Modal Coefficients and Density

I see that we need a modal coefficient for the analysis. I was wondering if we are supposed to find these somewhere or if we were supposed to calculate our own beta and c values from the data that we obtained from the lab.

As for Density, we took measurements in lab the length and the mass of the ruler. Should we just use the density we can calculate from those measurements or compare them to a standard that we can find online?

You are in luck, because they are listed in the lecture slides. In fact, the first four modes of a vibrating cantilevered beam are given in a table in the lecture.

You should definitely calculate a density based on the measurements that you took, and perhaps compare that with a 'standard' for stainless steel just to make sure what you calculated matches up with the standard within experimental uncertainty. You should be able to get densities like that from a textbook.

I checked the slides and they just show the first 4 modes of the vibrating beam but how do we know what number mode is for what beam? Also, do we need to know the specific beta value? It says that the coeff. is a function of length so do the values change?

Or maybe, do we just use the first value of the table for each length if each different length only has one natural frequency? So if there were two peaks in the FFT, is it that we use the first two values in the table?

So, you've got the right idea about which row of the table to use for which peak in the FFT (a rather astute observation). Let me add a comment to provide some more clarification.

That table of values for Bi*L is for the first four modes of vibration of a cantilevered beam. These are 'canned' solutions that arise from the governing equations dictated by a transcendental equation that has infinite solutions (i is any integer). The mathematical details are beyond the scope of what you need to concern yourself with, but suffice it to say that you do not need to calculate beta or worry about exactly what L is in this case...all you need to do is figure out the appropriate value of i (based on the vibrational mode) and then just plug in the number for Bi*L to get your modal coefficient. The frequency dependence on length will come out elsewhere in your model.