Investigating Bifurcation and Chaos Phenomenon in Nonlinear Vibration of Gear System Using Approximation of Backlash with Smoothing Function
Many researchers have investigated gear system, using numerical and approximation methods such as piecewise linear technique. It should, however, be noted that these methods cannot predict some of the important nonlinear phenomena, such as sub-harmonic and chaotic responses. For specific values of parameters, the system becomes chaotic, when the system has chaotic response; vibrations are in large amplitude and unpredictable. In this paper, nonlinear vibration of gear system and their chaotic behavior are discussed. Gear models, assuming the presence of backlash and approximation with smoothing function, are represented, and bifurcation and chaos in these models are investigated. Most parameters, which affect the chaotic behavior of the system using bifurcation diagrams, have been established and based on these results, the system has been designed.
Chaos, Bifurcation, Backlash, Nonlinear Vibration of Gear
Wang, J. Li, R. Peng, X. (2003), "Survey of nonlinear vibration of gear transmission systems", ASME J. 56(3), 309-329.
Kahraman, A. Blankenship, G.W. (1997), "Experiments on nonlinear dynamic behavior of an oscillator with clearance and periodically time-varying parameters", ASME J. Appl. Mech. 64, 217–226.
Munro, R. G. (1962), "The dynamic behavior of spur gears", PhD dissertation, Cambridge Univ.
Kubo, A. Yamada, K. Aida, T. Sato, S. (1972), "Research on ultra high speed gear devices (reports 1–3)", Trans JSME 38, 2692–2715.
Umezawa, K. Sata, T. Ishikawa, J. (1984), "Simulation of rotational vibration of spur gears", Bull. JSME, 38, 102–109.
Comparin, R. J. Singh, R. (1990), "Frequency response characteristics of a multi-degree-of freedom system with clearances", J. Sound Vib. 142(1), 101–124.
Padmanabhan, C. Singh, R. (1992), "Spectral coupling issues in a two-degree-of freedom system with clearance non-linearities", J. Sound Vib. 155(2), 209–230.
Kahraman, A. Singh, R. (1990), "Non-linear dynamics of a spur gear pair", J. Sound Vib. 142(1), 49–75.
Gregory, R. W. Harris, S. L. Munro, R.G. (1963), "Dynamic behavior of spur gears", Proc. Inst. Mech. Eng., IMechE Conf. 178, 207–226.
Retting, H. (1965), "Zahnkrafte und schwingungen in stirnradgetrieben", Konstroruktion, 17, 41–53.
Terauchi, Y. Hidaka, T. Nagashima, M. (1967), "Dynamic loads on spur gear teeth", Trans JSME 33, 456–472.
Sato, K. Kamada, O. Takatsu, N. (1979), "Jump phenomena in geared system to random excitation", Bull. JSME, 28, 1271–1278.
Sato, K. Yamamoto, S. Kawakami, T. (1991), "Bifurcation sets and chaotic states of a geared system subjected to harmonic excitation", Computational Mech., Berlin 7, 173–182.
Wong, C. W. Zhang, W. S. Lau, S. L. (1991), "Periodic forced vibration of unsymmetric piecewise-linear system by incremental harmonic balance method", J. Sound Vib. 149, 91–105.
Lau, S.L. Zhang, W. S. (1992), "Non-linear vibrations of piecewise linear systems by incremental harmonic balance method", ASME J. Appl. Mech. 59, 153–160.
Raghothama, A. Narayanan, S. (1999), "Bifurcation and chaos in geared rotor bearing system by incremental harmonic balance method, J. Sound Vib. 226(3), 469–492.
Kim, T. C. Rook, T. E. Singh, R. (2002), "Effect of smoothening function on the frequency response of an oscillator with clearance non-linearity", J. Sound Vib. 263, 665–678.
Kim, T.C. Rook, T. E. Singh, R. (2005), "Super- and sub-harmonic response calculations for a torsional system with clearance nonlinearity using the harmonic balance method", J. Sound Vib. 281, 965–993.