Vibration analysis of nonlinear systems with the bilinear hysteretic oscillator by using incremental harmonic balance method

doi:10.1016/j.cnsns.2016.06.005

Communications in Nonlinear Science and Numerical Simulation

Volume 42, January 2017, Pages 437-450

https://doi.org/10.1016/j.cnsns.2016.06.005 Get rights and content

Highlights

•
The iteration formulation of the improved IHB method considered the bilinear hysteresis is presented. It is adequate for not only other hysteretic models but strongly nonlinearities.
•
The stability analysis of Floquet multipliers considered bilinear hysteresis is investigated.
•
Two points tracing algorithm modified from arc-length increment method is proposed.
•
The effects of different parameters of the bilinear hysteretic absorber are studied numerically.

Abstract

This paper considers dynamics of bilinear hysteretic systems, which are widely used for vibration control and vibration absorption such as magneto-rheological damper, metal-rubber. The method of incremental harmonic balance (IHB) technique that hysteresis is considered in the corrective term is improved in order to determine periodic solutions of bilinear hysteretic systems. The improved continuation method called two points tracing algorithm which is stable to the turning point makes the calculation more efficient for tracing amplitude-frequency response. Precise Hsu's method for analysing the stability of periodic solutions is introduced. The effects of different parameters of bilinear hysteretic oscillator on the response are discussed numerically. Some numerical simulations of considered bilinear hysteretic systems, including a single DOF and a 2DOF system, are effectively obtained by the modified IHB method and the results compare very well with the 4-oder Runge-Kutta method.

Section snippets

Introduction

Recent theoretical [1], [2], [3], [4] and experimental [5], [6], [7] researches of systems comprising substructures and nonlinear attachments provided validation of nonlinear vibration absorption and mitigation applied to mechanical engineering. Numerous novel designs [8], [9], [10] have been proposed to bring more efficient and robust nonlinear vibration absorbers by purposeful transformation of nonlinearities. Nonlinear systems that involve hysteresis such as magneto-rheological damper,

The mathematical models of the system with a metal-rubber absorption

The profile of instrument installation plate of spacecraft coupled to a metal-rubber vibration absorber is presented in Figs. 1 and 2, simplified model of Fig. 1 has an obvious hysteretic nonlinear structure. It is clear that the relationship between restoring force and displacement is two snap stress-strain structure. Meanwhile, it is obvious that the nonlinearity of metal-rubber absorber differs from NES which has pure cubic stiffness in references [2], [3].

Therefore, mechanical model

Single-DOF nonlinear system

One of the distinctive advantages of the IHB method can be used in a wide range of frequency and it is suitable for strongly nonlinear system to obtain the nonlinear property of dynamical system. Specially, amplitude-frequency plot and stability analysis of system can be obtained effectively with some tracing algorithms and Floquet theory.

In the current section, numerical verifications of the analysis are performed. Taking a simple nonlinear systems depicted in Fig. 2 as a numerical example of

Conclusions

In this paper, vibrations analysis of a single DOF bilinear hysteretic system was investigated by the improved IHB method that the hysteretic restoring force were considered in the iteration formulation. Similarly, the hysteretic term of nonlinear system was added for analysing the stability of the periodic solutions. Although Floquet multipliers of bilinear hysteretic system have become more complicated, it could be resolved effectively by the precise Hsu' method. The basic idea of improved

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Research paperVibration analysis of nonlinear systems with the bilinear hysteretic oscillator by using incremental harmonic balance method

Highlights

Abstract

Section snippets

Introduction

The mathematical models of the system with a metal-rubber absorption

Single-DOF nonlinear system

Conclusions

J Sound Vib

J Sound Vib

Mech Syst Signal Pr

J Sound Vib

J Sound Vib

Appl Math Comput

J Differ Eqs

Nonlinear Anal-Real World Appl

Automatica

Nonlinear Anal-Real World Appl

J Sound Vib

J Sound Vib

Commun Nonlinear Sci Numer Simul

J Sound Vib

J Sound Vib

J Sound Vib

J Sound Vib

Eng Struct

Physica D

Research paper
Vibration analysis of nonlinear systems with the bilinear hysteretic oscillator by using incremental harmonic balance method