Skip to main content
SpringerLink
Log in

Vibration response analysis of a gear-rotor-bearing system considering steady-state temperature

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Thermal effect is a crucial factor leading to failure of gear system. However, the lack of comprehensive nonlinear dynamics model limits the further study of thermal effects. The constitutive relation of beam element considering steady-state temperature is reconstructed, and thermal node load is formulated. Considering the influences of thermal expansion and temperature on material properties, a more comprehensive dynamic model of gear-rotor-bearing system is established based on the finite element node method. Nonlinear friction, high-speed gyroscopic effect, thermal-related time-varying meshing stiffness (TVMS) and thermal backlash are included in the model. The effects of temperature and bearing type on the vibration response of gear system are analyzed. The results show that the system motion changes from period to chaos with the temperature increase in part of the speed range. The appropriate backlash could restrain the chaotic motion caused by temperature rise. Moreover, the temperature significantly increases the axial bearing force, and the appropriate bearing could reduce the axial displacement. This research can further understand the influence of temperature on the dynamic response of gear system and guide the design of gear system.

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

Similar content being viewed by others

Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Abbreviations

A :

Beam sectional area

a :

Coefficient of linear expansion

B :

Transformation matrix between strain vector and displacement vector

B :

Tooth face width

b, b * :

Half of the backlash and the thermal backlash

D :

Transformation matrix between strain matrix and stress matrix

E * :

Elastic modulus obtaining by fitting the values at different temperatures

e(t):

Static transmission error

F T :

Thermal node load

G * :

Shear modulus obtaining by fitting the values at different temperatures

I s :

Second area moment of inertia

k h, k a, k b, k s, k f :

Non-linear Hertzian contact stiffness, axial compressive stiffness, bending stiffness, shear stiffness and fillet-foundation stiffness

k m *, c m :

Thermal TVMS and damping

K * :

Thermal related time-varying meshing stiffness

l :

Length of the beam element

N :

Shape function matrix of beam element

r, r b, r k :

Dividing circle radius, base circle radius and radius of meshing point

s:

Tooth thickness

T 2 :

Torque applied to the output shaft

U:

Total potential energy of linear elastic solids

Z p,g :

Teeth number of the driving wheel and driven wheel

M, G, K, F, q :

Mass, gyroscopic, stiffness, load and displacement matrix. (Superscript d, g, e, b and s denote rigid rotor, gear, shaft , bearing and assemble model)

\(\alpha , \alpha _k\) :

Pressure angle and pressure angle at point k

\(\Delta ej(j=p, g)\) :

Thermal profile error of the driving wheel and driven wheel

\(\Delta T\) :

Amount of temperature change

\(\delta , \delta _m\) :

Dynamic transmission error and dynamic deformation along the meshing line

\(\varepsilon \) :

Strain matrix

\(\gamma _{m0}, \gamma _{m1}\) :

Backlash functions

\(\lambda \) :

Thermal expansion coefficient

\(\varvec{\sigma }\) :

Stress matrix

\(\Omega \) :

Spin speed of the gear rotor

\(\xi _i (i=1,2)\) :

Correction coefficient of the fillet-foundation stiffness of tooth pair i

References

  1. Long, H.: Operating temperatures of oil-lubricated medium-speed gears numerical models and experimental results. J. Aerosp. Eng. 217, 87 (2003)

    Google Scholar 

  2. Kleemola, J., Lehtovaara, A.: Evaluation of lubrication conditions in gear contacts using contact resistance and bulk temperature measurements. Proc. Inst. Mech. Eng. Part J. J. Eng. Tribol. 224, 367 (2010)

    Article  Google Scholar 

  3. Sutter, G., Ranc, N.: Flash temperature measurement during dry friction process at high sliding speed. Wear 268, 1237 (2010)

    Article  Google Scholar 

  4. Jihua, C., Shaojun, L., Xiaozhou, H.: A temperature measurement method for testing lubrication system or revealing scuffing failure mechanism of spur gear. Pro.c Inst. Mech. Eng. Part J. J. Eng. Tribol. 233, 831 (2018)

    Article  Google Scholar 

  5. Davis J. R.: Tensile Testing. (2004)

  6. Kahraman, A., Singh, R.: Non-linear dynamics of a spur gear pair. J. Sound Vib. 142, 49 (1990)

    Article  Google Scholar 

  7. Kahraman, A., Singh, R.: Non-linear dynamics of a geared rotor-bearing system with multiple clearances. J. Sound Vib. 144, 469 (1991)

    Article  Google Scholar 

  8. Wang, Y., Zhang, W.J.: Stochastic Vibration Model of Gear Transmission Systems Considering Speed-Dependent Random Errors. Nonlinear Dyn. 17, 187 (1998)

    Article  Google Scholar 

  9. Vaishya, M., Singh, R.: Analysis of periodically varying gear mesh systems with coulomb friction using floquet theory. J. Sound Vib. 243, 525 (2001)

    Article  Google Scholar 

  10. Li, Y., Chen, T., Wang, X.: Non-linear dynamics of gear pair with dynamic backlash subjected to combined internal and external periodic excitations. J. Vib. Control. 22, 1693 (2014)

    Article  MathSciNet  Google Scholar 

  11. Cao, Z., Chen, Z., Jiang, H.: Nonlinear dynamics of a spur gear pair with force-dependent mesh stiffness. Nonlinear Dyn. 99, 1227 (2019)

    Article  Google Scholar 

  12. Velex, P., Maatar, M.: A mathematical model for analyzing the influence of shape deviations and mounting errors on gear dynamic behavior. J. Sound Vib. 191, 629 (1996)

    Article  Google Scholar 

  13. Kim, W., Yoo, H.H., Chung, J.: Dynamic analysis for a pair of spur gears with translational motion due to bearing deformation. J. Sound Vib. 329, 4409 (2010)

    Article  Google Scholar 

  14. Ma, H., Yang, J., Song, R., et al.: Effects of tip relief on vibration responses of a geared rotor system. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 228, 1132 (2013)

    Article  Google Scholar 

  15. Ma, H., Feng, R., Pang, X., et al.: Effects of tooth crack on vibration responses of a profile shifted gear rotor system. J. Mech. Sci. Technol. 29, 4093 (2015)

    Article  Google Scholar 

  16. Ma, H., Pang, X., Feng, R., et al.: Fault features analysis of cracked gear considering the effects of the extended tooth contact. Eng. Fail Anal. 48, 105 (2015)

    Article  Google Scholar 

  17. Inalpolat, M., Handschuh, M., Kahraman, A.: Influence of indexing errors on dynamic response of spur gear pairs. Mech. Syst. Signal Process. 60–61, 391 (2015)

    Article  Google Scholar 

  18. Hu, Z., Tang, J., Zhong, J., et al.: Effects of tooth profile modification on dynamic responses of a high speed gear-rotor-bearing system. Mech. Syst. Signal Process. 76–77, 294 (2016)

    Article  Google Scholar 

  19. Hu, Z., Tang, J., Zhong, J., et al.: Frequency spectrum and vibration analysis of high speed gear-rotor system with tooth root crack considering transmission error excitation. Eng. Fail Anal. 60, 405 (2016)

    Article  Google Scholar 

  20. Chen, W., Chen, S., Hu, Z., et al.: A novel dynamic model for the spiral bevel gear drive with elastic ring squeeze film dampers. Nonlinear Dyn. 98, 1081 (2019)

    Article  Google Scholar 

  21. Liu, G., Hong, J., Parker, R.G.: Influence of simultaneous time-varying bearing and tooth mesh stiffness fluctuations on spur gear pair vibration. Nonlinear Dyn. 97, 1403 (2019)

    Article  Google Scholar 

  22. Peng, T., Lim, T.C.: Influence of gyroscopic effect on hypoid and bevel geared system dynamics SAE Int. J. Passeng. Cars Mech. Syst. 2, 1377 (2009)

    Article  Google Scholar 

  23. Chen, S., Tang, J., Li, Y., et al.: Rotordynamics analysis of a double-helical gear transmission system. Meccanica 51, 251 (2015)

    Article  MathSciNet  Google Scholar 

  24. Hu, Z., Tang, J., Chen, S.: Analysis of coupled lateral-torsional vibration response of a geared shaft rotor system with and without gyroscopic effect. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 232, 4550 (2018)

    Article  Google Scholar 

  25. Blok, H.: The flash temperature concept. Wear 6, 483 (1963)

    Article  Google Scholar 

  26. Li G.: Theoretical and Applied Research of Thermal Deformation of Complicated Components with Regular Camber. Hefei, China, (2006)

  27. Höhn, B.-R., Michaelis, K., Otto, H.-P.: Influence of immersion depth of dip lubricated gears on power loss, bulk temperature and scuffing load carrying capacity. Int. J. Mech. Mater. Des. 4, 145 (2007)

    Article  Google Scholar 

  28. Diogo M.: Bulk Temperature in Polymer-Metal Hybrid Gears. Porto, Portugal, Unibersidade do porto, (2017)

  29. Zhou C., Xing M., Wang H., et al.: A novel thermal network model for predicting the contact temperature of spur gears. Int. J. Therm. Sci. (2020)

  30. Gou, X., Zhu, L., Qi, C.: Nonlinear dynamic model of a gear-rotor-bearing system considering the flash temperature. J. Sound Vib. 410, 187 (2017)

    Article  Google Scholar 

  31. Shi, J., Gou, X., Zhu, L., et al.: Nonlinear dynamic modeling of 3-DOF gear system with tooth contact temperature. Adv. Mech. Des. 55, 213 (2018)

    Google Scholar 

  32. Pan, W., Li, X., Wang, L., et al.: Nonlinear response analysis of gear-shaft-bearing system considering tooth contact temperature and random excitations. Appl. Math. Model. 68, 113 (2019)

    Article  MathSciNet  Google Scholar 

  33. Luo, B., Li, W.: Experimental study on thermal dynamic characteristics of gear transmission system. Measurement 136, 154 (2019)

    Article  Google Scholar 

  34. Luo B., Li W., Fu C., et al.: Investigation of the thermal stiffness and thermal tooth profile modification of spur gears. J. Braz. Soc. Mech. Sci. Eng. 42, (2020)

  35. Sun Z., Chen S., Tao X., et al.: Research on dynamic characteristics of gear system considering the influence of temperature on material performance. J. Vib. Control. (2021)

  36. Handschuh R. F., Gargano L.: Test Facility Simulation Results for Aerospace Loss-of-Lubrication of Spur Gears. NASA Technical Memorandum, Glenn Research Center, Cleveland. (2014)

  37. Hu B., Zhou C., Wang H., et al.: Nonlinear tribo-dynamic model and experimental verification of a spur gear drive under loss-of-lubrication condition. Mech. Syst. Signal Process. 153, (2021)

Download references

Acknowledgements

This work is supported by the National Key R&D Program of China through Grant No. 2020YFB2008200 and the National Natural Science Foundation of China (NSFC) through Grant No. 52005515. The authors also gratefully acknowledge the supports of the Project of State Key Laboratory of High Performance Complex Manufacturing, Central South University, through Grant No. ZZYJKT2021-06.

Author information

Authors and Affiliations

  1. State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha, 410083, China

    Zhou Sun, Siyu Chen, Zehua Hu & Duncai Lei

  2. College of Mechanical and Electrical Engineering, Central South University, Changsha, 410083, China

    Zhou Sun, Siyu Chen, Zehua Hu & Duncai Lei

  3. AECC Hunan Aviation Powerplant Research Institute, Zhuzhou, 412002, China

    Duncai Lei

Authors
  1. Zhou Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Siyu Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zehua Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Duncai Lei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Siyu Chen.

Ethics declarations

Conflicts of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. The authors declare the following financial interests/personal relationships which may be considered as potential competing interests

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sun, Z., Chen, S., Hu, Z. et al. Vibration response analysis of a gear-rotor-bearing system considering steady-state temperature. Nonlinear Dyn 107, 477–493 (2022). https://doi.org/10.1007/s11071-021-07024-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-07024-8

Keywords

Search

Navigation