PD Control for Vibration Attenuation in a Physical Pendulum with Moving Mass

Abstract

This paper proposes a Proportional Derivative controller plus gravity compensation to damp out the oscillations of a frictionless physical pendulum with moving mass. A mass slides along the pendulum main axis and operates as an active vibration-damping element. The Lyapunov method together with the LaSalle's theorem allows concluding closed-loop asymptotic stability. The proposed approach only uses measurements of the moving mass position and velocity and it does not require synchronization of the pendulum and moving mass movements. Numerical simulations assess the performance of the closed-loop system.

An Improved Principle of Rapid Oscillation Suppression of a Pendulum by a Controllable Moving Mass: Theory and Simulation

“…ω 0 is the initial motion frequency of the oscillating pendulum, and ω m π/2∆t. In equation (8), the rst and fth equations represent the mass upward process, the third equation represents the mass downward process, and the second and fourth equations represent that the position of the mass remains stationary at l 0 (1 − ε) and l 0 (1 + ε), respectively.…”

“…Figure 4 indicates that the pendulum oscillates for one cycle corresponding to two cycles of the mass motion. e mass motions in the following analysis are equation (8) and the adjustment of equation 8.…”

“…e equivalent damping ratio of these two processes can be obtained by substituting equations (8) and (11) into equation (10). e equivalent damping ratio of the mass upward process ζ up and the mass downward process ζ down is obtained as follows:…”

“…Mass motion l (t) (m) Figure 4: e rst two cycles of the mass motion represented by equation (8). and the suppression e ect increases both with the decrease of ∆t.…”

“…e effectiveness of the control law was also discussed when the system had linear damping. Gutiérrez-Frias et al [8] proposed a proportional derivative controller plus gravity compensation based on the Lyapunov method together with the LaSalle's theorem to suppress the oscillation of the pendulum by controlling the moving mass.…”