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$                  RIGID FORMAT No. 3, Real Eigenvalue Analysis
$           Vibration of a Liquid in a Half-Filled Rigid Sphere (3-3-1)
$ 
$ A. Description
$ 
$ The model is similar to Demonstration Problem No. 3-2-1 except that a
$ hemispherical fluid model with a free surface is analyzed. Additional cards
$ demonstrated are the free surface list (FSLIST) free surface points (FREEPT).
$ The effective gravity for the fluid is found on the AXIF card. The fluid is
$ considered incompressible.
$ 
$ The lowest three eigenvalues and eigenvectors for the cosine and sine series
$ of n = 1 are analyzed, where n is the harmonic order.
$ 
$ B. Input
$ 
$ 1. Parameters
$ 
$                     2
$    g  =  10.0 ft/sec           (Gravity)
$ 
$    R  =  10.0 ft               (Radius of hemisphere)
$ 
$                         2   4
$    p  =  1.255014 lb-sec /ft   (Fluid mass density]
$ 
$    B  =  infinity              (Bulk modulus of fluid, incompressible)
$ 
$ C. Results
$ 
$ Reference 17 gives the derivations and analytical results. In particular, the
$ parameters used in the reference are:
$ 
$    e = 0  (half-filled sphere)
$ 
$              2                                                             (1)
$             w R
$    lambda = ----  (dimensionless eigenvalue)
$              g
$ 
$ Table 2 of Reference 17 lists the eigenvalues, lambda , lambda , and
$                                                      1        2
$ lambda , for the first three modes. Figure 13 of Reference 17 shows the mode
$       3
$ shapes.
$ 
$ The analytic and NASTRAN results are compared in Table 1. The frequencies are
$ listed and the resulting percentage errors are given. The maximum percent
$ error of the surface displacement, relative to the largest displacement, is
$ tabulated for each mode.
$ 
$ The free surface displacements may be obtained by the equation:
$ 
$        p
$    u = --                                                                  (2)
$        pg
$ 
$ where p is the pressure at the free surface recorded in the NASTRAN output.
$ Note that, since an Eulerian reference frame is used, the pressure at the
$ original (undisturbed) surface is equal to the gravity head produced by
$ motions of the surface. Special FREEPT data cards could also have been used
$ for output. Since the results are scaled for normalization anyway, the
$ harmonic pressures may be used directly as displacements.
$ 
$ 
$ Because the cosine series and the sine series produce identical eigenvalues,
$ the resulting eigenvectors may be linear combinations of both series. In other
$ words the points of maximum displacement will not necessarily occur at phi = 0
$ degrees or phi = 90 degrees. Since the results are scaled, however, and the
$ results at phi = 0 are proportional to the results at any other angle, the
$ results at phi = 0 were used.
$ 
$     Table 1. Comparison of Natural Frequencies and Free Surface Mode Shapes
$                         from the Reference and NASTRAN
$ 
$            ---------------------------------------------------------
$                          Natural Frequency (Hertz)
$                       --------------------------------  Mode Shape
$            Mode                              NASTRAN    Max. % Error
$            Number     Reference   NASTRAN    % Error    epsilon
$            ---------------------------------------------------------
$              1        0.1991      0.1988     -0.1        < 1
$ 
$              2        0.3678      0.3691      0.3        < 2.6
$ 
$              3        0.4684      0.4766      1.8        < 4
$            ---------------------------------------------------------
$ 
$ APPLICABLE REFERENCES
$ 
$ 17. B. Budiansky, "Sloshing of Liquids in Circular Canals and Spherical
$     Tanks", Journal of the Aerospace Sciences, Vol. 27, No. 3, pp 161-173,
$     March 1960.
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