Dynamics of a mass-spring-pendulum system with vastly different frequencies

Hiba Sheheitli, Richard H. Rand

Research output: Contribution to journalArticlepeer-review

23 Scopus citations

Abstract

We investigate the dynamics of a simple pendulum coupled to a horizontal mass-spring system. The spring is assumed to have a very large stiffness value such that the natural frequency of the mass-spring oscillator, when uncoupled from the pendulum, is an order of magnitude larger than that of the oscillations of the pendulum. The leading order dynamics of the autonomous coupled system is studied using the method of Direct Partition of Motion (DPM), in conjunction with a rescaling of fast time in a manner that is inspired by the WKB method. We particularly study the motions in which the amplitude of the motion of the harmonic oscillator is an order of magnitude smaller than that of the pendulum. In this regime, a pitchfork bifurcation of periodic orbits is found to occur for energy values larger that a critical value. The bifurcation gives rise to nonlocal periodic and quasi-periodic orbits in which the pendulum oscillates about an angle between zero and π/2 from the down right position. The bifurcating periodic orbits are nonlinear normal modes of the coupled system and correspond to fixed points of a Poincare map. An approximate expression for the value of the new fixed points of the map is obtained. These formal analytic results are confirmed by comparison with numerical integration.

Original languageEnglish (US)
Pages (from-to)25-41
Number of pages17
JournalNonlinear Dynamics
Volume70
Issue number1
DOIs
StatePublished - Oct 2012
Externally publishedYes

Keywords

  • Bifurcations
  • Coupled oscillators
  • DPM
  • Method of direct partition of motion
  • WKB method

Fingerprint

Dive into the research topics of 'Dynamics of a mass-spring-pendulum system with vastly different frequencies'. Together they form a unique fingerprint.

Cite this