Almost-sure stability of linear gyroscopic systems

This paper studies the stability behaviour of a linear gyroscopic system parametrically perturbed by a (multiplicative) real noise of small intensity. To this end, its maximal Lyapunov exponent is calculated using the method of Sri Namachchivaya et al.[1]. The results derived are suitable for cases where the response frequenciesω1 , ω2are non-commensurable and the infinitesimal generator associated with the noise process, ξ (t) has a simple zero eigenvalue. These results are then employed to determine the almost-sure stability boundaries of a rotating shaft subjected to random axial loading.

[1]  F. Dimentberg.  Flexural vibrations of rotating shafts , 1961 .

[2]  Yu-Kweng Michael Lin,et al.  Probabilistic Structural Dynamics: Advanced Theory and Applications , 1967 .

[3]  K. Huseyin,et al.  Vibrations and Stability of Multiple Parameter Systems , 1978 .

[4]  L. Arnold,et al.  A Formula Connecting Sample and Moment Stability of Linear Stochastic Systems , 1984 .

[5]  Etienne Pardoux,et al.  Almost sure and moment stability for linear ito equations , 1986 .

[6]  S A Molčanov,et al.  THE STRUCTURE OF EIGENFUNCTIONS OF ONE-DIMENSIONAL UNORDERED STRUCTURES , 1978 .

[7]  S. Karlin,et al.  A second course in stochastic processes , 1981 .

[8]  N. Sri Namachchivaya,et al.  Moment Lyapunov Exponent for Two Coupled Oscillators Driven by Real Noise , 1996, SIAM J. Appl. Math..

[9]  N. Namachchivaya,et al.  Bifurcation behavior of nonlinear pipes conveying pulsating flow , 1988 .

[10]  N. Sri Namachchivaya,et al.  Almost-sure asymptotic stability of a general four-dimensional system driven by real noise , 1994 .

[11]  Morton Nadler,et al.  The stability of motion , 1961 .

[12]  N. Namachchivaya,et al.  Stochastically Perturbed Linear Gyroscopic Systems , 1987 .

[13]  S. Ariaratnam,et al.  Lyapunov exponents and stochastic stability of coupled linear systems under white noise excitation , 1991 .

[14]  S. Ariaratnam,et al.  Lyapunov Exponents and Stochastic Stability of Coupled Linear Systems Under Real Noise Excitation , 1992 .

[15]  N. Sri Namachchivaya,et al.  Maximal Lyapunov exponent and rotation numbers for two coupled oscillators driven by real noise , 1993 .

[16]  W. Feller.  Diffusion processes in one dimension , 1954 .