Fig. 3 A two-dimensional solid problem decomposed in 9 substructures 7 Interpretation of static modes of the dual interface problem In the Dual Craig-Bampton, interpretation of modes and frequencies is a bit different from normal reduction techniques and we will further discuss this point here. If we consider again the initial dually assembled problem before reduction, equation (1), there exist 2Nλ infinite eigenvalues related to the “dynamics” of the Lagrange multipliers.2 This can be understood by the fact that the Lagrange Multipliers have associated coefficients inKa, but they have no contribution to the mass Ma. See [6] for further details. In the Dual Craig-Bampton approach the compatibility condition is weakened, which had as consequence to introduce contribution of the Lagrange multipliers to the reduced mass matrix (see section 3). The infinite eigenvalues of the original system therefore become finite and have negative values due to the fact that the dually assembled problem originates from a saddle point problem (one could say that the problem is positive definite for the physical displacements, and negative definite for the Lagrange multipliers). Hence it is understood that the modes associated to negative eigenvalues of the reduced system will be related to strong incompatibilities. To illustrate this discussion, we show in figure 5 the modes 1, 2 and 30 obtained with the Dual Craig Bampton method (without interface reduction and with 10 vibration modes per substructure). Clearly we see that the first modes are the correct modes of the system. The last mode, mode 30, looks fine: this mode had an accurate frequency (see previous section), but slight incompatibilities appear at some location on the interface. As a matter of fact, its MAC when compared to mode 30 of the non-reduced problem is about 0.8. 2 Here we assume that no redundant constraints are in the equations. If this would be the case, then one has as many zero frequencies as there are redundant constraints inB. Daniel J. Rixen 324
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