Synchronization of two forced double-well duffing oscillators with attached pendulums.

We investigate the dynamics of two coupled Duffing oscillators with attached pendulums forced kinematically by a common signal. Our attention is focused on different kinds of synchronization which can appear in the considered system. Different types of coupling (spring, damper and spring and damper simultaneously) are taken into account. We show in a two-parameters space (amplitude and frequency of excitation) existence of complete and phase synchronization and asynchronous ranges

Bifurcations of Oscillatory and Rotational Solutions of Double Pendulum with Parametric Vertical Excitation

This paper investigates dynamics of double pendulum subjected to vertical parametric kinematic excitation. It includes detailed bifurcation diagrams in two-parameter space (excitation’s frequency and amplitude) for both oscillations and rotations in the domain of periodic solutions.This work has been supported by the Foundation for Polish Science, Team Programme, under Project TEAM/2010/5/5

The optimization of the TMDI for efficient mitigation of the vibration

The paper concerns the optimization of a tuned mass damper with inerter (TMDI) based on two strategies, i.e., the minimum amplitude in the resonance peak and minimum area under the frequency response curve. The optimization is based on real, accessible parameters. Both optimization procedures are presented in two steps. In the first one, two parameters of the TMDI are tuned (inertance and damping coefficient), while in the second one, three parameters (mass, inertance, and damping coefficient). We show that both strategies give the optimum sets of parameters and allow the reduction of the amplitude of the damped system

Amplitude equations for collective spatio-temporal dynamics in arrays of coupled systems

We study the coupling induced destabilization in an array of identical oscillators coupled in a ring structure where the number of oscillators in the ring is large. The coupling structure includes different types of interactions with several next neighbors. We derive an amplitude equation of Ginzburg-Landau type, which describes the destabilization of a uniform stationary state and close-by solutions in the limit of a large number of nodes. Studying numerically an example of unidirectionally coupled Duffing oscillators, we observe a coupling induced transition to collective spatio-temporal chaos, which can be understood using the derived amplitude equations

Synchronization of clocks

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Synchronizacja oscylatorów mechanicznych wymuszanych kinematycznie.

In this paper, the behaviour of a system of coupled mechanical oscillators excited kinematically was studied numerically. Several methods of detection of synchronization were shown and advantages of each were mentioned. The relation between Lyapunov exponents and the synchronization state under different kinds of the excitation signal (harmonic, periodic and chaotic ones) was presented. It was demonstrated that the mode locking with excitation was dependent only on inner damping and completely uncorrelated with the connection between oscillators.W artykule pokazano numeryczną analizę dynamiki sprzężonych oscylatorów mechanicznych wymuszonych kinematycznie. Przedstawiono przegląd najważniejszych metod detekcji synchronizacji, zwracając szczególną uwagę na ich własności. Następnie opisano powiązania między widmem wykładników Lapunowa, a różnymi typami synchronizacji. Zbadano odpowiedź układu dla różnych typów sygnału wymuszającego (harmonicznego, okresowego i chaotycznego). Udowodniono, że zamykanie modów pomiędzy sygnałem oscylatora a wymuszeniem zależne jest tylko od tłumienia wewnętrznego oscylatorów, natomiast jest niezależne od sprzężeń pomiędzy nimi

Bilans energii dwóch zsynchronizowanych wahadeł samowzbudnych o różnych masach.

We 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.Artykuł prezentuje analizę zjawiska synchronizacji dwóch wahadeł samowzbudnych o różnych masach. Pokazano, że jeśli takie wahadła zostaną zawieszone na wspólnej, ruchomej podstawie, zachodzi zjawisko ich (prawie) zupełnej lub (prawie) antyfazowej synchronizacji. Analiza bilansu energetycznego układu pozwala na określenie parametrów układu w stanie synchronizacji (amplitudy drgań i przesunięcia fazowe). Analiza bilansu energetycznego wyjaśnia także mechanizm synchronizowania się ruchu wahadeł: stały przepływ strumienia energii od jednego wahadła, via wspólna ruchoma podstawa, do drugiego wahadła powoduje, że ruch układu jest okresowy, a przesunięcia fazowe pomiędzy wahadłami przyjmują stałe, charakterystyczne wartości

Amplitude equations for collective spatio-temporal dynamics in arrays of coupled systems

We study the coupling induced destabilization in an array of identical oscillators coupled in a ring structure where the number of oscillators in the ring is large. The coupling structure includes different types of interactions with several next neighbors. We derive an amplitude equation of Ginzburg-Landau type, which describes the destabilization of a uniform stationary state and close-by solutions in the limit of a large number of nodes. Studying numerically an example of unidirectionally coupled Duffing oscillators, we observe a coupling induced transition to collective spatio-temporal chaos, which can be understood using the derived amplitude equations