Abstract
The almost-sure asymptotic stability of a pair of coupled linear oscillators whose frequencies are equal or nearly equal and parametrically excited by a wide-band stochastic process is investigated. Using the method of stochastic averaging and a Fourier expansion technique for solving the Fokker-Planck equation associated with the response angles, the maximal Lyapunov exponent is evaluated. The sign of the Lyapunov exponent determines the stability or instability of the oscillators. The cases when the frequency matrix is of diagonal form and of Jordan-canonical form are considered.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
S. T. Ariaratnam, D. S. F. Tarn, and W.-C. Xie, “Lyapunov exponents and stochastic stability of coupled linear systems,” Probabilistic Eng. Meek, vol. 6, no. 2, pp. 51–56, 1990.
S. T. Ariaratnam and W.-C. Xie, Wie-Chau (1992) “Lyapunov exponents and stochastic stability of coupled linear systems under real noise excitation,” Journal of Applied Mechanics, vol. 59, pp. 664–673, 1992.
R. Griesbaum and W. Wedig, “Generalized Fourier analysis for stochastic stability problems,” Proceedings of the IUTAM Symposium on Nonlineariry and Stochastic Structural Dynamics, ed. S. Narayanan, R. N. Iyengar, Kluwer Academic Publishers, pp. 77–94, 1999.
R. Z. Khasminskii, “A limit theorem for the solutions of differential equations with random right-hand sides,” Theory of Probability and its Application, vol. 11, no. 3, pp. 390–406, 1966. (English Translation).
R. Z. Khasminskii, “Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems,” Theory of Probability and its Application, vol. 12, no. 1, pp. 144–147, 1967. (English Translation).
R. L. Stratonovich, Topics in the Theory of Random Noise, vol. 1, Gordon and Breach, New York, 1963.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media Dordrecht
About this paper
Cite this paper
Ariaratnam, S.T., Abdelrahman, N.M. (2003). Stochastic Stability of Coupled Oscillators in Internal Resonance. In: Namachchivaya, N.S., Lin, Y.K. (eds) IUTAM Symposium on Nonlinear Stochastic Dynamics. Solid Mechanics and Its Applications, vol 110. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0179-3_8
Download citation
DOI: https://doi.org/10.1007/978-94-010-0179-3_8
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-3985-7
Online ISBN: 978-94-010-0179-3
eBook Packages: Springer Book Archive