This paper develops an electromechanical TWUSM (traveling-wave ultrasonic motor) model combining the driving circuit with the motor itself. An equivalent circuit model substituted for the piezoelectric ceramics is designed in the driving circuit model to obtain accurate input currents and powers. Then teeth discretization is implemented in the stator–rotor contact model, which can calculate the interaction forces more accurately. After building the complete model of TWUSM, a typical startup–stopping process is divided into five stages by evaluating the changes in contact status and driving forces. Finally, the fitness of transient responses of the rotor speed increasing from 55 % to 89 % shows that the proposed model fits better than the one without teeth discretization, and the experimental tests under various driving parameters verify the effectiveness of the model.
The TWUSM (traveling-wave ultrasonic motor) is the piezoelectric actuator which is excited near the resonant frequencies. The polarized piezoelectric ceramics of TWUSM are usually actuated by two alternating singular voltages. The stator particles change their displacements in space elliptical motion (Liu et al., 2019; Chen et al., 2018; Shi et al., 2018) with the deformations of piezoelectric elements. Several teeth are distributed along the circumferential direction to amplify the driving effect and deformed in accordance with the traveling wave (Renteria-Marquez et al., 2018). Therefore these motors present many significant merits compared to electromagnetic motors, for instance, rapid response, simple structure, and the capability of miniaturization (Zhang et al., 2016).
Because of the combination of piezoelectric actuation and friction drive,
the TWUSM model attracts many researchers' attention. In general, their
models can be divided into three types. First and most typical is the model stemming from Hagood and McFarland (1995), who assume the vibrating stator to be a two-freedom lumped spring-mass-damping system. However, the teeth
discretization is ignored by only assuming the contact model to be continuous springs covering the ring area. The second type is proposed by Giraud et al. (2004), who imitate the
Besides these deficiencies, the above researchers simplify the model by idealizing the input signals as the ideal sinusoidal signals instead of deriving them from the driving circuit. The simplification not only deviates from reality, but also fails to detect the real-time currents. In this paper, a hybrid mechatronic model is proposed by combining the electrical system with the mechanical system. Besides, a more straightforward discretization strategy is adopted in the hybrid model to solve the problems of insufficient or excessive discretization, which can not only guarantee the accuracy, but also reduce the simulation nodes and calculation sources.
This paper is organized as follows. In Sect. 2, the principle of the ultrasonic motor is introduced. Then a hybrid model including the electrical circuit and the ultrasonic motor is built in the third section. In Sect. 4, the transient response and the contact status are studied under different torques to explore the microcosmic law. In Sect. 5, the integrated test system is built, and the simulation results under different parameters are verified from the comparisons of the experimental results.
Figure 1 illustrates the working mechanism of the
TWUSM. Piezoelectric ceramics are actuated by the two-phase sinusoidal
voltages (
The block diagram of the TWUSM:
The TWUSM prototype investigated in this paper is USR60-S3 (Shinsei Corp.
Ltd, Japan), which works in the ninth vibration mode (
The driving circuit is supposed to generate pure sinusoidal waves for each
fragment of the piezoelectric ceramics. Figure 2
illustrates the framework of the driving circuit actuated by three
parameters (
The framework of the driving circuit.
Equivalent circuit of the piezoelectric ceramics (one phase).
As shown in Fig. 3,
In order to simplify the calculation, the vibration system (the
piezoelectric elements and the stator) can be characterized as a
2-degree-of-freedom spring-mass-damping system. The modal coordinates (
Figure 4 illustrates the contact schematics of the
motor with teeth discretization, and the stator–rotor contact model is built on the assumption that the rotor is rigid over all the contact areas,
whereas the friction layer can be modeled as a series of linear springs. The
yellow points mean the contact borders with the zone [
The contact schematic of the TWUSM.
According to Fig. 4 and Eq. (9), the coordinates
of contact border
The teeth discretization implemented in the stator.
In succession, the left teeth edges are recorded as If If If If If If If If
However, not all the particles of the contact region serve as the valid
driving points. Whether the particles play the driving role needs the
following detailed further discussions of stick–slip regional distribution, which also can be divided into four cases via the comparison between the
contact borders and stick points.
Based on the above classifications, the interaction forces between the
stator and the rotor become more accurate. First of all, the friction force
In terms of the whole TWUSM, the vertical displacement and the rotational
velocity of the rotor can be described in the third and fourth functions in Eq. (27), where
The total model covering the elements in Fig. 1 is built in the Simulink platform, with the parameters listed in Appendix A. In order to obtain more microscopic properties, simulations are implemented on the transient response without load and the contact status under different torques.
Figure 6 displays the startup–stopping response of the TWUSM when the input signals are the sinusoid waves containing 800
periods and the amplitude
Transient response of the unloaded motor with the driving
signals containing 800 periods of activation:
There are five stages, including the pre-static stage [
Figure 7 proposes the contact status of the teeth in
a 40
The contact status:
Figure 8 displays the integrated measurement system
that consists of the driving circuit, the mechanical platform, the FPGA
board (National Instruments Corp, USA), and the dSPACE1103 control board (dSPACE Corp, Germany). There is an incremental encoder AFS60A (SICK Corp,
Germany) which has 65 536 lines. The external load is generated by a torque
motor with the maximum torque 1 N m. The dSPACE1103 also generates the input
parameters (
The integrated measurement and control system of USR60-S3.
As shown in Fig. 9, when the amplitude
The transient characteristics comparing the experiment results and simulation results with or without the teeth discretization.
Comparison between simulation results and experimental ones when the amplitude is 1 V:
Due to the complexity of piezoelectric ceramics, the vibration conditions are different in diverse regions before or after the resonant points. In order to obtain the ideal working range, especially the frequency interval of the proposed integrated model, the rotor velocities of different amplitudes and frequencies are displayed in Fig. 11a and b.
Velocity response under different driving parameters:
It can be seen from Fig. 11a that the linearity between the driving amplitudes and the rotor speed is evident on the whole. And Fig. 11b demonstrates that the speed gradually increases and then drops to a lower value when the frequency is changed from 40 to 44 kHz. Moreover, the frequencies located on the peak velocity decrease as the amplitude increases. This is due to the variation of the natural frequency of the stator caused by the softening nonlinearity. We can conclude that when the frequency comes close to the resonant frequency, the simulation model cannot describe the speed adequately, which may result from the simplification of the stator modal model, as the frequency higher than the resonant peak is usually chosen as the working range, where the fitness becomes better especially from 41 and 44 kHz. The effective fitting results further prove the feasibility of the proposed model.
In all, the above test of speed performances with two driving parameters verifies the validity of the model located in a specific working area, where the amplitude is higher than 0.7 V and the frequency is higher than 41 kHz.
In order to verify the model in the functions of external load, Fig. 12 illustrates the comparison of mechanical characteristics from the simulations and the experiments. As can be seen, the velocity–torque curve exists in good agreement within the frequencies from 42 to 43 kHz, while the results occur with a little error with the lower frequency. Also, we can observe that the efficiency becomes higher when the frequency is near the resonant value, and the optimal torque we should impose on the motor will be less than 0.5 N m.
Comparison of the mechanical characteristics between
simulation and experimentation:
This paper presents an electromechanical hybrid model that combines the
driving circuit and the TWUSM itself. The main work and contributions can be
listed as follows.
A driving circuit model combining the circuit components with the load-dependent equivalent circuit model is proposed to simulate the real
electric network, which not only supports the TWUSM model, but also helps to gain the input voltages and currents for the calculation of input power. The teeth discretized method is employed to refine the contact status and interaction forces limited to the teeth space, which improves the accuracy
of the rotor step response not only in the rising time, but also the steady value. Model agreements are tested on the rotor speed under different parameters (amplitude, frequency, and torque) based on a multi-parameter
test system, which demonstrates the feasibility and effectiveness within the
ideal frequency working range higher than the resonant frequency.
Moreover, the proposed model achieves the observation of the microscopic
characteristics like input currents and vibration response of the transient
startup–stopping operation, which are of great significance for precise control of micro-stepping of the TWUSM in the future.
The key simulation parameters of USR60.
The data generated during this study are available from the corresponding author on reasonable request.
NC contributed to this work with the building of the integrated measurement system and the analysis from simulation and experiment; DF contributed to the guidance of the research and the revision of the manuscript.
The authors declare that they have no conflict of interest.
The authors are grateful for the financial support from the National Basic Research Program of China (973 Program, grant no. 2015CB057503).
This research has been supported by the National Basic Research Program of China (973 Program (grant no. 2015CB057503)).
This paper was edited by Daniel Condurache and reviewed by two anonymous referees.