Перегляд за Автор "Mikhlin, Yuri V."
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Документ Algebraization in stability problem for stationary waves of the Klein-Gordon equation(Харківський національний університет імені В. Н. Каразіна, 2019) Goloskubova, Nataliia; Mikhlin, Yuri V.Nonlinear traveling waves of the Klein-Gordon equation with cubic nonlinearity are considered. These waves are described by the nonlinear ordinary differential equation of the second order having the energy integral. Linearized equation for variation obtained for such waves is transformed to the ordinary one using separation of variables. Then so-called algebraization by Ince is used. Namely, a new independent variable associated with the solution under consideration is introduced to the equation in variations. Integral of energy for the stationary waves is used in this transformation. An advantage of this approach is that an analysis of the stability problem does no need to use the specific form of the solution under consideration. As a result of the algebraization, the equation in variations with variable in time coefficients is transformed to equation with singular points. Indices of the singularities are found. Necessary conditions of the waves stability are obtained. Solutions of the variational equation, corresponding to boundaries of the stability/instability regions in the system parameter space, are constructed in power series by the new independent variable. Infinite recurrent systems of linear homogeneous algebraic equations to determine coefficients of the series can be written. Non-trivial solutions of these systems can be obtained if their determinants are equal to zero. These determinants are calculated up to the fifth order inclusively, then relations connecting the system parameters and corresponding to boundaries of the stability/ instability regions in the system parameter place are obtained. Namely, the relation between parameters of anharmonicity and energy of the waves are constructed. Analytical results are illustrated by numerical simulation by using the Runge-Kutta procedure for some chosen parameters of the system. A correspondence of the numerical and analytical results is observed.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.Документ Forced Nonlinear Normal Modes in the One Disk Rotor Dynamics(Institute of Mechanics and Mechatronics, Technical University of Vienna, 2014) Perepelkin, Nikolay V.; Mikhlin, Yuri V.; Pierre, ChristopheA new approach combining both the nonlinear normal modes approach and the Rauscher method is proposed to construct forced vibrations in non-autonomous systems with an internal resonance. Forced vibrations of a one-disk unbalanced rotor with the nonlinear elastic bearings are considered. Gyroscopic effects, an asymmetrical disposition of the disk in the isotropic elastic shaft and internal resonance are taken into account.Документ Forced nonlinear normal modes in the one disk rotor dynamics(Дніпровський національний університет імені Олеся Гончара, 2019) Mikhlin, Yuri V.; Perepelkin, NikolayЗ використання концепції нелінійних нормальних мод коливань розглядаються резонансні вимушені коливання однодискового ротору. Гіроскопічні ефекти, нелінійність у пружних опорах та внутрішні резонанси взято до уваги. Отримано амплітудно-частотні характеристики в околі першого резонансу.Документ Forced Resonance Vibrations of the Dissipative Spring-Pendulum System(Sapienza University of Rome, 2017) Mikhlin, Yuri V.; Plaksiy, KaterynaDynamics of the dissipative spring-pendulum system under periodic external excitation in the vicinity of external resonance and simultaneous external and internal resonances is studied. The concept of nonlinear normal vibration modes is used in this analysis. 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. Transient nonlinear normal modes, which exist only for some particular levels of the system energy, are obtained. In the vicinity of values of time, corresponding to these energy levels, the transient modes temporarily attract other system motions. Interaction of nonlinear vibration modes under resonance conditions is also analysedДокумент Non-classical nonlinear normal vibration modes in mechanical systems(Sapienza University of Rome, 2017) Mikhlin, Yuri V.; Plaksiy, Kateryna; Shmatko, Tatyana; Rudneva, GayaneNonlinear normal modes (NNMs) of forced chaotic vibrations can be found in models which are obtained by digitization of some elastic systems that have lost stability under external compressive force. A system of non-autonomous Duffing equations can be obtained; chaotic motions appear as the force amplitude is slowly increased. A stability of periodic or chaotic vibration mode in a space with a greater dimension is studied. In non-ideal dissipative systems NNMs by Kauderer-Rosenberg cannot be realized due to exponential decrease of vibration amplitude. In such systems under the resonance conditions the transient nonlinear normal modes (TNNMs), which exist only for some levels of energy, can be found. These TNNMs temporarily attract other motions of the system near values of time, corresponding to the mentioned energy levels.Документ Nonlinear Dissipative Systems in Vicinity of Internal and Forced Resonances(Institute of Mechani cs and Mechatronics, Technical University of Vienna, 2014) Plaksiy, Katerina; Mikhlin, Yuri V.Free and forced dynamics of some nonlinear dissipative systems in vicinity of internal resonance is considered. A reduced system with respect to the system energy, an arctangent of the vibration amplitudes ratio, and the phase difference is used in the analysis.Документ Nonlinear normal modes and their interaction in nonideal systems with vibration absorber(Institute of Mechanics and Mechatronics, Technical University of Vienna, 2014) Mikhlin, Yuri V.; Klimenko, A. A.; Plaksiy, K. Y.Nonlinear normal vibration modes (NNMs) of the non-ideal systems, where an interaction of source of energy and linear elastic subsystem takes place, are investigated. Systems under consideration contain the nonlinear absorber, which permits to decrease amplitudes of the elastic subsystem vibrations. Interaction of NNMs in vicinity of resonances is analyzed by using the multiple scales method and transformation to a reduced system.Документ Nonlinear normal modes of vibrating mechanical systems and their applications(Institute of Mechanics and Mechatronics, Technical University of Vienna, 2014) Mikhlin, Yuri V.; Avramov, Konstantin V.; Pierre, ChristopheThe principal concepts of nonlinear normal vibration modes (NNMs) and methods of their analysis are presented. NNMs for forced and parametric vibrations and generalization of the NNMs to continuous systems are considered. Nonlinear localization and transfer of energy are discussed in the light of NNMs. Different engineering applications of NNMs are analyzed.Документ Nonlinear normal vibration modes and associated problems(Wydawnictwo Politechniki Łódzkiej, 2019) Mikhlin, Yuri V.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 Resonance behavior of the non-ideal system which contains a snap-trough truss as absorber(Sapienza University of Rome, 2019) Mikhlin, Yuri V.; Onizhuk, Anton A.A resonance behavior of a system containing the linear oscillator, the Mises girder as absorber of elastic vibrations and the source of energy with a limited power-supply is analyzed. Stationary resonance regimes of vibrations near stable equilibrium position are considered, namely, vibrations near the resonance 1:1 between the linear oscillator and the motor, vibrations near the resonance 1:1 between the absorber and the motor. The stationary regime of snap through motion is also considered.Документ Resonance behavior of the system with limited power supply having nonlinear absorbers(Дніпровський національний університет імені Олеся Гончара, 2019) Mikhlin, Yuri V.; Onizhuk, AntonДокумент Special Issue on Nonlinear Dynamics(Sage Publications від імені Інституту інженерів-механіків, 2016) Mikhlin, Yuri V.; Cartmell, Matthew P.; Warminski, JerzyThe papers published in the Special Issue were selected by the Guest Editors, Professor Yuri Mikhlin, Professor Matthew Cartmell, and Professor Jerzy Warminski from papers presented at the mini symposia ‘‘Nonlinear Dynamics of Structural and Machine Elements’’ and ‘‘Nonlinear Phenomena in Mechanical and Structural Systems’’ in the frame work of the Eighth European Nonlinear Dynamics Conference (ENOC 2014) which took place at the Vienna University of Technology, Vienna, Austria, on 6–11 July 2014.Документ Stability of stationary regimes in nonlinear systems: analytical and numerical approaches(Sapienza University of Rome, 2019) Mikhlin, Yuri V.; Shmatko, Tatyana; Rudneva, Gayane; Goloskubova, Natalyia S.A stability of stationary regimes in the form of nonlinear normal modes (NNMs) with rectilinear or nearrectilinear trajectories is analysed by using the Ince algebraization when a variable associated with the vibration mode is chosen as the new independent argument. In this case the variational equations are transformed to equations with singular points. Other approach is realized for NNMs with regular or chaotic behavior in time. Namely, a test which is a consequence of the well-known Lyapunov criterion of stability is used. Both approaches are also used in analysis of stability of other stationary regimes, namely, standing or traveling nonlinear waves.Документ Stability of steady states with regular or chaotic behaviour in time(Wydawnictwo Politechniki Łódzkiej, 2019) Mikhlin, Yuri V.; Goloskubova, Natalyia S.; Shmatko, Tatyana V.