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.
Highlights
In this paper, we investigate the dynamics of the following system of n nonlinear Mathieu equations: x1 + (δ + cos t)x1 + γx31 + μx 1 = 0 (1)xi + (δ + cos t)xi + γx3i + μxi = αxi−1, 2 ≤ i ≤ n (2)1874-155X/18 2018 Bentham OpenThe Dynamics of One Way CouplingThe Open Mechanical Engineering Journal, 2018, Volume 12 109The Mathieu equation is a Hill equation with only one harmonic mode and is an example of parametric excitation
The perturbation method reveals the existence of an upper bound on the amplitude of steady state oscillations
We investigate the dynamics of the following system of n nonlinear Mathieu equations: x1 + (δ + cos t)x1 + γx31 + μx 1 = 0 (1)
Summary
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 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
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