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.

中文翻译：

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通过基于 Caputo 的导数对两耦合弹簧质量系统的机械振动进行分析和实验研究

这项工作研究了由两个弹簧耦合质量排列组成的无阻尼机械系统，使用卡普托意义上的分数导数。我们确定并实现了基于 Caputo-Fabrizio 算子的显式模型。为了实现这个模型，我们详细介绍了一种避免维度不一致的系统方法。然后，我们使用所提出的模型来解释从实验（相当基本）设置获得的系统动态。我们还将我们的结果与普通模型和使用 Caputo 导数定义的数值实现所获得的结果进行比较。结果证明，所提出的模型比普通模型和数值模型更好地描述了该机械系统的动态。我们还表明，分数阶允许模型（从理想化无阻尼系统获得）描述从完全耗散到完全振荡的动力学。此外，我们还讨论了所获得的响应的其他令人兴奋的特征。