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$           RIGID FORMAT No. 9, Transient Analysis - Direct Formulation
$    Transient Analysis of a 1000 Cell String, Traveling Wave Problem (9-2-1)
$        Transient Analysis of a 1000 Cell String, Traveling Wave Problem
$                                 (INPUT, 9-2-2)
$ 
$ A. Description
$ 
$ This problem illustrates the ability of NASTRAN to perform time integration
$ studies using the structural matrices directly. At each time step the applied
$ loads, the structural matrices, and the previous displacements are used to
$ calculate a new set of displacements, velocities, and accelerations. Initial
$ displacements and velocities are also allowed for all unconstrained
$ coordinates. The INPUT module is used to generate the scalar springs and
$ masses.
$ 
$ The structural model consists of a 1000 cell string under constant tension
$ modeled by scalar elements. The string is given an initial condition at one
$ end consisting of a triangular shaped set of initial displacements. The wave
$ will then travel along the string, retaining its initial shape. The ends of
$ the string are fixed, causing the wave to reflect with a sign reversal.
$ 
$ B. Input
$ 
$ 1. Parameters:
$ 
$                T          7
$      k   =  -------  =  10      - scalar spring rates
$       i     delta x
$ 
$      m   =  mu delta x  =  10   - scalar masses
$       i
$ 
$      N   =  1000                - number of cells
$ 
$    where
$ 
$      T is the tension
$ 
$      delta x is the incremental length
$ 
$      mu is the mass per unit length
$ 
$ 2. Loads:
$ 
$    The initial displacements are;
$ 
$      u   =  .2               u   =  1.8
$       2                       12
$ 
$      u   =  .4               u   =  1.6
$       3                       13
$                                 :
$      u   =  .6                  :
$       4                      u   = 0.0
$         :                     21
$         :
$      u   =  2.0              u   = 0, i > 21
$       11                      i
$ 
$ C. Theory
$ 
$ As shown in Reference 11, Chapter 6, the wave velocity c is,
$ 
$    c  =  +/- sqrt(T/mu) = +/- x sqrt(k /m ) = +/-1000 points/unit time     (1)
$                                       i  i
$ 
$ The initial displacement may be divided into two waves, traveling in opposite
$ directions. The first wave travels outward; the second wave travels toward the
$ fixed support and reflects with a sign change.
$ 
$ D. Results
$ 
$ The theoretical and NASTRAN results are quite close, when both waves have
$ traveled their complete width.
$ 
$ APPLICABLE REFERENCES
$ 
$ 11. I. S. Sokolnikoff and R. M. Redheffer, MATHEMATICS OF PHYSICS AND MODERN
$     ENGINEERING. McGraw-Hill, 1958.
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