On the equivalence between fractional and classical oscillators

Recently, much research has been conducted on fractional-order differential calculus to generalise existing mathematical models or propose new ones to describe different phenomena. Fractional models are not as well researched as the classical models, and it is recommended that they should be compared to the classical well-established models. However, questions arise about the relationships between fractional-order and classical integer-order models. The damped harmonic oscillator is the basis for many mathematical models in many areas of research. The physical interpretation and ranges of variability of its parameters are well known. Some fractional models lead to fractional -order differential equations similar in structure to the classical harmonic oscillator equation. Such models are called fractional oscillators.This article considers one of the fractional models based on the Scott Blair consti-tutive equation for viscoelastic materials, which, when applied to describe mechanical vibrations, leads to a fractional-order differential equation, i.e. the fractional oscillator equation. The aim of this analysis is first to find out whether and in what ranges of the parameters of harmonic oscillator there is a fractional oscillator whose behaviour, with good approximation, is the same as that of the classical harmonic oscillator, then whether there is only one such fractional oscillator, and finally what relationships exist between the parameters of the two oscillators.To answer these questions, we introduce the so-called divergence coefficient between fractional and classical oscillators. By minimising this coefficient, we aim to numerically find a fractional oscillator corresponding to the fixed classical oscillator. The relation-ships between the parameters of the models were determined for some of their ranges. The formulae were then used to calculate the coefficients of the fractional oscillator equation with a sinusoidal driving force; a good agreement with the classical equation was obtained. The attempt to use the formulae in the model resulting from the fractional Kelvin-Voigt relationship turned out to be only partially successful.(c) 2022 Elsevier B.V. All rights reserved.

Automated summary

This article discusses research on fractional-order differential calculus and the analysis of fractional oscillators, exploring the relationships between fractional oscillators and classical oscillators, and proposing a method for calculating the coefficients of fractional oscillator equations.