$-------------------------------------------------------------------------------
$      RIGID FORMAT No. 11, Frequency Response Analysis - Modal Formulation
$                Frequency Response of a 500-Cell String (11-2-1)
$             Frequency Response of a 500-Cell String (INPUT, 11-2-2)
$ 
$ A. Description
$ 
$ This problem illustrates the solution of a large frequency response problem
$ using modal coordinates. When large numbers of frequency steps are used, or
$ the problem is very large, the relative efficiency of the modal formulation is
$ more attractive than the direct formulation. The structural model consists of
$ scalar points, springs, and masses which simulate the transverse motions of a
$ string under tension, T, with a mass per length of mu. A duplicate model is
$ obtained via the INPUT module to generate the scalar springs and masses.
$ 
$ Selected scalar point displacements and scalar element forces are plotted
$ versus frequency. The magnitude and phase of the displacements are plotted
$ separately, each on one-half of the plotter frame. The magnitude plots for the
$ selected points are all drawn on a whole plotter frame for comparisons. The
$ center spring element has the magnitude of its internal force plotted versus
$ frequency.
$ 
$ B. Input
$ 
$ 1. Parameters:
$ 
$    m   =  10  - mass
$     i
$             7
$    K   =  10  - spring rate
$     i
$ 
$    N   =  500 - number of cells
$ 
$    where
$               T
$      K   =  -------  ,  m   =  mu delta x
$       i     delta x      i
$ 
$ 2. Loads
$ 
$    The load on each point is:
$ 
$                                   3
$      P (w)  =  delta xp   =  10 pi
$       i                x
$ 
$    where p  is the load per length of string.
$           x
$ 
$    The steady state frequency response is desired from .1 to 10 cycles per
$    second in 15 logarithmic increments.
$ 
$ 3. Real Eigenvalue Data
$ 
$    Method: FEER
$ 
$    Center of neighborhood: 10.5
$ 
$    Normalization: maximum deflection
$ 
$    Number of modes used in formulation: 20
$ 
$ C. Theory
$ 
$ The analysis of the string is given in Reference 11, Chapter 6. The response,
$ xi , of mode number n is given by the equation:
$   n
$ 
$                                     n pi x
$            integral o to l P(x) sin(------) dx
$                                        l
$    xi   =  -----------------------------------                             (1)
$      n                            2 n pi x   2    2
$            [integral o to l mu sin (------)[w  - w ]
$                                       l      n
$ 
$ where w , the natural frequencies, are (n pi/N)sqrt(K /m ) for the theoretical
$        n                                             i  i
$ continuous string.
$ 
$ For a uniform load:
$                                          2p l    2P N      4  2
$                             n pi x         x       i     10 pi
$    integral o to l P(x) sin(------) dx = ----  = ----  = ------            (2)
$                               l          n pi    n pi      n
$ 
$                                                Nm
$                          2 n pi x       ul       i           3
$    integral o to l mu sin (------) dx = --  =  --- = 2.5 x 10              (3)
$                              l          2       2
$ 
$ The displacement of the center point is:
$ 
$      l                  n pi
$    u(-)  =  sum xi  sin ----  = xi  - xi  + xi  -xi  + ...                 (4)
$      2            n      2        1     3     5    7
$ 
$ D. Results
$ 
$      At f = 0.1, the response due to 20 modes is:
$ 
$        l
$      u(-)  =  .97895 (Theory)
$        2
$ 
$      u     =  .97888 (NASTRAN)
$       251
$ 
$ APPLICABLE REFERENCES
$ 
$ 11. I. S. Sokolnikoff and R. M. Redheffer, MATHEMATICS OF PHYSICS AND MODERN
$     ENGINEERING. McGraw-Hill, 1958.
$-------------------------------------------------------------------------------
