Analytical and numerical studies of multi-degree-of-freedom(MDOF) nonlinear stochastic or deterministic dynamic systems have long been a technical challenge.This paper presents a highly-efficient method for determining the stationary probability density functions(PDFs) of MDOF nonlinear systems subjected to both additive and multiplicative Gaussian white noises. The proposed method takes advantages of the sufficient conditions of the reduced Fokker-Planck-Kolmogorov(FPK) equation when constructing the trial solution. The assumed solution consists of the analytically constructed trial solutions satisfying the sufficient conditions and an exponential polynomial of the state variables, and delivers a high accuracy of the solution because the analytically constructed trial solutions capture the main characteristics of the nonlinear system. We also make use of the concept from the data-science and propose a symbolic integration over a hypercube to replace the numerical integrations in a higher-dimensional space, which has been regarded as the insurmountable difficulty in the classical method of weighted residuals or stochastic averaging for high-dimensional dynamic systems. Three illustrative examples of MDOF nonlinear systems are analyzed in detail. The accuracy of the numerical results is validated by comparison with the Monte Carlo simulation(MCS) or the available exact solution. Furthermore, we also show the substantial gain in the computational efficiency of the proposed method compared with the MCS.
A disc-pad friction system is modelled as that two moving pads act symmetrically on an annular beam with flexible boundary condition.Simulation procedure is proposed to deal with the moving interactions and calculation is carried out by using the finite difference method,which shows that only the first-order mode vibration of the beam can be induced.Then the partial differential equation of motion of the disk is reduced to a first-order mode vibration system with time-varying stiffness.As the disk speed is decreased below the critical speeds,the relative equilibrium of the pad on the disk loses its stability and stick-slip type limit cycle vibrations are resulted in all directions′movements.Acceleration of the disk motion on the frictional instability is also investigated.The period of stick-slip vibration with large amplitude will be shortened with higher moving deceleration.
