Observing Vibrational Modes of Metal Plates

Once you have a working speckle interferometer, you will certainly want to use it to analyze something more interesting than a bent plate. One good option you have is to use a loudspeaker connected to a function generator to vibrate a thin metal plate. What makes this option so good is that these plates are easy to mount and have many vibrational modes that can be mathematically predicted. In addition, once you have completed your analysis of one plate, you can easily get a new plate of a different shape and repeat the experiment (although, this page guide will focus on rectangular plates).

Predicting Vibrational Modes

The goal of this section is not to completely work through a derivation for the natural frequencies of a metal plate; that is an exercise for the students to do. Rather, I will give a starting point for the derivation along with a couple of hints.

The fundamental equation that governs vibrations in a plate is[1]:

where w = w(x,y,t) is the displacement in the direction out of the plate (note that w could also take skew or polar coordinates, but if you are working with a rectangular plate, Cartesian is the obvious choice), ρ is mass density, and D is the “flexural rigidity.” D is defined as[1]:

where υ is Poisson Ratio’s, h is the thickness of the plate, and E is Young’s Modulus.

I will stop here and let the students take over. I advise doing the math for a rectangular plate first and moving onto other shapes later.

hints:

-Make equation (1) time-independent.

-Fourier's theorems are your friends.

Setup and Methods

The materials (excluding basic optics equipment) needed for this setup are:

1x translational stage

1x loudspeaker (I used a Realistic Amplified Speaker System)

1x function generator (I used a Keysight 33500B)

1x aluminum plate with hole punched in center (shape and size are up to you)

If the thicker metal plate that was bent in order to test the interferometer is still in place, remove it and put the setup shown in Figure 1 in its place.

There are a few things to keep in mind:

1) Even though you got rid of the thicker plate used to test the interferometer, leaving one of its posts in place is a good idea as seen in Figure 1. It serves as a reference point so that you have something to line the aluminum plate up against.

2) Where the horizontal post (the one that has the plate attached) is clamped to the vertical post (the one on top of the translational stage) is important. You will find that your images of vibrational modes have better quality when the horizontal post is clamped near its end, on the opposite side as the aluminum plate.

3) You may feel that the translational stage is not necessary, but it is very useful for getting the aluminum plate in position.

Figure 1: an image of the mounted aluminum plate. The function generator is not shown in this image, but is connected to the loudspeaker via BNC cable. Note that the post in front of the plate is not necessary to the setup, but was used as a reference. Lining up the plate with the post ensured the optical path lengths in both arms of the interferometer remained the same.

Figure 2: an image of the mounted aluminum plate view from just above the interferometer's camera.

With everything in place, you are ready to observe some vibrational modes. First, you will want to increase the exposure time of the camera (add polarizers to attenuate the beam's intensity if need be). I set the exposure time to 2 seconds. Keep the amplitude of the signal generator at 1mV for now; the loudspeaker has a natural frequency around 300Hz which gets very loud and you don't want to accidentally damage your hearing. Using the equation you derived for the natural frequencies of the aluminum plate, select a frequency for the signal generator. Now turn up the amplitude as you see fit. If your frequency is around 200Hz, an amplitude of ~10mV should suffice.

Turn off the room lights and take some photos of the plate being driven by the loudspeaker. Important: this is a little bit different than what you did before. With the vibrating plate, I got more reliable results by subtracting two images of the plate being driven by the loudspeaker. Before, you took an image of the metal plate without deformities or displacements and then another with the deformity or displacement.

If your subtracted images have poor quality, try the following:

1) Adjust where the horizontal post is clamped to the vertical one.

2) Within reason, turn up the signal amplitude. Some of the vibrational modes will have better quality at higher amplitudes. Use noise-blocking headphones if you deem them necessary, but do keep in mind that other people nearby don't have headphones on hand to protect their own hearing.

3) Reduce the amplitude. Louder is not always better and vibrational modes tend to be lost when the amplitude is too high.

4) Try a different vibrational mode, repeat the prior two steps as necessary, and then go back to the original mode.

Example Images

Below are a few images that I took. I will not tell you the frequencies at which I drove the aluminum plate, but they will give you an idea of the image quality you can aim for.

Figure 3: Sample image 1.

Figure 4: Sample image 2.

Figure 5: Sample image 3.

References

[1] Leissa, A. Vibration of Plates. 1970, NASA SP-160. https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19700009156.pdf