10 research outputs found

    Non-conventional phase attractors and repellers in weakly coupled autogenerators with hard excitation

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    In our earlier studies, we found the effect of non-conventional synchronization, which is a specific type of nonlinear stable beating in the system of two weakly coupled autogenerators with hard excitation given by generalized van der Pol-Duffing characteristics. The corresponding synchronized dynamics are due to a new type of attractor in a reduced phase space of the system. In the present work, we show that, as the strength of nonlinear stiffness and dissipation are changing, the phase portrait undergoes a complicated evolution leading to a quite unexpected appearance of difficult to detect repellers separating a stable limit cycle and equilibrium points in the phase plane. In terms of the original coordinates, the limit cycle associates with nonlinear beatings while the stationary points correspond to the stationary synchronous dynamics similar to the so-called nonlinear local modes

    Guidance of the resonance energy flow in the mechanism of coupled magnetic pendulums

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    This paper presents a methodology of controlling the resonance energy exchange in mechanical system consisting of two weakly coupled magnetic pendulums interacting with the magnetic field generated by coils placed underneath. It is shown that properly guided magnetic fields can effectively change mechanical potentials in a way that the energy flow between the oscillators takes the desired direction. Studies were considered by using a specific set of descriptive functions characterizing the total excitation level, its distribution between the pendulums, and the phase shift. The developed control strategies are based on the observation that, in the case of antiphase oscillation, the energy is moving from the pendulum subjected to the repelling magnetic field, to the oscillator under the attracting field. In contrast, during the inphase oscillations, the energy flow is reversed. Therefore, closed-loop controller requires only the information about phase shift, which is easily estimated from dynamic state signals through the coherency index. Advantage of suggested control strategy is that the temporal rate of inputs is dictated by the speed of beating, which is relatively slow compared to the carrying oscillations

    Nonlinear dynamics: between linear and impact limits

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    This book examines nonlinear dynamic analyses based on the existence of strongly nonlinear but simple counterparts to the linear models and tools. Discusses possible application to periodic elastic structures with non-smooth or discontinuous characteristics

    Impact modes in discrete vibrating systems with rigid barriers

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    International audienceA class of strongly non-linear vibrating systems composed of linear elastic structures under absolutely rigid constraints condition is considered. Impact mode exact solutions are expressed through a saw-tooth time argument and, as a result, represented in a closed unit form. Based on this special representation, su$cient conditions of existence and also non-existence for the impact modes are formulated and interpreted on the spectral axes. In particular, it was shown that impact modes exist when their basic frequencies are shifted into the right small neighborhood of any natural frequency of the linearized (no barriers) system. The frequencies of the localized impact mode solutions are located at the right of the linear spectrum and have no upper boundary

    Design of Energy Absorbing Metamaterials Using Stochastic Soft-Wall Billiards

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    Physical principles for designing cellwise artificial materials with energy-absorbing/harvesting and wave guiding properties are discussed in the present work. We analyzed the evolution of waves in a one-dimensional lattice of 3D massive potential wells with light particles inside. The potential wells were coupled with elastic springs and represented soft-wall versions of the so-called stochastic billiards. A billiard could switch from repelling to the stadium type as the parameter of shape changed its sign from positive to negative. We found that certain shapes of the potential wells/containers provided a one-directional trend of the energy flow from the chain of containers into the chaotically moving light inclusions by increasing their total kinetic energy. As a result, propagating waves became trapped by giving rise to standing waves with chaotic mode shapes with decaying amplitudes

    Closed Form Solutions for Nonlinear Oscillators Under Discontinuous and Impulsive Periodic Excitations

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    Periodic responses of linear and nonlinear systems under discontinuous and impulsive excitations are analyzed with non-smooth temporal transformations incorporating temporal symmetries of periodic processes. The related analytical manipulations are illustrated on a strongly nonlinear oscillator whose free vibrations admit an exact description in terms of elementary functions. As a result, closed form analytical solutions for the non-autonomous strongly nonlinear case are obtained. Conditions of existence for such solutions are represented as a family of period-amplitude curves. The family is represented by different couples of solutions associated with different numbers of vibration half cycles between any two consecutive pulses. Poincaré sections showed that the oscillator can respond quite chaotically when shifting from the period-amplitude curves

    Normal oscillations of a string with concentrated masses on non-linearly elastic supports

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    International audienceThe problem of constructing and classifying normal oscillations of a string with concentrated masses on non-linearly elastic supports is considered (special limit cases are linear and vibro-impact systems). It is shown that in the limit of intensive impact action, the non-linear system has supplementary properties of symmetry which enables this problem to be solved effectively. On the other hand, the normal oscillations of a vibro-impact system can be used as the generating solutions for dynamic calculations of essentially non-linear systems that are close to them. The connection between localized normal oscillations and solutions of the soliton type are discussed
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