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Study on tooth profile design of harmonic drive with deformation model of flexspline

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Abstract

The neutral curve of a flexspline (FS) is usually represented by the function curve of cam-shape curve in the traditional harmonic drive (HD) model. This representation may induce tooth tip interference and excessive deformation. In this study, a novel mathematical model of the FS deformation is established to accurately depict the neutral curve. Then, a tooth profile design method is proposed based on this model, which can obtain an exact conjugate tooth profile when giving an initial tooth profile. In addition, another design method that does not demand the initial tooth profile is proposed for simplicity of design. The HD designed by this method has a larger number of meshing teeth and a larger depth of engagement compared to traditional HDs. Moreover, a meshing model of a tooth pair is employed to verify the feasibility of the two design methods. Next, one of them is used to investigate the effect of the wave generator shape on the engagement of a couple tooth profiles, and the results show that the wave generator shapes with different parallel distances have no effect on the continuous engagement. Furthermore, an instantaneous transmission ratio calculation method of a tooth pair is developed for future calculations of the instantaneous transmission ratio between the input and output shafts of the HD. The calculation results manifest that the instantaneous transmission ratio of the tooth pair belongs to a variable transmission ratio.

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Abbreviations

φ, φ K −1 :

Eccentric angle of the cam-shape curve

n :

Positive integer

Δ S :

Line element

ϕ 1 :

Included angle between the positive axis of x1 and the positive axis of X

ϕ 2 :

Included angle between the positive axis of x2 and the positive axis of X

X i, Y i :

Trajectory coordinates of a point on the neutral curve

Φ ( i ) :

Tilt angle of the normal of a point on the neutral curve

u :

Variable of the FS tooth profile equation

r m :

Radius of the neutral curve before the FS is deformed

S :

Overall length of the neutral curve

\({\phi}_{{G}}^{\text{(}{1}\text{)}}\) :

Rotation angle of the CS tooth in the first operation type

\({\phi}_{{G}}^{\text{(}{2}\text{)}}\) :

Rotation angle of the CS tooth in the second operation type

ω F :

Angular velocity at the meshing point on the FS tooth

ϕ 3 :

Rotation angle at the meshing point on the CS tooth around its center

\({{i}}_{{FC}}^{{(}{{H}}{)}}\) :

Instantaneous transmission ratio of a tooth pair in the first operation type

ω H :

Angular velocity of the WG

\(i_{FH}^{(C)}\) :

Instantaneous transmission ratio of a tooth pair in the third operation type

N :

Common normal vector at the contact point

V 12 :

Relative velocity vector at the contact point

M 21 :

Coordinate transformation matrix from {x2o2y2} to {XOY}

W 21 :

Normal vector transformation matrix from {x2o2y2} to {XOY}

r (1) :

Radius vector of the FS tooth profile

N (1) :

Normal vector of the FS tooth profile

r (2) :

Radius vector of the CS tooth profile

N (2) :

Normal vector of the CS tooth profile

r c :

Original circle radius of the CS

a 0 :

Semi-major axis of the neutral curve of the deformed FS

δ :

Correlation coefficient

R :

Offset distance

r b :

Base circle radius of the FS

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Acknowledgements

This work was supported by the National Natural Science Foundation Regional Joint Project through a National Natural Science Foundation of China (Grant No.U1508211) and the National Key Research Development Program through Ministry of Science and Technology of the People’s Republic of China (Grant 2018YFB2001400).

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Authors and Affiliations

  1. School of Mechanical Engineering, Dalian University of Technology, Dalian, 116000, China

    Zhenjiang Yu

  2. Key Laboratory for Precision & Non-Traditional Machining of Ministry of Education, Dalian University of Technology, Dalian, 116000, China

    Siying Ling, Xiaodong Wang & Liding Wang

  3. Key Laboratory for Micro/Nano Technology and System of Liaoning Province, Dalian University of Technology, Dalian, 116000, China

    Siying Ling, Xiaodong Wang & Liding Wang

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  1. Zhenjiang Yu
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  2. Siying Ling
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  4. Liding Wang
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Correspondence to Xiaodong Wang.

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Yu, Z., Ling, S., Wang, X. et al. Study on tooth profile design of harmonic drive with deformation model of flexspline. Meccanica 56, 1883–1904 (2021). https://doi.org/10.1007/s11012-021-01372-w

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