Chapter 21 Second Order Systems and Aliasing
A second order system undergoing free motion will have dynamics that look like this
\begin{equation}
\ddot{\theta} + 2\zeta \omega_n \dot{\theta} + {\omega_n}^2\tag{21.0.1}
\end{equation}
In this case, the solution to the above equation can be solved using standard second order differential equation techniques to obtain the solution below [50].
\begin{equation}
\theta(t) = \theta_0e^{-\sigma t}cos(\omega_d t)\tag{21.0.2}
\end{equation}
\begin{equation}
\omega_n = \sqrt{\sigma^2 + {\omega_d}^2}\tag{21.0.3}
\end{equation}
\begin{equation}
\zeta = \frac{\sigma}{\omega_n}\tag{21.0.4}
\end{equation}
Looking at the equations you can see that if the time series of the oscillations are known, the damping constant and damped natural frequency can be obtained. These two values can be combined to obtain the natural frequency and the damping ratio. So let’s get some data.
What I’d like you to do for this lab is to find some sort of parameter that varies in some sort of sinusoidal way. I’ve come up with a few ideas below. You may pick anyone you want although some are easier than others.
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Drive over a speed bump - If you drive over a speed bump slowly and place your CPX on the dashboard your car will hopefully vibrate for a few seconds after you drive over it. Your acceleration will look somewhat like a sine wave.
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Vibrate a ruler - If you place a ruler on the edge of a desk and deflect it, it will vibrate. If you place the CPX on the ruler you’ll be able to measure the vibration of the ruler.
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Attach CPX to a spring or a weight to the end of rubber bands - Take the CPX and attach it to a weight of some kind and attach the weight to a spring or a set of rubber bands. This is a mass spring damper simple and will vibrate at the natural frequency of square root of stiffness divided by mass. I chose to do this example.
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Build a Pendulum: If you decide to build a pendulum, you need to hang a weight on the CPX or attach the CPX to something heavy. This will make the ratio between cable to end mass much smaller and thus better for data and fitting. An example includes duct taping your CPX to a water bottle or something. It would also be better to use a string and mount the CPX to the string with an external battery pack and have the CPX log data internally. This way most of the mass would be concentrated at the tip of the pendulum. Another idea is to take a paper towel cardboard tube and tape the cpx inside it with the cable running through it. Then duct a large weight on the end of the paper towel tube and then hinge the top of the tube by skewering a screwdriver through it. This will allow the tube to swing like a pendulum rather than the string.
There are most likely many other options so try and get creative and find something oscillatory or dynamic in some way. Try to find something that changes relatively quickly. The temperature outside changes in a sinusoidal fashion but it’s so slow it would take you days to do this experiment.
