Page 11 - Spring 2007

STRUCTURAL ACOUSTICS TUTORIAL—PART 2: SOUND—STRUCTURE INTERACTION
Stephen A. Hambric
and
John B. Fahnline
Applied Research Laboratory, The Pennsylvania State University State College, Pennsylvania 16804
Introduction
This is the second of a two-part tutorial on structural acoustics written for Acoustics Today. The first appeared in the October 2006 issue, and focused on vibrations in structures. In that article, I explained
• the various waves that can propagate through structures, and how bending waves are dispersive (their wavespeeds increase with frequency);
• the modes of vibration of finite structures;
• mobility and impedance, and how they are simply sum-
mations of individual modal responses;
• structural damping;
• how the mobility of an infinite structure is the mean
mobility of a finite one; and
• modeling vibrations with finite ele-
ment (FE) analysis.
Some of the feedback I received
from readers of the first article pointed
out that it was more of a ‘cliffs notes’
summary of structural acoustics than a
tutorial, but found the summary quite
useful nevertheless. After thinking about
it, I suppose that is true, and I thought
about renaming the second part of this
article ‘A Structural Acoustics
Cookbook,’ rather than a tutorial. For
continuity, though, I have retained the
original title, but hope the two articles will be a useful short reference on the subject, where handy formulas and concepts can be easily found.
In this article, I have added a co-author—Dr. John Fahnline—who specializes in analyzing sound–structure interaction using boundary element (BE) modeling tech- niques. John has already written one book1 on acoustic BE analysis (with Gary Koopmann), and is working on a second.
In this article we will explain:
• what structural vibrations do to neighboring acoustic flu-
ids, and
• what sound fields do to neighboring structures.
These problems are complementary (and reciprocal), and we will use analytic, numerical, and experimental data to demonstrate their basic concepts. As with Part 1 of the arti- cle, we will supply plenty of useful terms and equations.
The overarching concept of linear sound–structure inter- action is simple: the normal particle velocity in the structure and fluid along the fluid–structure interaction boundary must be the same. This means that when a structure vibrates against a fluid, the component of the vibration normal to the structur-
al surface must be identical to the corresponding particle velocity in the neighboring fluid. This simple equality allows us to couple the equations that define structural and fluid motion at the fluid–structure interface and solve for the total sound–structure behavior. While the normal particle velocity is identical in the structure and fluid, the in-plane, or tangent particle velocity is not. In fact, we allow a ‘slip condition’ between the structure and fluid, so that a structure can slide along a neighboring fluid without inducing any sound.
Of course, with any simple concept there are inevitably several assumptions. Here are ours:
• homogeneity (the fluid properties are the same every-
where),
• isotropy (the fluid properties are the same no matter what direction the wave propagates), and
• linearity (the fluid properties do not depend on the fluctuating pressure amplitude or phase).
With these simplifying assumptions, it is straightforward to couple the vibra- tions of structures with those in acoustic fluids. Incidentally, all of the information in Part 1 of this article made the same assumptions, but for the structural materials!
We will start by explaining what a structure’s vibrations do to a fluid—they compress and expand it. The spatial pattern of structural vibrations and their frequencies determines how much sound is radiated, and in what directions. It may be helpful to think of the acoustic fluid as an elastic blob surrounding the structure, being pushed and pulled over time by the motion of the structural boundary (in the normal direction only, of course). When the fluid’s mass density is comparable to the structure’s, the fluid not only absorbs sound, but also mass- loads the structure. We will explain how two important structures—a circular baffled piston, and a flat rectangular flexible finite plate—radiate sound and are fluid-loaded by
the impedance of the surrounding acoustic fluid.
Next, we will consider the complementary problem— how acoustic waves induce vibration in a structure. The same physics are at work in this reciprocal problem, as we shall see later. We will conclude by discussing how to make measure- ments of sound–structure interaction, and how to model it
using boundary elements.
A clarifying note: in this article we consider
sound–structure interaction, not fluid–structure interaction
“In this article we will explain: what structural vibrations do to neighboring acoustic fluids, and what sound fields do to neighboring structures.”
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