Ball in Tube Model¶
A fan at the bottom of a tube produces an air stream moving upwards. A ball levitates in the air stream.
The fan rotates with the rotational speed . It produces an air stream with the velocity . The factor describes the proportionality between the air’s volume flow rate and the fan’s rotational speed. The motor driving the fan is modeled as a PT2-element with the amplification , the damping and the time constant . An Arduino Uno controls the motor, its discrete control signal has a range of and amplifies the supply voltage .
The ball covers an area and has a mass . Its distance to the tube’s bottom is the position . The gap between the ball and the tube covers an area . The factor describes the proportionality between the force of flow resistance and the velocity of the air streaming through the gap.
The tube has a height .
The task is to control the ball’s position . Actuating variable is the motor’s control signal .
TWith the state vector
the model equations are given by
In case of the ball falling down and reaching a position below the fan, the root function of the model overrides the ball’s position and velocity .
The model’s boundary condition is violated if the ball leaves the tube on the upper end:
The ball’s position
is chosen as output.