196 R. Schultz et al. 17.2.1.3 Parameterizing the Eigenvalues and Eigenvectors • Generate full DOF component matrices at five parameter values. • Compute the eigenvalues and vectors for each of these five models. • Compute derivatives of these eigenvalues and vectors. • For each parameterized variable of interest, estimate the eigenvalues and vectors using a Taylor approximation using the derivatives of the eigenvalues and vectors. • Use the estimated eigenvalues and vectors either directly (modal) or in a subsequent analysis (e.g. modal transient). 17.3 Application of Parameterization Methods to an Example Finite Element Model To demonstrate the parameterization methods described above and explore the accuracy when parameterizing on different types of variables a finite element model was developed. The system model is composed of two identical steel beams (Beam 1 and Beam 2) connected through a lap joint and connected with three bolts. The beam system is analyzed in a free–free condition. Figure 17.2 is a diagram of the beam system and Fig. 17.3 shows the mesh. Matrices of each of the two component beams are parameterized in full DOF and reduced order CMS form. While this model is of simple form, it is meshed such that the DOF count was non-trivial; the system matrices are 18576 by 18576 (6,192 nodes with 3 DOF each). A modest model size was chosen to show that these methods can be applied to “real” models. To study the effects of parameterized variable type on the quality of the resulting approximated system, several variables of the beam components are parameterized: length, thickness, density, and Young’s modulus. Note that parameterizing the density and modulus do not require a change of mesh while parameterizing the length and thickness do require a new mesh for each of the five points used in the derivatives calculation. The parameterization range was chosen to be quite high: plus or minus 25 % of the nominal value for each variable except the length which uses a range of plus or minus 20 %. The calibration range, the range parameter values used for the computation of derivatives, is chosen as something smaller than the full range: plus or minus 15 %. This allows for insight into how the method works to extrapolate component matrices. This range is perhaps much larger than would normally be explored in the context of manufacturing uncertainty, but is chosen to explore the method over a very wide perturbation range, as would be necessary for design optimization. Predicted eigenvalues from each method are compared with the eigenvalue results from a solution of a full DOF model at each perturbation value. Fig. 17.2 Beam system composed of two components (Beam 1 and Beam 2) joined with three bolts Fig. 17.3 Beam system Hex mesh
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