$-------------------------------------------------------------------------------
$      RIGID FORMAT No. 8, Frequency Response Analysis - Direct Formulation
$                   Frequency Response of a 10x10 Plate (8-1-1)
$                   Frequency Response of a 20x20 Plate (8-1-2)
$               Frequency Response of a 10x10 Plate (INPUT, 8-1-3)
$               Frequency Response of a 20x20 Plate (INPUT, 8-1-4)
$ 
$ A. Description
$ 
$ This problem illustrates the use of the direct method of determining
$ structural response to steady-state sinusoidal loads, The applied load is
$ given in terms of complex numbers which reflect the amplitudes and phases at
$ each selected frequency. The steady-state response of the structure at each
$ frequency is calculated in terms of complex numbers which reflect the
$ magnitudes and phases of the results. Both configurations are duplicated via
$ the INPUT module to generate the QUAD1 elements.
$ 
$ The particular model for this analysis is a square plate composed of
$ quadrilateral plate elements. The exterior edges are supported on hinged
$ supports and symmetric boundaries are used along x = 0 and y = 0. The applied
$ load is sinusoidally distributed over the panel and increases with respect to
$ frequency. Although the applied load excites only the first node, the direct
$ formulation algorithm does not use this shortcut and solves the problem as
$ though the load were completely general.
$ 
$ B. Input
$ 
$ 1. Parameters:
$ 
$    a =  b = 10    - length and width of quarter model
$ 
$    t =  2.0       - thickness
$ 
$                 7
$    E =  3.0 x 10  - Young's Modulus
$ 
$    v =  0.3       - Poisson's Ratio
$ 
$    mu = 13.55715  - nonstructural mass per area
$ 
$ 2. Loads:
$ 
$    The frequency dependent pressure function is:
$ 
$                            pi x      pi y
$      P(x,y,f)  =  F(f) cos ----  cos ----                                  (1)
$                             2a        2b
$ 
$    where
$ 
$      F(f)  =  10. + 0.3f                                                   (2)
$ 
$ 3. Constraints:
$ 
$    Only vertical notions and bending rotations are allowed, The exterior
$    edges are hinged supports. The interior edges are planes of symmetry, This
$    implies:
$ 
$      along x = 0, theta  = 0
$                        y
$ 
$      along y = 0, theta  = 0
$                        x
$ 
$      along x = a, u  = theta  = 0
$                    z        x
$ 
$      along y = b, u  = theta  = 0
$                    z        y
$ 
$      all points, u  = u  = theta  = 0
$                   x    y        z
$ 
$ C. Theory
$ 
$ The excitation of the plate is orthogonal to the theoretical first mode, An
$ explanation of the equations is given in Reference 8. The equations of
$ response are:
$ 
$                     F(f)
$    u (f)  =  ------------------                                            (3)
$     z              2    2    2
$              (2 pi) mu(f  - f )
$                         1
$ 
$ where f  is the first natural frequency (10 Hz).
$        1
$ 
$ D. Results
$ 
$ The following table gives the theoretical and NASTRAN results:
$ 
$                ---------------------------------------------------
$                                                  4
$                                         u    x 10
$                                          z,1
$                Frequency    --------------------------------------
$                   Hz        Theory  10x10 NASTRAN   20x20 NASTRAN
$                ---------------------------------------------------
$                    0         1.868        1.874       1.869
$ 
$                    8         6.435        6.49        6.45
$ 
$                    9        12.489       12.69        12.53
$ 
$                   10        infinite    -824.92      -3284.4
$ 
$                   11       -11.833      -11.67       -11.79
$                ---------------------------------------------------
$ 
$ 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.
$-------------------------------------------------------------------------------
