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ISSN: 2475-4811 (print)
ISSN: 2475-482X (online)
Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland

Email: pawel.olejnik@p.lodz.pl

C. Steve Suh (editor)

Texas A&M University, USA

Email: ssuh@tamu.edu

Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China

Email: tuoxianguo@suse.edu.cn

Complete Bifurcation Trees of a Parametrically Driven Pendulum

Journal of Vibration Testing and System Dynamics 1(2) (2017) 93--134 | DOI:10.5890/JVTSD.2017.06.001

Yu Guo$^{1}$; Albert C.J. Luo$^{2}$

$^{1}$ McCoy School of Engineering, Midwestern State University, Wichita Falls, TX 76308, USA

$^{2}$ Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, IL 62026-1805, USA

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Abstract

In this paper, the complete bifurcation trees of a parametrically driven pendulum are investigated using discrete implicit maps obtained through a mid-point scheme. Based on the discrete maps, mapping structures are developed for periodic motions in such a parametric system. Analytical bifurcation trees of periodic motions to chaos are developed through the nonlinear algebraic equations of such implicit maps. The stability and bifurcation of periodic motions is carried out through eigenvalue analysis. For a better understanding of the motion complexity in such a system, the corresponding frequency-amplitude characteristics are presented. Finally, numerical results of periodic motions are illustrated in verification. Many new periodic motions in the parametrically excited pendulum are discovered.

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