second system is part of the three-dimensional assembly studied in [2], and it illustrates the complexities that can arise in a more complicated problem. 3.1. Two Dimensional T-beam System Consider the substructure uncoupling problem pictured in Fig. 1, which was described in Section 2. For the following example all of the subcomponents were modeled with finite elements in order to eliminate any measurement uncertainties. Beam A was 152 mm long, 25 mm wide and 19 mm thick, while Beam B was 305 mm long with the same cross section. The finite element model was set up so that only in-plane motion, both axial and bending, was possible. The mesh for beams A and B consisted of 21 and 30 nodes respectively. The first fifteen modes of C will be used in the uncoupling, corresponding to a modal test in which all modes out to 20kHz have been extracted. Each system has three rigid body modes with zero natural frequencies and the natural frequencies of the elastic modes are shown in Table 1. The corresponding mode shapes are not shown, but the lower modes were all observed to involve bending of the horizontal beam, B, while the vertical one (transmission simulator A) undergoes rigid body rotation. Some of the higher frequency modes show the horizontal beam vibrating axially as the transmission simulator bends. The first six free-modes of A are used in the transmission simulator model, three of which are rigid body modes, the 4th and 5th involve bending of beam A and mode 6 involves axial motion of beam A. Six modal constraints are used to join the negative transmission simulator A to C. Displacement in both the axial and bending directions at all 21 nodes of the finite element model of A are used in forming the modal constraints, although in an experiment one would likely not have such a detailed set of measurements. The rotations at those nodes are not used, since one cannot usually measure rotations in practice. Table 1 shows the natural frequencies of the B system estimated by the modal substructure uncoupling procedure. The actual FEA natural frequencies of the B system are also shown as well as the percent difference. All of the natural frequencies below 17kHz are very accurately predicted, having less than 3% error. However, the modal substructuring procedure returns three natural frequencies which are purely imaginary and do not correspond to any of the analytical natural frequencies. Recall that the negative transmission simulator model is still part of the B system even after substructure uncoupling. Its effect is to cancel the force exerted on B by the actual transmission simulator A (see [2]), but each of the nodes on the transmission simulator are still valid points on the B system and one can determine how the negative transmission simulator moves in each of B’s modes. The deformation shape of each of the modes corresponding to the three imaginary natural frequencies was observed and they were found to involve motion primarily on the negative transmission simulator model subsystem A; the motion was three orders of magnitude smaller on component B in each of these modes. A few FRFs of the B system were reconstructed (not shown here) in the axial and bending directions, and they were seen to overlay the analytical FRFs out to 17kHz, confirming that the spurious modes did not have a large effect on the FRFs. In some applications there would be no need to eliminate these spurious modes since they do not seem to affect the model for B. However, one may not always be so fortunate and in any event these spurious modes are problematic since they cannot be imported into most finite element packages. 123
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