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On methods for continuous systems with quadratic, cubic and quantic nonlinearities

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  • El-Bassiouny, A.F.

Abstract

Methods for study of weakly nonlinear continuous systems are discussed. The method of multiple scales is used to analyze the nonlinear response of a relief valve under combined static and dynamic loadings. We determine a second-order approximation to the response of the system for the case of primary resonance. Second, we derive a second-order nonlinear ordinary differential equation that describes the time evolution of a single-mode, the so-called single-mode discretization. Then, we use the multiple scales method to determine second-order approximate solutions of this equation, thereby obtaining the equations describe the modulations of the amplitude and phase of the response. We show that the results of the second approach are erroneous.

Suggested Citation

  • El-Bassiouny, A.F., 2009. "On methods for continuous systems with quadratic, cubic and quantic nonlinearities," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1308-1316.
    • Handle: RePEc:eee:chsofr:v:39:y:2009:i:3:p:1308-1316
    DOI: 10.1016/j.chaos.2007.06.021
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    1. El-Bassiouny, A.F., 2006. "Vibration and chaos control of non-linear torsional vibrating systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 366(C), pages 167-186.
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    3. Gendelman, O.V. & Lamarque, C.H., 2005. "Dynamics of linear oscillator coupled to strongly nonlinear attachment with multiple states of equilibrium," Chaos, Solitons & Fractals, Elsevier, vol. 24(2), pages 501-509.
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    5. Alasty, Aria & Salarieh, Hassan, 2007. "Nonlinear feedback control of chaotic pendulum in presence of saturation effect," Chaos, Solitons & Fractals, Elsevier, vol. 31(2), pages 292-304.
    6. Cao, Hongjun, 2005. "Primary resonant optimal control for homoclinic bifurcations in single-degree-of-freedom nonlinear oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 24(5), pages 1387-1398.
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