Transition Curves Analysis for Fractional Mathieu Equation

Abstract

In this study, the influence of the fractional order derivative on the location of stability domains and resonant curves for fractional Mathieu equation is considered. In fractional undamped Mathieu equation, an extra damping characteristic is observed when the inertia term has a fractional order derivative with no changes in the resonant values. This means that, the fractional derivative has the ability to generate damping characteristics in the stability domains, even though their damping terms are not present in the equation. Using the analogy of the classical damped Mathieu equation with the fractional undamped Mathieu equation via the harmonic balance method, a mathematical form to express the induced damping characteristic is derived in terms of the fractional order derivative. The analytical results are confirmed by the corresponding numerical ones for some special cases which are considered at some values of fractional order derivatives in the fractional undamped Mathieu equation. Notably, the Numerical results indicate the versatility and effectiveness of the corresponding analytical ones.