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The conservative and dissipative dynamics of a two-degree-of-freedom system composed of a grounded linear oscillator coupled to a lightweight mass by means of both strongly nonlinear and linear negative stiffnesses is investigated. Numerical studies are presented aiming to assess the influence of this combined coupling on the transient dynamics. In particular, these studies are focused on passive nonlinear targeted energy transfer from the impulsively excited linear oscillator to the nonlinear bistable lightweight attachment. It is shown that the main feature of the proposed configuration is the ability of assuring broadband efficient energy transfer over a broad range of input energy. Due to the bistability of the attachment, such favorable behavior is triggered by different nonlinear dynamic mechanisms depending on the energy level. For high energy levels, strongly modulated oscillations occur, and the dynamics is governed by fundamental (1:1) and superharmonic (1:3) resonances; for low energy levels, chaotic cross-well oscillations of the nonlinear attachment as well as subharmonic resonances lead to strong energy exchanges between the two oscillators. The results reported in this work indicate that properly designed attachments of this type can be efficient absorbers and dissipators of impulsively induced vibration energy

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