Transfer function of an air levitation system

[zacharoni16]

I'm building an air levitation system that will levitate a ball in a closed tube with a DC motor fan at one end of it. I'm having a hard time thinking of a way to obtain the transfer function of the system. I know the net vertical force will be the force of gravity the gradients of air pressure from the fan, and the drag force. Also I believe the Bernoulli principle would apply here?

I'm having a hard time relating the fan speed (changed with PWM) to the levitated position of the ball in the tube. I have an ultrasonic sensor to sense the position of the ball, I'm just having trouble modeling the system to get a transfer function and use PID controller.

I think I derived the equation to give height from a jet of air:

1/(height)^2 * ((1-4a)/3*height) = 4a(density of ball)G/3A^2(density of air)

it seems like the height of the ball has nothing to do with air velocity so that's good news, but it completely confuses me on what and how to control its height and develop a transfer function

With the complexity of this system or similar system, do you have to do it from a model like different operating points? Not sure, hopeful for someone to give advice, insight, and direction.

Thank you!

 

[Navin A S]

The best way to obtain the transfer function of this system is to use a combination of experimental data and mathematical modelling. You can start by measuring the position of the ball with the ultrasonic sensor, then plotting the response of the ball as you vary the PWM duty cycle of the fan motor. This will give you a good idea of how the two quantities are related. You can then use this data to construct a mathematical model that describes the relationship between the two variables, and from this model, you can derive the transfer function of the system.