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$           RIGID FORMAT No. 12, Transient Analysis - Modal Formulation
$           Transient Analysis of a Free One Hundred Cell Beam (12-1-1)
$ 
$ A. Description
$ 
$ 
$ The problem demonstrates the transient analysis of a free-body using the
$ integration algorithm for uncoupled modal formulations. The model is a
$ hundred-cell beam with a very large mass attached to one end. Modal damping is
$ included as a function of natural frequency. It does not affect the free-body
$ (zero frequency) modes. The omitted coordinate feature was used to reduce the
$ analysis set of displacements to correspond to eleven grid points.
$ 
$ Both structure plots and curve plots are requested. The types are as follows:
$ 
$ 1. Stereoscopic structure plots of the deformed structure are drawn for a
$    specified time step.
$ 
$ 2. Orthographic projections of the deformed structure are plotted. However,
$    two variations are plotted on each frame. The bottom region of the frame
$    shows the deformed shape and the top region shows vectors at every tenth
$    grid point which are proportional to the z-displacement at each specified
$    time step.
$ 
$ 3. Curve plots and printout of displacement versus time and of acceleration
$    versus time are requested.
$ 
$ When a structure is used without additional transfer functions or direct
$ matrix inputs, the transient analysis solves exact equations for the uncoupled
$ modes. The only errors will be in the discarded modes and the straight line
$ approximation of the loads between time steps. The speed of this solution is
$ offset by the fact that the eigenvalue calculation is relatively costly and
$ the transformation of the vectors to and from modal coordinates could be time
$ consuming.
$ 
$ The mass and inertia on point (1) were selected to be much larger than values
$ of the beam. The answers will therefore approximate a beam with a fixed end.
$ 
$ B. Input
$ 
$ 1. Parameters
$ 
$    Beam:
$ 
$      l =  20             (Length)
$ 
$      I =  .083           (Bending inertia)
$ 
$      A =  1.0            (Cross sectional area)
$                    6
$      E =  10.4 x 10      (Modulus of elasticity)
$                     -3
$      p =  .2523 x 10     (Mass density)
$ 
$    Lumped Mass:
$ 
$      m   =  10.0, I      =  1666.66
$       1            22,1
$ 
$ 2. Damping:
$ 
$    The damping coefficient for each mode is a function of the natural
$    frequency. The function is:
$ 
$              -3
$      g  =  10  f
$ 
$ 3. Load:
$ 
$      P       =  100 sin(2 pi 60t)
$       z,101
$ 
$ 4. Real Eigenvalue Data
$ 
$    Method: Inverse Power
$ 
$    Region of Interest: 0 < f < 1000
$ 
$    Normalization: Mass
$ 
$ D. Results
$ 
$ The modal mass may be calculated using the formula for the mode shape given in
$ Reference 8. The modal displacement is a single degree of freedom response
$ with a closed form solution.
$ 
$ 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.
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