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$       RIGID FORMAT No. 7, Complex Eigenvalue Analysis - Direct Formulation
$            Third Harmonic Complex Eigenvalue Analysis of a Gas-Filled
$                           Thin Elastic Cylinder (7-2-1)
$            Fifth Harmonic Complex Eigenvalue Analysis of a Gas-Filled
$                           Thin Elastic Cylinder (7-2-2)
$ 
$ A. Description
$ 
$ This problem demonstrates the use of symmetry to analyze specific harmonics of
$ a fluid-filled structure. The problem to be solved consists of a cylindrical
$ section filled with a compressible fluid. The end conditions for the cylinder
$ and the fluid are two planes of antisymmetry, perpendicular to the axis. These
$ end conditions correspond to the conditions that exist at periodic intervals
$ along a long, fluid-filled pipe vibrating in one of its vibration modes. The
$ antisymmetric boundary for the structure is defined by constraining the
$ motions which lie in the plane. An antisymmetric boundary for the fluid
$ corresponds to zero pressure. This may be modeled, in NASTRAN, by defining the
$ plane of antisymmetry as a free surface with zero gravity.
$ 
$ The lowest natural frequencies and mode shapes for the third and fifth
$ harmonics are analyzed separately. For the third harmonic, the structure is
$ defined by a section of a cylinder having an arc of 30 degrees or 1/12 of a
$ circle. The fifth harmonic analysis uses a section having an arc of 18 degrees
$ or 1/20 of a circle. The longitudinal edges, which were cut, are planes of
$ symmetry and antisymmetry in order to model a quarter cosine wave length.
$ 
$ The bulk data cards used are; AXIF, BDYLIST, CFLUID2, CFLUID4, CORD2C, CQUAD1,
$ EIGC, FLSYM, FSLIST, GRIDB, MAT1, PQUAD1, RINGFL, and SPC1.
$ 
$ B. Input
$ 
$ The finite element model for the third harmonic uses the following parameters:
$ 
$                 3      2
$    B = 2.88 x 10  lb/in             (Bulk modulus of fluid)
$ 
$                 -2       2   4
$    p  = 1.8 x 10   lb-sec /in       (Fluid mass density)
$     f
$                 -2       2   4
$    p  = 6.0 x 10   lb-sec /in       (Structure mass density)
$     s
$ 
$                 5      2
$    E  = 1.6 x 10  lb/in             (Young's modulus for structure)
$ 
$                4      2
$    G = 6.0 x 10  lb/in              (Shear modulus for structure)
$ 
$    a  =  10.0 inch                  (Radius of cylinder)
$ 
$    l  =  10.0 inch                  (Length of cylinder)
$ 
$    h  =  0.01 inch                  (Thickness of cylinder)
$ 
$ The model for the fifth harmonic is similar to the third harmonic model except
$ that the angle covered by the structure is 18 degrees instead of 30 degrees.
$ This is accomplished by simply removing the structural elements and boundary
$ GRIDB points corresponding to the two right-hand layers of structure (between
$ 18 degrees and 30 degrees). The FLSYM, FSLIST, and SPC1 cards are changed to
$ reflect the 1/20 symmetry.
$ 
$ C. Theory
$ 
$ The derivations and results for this problem are described in Reference 16.
$ The results for various dimensionless parameters are listed. The particular
$ parameters for the problem at hand are:
$ 
$          p a
$           f
$    n  =  ---  =  300.0
$          p h
$           s
$ 
$               Gp
$                 f
$    C  =  sqrt(---)  =  2.5
$               Bp
$                 x
$ 
$              P a
$               o
$    omega  =  ---  =  0.0
$              Gh
$ 
$ where n is the ratio of fluid mass to structure mass. C is the ratio of the
$ wave velocity in the structure material to the wave velocity in the fluid.
$ Omega is the factor describing static pressurization, P .
$                                                        o
$ 
$    The basic assumptions for this analysis are:
$ 
$    1. Thin shell theory is used for the structure. The bending moment terms in
$       the force equilibrium equations are ignored in the results.
$ 
$    2. The fluid is nonviscous and irrotational, and small motions are only
$       considered.
$ 
$ This particular problem becomes relatively easy to solve since the mode shapes
$ for the fluid in a rigid container and the modes of the structure with no
$ enclosed fluid have the same spatial function at the interface. Each mode of
$ the fluid is excited by only one mode of the structure and each mode of the
$ structure is excited by one mode of the fluid. The pressure in the fluid is
$ assumed to be a series of functions:
$ 
$              iwt               pi z
$    p  =  p  e    cos n phi sin ----  Q  (r,w)                              (1)
$           n                     l     n
$ 
$ where Q  is a Bessel Function or a modified Bessel Function of the first kind.
$        n
$ 
$ The characteristic shapes of the structure are a series of the form:
$ 
$              iwt               pi z
$    u  =  A  e    cos n phi sin ----                                        (2)
$                                 l
$ 
$ where u is the displacement normal to the surface. The fundamental momentum
$ equation for the fluid flow at the boundary is:
$ 
$                                ..
$    gradient (P(r)) e   =  - p  u                                           (3)
$                     r        f
$ 
$ 
$ where e  is a unit vector in the radial direction.
$        r
$ 
$ The forces on the structure at the boundary are:
$ 
$                 2
$                a F
$             1     1        ..
$    P(a)  =  -  ----  - p  hu                                               (4)
$             a    2      s
$                az
$ 
$ where the function F  is defined by the differential equation on the surface:
$                     1
$ 
$                          2
$            4        Eh  a u
$    gradient  F   =  --  ---                                                (5)
$               1     a     2
$                         az
$ 
$ The solution for F  is obtained by assuming that
$                   1
$ 
$              iwt               pi z
$    F   =  B e    cos n phi sin ----                                        (6)
$     1                           l
$ 
$ Combining Equations 1 through 6 results in the relationships:
$ 
$                  aQ (r,w)  |
$      2             n       |
$    pw  A  =  P   --------  |                                               (7)
$               n    ar      |r=a
$ 
$                  +                            +
$                  |  2  4                      |
$                  | a pi Eh                  2 |
$    Q (a,w) P  =  | ---------------  +  p  hw  | A                          (8)
$     n       n    |      2 2             s     |
$                  |  4 pi a     2 2            |
$                  | l (----- + n )             |
$                  +       2                    +
$                         l
$ 
$ Equation (7) is a statement of the continuity of displacement. Equation (8)
$ states the balance of the pressures. The above equations may be solved by
$ iterating on w. Reference 16 provides solutions for w over a wide range of
$ parameters.
$ 
$ D. Results
$ 
$ The analytic and NASTRAN eigenvalues are listed in Table 1. The corresponding
$ errors in the eigenvalues are tabulated and the maximum errors in displacement
$ at the container wall are given as the percentage of the maximum value.
$ 
$               Table 1. Comparison of Analytical and NASTRAN Results
$      --------------------------------------------------------------------
$                    Natural Frequency (Hz)           Mode Shape
$                 ---------------------------   ---------------------------
$      Harmonic   Analytical   NASTRAN  Error   Max. Error in Radial Displ.
$      --------------------------------------------------------------------
$         3         1.579       1.595    1.0              0.0
$ 
$         5         1.011       1.049    3.4              0.5
$      --------------------------------------------------------------------
$ 
$ APPLICABLE REFERENCES
$ 
$ 16. J. G. Berry and E. Reissner, "The Effect of an Internal Compressible Fluid
$     Column on the Breathing Vibrations of a Thin Pressurized Cylindrical
$     Shell", Journal of the Aeronautical Sciences, Vol. 25, No. 5, pp 288-294,
$     May 1958.
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