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Radiation from Vibrating Bodies Read chapter Read first chapter

2018 | OriginalPaper | Chapter

10. Modal Analysis of Enclosures

Author : Jerry H. Ginsberg

Published in: Acoustics-A Textbook for Engineers and Physicists

Publisher: Springer International Publishing

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Abstract

To say that a region containing an ideal fluid is a “cavity” is to imply that it is a void in some solid region. The synonymous term “enclosure” is more descriptive, in the sense that it conveys the notion that the domain is surrounded by a solid boundary. A tube terminated at both ends is the simplest configuration to analyze. Sections 2.​5.​3 and 3.​2.​2 addressed planar waves in such a system, but our concern here will extend to higher order waveguide modes. As is true for planar waves, closure results in reflection of the incident waves, thereby setting up a field that features standing waves in all directions. The developments that follow will primarily deal with regular geometries. The chapter will close by developing an approximate method for irregular cavities. The closure will be an approximate method for cavities and an elastic structures.

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previous chapter Modal Analysis of Waveguides
next chapter Geometrical Acoustics
Appendix
Available only for authorised users
Footnotes
1
It is possible to use the forced mode formulation to determine the field of a source within an enclosed regions. The method follows the development in Sect. 6.​4.​2, in which a solution is constructed by adding a function \(F\left( \bar{x}\right) \) to the free-space source solution \(G\left( \bar{x}\right) \). Similarly, a natural mode formulation can be used to address boundary excitation by adding to the series a term that satisfies all boundary conditions. Equations for the modal coefficients would be obtained by requiring that the sum of the added term and the natural mode series satisfy the inhomogeneous Helmholtz equation.
 
2
M.I. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, p. 440, Eq. 10.1.47 (1965).
 
3
M.I. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, (1965) Eq. 11.3.20, p. 484.
 
4
M.I. Abramowitz and I.A. Stegun, ibid.
 
5
J.H. Ginsberg, “Derivation of a Ritz series modeling technique for acoustic cavity-structural systems based on a constrained Hamilton’s principle,” J. Acoust. Soc. Am. 127, 2749–2758 (2010).
 
6
E. Chaladni, Entdeckungen über die Theorie des Klanges (“Discoveries in the Theory of Sound”) 1787.
 
7
M.I. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, p. 485, Eq. 11.4.5 (1965).
 
8
M.I. Abramowitz and I.A. Stegun, ibid, p. 485, Eq. 11.4.5.
 
9
J.W. Strutt Lord Rayleigh, Theory of Sound, Vol. 2, Dover (1945 reprint) Sect. 88.
 
10
Dowell, E.H., and Voss, H.M. (1962). “The effect of a cavity on panel vibration,” AIAA J., 1, 476–477.
Dowell, E.H., Gorman, G.F., and Smith, D.A. (1977). “Acoustoelasticity: General theory, acoustic natural modes and forced response to sinusoidal excitation, including comparisons with experiment” J. Sound Vib., 52, 519–542.
 
11
Weinstock, Calculus of Variations, Dover (1974).
 
12
Rayleigh is widely recognized for this formulation because it appears in his monumental texts on The Theory of Sound. Walther Ritz was a pioneering nuclear physicist who independently developed the formulation. He came to it by applying variational calculus concepts he first developed for quantum mechanics. The Rayleigh–Ritz formulation actually is a special application of Ritz’ general approach, which is known as the Ritz series method, see J.H. Ginsberg, Mechanical and Structural Vibrations, John Wiley and Sons (2000), Chap. 6. This approach has been used for many analyses in physics and engineering. It also is the foundation for the finite element method. Thus, it is remarkable that Ritz wrote only two papers that discussed acoustic-type systems: “On the new method for solving some variational problems of mathematical physics” (1908) and “Theory of the transverse oscillations of a square plate with free boundaries” (1909). These were the first to explain mathematically the occurrence of Chaladni lines in plate vibrations.
 
13
L. Meirovitch, Principles and Techniques of Vibrations, Prentice-Hall (1997) Chap. 8.
 
14
J.H. Ginsberg, Engineering Dynamics, Cambridge University Press (2008) Chap. 7
 
15
Ginsberg, Mechanical and Structural Vibrations (2001), Ch. 11.
 
Metadata
Title
Modal Analysis of Enclosures
Author
Jerry H. Ginsberg
Copyright Year
2018
Book
Acoustics-A Textbook for Engineers and Physicists

Print ISBN: 978-3-319-56846-1

Electronic ISBN: 978-3-319-56847-8

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-56847-8_10

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