$-------------------------------------------------------------------------------
$                   RIGID FORMAT No. 2, Inertia Relief Analysis
$     Truss Dynamic Analysis Using Automated Modal Synthesis, Run 1, (2-3-1)
$     Truss Dynamic Analysis Using Automated Nodal Synthesis, Run 2, (2-3-2)
$     Truss Dynamic Analysis Using Automated Hodal Synthesis, Run 3, (2-3-3)
$     Truss Dynamic Analysis Using Automated Modal Synthesis, Run 4, (2-3-4)
$ 
$ A. Description
$ 
$ This problem illustrates the automated substructuring and modal synthesis
$ procedures which provide accurate, efficient solutions in dynamic analysis.
$ Each component substructure is reduced to modal and boundary degrees of
$ freedom prior to the substructure combine operation. The combination
$ structure, formulated in terms of the component modes, is also reduced to
$ modal degrees of freedom for solution by the transient analysis rigid format
$ (Reference 37).
$ 
$ Four separate runs are performed.
$ 
$      In Runs 1 and 2, the substructuring Phase 1 operations formulate the
$      finite element matrices and basic static loads using Rigid Format 2.
$ 
$      In Run 3, the basic substructures are reduced to modal coordinates and
$      combined. Another modal synthesis reduction is performed on the
$      combination and the resulting eigenvectors are printed.
$ 
$      In Run 4, the second Phase 2 operation is performed on the reduced
$      structure with the SOLVE operation using Rigid Format 9 (transient
$      analysis). The transient output data is transformed back to the original
$      grid point definitions with the RECOVER command.
$ 
$ B. Input
$ 
$ All members are rod elements. All grid points are constrained to include only
$ in-plane displacements. The basic input and the substructure control data are
$ described below.
$ 
$ 1. Parameters:
$ 
$    a = 40                   (typical frame width)
$ 
$    h = 30                   (typical frame height)
$ 
$    A = 0.3                  (cross section of members)
$ 
$          6
$    E = 10                   (Young's Modulus)
$ 
$                -3
$    P = 2.5 x 10             (density)
$ 
$ 2. Constraints:
$ 
$    u  = theta  = theta  = theta  = 0 all points
$     z        x        y        z
$ 
$    (Boundary conditions are applied only during solution.)
$ 
$ 3. Loads:
$             3
$    P    = 10                load on substructure BBASIC
$     y42
$ 
$ 4. Transient Loads:
$ 
$    U    = .143  @t = 0      initial condition
$     y42
$ 
$             3
$    P    = 10   0 < t < .12  load history
$     y42
$ 
$ 5. Substructuring Parameters:
$ 
$    SOF(1) = SOF1,500 $ CDC
$ 
$    SOF(1) = FT19,500 $ IBM
$ 
$    SOF(1) = INP1,500 $ UNIVAC
$ 
$    PASSWORD = MDLSYN
$ 
$ C. Results
$ 
$ For assessing the accuracy of the modal synthesis, the whole structure is run
$ in Rigid Format 3 to determine natural frequency and mode shapes. ThIs
$ procedure eliminates the effects of finite element errors common to both
$ methods.
$ 
$ Natural frequencies for the combined system are shown in Table 1 along with
$ the error ratios of the difference. Note that the lowest frequency component
$ mode omitted from the analysis was 197.2 Hz. Below this frequency, the
$ resulting modes are excellent.
$ 
$      Table 1. Natural Frequencies and Differences for 20 Degrees of Freedom
$                                  Mode Synthesis
$ 
$                   -------------------------------------------
$                   Mode No.   Frequency (Hz)   % Difference
$                                               From Full Model
$                   -------------------------------------------
$                      1            3.596           .0085
$                      2           17.564           .0012
$                      3           28.492           .0639
$                      4           39.507           .0011
$                      5           61.099           .0051
$                      6           80.280           .0066
$                      7           84.454           .379
$                      8           98.898           .039
$                      9          111.764           .018
$                     10          123.348           .345
$                     11          127.556           .548
$                     12          130.359           .155
$                     13          134.922          2.220
$                     14          153.606           .174
$                     15          162.019          2.98
$                     16          180.321          5.27
$                     17          200.702         14.89
$                   -------------------------------------------
$ 
$ APPLICABLE REFERENCES
$ 
$ 37. Herting, D. N.: "Accuracy of Results with NASTRAN Modal Synthesis", NASA
$     CP-2062, October, 1978, pp. 389-404.
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