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In the present paper we report the existence of a new family of solitary waves in general one-dimensional dimer chains with elastic interactions between beads obeying a strongly nonlinear Hertzian force law. These dimers consist of pairs of "heavy" and "light" beads with no precompression. The solitary waves reported herein can be considered as analogous to the solitary waves in general homogeneous granular chains studied by Nesterenko, in the sense that they do not involve separations between beads, but rather satisfy special symmetries or, equivalently antiresonances in their intrinsic dynamics. We conjecture that these solitary waves are the direct products of a countable infinity of antiresonances in the dimer. An interesting finding is that the solitary waves in the dimer propagate faster than solitary waves in the corresponding homogeneous granular chain obtained in the limit of no mass mismatch between beads (i.e., composed of only heavy beads). This finding, which might seem counterintuitive, indicates that under certain conditions nonlinear antiresonances can increase the speed of disturbance transmission in periodic granular media, through the generation of different ways for transferring energy to the far field of these media. From a practical point of view, this result can have interesting implications in applications where granular media are employed as shock transmitters or attenuators.

A series of two papers is devoted to detailed investigation of the response regimes of linear oscillator with attached nonlinear energy sink (NES) under harmonic external forcing and assessment of possible application of the NES for vibration absorption and mitigation. The first paper of the series is devoted to analytic and numeric description of the attractors (response regimes) of the system. Analytic approach is based on averaging and multiple-scales analysis, the mass ratio being used as the small parameter. The problem of possible coexistence of different attractors is reduced to analysis of flow on slow invariant manifolds (SIM) of the system. Numeric simulation confirms the predictions of the analytic model concerning the number, the shape, and the structure of the response regimes and reveals some other features of these attractors.

This paper is the second one in the series of two papers devoted to detailed investigation of the response regimes of a linear oscillator with attached nonlinear energy sink (NES) under harmonic external forcing and assessment of possible application of the NES for vibration absorption and mitigation. In this paper, we study the performance of a strongly nonlinear, damped vibration absorber with relatively small mass attached to a periodically excited linear oscillator. We present a nonlinear absorber tuning procedure in the vicinity of (1:1) resonance which provides the best total system energy suppression, using analytical and numerical tools. A linear absorber is also tuned according to the same criterion of total system energy suppression as the nonlinear one. Both optimally tuned absorbers are compared under common parameters of damping, external forcing but different absorber stiffness characteristics; certain cases for which nonlinear absorber is preferable over the linear one are revealed and confirmed numerically.

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