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TECHNICAL PAPERS

Exact Stationary Solutions of Stochastically Excited and Dissipated Integrable Hamiltonian Systems

[+] Author and Article Information
W. Q. Zhu, Y. Q. Yang

Department of Mechanics, Zhejiang University, Hangzhou 310027, P. R. China

J. Appl. Mech 63(2), 493-500 (Jun 01, 1996) (8 pages) doi:10.1115/1.2788895 History: Received February 08, 1995; Revised May 10, 1995; Online October 26, 2007

Abstract

It is shown that the structure and property of the exact stationary solution of a stochastically excited and dissipated n -degree-of-freedom Hamiltonian system depend upon the integrability and resonant property of the Hamiltonian system modified by the Wong-Zakai correct terms. For a stochastically excited and dissipated nonintegrable Hamiltonian system, the exact stationary solution is a functional of the Hamiltonian and has the property of equipartition of energy. For a stochastically excited and dissipated integrable Hamiltonian system, the exact stationary solution is a functional of n independent integrals of motion or n action variables of the modified Hamiltonian system in nonresonant case, or a functional of both n action variables and α combinations of phase angles in resonant case with α (1 ≤ α ≤ n – 1) resonant relations, and has the property that the partition of the energy among n degrees-of-freedom can be adjusted by the magnitudes and distributions of dampings and stochastic excitations. All the exact stationary solutions obtained to date for nonlinear stochastic systems are those for stochastically excited and dissipated nonintegrable Hamiltonian systems, which are further generalized to account for the modification of the Hamiltonian by Wong-Zakai correct terms. Procedures to obtain the exact stationary solutions of stochastically excited and dissipated integrable Hamiltonian systems in both resonant and nonresonant cases are proposed and the conditions for such solutions to exist are deduced. The above procedures and results are further extended to a more general class of systems, which include the stochastically excited and dissipated Hamiltonian systems as special cases. Examples are given to illustrate the applications of the procedures.

Copyright © 1996 by The American Society of Mechanical Engineers
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