LIMITING PHASE TRAJECTORIES AND TRANSIENT RESONANCE OSCILLATIONS IN 1 AND 2 DOF ASYMMETRIC NONLINEAR SYSTEMS

The concept of limiting phase trajectories (LPT) has been introduced by one of the authors to describe intensive energy exchange between weakly coupled oscillators or oscillatory chains. It turns out that LPT can be considered as an alternative to nonlinear normal modes (NNMs), which are characterized by conservation of energy. LPT (in the introduced coordinates) describes the vibroimpact-type process with saw-tooth amplitude and a discontinuous derivative. It was shown earlier that this concept could also be extended to systems with one degree of freedom (DoF). In this case energy exchange between the oscillator and the source of energy can occur. In this paper, we generalize the above results in several ways, namely: (1) a consideration of the asymmetry of elastic potential; (2) a detailed description of the origin of vibroimpact-type behavior and the transition from nonresonant nonstationary oscillations to resonant ones (3) a direct application of obtained results to transient vibrations in strongly asymmetric 2DoF systems.

Extended Lindstedt-Poincare Method With Multiple Time Scales for Nonstationary Oscillations

The paper presents the extended Lindstedt-Poincare (ELP) method, which applies multiple time scales to treat nonstationary oscillations arising in dynamical systems with cubic non-linearities. The passage through the resonance is conducted to study deviations from the stationary response. The method is applied to the dynamical systems such as Duffing oscillator and van der Pol oscillator, whereat effects of varying the excitation frequency and varying the excitation amplitudes, respectively are studied. It is shown that application of multiple scales benefits to find more accurate expressions of stationary responses in comparison to the conventional Lindstedt-Poincare method and consequently contributes to the versatile and effective calculation of the nonstationary frequency response curves.