Dynamical interaction of an elastic system and essentially nonlinear absorber

Yuri V Mikhlin,S.N Reshetnikova

doi:10.1016/j.jsv.2004.03.061

Abstract

The 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.

Highlights

Numerous scientific publications contain a description and analysis of different devices for the vibration absorption of machines and mechanisms due to the importance of these problems in engineering
The localized normal vibration modes (NNMs) in the two-dof nonlinear system under consideration is appropriate for the absorption, when the main linear system and absorber have small and large amplitudes, respectively
Results of the NNMs stability analysis are approximate because only the harmonic approximation was used

Summary

Introduction

Numerous scientific publications contain a description and analysis of different devices for the vibration absorption of machines and mechanisms due to the importance of these problems in engineering. The localized NNM in the two-dof nonlinear system under consideration is appropriate for the absorption, when the main linear system and absorber have small and large amplitudes, respectively. The zero approximation with respect to ð 1⁄4 0Þ gives us y0 1⁄4 x þ ðc=gÞx3: This is the nonlocalized vibration mode In this regime the vibration energy is distributed both in the linear oscillator, and in the essentially nonlinear absorber, that is, the vibration amplitudes of the subsystems are comparable. By using the first approximation equation with respect to and the corresponding boundary conditions at the maximal isoenergetic surface, one obtains a solution of the form of the following power series by x: y1 1⁄4 a1x þ a3x3 þ a5x5 þ a7x7 þ Á Á Á ;.

Analysis of stability by using the Mathieu equation

Forced vibrations of system containing the essentially nonlinear absorber

Damped forced oscillations

Biharmonic approximation of the forced oscillations

Absorption of elastic waves by using the essentially nonlinear absorber

Construction of the approximate analytical solution

Conclusions