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Dynamic analysis and chaos control of spur gear transmission system with idler. (English) Zbl 1485.74037

Summary: This study aims to analyze the chaotic dynamics and present a chaos controller for spur gear transmission systems with idler. The chaotic dynamics of the spur gear mechanism has already been investigated. However, the presence of idler gears affects the chaotic behavior and the route to chaos for the nonlinear model of spur gears. This is investigated through the derivation of dimensionless dynamics, defining a Poincare’ section, and extracting the bifurcation diagrams of the system for variations of several parameter models. A nonlinear time-varying dynamic model of a spur gear transmission system with idler is established where backlash, time-varying stiffness, static transmission error, and external excitation are included and a region for the occurrence of chaos is found. The chaotic vibration suppression of the system is done by detecting the unstable periodic orbits embedded in the strange attractors and developing control law by employing sliding mode and adaptive sliding mode control strategy. The controller transmits a chaotic trajectory into the detected unstable periodic orbits. Numerical simulations including phase plane portrait, time histories diagrams, Poincare’ sections, and bifurcation diagrams demonstrate the behavior of the system and confirm the performance of the proposed controller.

MSC:

74H60 Dynamical bifurcation of solutions to dynamical problems in solid mechanics
74H65 Chaotic behavior of solutions to dynamical problems in solid mechanics
70K50 Bifurcations and instability for nonlinear problems in mechanics
70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
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[1] Atanasovska, I., The mathematical phenomenological mapping in non-linear dynamics of spur gear pair and radial ball bearing due to the variable stiffness, Int. J. Non Lin. Mech., 73, 114-120 (2015)
[2] Bu, S.; Wang, B.-H.; Jiang, P.-Q., Detecting unstable periodic orbits in chaotic systems by using an efficient algorithm, Chaos, Solit. Fractals, 22, 1, 237-241 (2004) · Zbl 1067.37039
[3] Chang-Jian, C.-W., Strong nonlinearity analysis for gear-bearing system under nonlinear suspension—bifurcation and chaos, Nonlinear Anal. R. World Appl., 11, 3, 1760-1774 (2010) · Zbl 1188.93034
[4] Chang-Jian, C.-W.; Chang, S.-M., Bifurcation and chaos analysis of spur gear pair with and without nonlinear suspension, Nonlinear Anal. R. World Appl., 12, 2, 979-989 (2011) · Zbl 1269.70038
[5] Chang-Jian, C.; Chen, C.o.-K., Bifurcation and chaos of a flexible rotor supported by turbulent journal bearings with non-linear suspension, Proc. IME J. J. Eng. Tribol., 220, 6, 549-561 (2006)
[6] Farshidianfar, A.; Saghafi, A., Global bifurcation and chaos analysis in nonlinear vibration of spur gear systems, Nonlinear Dynam., 75, 4, 783-806 (2014)
[7] Farshidianfar, A.; Saghafi, A., Identification and control of chaos in nonlinear gear dynamic systems using Melnikov analysis, Phys. Lett., 378, 46, 3457-3463 (2014)
[8] Ghamati, M.; Balochian, S., Design of adaptive sliding mode control for synchronization Genesio-Tesi chaotic system, Chaos, Solit. Fractals, 75, 111-117 (2015) · Zbl 1352.93059
[9] Hilborn, R. C., Chaos and Nonlinear Dynamics: an Introduction for Scientists and Engineers (2000), Oxford University Press on Demand · Zbl 1015.37001
[10] Huang, Y.-J.; Kuo, T.-C.; Chang, S.-H., Adaptive sliding-mode control for NonlinearSystems with uncertain parameters, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 38, 2, 534-539 (2008)
[11] Kahraman, A.; Blankenship, G. W., Experiments on nonlinear dynamic behavior of an oscillator with clearance and periodically time-varying parameters, J. Appl. Mech., 64, 1, 217-226 (1997)
[12] Lakshmanan, M.; Murali, K., Chaos in Nonlinear Oscillators: Controlling and Synchronization (1996), World scientific · Zbl 0868.58058
[13] Liu, G.; Parker, R. G., Nonlinear dynamics of idler gear systems, Nonlinear Dynam., 53, 4, 345-367 (2008) · Zbl 1170.70341
[14] Nazzal, J. M.; Natsheh, A. N., Chaos control using sliding-mode theory, Chaos, Solit. Fractals, 33, 2, 695-702 (2007)
[15] Ott, E.; Grebogi, C.; Yorke, J. A., Controlling chaos, Phys. Rev. Lett., 64, 11, 1196 (1990) · Zbl 0964.37501
[16] Özgüven, H. N.; Houser, D. R., Mathematical models used in gear dynamics—a review, J. Sound Vib., 121, 3, 383-411 (1988)
[17] Saghafi, A.; Farshidianfar, A., An analytical study of controlling chaotic dynamics in a spur gear system, Mech. Mach. Theor., 96, 179-191 (2016)
[18] Sato, K.; Yamamoto, S., Bifurcation sets and chaotic states of a gear system subjected to harmonic excitation, Comput. Mech., 7, 3, 173-182 (1991) · Zbl 0737.70013
[19] Slotine, J.-J. E.; Li, W., Applied nonlinear Control (No. 1) (1991), prentice-Hall: prentice-Hall Englewood Cliffs, NJ · Zbl 0753.93036
[20] Song, Z.; Sun, K.; Ling, S., Stabilization and synchronization for a mechanical system via adaptive sliding mode control, ISA Trans., 68, 353-366 (2017)
[21] Taghvaei, S.; Vatankhah, R., Detection of unstable periodic orbits and chaos control in a passive biped model, Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, 40, 4, 303-313 (2016)
[22] Theodossiades, S.; Natsiavas, S., Non-linear dynamics of gear-pair systems with periodic stiffness and backlash, J. Sound Vib., 229, 2, 287-310 (2000)
[23] Wang, J.; Wang, H.; Guo, L., Analysis of effect of random perturbation on dynamic response of gear transmission system, Chaos, Solit. Fractals, 68, 78-88 (2014) · Zbl 1354.74084
[24] Wang, J.-g.; Lv, B.; Sun, R.; Zhao, Y.-x., Resonance and stability analysis of a cracked gear system for railway locomotive, Appl. Math. Model., 77, 253-266 (2020/01/01/2020) · Zbl 1443.70020
[25] Yang, J., Vibration analysis on multi-mesh gear-trains under combined deterministic and random excitations, Mech. Mach. Theor., 59, 20-33 (2013)
[26] Yang, Y.; Xia, W.; Han, J.; Song, Y.; Wang, J.; Dai, Y., Vibration analysis for tooth crack detection in a spur gear system with clearance nonlinearity, Int. J. Mech. Sci., 157-158, 648-661 (2019/07/01/2019)
[27] Yang, Y.; Xu, M.; Du, Y.; Zhao, P.; Dai, Y., Dynamic analysis of nonlinear time-varying spur gear system subjected to multi-frequency excitation, J. Vib. Contr., 25, 6, 1210-1226 (2019)
[28] Yang, Y.; Li, H.; Dai, Y., Nonlinear vibration characteristics of spur gear system subjected to multiple harmonic excitations, Proc. IME C J. Mech. Eng. Sci., 233, 17, 6026-6050 (2019)
[29] Zhang, Y.; Spanos, P. D., Efficient response determination of a M-D-O-F gear model subject to combined periodic and stochastic excitations, Int. J. Non Lin. Mech., 120, 103378 (2020/04/01/2020)
[30] Zhou, S.; Song, G.; Ren, Z.; Wen, B., Nonlinear dynamic analysis of coupled gear-rotor-bearing system with the effect of internal and external excitations, Chin. J. Mech. Eng., 29, 2, 281-292 (2016)
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