Fractional-order analysis of nonlinear oscillations in a time-dependent mass pendulum

This work investigates the dynamic behavior of a fundamental oscillatory system in which the mass decreases continuously with time. The analysis begins with the derivation of the classical formulation based on the Lagrangian approach, leading to the corresponding Euler–Lagrange equation of motion. To extend this framework, fractional-order calculus is incorporated, yielding a generalized fractional Euler–Lagrange model. The system response is then examined numerically for different values of the fractional derivative order. The simulations reveal that a pendulum with a diminishing mass exhibits a variety of motion types, ranging from regular oscillations to irregular and aperiodic patterns. The results highlight the sensitivity of the system to variations in the fractional order α, which directly affects dynamic properties such as overshoot, rise time, and damping. Importantly, the fractional approach allows these parameters to be tuned in a controlled manner. When α approaches unity, the model converges smoothly to the classical case, yet the dynamical characteristics remain distinguishable, reflecting the unique influence of fractional memory effects. This study formulates a fractional-order dynamic model for a pendulum with exponentially decaying mass, a configuration that appears to be less explored in the existing literature compared to constant-mass systems... The results provide insights into the behavior of non-standard oscillatory systems and suggest that fractional formulations can be effective in capturing complex time-evolving phenomena that may be challenging for conventional methods to represent fully. Overall, the proposed fractional framework offers an adaptive and robust analytical tool, with promising potential for advanced control strategies in engineering and robotics, where tunability and precision in modeling are crucial.