Fractional investigation of time-dependent mass pendulum

In this paper, we aim to study the dynamical behaviour of motion for a simple pendulum with mass decreasing exponentially in time. To examine interesting system, firstly obtain classical Lagrangian and Euler-Lagrange equation accordingly. Later, generalized is constructed via non-integer order derivative operators. The corresponding derived, calculated approximate results are simulated respect different orders. Simulation show that time-dependent exhibits behaviours, such as oscillatory non-oscillatory motions, nature depends on derivative; they also demonstrate utilizing fractional approach yields model both valid flexible, displaying various properties physical system under investigation. This provides significant advantage understanding complex phenomena, which cannot be achieved through methods. Indeed, characteristics, overshoot, settling time, peak vary case by changing value α. Also, formulation recovered when α tends 1, while their output specifications completely different. These successful achievements diverse systems, enhancing adaptability effectiveness proposed scheme modelling dynamics.