Sub-Harmonic Response Analysis of Nonlinear Dynamic Behaviors Induced by Piecewise-Type Nonlinearities in a Torsional Vibratory System

Jong-Yun Yoon,Byeongil Kim

doi:10.3390/app12041845

Abstract

Piecewise-type nonlinearities, such as clutch dampers in a torsional system, induce complex nonlinear dynamic behaviors that resemble super- and sub-harmonic responses. This study focuses on investigating the sub-harmonic responses induced by piecewise-type nonlinearities in the middle of various dynamic behaviors in a torsional vibratory system. To examine the dynamic characteristics in a sub-harmonic regime, the harmonic balance method (HBM) was implemented. Its results were compared with the numerical simulation (NS). To reveal the sub-harmonic responses, the input conditions of the HBM were modified with a small number of input values. In addition, bifurcation diagrams were numerically determined and projected onto stable and unstable solutions of the HBM to examine the effective dynamic behaviors within the unstable regimes. The results of the HBM with the modified input conditions reveal the sub-harmonic effects well, and the comparisons of bifurcation diagrams under unstable conditions lead to an understanding of the complex dynamic behaviors. Overall, this study suggests the first analytical technique to determine the sub-harmonic responses with the HBM, and second investigates the complex dynamic behaviors in a practical vibratory system by considering the bifurcations in the unstable regimes.

Highlights

Piecewise-type nonlinearities such as multi-staged clutch dampers used in a practical torsional system induce highly complex dynamic responses
In the middle of these nonlinear dynamic behaviors, sub-harmonic responses are relatively difficult to detect by employing the harmonic balance method (HBM), the basic matrix of which is constructed by the integer-based, incremental formulations [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]
Nonlinear frequency response calculations of a torsional system with clearance-type nonlinearity have been developed by employing a multiterm HBM [1,2,5,8,10,12]

Summary

Introduction

Piecewise-type nonlinearities such as multi-staged clutch dampers used in a practical torsional system induce highly complex dynamic responses. Various modifications have been implemented to calculate sub-harmonic responses. Nonlinear frequency response calculations of a torsional system with clearance-type nonlinearity have been developed by employing a multiterm HBM [1,2,5,8,10,12]. Various prior studies have discussed nonlinear problems by employing the HBM with respect to super- and sub-harmonic responses [2,7,10,12]. Despite the successful determination of the sub-harmonic responses, the stability conditions of the HBM often cannot adequately explain the practical dynamic conditions. AA ssiinnggllee--ddeeggrreeee--ooff--ffrreeeeddoomm ssyysstteemm wwiitthh ppiieecceewwiissee ttyyppee nnoonnlliinneeaarriittiieess:: ((aa)) aa nnoonnlliinneeaarr ttoorrssiioonnaall ssyysstteemm mmooddeell wwiitthh 11DDOOFF;; ((bb)) TToorrqquueeTTCC((δδ11)) pprrooffiillee ffoorr aa mmuullttii--ssttaaggeedd cclluuttcchh ddaammppeerr [[11,,22]]

Problem Formulation with Multi-Staged Clutch Dampers

Initial Results of the Basic HBM and Modification of the Input Conditions

Dynamic Characteristics with Bifurcation Diagrams in Sub-Harmonic Regimes