Torque required to stop the pendulum
Motion of the Pendulum
The position of the center of mass of the system can be expressed as follows:
The derivatives are given as follows:
The potential energy is given as:
X, Y is the inertial reference frame, is the ith local coordinate, is the angle measured from the frame, is the distance from the inertial reference to the center of the mass.
The kinetic energy is given as:
The kinetic energy is thus given as:
The equation of motion is now thus given as:
Therefore, the equation of motion becomes;
Methods of Improving the Control of the Pendulum
The control of the pendulum can be improved by adding a rigid rod from the center of rotation. The rod should then be attached to the spring-damper mechanism, this way, the vibrations about the center of rotation will minimize hence improving control of the system.