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The Effects of Strong Cubic Nonlinearity on the Existence of Periodic Solutions of the Mathieu–Duffing Equation

[+] Author and Article Information
Ivana Kovacic

Department of Mechanics, Faculty of Technical Sciences, University of Novi Sad, 21121 Novi Sad, Serbiaivanakov@uns.ac.rs

Livija Cveticanin

Department of Mechanics, Faculty of Technical Sciences, University of Novi Sad, 21121 Novi Sad, Serbiacveticanin@uns.ns.ac.yu

J. Appl. Mech 76(5), 054501 (Jun 16, 2009) (3 pages) doi:10.1115/1.3112722 History: Received March 12, 2008; Revised February 04, 2009; Published June 16, 2009

This paper deals with systems governed by the Mathieu–Duffing equation, with a time-dependent coefficient of the linear term and a constant, not necessarily small coefficient of the cubic term. This coefficient can be positive or negative. The method of strained parameters applied to a linear system governed by the Mathieu equation is extended to a strongly nonlinear system. As a result, the curves corresponding to the parameter values at which periodic solutions exist are obtained. It is shown that they strongly depend on the value of the coefficient of nonlinearity and the initial conditions. The corresponding parameter planes are plotted. Numerical integrations are carried out to confirm the analytical results.

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Copyright © 2009 by American Society of Mechanical Engineers
Topics: Equations , Hardening
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Figures

Grahic Jump Location
Figure 1

Parameter δε-plane for (a) softening nonlinearity ε1=0.5 (solid line), ε1=0.12 (dashed line), and (b) hardening nonlinearity ε1=0.5 (solid line), ε1=0.06 (dashed line)

Grahic Jump Location
Figure 2

Phase projections for |c|=1 and ε=0.1: (a) point P1, (b) point P2, (c) point P3, and (d) point P4

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