The paper proposes a criterion for determining transient behaviour in a nonlinear Duffing oscillator. For this purpose studies of specific attractors typical of the system have been conducted. Exactly defined deviation value of Δ with respect to the mean value of the surface areas bounded by the successive trajectory cycles has been assumed as the termination of the transient behaviour
Słowa kluczowe: transient trajectory, transient behavior, Duffing oscillator, nonlinear oscillations, criterion of transient behaviour.
W pracy zaproponowano kryterium wyznaczania czasu trwania procesu przejściowego w nieliniowym oscylatorze Duffinga. W tym celu badano specyficzne atraktory charakteryzujące ten układ. Za kryterium końca procesu przejściowego przyjęto ściśle zdefiniowaną wartość odchyłki Δ od wartości średniej pól powierzchni ograniczonych kolejnymi cyklami trajektorii.
Keywords: trajektoria przejściowa, proces przejściowy, oscylator Duffinga, drgania nieliniowe, kryterium stanu przejściowego
From a practical point of view, to determine the duration of transient behavior in nonlinear physical systems constitutes both an important and interesting problem. Transient behaviour occurs both in mechanical systems as well as electrical and electronic ones. In stable linear systems, it is assumed that the transient process fades after a time equal to five time constants. For an autonomous system analysis a more exact assessment of the duration of transients was presented in . It used a Lyapunov function to define time ttr, in which the trajectory of an autonomous system, starting from the initial state, reaches a specific area including the origin of the coordinate system. In that case, ttr≤-ϑlnV[x(ttr)]/V[x(0)], where V[x] is a Lyapunov function, while ϑ is the largest eigenvalue of a matrix determined from dependency . max( ( ) / ( )) x V x V x . The analysis of transient behaviour in nonlinear systems is more complicated as the superposition principle cannot be used here and, as a result, transient and steady components cannot be separated. In harmonic enforcement, nonlinear systems are characterized by the occurrence of a non-sinusoidal response. In many cases, they are systems with chaotic dynamics in which transient chaos are distinguished [2, 11]. The duration of the transient behaviour depends, among others, on the choice of initial conditions and the values of the system parameters. A great number of physical phenomena are modeled by basic differential equations. For example, vibrations in electrical, electronic and mechanical systems can be analyzed using the Duffing equation [3, 4]. The system of three Lorenz equations has been used to characterize the convective movement occurring, for instance, in the Earth's atmosphere [5, 6]. Electronic oscillators with non-linear damping are described by Van der Pol's equation [7, 8]. Below, we present a brief overview of [...]
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