An Analysis of Periodic Motion Using Fractional Calculus

Abstract

Fractional calculus has gained significant attention from engineers because of its ability to generalise
the concept of derivatives to non-integer orders. This study explores the applications of fractional calculus in engineering mathematics, particularly focusing on the analysis of periodic motion. Although extensive research has been conducted in this domain, the proposed models and algorithms are still in their early stages of development. This study examines the harmonic oscillator problem using a fractional derivative damping term, which is proportional to the velocity, instead of the conventional damping term. This paper presents a series of solutions comparing fractional-order solutions and damping ratios, not only for semi-derivatives but also for a range of fractional orders. An association between the fractional order (α) and damping ratio (η) has been elucidated to minimise the computational duration necessary for resolving the fractional equation of motion pertaining to a one-dimensional simple harmonic oscillator. The roots obtained using this method can be applied to solve the simple harmonic oscillations of a mass between two springs with transverse oscillations. This investigation’s outcomes advance our understanding of fractional harmonic oscillator behaviour and
highlight the efficacy of fractional calculus in tackling intricate engineering challenges.