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 constitutive 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 relationships 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.

最近，对分数阶微积分进行了大量研究，以推广现有的数学模型或提出新的模型来描述不同的现象。分数模型的研究不如经典模型，建议将它们与经典的成熟模型进行比较。然而，关于分数阶模型和经典整数阶模型之间的关系的问题出现了。阻尼谐振子是许多研究领域中许多数学模型的基础。其参数的物理解释和可变范围是众所周知的。一些分数模型导致分数阶微分方程在结构上与经典谐振子方程相似。这种模型称为分数振荡器。

本文考虑了一种基于粘弹性材料Scott Blair本构方程的分数模型，当应用于描述机械振动时，会产生分数阶微分方程，即分数振子方程。本次分析的目的是首先找出谐波振子的参数范围内是否存在分数振子，其行为与经典谐振子具有良好的近似性，然后是否只有一个分数振子。这样的分数振荡器，最后两个振荡器的参数之间存在什么关系。

为了回答这些问题，我们引入了分数振荡器和经典振荡器之间的散度系数。通过最小化这个系数，我们的目标是在数值上找到一个与固定经典振子相对应的分数振子。模型参数之间的关系是针对它们的一些范围确定的。然后使用这些公式计算具有正弦驱动力的分数振荡器方程的系数；得到了与经典方程的良好一致性。在模型中使用由分数 Kelvin-Voigt 关系得出的公式的尝试被证明只是部分成功。