The Open Mechanical Engineering Journal

2018, 12 : 108-123
Published online 2018 April 30. DOI: 10.2174/1874155X01812010108
Publisher ID: TOMEJ-12-108

RESEARCH ARTICLE
The Dynamics of One Way Coupling in a System of Nonlinear Mathieu Equations

Alexander Bernstein1 , Richard Rand2, * and Robert Meller3
1 Center for Applied Mathematics, Cornell University, Ithaca, NY 14850, USA
2 Dept. of Mathematics and Dept. of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14850, USA
3 Cornell Lab for Accelerator-based Sciences and Education, Cornell University, Ithaca, NY 14850, USA

* Address correspondence to this authors at the Dept. of Mathematics, Cornell University, Ithaca, NY 14850, USA; Tel: 607 255 8198; E-mails: , rhr2@cornell.edu

ABSTRACT

Background:

This paper extends earlier research on the dynamics of two coupled Mathieu equations by introducing nonlinear terms and focusing on the effect of one-way coupling. The studied system of n equations models the motion of a train of n particle bunches in a circular particle accelerator.

Objective:

The goal is to determine (a) the system parameters which correspond to bounded motion, and (b) the resulting amplitudes of vibration for parameters in (a).

Method:

We start the investigation by examining two coupled equations and then generalize the results to any number of coupled equations. We use a perturbation method to obtain a slow flow and calculate its nontrivial fixed points to determine steady state oscillations.

Results:

The perturbation method reveals the existence of an upper bound on the amplitude of steady state oscillations.

Conclusion:

The model predicts how many bunches may be included in a train before instability occurs.

Keywords:

Parametric vibrations, Coupled oscillators, Mathieu’s equation, Synchrotron, Bifurcation theory, Perturbation methods.