# Development of Harmonic Drive Combining Four Arcs for Conventional Kinematic Application

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## Abstract

**:**

## 1. Introduction

## 2. Harmonic Cam Using Four-Arc Combination

#### 2.1. Kinetic Characteristics of Harmonic Drives

#### 2.2. Four-Arc Combination Cam

## 3. Harmonic Drive Design Guide

#### 3.1. Proposed Guide for Harmonic Drive Design with Cycloid Gear

- 1.
- Define the number ratio of lobes of the flex and circular splines, $i=\frac{{Z}_{c}}{{Z}_{f}}$
- 2.
- Define $\lambda $Calculate the equivalent number of teeth ${Z}_{t}$ in Equation (18)
- 3.
- Define the neutral line diameter ${d}_{f}$ and offset = $h$Calculate ${R}_{0}$ in Equation (9)Calculate the equivalent module in Equation (19)$${m}_{t}=\frac{2\left({R}_{0}+h\right)}{{Z}_{t}}\text{}$$

- (1)
- Arc AB: Arrange ${m}_{t}$ teeth into the circumferential pitch ${p}_{t}$
- (2)
- Arc BC: Arrange ${m}_{t}$ teeth into the circumferential pitch ${p}_{1}$If the teeth overlap in section B (Figure 6), arrange the teeth using Equation (20).$${\theta}_{1}={p}_{1}\left(1-\frac{{\theta}_{t}}{{p}_{t}}\right)$$

#### 3.2. Proposed Guide for Harmonic Drive Design with Involute Gear

- (1)
- Define the number ratio of lobes of the flex and circular splines, $i=\frac{{Z}_{c}}{{Z}_{f}}$
- (2)
- Define $\lambda $Calculate the equivalent number of teeth ${Z}_{t}$ in Equation (18)
- (3)
- Define neutral line diameter ${d}_{f}$ and offset = $h$Calculate ${R}_{0}$ in Equation (9)Calculate equivalent module in Equation (19)
- (4)
- Adjust the involute gearThe internal gear equation is$$inv{\alpha}_{b}=2\mathrm{tan}\alpha \frac{{x}_{1}+{x}_{2}}{{z}_{1}+{z}_{2}}+inv{\alpha}_{c}$$$$E=\frac{{z}_{2}-{z}_{1}}{2}m+\frac{{z}_{2}-{z}_{1}}{2}m\left(\frac{\mathrm{cos}{\alpha}_{c}}{\mathrm{cos}{\alpha}_{b}}-1\right)$$If the profile shifted gear is not used, $E$ becomes$$E=\frac{{z}_{2}-{z}_{1}}{2}m$$According to the harmonic drive design guide with the cycloid gear, $E$ is$$E={R}_{c}-\left({R}_{0}+h\right)$$$$E=\frac{{m}_{t}{z}_{c}}{2}-\frac{{m}_{t}{z}_{t}}{2}\text{}$$

## 4. Gear Design

## 5. Design of Harmonic Drive

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Musser, C.W. Strain Wave Gearing. U.S. Patent No. 2.906, 1959. [Google Scholar]
- Musser, C.W. Spline and Rotary Table. U.S. Patent No. 2,959,065, 1960. [Google Scholar]
- Musser, C.W. The harmonic drive. Mach. Des.
**1960**, 14, 160–173. [Google Scholar] - Brighton, D.K. Harmonic Drives. U.S. Patent No. 3,996,816, 1976. [Google Scholar]
- Kayabasi, O.; Fehmi, E. Shape optimization of tooth profile of a flexspline for a harmonic drive by finite element modelling. Mater. Des.
**2007**, 28, 441–447. [Google Scholar] [CrossRef] - Chen, X.; Liu, Y.; Xing, J.; Lin, S.; Ma, M. A novel method based on mechanical analysis for the stretch of the neutral line of the flexspline cup of a harmonic drive. Mech. Mach. Theory
**2014**, 76, 1–19. [Google Scholar] [CrossRef] - Jeon, H.S.; Oh, S.H. A study on stress and vibration analysis of a steel and hybrid flexspline for harmonic drive. Compos. Struct.
**1999**, 47, 827–833. [Google Scholar] [CrossRef] - Maiti, R. A Novel Harmonic Drive With Pure Involute Tooth Gear Pair. J. Mech. Des.
**2004**, 126, 178–182. [Google Scholar] [CrossRef] - Bamnote, A.J.; Mahale, P.; Gulhane, R. Meshing Analysis of Teeth of Harmonic Drives: A Computer Based Approach; Dept. of Mechanical Engg., YC College of Engg.: Nagpur, India, 2000; pp. 1–8. [Google Scholar]
- Ishikawa, S. The gear geometry of tooth engagement in harmonic drive. Proceedings of [the JSME] 1967 Semi-international Symposium; 1967. [Google Scholar]
- Ishikawa, S. Tooth Profile of Spline of Strain Wave Gearing. U.S. Patent No. 4,823,638, 1989. [Google Scholar]
- Kiyosawa, Y. Performance of a strain wave gearing using a new tooth profile. In ASME International Power Transmission and Gearing Conference; American Society of Mechanical Engineers: New York, NY, USA, 1989; Volume 11, p. 607. [Google Scholar]
- Xin, H.B.; Mo, H.N.; Gao, J.C.; Wang, W.J.; Cui, D.Q.; Liu, L.; Wang, T.; Xin, Y.F. Study on the Gear Tooth Influence Coefficients of Flexspline of Harmonic Drive. Adv. Mater. Res.
**2013**, 774–776, 144–147. [Google Scholar] [CrossRef] - Chen, X.; Liu, Y.; Xing, J.; Lin, S.; Xu, W. The parametric design of double-circular-arc tooth profile and its influence on the functional backlash of harmonic drive. Mech. Mach. Theory
**2014**, 73, 1–24. [Google Scholar] [CrossRef] - Dong, H.; Ting, K.-L.; Wang, D. Kinematic Fundamentals of Planar Harmonic Drives. J. Mech. Des.
**2011**, 133, 011007. [Google Scholar] [CrossRef] - Sahoo, V.; Maiti, R. Load sharing by tooth pairs in involute toothed harmonic drive with conventional wave generator cam. Meccanica
**2018**, 53, 373–394. [Google Scholar] [CrossRef] - Song, C.; Li, X.; Yang, Y.; Sun, J. Parameter design of double-circular-arc tooth profile and its influence on meshing characteristics of harmonic drive. Mech. Mach. Theory
**2021**, 167, 104567. [Google Scholar] [CrossRef] - Wang, S.; Li, D.; Mao, S.; Chen, B. Design and Analysis of Cam Wave Generator Based on Free Deformation in Non-Working Area of the Flexspline. Appl. Sci.
**2021**, 11, 6049. [Google Scholar] [CrossRef] - Gravagno, F.; Mucino, V.H.; Pennestrì, E. Influence of wave generator profile on the pure kinematic error and centrodes of harmonic drive. Mech. Mach. Theory
**2016**, 104, 100–117. [Google Scholar] [CrossRef] - Li, X.; Song, C.; Yang, Y.; Zhu, C.; Liao, D. Optimal design of wave generator profile for harmonic gear drive using support function. Mech. Mach. Theory
**2020**, 152, 103941. [Google Scholar] [CrossRef] - Mahanto, B.S.; Sahoo, V.; Maiti, R. Effect of Cam Insertion on Stresses in Harmonic Drive in Industrial Robotic Joints. Procedia Comput. Sci.
**2018**, 133, 432–439. [Google Scholar] [CrossRef] - Jia, H.; Li, J.; Xiang, G.; Wang, J.; Xiao, K.; Han, Y. Modeling and analysis of pure kinematic error in harmonic drive. Mech. Mach. Theory
**2020**, 155, 104122. [Google Scholar] [CrossRef] - Lee, C.W.; Oh, S.H.; Kim, J.C.; Jeon, H.S. Development of Harmonic Drive Using Cycloide Tooth Profile. Trans. Korean Soc. Mech. Eng. A
**1997**, 21, 1166–1173. [Google Scholar] - Jang, D.-J.; Kim, Y.-C.; Hong, E.-P.; Kim, G.-S. Geometry design and dynamic analysis of a modified cycloid reducer with epitrochoid tooth profile. Mech. Mach. Theory
**2021**, 164, 104399. [Google Scholar] [CrossRef]

Parameter | Value | |
---|---|---|

Input data | Lobes of circular spline, ${Z}_{c}$ | 74 |

Lobes of flex spline, ${Z}_{f}$ | 72 | |

Deviation coefficient, $\lambda $ | 0.99 | |

Diameter of flex spline, ${d}_{f}$ (mm) | 100 | |

Offset distance, $h$ (mm) | 1 | |

Results | Radii ${R}_{0}$, ${R}_{0}$ (mm) | 47.8876 |

Radii ${R}_{1}$, ${R}_{1}$ (mm) | 52.1124 | |

Eccentricity distance, $E$ (mm) | 2.9874 | |

Module, ${m}_{t}$ | 1.4179 |

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**MDPI and ACS Style**

Kim, T.; Lee, J.; Jeong, W.; Jeon, H.; Oh, S.
Development of Harmonic Drive Combining Four Arcs for Conventional Kinematic Application. *Machines* **2022**, *10*, 1058.
https://doi.org/10.3390/machines10111058

**AMA Style**

Kim T, Lee J, Jeong W, Jeon H, Oh S.
Development of Harmonic Drive Combining Four Arcs for Conventional Kinematic Application. *Machines*. 2022; 10(11):1058.
https://doi.org/10.3390/machines10111058

**Chicago/Turabian Style**

Kim, Taesu, Jongseok Lee, Wonhyeong Jeong, Hansu Jeon, and Sehoon Oh.
2022. "Development of Harmonic Drive Combining Four Arcs for Conventional Kinematic Application" *Machines* 10, no. 11: 1058.
https://doi.org/10.3390/machines10111058