of 2.0 c/m², 2.5 c/m², and 3.0 c/m², respectively. As the piezoelectric constant of the piezoelectric element increases, the response amplitude of the cantilever touch beam gradually increases. It can be seen from the image that when the excitation frequency is at the sixth-order natural frequency, the piezoelectric constant is increased from 2.0 to 3.0, and the vibration amplitude increases from 1.44 × 10⁻⁶ m to 2.37 × 10⁻⁶ m, and the variation is also obvious. We recommend using a high-quality piezoelectric material with a large piezoelectric constant to enhance the perceptibility of the touch beam.

4.2. Analysis for Frequency Domain Response of Touch Beams

Using Equation (22) of the theoretical derivation results, the frequency-domain response of the cantilever touching beam is solved by using MATLAB and the related curves are drawn. The 3rd–6th order frequency domain response of the touch beam is shown in Figure 8. From left to right are the 3rd, 4th, 5th, and 6th order frequency domain responses.

Available from Figure 8, the following can be noted.

• The touch beam has a significantly increased amplitude of the forced response when the excitation frequency is close to its natural frequencies, and the magnitude of the forced response of the touch beam becomes small when it is far from the natural frequency of the touch beam itself. Therefore, the touch beam needs to operate at its resonance frequency to be effective. As the order of the natural frequency increases, the displacement response of forced vibration will gradually decrease, and the displacement response of the sixth order decreases to about one-sixth of that of the third order. However, the frequency of the sixth order is 24,221 Hz, which is beyond the sound range that humans can recognize and so will not make the subject uncomfortable due to noise, showing that this excitation frequency is more suitable as a working frequency.

Meanwhile, the effect of changing electromechanical parameters is also needed to be investigated in the frequency domain, for observing the influence of parameters on the forced response when the frequency changes. For this, the 6th-order forced response is the working frequency used as the research subject. Keeping the other parameters unchanged, and changing the four parameters of piezoelectric constant, excitation voltage, modal stiffness, and vibration mode damping ratio, provides the relevant curves from three different values as shown in Figure 9.

From Figure 9, the following points are worth noting.

• With the increase of excitation voltage and piezoelectric constant, the displacement of the forced response of the touch beam gradually increases. Within the interval of resonance, especially, the increase values of the forced response are more significant. However, with the increase of the elasticity modulus, the modal stiffness increases, and therefore the displacement of forced response gradually decreases. Within the intervals of resonance, the decrease values of the forced response caused by the elasticity modulus are more significant. In addition, increasing the modal damping ratio can reduce the displacement of forced response in the resonance intervals significantly. However, out of the resonance intervals, this kind of influence on forced response is not obvious.

4.3. 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

$$a_c=\frac{2}{T}{\displaystyle \underset{0}{\overset{\frac{T}{2}}{\int }}\left|w^{(i)}{}^{″}(x_j,t)\right|\mathrm{d}t}$$

(23)

where x_j is the position coordinate of the j ciliary-like body.

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 Table 1 and Table 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 ω, the excitation voltage amplitude V₃, the piezoelectric coefficient e₃₁, 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/s², 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/s².
• 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/s². When the excitation voltage is 200 V, the vibration acceleration doubles to 13,8648 m/s².
• 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/m², the vibration acceleration is 10,864 m/s².
• 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/s², and the acceleration at some points is zero.

5. Experimental Analysis

The touch beam with the ciliary-like body structure of the groove ε = 1 mm is taken as the experimental object. In order to determine the vibration behavior of the whole touch beam as a function of the position coordinate, the vibration modes of the touching beam were measured for various locations of the sampling point. The touch beam was driven by a sinusoidal signal with a voltage peak of 100 V. The experimental environment is shown in Figure 12.

The experimental steps are as follows.

• Fix the ciliary-like body touch beam to the bracket by the cantilever beam and tighten the screws on one side, so the other side is free.
• The driving signal was supplied by an arbitrary signal generator. We then used the driving signal, which was modulated by the HFVA-42 power amplifier, to drive the touch beam. The voltage was adjusted to ~100 V and the signal frequency adjusted to the sixth-order natural frequency which was ~24,000 Hz.
• There existed some error between the theoretical calculation and the experiment, so the frequency was fine-tuned between 20,000 Hz and 24,000 Hz, while continuously touching and feeling the changing state of the surface of the touch beam until it could be clearly felt that the touch beam became smooth. After debugging, the touch beam was obviously smooth when the experimental prototype was excited at the frequency of ~23,200 Hz.
• The laser vibrometer was mounted vertically to measure the cantilever touch beam from directly above, and the measurement data was transmitted through the acquisition card to the supporting software of the computer. On the left and right touch beam segments of the piezoelectric sheets, point-by-point measurement was performed from left to right. The waveform data displayed by the computer’s companion software eZ-Analyst was continuously observed. The average amplitude of the real-time waveform of the measurement points was recorded.

After several measurements, five test points with larger vibration amplitudes were obtained as shown in Figure 13. Repeat measurements for each point of the larger vibration amplitudes were then taken. The vibration waveform measured results of the five points from one measurement are shown in Figure 14. The average amplitude data of the five test points, with the position coordinates, are shown in

Table 3. Position coordinates and average amplitude of five test points.

Test Point	①	②	③	④	⑤
Position coordinates (mm)	13.25	29.25	47.25	74.70	90.75
Average amplitude (m)	1.24 × 10−6	1.24 × 10−6	1.24 × 10−6	1.23 × 10−6	1.25 × 10−6

.

According to the test data in Table 3 and the test results in Figure 14, the fitting of vibration shapes under the actual measurement conditions are drawn through the tracing points. The fitting vibration shapes are compared with the theoretically calculated vibration shapes. We also adjust the excitation voltage to 200 V to repeat the above steps.

Figure 15 shows the comparison of the theoretical calculation and actual measurement of vibration response of the touch beam under the excitation voltage peaks of 100 V and 200 V. From these results, the following conclusions can be drawn.

• The amplitude is larger when the excitation voltage is 200 V, but the positions of the five larger vibration amplitude points under the two voltage excitations are basically the same, so increasing the excitation voltage will not change the position of the touch beam peak and valley.
• Overall, the amplitude of vibration shapes from the experiment is slightly larger than the theoretical calculation, and meanwhile the measured position of each peak and trough has a certain deviation from their theoretical values. However, the deviation is relatively small and completely acceptable, in addition the trend of consistency does not change under the two voltages.
• The maximum deviation of the 200 V excitation voltage is 9 mm, which is slightly larger than that of the 100 V excitation voltage of 8 mm. The two experimental curves are basically in accordance with the theoretical calculation. The experimental deviation value increases slightly with the increase of the excitation voltage.

6. Discussion

This paper introduces a piezoelectric ciliary-like body tactile feedback device, which aims to improve the tactile richness under a simple control system. Previous research results showed that the sensing of the acceleration of the forced vibration of the touch beam played a crucial role in the prediction of the equivalent friction coefficient. This work has established the dynamic equation of the forced vibration response of the touch beam and the effects of system electromechanical parameters on vibration acceleration were investigated. The study results show that larger excitation voltage, larger piezoelectric constants, and smaller elasticity modulus can be chosen to enhance the displacement and acceleration of the forced response of the touch beam. For proving the theoretical analysis, an experiment was conducted. The experimental results showed that when the touch beam was in the working mode, the driving frequency was close to the results of the theoretical analysis, and only a slight deviation occurred between the measured and theoretical vibration shapes, which was considered acceptable. Future work can explore more accurate solutions for the forced response of the touch beam.