Nonlinear Dynamics

, Volume 84, Issue 1, pp 203–222 | Cite as

The nonlinear dynamics response of cracked gear system in a coal cutter taking environmental multi-frequency excitation forces into consideration

  • Yu Jiang
  • Hua Zhu
  • Z. Li
  • Z. Peng
Original Paper


The normal operation of gear systems ensures the power transmission in coal cutters. Due to harsh work environment, the gears in coal cutters are prone to crack. By analyzing the gear dynamic response, useful crack detection indicators could be obtained. The literature review indicates that most of the gear dynamic models have been built with single-frequency excitation, but very limited work has considered multi-frequency excitations. The coal cutter gear systems are always subjected to multi-frequency excitations. Hence, a nonlinear gear dynamic model taking multi-frequency excitations into account was established in this work. A meshing stiffness coefficient was defined to model the crack fault influence on the gear model. The amplitude–frequency of the model main resonance was analyzed by means of multi-scale method, and the model dynamic response was calculated by the incremental harmonic balance method. Numerical simulation results show that the dynamic response of the presented gear model provided strong chaotic characteristics and the chaotic degree of the gear dynamics increased with the deterioration of the crack level. Experimental validation was carried out on a real gearbox, and the analysis results were consistent with the simulations. Hence, the simulation and experimental analysis demonstrates that the multi-frequency excitation-based gear dynamic model was more correct than the single-frequency excitation-based model in representing the system dynamics.


Dynamic model Nonlinear response Gear system  Gear crack Choas 



This research was funded by the National Natural Sciences Foundation of China (NSFC) (Nos. 51375480; 51505475), the Science Foundation of Jiangsu Province (No. BK20140200) and the Priority Academic Program Development of Jiangsu Higher Education Institutions.


  1. 1.
    Li, Z., Ge, S., Zhu, H.: Key issues in the wear fault monitoring and diagnosis for critical components of coal cutters under deep coal seam. Tribology 34(6), 729–730 (2014)Google Scholar
  2. 2.
    Qian, P.: Fault diagnosis and reliability analysis for transmission system of shearer cutting part. Ph. D. Thesis, China University of Mining and Technology, Xuzhou (2015)Google Scholar
  3. 3.
    Baydar, N., Ball, A.: Detection of gear deterioration under varying load conditions by using the instantaneous power spectrum. Mech. Syst. Signal Process. 14, 907–921 (2000)CrossRefGoogle Scholar
  4. 4.
    Randall, B.: Vibration-Based Condition Monitoring: Industrial, Aerospace and Automotive Applications. Wiley, New York (2011)CrossRefGoogle Scholar
  5. 5.
    McFadden, P.: Examination of a technique for the early detection of failure in gears by signal processing of the time domain average of the meshing vibration. Mech. Syst. Signal Process. 1, 173–183 (1987)CrossRefGoogle Scholar
  6. 6.
    Jardine, A., Lin, D., Banjevic, D.: A review on machinery diagnostics and prognostics implementing condition-based maintenance. Mech. Syst. Signal Process. 20, 1483–1510 (2006)CrossRefGoogle Scholar
  7. 7.
    Ma, H., Zeng, J., Feng, R., Pang, X., Wang, Q., Wen, B.: Review on dynamics of cracked gear systems. Eng. Fail. Anal. 55, 224–245 (2015)CrossRefGoogle Scholar
  8. 8.
    Mark, D., Reagor, P., McPherson, R.: Assessing the role of plastic deformation in gear-health monitoring by precision measurement of failed gears. Mech. Syst. Signal Process. 21, 177–192 (2007)CrossRefGoogle Scholar
  9. 9.
    Pandya, Y., Parey, A.: Simulation of crack propagation in spur gear tooth for different gear parameter and its influence on mesh stiffness. Eng. Fail. Anal. 30, 124–137 (2013)CrossRefGoogle Scholar
  10. 10.
    Pandya, Y., Parey, A.: Failure path based modified gear mesh stiffness for spur gear pair with tooth root crack. Eng. Fail. Anal. 27, 286–96 (2013)CrossRefGoogle Scholar
  11. 11.
    Pandya, Y., Parey, A.: Crack behavior in a high contact ratio spur gear tooth and its effect on mesh stiffness. Eng. Fail. Anal. 34, 69–78 (2013)CrossRefGoogle Scholar
  12. 12.
    Yin, J., Wang, W., Man, Z.: Modeling and analysis of gear tooth crack growth under variable-amplitude loading. Mech. Syst. Signal Process. 40, 105–113 (2013)CrossRefGoogle Scholar
  13. 13.
    Guilbault, R., Lalonde, S., Thomas, M.: Modeling and monitoring of tooth fillet crack growth in dynamic simulation of spur gear set. J. Vib. Sound 343, 144–165 (2015)CrossRefGoogle Scholar
  14. 14.
    Rad, A., Forouzan, M., Dolatabad, A.: Three-dimensional fatigue crack growth modelling in a helical gear using extended finite element method. Fatigue Fract. Eng. Mater. Struct. 37, 581–591 (2014)CrossRefGoogle Scholar
  15. 15.
    Ghaffari, M., Pahl, E., Xiao, S.: Three dimensional fatigue crack initiation and propagation analysis of a gear tooth under various load conditions and fatigue life extension with boron/epoxy patches. Eng. Fract. Mech. 135, 126–146 (2015)CrossRefGoogle Scholar
  16. 16.
    Wan, Z., Cao, H., Zi, Y., He, W., He, Z.: An improved time-varying mesh stiffness algorithm and dynamic modeling of gear-rotor system with tooth root crack. Eng. Fail. Anal. 42, 157–177 (2014)CrossRefGoogle Scholar
  17. 17.
    Yu, W., Shao, Y., Mechefske, K.: The effects of spur gear tooth spatial crack propagation on gear mesh stiffness. Eng. Fail. Anal. 54, 103–119 (2015)CrossRefGoogle Scholar
  18. 18.
    Sheng, D., Jin, G., Lu, F., Bao, H.: Dynamic load sharing behavior of transverse-torsional coupled planetary gear train with multiple clearances. J. Cent. South Univ. 22(7), 2521–2532 (2015)CrossRefGoogle Scholar
  19. 19.
    Cho, S., Choi, J., Choi, J., Rhim, S.: Numerical estimation of dynamic transmission error of gear by using quasi-flexible-body modeling method. J. Mech. Sci. Technol. 29(7), 2713–2719 (2015)CrossRefGoogle Scholar
  20. 20.
    Litak, G., Friswell, M.: Dynamics of a gear system with faults in meshing stiffness. Nonlinear Dyn. 41, 415–421 (2005)CrossRefzbMATHGoogle Scholar
  21. 21.
    Ma, H., Pang, X., Feng, R., Song, R., Wen, B.: Fault features analysis of cracked gear considering the effects of the extended tooth contact. Eng. Fail. Anal. 48, 105–20 (2015)CrossRefGoogle Scholar
  22. 22.
    Yang, D., Dong, L., Shi, J., Lan, C.: Duffing system vibration behavior under multi-frequency excitation. J. Vib. Shock 30(12), 19–22 (2011)Google Scholar
  23. 23.
    Lou, J., He, Q., Zhu, S.: Chaos in the softening Duffing system under multi-frequency periodic forces. Appl. Math. Mech. 25(12), 1421–1427 (2004)CrossRefzbMATHGoogle Scholar
  24. 24.
    Bi, Q., Chen, Y., Wu, Z.: Bifurcation in a nonlinear Duffing system with multi-frequency external periodic forces. Appl. Math. Mech. 19(2), 121–128 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Qin, Z., Chen, Y.: Singularity analysis of Duffing–van der Pol system with two bifurcation parameters under multi-frequency excitations. Appl. Math. Mech. 31(8), 1019–1026 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhao, Z., Zhang, M., Yang, S., Xing, H.: A Duffing–van der Pol system’s vibration behavior under multi-frequency excitation. J. Vib. Shock 32(19), 76–79 (2013)Google Scholar
  27. 27.
    Rong, H., Wang, X., Luo, Q., Xu, W., Fang, T.: Bifurcation of safe basins and chaos in nonlinear vibroimpact oscillator under harmonic and bounded noise excitations. J. Appl. Math. 967395, 10 (2014)MathSciNetGoogle Scholar
  28. 28.
    Park, H., Liu, W.: An introduction and tutorial on multiple-scale analysis in solids. Comput. Methods Appl. Mech. Eng. 193(1720), 1733–1772 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Li, Y., Chen, T., Wang, X., Yu, K., Zhou, H., Zhang, Z.: Nonlinear dynamics of spur gear pair under external periodic excitation. J. Xian Jiaotong Univ. 48(1), 1014–1105 (2014)Google Scholar
  30. 30.
    Zhu, W., Wu, S., Wang, X., Peng, Z.: Harmonic balance method implementation of nonlinear dynamic characteristics for compound planetary gear sets. Nonlinear Dyn. 81(3), 1511–1522 (2015)CrossRefGoogle Scholar
  31. 31.
    Moradi, H., Salarieh, H.: Analysis of nonlinear oscillations in spur gear pairs with approximated modelling of backlash nonlinearity. Mech. Mach. Theory 51, 14–31 (2012)CrossRefGoogle Scholar
  32. 32.
    Chen, Q., Ma, Y., Huang, S., Zhai, H.: Research on gears dynamic performance influenced by gear backlashbased on fractal theory. Appl. Surf. Sci. 313, 325–332 (2014)CrossRefGoogle Scholar
  33. 33.
    Fakher, C., Tahar, F.: Analytical modeling of spur gear tooth crack and influence on gear mesh stiffness. Eur. J. Mech. A Solids 28, 461–468 (2009)CrossRefzbMATHGoogle Scholar
  34. 34.
    Theodossiades, S., Natsiavas, S.: Non-linear dynamics of gear-pair systems with periodic stiffness and backlash. J. Sound Vib. 229(2), 287–310 (2000)CrossRefGoogle Scholar
  35. 35.
    Ma, R., Chen, Y.: Nonlinear dynamic research on gear system with cracked failure. J. Mech. Eng. 47(21), 85–91 (2011)CrossRefGoogle Scholar
  36. 36.
    Gu, X., Liu, Y., Yang, S., Liao, Y.: Fault diagnosis of rolling bearing based on characteristic auantities of chaotic attractor. J. Shijiazhuang Tiedao Univ. 28, 91–95 (2015)Google Scholar
  37. 37.
    Cencini, M., Cecconi, F., Vulpiani, A.: Chaos: From Simple Models to Complex Systems. World Scientific Publishing Company, Singapore (2010)Google Scholar
  38. 38.
    Fakhfakh, T., Chaari, F., Haddar, M.: Numerical and experimental analysis of a gear system with teeth defects. Int. J. Adv. Manuf. Technol. 25, 542–550 (2005)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.School of Mechatronic Engineering, Jiangsu Key Laboratory of Mine Mechanical and Electrical EquipmentChina University of Mining and TechnologyXuzhouChina
  2. 2.School of Mechanical and Manufacturing EngineeringUniversity of New South WalesSydneyAustralia

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