Chaotic Oscillations of a Kinematically Excited Flat Shell During Geometrically Non-linear Deformation

Abstract

We study the forced oscillations of a cantilevered flat shell of constant curvature. These movements are excited by a kinematic periodic embedding motion. To describe geometrically non-linear deformation, the non-linear theory of Donel shells is used. To build a non-linear dynamic system with a finite number of degrees of freedom, the method of specified forms is used. Since the eigen frequencies of longitudinal and torsional oscillations are much higher than bending ones, the inertial forces in the longitudinal and torsional directions are not taken into account. Therefore, the generalized coordinates of longitudinal and torsional oscillations are expressed in terms of bending ones. As a result, a non-linear dynamic system with respect to bending generalized coordinates is obtained. To calculate the eigen forms of linear oscillations, by using which the non-linear dynamic problem decomposes, the Rayleigh-Ritz method is used. Then only kinematic boundary conditions are satisfied. When the solution converges, the force boundary conditions are automatically satisfied. To study the convergence of eigen frequencies, calculations were performed with a different number of basis functions, which are B-splines. A comparison is made with the experimental data on the analysis of eigen frequencies, with the data published in authors' previous article. To numerically analyze the non-linear periodic oscillations, a two-point boundary value problem is solved for ordinary differential equations by the shooting method. The stability of periodic motions and their bifurcations are estimated using multipliers. To study the bifurcations of periodic oscillations, the parameter continuation method is applied. In the region of the main resonance, saddle-node bifurcations, period-doubling bifurcations, and Neimark-Sacker bifurcations are found. To study the steady-state almost periodic and chaotic oscillations, Poincaré sections, spectra of Lyapunov characteristic exponents, and spectral densities are calculated, with the stroboscopic phase portrait used as Poincaré sections. The properties of steady-state oscillations are investigated with a quasistatic change in the frequency of the disturbing action.