An Advanced Method of Modeling the Dynamics of a Suspended Monorail Using Fractal Analysis

Fractional differential operators provide an effective approach for modeling complex technological processes, particularly physical phenomena in continuum mechanics characterized by memory and non-local effects. Different types of fractional derivatives require different numerical approximation schemes; in this study, the Caputo and Grünwald–Letnikov derivatives are considered. The aim of this work was to develop and validate a fractional differential model of longitudinal oscillations in a suspended monorail system that accounts for nonlinear and memory-dependent effects. In contrast to classical integer-order approaches, the proposed framework incorporates multiscale surface irregularity effects, including rail roughness, friction, and other disturbances influencing system dynamics, through a fractional-order formulation. A fractional differential mathematical model describing the motion of longitudinal oscillations of a large-sized cargo transported along a suspended monorail is proposed. A numerical algorithm based on finite-difference approximation of fractional operators was developed for its implementation. The scientific contribution lies in integrating multiscale surface irregularity effects into a fractional-order modeling framework to improve the accuracy of dynamic response prediction. Numerical experiments demonstrated the effectiveness of the approach, and the results were validated through comparison with existing models of monorail dynamics. Additionally, statistical validation based on correlation analysis confirmed good agreement with the experimental data. The proposed model can be applied to the design and optimization of suspended transport systems, improving vibration control, reliability, and operational safety under real dynamic loading conditions.