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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Stability Boundaries of Period-1 Rotation for a Pendulum Under Combined Vertical and Horizontal Excitation

Journal of Applied Nonlinear Dynamics 2(2) (2013) 103--126 | DOI:10.5890/JAND.2013.04.001

B. Horton$^{1}$, S. Lenci$^{2}$, E. Pavlovskaia$^{1}$, F. Romeo$^{3}$, G. Rega$^{3}$, M. Wiercigroch$^{1}$

$^{1}$ Centre for Applied Dynamics Research, School of Engineering, University of Aberdeen, Kings College, Aberdeen, AB24 3UE, UK

$^{2}$ Università Politecnica delle Marche, Dipartimento di Ingegneria Civile, Edile e Architettura (DICEA), Via Brecce Bianche, 60131 Ancona, Italy

$^{3}$ Sapienza Università di Roma, Dipartimento di Ingegneria Strutturale e Geotecnica (DISG), Via A. Gramsci 53, 00197 Roma, Italy

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Abstract

The aim of this work is to study the dynamics of pendulum driven through its pivot moving in both horizontal and vertical directions. It expands the results obtained for the parametric pendulum by Lenci et al. to two other cases, i.e. the elliptically excited pendulum and the pendulum, with an inclined rectilinear base motion (the tilted pendulum). Here we derive approximate analytical expressions representing the position of the saddle-node bifurcation associated with period-1 rotations in the excitation amplitude/frequency plane in the presence of damping by using the perturbation method proposed by Lenci et al. This includes development of a procedure for deducing expressions for the period doubling, creating a pair of stable period-2 rotational attractors. The obtained approximations are plotted on the excitation parameters plane and compared with numerical results. Simple Padé approximations for the analytical expressions relating to the position of the saddle-node bifurcation are also obtained.

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