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    Leonid Manevitch

    Kosygina st. 4, Moscow, 117977, Russia
    N. N. Semenov Institute of Chemical Physics, Russian Academy of Sciences


    Smirnov V., Kovaleva M., Manevitch L. I.
    Nonlinear Dynamics of Torsion Lattices
    2018, Vol. 14, no. 2, pp.  179-193
    We present an analysis of torsion oscillations in quasi-one-dimensional lattices with periodic potentials of the nearest neighbor interaction. A one-dimensional chain of point dipoles (spins) under an external field and without the latter is the simplest realization of such a system. We obtained dispersion relations for the nonlinear normal modes for a wide range of oscillation amplitudes and wave numbers. The features of the short wavelength part of the spectrum at large-amplitude oscillations are discussed. The problem of localized excitations near the edges of the spectrum is studied by the asymptotic method. We show that the localized oscillations (breathers) appear near the long wavelength edge, while the short wavelength edge of the spectrum contains only dark solitons. The continuum limit of the dynamic equations leads to a generalization of the nonlinear Schrödinger equation and can be considered as a complex representation of the sine-Gordon equation.
    Keywords: essentially nonlinear systems, coupled pendulums, nonlinear normal modes, limiting phase trajectories
    Citation: Smirnov V., Kovaleva M., Manevitch L. I.,  Nonlinear Dynamics of Torsion Lattices, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 2, pp.  179-193
    Kovaleva M., Smirnov V., Manevitch L. I.
    An analysis is presented of the nonlinear dynamics of harmonically coupled pendulums without restrictions to oscillation amplitudes. This is a basic model in many areas of mechanics and physics (paraffin crystals, DNA molecules etc.). Stationary solutions of equations of motion corresponding to nonlinear normal modes (NNMs) are obtained. The inversion of the NNM frequencies with increasing oscillation amplitude is found. An essentially nonstationary process of the resonant energy exchange is described in terms of limiting phase trajectories (LPTs), for which an effective analytic representation is obtained in slow time-scale. Explicit expressions of threshold values of dimensionless parameters are found which correspond to the instability of NNMs and to the transition (in parametric space) from the full energy exchange between the pendulums to the localization of energy. The analytic results obtained are verified by analysis of the Poincar´e sections describing evolution of the initial system.
    Keywords: essentially nonlinear systems, coupled pendulums, nonlinear normal modes, limiting phase trajectories
    Citation: Kovaleva M., Smirnov V., Manevitch L. I.,  Stationary and nonstationary dynamics of the system of two harmonically coupled pendulums, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 1, pp.  105-115
    Koroleva (Kikot) I. P., Manevitch L. I.
    We present results of analytical and numerical investigation of the nonstationary planar dynamics of a string with uniformly distributed discrete masses without preliminary tension and taking into account the bending stiffness. Each mass is coupled to the ground by lateral springs without tension which have (effectively) a characteristic that is nonlinearizable in the case of planar motion. The most important limiting case corresponding to low-energy transversal motions is considered taking into account geometrical nonlinearity. Since such excitations are described by approximate equations where cubic elastic forces contribute the most, oscillations take place under conditions close to the acoustic vacuum. We obtain an adequate analytical description of resonant nonstationary processes in the system under consideration, which correspond to an intensive energy exchange between its parts (clusters) in the domain of low frequencies. Conditions of energy localization are given. The analytical results obtained are supported by computer numerical simulations. The system considered may be used as an energy sink of enhanced effectiveness.
    Keywords: nonlinear dynamics, nonlinear normal mode, limiting phase trajectory, energy exchange, localization
    Citation: Koroleva (Kikot) I. P., Manevitch L. I.,  Oscillatory chain with elastic supports and bending stiffness under conditions close to acoustic vacuum, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 3, pp.  311-325
    Koroleva (Kikot) I. P., Manevitch L. I.
    In this work we investigate dynamics of a string with uniformly distributed discrete masses without tension both analytically and numerically. Each mass is also coupled to the ground through lateral spring which provides effect of cubic grounding support. The most important limiting case of low-energy transversal oscillations is considered accounting for geometric nonlinearity. Since such oscillations are governed by motion equations with purely cubic stiffness nonlinearities, the chain behaves as a nonlinear acoustic vacuum.We obtained adequate analytical description of resonance non-stationary processes in the system which correspond to intensive energy exchange between parts (clusters) of the chain in low-frequency domain. Conditions of energy localization are given. Obtained analytical results agree well with results of computer simulations. The considered system is shown to be able to be used as very effective energy sink.
    Keywords: nonlinear dynamics, nonlinear normal mode, limiting phase trajectory, energy exchange, localization
    Citation: Koroleva (Kikot) I. P., Manevitch L. I.,  Oscillatory chain with grounding support in conditions of acoustic vacuum, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 3, pp.  487-502
    Koroleva (Kikot) I. P., Manevitch L. I.
    A weightless string without preliminary tension with two symmetric discrete masses, which are influenced by elastic supports with cubic characteristics, is investigated both by numerical and analytical methods. The most important limit case corresponding to domination of resonance lowenergy transversal oscillations is considered. Since such oscillations are described by approximate equations only with cubic terms (without linear ones), the transversal dynamics occurs n the conditions of acoustic vakuum. If there is no elastic supports nonlinear normal modes of the system under investigation coincide with (or are close to) those of corresponding linear oscillator system. However within the presence of elastic supports one of NNM can be unstable, that causes formation of two another assymmetric modes and a separatrix which divides them. Such dynamical transition which is observed under certain relation between elastic constants of the string and of the support, relates to stationary resonance dynamics. This transition determines also a possibility of the second dynamical transition which occurs when the supports contribution grows. It relates already to non-stationary resonance dynamics when the modal approach turns out to be inadequate. Effective description of both dynamical transitions can be attained in terms of weakly interacting oscillators and limiting phase trajectories, corresponding to complete energy echange between the oscillators.
    Keywords: string with discrete masses, elastic support, nonlinear dynamics, asymptotical method, complete energy exchange, limiting phase trajectory, energy localization
    Citation: Koroleva (Kikot) I. P., Manevitch L. I.,  Weakly coupled oscillators in the presence of elactic support in the conditions of acoustic vacuum, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 3, pp.  245-263

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