Exact solution for the unforced Duffing oscillator with cubic and quintic nonlinearities

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• 作者：A. Beléndez ; T. Beléndez ; F. J. Martínez ; C. Pascual ; M. L. Alvarez…
• 关键词：Dynamical systems ; Nonlinear oscillators ; Conservative systems ; Cubic–quintic nonlinear oscillator ; Exact solution ; Jacobian elliptic functions ; Symbolic computation
• 刊名：Nonlinear Dynamics
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：86
• 期：3
• 页码：1687-1700
• 全文大小：1,087 KB
• 刊物类别：Engineering
• 刊物主题：Vibration, Dynamical Systems and Control
  Mechanics
  Mechanical Engineering
  Automotive and Aerospace Engineering and Traffic
  
• 出版者：Springer Netherlands
• ISSN：1573-269X
• 卷排序：86

文摘

The nonlinear differential equation governing the periodic motion of the one-dimensional, undamped, unforced cubic–quintic Duffing oscillator is solved exactly, providing exact expressions for the period and the solution. The period is given in terms of the complete elliptic integral of the first kind and the solution involves Jacobi elliptic functions. Some particular cases obtained varying the parameters that characterize this oscillator are presented and discussed. The behaviour of the period as a function of the initial amplitude is analysed, the exact solutions and velocities for several values of the initial amplitude are plotted, and the Fourier series expansions for the exact solutions are also obtained. All this allows us to conclude that the quintic term appearing in the cubic–quintic Duffing equation makes this nonlinear oscillator not only more complex but also more interesting to study.