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

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.