SPATIO-TEMPORAL CHAOS AND SOLITONS EXHIBITED BY VON KÁRMÁN MODEL

Abstract

Forced oscillations of flexible plates with a longitudinal, time dependent load acting on one plate side are investigated. Regular (harmonic, subharmonic and quasi-periodic) and irregular (chaotic) oscillations appear depending on the system parameters as well as initial and boundary conditions. In order to achieve highly reliable results, an effective algorithm has been applied to convert a problem of finding solutions to the hybrid type partial differential equations (the so-called von Kármán form) to that of the ordinary differential equations (ODEs) and algebraic equations (AEs). The obtained equations are solved using finite difference method with the approximations 0(h4) and 0(h2) (in respect to the spatial coordinates). The ODEs are solved using the Runge–Kutta fourth order method, whereas the AEs are solved using either the Gauss or relaxation methods. The analysis and identification of spatio-temporal oscillations are carried out by investigation of the series wij(t), wt,ij(t), phase portraits wt,ij (wij) and wtt,ij(wt,ij, wij) and the mode portraits in the planes wx,ij(wij), wy,ij (wij) and in the space wxx(wx,ij,wij), FFT as well as the Poincaré sections and pseudo-sections.