Free vibration analysis for a non-contact piezoelectric motor modulated by electromagnetic field

Abstract

For a non-contact piezoelectric motor modulated by electromagnetic field, the equation of the free vibration for the driving system of the motor is given. Using the boundary conditions and continuity conditions, the natural frequencies and mode functions of the driving system are obtained. The variation law of the natural frequency is investigated, when the system parameters of the driving system are changed in a certain range. The results show that the change of the system parameters of piezoelectric stacks and driving beams has obvious effects on odd order natural frequencies of the driving system; the change of the system parameters of the center shaft have obvious effects on even order natural frequencies; In the odd order modes, the maximum vibration amplitude of the driving system occurs at the junction of the driving beam and the center shaft. It is conducive to output of the motor rotation angle. All the results are useful for design and control of operating performance of the motor.

Keywords

Free vibration non-contact modulated by electromagnetic field natural frequency mode function

1. Introduction

According to the contact state of the stator and rotor, the piezoelectric precision actuator can be divided into contact and non-contact [1, 2, 3]. There is friction between the stator and the rotor of the contact piezoelectric actuator, so this type of actuator is not suitable for working a long time, because of the serious wear of the contact surface between the stator and the rotor. Therefore, the contact type piezoelectric actuator has the disadvantages of short life and low reliability [4, 5]. To overcome the disadvantages of the contact type piezoelectric actuator, in 90s of last century, the non-contact piezoelectric actuator is proposed. This type of actuators between the stator and rotor is non-contact or indirect contact, so it has a long life, and no self-locking phenomenon [6]. At present, the driving principle of the non-contact type piezoelectric actuator mainly is based on ultrasonic levitating and electrorheological fluid modulation [7, 8]. With these principles, many applications have been proposed and fabricated [9, 10]. The ultrasonic levitating piezoelectric actuator has the characteristics of high speed and low output torque. However, the most attractive feature of the piezoelectric actuator is the low speed and large torque is lost, and the application field of this kind of motor is limited. Another type of non-contact piezoelectric motor which is modulated by electrorheological fluid has the characteristics of high output torque and high resolution [11]. However, using the electrorheological fluid as the clamping medium, the difficulty to design a wide speed range motor is increased, because of the long response time and poor controllability of electrorheological fluid [12, 13].

To solve the above problems existing in the non-contact piezoelectric drive system, we propose a novel non-contact piezoelectric motor modulated by electromagnetic field. The motor mainly consists of a stator (electromagnetic coupling mechanism), a rotor, driving mechanism and two piezoelectric stacks, as shown in Fig. 1. There are two short beams in the driving mechanism. One end of each beam is connected with the frame through a flexible hinge; the other end is fixed on the center shaft. Two piezoelectric stacks are mounted on the driving mechanism to bend short beams respectively. As soon as the piezoelectric stacks are excited by a sinusoidal signal, and the stator is swung. Then, the rotor modulated by the electromagnetic coupling mechanism, which is controlled by square wave signal, can rotate one step by one.

However, the free vibration analysis for driving system of the motor has not been investigated yet. It is unfavorable to design the motor structure size and choose the system parameters. In this paper, the structure and working principle of the motor is presented. The dynamics equation of driving system is proposed. Using the boundary conditions and continuity conditions, the natural frequencies and mode functions of the driving system are obtained. The effect of system parameters on the structural design is investigated. All the results are helpful to design a high-performance non-contact piezoelectric motor.

Figure 1.

The non-contact piezoelectric motor modulated by electromagnetic field.

Figure 2.

Schematic sketch of driving principle.

Figure 3.

Driving signal diagram.

2. Working principle

The driving system is extracted from the motor. They are composed of a stator, two driving beams, two piezoelectric stacks and center shaft, as shown in Fig. 2. The working procedure of the motor is as follows:

As shown in Fig. 3, the sine signal is supplied in two piezoelectric stacks, and the square wave signal is supplied in the stator (electromagnetic coupling mechanism). When the drive voltage signal is in 0 $\sim T$ /2, the two piezoelectric stacks occur elongation deformation to bend the driving beams. Then, the center shaft and stator turn a certain angle $\theta$ with the driving beams. As the same time, the electromagnetic coupling mechanism produces magnetic force and grip the rotor with the stator rotation.

When the drive voltage signal is in $T$ /2 $\sim T$ , the voltage of the sine signal is reduced from $U_{1}$ to 0, and the driving beams restore deformation. Then, the stator reverses rotation a certain angle $\theta$ with the center shaft. Due to this period of time the stator electromagnetic coupling mechanism is not provided with the voltage, the magnetic force disappears. The rotor keeps the angle $\theta$ turned during the first half cycle (see Fig. 2b). Repeat the above process, the rotor can continuously rotate one step by one.

3. Dynamic model

For establishing the dynamic model of the motor to solve the natural frequencies and mode functions, it is necessary to simplify the motor structure. Therefore, the dynamic model of the continuous elastic system is established by considering the longitudinal deformation of the piezoelectric stack, the bending deformation of the driving beams and torsional deformation of the center axis. Moreover, the simplified dynamic model is based on the following assumptions:

Figure 4.

Schematic sketch of dynamic model.

As shown in Fig. 4, the length of two driving beams is less than 5 times the height of cross section. So they are considered Timoshenko beams.

There is a flexible hinge in the driving beam end. Its tensile deformation is ignored; the flexure hinge is simplified for the combination of a hinge with a coil spring, which stiffness is $K$ [14],

$\displaystyle K=\frac{Eh_{j}R^{2}}{12\eta},$ $\displaystyle\eta=\frac{2s^{3}\left({6s^{2}+4s+1}\right)}{\left({2s+1}\right)% \left({4s+1}\right)^{2}}+\frac{12s^{4}\left({2s+1}\right)}{\left({4s+1}\right)% ^{\frac{5}{2}}}\arctan\sqrt{4s+1},$ $\displaystyle s=R/{t_{h}}.$

where $E$ is the elasticity modulus of the driving system, $h_{j}$ is the width of the flexible hinge, $t_{h}$ is the minimum thickness of the flexible hinge recess, and $R$ is the notch radius of the flexible hinge.

The electromagnet is simplified into a mass block and is mounted the upper end of the center shaft; the lower end of the shaft is fixedly connected with the ends of the two driving beams.

3.1 Dynamic model of a piezoelectric stack

As shown in Fig. 5, the local coordinate system of a piezoelectric stack is established. Piezoelectric stacks are made by bonding multilayer piezoelectric ceramic sheets. Here, we consider the piezoelectric stack as a continuous system, and ignore the energy loss between the layer and the layer. We also consider the charge cannot be generated by the preload. The effect of preload force on the dynamic model is ignored. The vibration direction of the piezoelectric stack is only longitudinal.

Figure 5.

The dynamic model of a piezoelectric stack.

From the model, the dynamics equation of the piezoelectric stack micro unit can be given as follows

$\displaystyle\rho_{p}A_{p}\text{d}y_{p}\frac{\partial^{2}v\left({y_{p},t}% \right)}{\partial t^{2}}=\left({F+\frac{\partial F}{\partial y_{p}}\text{d}y_{% p}}\right)-F+f\text{d}y_{p}$ (1)

where $v(y_{p},t)$ is the longitudinal displacement of a piezoelectric stack, $y_{p}$ is the vertical coordinate, $t$ is the time, $f$ is the external force on unit length, $\rho_{p}$ is the material density per unit length, $F$ is the internal force, $A_{p}$ is the sectional area of the piezoelectric stack.

Using the piezoelectric equation, the internal force can be given as [14]

$\displaystyle F=c_{33}^{E}Ap\frac{\partial v}{\partial y_{p}}-c_{33}^{E}d_{33}% Ap\frac{U_{1}(t)}{lp}$ (2)

where $d_{33}$ is the piezoelectric strain constant, $c_{33}^{E}$ is the piezoelectric stress constant. When driving voltage $U_{1}(t)=$ 0, let $f=$ 0, and substituting Eq. (2) into Eq. (1) yields

$\displaystyle\frac{\partial^{2}v}{\partial t^{2}}=a^{2}\frac{\partial^{2}v}{% \partial y_{p}^{2}}$ (3)

where $a=\sqrt{\frac{c_{33}^{E}}{\rho_{p}}}$ .

Let $v(y_{pj},t)=\varphi_{pj}(y_{pj})q(t)$ . Then, the longitudinal vibration mode function of the piezoelectric stack is

$\displaystyle\varphi_{jp}(y_{jp})=A_{j1}\sin\frac{\omega y_{jp}}{a}+A_{j2}\cos% \frac{\omega y_{jp}}{a}(j=1,2)$ (4)

where $\omega$ is the natural frequency of the system, $A_{ji}(i=1,2)$ is the coefficient of $\varphi_{pj}$ .

3.2 Bending vibration of the driving beams

The driving beam is divided into two sections. In order to obtain the equal amplitude oscillation of the electromagnetic coupling mechanism, we designed the driving beam as a symmetrical structure about the axis of the central axis, so the two beams are equal $l_{1}=l_{2}=l$ . Then the local coordinate system of the two beams is established with the center as the origin. The dynamic model is shown as Fig. 6.

Figure 6.

The dynamic model of the driving beams.

Considering the driving beam as a Timoshenko beam, the dynamic equation of the bending vibration of the beam can be given as follows

$\displaystyle\rho S_{j}\frac{\partial^{2}y_{j}}{\partial t^{2}}+EI_{j}\frac{% \partial^{4}y_{j}}{\partial x_{j}^{4}}-\rho I_{j}\left({1+\frac{E}{\kappa G}}% \right)\frac{\partial^{4}y_{j}}{\partial x_{j}^{2}\partial t^{2}}+\frac{\rho^{% 2}I_{j}\partial^{4}y_{j}}{\kappa G\partial t^{4}}=0$ (5)

where $y_{j}(x_{j},t)$ is the transverse displacement of the beam, $\kappa$ is the section factor, $G$ is the shear modulus, $I_{j}$ is the second moment of the area of a beam, and $S_{j}$ is the section area of a beam, $\rho$ is the material density per unit length of the driving beams. The effect of section moment of inertia is considered, and shear deformation is ignored. Thus, we obtain

$\displaystyle\frac{\partial^{2}y_{j}}{\partial t^{2}}+\left({\frac{EI_{j}}{% \rho S_{j}}}\right)\frac{\partial^{4}y_{j}}{\partial x_{j}^{4}}-\left({\frac{I% _{1}}{S}}\right)\frac{\partial^{4}y_{j}}{\partial x_{j}^{2}\partial t^{2}}=0$ (6)

Let $y_{j}(x_{j},t)=\varphi_{j}(x_{j})q(t)$ , and substituting it into Eq. (6) yields

$\displaystyle\ddot{q}(t)+\omega^{2}q(t)=0$ (7) $\displaystyle\varphi_{j}^{(4)}(x_{j})+\delta^{2}{\varphi}^{\prime\prime}_{j}(x% _{j})-\beta_{j}^{4}(x_{j})=0$ (8)

where $\beta_{j}^{4}=\frac{\rho S_{j}}{EI_{j}}\omega^{2}$ , $\delta^{2}=\frac{\rho\omega^{2}}{E}$ .

Thus, the general solution of Eq. (8) is given by

$\displaystyle\varphi_{j}(x_{j})=B_{j1}\cos(\lambda_{j1}x_{j})+B_{j2}\sin(% \lambda_{j1}x_{j})+B_{j3}\cosh(\lambda_{j2}x_{j})+B_{j4}\sinh(\lambda_{j2}x_{j})$ (9)

where $\lambda_{j1}=\sqrt{\sqrt{\beta_{j}^{4}+\frac{\delta^{4}}{4}}+\frac{\delta^{2}}% {2}},\lambda_{j2}=\sqrt{\sqrt{\beta_{j}^{4}+\frac{\delta^{4}}{4}}-\frac{\delta% ^{2}}{2}}$ , $B_{ji}$ ( $j=1,2;i=1,2,3,4$ ) is the coefficient of $\varphi_{j}$ .

3.3 Torsional vibration of the center shaft

With the coupling end of the central axis and the driving beam as the origin, the local coordinate system is established. $y_{s}$ is the vertical coordinates, $\theta$ ( $y_{s}$ , $t$ ) is the angular displacement of the section. As shown in Fig. 7, the relation between the torque $M_{T}$ and angular displacement on the section is

$\displaystyle M_{T}=GI_{s}\frac{\partial\theta}{\partial y_{s}}$ (10)

where $I_{s}$ is the torsional inertial moment of the cross section.

Figure 7.

The dynamic model of the center shaft.

From the model, the dynamic equation of the center shaft can be described as follows

$\displaystyle\rho I_{s}\text{d}x\frac{\partial^{2}\theta}{\partial t^{2}}=% \left({M_{T}+\frac{\partial M_{T}}{\partial y_{s}}\text{d}y_{s}}\right)-M_{T}$ (11)

Substituting Eq. (10) into Eq. (11) yields

$\displaystyle\frac{\partial^{2}\theta}{\partial t^{2}}=c^{2}\frac{\partial^{2}% \theta}{\partial y_{s}^{2}}$ (12)

where $c=\sqrt{\frac{G}{\rho}}$ .

Let $\theta(y_{s},t)=\phi(y_{s})q(t)$ . Thus, the torsional vibration mode function of the center shaft is

$\displaystyle\phi(y_{s})=C_{1}\sin\frac{\omega y_{s}}{c}+C_{2}\cos\frac{\omega y% _{s}}{c}$ (13)

where $C_{i}(i=1,2)$ is the coefficient of $\phi$ .

3.4 Boundary conditions and continuity conditions

As a continuous system, the relation of each component should be in accordance with the boundary conditions, force balance conditions and the deformation compatibility conditions as follows:

When $y$ coordinates of a piezoelectric stack is 0, the point is the fixed end. So the longitudinal deformation is 0. Thus

$\displaystyle\left\{{\begin{array}[]{l}\varphi_{1p}{(0)=0}\\ \varphi_{2p}{(0)=0}\\ \end{array}}\right.$ (14)

The end of the driving beams is considered as the combination of a hinge with a coil spring. So the displacement response of the ends is 0, and the moments of the ends are equal to the negative product of the ends angle and the stiffness of the flexible hinge. The boundary conditions of displacement and rotation are as follows

$\displaystyle\left\{{\begin{array}[]{l}\varphi_{1}\left(l\right)=0\\ \varphi_{2}\left(l\right)=0\\ EI_{1}{\varphi}^{\prime\prime}_{1}(l)=-K{\varphi}^{\prime}_{1}(l)\\ EI_{2}{\varphi}^{\prime\prime}_{2}(l)=-K{\varphi}^{\prime}_{2}(l)\\ \end{array}}\right.$ (15)

Table 1

The parameters of the piezoelectric stack

$l_{np}$ (mm)	$b_{np}$ (mm)	$h_{np}$ (mm)	$c_{33}^{E}$ (GPa)	$\rho_{p}$ (kg/m ${}^{3}$ )
20	5	5	55.6	8 $\times$ 10 ${}^{3}$

Figure 8.

Mode 1.

At the coupling point of a piezoelectric stack and s driving beam, the longitudinal displacement of the piezoelectric stack is equal to the bending displacement of the driving beam, and the output force of the piezoelectric stack is equal to the shear force of the beam. Thus

$\displaystyle\left\{{\begin{array}[]{l}\varphi_{1p}(l_{np})=\varphi_{1}(l_{c})% \\ \varphi_{1p}(l_{np})=\varphi_{1}(l_{c})\\ c_{33}^{E}A_{1p}{\varphi}^{\prime}_{1p}(l_{np})=EI_{1}\varphi_{1}^{\prime% \prime\prime}(l_{c})\\ c_{33}^{E}A_{2p}{\varphi}^{\prime}_{2p}(l_{np})=EI_{2}\varphi_{2}^{\prime% \prime\prime}(l_{c})\\ \end{array}}\right.$ (16)

where $l_{c}$ is the contact position of a driving beam and a piezoelectric stack.

There is a mass on the upper end of the center shaft. So the upper end torsion of central shaft is equal to the torsional inertia force of the mass. Thus, the boundary conditions is

$\displaystyle GI_{s}{\phi}^{\prime}\left({l_{s}}\right)=-J\omega^{2}\phi\left(% {l_{s}}\right)$ (17)

Table 2

The parameters of the driving beam

$l$ (mm)	$b_{q}$ (mm)	$h_{q}$ (mm)	$E$ (GPa)	$\rho$ (kg/m ${}^{3}$ )	$h_{j}$ (mm)	$t$ (mm)	$R$ (mm)
16	4	6.5	206	7.7 $\times$ 10 ${}^{3}$	6.5	1	1.5

Figure 9.

Mode 2.

where $J$ is the inertial moment of the mass, $l_{s}$ the length of the center shaft.

The lower end of the center shaft is coupling with the driving beams. Because the lower end face of center shaft is free, the torsion is 0. But the torsional linear displacement of the lower shaft is equal to the lateral displacement of the junction of the driving beams and the center shaft. Thus, the boundary conditions is

$\displaystyle\left\{{\begin{array}[]{l}{\phi}^{\prime}\left(0\right)=0\\ \phi\left(0\right)r=\varphi_{1}\left(0\right)\\ \phi\left(0\right)r=-\varphi_{2}\left(0\right)\\ \end{array}}\right.$ (18)

The coefficients $A_{ji}$ , $B_{ji}$ , $C_{i}$ , and $\omega$ can be obtained through substitute of boundary conditions. Thus, we can obtain the natural frequencies of the driving system, and the mode functions can be established.

Table 3

The parameters of the electromagnetic coupling mechanism

$l_{s}$ (mm)	$d$ (mm)	$J$ (kg $\cdot$ m ${}^{2}$ )	$E$ (GPa)	$\rho$ (kg/m ${}^{3}$ )
20	8	56.25 $\times$ 10 ${}^{-7}$	206	7.7 $\times$ 10 ${}^{3}$

Table 4

Natural frequency of the driving system (rad/s)

$\omega_{1}$	$\omega_{2}$	$\omega_{3}$	$\omega_{4}$	$\omega_{5}$	$\omega_{6}$	$\omega_{7}$	$\omega_{8}$
166457	270742	686782	813701	1222809	1356364	1692043	1898985

Figure 10.

Mode 3.

Figure 11.

Mode 4.

Figure 12.

Mode 5.

4. Results and discussions

4.1 Modal analysis

Spring steel is chosen as the material of the driving system, and the other system parameter is listed in Tables 1–3. The equations of the boundary conditions are utilized for the free vibration analysis of the driving system to obtain the natural frequencies. By substituting these natural frequencies into Eqs (4), (9) and (13), the vibration modes of the driving system can be obtained. Table 4 lists the first eight natural frequencies, and the first six vibration modes are shown in Figs 8–13. From the results of the natural frequencies and vibration modes, we can obtain the first eight natural frequency of the driving system increase with the increase of the orders. Especially, the linear degree is remarkable in the even order natural frequency.

In the odd order modes, the amplitude of driving beams is relatively large, and the amplitude of the center shaft and pizoelectric stacks is relatively small. With mode order of driving system increases, the peak number of the mode function of each subsystem increases. Therefore, when the excitation frequency is close to odd order natural frequency, the vibration of the driving beams is easy to occur.

In the even order modes, the amplitude of the center shaft is relatively large, and the amplitude of the driving beams and piezoelectric stacks is relatively small. With mode order of driving system increases, the peak number of the mode function of the center shaft and piezoelectric stacks increases, but the change of the driving beam is not remarkable. The maximum amplitude of the vibration of the driving system occurs at the junction of a drivding beam and the center shaft. Therefore, when the excitation frequency is close to even order natural frequency, the vibration of the center shaft is easy to occur.

4.2 Sensibility analysis

The effect of system parameters on the inherent characteristics is necessary to be investigated, because this effect can provide a theoretical basis for the design and parameter selection of drive system. Here, five kinds of different materials are selected to make the drive system. Using the dynamic equation of the driving system, the first three natural frequencies can be obtained, and the results are shown in the Table 5. As seen, in the 5 kind of materials, the first-order natural frequency of alloy steel is lowest. The second-order and third-order natural frequency of brass is lowest. In the first-order natural frequency, the brass and the spring steel is closer; in the second-order natural frequency, the aluminium alloy, alloy steel and spring steel is closer; in the third-order natural frequency, the alloy steel and spring steel is closer.

Table 5
The first – three natural frequency of different materials

Material	Density $E$ (GPa)	Elastic modulus $\rho$ (kg/m ${}^{3}$ )	Poisson ratio ${\mu}$	Frequency (rad/s)
				${\omega_{1}}$	${\omega_{2}}$	${\omega_{3}}$
Aluminium alloy	${2.7\times 10}^{3}$	${\phantom{0}70}$	${0.3\phantom{0}}$	${195180}$	${275374}$	${640295}$
Ductile iron	${7.7\times 10}^{3}$	${120}$	${0.25}$	${190049}$	${217445}$	${653503}$
Brass	${8.9\times 10}^{3}$	${\phantom{0}90}$	${0.25}$	${168493}$	${201410}$	${506526}$
Alloy steel	${7.9\times 10}^{3}$	${206}$	${0.25}$	${163126}$	${281255}$	${684871}$
Spring steel	${7.8\times 10}^{3}$	${195}$	${0.3\phantom{0}}$	${166457}$	${270742}$	${686782}$

Figure 13.

Mode 6.

Figure 14.

Changes of natural frequencies with respect to various parameters.

Spring steel is chosen as the material of driving system. When the system parameters are changed in a certain range, the change curves of the natural frequencies are shown in the Fig. 14. From the Fig. 14, the following observations were worth noting.

When the parameters of the piezoelectric stacks ( $l_{np}$ , $A_{p}$ , $c_{33})$ are changed in a certain range, the effect of these parameters on the second-order natural frequency is very small. With the length of the piezoelectric stacks ( $l_{np}$ ) increases, the first-order and the third-order natural frequency nonlinearly reduce; with the coefficient of piezoelectric elasticity ( $c_{33}$ ) increases, the first-order and the third-order natural frequency nonlinearly increase; with the area of the sectional area of the piezoelectric stack increases, the first-order natural frequency nonlinearly increases, and the third-order natural frequency nonlinearly reduces.

When the length of the driving beams ( $l$ ) is changed in 12 mm–20 mm, the changing curves of natural frequency produce mutation in 14 mm. The phenomenon may be caused by the modal mutation. Subsequently, with the length of the driving beam ( $l$ ) increases, the first-order and the third-order natural frequency increase, but the second-order natural frequency remains unchanged.

When the width of the driving beams ( $b_{q}$ ) is changed in 4 mm–8 mm, the first-order natural frequency reduces; second-order natural frequency remains unchanged; third-order natural frequency increases.

When the thickness of driving beams ( $h_{q}$ ) is changed in 2 mm–5 mm, the first-order and the third-order natural frequency reduce and the second-order natural frequency remains unchanged. The changing curves of natural frequency produce mutation in 5 mm.

When the contact position of the beams and the piezoelectric stack ( $l_{c}$ ) is changed nearby the origin 8 mm–12 mm, the first-order and third-order natural frequency nonlinearly reduce. But it has no effect on the second-order natural frequency.

When the length of the center shaft ( $l_{s}$ ) is changed in 10 mm–30 mm, the first-order natural frequency is not sensitive to the change of $l_{s}$ . With $l_{s}$ increases, the second-order natural frequency reduces. When the $l_{s}$ is changed in 10 mm–20 mm, the third-order natural frequency remains unchanged. However, with the $l_{s}$ increases in 20 mm–30 mm, the third-order natural frequency reduces.

The diameter of the center shaft ( $d$ ), the diameter of electromagnet ( $d_{d}$ ) and the mass of the electromagnet ( $M$ ) are also investigated. These three parameters mainly affect the second-order natural frequency, but the system is less sensitive to their change.

5. Conclusions

A non-contact piezoelectric motor modulated by electromagnetic field is proposed. The dynamics equation of the driving system of the motor is presented. Then, the natural frequencies and mode functions are obtained. The sensitivities of inherent characteristics to system parameters are investigated. The results show:

The structural parameters and materials of piezoelectric stacks and driving beams have obvious effects on odd order natural frequencies of the driving system. The system parameters of the center shaft have obvious effects on even order natural frequencies.

In the odd order modes, the maximum vibration amplitude of the driving system occurs at the junction of the driving beams and the center shaft. It is conducive to output of the motor rotation angle. All the results are useful for design and control of operating performance of the motor.

Notation

$T$	excitation signal period
$U_{i}$	voltage amplitude of the excition signal
$E$	modulus of elasticity
$h_{j}$	width of the flexible hinge
$t_{h}$	thickness of the flexible hinge recess
$R$	notch radius of the flexible hinge
$v$	longitudinal displacement of a piezoelectric stack
$y_{p}$	vertical coordinate of a piezoelectric stack

$t$	time
$f$	external force on unit length
$\rho_{p}$	material density per unit length of a piezoelectric stack
$F$	internal force
$A_{p}$	sectional area of a piezoelectric stack
$d_{33}$	piezoelectric strain constant
$c_{33}^{E}$	piezoelectric stress constant
$q(t)$	time variant function
$A_{ji}$	coefficient of $\phi_{pj}$
$\omega_{i}$	natural frequency
$l$	length of the driving beams
$y_{i}(x_{i},t)$	transverse displacement of the driving beams
$\kappa$	section factor of the driving beams
$G$	shear modulus
$I_{j}$	polar inertia moment of area of the driving beams
$S_{j}$	section area of the driving beams
$\rho$	material density per unit length of the driving beams
$B_{ji}$	coefficient of $\varphi_{j}$
$y_{s}$	vertical coordinates of the center shaft
$\theta$	angular displacement of the section
$I_{s}$	torsional inertial moment of the cross section
C ${}_{i}$	constants of $\phi$
$l_{c}$	contact position of the beams and the piezoelectric stack
$J$	moment of inertia of the mass
$l_{s}$	length of the center shaft
$b_{np}$	width of a piezoelectric stack
$h_{np}$	height of a piezoelectric stack
$b_{q}$	width of the driving beams
$h_{q}$	height of the driving beams

Footnotes

Acknowledgments

This project is supported by National Natural Science Foundation of China (51605423).

References

Sun

and Chen

, Non-contact linear piezoelectric actuator based on near-field acoustic levitation, China Mechanical Engineering 21(24) (2010), 2952–2956.

Yang

et al., Simulation and experimental study of non-contact piezoelectric micromotor, Optics & Precision Engineering 13(2) (2005), 165–170.

and Li

, Dynamics for an Electromechanical Integrated Harmonic Piezodrive System, International Journal of Applied Electromagnetics & Mechanics 46(3) (2014), 709–726.

Isobe

and Kyusojin

, Frequency characteristics of non-contact ultrasonic motor with motion error correction, Precision Engineering 31 (2007), 351–357.

Jung

Lee

and Jung

, Survey of a contact model and characteristic analysis method for a travelling wave ultrasonic motor, International Journal of Applied Electromagnetics & Mechanics 46(3) (2014), 437–453.

Chen

Gao S

Pan

et al., Self-running and self-floating two-dimensional actuator using near-field acoustic levitation, Applied Physics Letters 109(12) (2016), 123504.

Stepanenko

D.A.

and Minchenya

V.T.

, Development and study of novel non-contact ultrasonic motor based on principle of structural asymmetry, Ultrasonics 52 (2012), 866–872.

Qiu

Hong

Mizuno

et al., 3P4-2 A Rotary Motor Using Giant Electrorheological Fluid and Piezoelectric Torsional Vibrator (Poster Session), in: Symposium on Ultrasonic Electronics Steering Committee of Symposium on Ultrasonic Electronics, 2013, pp. 433–434.

Chen

Jia

and Wang

, Study on the axial/radial coupling non-contact ultrasonic piezoelectric actuator with a spherical rotor, Symposium on Piezoelectricity 8204(1) (2011), 36–39.

10.

Qiu

et al., Non-contact piezoelectric rotary motor modulated by giant electrorheological fluid, Sensors and Actuators 217 (2014), 124–128.

11.

et al., A novel noncontact ultrasonic levitating bearing excited by piezoelectric ceramics, Applied Sciences 6(10) (2016), 280; doi: 10.3390/app6100280.

12.

Meng

and Tian

, Recession in a linear stepper motor based on piezoelectric actuator and electrorheological clampers, Smart Materials & Structures 21(12) (2012), 125008–125015(8).

13.

Sheng

and Wen

, Electrorheological Fluids: Mechanisms, Dynamics, and Microfluidics Applications, Fluid Mechanics 44(1) (2011), 143–174.

14.

and Zhou

, Design calculations for flexure hinge, Review of Scientific Instruments 73 (2002), 3101–3106.

15.