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Oscillators with the Time Variable Parameters

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Strong Nonlinear Oscillators

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Abstract

In this chapter the motion of the oscillator with time variable parameters is considered.

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References

  • Abdalla, M. S. (1986a). Canonical treatment of harmonic oscillator with variable mass. Physical Review A, 33, 2870–2876.

    Article  MathSciNet  Google Scholar 

  • Abdalla, M. S. (1986b). Time-dependent harmonic oscillator with variable mass under the action of a driving force. Physical Review A, 34, 4598–4605.

    Article  Google Scholar 

  • Abraham, G. T., & Chatterjee, A. (2003). Approximate asymptotic solution for nonlinear Mathieu equation using harmonic balance based averaging. Nonlinear Dynamics, 31, 347–365.

    Article  MathSciNet  MATH  Google Scholar 

  • Abramowitz, M., & Stegun, I. A. (1979). Handbook of mathematical functions with formulas, graphs and mathematical tables. Moscow: Nauka.

    MATH  Google Scholar 

  • Belovodsky, V. N., Tsyfansky, S. L., & Beresnevich, V. I. (2002). The dynamics of a vibromachine with parametric excitation. Journal of Sound and Vibration, 254, 897–910.

    Article  Google Scholar 

  • Bessonov, A. P. (1967). Osnovji dinamiki mehanizmov s peremennoj massoj zvenjev. Moscow: Nauka.

    Google Scholar 

  • Bogolubov, N., & Mitropolski, J. A. (1963). Asymptotical methods in the theory of nonlinear oscillations. Delhi: Hindustan Publishing Co.

    Google Scholar 

  • Byrd, P. F., & Friedman, M. D. (1954). Handbook of elliptic integrals for engineers and physicists. Berlin: Springer.

    Book  MATH  Google Scholar 

  • Crespo, G., Proto, A. N., Plastino, A., & Otero, D. (1990). Information-theory approach to the variable-mass harmonic oscillator. Physical Review A, 42, 3608–3617.

    Article  MathSciNet  Google Scholar 

  • Cveticanin, L. (1992). The influence of the reactive force on a nonlinear oscillator with variable parameter. Transaction of ASME Journal of Vibration and Acoustics, 114, 578–580.

    Article  Google Scholar 

  • Cveticanin, L. (1995). Approximate solution of a time-dependent differential equation. Meccanica, 30, 665–671.

    Article  MathSciNet  MATH  Google Scholar 

  • Cveticanin, L. (1998). Dynamics of machines with variable mass. London: Gordon and Breach.

    MATH  Google Scholar 

  • Cveticanin, L. (2000). Vibrations in a parametrically excited system. Journal of Sound and Vibra- tion, 229, 245–271.

    Google Scholar 

  • Cveticanin, L. (2012). Oscillator with non-integer order nonlinearity and time variable parameters. Acta Mechanica, 223, 1417–1429.

    Article  MathSciNet  MATH  Google Scholar 

  • Cveticanin, L. (2013). Van der Pol oscillator with time-variable parameters. Acta Mechanica, 224, 945–955.

    Article  MathSciNet  MATH  Google Scholar 

  • Cveticanin, L., & Kovacic, I. (2007). Parametrically excited vibrations of the oscillator with strong cubic negative non-linearity. Journal of Sound and Vibration, 304, 201–212.

    Article  Google Scholar 

  • Drogomirecka, H. T. (1997). On integration of a special Ateb-function. Visnik Lvivskogo Universitetu, Serija mehaniko-matematichna, 46, 108–110. (in Ukranian).

    Google Scholar 

  • El-Dib, Y. O. (2001). Nonlinear Mathieu equation and coupled resonance mechanism. Chaos, Solitons and Fractals, 12, 705–720.

    Article  MathSciNet  MATH  Google Scholar 

  • Esmailzadeh, E., & Goodarzi, A. (2001). Stability analysis of a CALM floating offshore structure. International Journal of Non-Linear Mechanics, 36, 917–926.

    Article  MATH  Google Scholar 

  • Esmailzadeh, E., & Jalili, N. (1998). Parametric response of cantilever Timoshenko beams with tip mass under harmonic support motion. International Journal of Non-Linear Mechanics, 33, 765–781.

    Article  MathSciNet  MATH  Google Scholar 

  • Esmailzadeh, E., Jazar, G. N., & Mehri, B. (1997). Existence of periodic solution for beams with harmonically variable length. ASME Journal of Vibration and Acoustics, 119, 485–488.

    Article  Google Scholar 

  • Flores, J., Solovey, G., & Gill, S. (2003). Variable mass oscillator. American Journal of Physics, 71, 721–725.

    Article  Google Scholar 

  • Gricik, V. V., & Nazarkevich, M. A. (2007). Mathematical models algorythms and computation of Ateb-functions. Dopovidi NAN Ukraini Series A, 12, 37–43. (in Ukranian).

    Google Scholar 

  • Irschik, H., & Holl, H. J. (2004). Mechanics of variable-mass systems—Part 1: Balance of mass and linear momentum. Applied Mechanics Reviews, 57, 145–161.

    Article  Google Scholar 

  • Jazar, G. N. (2004). Stability chart of parametric vibrating systems using energy-rate method. International Journal of Non-Linear Mechanics, 39, 1319–1331.

    Article  MATH  Google Scholar 

  • Kamke, E. (1971). Spravocnik po objiknovennjim differencialjnjim uravnenijam. Moscow: Nauka.

    Google Scholar 

  • Krylov, N., & Bogolubov, N. (1943). Introduction to nonlinear mechanics. New York: Princenton University Press.

    Google Scholar 

  • Leach, P. G. L. (1983). Harmonic oscillator with variable mass. Journal of Physics A: General Physics, 16, Art. No. 019, 3261–3269.

    Google Scholar 

  • Levi-Civita, T. (1928). Sul moto di un corpo di massa variabile. Rendiconti del Linci, 329–333, 621–622.

    MATH  Google Scholar 

  • Li, J., Xu, W., Ren, Z., & Lei, Y. (2005). Maximal Lyapunov exponent and almost-sure stability for stochastic Mathieu–Duffing systems. Journal of Sound and Vibration, 286, 395–402.

    Article  MathSciNet  MATH  Google Scholar 

  • Meshcherskij, I. V. (1952). Rabotji po mehanike tel peremennoj massji. Moscow: Gos. Izd. tehniko-teoret.lit.

    Google Scholar 

  • Mickens, R. E. (2010). Truly nonlinear oscillations. Singapore: World Scientific.

    Book  MATH  Google Scholar 

  • Mond, M., & Cederbaum, G. (1993). Stability analysis of the non-linear Mathieu equation. Journal of Sound and Vibration, 167, 77–89.

    Article  MathSciNet  MATH  Google Scholar 

  • Nayfeh, A. H., & Mook, D. T. (1979). Nonlinear oscillations. New York: Wiley.

    MATH  Google Scholar 

  • Ng, L., & Rand, R. (2002a). Bifurcations in a Mathieu equation with cubic nonlinearities. Chaos, Solitons and Fractals, 14, 173–181.

    Article  MathSciNet  MATH  Google Scholar 

  • Ng, L., & Rand, R. (2002b). Bifurcations in a Mathieu equation with cubic nonlinearities (Part II). Communications in Nonlinear Science and Numerical Simulation, 7, 107–121.

    Article  MathSciNet  MATH  Google Scholar 

  • Rhoads, J. F., Shaw, S. W., Turner, K. L., Moehlis, J., DeMartini, B. E., & Zhang, W. (2006). Generalized parametric resonance in electrostatically actuated microelectromechanical oscillators. Journal of Sound and Vibration, 296, 797–829.

    Article  Google Scholar 

  • Sanchez-Ortiz, G. I., & Salas-Brito, A. L. (1995). Chaos in a variable mass relaxation oscillator model for the leaky tap. Physica D: Nonlinear Phenomena, 89, 151–168.

    Article  MathSciNet  MATH  Google Scholar 

  • Wiegand, M., Scheithauer, S., & Theil, S. (2004). Step proof mass dynamics. Acta Astronautica, 54, 631–638.

    Article  Google Scholar 

  • Xie, G.-Q., Qian, S.-W., & Gu, Z.-Y. (1995). Separation of variables treatment of the time-dependent damped harmonic oscillator with an arbitrary varying mass and with a force quadratic in the velocity under the action of an arbitrary time-varying force. Physics Letters A, 207, 11–16.

    Article  MathSciNet  MATH  Google Scholar 

  • Younesian, D., Esmailzadeh, E., & Sedaghati, R. (2005). Existence of periodic solutions for the generalized form of Mathieu Equation. Nonlinear Dynamics, 39, 335–348.

    Article  MathSciNet  MATH  Google Scholar 

  • Yuste, B. S. (1991). On Duffing oscillators with slowly varying parameters. International Journal of Non-Linear Mechanics, 26, 671–677.

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang, W., Baskaran, R., & Turner, K. L. (2002). Effect of cubic nonlinearity on auto-parametrically amplified resonant MEMS mass sensor. Sensors and Actuators A, 102, 139–150.

    Article  Google Scholar 

  • Zounes, R. S., & Rand, R. H. (2002). Subharmonic resonance in the non-linear Mathieu equation. International Journal of Non-Linear Mechanics, 37, 43–73.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia

    Livija Cveticanin

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  1. Livija Cveticanin
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Correspondence to Livija Cveticanin .

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Cveticanin, L. (2018). Oscillators with the Time Variable Parameters. In: Strong Nonlinear Oscillators. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-58826-1_5

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  • DOI: https://doi.org/10.1007/978-3-319-58826-1_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-58825-4

  • Online ISBN: 978-3-319-58826-1

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