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Research on Tolerance Design of 2K-V Reducer with Beveloid Gear Considering the Effect of Anti-Backlash

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Abstract

A mathematical model for calculating the lost motion of the 2K-V reducer with beveloid gear was established, considering the influence of assembly-induced error factors and the effect of backlash elimination. Sensitivity analysis was conducted on each error factor, and in combination with actual manufacturing processes, a set of tolerance correction values was designed using the Monte Carlo method. Based on these tolerance values, prototype manufacturing and performance testing were carried out. The validity of the proposed theoretical model was verified through experimental validation. The results indicate that three significant factors affecting the overall lost motion are the radial clearance of the tapered roller bearing, the backlash of the involute spline, and the radial runout of the small tooth difference gear. The tolerance correction reduced the difficulty of actual manufacturing, allowing the three highly sensitive error ranges to be relaxed to 757.58 times, 1.26 times, and 8.63 times of their original values, when the lost motion of the whole machine is less than 1arcmin.

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Abbreviations

\({\alpha }_{n}\) :

Tooth angle (indexing circle pressure angle)

B :

Lost motion

\(jt\) :

Circumferential side backlash

\(\Delta jt\) :

Circumferential side backlash

\(\Delta j_{n}\) :

Normal phase side backlash

\(\Delta j_{r}\) :

Radial side backlash

\(m_{n}\) :

Normal phase modulus

\(Z\) :

Number of teeth

\(K_{\alpha }\) :

Conversion factor

\(\alpha_{t}{\prime}\) :

Pressure angle at any circle of the gear blank

\({\upalpha }_{{\text{t}}}\) :

Pressure angle of transverse section

\(j_{{tF_{r} }}\) :

Amount of circumferential side backlash caused by the radial runout of the gear

\(F_{r}\) :

Radial runout value of the gear

\(\theta\) :

Radial runout phase angle of the gear

\(j_{{t\delta_{w1} }}\) :

External gear circumferential side backlash caused by the deviation of external gear bearing hole position

\(\delta_{w1}\) :

Value of external gear bearing hole position deviation

\(\theta_{1}\) :

Phase angle of external gear bearing hole position error

\(j_{t\delta w2}\) :

Amount of external gear circumferential side clearance caused by the deviation of external gear bearing hole position

\(\delta w2\) :

Value of deviation of planetary frame bearing hole position

\(\theta_{2}\) :

Phase angle of planetary frame bearing hole position error

\(B_{\Delta ry}\) :

Output shaft lost motion generated by the radial backlash of the tapered roller bearing

\(\Delta_{ry}\) :

Radial backlash of the tapered roller bearing

\(a_{0}\) :

Crankshaft center distance

\(j_{t\Delta rj}\) :

Gear side backlash generated by the radial backlash of the angular contact ball bearing

\(\Delta_{rj}\) :

Radial backlash of the angular contact ball bearing

\(j_{{t\delta_{t2} }}\) :

Gear side clearance generated by the radial runout of the journal on the planetary carrier

\(\delta_{t2}\) :

Radial runout error of the journal on the planetary carrier

\(\theta_{3}\) :

Phase angle of the radial runout error of the journal on the planetary carrier

\(j_{{t\delta_{t3} }}\) :

Amount of external gear circumferential side backlash caused by the position degree error of the housings bearing hole

\(\delta_{t3}\) :

Coaxiality error of the housings bearing hole

\(\theta_{4}\) :

Phase angle of the coaxiality error of the box bearing hole

\(j_{tck}\) :

Amount of circumferential side backlash of the planetary wheel caused by the involute spline backlash

\(ck\) :

Involute spline backlash

\(m_{k}\) :

Spline modulus

\(z_{k}\) :

Number of spline teeth

\(D_{k}\) :

Large diameter of the involute spline

\(i_{3w}\) :

Ratio of the small tooth difference stage to the output

\(s_{i}\) :

Sensitivity index of parameter i

\(w_{i}\) :

Weight coefficient of a design parameter

\(\varepsilon_{i}\) :

Permissible lost motion of this design parameter

\(\varepsilon_{0}\) :

Total permissible lost motion of the reducer

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Acknowledgments

The authors would like to thank the National Key R&D Program of China (Grant No. 2019YFB2004700).

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Authors and Affiliations

  1. State Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing, 400030, China

    Zhiying Cui, Chaosheng Song, Feihong Zhu & Caichao Zhu

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  1. Zhiying Cui
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Correspondence to Chaosheng Song.

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Cui, Z., Song, C., Zhu, F. et al. Research on Tolerance Design of 2K-V Reducer with Beveloid Gear Considering the Effect of Anti-Backlash. Int. J. Precis. Eng. Manuf. 25, 349–362 (2024). https://doi.org/10.1007/s12541-023-00916-2

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  • DOI: https://doi.org/10.1007/s12541-023-00916-2

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