Energy transfer in archetipal coupled mechanical systems

with L. Manevitch, V. Smirnov, M. Kovaleva

Models of coupled pendula and some of its modifications play a significant role in mechanics, solid-state physics, photonics and biophysics. The majority of the studies in all these fields relate to stationary Hamiltonian dynamics or its extension to damped and forced models in which several peculiar dynamical phenomena can arise, such as synchronised and chimera states. They are based on the system fundamental regimes, namely the Nonlinear Normal Modes (NNMs) in finite systems and solitons (breathers) in infinite models. The recently developed concept of Limiting Phase Trajectories (LPTs) allowed for a systematic analytical approach to describe non-stationary resonance regimes. This concept introduces a fundamental non-stationary process of new type which corresponds to the maximum possible energy exchange between the oscillators (here pendula) or clusters of oscillators. In essence, the LPTs play in the nonstationary resonance dynamics of finite systems a role similar to that of the NNMs in the stationary theory and in the study of non-stationary yet non-resonant regimes.

For two weakly coupled pendula, the analytical description of highly non-stationary Hamiltonian dynamics without any restrictions on the amplitude of oscillations is presented in [1]. It is shown that such processes can be adequately described by LPTs corresponding to the maximum possible energy exchange between pendula. These regimes encircle the domains of regular motion which are determined for all initial angles in the oscillation dynamic regime. The threshold coupling governing the transition from intense inter-pendulum energy exchange to predominant energy localization in one of the two pendula was identified. The manifestation of chaotic behavior in the considered model is also shown to be strongly connected with this, purely non-stationary, dynamic transition.

For the harmonically forced pendulum, in [2], the analytical representation of the dynamical pendulum states was found without any restrictions on the oscillations amplitude. Two qualitative transitions (in parametric space) were revealed. One of them corresponds to the birth of a pair of additional, one stable and one unstable, stationary states and heteroclinic of the other transition

is connected with coalescence of dynamic separatrix and LPT entailing an abrupt growth of oscillation amplitude. As a result, at this purely non-stationary transition, the conditions of maximum energy that can be drawn from the source are realized. Furthermore, we have shown that chaotization of dynamic trajectories is clearly manifested in the vicinity of the non-stationary transition.

New phenomena of energy localization and transition to chaos in the finite system of coupled pendula (which is a particular case of the Frenkel-Kontorova model), without any restrictions on the amplitudes of oscillations, were discussed in [3]. Group of pendula with similar displacements represent the so called “coherent domains”. The periodic change of the predominant displacement of the pendula from one coherent domain to another is the consequence of the eigenfrequency difference in the exactly linear chain, in analogy with the classic beating process in a system of two weakly coupled oscillators. In the nonlinear system this process is not ordinary and may depend on the system parameters and vibration amplitudes. The instability and localization “effective” thresholds for the chains with various lengths are analytically identified.

By combining complexification and LPT concepts, in [4] is shown that the non-stationary as well as the stationary dynamics for arbitrary pendulum oscillation amplitude can be described. In the quasi-linear approximation, the conditions for the appearance of unstable stationary regimes are determined; for large oscillations, the frequency-amplitude dependence for parametrically forced pendulum are described. The nonstationary analysis for large amplitude oscillations is eventually extended to the non-conservative case. For large amplitude regimes we show the non-applicability of the quasilinear case, while the proposed semi-inverse method yields an adequate description of the resonant process.

[1] Manevitch L.I., Romeo F., Non-stationary resonance dynamics of weakly coupled pendula, EPL (Europhysics Letters), vol. 112, 30005, 2015.

[2] Manevitch L.I., Smirnov V.V., Romeo F., Non-stationary resonance dynamics of the harmonically forced pendulum, Cybernetics and Physics, vol. 5, 3, 91-95, 2016.

[3] Manevitch L.I., Smirnov V.V., Romeo F., Stationary and non-stationary resonance dynamics of the finite chain of coupled pendula, Cybernetics and Physics, vol. 5, 4, 130-135, 2016.

[4] Kovaleva M., Manevitch L.I., Romeo F., Stationary and Non-stationary Oscillatory Dynamics of the Parametric Pendulum, Comm. Nonl. Sc. Numer. Simul., in press.

Pendulum systems: a) two weakly coupled pendula; b) chain of N weakly coupled pendula; c) pendulum under harmonic torque; d) parametrically excited pendulum.

Evolution of the dynamic transitions in the (ω-ε) parameter space. The upper horizontal axis shows the maximum angle q for the corresponding ω in the lower axis. (a) First and second dynamic transitions, analytical prediction (I, II) and numerical observation (I*, II*); (b) LPTs phase plane; (c) temporal behavior of θ colored according to the trajectories shown in (b).

Evolution of the Δ-Q phase portrait for f = 0.06 and s = 0.1 for increasing forcing frequency ω. (a) Before the non-stationary transition, ω = 0.79; (b) at the non-stationary transition, ω = 0.8106; (c) after the non-stationary transition, ω = 0.82; (d) after the stationary transition, ω = 0.85.

Poincaré sections of the fully nonlinear pendulum system in the neighborhood of the main parametric resonance: a) frequency is below the resonant area; b) frequency is inside the resonant area; c) field frequency is above the resonant area.

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