### Abstract

Using a series of functional transformations we reduce the unforced, damped Duffing oscillator to equivalent equations of the Abel and Emden-Fowler classes. Taking into account the known exact analytic solutions of these equivalent equations we prove that there does not exist an exact analytic solution of the damped, unforced Duffing oscillator in terms of known (tabulated) analytic functions. It follows that a new class of solutions must be defined for solving this problem 'exactly'. Finally, a new approximate solution of the intermediate integral of the damped Duffing oscillator with weak damping is constructed.

Original language | English (US) |
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Pages (from-to) | 1-16 |

Number of pages | 16 |

Journal | Nonlinear Dynamics |

Volume | 28 |

Issue number | 1 |

DOIs | |

State | Published - Apr 2002 |

Externally published | Yes |

### Keywords

- Asymptotic solutions
- Duffing oscillator
- Nonlinear ordinary differential equations

### ASJC Scopus subject areas

- Mechanical Engineering
- Mechanics of Materials
- Computational Mechanics

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## Cite this

Panayotounakos, D. E., Panayotounakou, A. D., & Vakakis, A. F. (2002). On the solution of the unforced damped duffing oscillator with no linear stiffness term.

*Nonlinear Dynamics*,*28*(1), 1-16. https://doi.org/10.1023/A:1014925032022