$-------------------------------------------------------------------------------
$      RIGID FORMAT No. 7, Complex Eigenvalue Analysis - Direct Formulation
$            Complex Eigenvalue Analysis of a 500-Cell String (7-1-1)
$         Complex Eigenvalue Analysis of a 500-Cell String (INPUT, 7-1-2)
$ 
$ A. Description
$ 
$ This problem demonstrates both the use of direct complex eigenvalue analysis
$ and the various methods of supplying damping to the structure. The simulated
$ model is a string under tension having uniform viscous and structural damping.
$ The stiffness due to tension is modeled with scalar springs, the mass is
$ represented by scalar masses, and the viscous damping is provided by scalar
$ dampers connected on one end to the points and fixed on the other end. The
$ structural damping is provided by the scalar springs and an overall damping
$ factor, g . The INPUT module is used to generate the scalar springs, dampers,
$          3
$ and masses.
$ 
$ B. Input
$ 
$ 1. Parameters:
$ 
$                   7
$    k   =  1.0 x 10    (scalar springs)
$     i
$ 
$    m   =  10.0        (scalar masses)
$     i
$ 
$    b   =  6.28318     (scalar dampers)
$     i
$ 
$    g   =  0.05        (structural element damping)
$     s
$ 
$    g   =  0.05        (overall damping parameter)
$     3
$ 
$    N   =  500         (number of scalar springs)
$ 
$ 2. Constraints:
$ 
$    The end scalar springs are fixed on the outer ends so constraints are
$    unnecessary.
$ 
$ 3. Eigenvalue ExtractIon Data:
$ 
$    Method: FEER
$ 
$    Center Point: (r,i) = (-1.0, 15.0)
$ 
$    Normalization: Maximum deflection
$ 
$ C. Theory
$ 
$ Complex Eigenvalue Analysis is used to solve the following general matrix
$ equation:
$ 
$         2
$    ([M]p  + [B]p + [K]){u}  =  0                                           (1)
$ 
$ where
$ 
$    p     is the complex root
$ 
$    [M]   is the complex mass matrix
$ 
$    [B]   is the complex damping matrix for viscous damping
$ 
$    [K]   is the complex stiffness matrix which contains imaginary components
$          representing the structural damping.
$ 
$ According to Reference 11, Chapter 6, the differential equation for a string
$ under tension is
$ 
$       2           2
$      a u         a u         au
$    T ---  =  -mu ---  - beta --                                            (2)
$        2            2        at
$      ax           at
$ 
$ where
$ 
$     T    is the string tension
$ 
$     mu   is the mass per unit length
$ 
$     beta is the viscous damping per unit length
$ 
$ The finite difference representation for this equation is
$ 
$                                       2
$                                      d  u         du
$       T                                  i          i
$    -------- (u   - 2u  + u   ) = -mu -----  -beta ---                      (3)
$           2   i-1    i    i+1           2         dt
$    delta x                            dt
$ 
$ The equation of the finite element model which corresponds to this equation is
$ 
$      ..     .
$    m u  + b u  + (1 + ig)k  (u    - 2u  + u   )  =  F                      (4)
$     i i    i i            i   i-1     i    i+1       i
$ 
$ where
$ 
$    g  =  g  + g         (additional structural damping defined for the
$           3    s         NASTRAN solution)
$ 
$    m   =  mu delta x    (element mass)
$     i
$ 
$    b   =  beta delta x  (element vi scorns damping)
$     i
$ 
$              T
$    k   =  --------      (element stiffness)
$     i            2
$           delta x
$ 
$ The natural frequency for an undamped string, according to Reference 11, is
$ 
$           pi n              pi n
$    w   =  ---- sqrt(T/mu) = ---- sqrt(k /m )  = 2 pi n                     (5)
$     n      l                  N        i  i
$ 
$ Its deflection shape is
$ 
$                 n pi x
$    u(x)  =  sin -------                                                    (6)
$                    l
$ 
$ and
$ 
$                  n pi i
$    phi    =  sin ------                                                    (7)
$       in            N
$ 
$ The modal masses are
$ 
$                                                        m N
$                                    2          mu l      i
$    M   =  integral from o to l mu u (x)dx  =  ----  =  ---                 (8)
$     n                                           2       2
$ 
$ Substituting the real eigenvectors and eigenvalues into the complex equation
$ for complex roots we obtain for each mode, n,
$ 
$       2                           2
$    M p  + (b /m ) M p + (1 + ig) w  M  = 0                                 (9)
$     n       i  i   n              n  n
$ 
$ The solution is
$ 
$            b
$             i                     2          2
$    p  =  - ----  +/- sqrt((b /2m )  - (1+ig)w  )                          (10)
$            2m               i   i            n
$              i
$ 
$ D. Results
$ 
$ The theoretical and NASTRAN complex roots are presented below in Table 1. The
$ eigenvectors, which are the same as the real eigenvectors, are nearly exact
$ for the finite element model.
$ 
$                   Table 1. NASTRAN and Analytical Complex Roots
$        -------------------------------------------------------------------
$             Real Natural
$              Frequency        Theoretical Roots          NASTRAN Roots
$        n        (Hz)        (radians per second)     (radians per second)
$        -------------------------------------------------------------------
$        1        1.0          - .6283 +/- 6.2832i      - .6283 +/- 6.2832i
$ 
$        2        2.0          - .9419 +/- 12.578i      - .9419 +/- 12.578i
$ 
$        3        3.0          -1.2556 +/- 18.870i      -1.2556 +/- 18.870i
$ 
$        4        4.0          -1.5693 +/- 25.162i      -1.5693 +/- 25.161i
$        -------------------------------------------------------------------
$ 
$ APPLICABLE REFERENCES
$ 
$ 11. I. S. Sokolnikoff and R. M. Redheffer, MATHEMATICS OF PHYSICS AND MODERN
$     ENGINEERING. McGraw-Hill, 1958.
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