u λ = ¯TdualTI η ˜λ = Θ R˜Xα −GBT ˜Xλ 0 ˜Xλ η ˜λ (36) 5 Numerical example of a truss frame To illustrate the Dual Craig-Bampton method with interface reduction, let us consider the truss frame used in [12, 13] and depicted in figure 1. The frame is composed of repeated cells, each cell having dimension 0.35×0.35×0.5 m. The outer beams of the cells have a hollow circular cross-section (outside/inside diameters are 0.02 m and 0.018 m). The diagonal members inside the cells have plain circular section of diameter 0.008 m. We consider five substructures for this system, one per arm and one for the central part. The arms are connected to the central part through 12 points: 4 points at the center of the faces of the central part, and 8 points it the vertices. The dofs of on the mid-face points must be connected only with one neighbor, whereas the vertices have two neighbors and thus two Lagrange multipliers are needed for those dofs. Therefore the compatibility conditions defined byBrequires 4×6+8×6×2=120 Lagrange multipliers λ. For practical implementation reasons, we treat all substructures in an equal way and therefore redundant constraints are defined on the vertices by stating the equality between dofs for all combination of neighbors. This leads to using 3 Lagrange multipliers for the dofs on the vertices. So in our implementation we will have 4×6+8×6×3=168 Lagrange multipliers, out of which 48 are redundant. By trial and error, we have chosen the minimal number of modes per substructure such that, without interface reduction, the first 30 eigenfrequencies computed for the Dual Craig-Bampton have an absolute relative error of no more than 1.5 % when compared to the eigenfrequencies of the non-reduced problem. We find that to have max k=1...30 ' ' ' ωfullj −ωDCBj ' ' ' ωfullj <1% we need must take 29 modes in the arms of the frame that are not attached to the ground, 27 modes in the arm attached to the ground, and 1 mode for the central part. Since the total number of rigid body modes for the substructures is 4×6=24 and since there are 120 effective Lagrange multipliers, the effective size of the reduced model is 3×29+27+1+24+120=259. Applying now the interface reduction as proposed in the previous section, we find by trial and error that a minimum of 74 interface modes. So the effective size of the reduced system is now 213. The relative frequency errors are plotted in figure 2. Note that the first negative eigenvalue for the Dual Craig- Bampton appears for mode 115 in the case no interface reduction is applied, and for mode 111 in case of interface reduction. Let us note in figure 2 that for some frequencies the error decreases when interface reduction is applied (see for instance errors on frequencies of modes 4, 5, Interface Reduction in the Dual Craig-Bampton method 321
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