$-------------------------------------------------------------------------------
$                  RIGID FORMAT No. 3, Real Eigenvalue Analysis
$             Vibrations of a Linear Tapered Cantilever Plate (3-7-1)
$ 
$ A. Description
$ 
$ This problem demonstrates the use of the higher order triangular bending
$ element TRPLT1 to solve a normal modes analysis. The structural model is that
$ of a thin isotropic plate with tapered cross section, cantilevered at one end.
$ 
$ B. Input
$ 
$                7      2
$    E = 3.0 x 10  lb/in      (Modulus of elasticity)
$ 
$                    -5   4
$    I  = 4.3877 x 10   in    (Maximum bending inertia)
$     o
$ 
$    t  = 0.0807 in           (Maximum thickness)
$     o
$ 
$    a  = 5.0 in              (Length)
$ 
$    v  = .3                  (Poisson's ratio)
$ 
$                       2   4
$    p  =  7.3698 lb sec /in  (Mass density)
$ 
$ C. Theory
$ 
$ The theory for the tapered plate elements is developed in Reference 33. In
$ this reference, a frequency parameter is defined as
$ 
$               2
$    omega = w a  sqrt (pt  / D )                                            (1)
$                         o    o
$ 
$ where
$ 
$    a   =  length
$ 
$    p   =  mass density
$ 
$    w   =  circular frequency
$ 
$    t   =  thickness
$     o
$ 
$ The bending rigidity, D , is defined as
$                        o
$ 
$              3
$            Et
$              o
$    D  = ----------                                                         (2)
$     o           2
$         12(1 - v )
$ 
$ D. Results
$ 
$ The results of the NASTRAN analysis using the TRPLT1 element are presented in
$ Table 1. For purposes of comparison, results are presented from an experiment
$ described by Plunkett in Reference 34. In this table the modes are identified
$ by m and n, where m represents the number of nodal lines perpendicular to the
$ support and n represents the number of nodal limes parallel to the support.
$ 
$               Table 1. Frequency Parameters for a Linearly Tapered
$                       Rectangular Cantilever Plate; v = 0.3
$                  ----------------------------------------------
$                                        Frequency Parameter
$                                                   2        1/2
$                                       omega  =w  a (pt /D )
$                     Mode                   mn  mn     o  o
$                  ----------------------------------------------
$                  m         n        TRPLT1         Experiment
$                  ----------------------------------------------
$                  0         0         2.25             2.47
$ 
$                  1         0        10.0             10.6
$ 
$                  0         1        13.6             14.5
$ 
$                  1         1        27.0             28.7
$ 
$                  0         2        32.8             34.4
$ 
$                  0         3        47.3             47.4
$ 
$                  2         0        53.3             52.5
$ 
$                  1         2        57.7             54.0
$                  ----------------------------------------------
$ 
$ APPLICABLE REFERENCES
$ 
$ 33. Leissa, A. W.: Vibration of Plates, NASA SP-160, 1969, Chapter 11.
$ 
$ 34. Plunkett, R.: "Natural Frequencies of Uniform and Non-Uniform Rectangular
$     Cantilever Plates", J. Mech. Engr. Sci., Vol 5, 1963, pp. 146-156.
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