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Investigation of the transmission accuracy of ball screw considering errors and preloading level

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Abstract

Transmission accuracy is one of the most important parameters in ball screw mechanism (BSM); however, very few researches can be found on the transmission error modeling for BSM. Therefore, on the basis of the converting principle of the errors in the normal and axial direction proposed in this paper, this paper proposes a new model to predicate the transmission accuracy of BSM considering the manufacturing errors, installation errors, as well as the transmission error due to different loading status. After the error analysis and calibration of a transmission accuracy measuring system, the transmission accuracy measurement of a typical BSM under five different preloading levels is performed. The experimental results show that the difference compared with the analytical solution is 21.6% under no preload condition, and is less than 11% under preload condition, largely owing to the uneven distribution of clearance can increase the travel deviation. Further analysis shows that the eccentricity error is the dominant factor leading to the periodic fluctuation of the transmission error. More importantly, the travel deviation increases with increasing preload, which indicates the transmission accuracy of the BSM deteriorates when the load increases.

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Abbreviations

\({F}_{a}\) :

The applied axial force on the ball screw

\({Q}_{i}\) :

The contact force between the ith ball and the raceway

\({F}_{ni}\) :

The load of the nut between the ith and the (i + 1)th circle

\({F}_{si}\) :

The load of the screw between the ith and the (i + 1)th circle

\({\delta }_{i}\) :

The normal deformation of the ith ball with screw and nut raceway corresponding to the ith ball

\({\delta }_{bi}\) :

The variation of the ball center relative to the initial position

\({\Delta L}_{si}\) :

The axial distance of screw between the adjacent balls

\({\Delta L}_{ni}\) :

The axial distance of nut between the adjacent balls

\(\Delta {n}_{i}\) :

The deformation of the nut within the ith and (i + 1)th circle along the axial direction

\(\Delta {s}_{i}\) :

The deformation of the screw within the ith and (i + 1)th circle along the axial direction

\({\delta }_{s,i}\) :

Axial displacement of the ith ball with screw

\({\delta }_{n,i}\) :

Axial displacement of the ith ball with nut

\({\delta }_{i\_a}\) :

The axial deformation of the ith ball with screw and nut raceway

q :

The gravity of the ball screw per unit length

\(\Delta {d}_{p}\) :

The travel error caused by the profile error

\(\Delta {d}_{l}\) :

The travel error caused by the lead error

\(\Delta {d}_{e}\) :

The travel error caused by the eccentricity error

\(\Delta {d}_{r}\) :

The travel error caused by the roundness error of raceway

\({\delta }_{sg}\) :

The travel error caused by the support unit

\({\delta }_{in}\) :

The travel error caused by the inclined installation error

\({x}_{g}\) :

The deflection caused by the weight of ball screw

I :

The polar moment of inertia

\({x}_{s}\) :

The travel caused by the screw support and weight

\({x}_{b}\) :

The jacking height of the middle support unit

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

Elastic modulus of the nut

\({E}_{s}\) :

Elastic modulus of the screw

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

The superficial area of the cross-section of the nut

\({A}_{s}\) :

The superficial area of the cross-section of the screw

M :

Total ball number of ball screw

\({z}_{b}\) :

The number of the balls in a circle

φ :

The helix angle of the ball screw

\({P}_{h}\) :

Lead of ball screw

θ :

The central angle corresponding to the point on the normal profile

λ :

The phase angle of the ball along the raceway

\({\lambda }_{s}\) :

The circumference angle of the screw around the axis

\({\theta }_{l}\) :

The inclination angle of the sinusoidal curve caused by the eccentricity

\({\alpha }^{0}\) :

The initial contact angle of ball screw

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

The contact angle of ith ball

e :

The eccentricity of the arc on the normal section

\({e}_{sc}\) :

The eccentricity of the screw

\({\theta }_{t}\) :

The inclined angle of installation of screw

\({L}_{ns}\) :

The center distance between the center of screw and nut

\({V}_{n0}\) :

The projection distances of the initial center distance in n direction

\({V}_{b0}\) :

The projection distances of the initial center distance in b direction

\({\delta }_{ni}\) :

The displacement of screw center in n direction

\({\delta }_{bi}\) :

The displacement of screw center in b direction

\({f}_{s}\) :

Conformity of screw

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

Conformity of nut

\({r}_{Gs}\) :

The radius of the screw raceway

\({r}_{Gn}\) :

The radius of the nut raceway

\({r}_{LG}\) :

The radius of left arc of the raceway

\({r}_{RG}\) :

The radius of right arc of the raceway

\({Y}_{s}\) :

Intermediate variable of screw

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

Intermediate variable of nut

\({\rho }_{s}\) :

Reciprocal of radius of curvature on the contact between ball and screw raceway

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

Reciprocal of radius of curvature on the contact between ball and nut raceway

\({\tau }_{s}\) :

Intermediate variable of screw

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

Intermediate variable of nut

\({c}_{E}\) :

Material constant

\({c}_{K}\) :

Geometry factor

\({D}_{pw}\) :

Pitch circle diameter of ball screw

\({r}_{m}\) :

Half of the pitch diameter of the screw

\({D}_{b}\) :

The diameter of the ball

\({r}_{b}\) :

The radius of the ball

\({\Delta L}_{\mathrm{m}}\) :

The relative displacement under preload

\({L}_{ns}\) :

The distance between the center of the nut and screw raceway

\({\delta }_{c}\) :

The thermal deformation error

\({\delta }_{p}\) :

The periodic error

\({\delta }_{r}\) :

The accidental error

\({\theta }_{AD}\) :

The angular displacement of the screw

\({E}_{bs}\) :

The transmission error of the ball screw

\({E}_{s}\) :

The transmission error of the screw

\({O}_{si}\) :

The initial center of the screw raceway corresponding to the ith ball

\({O}_{ni}\) :

The initial center of the nut raceway corresponding to the ith ball

\({O}_{bi}\) :

The initial ball center corresponding to the ith ball

\({O}_{GL}\) :

The center of left arc

\({O}_{GR}\) :

The center of right arc

\({O}_{bi}{\prime}\) :

The ball center after deformation corresponding to the ith ball

\({O}_{si}{\prime}\) :

The center of the screw raceway after deformation corresponding to the ith ball

\({O}_{ni}{\prime}\) :

The center of the nut raceway after deformation corresponding to the ith ball

\({e}_{1}\) :

The straightness error of the mobile platform

\({e}_{2}\) :

The abbe error during measurement

\({e}_{3}\) :

The laser measuring system error

\({e}_{4}\) :

The circular grating error

\({V}_{ni}\) :

Radial distance of raceway centers after deformation

\({V}_{bi}\) :

Axial distance of raceway centers after deformation

\({V}_{n0}\) :

Initial radial distance of raceway centers

\({V}_{b0}\) :

Initial axial distance of raceway centers

\({e}_{p}\) :

The travel deviation

T :

The temperature compensation of the travel

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Funding

This project is supported by the National Natural Science Foundation of China (Grant No. 51905274, 51705252), National Science and Technology Major Projects of China (No. 2018ZX04039001).

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Authors and Affiliations

  1. Department of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing, 210094, China

    Kai Wang, Chang-Guang Zhou, Yi Ou & Hu-Tian Feng

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  1. Kai Wang
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Correspondence to Yi Ou.

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Wang, K., Zhou, CG., Ou, Y. et al. Investigation of the transmission accuracy of ball screw considering errors and preloading level. Int J Adv Manuf Technol 118, 3917–3932 (2022). https://doi.org/10.1007/s00170-021-08088-x

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