15 Craig-Bampton Substructuring for Geometrically Nonlinear Subcomponents 173 Table 15.1 Exact linear natural frequencies of finite element models in Fig. 15.2 Simply support beam A: 9 in. Simply support beam B: 6 in. Assembly (C) Mode number Free-interface (Hz) Fixed-interface (Hz) Free-interface (Hz) Fixed-interface (Hz) Free-interface (Hz) 1 34.85 54.44 78.41 122.5 42:50 2 139.4 176.5 313.8 397.3 97:73 3 313.8 368.4 706.4 829.5 162:7 4 558.2 630.4 1,257 1,420 313:8 5 872.7 962.8 1,966 2,170 382:1 6 1,258 1,366 2,835 3,080 588:0 7 1,714 1,840 3,866 4,153 762:5 8 2,241 2,385 5,059 5,389 930:0 Table 15.2 Percent error of predicted linear frequencies of assembly (C) using modes up to 1,000Hz % Error of linear substructuring Assembly (C) mode number Free-interface modes CB modes 1 3.2 5.5 10 4 2 5.5 1.6 10 4 3 9.3 1.9 10 3 4a 2.8 10 6 2.5 10 2 5 15 3.4 10 3 aInterestingly, because the lengths of the two beams have a ratio 1.5:1, the 4th mode of the assembly is perfectly described by combining one free-interface mode of each, so its natural frequency is estimated very precisely using this basis Table 15.1, along with the exact natural frequencies of the full finite element model of assembly (C) in Fig. 15.2. The first five modes of the assembly are taken to be the modes of interest covering a frequency range from 0 to 500 Hz. A rule of thumb for substructuring with linear subcomponents is to include modes up to 1.5–2.0 times the maximum frequency of interest. To be conservative with the target range of 500 Hz, each subcomponent includes modes (either freeinterface or fixed-interface) with frequencies up to 1,000 Hz. Then that set of modes was used to predict the modes of the assembly and the errors in each of the predicted natural frequencies are given in Table 15.2. (The exact frequencies of the FEA assembly are given in the last column of Table 15.1.) It happens that this frequency range includes the same number of modes, 5 for A and 3 for B, whether fixed-interface or free-interface modes were used. Using the free-interface substructuring approach, the maximum frequency error of 15 % occurs with the 5th assembly mode. As expected, this modal basis is not well suited for substructuring since the kinematics of the subcomponent do not account for deformation at the interface [11], and one would need far more free-interface modes before the basis could begin to properly describe this motion. In fact, in order to get the frequency error of the free-interface approach below 1 % for the first 5 modes of the assembly, each beam needed a total of 72 modes. The CB approach achieves the same performance with each beam having three fixed-interface modes, and one constraint mode. The CB approach clearly produces a more efficient substructure model because of the kinematics supplied by the constraint modes. 15.3.2 Validate Nonlinear Reduced Order Models Next, the NLROMs and CB-NLROMs were generated for each geometrically nonlinear beam in Fig. 15.2. In the linear case, the CB method needed five and three modes, respectively, for the 9-in. beam and 6-in. beam in order to predict the assembly modes out to 500 Hz. However, due to the geometric nonlinearity in the models, higher frequency modes can be coupled to lower frequency modes, meaning that more modes are needed in each subcomponent to represent the higher frequency modes of the assembly. Because of this, each subcomponent used a total of 10 modes in the basis. The NLROMs of each beam included the first 10 linear free-interface modes (all bending) using the ICE approach in [15]. The load scale factors in Eq. (15.9) were determined such that the amplitude of each force deformed the linear structure to a maximum displacement of 0.25 times the beam thickness (or 7.75 10 3 in.). The CB-NLROMs were generated in a similar manner, but instead included the first 10 fixed-interface modes and 1 constraint mode. The load scale factors were chosen such that each force, when applied to the linear structure with no constraints at the interface, produced a maximum displacement of 1.0 times the beam thickness (or 0.031 in.). Both the CB-NLROM and NLROMs were fit using the constrained approach
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