Exact Solutions for Strong Nonlinear Oscillators with Linear Damping

Livija Cveticanin ()
Additional contact information
Livija Cveticanin: Faculty of Technical Sciences, University of Novi Sad, Trg D.Obradovica 6, 21000 Novi Sad, Serbia

Mathematics, 2025, vol. 13, issue 10, 1-25

Abstract: This paper presents the derivation of an exact solution for a damped nonlinear oscillator of arbitrary order (both integer and non-integer). A coefficient relationship was defined under which such a solution exists. The analytical procedure was developed based on the application of the Ateb (inverse beta) function. It has been shown that an exact solution exists for a specific relationship between the damping coefficient and the coefficient of the linear elastic term, and that this relationship depends on the order of nonlinearity. The exact amplitude of vibration was found to be a time-decreasing function, depending on the initial amplitude, damping coefficient, and the order of nonlinearity. The period of vibration was also shown to depend not only on the amplitude but also on both the nonlinearity coefficient and its order. For cases where the damping coefficient of the exact oscillator is slightly perturbed, an approximate solution based on the exact one was proposed. Three illustrative examples of oscillators with different orders of nonlinearity were considered: a nearly linear oscillator, a Duffing oscillator, and one with strong nonlinearity. For all cases, the high accuracy of the asymptotic solution was confirmed. Since no exact analytic solution exists for a purely nonlinear damped oscillator, an approximate solution was constructed using the solution of the corresponding undamped oscillator with a time-varying amplitude and phase. In the case of a purely cubic damped oscillator, the approximate solution was compared with numerical results, and good agreement was demonstrated.

Keywords: exact analytic solution; linear damping; pure nonlinear oscillator; first integral; amplitude variation; variable period of vibration; Ateb function (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/13/10/1662/pdf (application/pdf)
https://www.mdpi.com/2227-7390/13/10/1662/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:13:y:2025:i:10:p:1662-:d:1659205

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().