How do I get the damping coefficient experimentally

[williamshipman]

Homework Statement

This problem is part of my project for Controls 3B. The project revolves around modeling a bouncing ball that is dropped from a certain height and bounces of the ground. I have been told to use the mass-spring-damper model to simulate the ball while it is in contact with the ground. The model for this system therefore consists of two equations: one for when it is falling through the air, with only gravity affecting it, and the second for when the ball is in contact with the ground.

My problem is that none of the physical properties of the ball are known before hand. Finding out the spring constant is easy enough as you simply need to squash the ball and measure how much force is being applied and the new size of the ball.

The damping coefficient is the one that is giving me problems.

Homework Equations

The second order differential equation describing the ball during the collision is:

$$\ddot{x}+\frac{b}{m}\dot{x}-\frac{k}{m}x=g-\frac{k}{m}r$$​

k is the spring constant, b is the damping coefficient, m is the mass of the ball, g is 9.81 - gravity, x is the height above the ground. I defined down to be the positive direction for my forces and velocity.

I have been thinking for this for a while and have put all of my thoughts into "bounce model idea.pdf". The file "prac1.pdf" is the project description and requirements given by the lecturer.

The Attempt at a Solution

From the pdf, I have some possible leads to follow. The only problem is that at each turn, I am limited by what I can measure. The easiest value to measure is the maximum height of each bounce. The time spent in contact with the ground is so small that I can't measure it without expensive equipment (which I don't have). The same problem exists for measuring the amount of deformation in the ball during the collision. My Internet research has turned up various terms such as damping ratios or damping factors, but only gives these in terms of the damping coefficient and the spring constant. No physical link is given, making it difficult for me to derive an experiment to determine b.

Any help would be greatly appreciated. Thanks.

 

[DaveE]

The maximum height of each bounce is determined by the damping. If you don't know the initial height or velocity, you can compare the height of the peaks normalized to the first bounce since you know the height of that and that the velocity is zero then (pretend the first bounce is where it was dropped with zero velocity). So if you solve for the maxima of ball height you should get a solution. i.e. if the 3rd peak is 57% of the 2nd peak value that will be associated with a unique damping coefficient.

I'm glad I don't have to do the math though, this method sounds tedious.