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Linkage model and interpolation analysis of helical non-circular gear hobbing

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Abstract

The design flexibility and transmission stability of the non-circular gears can be improved using the helical tooth scheme. Herein, a linkage model was derived for hobbing the helical non-circular gears based on the influence of the axial feed motion of the hob on the motion of the projecting rack on the gear-blank end face. This axial feed motion produces additional motion effects on the rotary axis of the gear-blank and the moving axis of the hob. Further, the accuracy of the linkage model was verified by kinematic simulations. The global convergence characteristics of the transcendental equation used for obtaining the polar angle of the pitch curve were ascertained to derive the interpolation calculation process for the linkage model-based electronic gearbox. The cause of cumulative error during the multi-turn hobbing process of the gear blank was analyzed. The error accumulation was effectively controlled by optimizing the interpolation algorithm. The hobbing experiments and meshing transmission test were conducted using the self-developed non-circular gear hobbing system to verify the effectiveness of the linkage model and interpolation algorithm.

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All data generated or analyzed during this study are included in this published article.

Abbreviations

ω c :

The angular velocity of the gear-blank axis

ω b :

The angular velocity of the hob axis

v x :

The linear velocity of the hob axis in x-axis

v z :

The linear velocity of the hob axis in z-axis

v y :

The linear velocity of the virtual rack in y-axis

A and B:

The points on pitch curve of non-circular gear

v B :

The velocity of point B

θ :

The rotation angle of gear blank

φ :

The polar angle of pitch curve

μ :

The angle between the polar radius and the tangent

δ :

The angle between vB and vy

r :

The polar radius of pitch curve

β :

The helical angle of non-circular gear

∆ω c :

The additional angle of gear blank

M-M:

The section parallel to the end face of the gear blank

N–N:

The section parallel to the end face of the gear blank

Q1 :

The point located on the centerline of the equivalent rack in the M–M section

Q2 :

The point located on the centerline of the equivalent rack in the N–N section

P1 and P2 :

The points located on the pitch plane of the equivalent helical rack

Q3 :

The point located on the segment P1P2 and the section N–N

∆z:

The distance between sections M-M and N–N

v y*:

The moving speed of the projection rack in the y-axis direction

T:

The number of hob threads

m n :

The normal modulus of non-circular gear

K z and K c :

The sign coefficients

A :

The length of the elliptical semimajor axis

e :

The eccentricity of ellipse

tz :

The total simulation time

∆t :

The time step of simulation

∆b and ∆z:

The displacements of the b- and z-axes during the time ∆t

s :

The displacement of the hob in the x-axis direction

a and b:

The interval value in the solution of φ

φ 0 and φ * :

The initial value and convergence value of φ

L :

The positive number in the solution of φ

i 12 :

The gear ratio of the non-circular gear pair

D :

The transmission center distance

ε :

The error limit of Steffensen’s iterative method

N :

The number of Steffensen iterations

l :

The displacement of the virtual rack in the y-axis direction

eθ and es :

The interpolation errors

∆l i :

The displacement of the projected rack in the y-axis direction during each interpolation period

f 1 and f 2 :

The coefficients for evaluating error accumulation

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Funding

This work was financially supported by the National Natural Science Foundation of China (Grant No. 51875161 & 51575154).

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Authors and Affiliations

  1. CIMS Institute, Hefei University of Technology, Hefei, 230009, People’s Republic of China

    Jiang Han, Dazhu Li, Xiaoqing Tian & Lian Xia

Authors
  1. Jiang Han
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  2. Dazhu Li
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  3. Xiaoqing Tian
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  4. Lian Xia
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Contributions

JH and LX were in charge of the whole trial; DL wrote the manuscript; and XT assisted with design and experiment analyses. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Dazhu Li.

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The authors declare that they have no competing interests.

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Technical Editor: Marcelo Areias Trindade.

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Han, J., Li, D., Tian, X. et al. Linkage model and interpolation analysis of helical non-circular gear hobbing. J Braz. Soc. Mech. Sci. Eng. 42, 582 (2020). https://doi.org/10.1007/s40430-020-02663-1

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  • DOI: https://doi.org/10.1007/s40430-020-02663-1

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