2:2:1 Resonance in the Quasiperiodic Mathieu Equation

Abstract

In this work, we investigate regions of stability in the vicinity of 2:2:1 resonance in the quasiperiodic Mathieu equation: d2xdt2+(δ+εcost+εμcos(1+εΔ)t)x=0, using two successive perturbation methods. The parameters ε and μ are assumed to be small. The parameter ε serves for deriving the corresponding slow flow differential system and μ serves to implement a second perturbation analysis on the slow flow system near its proper resonance. This strategy allows us to obtain analytical expressions for the transition curves in the resonant quasiperiodic Mathieu equation. We compare the analytical results with those of direct numerical integration. This work has application to parametrically excited systems in which there are two periodic drivers, each with frequency close to twice the frequency of the unforced system.

Cross-Coupling Errors of Micromachined Gyroscopes

“…In (9), are the components of the total moment applied to the rotor, and are the principal components of the rotor moment of inertia tensor, with (10) , for a perfectly symmetric rotor. 2 It is worth noting that the rotor is driven into a sinusoidal oscillation whose amplitude is typically 0.08 (rad) while the tilt angles and and the bounce motion are limited by the gap between the rotor and the substrate, which means that .…”

“…The assumption that the bounce motion is prescribed is valid because, theoretically, one can adjust by varying the electro-magnetic field between the rotor and Equation (41) is a quasiperiodically driven Mathieu equation with cubic nonlinearity and external forcing. This equation has been extensively studied [7]- [10] for the case of no substrate motion, that is, when . In [7], it is shown that in the nonlinear Mathieu equation, one obtains finite-amplitude motion, which may be quasiperiodic or chaotic, due to the nonlinear resonance in contrast to the linear Mathieu equation where instability implies unboundedness.…”

The Damped Nonlinear Quasiperiodic Mathieu Equation Near 2:2:1 Resonance

“…This treatment yields a global analysis for the existence and bifurcation of QP motions near the resonant curve lying in the largest instability region. Note that in Rand et al [3], analytical approximations to this largest instability region near 2:2:1 resonance were obtained for the linear QP Mathieu Equation (2).…”

“…This represents the 2:2:1 resonance case. Analytic expressions for the transitions curves separating stable and unstable regions were obtained in Rand et al [3] by using a double perturbation method valid for small ε and μ. In particular, the analytic approximation of the transition curves bounding the largest of the local instability regions near δ = 0.25 and ω = 1 in Equation (1) were obtained in the form…”

“…Figure 1a illustrates these curves in the case ε = 0.1. In order to understand this complicated figure through analytical treatment, Rand et al [3] applied a double perturbation procedure, which consists of applying two successive perturbation methods, and determined the transition curves of the QP Mathieu Equation (1) in the vicinity of 2:2:1 resonance. In Rand et al [3], Equation (1) was studied in the form…”