# Meshing Drive Mechanism of Double Traveling Waves for Rotary Piezoelectric Motors

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Input Parameters and Boundary Conditions for FEM

#### 2.2. Theoretical Derivation for Wave Generating

## 3. Results and Discussion

#### 3.1. Numerical Results of FEM

#### 3.2. Prototype Experimental Results

#### 3.3. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Comparison of mode shapes for stator and rotors. (

**a**) 9th modal shape of stator. (

**b**) 9th modal shape of resonance rotor. (

**c**) Adjacent modal shape of reference rotor. (

**d**) 9th modal shape of reference rotor. (

**e**) Adjacent modal shape of flexible rotor. (

**f**) The other adjacent modal shape of flexible rotor.

**Figure 2.**Amplitude response of the stator in the peripheral, axial and radial directions. (

**a**) Modal response nephogram of stator. (

**b**) Displacement amplitude versus sweeping drive frequency. (

**c**) Velocity amplitude versus sweeping drive frequency. (

**d**) Acceleration amplitude versus sweeping drive frequency.

**Figure 3.**Amplitude response of resonance rotor in the peripheral, axial and radial directions. (

**a**) Modal response nephogram of resonance rotor. (

**b**) Displacement amplitude versus sweeping drive frequency. (

**c**) Velocity amplitude versus sweeping drive frequency. (

**d**) Acceleration amplitude versus sweeping drive frequency.

**Figure 4.**Modal response of stator and resonance rotor within one oscillatory period. (

**a**) Periodical displacement response of stator. (

**b**) Periodical velocity response of stator. (

**c**) Periodical acceleration response of stator. (

**d**) Periodical displacement response of resonance rotor. (

**e**) Periodical velocity response of resonance rotor. (

**f**) Periodical acceleration response of resonance rotor.

**Figure 5.**Variation of phase difference of stator and resonance rotor as sweeping frequency. (

**a**) Peripheral phase difference of stator. (

**b**) Axial phase difference of stator. (

**c**) Radial phase difference of stator. (

**a**) Peripheral phase difference of resonance rotor. (

**b**) Axial phase difference of resonance rotor. (

**c**) Radial phase difference of resonance rotor.

**Figure 6.**Amplitude response of reference rotor in the peripheral, axial and radial directions. (

**a**) Modal response nephogram of reference rotor. (

**b**) Displacement amplitude versus sweeping drive frequency. (

**c**) Velocity amplitude versus sweeping drive frequency. (

**d**) Acceleration amplitude versus sweeping drive frequency.

**Figure 7.**Amplitude response of the flexible rotor in peripheral, axial and radial directions. (

**a**) Modal response nephogram of flexible rotor. (

**b**) Displacement amplitude versus sweeping drive frequency. (

**c**) Velocity amplitude versus sweeping drive frequency. (

**d**) Acceleration amplitude versus sweeping drive frequency.

**Figure 8.**Modal response of reference rotor and flexible rotor within one oscillatory period. (

**a**) Periodical displacement response of reference rotor. (

**b**) Periodical velocity response of reference rotor. (

**c**) Periodical acceleration response of reference rotor. (

**d**) Periodical displacement response of flexible rotor. (

**e**) Periodical velocity response of flexible rotor. (

**f**) Periodical acceleration response of flexible rotor.

**Figure 9.**Variation of phase difference of reference rotor and flexible rotor as sweeping frequency. (

**a**) Peripheral phase difference of reference rotor. (

**b**) Axial phase difference of reference rotor. (

**c**) Radial phase difference of reference rotor. (

**d**) Peripheral phase difference of flexible rotor. (

**e**) Axial phase difference of flexible rotor. (

**f**) Radial phase difference of flexible rotor.

**Figure 10.**Prototype experiments and output performance curves. (

**a**) Stator. (

**b**) Resonance rotor. (

**c**) Reference rotor. (

**d**) Flexible rotor. (

**e**) Experimental device. (

**f**) Speed-torque curve. (

**g**) Power-torque curve.

Vibrators | Materials | Outer Diameter (mm) | Inner Diameter (mm) | Total Height (mm) |
---|---|---|---|---|

Stator | Phosphor bronze & PZT-8H | 60 | 18 | 4.5 + 0.5 |

Resonance rotor | 18 | 4.5 + 0.5 | ||

Reference rotor | Phosphor bronze | 8 | 5 | |

Flexible rotor | Aluminum | 8 | 5 |

Parameters | Phosphor Bronze | Aluminum | PZT-8H |
---|---|---|---|

Mass density (kg/m^{3}) | 8780 | 2770 | 7600 |

Poisson’s ratio | 0.341 | 0.3 | — |

Young’s modulus (N/m^{2}) | 1.1E11 | 7.17E10 | — |

Calculated Results | Stator | Resonance Rotor | Reference Rotor | Flexible Rotor | |
---|---|---|---|---|---|

9th modal frequency (Hz) | 39,535 | 39,514 | 38,419 | — | |

Adjacent modal frequency (Hz) | — | — | 38,756 | 37,912 41,136 | |

Displacement amplitude (μm) ^{1} | Peripheral | 3.608 | 3.716 | 2.139 | 0.466 |

Axial | 3.258 | 2.255 | 0.243 | 0.085 | |

Radial | 1.691 | 0.699 | 0.105 | 0.020 | |

Velocity amplitude (mm/s) ^{1} | Peripheral | 896.5 | 923.1 | 531.3 | 115.9 |

Axial | 809.6 | 560.3 | 60.49 | 21.27 | |

Radial | 420.3 | 173.8 | 26.10 | 5.035 | |

Acceleration amplitude (mm/s^{2}) ^{1} | Peripheral | 2.227 × 10^{8} | 2.293 | 1.319 × 10^{8} | 0.288 × 10^{8} |

Axial | 2.011 × 10^{8} | 1.391 | 0.150 × 10^{8} | 0.052 × 10^{8} | |

Radial | 1.044 × 10^{8} | 0.431 | 0.064 × 10^{8} | 0.012 × 10^{8} |

^{1}Driving frequency is equal to 9th modal frequency of stator.

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**MDPI and ACS Style**

An, D.; Huang, W.; Liu, W.; Xiao, J.; Liu, X.; Liang, Z.
Meshing Drive Mechanism of Double Traveling Waves for Rotary Piezoelectric Motors. *Mathematics* **2021**, *9*, 445.
https://doi.org/10.3390/math9040445

**AMA Style**

An D, Huang W, Liu W, Xiao J, Liu X, Liang Z.
Meshing Drive Mechanism of Double Traveling Waves for Rotary Piezoelectric Motors. *Mathematics*. 2021; 9(4):445.
https://doi.org/10.3390/math9040445

**Chicago/Turabian Style**

An, Dawei, Weiqing Huang, Weiquan Liu, Jinrui Xiao, Xiaochu Liu, and Zhongwei Liang.
2021. "Meshing Drive Mechanism of Double Traveling Waves for Rotary Piezoelectric Motors" *Mathematics* 9, no. 4: 445.
https://doi.org/10.3390/math9040445