SpringerLink
Log in

A mathematical model for load distribution of planetary roller screw with pitch deviation

  • Technical Paper
  • Published:
Journal of the Brazilian Society of Mechanical Sciences and Engineering Aims and scope Submit manuscript

Abstract

In this work, a mathematical model for load distribution of planetary roller screw mechanism with pitch deviation is proposed. The load distribution coefficient on the screw–roller and nut–roller sides is determined based on the relationship between axial contact stiffness, axial contact deformation and axial load. The presented model considers the variations of the axial clearance and contact state caused by the pitch deviation and is verified by comparing with the published model. Furthermore, the effects of one-thread pitch deviation, multi-thread pitch deviation and axial applied load on the load distribution coefficient of planetary roller screw mechanism are investigated. The results show that reducing the pitch deviation and increasing the axial applied load are conducive to improve the uniformity of load distribution and avoid the empty load of thread pairs. It is found that the increase of axial applied load is beneficial to reduce the load distribution fluctuation of planetary roller screw mechanism with pitch deviation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Abbreviations

A :

Cross-sectional area of threaded shaft (mm2)

a :

The thickness of thread root (mm)

b :

The thickness of thread (mm)

c :

The root height of thread (mm)

D 0 :

The outer diameter of hollow cylinder of internal thread (mm)

d :

The effective diameter of thread (mm)

E :

Elastic modulus (MPa)

E ij :

Difference of axial contact deformation between thread pairs i and j (mm)

\(\tilde{E}_{ij}^{{}}\) :

Difference of axial contact deformation between thread pairs i and j with pitch deviation (mm)

F a :

Axial applied load bearing by single roller (N)

F n i :

Normal contact force of thread pair i (N)

F r :

Radial load bearing by thread (N)

k b :

Threaded shaft section stiffness (N/mm)

k t :

Thread stiffness (N/mm)

k i :

Axial contact stiffness of thread without pitch deviation (N/mm)

\(k_{c}^{{{\text{sr}}}}\) :

Total equivalent axial contact stiffness in screw–roller side (N/mm)

P :

Thread pitch without pitch deviation (mm)

\({P}_{i}^{e}\) :

Thread pitch with pitch deviation (mm)

z :

Roller number

κ i :

Load distribution coefficient of thread pair i with pitch deviation

α :

Half thread angle (°)

β r :

Helical angle of roller thread (°)

δ e :

Axial contact deformation of thread pairs with pitch deviation (mm)

δ i :

Axial contact deformation of thread without pitch deviation (mm)

δ mr c :

Normal contact deformation of thread pair (mm)

δ rc - a :

Axial contact deformation (mm)

δ sr :

Total equivalent axial contact deformation in screw–roller side (mm)

δ y 1 :

Bending deformation (mm)

δ y 2 :

Shearing deformation (mm)

δ y 3 :

Root incline deformation (mm)

δ y 4 :

Root shear deformation (mm)

δ y 5 :

Deformation caused by radial load (mm)

δ y 5 - i :

Deformation caused by radial load for internal thread (mm)

δ y 5 - o :

Deformation caused by radial load for external thread (mm)

δ * :

Contact parameter

ɛ p :

Pitch deviation (mm)

μ :

Poisson’s ratio of material

Σρ :

Sum of curvatures of two contact surfaces

i,j :

The number of thread pairs

m(m = s, n):

Screw or nut

t(t = s, n, r):

Screw, nut or roller

sr:

Screw–roller side

nr:

Nut–roller side

References

  1. **e Z, Xue Q, Wu J, Song B (2019) Mixed-lubrication analysis of planetary roller screw. Tribol Int 140:105883

    Article  Google Scholar 

  2. Zhang W, Liu G, Ma S, Tong R (2019) Load distribution over threads of planetary roller screw mechanism with pitch deviation. Proc Inst Mech Eng C J Mech 233(13):4653–4666

    Article  Google Scholar 

  3. Yao Q, Zhang M, Liu Y, Ma S (2021) Multi-objective optimization of planetary roller screw mechanism based on improved mathematical modelling. Tribol Int 161:107095

    Article  Google Scholar 

  4. Ma S, Liu G, Zhou J, Tong R (2011) Optimal design and contact analysis for planetary roller screw. Appl Mech Mater 86:361–364

    Article  Google Scholar 

  5. Ma S, Liu G, Tong R, Zhang X (2012) A new study on the parameter relationships of planetary roller screws. Math Probl Eng 2012:123–132

    Article  Google Scholar 

  6. Ma S, Cai W, Wu L, Liu G, Peng C (2019) Modelling of transmission accuracy of a planetary roller screw mechanism considering errors and elastic deformations. Mech Mach Theory 134:151–168

    Article  Google Scholar 

  7. Zhang W, Li W, Zheng S, Zhang P, Qi X (2018) Research on motion relationships and transmission efficiency of planetary roller screw. In: Asia-pacific international symposium on aerospace technology, pp 2838–2848

  8. Velinsky SA, Chu B, Lasky TA (2009) Kinematics and efficiency analysis of the planetary roller screw mechanism. J Mech Des 131(1):011016

    Article  Google Scholar 

  9. Fu X, Liu G, Ma S, Tong R, Lim T (2018) Kinematic model of planetary roller screw mechanism with run-out and position errors. J Mech Des 140(3):032301

    Article  Google Scholar 

  10. Ma S, Zhang T, Liu G, Tong R, Fu X (2015) Kinematics of planetary roller screw mechanism considering helical directions of screw and roller threads. Math Probl Eng 2015:459462

    Article  Google Scholar 

  11. Jones MH, Velinsky SA (2012) Kinematics of roller migration in the planetary roller screw mechanism. J Mech Des 134(6):061006

    Article  Google Scholar 

  12. Jones MH, Velinsky SA (2013) Contact kinematics in the roller screw mechanism. J Mech Des 135(5):051003

    Article  Google Scholar 

  13. **ng M, Zhang B, Deng P, Xu J, Cui Y (2022) A comprehensive analysis of contact kinematics for planetary roller screw mechanism. Tribol Int 179(6):108127

    Google Scholar 

  14. Fu X, Liu G, Tong R, Ma S, Lim T (2018) A nonlinear six degrees of freedom dynamic model of planetary roller screw mechanism. Mech Mach Theory 119:22–36

    Article  Google Scholar 

  15. Fu X, Liu G, Ma S, Tong R, Li X (2020) An efficient method for the dynamic analysis of planetary roller screw mechanism. Mech Mach Theory 150:103851

    Article  Google Scholar 

  16. Fu X, Liu G, Li X, Ma S, Qiao G (2021) Dynamic modeling of the double-nut planetary roller screw mechanism considering elastic deformations. J Comput Nonlinear Dyn 16(5):051003

    Article  Google Scholar 

  17. Jones MH, Velinsky SA, Lasky TA (2016) Dynamics of the planetary roller screw mechanism. J Mech Robot 8(1):014503

    Article  Google Scholar 

  18. **ng M, Zhang B, Deng P, Xu J, Cui Y (2022) A novel wear prediction model for planetary roller screw based on universal sliding distance model. Tribol Int 175(1):107851

    Article  Google Scholar 

  19. Aurégan G, Vincent F, Philippe K, Fernand R (2016) Wear behavior of martensitic stainless steel in rolling-sliding contact for planetary roller screw mechanism: study of the WC/C solution. Tribol Online 11(2):209–217

    Article  Google Scholar 

  20. Zhou G, Zhang Y, Wang Z, Pu W (2021) Analysis of transient mixed elastohydrodynamic lubrication in planetary roller screw mechanism. Tribol Int 163:107158

    Article  Google Scholar 

  21. Qiao G, Liu G, Ma S, Wang Y, Li P, Lim T (2019) Thermal characteristics analysis and experimental study of the planetary roller screw mechanism. Appl Therm Eng 149:1345–1358

    Article  Google Scholar 

  22. Du X, Li X, Yue Y, Chen F (2015) Thermal-structural Coupling Analysis of Planetary Roller Screw. Appl Mech Mater 757:55–59

    Article  Google Scholar 

  23. Abevi F, Daidie A, Chaussumier M, Sartor M (2016) Static load distribution and axial stiffness in a planetary roller screw mechanism. J Mech Des 138(1):012301

    Article  Google Scholar 

  24. Abevi F, Daidie A, Chaussumier M, Orieux S (2016) Static analysis of an inverted planetary roller screw mechanism. J Mech Robot 8(4):041020

    Article  Google Scholar 

  25. Lisowski F (2015) Numerical computation of stresses and deformations in the planetary roller screw components. Czas Tech 2015:141–148

    Google Scholar 

  26. Lisowski F (2014) The analysis of displacements and the load distribution between elements in a Planetary Roller Screw. Appl Mech Mater 680:326–329

    Article  Google Scholar 

  27. Jones MH, Velinsky SA (2014) Stiffness of the roller screw mechanism by the direct method. Mech Based Des Struct 42(1):17–34

    Article  Google Scholar 

  28. Guo J, Peng H, Huang H, Liu Z, Huang Y, Ding W (2017) Analytical and experimental of planetary roller screw axial stiffness. In: 2017 IEEE international conference on mechatronics and automation (ICMA), pp 752–757

  29. Zhang W, Liu G, Tong R, Ma S (2016) Load distribution of planetary roller screw mechanism and its improvement approach. Proc Inst Mech Eng C-J Mech 230(18):3304–3318

    Article  Google Scholar 

  30. Du X, Chen B, Zheng Z (2021) Investigation on mechanical behavior of planetary roller screw mechanism with the effects of external loads and machining errors. Tribol Int 154:106689

    Article  Google Scholar 

  31. Yamamoto S (1984) Theory and calculation of screw thread connection (trans: Guo KQ). Shanghai Science and Technology Literature Press, Shanghai, pp 45–49 (In Chinese)

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 52171315) and China State Shipbuilding Corporation Limited-Shanghai Jiaotong University Marine Equipment Prospective Innovation Joint Fund (Grant No. 2022-3-B1).

Author information

Authors and Affiliations

  1. School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai, 200240, People’s Republic of China

    Mingcai **ng

  2. Key Laboratory for Power Machinery and Engineering of Ministry of Education, School of Mechanical Engineering, Shanghai Jiaotong University, Shanghai, 200240, People’s Republic of China

    Mingcai **ng, Bohong Zhang & Yi Cui

  3. Shanghai Marine Equipment Research Institute, Shanghai, 200031, People’s Republic of China

    Pan Deng & **guo Lin

Authors
  1. Pan Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mingcai **ng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. **guo Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Bohong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Yi Cui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yi Cui.

Ethics declarations

Conflict of interest

The authors declare no potential conflict of interest with respect to the research, authorship and publication of this article.

Additional information

Technical Editor: Zilda de Castro Silveira.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Deng, P., **ng, M., Lin, J. et al. A mathematical model for load distribution of planetary roller screw with pitch deviation. J Braz. Soc. Mech. Sci. Eng. 45, 547 (2023). https://doi.org/10.1007/s40430-023-04453-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40430-023-04453-x

Keywords

Search

Navigation