ХАОТИЧЕСКИЕ КОЛЕБАНИЯ ГЕОМЕТРИЧЕСКИ НЕЛИНЕЙНЫХ НАНОРАЗМЕРНЫХ ПОЛОГИХ ОСЕСИММЕТРИЧНЫХ ОБОЛОЧЕК

Abstract

The Hamilton - Ostrogradsky principle is used in the paper to construct a mathematical model of vibrations of geometrically nonlinear nano-dimensional shallow axisymmetric spherical shells. The model is based on the following relations and assumptions: the body of the shell is elastic, homogeneous and isotropic, the Kirchhoff - Love hypothesis, a modified momentary theory of elasticity is used to account for the dependence of elastic behavior of the shell on the dimension-dependent parameter; the shallowness of the shell is determined based on the hypotheses of V.Z. Vlasov, the geometrical nonlinearity is determined using T. Karman's hypothesis. An approach for determining the “true” solution is proposed. The partial differential equation is reduced to a Cauchy problem with the help of the second-order accuracy finite difference method, which is solved using several Runge - Kutta-type methods: the 4th- and 2nd-order Runge - Kutta methods, the 4th-order Runge - Kutta - Fehlberg method, the 4th-order Cash - Karp method, the 8th-order Runge - Kutta - Prince - Dormand method, the implicit 2nd- and 4th-order Runge - Kutta method. An algorithm and a software complex for obtaining numerical results are developed. The convergence of the above methods for a space and time coordinate is studied. The study is based on the qualitative theory of nonlinear dynamics: signals, 2D and 3D phase-plane portraits, power spectrums, Morley's wavelets, deflection curves, Poincare sections and autocorrelation functions are analyzed, the sign of Lyapunov's index is analyzed using several methods: Wolf's, Kantz's, Rosenstein's. An example of analyzing plates and shallow shells is given. The analysis of the obtained results shows that with the increase of the dimension-dependent parameter the vibration character changes from chaotic to harmonic, and the value of the dynamic critical load increases.