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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 375913, 18 pages
http://dx.doi.org/10.1155/2012/375913
Research Article

Stochastic Stability of Damped Mathieu Oscillator Parametrically Excited by a Gaussian Noise

Department of Structural Engineering, Politecnico di Milano, 20133 Milan, Italy

Received 16 May 2011; Accepted 7 November 2011

Academic Editor: M. D. S. Aliyu

Copyright © 2012 Claudio Floris. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper analyzes the stochastic stability of a damped Mathieu oscillator subjected to a parametric excitation of the form of a stationary Gaussian process, which may be both white and coloured. By applying deterministic and stochastic averaging, two Itô’s differential equations are retrieved. Reference is made to stochastic stability in moments. The differential equations ruling the response statistical moment evolution are written by means of Itô’s differential rule. A necessary and sufficient condition of stability in the moments of order r is that the matrix of the coefficients of the ODE system ruling them has negative real eigenvalues and complex eigenvalues with negative real parts. Because of the linearity of the system the stability of the first two moments is the strongest condition of stability. In the case of the first moments (averages), critical values of the parameters are expressed analytically, while for the second moments the search for the critical values is made numerically. Some graphs are presented for representative cases.