Homelab 4

In this homelab we will measure the vibration frequencies of the bending modes of a nail, and compare the results to predictions from the Euler-Bernoulli (EB) theory for vibrations of a uniform beam. The nail (16d 3-1/2" bright box) and a piece of thread will be provided in class.

Step 1 Tie the piece of thread to the nail near its head. Hang the nail from the thread and gently strike it with something hard. You should hear a note sound.

Step 2 While watching the spectrogram on Raven Lite, gently strike the nail and make adjustments until you can clearly see the first four modes of the nail. You should find that the lowest mode is near 2 kHz and the highest one is roughly 9 times higher. These frequencies are much higher than we found for the string in homelabs 2 and 3, so be sure to adjust the Raven Lite frequency scale appropriately. Adjust the focus control, and the brightness and contrast as needed. When you are satisfied with the spectrogram, stop recording and measure the four mode frequencies as accurately as you can. The lower modes decay slowly and you should be able to measure their frequencies with an accuracy of a few Hertz. You will have to zoom in the frequency scale to do this. You might not be able to measure the higher mode frequencies as accurately because they decay more quickly, and you may find that it is better to use a different setting of the focus control for these modes. The highest modes might be split into two closely spaced frequencies. If so, record the average of the two frequencies for the highest mode. (We will explain below why some of the bending mode frequencies may be split into pairs.)

Step 3 Print a paper copy of this form and record your measured frequencies for the bending modes in the second column. We will compare our results to the Euler-Bernoulli theory, which predicts the frequency ratios already entered in the third column. Multiply your measured first mode frequency by these ratios and enter the results in the fourth column. (For example, the fourth column for mode 2 should be 2.757 times your measured first mode frequency.) Since we are only taking frequency ratios from the EB theory, the predicted first mode frequency will agree with your measured frequency exactly, but the others may deviate for many different reasons.

We won't be concerned about the details of the EB theory. Besides the frequency ratios of the different modes, it also predicts the absolute frequencies of the modes and the mode shapes. Figure 1 in this description of the EB theory shows the mode shapes. They may look superficially similar to the modes of a string, but there are several important differences. The beam is free to move at both ends, and in fact the motions are largest at the ends. The shapes are not sine functions, but sums of sine and exponential functions. The theory applies to a beam with uniform cross-section, so it does not account for the head or point at the ends of our nail.

A beam can have bending vibrations in two different directions. If you imagine lining up the nail with the x-axis of Cartesian coordinates, it could vibrate in the + or - y direction, or in the + or - z direction. The modes for the two different directions would have exactly the same frequency if the nail were symmetric enough, for example if it were exactly a cylinder. If it is not quite symmetric we will see two closely spaced frequencies corresponding to two slightly different modes.

Step 4 Now we want to make a graph that will show at a glance how our data compares with the EB theory, and we'll also show how different the measured frequencies are from a harmonic series. Label the x-axis of the plot "Mode Number" (without the quotes) and place the mode numbers evenly spaced along the axis. Label the y axis "Frequency (kHz)" and choose a scale that will use most of the available space. As in homelab 2, it will be helpful to make the origin of the graph correspond to zero frequency and mode number zero. (If you are confused about this point, look at the solutions to homelab 2.) Plot your measured mode frequencies as black dots, locating them as accurately as you can on the graph. Draw a straight line that starts at the origin of the graph and passes through your point for the highest mode. (If the frequencies were from a harmonic series, they would all be on this line.) Now carefully plot the frequencies predicted by the EB theory, using open circles. For mode 1, the dot and circle will be in the same location, so make the circle large enough so we can see both symbols. (Some of the other circles may fall of top of or overlap with the dots.) Finally, add a key to the plot in the upper left of the grid where there is unused space. Draw your black dot symbol and write "experiment" next to it. Beneath the dot draw your circle symbol and write "theory" next to it. Try to make all of your plotting symbols (dots and circles) uniform so that it is clear that they refer to the same kind of information everywhere on the graph.

Step 5 We will not try to analyse the uncertainties in this experiment, but there are several issues worth thinking about. Your results are probably very far from a harmonic series. But is the disagreement significant? In other words, are your measurements accurate enough to conclusively prove that the mode frequencies of the nail do not form a harmonic series? Your results probably agree better with the prediction of the EB theory. Can you say that your mode frequencies agree with the EB theory? Are the deviations you see due to the limited accuracy of your data, or are they because a nail is not quite a uniform beam?

Step 6 Scan your homelab to a pdf file using a scanner or a smartphone app, and upload it to the dropbox on our Canvas site before it is due.