Nonlinear dynamics of single-stage gear transmission

  • Józef WojnarowskiEmail author
  • Jerzy Margielewicz
Conference paper
Part of the Mechanisms and Machine Science book series (Mechan. Machine Science, volume 73)


In this work, model tests were carried out, which are mainly focused on the identification of zones in which the motion of gear transmission occurs chaotically. To identify chaotic areas a procedure was used to estimate the largest Lyapunov exponent. The formulated gear model includes variable meshing stiffness and gear backlash characteristics, which was approximated by a logarithmic function. The method of bond graphs was used to derive differential equations of motion, on the basis of which the effect of the error of cooperation between gears and the average load transmitted by the gear was examined. It has been shown that the load conditions affecting the gears have a significant effect on the character of the model solution that maps the gear dynamics. This influence is particularly noticeable in the bifurcation diagrams of steady states. On this basis, the values of the control parameter ω have been determined, for which the Poincaré cross-sections assume the geometric structure of the chaotic attractor. When plotting Poincaré cross-sections, additionally a histogram was taken into account of where the trajectory most often intersects the control plane. Such a complemented image of the graphic cross-section provides important information on the evolution of chaotic attractors.


Nonlinear vibrations Chaos Bond graphs 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Lorenz E. N.: Deterministic non-periodic flow, Journal of the Atmospheric Sciences, 20, 130-141 (1963).MathSciNetCrossRefGoogle Scholar
  2. 2.
    Cai-Wan Chang-Jian, Shiuh-Ming Chang: Bifurcation and chaos analysis of spur gear pair with and without nonlinear suspension, Nonlinear Analysis: Real World Applications, 12, 979-989 (2011).MathSciNetCrossRefGoogle Scholar
  3. 3.
    Khomeriki G.: Parametric resonance induced chaos in magnetic damped driven pendulum, Physic Letters A, 380, 2382-2385 (2016).MathSciNetCrossRefGoogle Scholar
  4. 4.
    Awrejcewicz J., Krysko-jr V. A., Pakovleva T. V., Krysko V. A.: Alternating chaos versus synchronized vibrations of interacting plane with beams, International Journal of Non-Linear Mechanics, 88, 21-30 (2017).CrossRefGoogle Scholar
  5. 5.
    Armand Eyebe Fouda J. S., Bodo B., Djeufa G. M. D., Sabat S. L.: Experimenta chaos detection in the Duffung oscillator, Commun Nonlinear Sci Numer Simulat, 33, 259-269 (2016).MathSciNetCrossRefGoogle Scholar
  6. 6.
    Sajid I, Xizhe Z., Yanhe Z., Jie Z.: Bifurcation and chaos in passive dynamic walking, A review, Robotics and Autonomous System, 62, 889-909 (2014).CrossRefGoogle Scholar
  7. 7.
    Zhou J, Sun W, Yuan L.: Nonlinear Vibroimpact Characteristics of a Planetary GearTransmission System. Shock Vib 2016.Google Scholar
  8. 8.
    Ahmadian H, Hassan-Beygi SR, Ghobadian B, Najafi G. ANFIS modeling of vibration transmissibility of a power tiller to operator. Appl Acoust, 138, 39–51 (2018).CrossRefGoogle Scholar
  9. 9.
    Leban B, Fancello G, Fadda P, Pau M. Changes in trunk sway of quay crane operators during work shift: A possible marker for fatigue? Appl Ergon, 65, 105–11 (2017).CrossRefGoogle Scholar
  10. 10.
    Zhao Xin, Chen Changzheng, Liu Jie, Zhang Lei: Dynamics characteristics of a spur gear transmission system for a wind turbine, International Conference on Automation, Mechanical Control and Computational Engineering, 1985-1990, (2015).Google Scholar
  11. 11.
    Kokare D. K., Patil S. S.: Numerical analysis of vibration in mesh stiffness for spur gear pair with method of phasing, International Journal of Current Engineering and Technology, Special Issue-3, 156-159 (2014).Google Scholar
  12. 12.
    Xiong Y, Huang K, Wang T, Chen Q, Xu R. Dynamic modelling and analysis of the microsegment gear. Shock Vib 2016.Google Scholar
  13. 13.
    Li S, Huang H, Fan X, Miao Q, Road N. Nonlinear dynamic simulation of coupled lateral-torsional vibrations of a gear transmission system. IJCSNS Int J Comput Sci Netw Secur, 6, 27–35 (2006).Google Scholar
  14. 14.
    Wang J., Guo L., Wang H.: Analysis of bifurcation and nonlinear control for chaos in gear transmission system, Research Journal of Applied Sciences, Engineering and Technology, 6(10), 1818-1824 (2013).CrossRefGoogle Scholar
  15. 15.
    Xiang L., Yi J., Aijun H.: Bifurcation and chaos analysis for multi-freedom gear-bearing system with time-varying stiffness, Applied Mathematical Modelling, 40, 10506-10520 (2016).MathSciNetCrossRefGoogle Scholar
  16. 16.
    Litak G., Friswell M. I.: Vibration in gear system, Chaos, Solution and Fractals, 16, 795-800 (2003).CrossRefGoogle Scholar
  17. 17.
    Wang J., Zheng J., Yang A.: An analytical study of bifurcation and chaos in a spur gear pair with sliding friction, International Conference on Advances in Computational Modeling and Simulation, Procedia Engineering 31, 563-570 (2012).CrossRefGoogle Scholar
  18. 18.
    Zhang Y., Meng Z., Sun Y.: Dynamic modeling and chaotic analysis of gear transmission system in a braiding machine with or without random perturbation, Shock and Vibration,2016,Google Scholar
  19. 19.
    Lang S. Y. T.: Graph-theoretical modeling of epicyclic gear system, Mechanism and Machine Theory, 40, 511-529 (2005).MathSciNetCrossRefGoogle Scholar
  20. 20.
    Wojnarowski J., Kopeć J., Zawiślak S.: Gear and graphs, Journal of Theoretical and Applied Mechanics, 44(1), 139-151 (2006).Google Scholar
  21. 21.
    Drewniak J. Zawiślak S.: Linear-graph and comtour-graph-based models of planetary gear, Journal of Theoretical and Applied Mechanics, 48(2), 415-433 (2010).Google Scholar
  22. 22.
    Luo Y., Tam D.: Dynamics Modeling of Planetary gear set considering meshing stiffness based on bond graph, Procedia Engineering, 24, 850-855 (2011).CrossRefGoogle Scholar
  23. 23.
    Guo Y., Liu D., Yang S., Li X., Chen J.: Hydraulic-mechanical couping modeling by bond graph for impact system of a high frequency rock drill drifter with sleeve distributor, Automation in Construction, 63, 88-99 (2016).CrossRefGoogle Scholar
  24. 24.
    Farshidianfar A, Saghafi A. Identification and control of chaos in nonlinear gear dynamic systems using Melnikov analysis. Phys Lett Sect A Gen At Solid State Phys,378, 3457–63 (2014).MathSciNetCrossRefGoogle Scholar
  25. 25.
    Rashidifar MA, Rashidifar AA. Investigating Bifurcation and Chaos Phenomenon in Nonlinear Vibration of Gear System Using Approximation of Backlash with Smoothing Function. Am J Mech Eng Autom, 2, 48–54 (2015).Google Scholar
  26. 26.
    Zuo Z, Ju X, Ding Z. Control of Gear Transmission Servo Systems with Asymmetric Deadzone Nonlinearity. IEEE Trans Control Syst Technol, 24, 1472–9 (2016).CrossRefGoogle Scholar
  27. 27.
    He S, Jia Q, Chen G, Sun H. Modeling and dynamic analysis of planetary gear transmission joints with backlash. Int J Control Autom, 8, 153–62 (2015).CrossRefGoogle Scholar
  28. 28.
    Sandri M. Numerical Calculation of Lyapunov Exponents. Math J, 6, 78–84 (1996).Google Scholar
  29. 29.
    Stefanski A, Dabrowski A, Kapitaniak T. Evaluation of the largest Lyapunov exponent in dynamical systems with time delay. Chaos, Solitons and Fractals, 23, 1651–9 (2005).MathSciNetCrossRefGoogle Scholar
  30. 30.
    Wolf A, Swift JB, Swinney HL, Vastano JA. Determining Lyapunov exponents from a time series. Phys D Nonlinear Phenom, 16, 285–317 (1985).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Technical Institute of the State Higher Vocational SchoolNowy SączPoland
  2. 2.Silesian University of TechnologyKatowicePoland

Personalised recommendations