Page 31 - Fall 2006

energy loss occurs for every spatial cycle of the wave, so as frequency increases, the number of spatial cycles, and there- fore energy loss, increases. For high structural damping and frequency, the amplitude of the wave that eventually returns to the source is so small it is barely noticeable, and the source mobility resembles that of an infinite structure.
Why is infinite structure theory useful? The mobility equa- tions (we will see more of them later—for beams and shells) can and should be used to check mobility measurements. I have often looked at measured mobility plots and noticed that the levels seemed strange, confirming my suspicions by performing a simple infinite structure mobility calculation. Usually the dis- crepancy is a neglected gain factor applied to instrumentation, or a forgotten units conversion factor.
Infinite structure theory can also be used to make cost- effective back-of-the-envelope estimates. An engineer trying to decide between various structural materials can use the simple equations to conduct tradeoff studies prior to invest- ing time in more rigorous and costly analyses. Finally, infinite structure theory is useful for scaling the response of a struc- ture to that of another geometrically identical structure con- structed out of a different material.
In Fig. 15, the measured corner drive point mobilities of an Aluminum and Lexan ribbed panel of nominally identical geometry are compared (the plots are adapted from Reference 23). Note that in the top plot the infinite plate mobility is included. In the bottom plot, the mobility of the
Aluminum plate is scaled to that of Lexan using the ratio of the infinite plate mobilities. Also, the frequency of the Aluminum mobility is scaled to that of Lexan using the lon- gitudinal wave speed ratio:
. (24)
The agreement between the two sets of measurements, when scaled, is striking, with the differences in the peak responses due to differences in the structural loss factor.
Vibrations in cylindrical shells
In most flat plates, bending and membrane (longitudinal) waves are uncoupled. Take a flat plate and bend it statically, and those waves become coupled. Take the flat plate and wrap it around a circle to form a cylinder, and the coupled bending- membrane waves become continuous around the circle. This is a circular cylindrical shell—a structure that has received almost as much attention over the years as flat plates have.
A typical vibration field in a short cylindrical shell is shown in Fig. 16. In the example, a bending wave wraps around the shell so that radial deformation (and slope) are continuous everywhere. Flexural waves also travel along the axis of the shell, and resemble those in flat plates. The vibra- tion field in a cylindrical shell is decomposed into its cir- cumferential and axial components:
(25) where n is the circumferential harmonic for both cosine and
Fig. 15. Drive point mobility measurements of two geometrically identical ribbed panels, one made of Aluminum and the other of Lexan. Top–raw mobility meas- urements; Middle–mobility and frequency scaled to those of Lexan; Bottom–Geometry of the ribbed panels.
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Structural Acoustics Tutorial 29