Standing wave modes of a violin.

Transcription

1 Introduction. Standing wave modes of a violin. The intent of this lab is to explore a few modes of a complicated object, and employ some of the tools used in physics labs to measure their properties and understand their origins. In the first part, you will explore standing waves on strings, and become comfortable with some basic techniques. We have written down step-by-step instructions for this part. In it you will learn more about operation of the digital scope and Fourier analysis of signals. In the second part, you will use these techniques and expand on them to explore these modes in greater detail. We want you to explore some question or phenomenon related to standing wave on these strings. Find something that interests you. Your TF can help you with details. In section 2A below, we have listed a few topics that came to mind as we were creating the lab. They are just examples don t limit yourselves to this list. The violin ( violin in this use refers to any bowed string instrument) can be understood as a system in which: 1. Bowing or plucking drives standing wave modes of the strings. The oscillating strings exert transverse forces on the bridge. 2. These forces drive modes of the violin bridge and body. These modes involve oscillation of the air enclosed in the box and bending of various parts of the instrument. 3. The oscillating modes drive sound waves, which propagate. The job of a violin maker is to build an instrument in which the frequency and response of these modes are what players have come to expect. One step in this process is to make sure that some of the most important components of the instrument have the desired properties, as measured before assembly. For many years violin builders have tuned their front and back plates before assembly by adjusting the thickness until the tap tones are at the desired pitch and have a clear ring. The tap tones are due to the oscillations of the modes of violin bellies and backs that are not yet (or not anymore) part of a violin. You will have a chance to measure some of these modes. Equipment Digital oscilloscope Function generator, with low impedance output Op-amp based preamplifier, inside aluminum box String Instrument, supported by foam blocks Cables Permanent magnets with supports Microphone, cable altered to end in BNC connection Oscillator (PASCO Wavedriver) driven by function generator.

2 Experimental: 1. String modes: Detailed instructions for standing waves. A. Acquire the spectrum of the sound produced by plucking a string. 1. Set up for sound collection. Place the violin on foam blocks so that it is suspended above the table. Mount a microphone in a likely spot. Open the aluminum box that contains the preamp, turn it on, admire it, and close the box. Connect the microphone to the preamp input and the preamp output to Ch 1 of the oscilloscope. 2. Finding signal on scope. a. Press the Save/Recall button and select Setup 1 to retrieve a saved setup that should work well for this signal. b. Pluck the string and find the signal. Adjust the vertical and horizontal scales of the scope until you are clear about their function. 3. Triggering. On the Trigger Menu, make sure that: a) Mode is Normal. The Scope waits to take data until a defined condition occurs. (for example, signal in Ch1 rises above 0.5 V) b) Source is Ch 1. Scope triggers on data in Ch 1. c) Coupling is DC.

3 4. Set the trigger level to about 500 mv and observe the signal. See what happens as you change the trigger level and pluck again. When the trigger level is low enough you will see that repeated triggering occurs for a given pluck. 5. Use the Acquire menu (Choose Horizontal Resolution ) to find out how many data points are acquired and what is the time between points. Change the horizontal scale and observe the change. Zoom in using the horizontal scale control and verify that the time between points and the data length are as advertised. 6. Get a Fourier Transform. Return the scope to setup 1 (SAVE/RECALL button). Set time scale to 40 ms per division and acquire the data from a pluck. Turn on the MATH channel and make sure that it is calculating the FT of the data in Ch 1. a) Explore the FT using the vertical and horizontal scale controls, and the horizontal position knob. Zoom in and out as far as possible. b) Turn on the cursor function (choose vertical bars). As you zoom in and out you can use the buttons on the cursor menu to relocate the cursors. Use the cursors to find: (1) How many peaks are there? (2) What is the maximum frequency? (3) What are the center frequencies and the full width at half-maximum (FWHM) of each peak? 7. If you change the time duration of the data taken in Ch 1, you will find that the peak resolution (FWHM) and the maximum frequency of the FT in the MATH channel change. Take a few minutes to understand what is happening. Take data with total duration of 40, 400, 4000 ms and record the sampling rate (ACQUIRE Menu), the FWHM of the fundamental, and the maximum FT frequency. 8. Finally, explain (to yourself and in the lab writeup) how it is possible to excite a large range of frequencies with a pluck. Sketch the force vs. time graph for the force that your finger exerts on the string. What are the frequency components of this force? B. Drive the string with a variable frequency sinusoidal force. Measure amplitude and phase of the microphone signal. 1. Set up function generator to provide current through the string. The low impedance output of the function generator should drive current through the string in series with a 3.9 Ω power resistor, as shown in the

4 drawing below. Keep the amplitude dial of the function generator below about half maximum to avoid overheating the string and creating annoying sounds. Set the sine wave frequency to that of the fundamental (lowest frequency) mode.

5 2. Place a small permanent magnet within about 1 mm of the string as shown in the figure below. The magnet should be near the center of the vibrating length of the string. 3. Place the microphone near an f-hole and connect through the preamplifier to Ch 1 of the scope. 4. You should be able to see a signal now with the scope triggering on the signal itself. We want you to be able to measure the relative phase of the sound and the sinusoidal driver, so you will need to trigger on a signal that is synchronized with that sine wave. Set the scope to trigger on the TTL output of the function generator. This TTL output gives a logic level (0 or 5V) square pulse at the same frequency as the sine wave output. Use a BNC cable to connect the TTL output to Ch 2 of the scope. From the trigger menu, select source as Ch 2 and adjust the trigger level with the trigger level knob to be about 1 V. You may turn on Ch 2 to see the TTL pulses. 5. Observe the microphone signal as you sweep the frequency around the resonance. What happens to the phase and amplitude as you move through resonance? In part II you may learn how to use the scope measuring and averaging functions to collect this data for a detailed picture of the resonant behavior. 6. Find higher frequency modes. Which modes can you drive well with the magnet in the center of the string length? Compare frequencies to those obtained in the plucked spectra.

6 C. Measure induced emf in the string. Here you will gain information about the string motion through measuring the emf induced in the string as it moves through the field of another permanent magnet. 1. Place leads on the non-vibrating ends of the string using a BNC cable with alligator clips. Connect this cable to the amplifier, with output into the scope Ch Place the magnet under string and attach to the violin body with tape. This allows greater freedom to move the instrument that a magnet suspended externally as in the previous step. 3. Find signal on the scope. (Trigger on the signal again. Use the SINGLE SEQ mode.) 4. What does the signal tell you about the string motion? 5. Record the Fourier Transform of the emf signal for a plucked string. Compare with the frequencies present in the sound spectrum. How are the two signals related? 2. Further exploration. A. Examples of other things you can do with string modes. 1. Calculate the wave speed on the string. Measure the string density and calculate the tension. 2. Measure decay time of the resonance and derive the Q.

7 Violin Lab for 15c, April, Sweep the driving frequency through a resonance. Measure and plot amplitude and phase as a function of frequency (See App. 1). Find Q from the resonance width. 4. When the string the string is plucked with a strong magnet in close proximity, there is an audible beating (amplitude modulation) that disappears when the magnet is moved away. What is going on? 5. Drive the string with a mechanical sinusoidal driver. This will allow you to measure the string velocity through induced emf and at the same time observe the microphone signal. 6. How does the spectrum depend on the location of the pluck? 7. How is the spectrum of the sound signal related to that of the induced emf signal? 8. How is the amplitude and phase of the sound signal related to that of the induced emf in the string? (see number 2 above.) 9. What is the string motion when the string is bowed? 10. Mass loading. Load the string with extra mass, small compared to the mass of the string. This can be done with small pieces of tape. Predict the effect this will have on the mode frequencies. Measure the frequencies. 11. Measure the effect of mass loading when the string is plucked, and when it is bowed. 12. Measure the effect of mass loading as the mass is moved along the string. B. Modes of free plates: glitter patterns. To do justice to this, we would need several more hours, but we think that it is so much fun that we want you to play with it for a few minutes.

8 As shown in the photos above, you can use the function generator and oscillator to excite the vibrational modes of flat plates, in this case a thin spruce plate. The plate is balanced on the driver shaft and held in place with a pair of magnets. When the oscillator frequency matches that of a mode of the plate, the amplitude builds up. Glitter sprinkled on the plate gathers at the nodes, revealing the mode pattern, as seen above for the Hz mode of this plate. If you suspend the plate with your fingers, touching it at the nodal lines only, you should hear a pitch corresponding to the frequency of this mode when you tap the plate. Try this for the violin bellies and backs that we have provided. Two of the modes that violin makers care about are the one shaped like an x that looks like that in the picture above, and one usually about an octave (factor of two) higher in frequency, shaped roughly like a ring 1. You may be able to find these modes by shaking and again by tapping and listening. 1 These two modes are formed from the coupling of two simple bending modes of the plates. See the book by Rossing: The Physics of Musical Instruments.

9 Appendix 1. Some notes on triggering and data acquisition for the TEK Scope. 1. The illustration below comes from the TEK scope user manual, which you have on the course website and your lab computer desktop. It explains the meaning of the hard-to-see but very useful marks on the top of the scope screen. Verify that when you change the horizontal scale the data expands around the expansion point. 2. The following steps will give you precise control over the data that the scope collects: a) Use the SINGLE SEQ button on the right side of the panel to put the scope into a mode in which it triggers only once. To trigger again, push the button again. To return to the original freetriggering mode, press the RUN/STOP button (just above SINGLE SEQ). b) The horizontal position control knob moves the trigger position within the screen display. Move the trigger about ¼ of the way into the screen and take data. You should see the data that was acquired some time before the signal. (See TEK manual, p. 3-31)

10 c) It is more useful in our case to be able to take data in a time period after a trigger event. You may want, for example, to eliminate the initial transient sounds, or you may be interested in the evolution of the sound during the time after the pluck. To do this, hit the horizontal delay button. (TEK manual, p. 3-33) Now the horizontal position knob adjusts the delay between the trigger and the data expansion point (now in the center of the screen). The delay value is shown at the bottom of the screen. Set the delay positive, set it negative, and observe what happens. To check your expertise, show on the screen a data set consisting of the 40 ms of sound that occurs starting 100 ms after a pluck. Appendix 2. How to use the scope functions to average and automatically measure amplitude and phase. 1. Driving with the function generator, obtain a signal that is triggered from the TTL pulse. Press the Channel button corresponding to the signal to make it active. 2. Select the MEASURE button. Scroll through the menu of measurements for the channel, and select amplitude and phase (relative to your TTL trigger pulse). 3. The measurements will not be very constant, especially for low signal levels, because of background sounds. In order to reduce the effect of the background, use the average function (ACQUIRE menu, Mode button) to average several sets of data. Select the number of data sets to average using the selection knob at the top of the scope.