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.
Issue Section:
Technical Briefs
1.
Faraday
, M.
, 1831, “On a Peculiar Class of Acoustical Figures; and on Certain Forms Assumed by a Group of Particles Upon Vibrating Elastic Surfaces
,” Philos. Trans. R. Soc. London
0370-2316, 121
, pp. 299
–318
.2.
Mond
, M.
, and Cederbaum
, G.
, 1993, “Stability Analysis of the Non-Linear Mathieu Equation
,” J. Sound Vib.
, 167
(1
), pp. 77
–89
. 0022-460X3.
Esmailzadeh
, E.
, Jazar
, G. N.
, and Mehri
, B.
, 1997, “Existence of Periodic Solution for Beams With Harmonically Variable Length
,” ASME J. Vibr. Acoust.
0739-3717, 119
(3
), pp. 485
–488
.4.
Esmailzadeh
, E.
, and Jalili
, N.
, 1998, “Parametric Response of Cantilever Timoshenko Beams With Tip Mass Under Harmonic Support Motion
,” Int. J. Non-Linear Mech.
0020-7462, 33
(5
), pp. 765
–781
.5.
Esmailzadeh
, E.
, and Goodarzi
, A.
, 2001, “Stability Analysis of a CALM Floating Offshore Structure
,” Int. J. Non-Linear Mech.
, 36
(6
), pp. 917
–926
. 0020-74626.
El-Dib
, Y. O.
, 2001, “Nonlinear Mathieu Equation and Coupled Resonance Mechanism
,” Chaos, Solitons Fractals
0960-0779, 12
(4
), pp. 705
–720
.7.
Ng
, L.
, and Rand
, R.
, 2002, “Bifurcations in a Mathieu Equation With Cubic Nonlinearities
,” Chaos, Solitons Fractals
, 14
, pp. 173
–181
. 0960-07798.
Abraham
, G. T.
, and Chatterjee
, A.
, 2003, “Approximate Asymptotic Solution for Nonlinear Mathieu Equation Using Harmonic Balance Based Averaging
,” Nonlinear Dyn.
0924-090X, 31
, pp. 347
–365
.9.
Rhoads
, J. F.
, Shaw
, S. W.
, Turner
, K. L.
, Moehlis
, J.
, De Martini
, B. E.
, and Zhang
, W.
, 2006, “Generalized Parametric Resonance in Electrostatically Actuated Microelectromechanical Oscillators
,” J. Sound Vib.
, 296
, pp. 797
–829
. 0022-460X10.
Nayfeh
, A. H.
, and Mook
, D. T.
, 1979, Nonlinear Oscillations
, Wiley
, New York
.11.
Rand
, R. H.
, “Lecture Notes on Non-Linear Vibrations
,” Version 45, http://tam.cornell.edu/randdocs/nlvibe45.pdfhttp://tam.cornell.edu/randdocs/nlvibe45.pdf.12.
Esmailzadeh
, E.
, and Nakhaie-Jazar
, G.
, 1997, “Periodic Solution of a Mathieu-Duffing Type Equation
,” Int. J. Non-Linear Mech.
, 32
(5
), pp. 905
–912
. 0020-746213.
Zounes
, R. S.
, and Rand
, R. H.
, 2002, “Subharmonic Resonance in the Non-Linear Mathieu Equation
,” Int. J. Non-Linear Mech.
0020-7462, 37
, pp. 43
–73
.14.
Byrd
, P. F.
, and Friedman
, M. D.
, 1954, Handbook of Elliptic Integrals for Engineers and Physicists
, Springer-Verlag
, Berlin
.15.
Abramowitz
, M.
, and Stegun
, I. A.
, 1979, Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables
, Nauka
, Moscow
.16.
Cveticanin
, L.
, and Kovacic
, I.
, 2007, “Parametrically Excited Vibrations of an Oscillator With Strong Cubic Negative Non-Linearity
,” J. Sound Vib.
, 304
(1–2
), pp. 201
–212
. 0022-460XCopyright © 2009
by American Society of Mechanical Engineers
You do not currently have access to this content.
