COMPARISON OF NUMERICAL AND SEMI-ANALYTICAL SOLUTION OF A SIMPLE NON-LINEAR SYSTEM IN STATE OF THE STOCHASTIC RESONANCE

Abstract eng:
Stochastic resonance (SR) is a phenomenon which can be observed at some nonlinear dynamic systems under combined excitation including deterministic harmonic force and random noise. The Duffing single degree of freedom oscillator is treated and the Gaussian white noise as the random excitation component is considered. Mathematical basis of this phenomenon follows from properties of the Duffing system with negative linear part of the stiffness. Under certain combinations of the system and excitation parameters the SR can emerge. It manifests by stable periodic hopping between two nearly constant limits perturbed by random noises. SR is observed and practically used in a number of disciplines in physics, biophysics, chemistry, etc. However it seems to be promising as a theoretical model of several aeroelastic post-critical effects arising at a prismatic beam in a cross flow. Three independent theoretical solution methods have been addressed and tested in order to compare results of the system response. The first kind is semi-analytic dealing with the relevant Fokker-Planck Equation (FP). It is solved by means of the stochastic moment procedure. The multiharmonic non-stationary solution of the Probability Density Function (PDF) is expected. The Galerkin approach is adopted. The second way is based on the FEM solution of the FP equation. It is analyzed in an original evolutionary form which enables an analysis of transition effects starting the Dirac type initial conditions. The last procedure represents simulation regarding the original Duffing or relevant Ito stochastic system. Comparison of results provided by the above three methods has revealed appropriate domains of their application to particular problems regarding a preliminary analysis or careful detailed inspection in specific small domains in final stage of an engineering system design.

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