Effects of Exponential Mass Variation on Pendulum Damping and Resonance

Abstract

The suggested paper presents a theoretical study to analyze the dynamical response of a simple pendulum carrying a time-dependent mass. The physical model introduced in this paper builds on the assumption that the mass of the hanging point particle varies exponentially with time. By using the extended lagrangian formulism which takes the reactive force (Meshchersky force) resulting from joining\losing the mass into consideration, the equation of motion of the system was derived. The analysis shows that the effect of the exponential variation of the mass leads to a appears a physical term which behaves like a linear damping term in the equation of motion defined as , hence because of this damping coefficient, the total energy of the pendulum was exponentially decaying. The cases of I: null attached mass's velocity ( ) which leads to a free damped case, II: constant attached mass's velocity ( ) which results a steady shift in the position of the equilibrium point, III: variable attached mass's velocity ( ) which behaves like an external force term that can leads to a resonant amplifications or changes the spectrum of the dynamical response were discussed. The analytical solutions like the Laplace transform, Green's function and spectral analysis were obtained. The power delivered from the external driving force to the oscillator and the quality factor of the studied system was introduced.