Exact Solutions for Stationary Responses of Several Classes of Nonlinear Systems to Parametric and/or External White Noise Excitations

Zhu Wei-qiu

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 1990 > 11(2): 155-164

Zhu Wei-qiu. Exact Solutions for Stationary Responses of Several Classes of Nonlinear Systems to Parametric and/or External White Noise Excitations[J]. Applied Mathematics and Mechanics, 1990, 11(2): 155-164.

Citation:

PDF( 2763 KB)

Exact Solutions for Stationary Responses of Several Classes of Nonlinear Systems to Parametric and/or External White Noise Excitations

Zhu Wei-qiu

Dept. of Mechanics, Zhejiang University, Hangzhou

Received Date: 1988-10-31
Publish Date: 1990-02-15

Abstract

Abstract

The exact solutions for stationary responses of one class of the second order and three classes of higher order nonlinear systems to parametric and/or external while noise excitations are constructed by using Fokkcr-Planck-Kolmogorov et/ualion approach.The conditions for the existence and uniqueness and the behavior of the solutions are discussed.All the systems under consideration are characterized by the dependence ofnonconservative fqrces on the first integrals of the corresponding conservative systems and arc catted generalized-energy-dependent f G.E.D.) systems.It is shown taht for each of the four classes of G.E.D.nonlinear stochastic systems there is a family of non-G.E.D.systems which are equivalent to the G.E.D.system in the sense of having identical stationary solution.The way to find the equivalent stochastic systems for a given G.E.D.system is indicated and.as an example,the equivalent stochastic systems for the second order G.E.D.nonlinear stochastic system are given.It is pointed out and illustrated with example that the exact stationary solutions for many non-G.E.D.nonlinear stochastic systems may he found by searching the equivalent G.E.D.systems.

FullText(HTML)

References(10)

References

[1]	Crandall,S.H.and W.Q.Zhu,Random vibration:A survey of recent developments,J.Appl.Mech.,50th Anniversary Volume,50(1983),953-962.
[2]	Caughey,T.K.,Nonlinear theory of random vibrations,Advances in Applied Mechanics,11(1971),209-253.
[3]	Roberts,J.B.,Response of nonlinear mechanical systems to random excitation,Part 1:Markov method,The Shock and Vibration Digest,13,4(1981),17-28.
[4]	Roberts,J.B.,Response of nonlinear mechanical systems to random excitation,Part 2:Equivalent linearization and other methods,The Shock and Vibration Digest,13,5(1981),15-29.
[5]	Roberts,J.B.,Techniques for non-linear random vibration problems,The Shock and Vibration Digest,16,9(1984),3-14.
[6]	Caughey,T.K.and F.Ma,The exact steady-state solution of a class of non-linear stochastic systems,Int.J.Non-Linear Mechanics,17(1982),137-142.
[7]	Caughey,T.K.and F.Ma,The steady-state response of a class of dynamical systems to stochastic excitation,J.Appl.Mech.,49(1982),629-632.
[8]	Dimentberg,M.F.,An exact solution to a certain non-linear random vibration problem,Int.J.Non-Linear Mechanics,17(1982),231-236.
[9]	Yong,Y.and Y.K.Lin,Exact stationary-response solution for second order nonlinear systems under parametric and external white-noise excitations,J.Appl.Mech.,54(1987),414-418.
[10]	Dimentberg,M.F.,Nonlinear Stochastic Problems of Mechanical Vibrations,Nauka,Moscow(1980).