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$                  RIGID FORMAT No. 3, Real Eigenvalue Analysis
$        Vibration of a Compressible Gas in a Rigid Spherical Tank (3-2-1)
$ 
$ A. Description
$ 
$ This problem demonstrates a compressible gas in a rigid spherical container.
$ In NASTRAN a rigid boundary is the default for the fluid and, as such, no
$ elements or boundary lists are necessary to model the container.
$ 
$ Aside from the NASTRAN bulk data cards currently implemented, this problem
$ demonstrates the use of the hydroelastic data cards: AXIF, CFLUID2, CFLUID3,
$ and RINGFL.
$ 
$ The lowest mode frequencies and their mode shapes for n = 0, 1, and 2 are
$ analyzed where n is the Fourier harmonic number. Only the cosine series is
$ analyzed.
$ 
$ B. Model
$ 
$ 1. Parameters
$ 
$    R = 10.0 m              (Radius of sphere)
$ 
$                -3     3
$    p = 1.0 x 10   Kg/m     (Mass density of fluid)
$ 
$                3         2
$    B = 1.0 x 10  Newton/m  (Bulk modulus of fluid)
$ 
$ 2. The last 3 digits of the RINGFL identification number correspond
$    approximately to the angle, theta, from the polar axis along a meridian.
$ 
$ C. Theory
$ 
$ From Reference 18, the pressure in the cylinder is proportional to a series
$ of functions:
$ 
$             J      (X)
$              m+1/2        n
$    Q     =  -----------  p  (cos theta) cos n phi ,   n <= m               (1)
$     n,m      sqrt(X)      m                           m = 0,1,2
$ 
$ 
$ where:
$ 
$    Q       Pressure coefficient for each mode
$     n,m
$                                     w
$                                      mk
$    X       Nondimensional radius  = ---- r
$                                      a
$ 
$    w       Natural frequency for the kth mode number and mth radial number in
$     mk     radlans per second
$ 
$ 
$    J       Bessel function of the first kind
$     m+1/2
$ 
$    r       radius
$ 
$    a = sqrt(B/p)  speed of sound in the gas
$ 
$     n
$    p       associated Legendre functions
$     m
$ 
$    theta   meridional angle
$ 
$    phi     circumferential angle
$ 
$    n       harmonic number
$ 
$    m       number of radial node lines
$ 
$ The solution for X and hence w   is found by the use of the boundary
$                               mk
$ condition that the flow through the container is zero.
$ 
$    +       +             + +
$    |       | J      (X)  | |
$    |  d    |  m+1/2      | |
$    | ---   | ----------- | |      = 0.0                                    (2)
$    |  dX   |  sqrt(X)    | |
$    +       +             + + r=R
$ 
$ where R is the outer radius.
$ 
$ This results in zero frequency for the first root. Multiple roots for other
$ modes can be seen in Table 1. The finite element model assumes different
$ pressure distributions in the two angular directions, which causes the
$ difference in frequencies.
$ 
$ D. Results
$ 
$ Table 1 summarizes the NASTRAN and analytic results for the lowest nonzero
$ root in each harmonic. Table 1 lists the theoretical natural frequencies, the
$ NASTRAN frequencies, the percent error in frequency, and the maximum percent
$ error in pressure at the wall as compared to the maximum value. Theoretical
$                                                              0
$ pressure distributions correspond to the Legendre functions P  (cos theta),
$                                                              o
$  1                   2
$ P  (cos theta), and P  (cos theta), which are proportiomal to cos theta,
$  o                   o
$                   2
$ sin theta, and sin theta, respectively.
$ 
$              Table 1. Comparison of NASTRAN and Analytical Results.
$ 
$             ---------------------------------------------------------
$                           Natural Frequency (Hertz)
$                        --------------------------------  Pressure
$                                                          Max. % Error
$             Harmonic   Analytical  NASTRAN    % Error    at Wall
$             ---------------------------------------------------------
$               0        33.1279     33.2383     0.33        1.19
$ 
$               1        33.1279     33.2060     0.24        0.47
$ 
$               2        53.1915     53.3352     0.27        0.91
$             ---------------------------------------------------------
$ 
$ APPLICABLE REFERENCES
$ 
$ 18. B. Raylelgh, THE THEORY OF SOUND. Section 330, 331, MacMillan Co., 1945.
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