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Pozycja Basic Study of Synchronization in a Two Degree of Freedom, Duffing Based System(Wydawnictwo Politechniki Łódzkiej ; Katedra Dynamiki Maszyn - Wydział Mechaniczny Politechniki Łódzkiej, 2015) Dąbrowski, Artur; Mnich, Konrad; Stefański, AndrzejThe issue of synchronization has been analyzed from many points of view since it was first described in 17th century. Nowadays the theory that stands behind it is so developed, it might be difficult to understand where it all comes from. This paper reminds the key concepts of synchronization by examination of two coupled Duffing oscillators. It describes also the origins of Duffing equation and proves its ability to behave chaotically.Pozycja Basic Study of Synchronization in a Two Degree of Freedom, Duffing Based System(Lodz University of Technology. Faculty of Mechanical Engineering. Department Division of Dynamics., 2015) Dąbrowski, Artur; Mnich, Konrad; Stefański, AndrzejThe issue of synchronization has been analyzed from many points of view since it was first described in 17th century. Nowadays the theory that stands behind it is so developed, it might be difficult to understand where it all comes from. This paper reminds the key concepts of synchronization by examination of two coupled Duffing oscillators. It describes also the origins of Duffing equation and proves its ability to behave chaotically.Pozycja Synchronization in arrays of coupled self-induced friction oscillators(Springer, 2016-11) Marszal, Michał; Saha, Ashesh; Jankowski, Krzysztof; Stefański, Andrzej; Katedra Dynamiki Maszyn. Wydział Mechaniczny. Politechnika Łódzka.; Division of Dynamics. Faculty of Mechanical Engineering. Lodz University of Technology.We investigate synchronization phenomena in systems of self-induced dry friction oscillators with kinematic excitation coupled by linear springs. Friction force is modelled according to exponential model. Initially, a single degree of freedom mass-spring system on a moving belt is considered to check the type of motion of the system (periodic, non-periodic). Then the system is coupled in chain of identical oscillators starting from two, up to four oscillators. A reference probe of two coupled oscillators is applied in order to detect synchronization thresholds for both periodic and non-periodic motion of the system. The master stability function is applied to predict the synchronization thresholds for longer chains of oscillators basing on two oscillator probe. It is shown that synchronization is possible both for three and four coupled oscillators under certain circumstances. Our results confirmed that this technique can be also applied for the systems with discontinuities.Pozycja Energy method for position stability analysis of critical points of robot manipulators.(Polskie Towarzystwo Mechaniki Teoretycznej i Stosowanej, 2004) Szumiński, Przemysław; Kapitaniak, Tomasz; Stefański, AndrzejAn energy method for the position stability analysis of critical points (equilibrium positions) of dynamical systems with an arbitrary finite number of degrees of freedom is presented. The method proposed consists in the analysis of the potential energy balance of a perturbation of an arbitrary number of generalised co-ordinates of the dynamical system in order to establish the energy stability of individual degrees of freedom and of the whole system. In the case of mechanical systems, the position stability criterion, understood as minimisation of loads of driving systems with conservative forces, is discussed. Some examples of application of the method to the position stability analysis of critical points of one- and two-degree-of-freedom systems are analysed.Pozycja Lyapunov exponents of systems with noise and fluctuating parameters.(Polskie Towarzystwo Mechaniki Teoretycznej i Stosowanej, 2008) Stefański, AndrzejThis paper deals with the problem of determination of Lyapunov exponents in dynamical systems with noise or fluctuating parameters. The method for identifying the character of motion in such systems is proposed. This approach is based on the phenomenon of complete synchro- nization in double-oscillator systems via diagonal, master-slave coupling between them. The idea of effective Lyapunov exponents is introduced for quantifying the local stability in the presence of noise. Examples of the method application and its comparison with bifurcation diagrams representing the system dynamics are demonstrated. Finally, the pro- perties of the method are discussed.Pozycja Energy balance of two synchronized self-excited pendulums with different masses.(Polskie Towarzystwo Mechaniki Teoretycznej i Stosowanej, 2012) Kapitaniak, Tomasz; Czołczyński, Krzysztof; Perlikowski, Przemysław; Stefański, AndrzejWe consider the synchronization of two self-excited pendulums with different masses. We show that such pendulums hanging on the same beam can show almost-complete (in-phase) and almost-antiphase synchronizations in which the difference of the pendulums displacements is small. Our approximate analytical analysis allows one to derive the synchronization conditions and explains the observed types of synchronizations as well as gives an approximate formula for amplitudes of both the pendulums and the phase shift between them. We consider the energy balance in the system and show how the energy is transferred between the pendulums via the oscillating beam allowing synchronization of the pendulums.