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$                  RIGID FORMAT No. 3, Real Eigenvalue Analysis
$                       Vibration of a 10x20 Plate (3-1-1)
$                       Vibration of a 20x40 Plate (3-1-2)
$                    Vibration of a 10x20 Plate (INPUT, 3-1-3)
$                    Vibration of a 20x40 Plate (INPUT, 3-1-4)
$ 
$ A. Description
$ 
$ This problem demonstrates the solution for natural frequencies of a large-
$ order problem. The structural model consists of a square plate with hinged
$ supports on all boundaries. The 10x20 model (Problem 3-1-1) represents one
$ half of the structure with symmetric boundary constraints on the mid-line to
$ reduce the order of the problem and the bandwidth by one half. The 20x40 model
$ (Problem 3-1-2) has the same dimensions, but with a finer mesh. Both
$ configurations are developed via the INPUT module (Problems 3-1-3 and 3-1-4
$ for coarse mesh and fine mesh, respectively) to generate the QUAD1 elements.
$ 
$ Because only the bending modes are desired, the in-plane deflections and
$ rotations normal to the plane are constrained. The inverse power method of
$ eigenvalue extraction is selected for the smaller model and the FEER method
$ (Reference 32) is selected for the larger model. Both structural mass density
$ and non-structural mass-per-area are used to define the mass matrix.
$ 
$ An undeformed structure plot is executed without plot elements. This is
$ expensive on most plotters since all four sides of each quadrilateral are
$ drawn. For the deformed plots of each eigenvector, plot elements are used to
$ draw an edge only once and to draw only selected coordinate lines (every
$ second or fourth line depending on the model used).
$ 
$ B. Input
$ 
$ 1. Parameters:
$ 
$      l = w = 20.0   (Length and width)
$ 
$      I = 1/12       (Moment of inertia)
$ 
$      t = 1.0        (Thickness)
$ 
$                  7
$      E = 3.0 x 10   (Modu1us of elasticity)
$ 
$      v = 0.30       (Poisson's ratio)
$ 
$      p = 206.0439  (Mass density, 200.0 structural and 6.0439 non-structural
$                    mass)
$ 
$ 2. Boundary constraints:
$ 
$    along x = 0, theta  =  0           Symmetric Boundary
$                      y
$                                    +
$    along y = 0, u  = theta  =  0   |
$                  z        y        |
$                                    |
$    along x = 10, u  = theta  = 0   |  Hinged Supports
$                   z        x       |
$                                    |
$    along y = 20, u  = theta  = 0   |
$                   z        y       |
$                                    +
$ 3. Eigenvalue extraction data:
$ 
$    Method: Inverse Power and FEER
$ 
$    Region of interest for inverse power: .89 <= f <= 1.0
$ 
$    Center point for FEER: .87
$ 
$    Number of desired roots: 3
$ 
$    Number of estimated roots: 1
$ 
$ C. Results
$ 
$ Table 1 lists the NASTRAN and theoretical natural frequencies as defined in
$ Reference 8.
$ 
$                       Table 1. Natural Frequencies, cps.
$ 
$                      -------------------------------------
$                                           NASTRAN  NASTRAN
$                      Mode   Theoretical   10x20    20x40
$                       No.                 (INV)    (FEER)
$                      -------------------------------------
$                       1        .9069      .9056     .9066
$ 
$                       2       2.2672     2.2634    2.2663
$ 
$                       3       4.5345     4.5329    4.5340
$                      -------------------------------------
$ 
$ APPLICABLE REFERENCES
$ 
$ 8. W. F. Stokey, "Vibration of Systems Having Distributed Mass and
$    Elasticity", Chap. 7, SHOCK AND VIBRATION HANDBOOK, C. M. Harris and C. E.
$    Crede, Editors, McGraw-Hill, 1961.
$ 
$ 32. Newman, Malcolm and Flanaga, Paul F.: Eigenvalue Extraction in NASTRAN by
$     the Tridiagonal Reduction (FEER) Method - Real Eigenvalue Analysis, NASA
$     CR-2731, August, 1976.
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