Dynamic Analysis of a Two-Mass System with Essentially Nonlinear Vibration Damping

Abstract

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...

Nonlinear forced vibrations of a cylindrical shell with two internal resonances

“…Note that system (15) is independent of Eqs. (16) and (17). Equation (16) is also independent, and Eq.…”

“…Free and forced nonlinear vibrations of cylindrical shells were also addressed in [15,16]. Damping of free vibrations in linear systems and the dynamic behavior of systems with nonlinear vibration damping were studied in [14,17].We will study forced nonlinear vibrations of a cylindrical shell without initial imperfections and with two internal resonances. The associated vibration equations have been derived in the monograph [7], but the vibrations themselves with two internal resonances have not been examined therein.…”

“…Free and forced nonlinear vibrations of cylindrical shells were also addressed in [15,16]. Damping of free vibrations in linear systems and the dynamic behavior of systems with nonlinear vibration damping were studied in [14,17].…”

On the nonlinear dynamics of elastically interacting asymmetric rigid bodies

“…The adapter modules are modeled by inertialess rectilinear elastic bars of certain length and cross section embedded in the bodies. Such dynamic models are of importance since they allow us to analyze the dynamics of an SV system with allowance for the elastic interaction between them during maneuvers and to estimate the dynamic loading on both the space vehicle [1] and the adapter module [2].Design models in the form of connected rigid bodies are widely used to solve applied dynamic problems for complex engineering structures [3,4,[11][12][13][14]. Such dynamic problems can mainly be solved by conducting a computational experiment [5].…”

“…Design models in the form of connected rigid bodies are widely used to solve applied dynamic problems for complex engineering structures [3,4,[11][12][13][14]. Such dynamic problems can mainly be solved by conducting a computational experiment [5].…”