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References and endnotes:
1 L. Cremer, M. Heckl, and E. Ungar, Structure-borne Sound, 2nd Edition (Springer Verlag, 1988).
2 D. Ross, Mechanics of Underwater Noise (Peninsula Publishing, 1987).
3 F. J. Fahy, Sound and Structural Vibration: Radiation, Transmission, and
Response (Academic Press, 1987).
4 M. C. Junger and D. Feit, Sound, Structures, and Their Interaction
(Acoustical Society of America, 1993).
5 Note we are considering only acoustic waves in our discussion here,
since fluid dynamicists will take great issue with ignoring a viscous
fluid’s ability to resist shear.
6 Note that there are two off-axis directions, so that a maximum
Poisson’s ratio of 0.5 implies the deformation in both off-axes sums to
the deformation in the main axis.
7 Thin beam theory is sometimes called Bernoulli-Euler theory.
8 Thick beam theory is usually attributed to Timoshenko.
9 Thick plate theory is usually attributed to Mindlin.
10 Exhibits like this for horizontal acoustic resonance tubes are
common in science museums, where the frequency of sound within the tube is adjusted until acoustic resonance of the air column occurs and the sound waves excite a shallow pool of water in the tube leading to strong water pulsations at the peaks of the sound waves.
11 The wave number of a free wave is the radial frequency, ω, divided by its sound speed. The wavenumber of a mode shape is the number of radians over the spatial vibration pattern divided by the length of the pattern.
12 A. Leissa, Vibration of Plates (Acoustical Society of America, 1993).
13 R. Blevins, Formulas for Natural Frequency and Mode Shape (Van
Nostrand Reinhold, 1979).
14 With these boundary conditions, low-frequency resonances exist
where the plate vibrates as a rigid body attached to soft springs. These
resonances do not affect those of the flexural modes in the plate.
15 Mobility is also called admittance.
16 D. J. Ewins, Modal testing: Theory and practice (J. Wiley & Sons, 1986).
17 P. Avitabile, “Experimental Modal Analysis–a Simple Non-
Mathematical Presentation,” Sound and Vibration, (January 2001).
18 This is only true for simply supported, homogeneous plates. However, the modal mass of any mode shape is always a fraction of the total stat- ic mass, and for high order mode shapes of plates with other bound- ary conditions, the 1/4 factor is a reasonable approximation. For beam-like flexural modes, the modal mass is about half of the static
mass.
19 A. D. Nashif, D. I. G. Jones, and J. P. Henderson, Vibration Damping (J.
Wiley and Sons, 1985).
20 L. L. Beranek, Noise and Vibration Control, Revised Edition (McGraw
Hill, 1988).
21 E. Ungar, “Damping by Viscoelastic Layers,” Appl. Mech. Rev. 53, 6
(2000).
22 S. A. Hambric, A. W. Jarrett, G. F. Lee, and J. F. Fedderly, “Inferring
Viscoelastic Dynamic Material Properties from Finite Element and Experimental Studies of Beams with Constrained Layer Damping,” ASME J. of Vib. and Acoust. (to be published in 2007).
23 S. A. Hambric and A. D. Munro, “Predicted and measured mobilities of the INCE standard ribbed panels,” Proceedings of NoiseCon 2001, Portland, ME (October 2001).
24 A. Leissa, Vibration of Shells (Acoustical Society of America, Melville, NY, 1993).
25 B. J. Doty, S. A. Hambric, S. C. Conlon, and J. B. Fahnline, “Structural- Acoustic Measurements of Pipes with Ninety-Degree Elbows, Under Water Loading,” Proceedings of NoiseCon 2005, Minneapolis, MN (October 2005).
26 C. Zienkiewicz, The Finite Element Method, 5th ed. (Butterworth- Heinemann, 2000).
Steve Hambric and his daughter, Lily
Stephen A. Hambric is head of the Structural Acoustics Department at the Applied Research Lab at Penn State Univ. and Associate Professor in the Graduate Program in Acoustics. Prior to joining Penn State nine years in the
in 1996, Dr. Hambric worked for
Computational Mechanics Office at the Naval Surface Warfare Center, Carderock Division. Dr. Hambric has directed many numerical and experimental flow and structural acoustics research and development programs for the Navy, U.S. indus- try, and the U.S. Nuclear Regulatory Commission. He has authored over 60 conference and journal articles and advised many graduate students at Penn State. He teaches courses in Structural Acoustics, and Writing for Acousticians on campus at Penn State, and also to off-campus students working in industry and government. He currently serves on the board of directors of the Institute for Noise Control Engineering (INCE), on the Executive Committee of the American Society of Mechanical Engineers (ASME) Noise Control and Acoustics Division, and as an associate editor of ASME’s Journal of Vibration and Acoustics.
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Structural Acoustics Tutorial 33