How do I get the damping coefficient experimentally

In summary, the conversation is about a problem in a project for Controls 3B that involves modeling a bouncing ball. The mass-spring-damper model is used to simulate the ball while it is in contact with the ground. The problem is that the damping coefficient is unknown and difficult to measure without expensive equipment. The conversation discusses possible methods for determining the damping coefficient, such as comparing the height of the peaks normalized to the first bounce.
  • #1
williamshipman
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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:
[tex]\ddot{x}+\frac{b}{m}\dot{x}-\frac{k}{m}x=g-\frac{k}{m}r[/tex]​

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.
 

Attachments

  • bounce model idea.pdf
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  • #2
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.
 

1. How do I measure the damping coefficient in an experiment?

The damping coefficient is typically measured by conducting a free vibration experiment. This involves setting a mass-spring system into motion and recording the amplitude of the oscillations over time. The damping coefficient can then be calculated using the logarithmic decrement method or by fitting the data to a damping model.

2. What equipment do I need to measure the damping coefficient?

To measure the damping coefficient, you will need a mass-spring system, a timer or stopwatch, and a device to record the amplitude of the oscillations (such as a ruler or motion sensor). Optional equipment may include a data logger and computer software for data analysis.

3. How do I ensure accurate results when measuring the damping coefficient?

To ensure accurate results, it is important to minimize external forces and disturbances on the mass-spring system during the experiment. This can be achieved by conducting the experiment in a controlled environment and carefully setting up the equipment. It is also important to take multiple measurements and average the results to reduce experimental error.

4. Can the damping coefficient be measured for any type of oscillating system?

Yes, the damping coefficient can be measured for any type of oscillating system, including mechanical, electrical, and fluid systems. However, the method for measuring the damping coefficient may vary depending on the system and its properties.

5. How can knowing the damping coefficient be useful in scientific research?

The damping coefficient is an important parameter in understanding the behavior of oscillating systems. It can be used to analyze the energy dissipation in a system, predict the damping effects on the motion of the system, and optimize the design of systems for specific applications. Knowing the damping coefficient can also aid in the development of mathematical models for predicting the behavior of complex systems.

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