Damping efficiency of the fractional Duffing system and an assessment of its solution accuracy

Dynamical systems with high damping efficiency in a wide frequency band can be useful on a small scale – for harvesting energy from ambient vibrations, and on a large scale – for damping harmful vibrations of mechanical structures. This repository contains datasets for assessment of the quality of solutions and damping efficiency of systems with fractional order derivatives. To simulate the fractional system the fourth-order Runge–Kutta method and Grünwald–Letnikov methods are used. We propose a coefficient for assessing the quality of solutions to fractional systems by reference to the quality of the calculated energy balance. As an exemplary system we study the Duffing model with embedded additional fractional-order derivative terms. Based on this coefficient, intervals of key numerical simulation parameters are determined to ensure the appropriate quality for the calculations of energy flows and energy balance. The determined values of these parameters are then used in tests of the damping efficiency of the studied system.Our results show that by modifying the fractional terms it is possible to configure a system that exhibits a ‘‘broadband effect’’, i.e. a system that is characterized by high-amplitude vibrations and, consequently, high energy efficiency in a wide range of excitation frequencies.Datasets in DAT and MAT files (Matalb formats) contain the source data presented in the figures in the article mentioned in the related work.在宽频段内具备高阻尼效率的动力学系统兼具微观与宏观应用价值：微观场景下可用于采集环境振动能量，宏观场景下则可用于抑制机械结构的有害振动。本数据集仓库收录了用于评估带分数阶导数（fractional order derivatives）系统的求解质量与阻尼效率的数据集。本研究采用四阶龙格-库塔法（fourth-order Runge–Kutta）与格林伍德-列特尼科夫法（Grünwald–Letnikov）对分数阶系统进行仿真。我们提出了一种基于计算所得能量平衡质量的系数，用于评估分数阶系统的求解质量。我们以嵌入额外分数阶导数项的达芬模型（Duffing model）作为示例系统展开研究。基于该系数，我们确定了关键数值仿真参数的取值区间，以确保能量流与能量平衡计算具备合格的求解质量。随后将这些参数的确定取值应用于目标系统的阻尼效率测试中。研究结果表明，通过调整分数阶项，可以构建出具备“宽带效应”的系统：该系统在宽泛的激励频率范围内均表现出高振幅振动特性，进而具备较高的能量利用效率。以DAT与MAT格式（Matlab格式）存储的数据集，包含了相关研究提及的论文中各图表所对应的原始数据。