# Ball and beam : model

This post deals describe the model of the ball and beam. The system consists of a long beam which can be tilted by an actuator together with a ball rolling back and forth on top of the beam.

We will make the following assumptions:

• $m$ is the mass of the ball

• $\Theta$ is the angle of the beam

• $x$ is the position of the ball on the beam

• $b$ is the friction coefficient of the ball

Based on the fundamental principle of dynamics and considering that the friction is proportional to the speed, we can write:

$m.a = m.\ddot{x} = \sum{\vec{F}} = m.g.sin(\Theta) - b.\dot{x}$

In conclusion, the differential equation of the system is:

$\ddot{x} = - \frac{b}{m}.\dot{x} + g.sin(\Theta)$