Analytical and experimental study for mechanical vibrations of a two‐coupled spring masses system via Caputo‐based derivatives

Abstract

This work studies a mechanical system composed of two‐spring coupled masses arrangement without damping using fractional derivatives under the Caputo sense. We determine and implement an explicit model based on the Caputo‐Fabrizio operator. To achieve this model, we detail a systematic methodology that avoids dimensional inconsistencies. Then, we use the proposed model to explain the system dynamic acquired from an experimental (quite rudimentary) setup. We also compare our results to those achieved from the ordinary model and the numerical implementation using the Caputo derivative definition. Results prove that the proposed model describes the dynamic of this mechanical system better than the ordinary and numerical models. We also show that the fractional order allows the model (obtained from an idealized undamped system) to describe dynamics from fully dissipative to wholly oscillatory. Moreover, we discuss other exciting characteristics of the obtained responses.

Editorial of the special issue on modelling, analysis, and applications

“…An efficient local meshless numerical method is proposed in another contribution, to study the variation, in option pricing, of the fractal transmission system, modelled by the time fractional Black‐Scholes equation [3]. A fractional mechanical system composed of two‐spring coupled masses [21], without damping, is analyzed in another paper. The effect of triple diffusion on convection of viscous liquid in a vertical channel, saturated with permeable material, subject to Robin conditions on the boundaries, is the topic of another research [22].…”