Stochastic stability of damped Mathieu oscillator parametrically excited by a Gaussian noise. (English) Zbl 1264.34117

Summary: We analyze 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 \(\mathbf A_r\) 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.


34F05 Ordinary differential equations and systems with randomness
60H25 Random operators and equations (aspects of stochastic analysis)
34D20 Stability of solutions to ordinary differential equations
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[1] R. Campbell, Théorie Générale de l’Equation de Mathieu et de Quelques Autres Equations Differentielles de la Mécanique, Masson, Paris, France, 1955. · Zbl 0066.31702
[2] N. W. McLachlan, Theory and Application of Mathieu Functions, Dover Publications, Mineola, NY, USA, 1962. · Zbl 0184.34504
[3] S. P. Timoshenko, Theory of Elastic Stability, Dover Publications, Mineola, NY, USA, 2009. · Zbl 1198.94004
[4] L. Meirovitch, Methods of Analytical Dynamics, Dover Publications, Mineola, NY, USA, 2003. · Zbl 1115.70002
[5] R. L. Stratonovic and Y. M. Romanovski, “Parametric effect of a random force on linear and nonlinear oscillatory systems,” in Non-Linear Transformation of Random Processes, P. I. Kuznetsov, R. L. Stratonovic, and V. I. Tikhonov, Eds., p. 341, Pergamon Press, New York, NY, USA, 1965.
[6] M. F. Dimentberg, Statistical Dynamics of Nonlinear and Time-Varying Systems, Research Studies Press, Taunton, UK, 1988. · Zbl 0713.70004
[7] G. O. Cai and Y. K. Lin, “Nonlinearly damped systems under simultaneous broad-band and harmonic excitations,” Nonlinear Dynamics, vol. 6, no. 2, pp. 163-177, 1994.
[8] H. Rong, G. Meng, X. Wang, W. Xu, and T. Fang, “Invariant measures and Lyapunov exponents for stochastic Mathieu system,” Nonlinear Dynamics, vol. 30, no. 4, pp. 313-321, 2002. · Zbl 1013.70019
[9] W.-C. Xie, Dynamic Stability of Structures, Cambridge University Press, New York, NY, USA, 2006. · Zbl 1116.70001
[10] Y. K. Lin and G. Q. Cai, “Stochastic stability of nonlinear systems,” International Journal of Non-Linear Mechanics, vol. 29, no. 4, pp. 539-553, 1994. · Zbl 0813.70020
[11] Y. K. Lin and G. Q. Cai, Probabilistic Structural Dynamics: Advanced Theory and Applications, McGraw-Hill, New York, NY, USA, 1995.
[12] C. Floris, “Stochastic stability of a viscoelastic column axially loaded by a white noise force,” Mechanics Research Communications, vol. 38, no. 1, pp. 57-61, 2011. · Zbl 1272.74311
[13] K. Itô, “On stochastic differential equations,” Memoirs of the American Mathematical Society, vol. 4, pp. 289-302, 1951. · Zbl 0054.05803
[14] K. Itô, “On a formula concerning stochastic differentials,” Nagoya Mathematical Journal, vol. 3, pp. 55-65, 1951. · Zbl 0045.07603
[15] M. Di Paola, “Stochastic differential calculus,” in Dynamic Motion: Chaotic and Stochastic Behaviour, F. Casciati, Ed., vol. 340, pp. 29-92, Springer, Vienna, Austria, 1993. · Zbl 0818.60057
[16] M. Grigoriu, Stochastic Calculus, Birkhäuser Boston, Boston, Mass, USA, 2002. · Zbl 1015.60001
[17] R. L. Stratonovich, Topics in the Theory of Random Noise, vol. 1, Gordon and Breach, New York, NY, USA, 1963. · Zbl 0119.14502
[18] R. L. Stratonovich, Topics in the Theory of Random Noise, vol. 2, Gordon and Breach, New York, NY, USA, 1967. · Zbl 0183.22007
[19] J. B. Roberts and P.-T. D. Spanos, “Stochastic averaging: an approximate method of solving random vibration problems,” International Journal of Non-Linear Mechanics, vol. 21, no. 2, pp. 111-134, 1986. · Zbl 0582.73077
[20] W. Q. Zhu, “Stochastic averaging methods in random vibration,” ASME Applied Mechanics Review, vol. 41, no. 5, pp. 189-197, 1988.
[21] N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations, Gordon and Breach, New York, NY, USA, 1961. · Zbl 0151.12201
[22] E. Mettler, “Stability and vibration problems of mechanics under harmonic excitation,” in Dynamic Stability of Structures, G. Herrmann, Ed., Pergamon Press, Oxford, UK, 1967. · Zbl 0153.27001
[23] R. Z. Khasminskii, “A limit theorem for the solution of differential equations with random right hand sides,” Theory of Probability and Applications, vol. 11, no. 3, pp. 390-405, 1966. · Zbl 0202.48601
[24] T. T. Soong and M. Grigoriu, Random Vibration of Mechanical and Structural Systems, Prentice Hall, Englewood Cliffs, NJ, USA, 1993. · Zbl 0788.73005
[25] E. Wong and M. Zakai, “On the relation between ordinary and stochastic differential equations,” International Journal of Engineering Science, vol. 3, no. 2, pp. 213-229, 1965. · Zbl 0131.16401
[26] R. L. Stratonovich, “A new representation for stochastic integrals and equations,” SIAM Journal on Control and Optimization, vol. 4, no. 2, pp. 362-371, 1966. · Zbl 0143.19002
[27] L. Arnold, “A formula connecting sample and moment stability of linear stochastic systems,” SIAM Journal on Applied Mathematics, vol. 44, no. 4, pp. 793-802, 1984. · Zbl 0561.93063
[28] M. Di Paola, “Linear systems excited by polynomials of filtered Poisson pulses,” ASME Journal of Applied Mechanics, vol. 64, no. 3, pp. 712-717, 1997. · Zbl 0913.70018
[29] N. G. Chetayev, The Stability of Motion, Pergamon Press, New York, NY, USA, 1971.
[30] H. Calisto and M. G. Clerc, “A new perspective on stochastic resonance in monostable systems,” New Journal of Physics, vol. 12, Article ID 113027, 2010.
[31] S. T. Ariaratnam and D. S. F. Tam, “Parametric random excitation of a damped Mathieu oscillator,” Journal of Applied Mathematics and Mechanics (ZAMM), vol. 56, pp. 449-452, 1976. · Zbl 0352.34028
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