In this paper a non-ideal mechanical system with clearance is considered. The mechanical model of the system is an oscillator connected with an unbalanced motor. Due to the existence of clearance the connecting force between motor and the fixed part of the system is discontinuous but linear. The mathematical model of the system is represented by two coupled second-order differential equations. The transient and steady-state motion and also the stability of the system are analyzed. The Sommerfeld effect is detected. For certain values of the system parameters the motion is chaotic. This is caused by the period doubling bifurcation. The existence of chaos is proved with maximal Lyapunov exponent. A new chaos control method based on the known energy analysis is introduced and the chaotic motion is transformed into a periodic one.
Alvarez-Ramirez, J., Espinosa-Paredes G., and Puebla, H., 2003, "Chaos control using small-amplitude damping signals ," Physics Letters A316, 196-205.
2.
Balthazar, J.L. and Brasil, R.M.L. R. F., 2004, "On saturation control of a non-ideal vibrating portal frame founded type shear - building," Journal of Vibration and Control10, 1739-1748.
3.
Balthazar, J.M., Mook, D.T., Weber, H.I., Brasil, R.M.I. R. F., Fenili, A., Beltano, D., and Felix, J.L.P., 2003, "An overview on non-ideal vibrations," Meccanica38, 613-621.
4.
Bogolyubov, N.N. and Mitropolskij, Ju. A., 1974, Asimptoticheskie metodi v teorii nelinejnih kolebanij, Nauka, Moscow.
5.
Dimentberg, M.F., McGovern, L., Norton, R.L., Chapdelaine, J., and Harrison, R., 1997, "Dynamics of an unbalanced shaft interacting with a limited power supply," Nonlinear Dynamics13, 171-187.
6.
Kononenko, V.O., 1969, Vibrating Systems with a Limited Power Supply, Iliffe Books Ltd, London.
Lin, R.M. and Ewins, D.J., 1993, "Chaotic vibration of mechanical systems with backlash ", Mechanical Systems and Signal Processing7, 257-272.
9.
Nayfeh, A.H. and Mook, D.T., 1979, Nonlinear Oscillations, John Wiley & Sons, New York.
10.
Ott, E., Grebogi, C., and Yorke, Y.A., 1990, "Controlling chaos," Physical Review Letters64, 1196-1999.
11.
Pyragas, K., 1992, "Continuous control of chaos by self controlling feedback ," Physics Letters A170, 421-428.
12.
Pyragas, K., 1996, "Continuous control of chaos by self-controlling feedback ," Controlling Chaos, Academic Press, San Diego, CA, pp. 118-123.
13.
Sandri, M., 1996, "Numerical calculation of Lyapunov exponents", The Mathematica Journal6, 78-84.
14.
Souza, S.L.T., Caldas, I.L., Viana, R.L., Balthazar, J.M., and Brasil, R.M.L. R. F., 2005a, "Impact dampers for controlling chaos in systems with limited power supply," Journal of Sound and Vibration279, 955-965.
15.
Souza, S.L.T., Caldas, I.L., Viana, R.L., Balthazar, J.M., and Brasil, R.M.L. R. F., 2005b, "Basins of attraction changes by amplitude constraining of oscillators with limited power supply," Chaos, Solitons and Fractals26, 1211-1220.
16.
Souza, Y.A.Caldas, I.L., Viana, R.L., Balthazar, J.M., and Brasil, R.M.L. R. F., 2007, "A simple feedback control for a chaotic oscillator with limited power supply," Journal of Sound and Vibration299, 664-671.
17.
Tereshko, V., Chacon, R., and Preciado, V., 2004, "Controlling chaotic oscillators by altering their energy ," Physics Letters A320, 408-416.
18.
Tsuchida, M., Guilherme, K.L., and Balthazar, J.M., 2005, "On chaotic vibrations of a non-ideal system with two degree of freedom: 1:2 resonance and Sommerfeld effect," Journal of Sound and Vibration282, 1201-1207.
19.
Warminski, J., Balthazar, J.M., and Brasil, R.M.L.R.F. , 2001, "Vibrations of a non-ideal parametrically and self-excited model," Journal of Sound and Vibration245, 363-374.
20.
Wolf, A., 1984, "Quantifying Chaos with Lyapunov Exponent", Nonlinear Science: Theory and Application, A. V. Holden (ed.), Manchester University Press, Manchester.
21.
Wolf, A., Swift, J., Swinney, H., and Vastano, J., 1985, "Determining Lyapunov exponents from a time series ," Physica D16, 285-317.
22.
Zukovic, M. and Cveticanin, L., 2007, "Chaotic responses in a stable Duffing system of non-ideal type," Journal of Vibration and Control13, 751-767.
