PHYS-482, Acoustics II: Sound & Vibration (Winter and Spring Terms 2006)

Photographs on this page are ©2006 Dan Russell and were taken when he taught this course in the Winter and Spring Terms of 2006

The greater part of this advanced elective course is lecture based - we discuss vibration of simple and complex systems, circuit analogies for mechanical and acoustical systems, loudspeaker design and performance, coupled oscillating systems, the vibrating string, acoustic waves in solids (longitudinal, torsional and bending waves), experimental modal analysis, sound transmission through walls, and other topics involving sound-structure interactions. However, while we do spend most of our time in the classroom, the enrollment in this course is usually small enough (during the Winter 2006 term we had 9 students) that roughly once a week we take a "field trip" to the acoustics lab and get some hands-on experience playing with the systems we've been discussing in the classroom. These field trips are not full-blown laboratory experiments, but are intended to provide students with the opportunity to see the theory in action, and to learn some of the ways in which theory is only able to approximately describe real systems. A full blown laboratory course (PHYS-484, Acoustical Measurements) is available for students who wish to gain more laboratory experience with acoustical equipment and phenomena.

Resonance, Damping, and Nonlinear Oscillations

During the first week of class we review the basic behavior of simple oscillation including damped oscillators, driven oscillators and oscillators with nonlinear elastic mechanisms.  A pop bottle with a small amount of cotton inserted into the neck acts as a very nice example of a damped acoustic oscillator (otherwise called a Helmholtz resonator). The volume of air in the cavity provides the springiness while the "plug" of air in the neck provides an inertia (mass) effect as it oscillates in and out of the neck opening. A microphone is inserted into the acoustic cavity of the pop bottle to measure the variation in pressure - analogous to measuring the motion of a mass-spring system, and the bottle is acoustically excited by sound waves from a loudspeaker. Sweeping through a range of frequencies and recording the resulting microhone response with a Frequency Analyzer the students are able to obtain a frequency response curve with a peak indicating the resonance frequency. The height and broadness of the peak indicate the amount of damping. By changing the amount of cotton in the neck, students are able to see the effect of various damping amounts on the resulting frequency response curve. By recording the frequency and amplitude of the peak, as well as the frequencies at which the amplitude is 3dB less than the peak, students are able to estimate the damping coefficient for this damped system.

In a second experimental exercise students extract the damping constant of a system by observing the amplitude as it decays with time. We use a pendulum which consists of a large board suspended from strings. Upon being released the pendulum swings back and forth, but loses energy to aerodynamic drag forces so that the amplitude decreases with each oscillation. Plotting the amplitude as a function of time results in a roughly exponential decay curve* and students can estimage the damping constant using the logarithmic decrement.  The departure from a purely exponential decay provides evidence that a realistic aerodynamic drag friction forces are not the same as the simpler viscous damping forces one usually encounters in a textbook.

A simple pendulum serves as an example of a system with a nonlinear elastic restoring force.  As long as the initial angle is small enough that the small angle approximation holds (roughly about 17o), the pendulum will swing with a period that is independent of the initial angle.  However, for larger angles, the restoring force depends on the sine of the angle, and the period now increases with increasing amplitude.
*Exponential decay is the result of a viscous damping force which is proportional to velocity. Aerodynamic drag forces are actually proportional to velocity. However, with an appropriate conversion factor it is still possible to extract the actual damping rate from the exponential decay curve.

Impedance (Electrical, Mechanical, and Acoustical)

Impedance is an essential tool for determining the frequency response of electrcial, mechanical, and acoustical systems. In general, the impedance of a system is a complex quantity, with the real part (resistance) indicating the presence of a damping mechanism which removes energy from the system, while the imaginary part (reactance) indicating the storage of energy in the system. The natural frequencies of system are those frequencies where the imaginary part of the impedance vanishes. The real part of the impedance, reveals how much power is dissapated by the system when driven.

In this lab exercise, students measured the input electrical impedance of a loudspeaker by simultaneously measuring the current into a loudspeaker and the voltage across the leads and taking the ratio of voltage to current. Using a two-channel FFT analyzer we looked at the magnitude, real part, and imaginary part of the impedance as a function of frequency.

We measured the driving point mechanical impedance of a loudspeaker by using a mechanical impedance head. This transducer consists of a piezoelectric force transducer and an accelerometer in the same housing. We fastned the impedance head to a mechanical shaker do drive the system and connected the impedance head to the front cone of the loudspeaker. On the FFT analyzer we integrated the acceleration signal to obtain the velocity and looked at the frequency response function consisting of the ratio of force divided to velocity, which provides the mechanical impedance.

We measured the acoustic impedance of a pop bottle (helmholtz resonator) by driving the resonator with a loudspeaker and simultaneously measuring the pressure and particle velocity at the neck opening of the bottle. The transducer we used to measure the acoustic impedance is a Micrflown p-u probe which consists of a tiny microphone to measure pressure and a tiny hot-wire animometer to measure particle velocity.

Resonance Frequencies of a Loudspeaker

First we measured the "free-air" resonance of the loudspeaker by measuring the input electrical impedance as a function of frequency. Then we added a known amount of mass to the speaker and observed how much the added mass lowered the free-air resonance. Armed with two frequencies (and two equations) we were able to calculate the mechanical stiffness of the speaker surround and spider.

Then we put the speaker in a bell jar and used a vacuum pump to suck the air out of the jar and determined the new natural frequency. Removing the air reduced the acoustic mass loading. By comparing the "free-air" resonance with the vacuum resonance frequency we can estimate the amount of mass loading due to the air.

Finally we put the speaker in a box to observe how presence of the box increases the natural frequency. This happens because the air in the closed volume of the box compresses and expands as the speaker oscillates. We measured the volume of the box in order to calculate the acoustic stiffness of the box and compared the result to the measured shift in frequency.

Then we stuffed the box with absorbing material, which lowers the amplitude of the speaker response somewhat and shifts the natural frequency lower because the volume is reduced.

Vibration of a Multiple-Degree-of-Freedom System

Whiffle golf balls and rubber bands make for an easy to assemble demonstration of a vibrating system with a finite number of degrees of freedom. Each whiffle ball provides local mass, while the rubber bands provide the elastic restoring forces. A single ball system has one resonance frequency. A two-ball system has two resonances, one in which both balls move in the same direction and a second mode in which both balls move in opposite directions. We have some animations illustrating the mode shapes for multiple-degree-of-freedom systems. The images below show the three modeshapes for a three-ball system. (Click on the images to download a movie of each mode). In this hands-on session students try to guess the mode shapes and relative frequencies as they continue adding balls to the system. At the end we look at the first several mode shapes and frequencies of a vibrating string (which represents an infinite number of masses connected by an infinite number of springs).

The Fixed-Fixed String and the Physics of Guitar Pickups

After spending several class lectures investigating the behavior of a fixed-fixed string and the effects of realistic boundary conditions, we spent day in the lab playing with real strings on an electric guitar. We looked at (and listened to) the time signals and frequency spectra for the low E string and compared the results when using both a magnetic pickup and an optical pickup to detect the string motion. When we measured the higher frequencies of the string's spectrum and compared the values to the harmonic series predicted for an elastic spring we found evidence of inharmonicity due to the stiffness of the steel string. We also looked at the frequencies of a string from which half of the winding had removed and it sounded very different. The change in the mass density of the string caused the higher frequencies to no longer be integer multiples, and instead of sounding like a plucked string, it sounded like a struck bell.

The Physics of the Piano

Today we spent the day in the auditorium exploring Kettering University's Yamaha Baby Grand piano. We took the keyboard action out of the piano to see how the complicated mechanical levers work.* The keyboard action has about 50,000 moving parts. We learned how the lever action throws the hammer upwards to strike the keys, and how the nonlinear properties of the wool felt affect the tonal quality of the instrument and how that tonal quality varies with loud and soft blows. We learned why the piano strings are very thick and heavily wound for bass notes, and why most hammers strike two or three strings. We learned what the three pedals do: the right pedal lifts the dampers for all keys so all strings vibrate longer, and the left pedal shifts the entire keyboard over so the hammers strike only one string instead of all three (thus it is called the "una corda" pedal). We heard the harmonic series by holding down keys corresponding to frequency multiples of 2,3,4,5,6,7,8,9 and then played the fundamental and heard all of the harmonics respond sympathetically. We also held the sustain pedal down and shouted into the piano - and our shout was "recorded" in the sympathetic vibration of the strings - kind of a temporary mechanical fourier transform. We learned how the bridge couples the string vibration to the soundboard, how the lid aids in the directionality of sound radiation. At the end of the class period, Dr. Russell put the piano back together and played a song.

*Don't try this at home! In addition to a Ph.D. in Acoustics, Dr. Russell has a B.Mus. in Piano Performance and his Master's Thesis involved studying the nonlinear behavior of piano hammer felt and the acoustics of the piano in general. He knows his way around a piano. He has done this demonstration more than a dozen times in the last 10 years. Removing the action without knowing what you are doing could result in damage (ie, breaking hammer shafts, bending springs, or damaging the fragile and complicated action mechanism).

Circular Membranes and the Acoustics of Drums

Today we found the first several mode shapes of a drumhead (single-head tom drum). We drove the drum into resonance using a loudspeaker, and sprinkled sand on the drum to find the Chladni patterns. When the drum was vibrating at one of its resonance frequencies the sand collected at the nodes (lines and circles that don't move) and we could detect the vibrational pattern. We also compared the frequencies of the modeshapes we found to the predictions for an ideal membrane and discussed the effects of mass loading by the air on the top of the drum, and the enclosed column of air inside the drum shell.

Longitudinal Waves

Forgot to bring the camera to the lab today, but we measured the frequency spectra for several rods of different materials (wood, aluminum, steel, brass, and plastic) by tapping the rod at one end and recording the longitudinal vibration with a small accelerometer at the other end. We used the frequencies of the first several harmonics to determine the speed of sound in each of the materials. We also measured the transit time - the time it takes an impulse signal to travel the length of the rod, and used this time to calculate the speed of sound in each material.

Bending and Torsional Waves in Beams

After spending several class sessions deriving the equation of motion for bending vibrations/waves in beams and applying various sets of boundary conditions (free-free, clamped-free, pinned-pinned) to determine resonance frequencies and vibrational mode shapes, we spent a couple of days in the lab investigating the vibratonal behavior of beams. In one experiment we determined the mode shapes of a large rectangular beam using a microphone scanning technique our professor learned while in grad school. * We attached a small NeFeB magnet to one corner of the beam, and placed a small electromagnetic coil above it, powered by a function generator. The alternating current in the coil provided a magnetic force pushing and pulling on the magnet, driving the beam into vibration. We scanned the near-field of the beam with a microphone, and looked at the Lissajous figures comparing the driving signal with the micophone response on an oscilloscope. When the beam was driven at resonance, the Lissajous figure was a narrow oval with either a large positive or large negative slope. As the microphone was moved over the surface of the beam, the slope (phase) of the Lissajous pattern changed sign each time the microphone crossed a nodal line. So, by scanning the surface of the beam with the microphone we were able to locate the nodes and then to identify the mode shape and the type of vibration (ie, either transverse bending or torsional). The image below right is a link to a movie (11.7 MB MOV file) showing how the Lissajous pattern slope changes sign as the microphone is scanned along the surface of the beam from one end to the other.

*T.D. Rossing and D. A. Russell, "Laboratory observation of elastic waves in solids," Am. J. Phys., 58(12), 1153-1162 (1990)

The other experiment we did was to observe the mode shapes and frequencies of a canilevered beam (clamped at one end and free at the other). We had a set of six thin metal strips, of varying lengths, driven at the clamped end by a mechanical shaker. We compared the ratios of the frequencies of the first two modes to the theory we had developed in class. Then we plotted the frequencies as a function of length and compared the data with theoretical expectations. We also measured the lengths of the tines of two sets of tuning forks (one made of aluminum, and the other of steel) and plotted the frequencies as a function of length to compare with theory from class.
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