# Analysis of Preload of Three-Stator Ultrasonic Motor

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Three-Stator Ultrasonic Motor Structure and Working Mechanism

#### 2.1. Motor Structure

_{1}, ω

_{2}, and ω

_{3}are the angular velocity scalar, and u

_{1}, u

_{2}, and u

_{3}are the corresponding direction vector. u

_{2}and u

_{3}are obtained by rotating u1 around the Z axis by 120° and 240°, respectively, so they are:

#### 2.2. Working Principle

_{n}is angular frequency.

## 3. The Effect of Pre-Pressure on the Contact Range and Isokinetic Point of the Stator and Rotor

_{N}along the z-axis is applied to the friction material. At the same time, the particle moves relative to the rotor. Within the contact range, there is a frictional driving force F between the two in the positive x-axis direction, the magnitude of which is f, which drives the rotor to move in the positive x-axis direction. The thickness of the friction layer is set to h, which can be ideally equivalent to a linear spring. From Hooke’s law, the normal force of the stator on the rotor per unit length can be obtained:

_{s}, C

_{s}and K

_{s}are the modal quality, damping, and modal rigidity of the stator, separately, x is the displacement of the neutral plane, $\Theta $ is the electromechanical coupling coefficient, V is the excitation voltage, and F

_{n}is the pressure of the contact interface that is positively correlated with the pre-pressure. According to this formula, when the given pre-pressure increases, the contact range increases correspondingly; that is, the longitudinal restraint effect of the contact interface on the stator increases, so the amplitude of the stator will decrease. The fitting numerical relationship between the amplitude of the particles and the pre-pressure [11] can be expressed as:

_{max}at point b, and runs at the minimum horizontal speed V

_{min}at the contact end x = a. The horizontal speed of the particles is also different at different contact positions [17,18,19,20,21]. When running with load, the rotor will run at a fixed speed, so there is a speed difference between the stator and the rotor outside the two constant velocity points. In the area [b, c], the stator surface particle speed is always greater than the rotor speed. In actual operation, this area can produce driving torque; in the [c, d] area, the rotor speed is always greater than the stator surface particle speed. In fact, the stator will obstruct the movement of the rotor will generate friction torque. The available symbolic functions are expressed as:

_{T}is the propagation speed of the bending wave in the beam.

_{m}is the bending stiffness of the rotor, and a is half the length of the contact area.

_{d}is the weight fraction of the reinforcing material, E

_{n}is the elastic modulus of the matrix material, n is the correlation coefficient between the matrix and the filler arrangement, $\beta $ is the correction coefficient, and E

_{d}is the elastic modulus of the reinforcing material. Among them, according to this formula, it can be seen that as the weight fraction of the reinforcing material increases, the elastic modulus of the composite material will also increase, but this must be increased within a certain reasonable range, otherwise the crystal structure properties of the material itself will be destroyed, resulting in the material itself losing elasticity and the performance of the elastic modulus of the composite material deteriorating.

_{c}should not exceed the distance between the axis of the traveling wave and the neutral plane, which will cause a sudden change in the curvature radius of the contact surface, and the forces in the tangential direction of the rotor will cancel each other out, thereby reducing the driving force obtained. The stator particles follow the law of conservation of momentum in the vertical direction, and the sum of the excitation force of the piezoelectric ceramic received by the particles and the driving force of the friction material is the momentum of the stator particles. According to the equivalent spring stiffness theory [27], it can be deduced that in the contact range a, the equivalent elastic modulus of the friction material is:

## 4. The Influence of Preload on Modal Frequency

_{d}is the equivalent elastic friction coefficient, and $w\left(x,t\right)$ is the displacement of the stator’s transverse bending vibration. From this formula, it can be seen that the pre-pressure is proportional to the equivalent elastic friction coefficient, and its effect on the equivalent elastic friction coefficient is equivalent to the effect on the modal frequency. As the pre-pressure increases, the modal frequency of the stator will also be rise accordingly.

## 5. Experimental Test of Pre-Pressure on Motor Output Performance

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Schematic diagram of motor structure and pre-pressure device. (

**a**) Motor structure. (

**b**) Pre-pressure device.

**Figure 2.**Angular velocity and torque distribution diagram of multi-degree-of-freedom spherical ultrasonic motor.

**Figure 5.**Schematic diagram of the velocity of the mass point on the stator superficies at different contact positions.

**Figure 6.**Schematic diagram of contact deformation of stator and rotor. (

**a**) Stator and rotor contact modeling. (

**b**) Stator and rotor contact simulation.

**Figure 9.**Stator modal frequency. (

**a**) B

_{09}mode frequency with a preload of 0 N. (

**b**) B

_{09}mode frequency with a preload of 50 N.

**Figure 12.**The effect of pre-pressure on frequency. (

**a**) Impedance diagram when the pre-pressure is 0 N. (

**b**) Impedance diagram when the preload is 500 N.

Outer diameter | 30 mm | Tooth circumference angle | 3° |

Internal diameter | 9 mm | Tooth width | 5.5 mm |

Tooth height | 2 mm | Elastic base thickness | 2.5 mm |

Slotting angle | 5° | Piezoelectric ceramic thickness | 0.5 mm |

Material | $\mathbf{Elastic}\mathbf{Modulus}\mathbf{\left(}\mathit{E}\mathbf{\right)}$ | $\mathbf{Poisson}\mathbf{\u2019}\mathbf{s}\mathbf{Ratio}\mathbf{\left(}\mathit{\mu}\mathbf{\right)}$ | $\mathbf{Density}\mathbf{\left(}\mathit{\rho}\mathbf{\right)}$ |

Phosphor bronze | $1.12\times {10}^{11}\mathrm{N}/{\mathrm{m}}^{2}$ | 0.33 | $8800\mathrm{kg}/{\mathrm{m}}^{2}$ |

PZT-8 | $8\times {10}^{10}\mathrm{N}/{\mathrm{m}}^{2}$ | 0.30 | $7600\mathrm{kg}/{\mathrm{m}}^{2}$ |

Pre-Pressure | Simulation Frequency | Experiment Frequency | Relative Error |
---|---|---|---|

150 N | 41,650 Hz | 40,895 Hz | 1.81% |

200 N | 42,362 Hz | 41,098 Hz | 2.98% |

250 N | 43,408 Hz | 41,343 Hz | 4.8% |

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**MDPI and ACS Style**

Li, Z.; Zhao, H.; Che, S.; Chen, X.; Sun, H.
Analysis of Preload of Three-Stator Ultrasonic Motor. *Micromachines* **2022**, *13*, 5.
https://doi.org/10.3390/mi13010005

**AMA Style**

Li Z, Zhao H, Che S, Chen X, Sun H.
Analysis of Preload of Three-Stator Ultrasonic Motor. *Micromachines*. 2022; 13(1):5.
https://doi.org/10.3390/mi13010005

**Chicago/Turabian Style**

Li, Zheng, Hui Zhao, Shuai Che, Xuetong Chen, and Hexu Sun.
2022. "Analysis of Preload of Three-Stator Ultrasonic Motor" *Micromachines* 13, no. 1: 5.
https://doi.org/10.3390/mi13010005