remove more mass from the first and second bending modes than it should. This might explain why too much mass is removed from the system when the transmission simulator model is subtracted, resulting in negative eigenvalues in the estimated mass matrix. The first eigenvalue of CM Q is 1.012, and the table shows that mode 12 contributes 0.04 to it, so if mode 12 were not present then this eigenvalue might reduce below 1.0 resulting in a positive definite model for B. -15 -10 -5 0 5 10 15 20 -50 0 50 x (mm) y (mm) Figure 3: Shape of Mode 12 of system C: (black/dots) undeformed structure, (blue/circles) mode of C, (red/dots) projection, ˆxCm , of C onto the free modes of A. Various modifications to the substructuring process were explored, revealing that the discrepancy in Figure 3 could be reduced greatly by increasing the number of modes in system A to seven. When that was done, the maximum discrepancy between ˆxCm and Cm x was found to reduce from 44.6% to 6.2%. The substructuring calculations were repeated and the corresponding negative eigenvalue of ˆBm had disappeared (although the other remained). Another alternative would be to reduce the number of modes used in C so that the sixmode model for A would adequately span the observed motion of C. Using six modes for system A and eleven modes for system C, a positive definite mass matrix was obtained with the smallest eigenvalue being 0.00034. However, since fewer modes were used for C, the model obtained for B was only accurate up to 14kHz, whereas the FRFs were accurately reconstructed out to 17kHz when 15 modes were used for C. 3.2. Three Dimensional Cylinder/Fixture System The next example considered is one of the system’s discussed in [2] where the concept of modal constraints was introduced. Substructure C, illustrated in Fig. 1, consists of a hollow cylinder and an attached ring-shaped fixture with tabs. The objective is to obtain an experimental representation of the cylinder alone, substructure B, by subtracting off a finite element representation of the fixture (or transmission simulator) using the MCFS approach. For this illustrative example, it is assumed that the response of the cylinder and transmission simulator is measured in three degrees of freedom at 12 points on the transmission simulator, as described in [2]. One hundred modes for substructure C were simulated with a finite element model, including six rigid body modes and elastic modes ranging between 433.8 and 6165 Hz. Fifty modes were calculated for the transmission simulator using finite element model, including six rigid body modes and elastic modes ranging between 200.4 and 9,382.0 Hz. 126
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