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:

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

- Dynamic model of an inverted pendulum (part 1)
- Dynamic model of an inverted pendulum (part 2)
- Dynamic model of an inverted pendulum (part 3)
- Dynamic model of an inverted pendulum (part 4)
- Dynamic model of an inverted pendulum (part 5)
- Dynamic model of an inverted pendulum (part 6)
- Modelling of a simple pendulum
- Equation of closed and open loop systems
- PI-based first-order controller

Last update : 12/17/2021