# Experimental Studies on Torsional Stiffness of Cycloid Gear Based on Machining Parameters of Tooth Surfaces

• Zhifeng Liu
• Tao Zhang
• Yida Wang
• Congbin Yang
• Yongsheng Zhao
Regular Paper

## Abstract

Estimating the torsional stiffness has always been the primary issue in analyzing the dynamic characteristics of cycloid gears. The traditional method of obtaining torsional rigidity involves calculating the ratio of the input torque and rotation angle, treating the deformation of cycloid gear as a black box. In order to thoroughly understand the rotation angle caused by the local contact deformation of each cycloid pin gear, a Majumdar–Bhushan contact model and the finite element method are combined to express the normal contact stiffness. By multiplying the normal contact stiffness of each pin gear and the arm of normal contact force, the torsional stiffness of the cycloidal pin wheel system can be calculated. Experiments are conducted to establish the relationship between the torsional stiffness and roughness parameters of the machined tooth surface. The effect of input torque on the torsional stiffness has also been analyzed. This study formulates a relationship between the torsional stiffness and surface characteristics of cycloid gears, which can help improve their design and manufacture in the future.

## Keywords

Cycloid gear Torsional stiffness Fractal theory M–B contact model

## List of Symbols

$$a^{\prime}$$

Truncated area of a single asperity

$$a^{\prime}_{l}$$

Maximum truncated area of a single asperity

$$a^{\prime}_{c}$$

Critical truncated area between the elastic and elastic–plastic deformation regions

$$A_{0}$$

Contact area of each small grid

$$D$$

Two-dimensional fractal dimension

$$D_{s}$$

Three-dimensional fractal dimension which can be expressed as Ds = D + 1

$$e$$

The eccentricity of the cycloidal gear

$$E$$

Equivalent elastic modulus

$$f_{e}$$

Normal contact force of a single aspect

$$F$$

Normal load of the contact area

$$F_{i}$$

Average contact stress of each contact zone

$$G$$

Fractal roughness parameter

$$H$$

Material hardness

$$k_{n}$$

Normal stiffness of contact area

$$k_{ne}$$

Normal stiffness of a single asperity

$$K_{i}$$

Normal contact stiffness of each contact grid

$$K_{1}$$

Short amplitude coefficient

$$K_{t}$$

Total equivalent torsional stiffness of the cycloidal gear

$$K_{ni}$$

Normal contact stiffness of the ith tooth pair

$$K_{ti}$$

Equivalent torsional stiffness of the ith tooth pair

$$L_{i}$$

Length of the contact force arm at position i

$$n(a^{\prime})$$

Size distribution function of asperities

$$N$$

Number of teeth engaged in the engagement of the needle wheel

$$r_{p}$$

Circular radius of the pin gear

$$z_{c}$$

Tooth number of the cycloidal gear

$$\updelta$$

Normal deformation of a single aspect

$$\psi$$

Domain extension factor for micro-contact size distribution

$$\gamma$$

Dimension parameter of the spectral density

$$\angle O_{i} O_{p} P$$

Angle between the crank and the two center points

## Notes

### Acknowledgements

The authors would like to thank the National Natural Science Fund No. 51575009, Jing-Hua Talents Project of Beijing University of Technology and Beijing Science and Technology Major Project coded D17110400590000 for supporting the research.

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© Korean Society for Precision Engineering 2019

## Authors and Affiliations

• Zhifeng Liu
• 1
• Tao Zhang
• 1
• Yida Wang
• 2
• Congbin Yang
• 1
Email author
• Yongsheng Zhao
• 1
1. 1.Key Laboratory of Advanced Manufacturing TechnologyBeijing University of TechnologyBeijingPeople’s Republic of China
2. 2.Shandong Institute of Space Electronic TechnologyYantaiPeople’s Republic of China