Over damped response transfer function (help pls)

[Uridan]

Hi,

Not sure on where to post this tread since it involved some fluid mechanics and Maths/ Control theory.

I have found the response of a ball flouting in a vertical jet stream of air and the result is a highly non linear system. That is, there is different open loop response as the ball goes higher and higher.

As always, there is the choice to select average linear parts within the prerequisite range and I have found 3 linear parts, form 0-5 cm 5- 15cm and 15-25cm, each part gave 3 different open loop response of the system which resulted in this way:

0-5cm = under damped response with small overshoot peak but a very large settling time.
5-15cm= under damped response with average overshoot peak and average settling time.
15-20cm=under damped response with High overshoot peak but very small settling time.

I think you get the picture and physics wise, it makes sens but since I am an electrical student I don't now that much about fluid mechanics.

As shown above each response is an under damped response, thus a graph that oscillated a lot before reaching settling time in the form of an enveloped shape. The problem is that with my system, I can never reach settling time since there are shedding vortex which will not allow the ball to stay at a steady vertical position and will keep oscillating at that particular position.

This means that the under damped response will still have an enveloped shaped form, but it never reaching the steady state position, since it keeps oscillating between the set point (constant amplitude and frequency).

Can anyone tell me how to calculate the transfer function of such under damped response? I have to calculate the order of the system and use z and w notation to find the G(s) (transfer function) of the system. A general example would be great :).

thanks
regards
Uridan

 

[Achy47]

Dear Uridan,

Thank you for your post. It sounds like you are working on an interesting project involving fluid mechanics and control theory. The behavior you have described, with the oscillating ball in the vertical jet stream, is indeed a highly nonlinear system.

To calculate the transfer function of such an underdamped response, you will need to first determine the order of the system. This can be done by analyzing the number of energy storage elements (such as capacitors or springs) and energy dissipating elements (such as resistors or dampers) in your system. Once you have determined the order of the system, you can use the z and w notation to find the transfer function.

To give a general example, let's consider a simple second-order system with a single energy storage element and a single energy dissipating element. Let's say we have a spring-mass-damper system, where a mass is attached to a spring and a damper. The input to this system is a force applied to the mass, and the output is the position of the mass.

Using the z and w notation, we can represent this system as:

G(s) = (w0^2)/(s^2 + 2w0z + w0^2)

Where w0 is the natural frequency of the system and z is the damping ratio. The natural frequency is a measure of how quickly the system will oscillate, while the damping ratio is a measure of how quickly the oscillations will decay.

To determine the values of w0 and z for your system, you will need to analyze the system's response at different input forces. This can be done using experimental data or through mathematical modeling.

I hope this helps you in your calculations. Good luck with your project!