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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.Документ Nonlinear normal vibration modes and associated problems(Wydawnictwo Politechniki Łódzkiej, 2019) Mikhlin, Yuri V.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 Resonance behavior of the system with limited power supply having nonlinear absorbers(Дніпровський національний університет імені Олеся Гончара, 2019) Mikhlin, Yuri V.; Onizhuk, AntonДокумент 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 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 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 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 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.