34 R. N. Coppolino Table 3.4 Mass-weighted self-orthogonality (absolute values) of linear least-squares refitted (Real) SFD-2018 modes Orthogonality of the linear least-squares refitted test modes does not exhibit the overall level of improvement noted for the case of modes approximated by the Aerospace Corporation practice (particularly for test modes 26 and 27). This issue warrants further examination in the next section of this chapter. 3.7.3 Further Analysis of the Aerospace Corporation Real Test Mode Approximation The practice adopted by the Aerospace Corporation for “real test mode” approximation warrants further analysis through study of analytically simulated modal test data for which exact, unreduced modal solutions are known. The simulated data selected for further examination relates to the 5616 DOF shell structure FEM (and simulated test mode summary) depicted in Fig. 3.9. The blue and green colored grid points denote locations associated with axisymmetric shell (mass and stiffness) elements, the green colored grid points denote locations associated with localized damping elements, and the red colored grid points denote locations associated with localized non-axisymmetric added mass coefficients. Three separate applied excitation forces, denoted in Fig. 3.9, are intended to stimulate lateral and axial responses. The (a) complex mode self-orthogonality and (b) RTMA self-orthogonality summaries are provided in Tables 3.5 and 3.6, respectively. Cross-orthogonality between real, undamped FEM modes and (a) complex test modes and (b) RTMA results are provided in Tables 3.7 and 3.8, respectively. The results summarized in Tables 3.5, 3.6, 3.7, and 3.8 indicate that the Aerospace Corporation real test mode approximation yields (a) the self-orthogonality matrix that satisfies US aerospace community orthogonality criteria, and (b) improved cross-orthogonality between simulated test and undamped FEM modes with respect to the simulated complex test mode results. The fact that the Aerospace Corporation method’s real test modes do not produce an “identity” crossorthogonality matrix indicates that the approximation does not exactly correspond to the “true” real modes. Moreover, while the approximation improves test-FEM cross-orthogonality, complete reliance on the approximation may not be universally appropriate for model updating and reconciliation operations. 3.7.4 Why the Real Test Mode Approximation (RTMA) Works (A Heuristic Reliability Proof) While RTMA has been employed by that organization for many years, an analytical “proof” of its appropriateness is unavailable in a key reference document [3]. This section of this chapter offers a heuristic proof of the approximation’s accuracy.
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