Investigation on the traveling wave and three-dimensional trajectory of motion on the stator of rotational ultrasonic motors

Abstract

Better understanding of the characteristics of the traveling wave and three-dimensional trajectory related to motion on the surface of the stator is very important for the design and performance improvement of the ultrasonic motors. In this paper, an accurate finite element model of a single stator with a fully coupled piezoelectric layer was established at a moderate computational cost. The finite element model was verified by experimental test at the inverse resonance point. Based on this model, the traveling wave and three-dimensional trajectory of stator surface, including the influence of the input voltage on the phase and amplitude of the displacements in three directions, are investigated. The results show that the trajectory of particles on the stator surface is an ellipse in three-dimensional space due to the phase differences between the three components of displacement in the radial, circumferential and axial directions. The amplitude of radial displacement is about 39.5% of that in the circumferential displacement, which should not be neglected.

Keywords

Traveling wave vibration characteristic ultrasonic motor anti-resonance point finite element model

1. Introduction

Traveling wave ultrasonic motors (TWUSM) have become a new type of widely used precision actuators in many engineering applications [1]. Traveling wave at ultrasonic frequency is generated on the stator of TWUSM by a bonded piezoelectric layer and used to drive the rotor through friction. Hence, the trajectory of motion on the contacting surface of the stator is critical for the performance of a TWUSM. Many studies have been carried out on dynamical modeling and analysis of TWUSM. Hagood et al. proposed a relatively complete analytical model of TWUSM [2]. Based on this model, the transmission mechanism of an ultrasonic motor was analyzed by Zhu [3] and Zhao [4], respectively. Renteria Marquez et al. analyzed the generation of the traveling wave in the stator surface using the finite volume method (FVM) [5]. However, the above models of TWUSM were simplified as a two-dimensional structure, ignoring the radial sliding between the rotor and the stator. Numerical models based finite element method (FEM) [6–9] the effect of the piezoelectric coupling was not considered in the model. Moreover, Ren et al. used FEM to calculate the ultrasonic motor model on the basis of considering the piezoelectric coupling effect, but did not directly verify the displacement amplitude of the stator surface in each direction [10].

Based on the substructure method, Zhao et al. concluded that the influence of the radial velocity of the stator at the interface should not be neglected because the radial velocity has the same order as the circumferential velocity [1]. It was also verified by experiments that the optimization of the stator teeth can significantly improve the output power and efficiency of the motor [11,12], which indicates that the displacement amplitudes of the stator at the driving surface in three directions has a significant influence on the performance of the motor. Hence, better understanding of the characteristics of the traveling wave and three-dimensional trajectory of motion on the surface of the stator is very important for the design and performance improvement of the ultrasonic motors.

In this paper, an accurate finite element model of a single stator with a fully coupled piezoelectric layer was established at a moderate computational cost. The finite element model was verified by experimental test at the inverse resonance point [1]. Based on this model, the traveling wave and three-dimensional trajectory of stator surface, including the influence of the input voltage on the phase and amplitude of the displacements in three directions, is investigated.

Fig. 1.

(a) Stator structure diagram of traveling wave motor and (b) The polarization of piezoelectric ceramic.

2. Generation of the travelling wave

Figure 1(a) illustrates the structure of the stator, including a metal disk with teeth on the outer ring and a piezoelectric ceramic ring bonded together. The electrodes on the piezoelectric ring is shown in Fig. 1(b). Two voltages with a phase difference of 𝛼, which is usually set to π∕2, are applied to areas A and B, respectively, to excite two orthogonal vibration modes of the same frequency. Both areas A and B are divided into small sub-areas, which are labeled “+” and “−”, have half-wave length and are poled in opposite direction alternatively.

The displacement of the modes excited by voltages A and B in the stator can be written as: $\begin{eqnarray}\displaystyle {\Omega}_{A}(r,{\theta},t)=\frac{1}{2}W_{A}R(r)\sin (n{\theta})\sin ({\omega}_{n}t),\quad {\Omega}_{B}(r,{\theta},t)=W_{B}R(r)\cos (n{\theta})\sin ({\omega}_{n}t+{\alpha}) & & \displaystyle\end{eqnarray}$ (1) where W_A and W_B are displacement amplitudes by two different voltage inputs; 𝛼 is phase differences between the two voltage inputs; R (r) is the transverse displacement distribution along the radial direction; n is the circumferential order of mode the stator. Summation of the two displacement gives $\begin{eqnarray}\displaystyle {\Omega}(r,{\theta},t) & = & \displaystyle {\Omega}_{A}(r,{\theta},t)+{\Omega}_{B}(r,{\theta},t)\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{2}R(r)\{(W_{A}-W_{B}\sin {\alpha})\sin (n{\theta}+{\omega}_{n}t)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼(W_{A}+W_{B}\sin {\alpha})\sin (\text{n}{\theta}-{\omega}_{n}t)+2W_{B}\cos {\alpha}\cos \text{n}{\theta}\cos {\omega}_{n}t\}.\end{eqnarray}$ (2) When W_A = W_B = W₀, 𝛼 = ±π∕2, a travelling wave is generated in the stator. The displacement of the travelling wave can be expressed as $\begin{eqnarray}\displaystyle {\Omega}(r,{\theta},t)={\Omega}_{A}(r,{\theta},t)+{\Omega}_{B}(r,{\theta},t)=W_{0}R(r)\sin (n{\theta}\mp {\omega}_{n}t). & & \displaystyle\end{eqnarray}$ (3)

Therefore, two conditions need to be satisfied to generate a travelling wave: one is the specific partition of piezoelectric ceramic ring, and the other is that two AC voltages with a temporal phase differences of π∕2 are applied at the same time.

Table 1

The material parameters

Material	Density (kg/m³)	Young modulus (GPa)	Poisson’s ratio
Stator (QSn-6.5 )	8800	113	0.33
Piezoelectric ring
(Unpolarized PZT-8)	7600	102	0.3

Table 2

The material properties of the piezoelectric ring (Polarized PZT-8)

Piezoelectric Polarized (PZT-8)
Young modulus	GPa	c₁₁ = c₂₂	146.87	c₁₂ = c₂₁	81.87
	GPa	c₃₃	131.71	c₁₃ = c₃₁	81.05
	GPa	c₂₃ = c₃₂	81.05
Density	kg/m³		7600
Charge constants	C/m²	d₃₁ = d₃₂	−3.87538	d₁₅ = d₂₄	10.3448
Dielectric constants	C/(Vm)	ϵ₁₁= ϵ ₂₂	8.004 × 10⁻⁹	ϵ₃₃	4.973 × 10⁻⁹

3. 3D finite element simulation of the single stator

A three-dimensional finite element model of the single stator with 60 mm in diameter was established. The model is composed of two substructures: the stator (127,820 elements) and the piezoelectric ceramic ring (15,840 elements). All the numerical simulations have been performed with the FE code ABAQUS Standard. According to the ABAQUS analysis manual [13], C3D8R and C3D8E is used for the stator and piezoelectric region, respectively. In this study, the stator is made of phosphor bronze, and the piezoelectric ring is made of PZT-8. The parameters are shown in Tables 1 and 2.

It should be noted that the properties of the piezoelectric materials in the finite element model was according to the actual partition of electrodes in piezoelectric ring. In the analysis of dynamic response of the stator, the damping effects must be taken into consideration. In this study, Rayleigh damping is adopted. The damping coefficient was identified experimentally.

4. Experimental setup of the single stator

As shown in Fig. 2, the experimental setup consists of a vibration measurement system based on a three-dimensional vibration meter (Polytec Ltd, PSV-500-3D-M) and PXI acquisition system (NI, PXIe-1082) and a current measurement system. The traveling wave in the stator is excited by two voltages with phase difference of 90° from two amplifiers. The voltage was given and the response of the stator and the electric current were measured.

Fig. 2.

3D vibration and power measurement system.

5. Results

5.1. Simulation model validation

According to literature [1], the operating frequency of the motor is greater than the frequency of the anti-resonance point, in which the motor input current is relatively small. Therefore, the anti-resonance point is chosen as the benchmark in this study to conduct the verification experiment. The stator is fixed on the base, and the numerical result of frequency of the mode with 9 peaks is 39.333 kHz. The numerical result of the traveling wave generated in the stator by a voltage of 80 V_p−p at a frequency of 39.6 kHz (100 Hz larger than the anti-resonance point) is shown in Fig. 3(a).

The displacement distribution of the mode with 9 peaks was measured by a Laser Doppler Vibrometer and the result is shown in Fig. 3(b) when the stator is excited by a voltage of 80 V_p−p at the anti-resonance frequency. The experimental value of the anti-resonance frequency is 40.075 kHz, which is 0.575 kHz higher than the simulation result. Taking the geometrical errors of the model and the influence of boundary conditions into account, an error of less than 1.5% in resonance frequency of the 9th mode is reasonable. In order to validate the simulation model, the frequency characteristics of the electrical currents and those of displacement at the same point on the driving surface of the stator were compared and the results are shown in Fig. 4. For a reasonable comparison, the simulation results have been shifted on the frequency axis to agree the experimental results. Figure 4 suggests that the simulation and experimental results are in good agreement.

Fig. 3.

Displacement distribution of single stator drive surface for (a) simulation and (b) Experiment.

Fig. 4.

Comparison between experimental and simulation results: (a) input current of single stator, (b) displacement amplitude of drive surface.

5.2. Influence of frequency and voltage on elliptic motion

Based on the finite element model, the traveling wave in the state and the three-dimensional elliptic trajectory of a point on the teeth of the stator are analyzed. Figure 5(a) and Fig. 5(b) show the elliptic trajectory of a point on the stator surface in three-dimensional space and the displacements in the three directions, respectively. With respect to the displacement in the axial direction, there is a phase delay of 86.2° and 177.4° in the circumferential direction and in the radial direction, respectively. Near the anti-resonance point, the amplitude of the radial displacement is 39.5% of the circumferential amplitude, which indicates that the radial displacement cannot be ignored. The orientation of the major and minor axes of the ellipse depends on the phase differences and the amplitudes of the three displacements.

The influence of excitation frequency and voltage on phase differences was further analyzed, and the results are shown in Fig. 6. The results show that the influence of the input voltage on the phase difference between the elliptic displacements is less than 5‰. With the increase of frequency, the phase difference between circumferential and radial decreases (up to 1.6%), while the phase differences between the axial and radial increases (less than 4.5‰).

5.3. Traveling wave vibration characteristic analysis

Figure 7 shows the distribution of the three displacements along the circumferential line at different instants, in which T is the period of excitation. Due to the phase differences between the three displacements in time domain, phase differences between them in space can also be found. Figure 8 shows the distribution of displacements along the radial coordinate on the stator tooth surface at different instants. The results indicate that the stator radius has the greatest influence on the axial component of the displacement, and the amplitude of axial displacement at the inner circle of the teeth is only about 66.9% of that at the outer circle.

Fig. 5.

The elliptic motion: (a) three dimensional elliptic trajectory, (b) time shift curve of the stator.

Fig. 6.

Influence of working parameters on phase differences: (a) frequency, (b) voltage.

Fig. 7.

Displacement distribution on the circumference at different moments: (a) 0, (b) T/4, (c) T/2, (d) 3T/4.

Fig. 8.

Displacement distribution on the radius at different moments: (a) 0, (b) T/4, (c) T/2, (d) 3T/4.

6. Conclusions

A three-dimensional finite element model of the single stator was been established and validated experimentally. Based on this model, the traveling wave and three-dimensional trajectory of stator surface, including the influence of the input voltage on the phase and amplitude of the displacements in three directions, are investigated. The results show that the trajectory of particles on the stator surface is an ellipse in three-dimensional space due to the phase differences between the three components of displacement in the radial, circumferential and axial directions. The amplitude of radial displacement is about 39.5% of that in the circumferential displacement, which should not be neglected. The stator’s radius has the greatest influence on the distribution of the axial displacement. The results show that the model has good accuracy and can be used as a base model for the study of interfacial friction.

Footnotes

Acknowledgements

The research in this paper is sponsored by The National Basic Research Program of China (973 Program, grant no. 2015CB057501).

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