International audienceThe nonlinear two-degrees-of-freedom system under consideration consists of a linear oscillator with a relatively big mass which is an approximation of some continuous elastic system, and an essentially nonlinear oscillator with a relatively small mass which is an absorber of the main linear system vibrations. Free and forced vibrations of the system are investigated. Analysis of nonlinear normal vibration modes shows that a stable localized vibration mode, which provides the vibration regime appropriate for an absorption, exists in a large region of the system parameters.In this regime amplitudes of vibrations of the main elastic system are small; simultaneously vibrations of the absorber are significant.Frequency response of the system under external periodic force is obtained. The dynamical interaction of elastic string under impact impulse and the essentially nonlinear absorber is considered too. Absorption of a longitudinal traveling wave in the system is analyzed

A nonlinear system with two degrees of freedom is considered. The system consists of an oscillator with relatively large mass, which approximates some continuous elastic system, and an oscillator with relatively small mass, which damps the vibrations of the elastic system. A modal analysis reveals a local stable mode that exists within a rather wide range of system parameters and favors vibration damping. In this mode, the vibration amplitudes of the elastic system and the damper are small and high, respectively Keywords: systems of two oscillators, vibration damper, normal modes, damping mode, stable local mode Introduction. Elastic vibration dampers (absorbers) are one of the most popular means of vibroprotection [1,3]. To create such devices, capable of protecting objects from vibrations and impacts and possessing, which is very important, small dimensions, it is necessary to solve sophisticated engineering and mathematical problems. Nonlinear passive elements can sometimes be used for vibroprotection [1]. There are a huge number of publications on vibration damping. We will analyze below only those concerned with passive dampers.Analyzing different models of machine elements, many authors (see, e.g., [3-5, 9, 13, 15, 23, etc.]) tried to find structural elements (or system parameters) that would favor damping vibrations or tuning the system away from the fundamental and sub-harmonic resonances. One of the ways to do that is to couple the system with a linear or nonlinear (which is highly efficient) one-degree-of-freedom oscillator set up so that the resonant frequency is beyond the frequency range of the system. The parameters of the system or the damper are selected using various optimization methods [2][3][4][5]9].A beam-like damper coupled with a mass-string system was examined in [6]. The resonances of a system with a nonlinear damper under forced vibrations were analyzed in [7]. A nonlinearly restrained oscillator was used in [8] to damp forced vibrations of a Duffing oscillator. In [10], a nonlinear oscillator was used to damp vibrations (auto-oscillations) in a one-degree-of-freedom system. The above-mentioned studies employed perturbation methods. The general theory of linear and nonlinear vibration dampers is outlined in [11]. The authors of [12,14] examined, theoretically and experimentally, the possibility of damping torsional vibrations in motors with pendulum dampers rotating like a centrifuge. The paper [16] addresses a semi-infinite linear chain with an essentially nonlinear spring intended to absorb vibrational energy. Vibroimpact oscillators as vibration dampers are analyzed in many publications [17, etc.]. Note that almost all results on vibration damping in elastic systems can be extended to distributed systems, since there are well-developed, more or less accurate methods for discretizing the latter [17, etc.].The present paper considers an essentially nonlinear oscillator with one degree of freedom as a passive vibration damper of some linear oscillator. The mass of the damper is assumed t...

The nonlinear two-degree-of-freedom system under consideration consists of the linear oscillator with a relatively big mass, which is an approximation of some continuous elastic system, and of the vibro-impact oscillator with a relatively small mass, which is an absorber of the linear system vibrations. Analysis of nonlinear normal vibration modes shows that a stable localized vibration mode, which provides the vibration regime appropriate for the elastic vibration absorption, exists in a large region of the system parameters. In this regime, amplitudes of vibrations of the linear system are small, simultaneously vibrations of the absorber are significant.

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