Empirical Rules for Oscillation and Harmonic Approximation of Fractional Kelvin–Voigt Oscillators

Fractional Kelvin–Voigt (FKV) oscillators describe vibrations in viscoelastic structures with memory effects, leading to dynamics that are often more complex than those of classical harmonic oscillators. Since the harmonic oscillator is a simple, widely known, and broadly applied model, it is natural to ask under which conditions the dynamics of an FKV oscillator can be reliably approximated by a classical harmonic oscillator. In this work, we develop practical tools for such analysis by deriving approximate formulas that relate the parameters of an FKV oscillator to those of a best-fitting harmonic oscillator. The fitting is performed by minimizing a so-called divergence coefficient, a discrepancy measure that quantifies the difference between the responses of the FKV oscillator and its harmonic counterpart, using a genetic algorithm. The resulting data are then used to identify functional relationships between FKV parameters and the corresponding frequency and damping ratio of the approximating harmonic oscillator. The quality of these approximations is evaluated across a broad range of FKV parameters, leading to the identification of parameter regions where the approximation is reliable. In addition, we establish an empirical criterion that separates oscillatory from non-oscillatory FKV systems and employ statistical tools to validate both this classification and the accuracy of the proposed formulas over a wide parameter space. The methodology supports simplified modeling of viscoelastic dynamics and may contribute to applications in structural vibration analysis and material characterization.