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Exact Solutions of the Cubic Duffing Equation by Leaf Functions under Free Vibration

Kazunori Shinohara1

Daido University, 10-3 Takiharucho, Minamiku, Nagoya 457-8530, Japan.

*Corresponding Author: Kazunori Shinohara. Email:

Computer Modeling in Engineering & Sciences 2018, 115(2), 149-215. 10.3970/cmes.2018.02179


Exact solutions of the cubic Duffing equation with the initial conditions are presented. These exact solutions are expressed in terms of leaf functions and trigonometric functions. The leaf function r=sleafn(t) or r=cleafn(t) satisfies the ordinary differential equation dx2/dt2=-nr2n-1. The second-order differential of the leaf function is equal to -n times the function raised to the (2n-1) power of the leaf function. By using the leaf functions, the exact solutions of the cubic Duffing equation can be derived under several conditions. These solutions are constructed using the integral functions of leaf functions sleaf2(t) and cleaf2(t) for the phase of a trigonometric function. Since the leaf function and the trigonometric function are used in combination, a highly accurate solution of the Duffing equation can be easily obtained based on the data of leaf functions. In this study, seven types of the exact solutions are derived from leaf functions; the derivation of the seven exact solutions is detailed in the paper. Finally, waves obtained by the exact solutions are graphically visualized with the numerical results.


Cite This Article

Shinohara, K. (2018). Exact Solutions of the Cubic Duffing Equation by Leaf Functions under Free Vibration. CMES-Computer Modeling in Engineering & Sciences, 115(2), 149–215.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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