Limiting phase trajectories and energy exchange between anharmonic oscillator and external force

Abstract

We present a simple analytical description of transient vibrations of forced anharmonic oscillator in terms of periodic non-smooth functions. Such a description (in chosen variables) is similar to that of a vibro-impact process. The main attention is paid to properties of the limiting phase trajectories (LPTs) describing intensive energy exchange between the oscillator and external force. We show that two dynamical transitions occur in the undamped system under consideration with increasing parameter of nonlinearity. The first transition consists in fast change of slow modulation period and drastic change of amplitude. This transition is characterized by appearance of dynamics resembling vibro-impact behavior in appropriate coordinates (amplitude and phase shift between the oscillator and external force). It corresponds to qualitative transformation of the LPT. The second transition corresponds to annihilation of two stationary points. Numerical study confirms our analytical results.

Limiting phase trajectories and non-stationary resonance oscillations of the Duffing oscillator. Part 1. A non-dissipative oscillator

“…Recently, the LPT ideas have been applied to the analysis of resonance-forced vibration [3,4] but an explicit expression of the LPT of a single oscillator in terms of elliptic functions is prohibitively difficult for practical utility. The purpose of the present paper is to show that a closed-form asymptotic solution can be obtained for a class of problems that are associated with the maximum energy exchange, and introduce relevant techniques.…”

Non-stationary resonance dynamics of a nonlinear sonic vacuum with grounding supports

“…This represents a second independent first integral of motion of the slow flow (15), in addition to the energy-conservation first integral (16). Before proceeding to a more detailed analytical treatment of the dynamics on the two-torus (19a) and (19b) we discuss the bifurcations that occur in this system as the parameter μ changes.…”

“…(9). As shown in [7] and then in [8,[15][16][17][18], a unified description of highly non-stationary resonance dynamics can be performed by introducing the concept of LPT. This concept is important for understanding resonant energy exchange and non-stationary energy localization phenomena caused by the strong geometric nonlinearity in the system.…”

Non-smooth approximations of the limiting phase trajectories for the Duffing oscillator near 1:1 resonance

“…Recent studies [2] have shown that the NNM approach is effective in the case of weak energy exchange, while the concept of the limiting phase trajectories (LPT) can be used to describe intense energy exchange between weakly coupled oscillators or oscillatory chains, or targeted energy transfer from a source of energy to an oscillator. Recently, the LPT concept has been applied to study resonance energy transfer from a source of harmonic excitation to the Duffing oscillator, and the precise solution of the LPT problem have been obtained [3,4]. However, the precise solution expressed in terms of elliptic functions is * Corresponding author.…”