Page 23 - Fall 2006

STRUCTURAL ACOUSTICS TUTORIAL—PART 1: VIBRATIONS IN STRUCTURES
Stephen A. Hambric
Applied Research Laboratory, The Pennsylvania State University State College, Pennsylvania 16804
Most sounds that you hear throughout the day are radiated by vibrating structures. Walls and win- dows radiate sound into your house and office building. Windows radiate sound into your automobile, or into other vehicles, like buses, trains, and airplanes. The cones on the speakers of your stereo are vibrating structures that radiate sound into the air around you.
However, these structures are usually not the original sources of the sounds you hear. For example, the walls and windows in your house are driven by acoustic pressure waves caused by passing vehicles, noisy neighbors (often with loud lawn and garden equipment such as leaf blowers), or by the wind through the trees. The pressures impinge on your win- dows, which in turn vibrate and pass some of the incident sound through to the interior. In air-
planes and high-speed trains, tiny pres-
sure waves within turbulence outside the
vehicles drive the walls, which then
vibrate and radiate sound. There are, of
course, many other sources of vibration
and the subsequent sound that we hear.
Although often the sounds radiated
by vibrating structures are annoying
(your neighbor’s leaf blower), sometimes they are pleasing, like the sounds radiated by musical instruments. Pianos, vio- lins, guitars, brass instruments, and the air within and around them are complex structural-acoustic systems. The sound from musical instruments (including the human voice) is often reproduced by audio equipment, such as CD players, amplifiers, and speakers. Speakers, with their multi- ple pulsating pistons mounted on the surfaces of boxes filled with air, are also very complex structural-acoustic systems, and engineers working for speaker companies spend entire careers trying to design systems that reproduce input signals exactly (they haven’t succeeded completely yet!).
Those of us who study how structures vibrate and radi- ate sound usually call ourselves Structural-Acousticians. As I have taught structural acoustics to members of Penn State’s Graduate Program in Acoustics (and others in industry and government), I find those taking my courses want answers to the following questions:
• How do structures vibrate?
• How do vibration patterns over a structure’s surface
radiate sound?
• Conversely, how do sound waves induce vibrations in
structures they impinge on?
Some of the inevitable follow-on questions are:
• How can I modify a structure to reduce how much sound it makes (noise control engineering)?
• Or, for designers of musical instruments and loud-
speakers, how can I increase how much sound my structure makes, and craft its frequency dependence to be more pleasing to human listeners?
The answers to all of these questions depend on a struc- ture’s shape and material properties, which define how fast and strongly different structural waves propagate through it. We will start by studying how structures vibrate in this first part of the article, and then consider in the second part of the article (to appear in a future issue of Acoustics Today) how a structur- al surface’s vibration patterns act on surrounding fluids, radi- ating acoustic sound fields. Also in part two, we will analyze how structures are excited by incident sound fields. I have written mostly about simple structures, like flat plates and cylindrical shells, and use plenty of examples to explain the
concepts of vibration and sound radiation. I have tried to make this tutorial gen- eral and interesting to the non-structural acoustician, while also including enough detailed information (yes, equations) for those interested in pursuing structural- acoustics further. I have drawn from the course I teach at Penn State, several out- standing textbooks1-4 and articles in the field (see the Reference list at the end of the article), and research performed by several members of the Penn State
Graduate Program in Acoustics.
Compressional and shear waves in isotropic, homogeneous structures
Structural materials, like metals, plastics, and rubbers, deform in ways far more complicated than air or water. This is because of one simple fact: structural materials can resist
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shear deformation, and fluids cannot. This means that both
dilatational (and compressive) and shear waves can co-exist in structures. By itself, this is not very exciting. However, one more attribute of nearly all practical structures makes the field of structural-acoustics so interesting and compli- cated: most structures have one or two dimensions that are very small with respect to internal wavelengths. We call these structures plates and beams, and they vibrate flexu- raly. Why is this so interesting? Because flexural waves are dispersive, meaning that their wave speeds increase with increasing frequency.
Dispersive waves are odd to those not familiar with structural vibrations. Imagine a long plate with two trans- verse sources at one end which excite flexural waves in the plate. One source drives the plate at a low frequency, while the other vibrates at a high frequency. The sources are turned on at the same time, and somehow the high frequency wave arrives at the other end of the plate faster than the low fre-
“How do structures vibrate? How do vibration patterns over a structure’s surface radiate sound?”
Structural Acoustics Tutorial 21