Dynamics of a Linear Oscillator Coupled to a Bistable Light Attachment: Analytical Study

We present an analytical study of the conservative and dissipative dynamics of a two-degree-of-freedom (DOF) system consisting of a linear oscillator coupled to a bistable light attachment. The main objective of the paper is to study the beneficial effect of the bistability on passive nonlinear targeted energy transfer from the impulsively excited linear oscillator to an appropriately designed attachment. As a numerical study of the problem has shown in a companion paper (Romeo, F., Sigalov, G., Bergman, L. A., and Vakakis, A. F., 2013, “Dynamics of a Linear Oscillator Coupled to a Bistable Light Attachment: Numerical Study,” J. Comput. Nonlinear Dyn. (submitted)) there is an essential difference in the system's behavior when compared to the conventional case of a monostable attachment. On the other hand, some similarity to the behavior of an oscillator with rotator attachment has been revealed. It relates, in particular, to the generation of nonconventional nonlinear normal modes and to the existence of two qualitatively different types of dynamics. We find that all numerical results can be explained in the framework of fundamental (1:1) and superharmonic (1:3) resonances (for large energies), as well as a subharmonic resonance (for small energies). This allows us to use the concept of limiting phase trajectories (LPTs) introduced earlier by one of the authors, and to derive accurate analytical approximations to the dynamics of the problem in terms of nonsmooth generating functions.

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.