Fractional-based nonlinear mechanical system modeling with FELEs: numerical analysis of oscillatory and nonoscillatory behavior of the inverted pendulum system

Abstract This paper examines the behavior of the inverted pendulum, a notably nonlinear system, in fractional dimensions using different fractional derivative types and order. The inverted pendulum, a two-degree-of-freedom system, exhibits both linear behavior due to the cart’s motion in the horizontal plane and oscillatory behavior due to the pendulum’s angular motion. Initially, the system’s equations of motion have been derived using the classical Euler–Lagrange equation (CELE), thereby obtaining the classical integer-order model. Subsequently, the fractional model has been developed using the fractional Euler–Lagrange equation (FELE) with the Riemann-Liouville and the Caputo–Fabrizio fractional derivatives. The results of the models obtained were shown in the simulation platform and presented comparatively. In this paper, the impact of fractional-order modeling on both oscillatory and nonoscillatory motions of mechanical systems is analyzed. This is achieved by introducing the inverted pendulum model and employing two different types of fractional-order derivatives.