Physics 3 Lab 5 Normal Modes and Resonance

Transcription

1 Physics 3 Lab 5 Normal Modes and Resonance 1 Physics 3 Lab 5 Normal Modes and Resonance INTRODUCTION Earlier in the semester you did an experiment with the simplest possible vibrating object, the simple harmonic oscillator. Real vibrating objects, such as are used to construct musical instruments, are more complicated, but all vibrating objects have some basic behavior in common which you re going to investigate in this experiment. BACKGROUND Nearly all musical instruments that have distinctive sounds (which is pretty much all of them) depend on the idea of normal modes: any vibrating object more complicated than a single mass on a spring or its equivalent has two or more simple motions, called normal modes, at which it wants to vibrate. Typically the more moving parts the object has, the more normal modes. A single mass on a spring has only one moving part and a single normal mode, while a vibrating string, which you can think of as a very large number of very small masses connected by very small springs, has a very large number of normal modes. We re going to keep the number of normal modes manageable in this experment by working with two or three gliders on an air track, where the number of modes will turn out to be equal to the number of gliders. Any actual vibration of the object can be regarded as a combination of those natural motions. Equally important to the operation of musical instruments is a property of vibrating objects called resonance: if you shake or drive an object at one of its normal frequencies, the object will vibrate much more strongly (specifically, its vibration will have a much larger amplitude) than if you drove the object at some randomly-chosen frequency. This phenomenon explains how the sound production in a musical instrument can start with something as disorganized as a brass player s buzzing lips and still produce tones built from a harmonic series. EQUIPMENT Air track with mounted metric scale and air compressor Glider connected with springs to the ends of the air track Additional gliders and springs Computer running Data Studio PROCEDURE Part 1. Measuring the natural frequency of a single glider. A schematic of the apparatus is given on the next page. Data Studio should already be set up and running, if it isn t, instructions for getting it started are given in the appendix at the end of this file. If necessary, add or remove gliders and springs so that you have a single glider connected to each end of the air track by a spring, the same as your setup for the simple harmonic oscillator and similar to the schematic above. One spring will actually be connected to

2 Physics 3 Lab 5 Normal Modes and Resonance 2 a piston that you will use to drive the glider at a frequency that you can set. The computer at your station should have a file called SHOFit.ds already loaded and the Sonic Ranger set up. You should see two open windows, one a graph of the position of the glider as a function of time, although it will probably be labelled something like Graph 1. The other window is labeled Curve Fit, and that s where the heavy lifting will take place. Schematic of air track Driver Spring Sail Spring Glider Intake for hose from compressor Start the glider oscillating and then click the Start button, which has a green rightpointing triangle on it. You will notice that Start now changes to Stop and the green triangle changes to a red square. Let it run for 5-10 seconds and then click the Stop button, which will display all the data you took for that run. Your data should look oscillatory, although the amplitude of oscillation will decrease over time because the air resistance of the sail is pretty substantial. We can t get rid of the sail, though, because that s what reflects the pulses from the Sonic Ranger back to its detector. You will need to know the frequency of your single-glider system to study resonance. In the earlier experiment on the simple harmonic oscillator, you measured the period off the graph, and you can calculate the frequency once you know the period. There s actually a more precise way to measure the period, which for devious reasons of my own I didn t have you use last time. Start by taking a data run with the single glider if you haven t already. If you do have a previous run with more than one glider, delete it by going to Experiment in the top menu, and choose Delete last data run. Then go to the Curve Fit box and choose Sine Fit from the pull-down menu. Although your run will show some drop in the amplitude because of air resistance, if your run is only a few oscillations long the effect of air resistance will be pretty minimal and you will get an accurate measurement of the period. From the period you can calculate the frequency of the oscillator (in Hz). Part 2. Resonance and the single glider. In the Introduction, I claimed that driving a vibrating system at one of its natural or normal frequencies will produce a much larger response than driving the system at some arbitrary frequency. This sensitivity to the driving frequency is what physicists call resonance, and you can test my claim quite easily. Notice that the function generator is wired to a cylindrical box with a sort of piston in it, and the piston is connected to one of the springs connected to the glider. Set the amplitude knob about one-quarter of the way between its

3 Physics 3 Lab 5 Normal Modes and Resonance 3 minimum and maximum settings, and notice that now the pistion moves in and out, stretching and compressing the spring, which in turn exerts a (sinusoically varying) force on the glider. How far the piston moves depends on the setting of the amplitude knob, and the frequency with which the piston is moving, and hence the frequency at which you are driving the oscillator, is determined by the frequency setting. Nothing else should be happening, though, because the compressed air is turned off. Take your measurement of the natural frequency that you found above and set the function generator attached to the driver to that frequency. Now you are driving the oscillator at its natural frequency. Finally you can turn on the compressed air! Start taking data right away so that you can see the transient, which is the initial motion of the glider, a combination of the oscillation you would get if you just gave it a shove and the motion you get from driving it. After a minute or so, you should see the motion settle down and become genuinely periodic, reaching its steady-state motion which is determined by the driving force but is independent of how the oscillator got started. Once the steady-state motion is established, stop the current data run, delete it from the graph, and then start a new one that you ll use to measure the amplitude and frequency of the oscillation. Get about 10 seconds of data for your fit. You can either measure the amplitude and period of the oscillation directly from the graph, or you can use the Fit button for this data. If you use the Fit button to find the period, choose Sine Fit. With either method, record the period and the amplitude of the oscillation. From the period, calculate the frequency of the oscillation. Then delete the run and the fit, and also turn off the compressed air and put the glider approximately at its equilbrium position. Next set your function generator to a driving frequency about 20% lower than the natural frequency, repeat the experiment above, and measure the frequency and amplitude of the new oscillation. (Remember that the driving frequency is the frequency you read off the function generator, while the natural frequency is the frequency you measured in Part 1 for the undriven oscillator.) Do this again for a driving frequency about 20% higher than the natural frequency. What happens to the amplitude of oscillation when you drive the system at a frequency noticeably different from the natural frequency of oscillation? How is the frequency of the resulting motion related to the natural frequency of the oscillator and/or the driving frequency? In answering the questions above (which appear on the data summary sheet), it s important to remember that there are conceptually three different frequencies you need to keep straight. (They may not all have different numerical values, but they have three different roles in this experiment.) The first is the natural frequency of the undriven oscillator: that s the frequency you measured in the first part of the experiment, with the function generator turned off (or at least the amplitude knob turned down to zero). The second is the driving frequency: that s the setting on the function generator. Since the natural frequency is set by the spring constant and the glider mass, while the driving frequency is set by you to whatever value you choose, those two frequencies need not have the same value (although of course if you set the value of the driving frequency value to the number you measured for the natural frequency, they will have the same value). The third frequency is the frequency of the driven oscillation, the frequency at which the oscillator actually oscillates, and one question you will want to think about and test experimentally is whether or not the frequency of the driven oscillation is always equal to one of the other two frequencies.

4 Physics 3 Lab 5 Normal Modes and Resonance 4 Part 3. The two-glider system. Now add a second glider and a spring so that you have two gliders and three connecting springs on the air track. The glider closer to the motion sensor should have the sail mounted on it. You will probably also have to move the ring stand which is connected to one of the springs so that all three springs are under tension. Turn on the compressed air to the air track and measure the equilibrium positions of both gliders, where each glider is at rest because the forces exerted on it by the two springs balance each other. It doesn t matter what point on the gliders you use as long as you re consistent, so pick some point that s convenient. Take one run (if you haven t already) that you start by displacing one of the gliders from equilibrium and then letting go. Get about ten seconds worth of data. Probably your graph will be more or less oscillatory but not really periodic, because you started the system randomly. Since the two gliders are connected by a spring, in addition to being connected to the two ends of the air track, when one of the gliders moves it will exert a force on the other glider and cause it to move as well. The result is motion that looks pretty complicated, at least to the casual observer. Just for laughs (and also to make a point), run a Sine Fit on your data. Data Studio will almost certainly come up with a set of fit parameters. The resulting sine wave just won t look very much like your data. The point to running this fit is to notice that you can (try to) fit any function you want to any data you have, and the procedure will come up with a best fit. There s no guarantee, though, that this best fit is actually any good. And in this case, the best fit single sine wave is probably a pretty terrible fit because the motion of your two-glider system is more complicated than a single sine wave. (One hazard to these built-in fitting routines is that finding a fit is so easy that you often forget that the fit may or may not mean anything. When people had to fit functions by hand, they were pretty careful to start with a sensible guess for the function they were trying to fit.) However, if you go one step more complicated, you can get a pretty good fit. Instead of fitting a single sine wave, try the next item in the list of fitting functions, Sine Series. The Sine Series will try to fit the sum of two sine waves, with different periods and probably different amplitudes, to your data. This time, you should get a function that looks quite a bit like the motion of the glider with the sail, and you can read the periods for both sine waves right off the fit. From the periods you can calculate the two normal frequencies for your system of oscillators. Once you ve found the two normal frequencies, one for each normal mode, see if you can get your two-glider system to go into resonance. Set the function generator to one of your two normal frequencies and turn on the compressed air. While in principle this should work no matter how the gliders get started, in practice you can save a lot of time by starting with the gliders at rest and close to their equilibrium positions. Even so, you may have to wait a minute or so for the transient motion (which does depend on how the gliders get started) to damp out. Do you get the single mode (or close to it) that you would expect from the frequency? Now suppose you set the frequency counter either 20% above or below the normal frequency. What does the motion look like now? You can use the Sine Series fit to measure the amplitudes of the two modes. If you really are driving the system at one of its resonant frequencies, one of the amplitudes should be much larger than the other. Is this the case? Is the frequency of the largeamplitude motion the one you would expect?

5 Physics 3 Lab 5 Normal Modes and Resonance 5 Repeat this process with the other normal frequency. (Turn off the compressed air and start the gliders over.) Do you now get motion at the mode associated with that frequency? What happens when you set the frequency 20% high or low? (Setting the frequency to some frequency different from one of the normal frequencies is sometimes called de-tuning, or going off-resonance. ) Measuring the amplitudes and frequencies in this case is trickier because you re really driving the system in both modes at once. The amplitudes in your Sine Series fit should be consistent with this statement. Appendix. Setting up Data Studio. A shortcut for Data Studio should appear on the desktop; double-click the icon to start. When the program starts you will be asked, How would you like to use Data Studio? Choose Create Experiment. You will get a window called Experiment Setup, which will include a picture of the black box (more precisely, the ScienceWorkshop 750 Interface) which connects your computer with the motion sensor. That black box should have two cables, one with a yellow plug and one with a black plug, plugged into it. Point the cursor at the picture of one of those two sockets and click on it. You should get a menu of sensors: choose the motion sensor and click the OK button. If you get a Help window that says something like, Data Studio could not connect, that probably means that the black box isn t turned on. Its on/off switch is in back on the right. After you ve turned it on, you ll get a yellow triangle with an exclamation point and the instructions to click to connect. After doing so, the Help window should now tell you, The interface is ready for use. After you ve connected the motion sensor you can close the Experiment Setup window. For displaying your data, you will want a graph of the position of the sail-equipped glider as a function of time. The left side of the field should contain two small tables, one labeled Data and the other labeled Displays. Select Position, Ch 1 & 2 from the Data table and drag it down to Graph in the Displays table. (If you don t have those two tables, click the Summary button near the top of the window on the left.) You now get a graphing area which at the moment has no data in it. With luck you ll have one or more gliders on your air track, one of them with a sail. Turn on the compressed air, get that glider moving, and then click the Start button, which you encountered earlier in the Simple Harmonic Oscillator experiment. The motion sensor will start to collect data and graph it. You can scale the graph so your data fill the screen by clicking the left-most button in the graphing window. That s the Scale to fit button but it looks like a tiny baseball field to me. Now you re ready to start the experiment.