Frequencies corresponding to the 15 lowest eigenvalues are extracted and tabulated in Table 2.2.1–1 for each model. To study the effect of retaining dynamic modes during substructure generation, the substructure models are run after extracting 0, 5, and 20 dynamic modes during substructure generation.

While the Guyan reduction technique (0) yields frequencies that are reasonable compared to those of the full model, the values obtained with 5 retained modes are much closer to full model predictions, especially for the higher eigenvalues. Increasing the number of retained modes to 20 does not yield a significant improvement in the results, consistent with the fact that in the Guyan reduction technique the choice of retained degrees of freedom affects accuracy, while for the restrained mode addition technique the modes corresponding to the lowest frequencies are by definition optimal.

When substructures are used in an eigenfrequency analysis, it is to be expected that the lowest eigenfrequency in the substructure model is higher than the lowest eigenfrequency in the corresponding model without substructures. This is indeed the case for the single-level substructure analysis, but for the multi-level substructure analysis the lowest eigenfrequency is below the one for the full model. This occurs because the stress and load stiffness for the lowest-level substructure (the blade) are generated with the root of the blade fixed, whereas in the full model the root of the blade will move radially due to the deformation of the hub under the applied centrifugal load. Hence, the substructure stiffness is somewhat inaccurate. Since the radial displacements at the blade root are small compared to the overall dimensions of the model (of order 10^–3), the resulting error should be small, as is observed from the results.

Table 2.2.1–2 shows what happens if the NLGEOM parameter is omitted during the preloading steps. It is clear that the results are significantly different from the ones that take the effect of the preload on the stiffness into account. In this case the lowest eigenfrequency in the substructure models is indeed above the lowest eigenfrequency in the model without substructures.

A static analysis of the fan is carried out about the preloaded base state by applying a pressure load of 10⁵ Pa normal to the blades of the fan. The axial displacement of the outer edge of the fan blade due to the pressure load is monitored at nodes along path , as shown in Figure 2.2.1–1. The results are shown in Figure 2.2.1–4; there is good agreement between the solutions for the substructure models and the full model.

While substructures can be generated from models that exhibit nonlinear response, it must be noted that, once created, a substructure always exhibits linear response at the usage level. Hence, a preloaded substructure will produce a response equivalent to that of the response to a linear perturbation load on a preloaded full model. Consequently, the full model is analyzed by applying the centrifugal preload in a general step and the pressure load in a linear perturbation step. Since an analysis using substructures is equivalent to a perturbation step, the results obtained do not incorporate the preload deformation. Thus, if the total displacement of the structure is desired, the results of this perturbation step need to be added to the base state solution of the structure.

A modal-based, steady-state analysis of the fan is carried out about the preloaded base state, as shown in fan_cyclicsymmodel_ss.inp. In the general static step, which includes nonlinear geometry, the centrifugal load is applied to the datum sector. Only symmetric loads can be applied in general static steps with the *CYCLIC SYMMETRY MODEL option. A sequence of three eigenvalue extraction and steady-state dynamics steps follows the preload step. Each eigenvalue extraction requests only one cyclic symmetry mode that is used in the load projection in the steady-state dynamic analysis that follows. The *SELECT CYCLIC SYMMETRY MODES option is used to extract modes belonging to the cyclic symmetry modes 0, 1, and 2, respectively. The computed eigenvalues are identical to those obtained for the entire 360° model, as shown in Table 2.2.1–1. The additional information obtained during the eigenvalue extraction is the cyclic symmetry mode number associated with each eigenvalue. In the case of 4 repetitive sectors, all the eigenvalues corresponding to cyclic symmetry mode 1 appear in pairs; the eigenvalues corresponding to modes 0 and 2 are single. The lowest two eigenvalues correspond to cyclic symmetry mode 1, followed by the single eigenvalues corresponding to cyclic symmetry modes 2 and 0. For a comparison with the cyclic symmetry model option, the eigenvalue problem is also modeled with *MPC type CYCLSYM (see fansubstr_mpc.inp). To verify the use of substructures with the cyclic symmetry model, it was determined that the eigenvalues obtained with fansubstr_cyclic.inp were identical to those obtained with fansubstr_1level_freq.inp. The last step is the modal-based, steady-state dynamic analysis. A pressure load is applied to the entire structure as projected onto three different cyclic symmetry modes using the CYCLIC MODE parameter on the *CLOAD, *DLOAD, and *DSLOAD options.