Nonlinear Dynamics : міжнародна конференція
Постійне посилання на розділhttps://repository.kpi.kharkov.ua/handle/KhPI-Press/24521
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Документ Resonance Behavior of the Forced Dissipative Spring-Pendulum System(NTU "KhPI", 2016) Plaksiy, Kateryna Yu.; Mikhlin, Yuri V.Dynamics of the dissipative spring-pendulum system under periodic external excitation in the vicinity of external resonance and simultaneous external and internal resonances is studied. Analysis of the system resonance behaviour is made on the base of the concept of nonlinear normal vibration modes (NNMs), which is generalized for systems with small dissipation. The multiple scales method and subsequent transformation to the reduced system with respect to the system energy, an arctangent of the amplitudes ratio and a difference of phases of required solutions are applied. Equilibrium positions of the reduced system correspond to nonlinear normal modes. So-called Transient nonlinear normal modes (TNNMs), which exist only for some certain levels of the system energy are selected. In the vicinity of values of time, corresponding to these energy levels, these TNNMs temporarily attract other system motions. Interaction of nonlinear vibration modes under resonance conditions is also analysed. Reliability of obtained analytical results is confirmed by numerical and numerical-analytical simulation.Документ Analysis of Traveling and Standing Waves in the DNA Model by Peyrard-Bishop-Dauxois(NTU "KhPI", 2016) Mikhlin, Yuri V.; Goloskubova, Natalia S.The model by Peyrard - Bishop - Dauxois (the PBD model), which describes the DNA molecule nonlinear dynamics, is considered. This model represents two chains of rigid disks connected by nonlinear springs. An interaction between opposite disks of different chains is modeled by the Morse potential. Solutions of equations of motion are obtained analytically in two approximations of the small parameter method for two limit cases. The first one is the long-wavelength limit of traveling waves, when frequencies of vibrations are small. Dispersion relations are obtained also for the long-wavelength limit by the small parameter method. The second case is a limit of high frequency standing waves in the form of out-of-phase vibration modes. Two such out-of-phase modes are obtained; it is selected one of them, which has the larger frequency. In both cases systems of nonlinear ODEs are obtained. Nonlinear terms are presented by the Tailor series expansion, where terms up to third degree by displacement are saved. The analytical solutions are compared with checking numerical simulation obtained by the Runge - Kutta method of the 4-th order. The comparison shows a good exactness of these approximate analytical solutions. Stability of the standing localized modes is analyzed by the numerical-analytical approach, which is connected with the Lyapunov definition of stability.