3 A Somewhat Comprehensive Critique of Experimental Modal Analysis 37 In the case of a lightly damped system, the eigenvalues are expressed as .λI =/λ 2 I −λ 2 R ≈ωn, λR ≈−ζnωn. (3.36 ) Substitution of the above result into Eq. 3.35 results in . |−ω 2 nϕR + ˜KϕR|+|− ˜BϕRξnωn − ˜BϕIωn +2ϕRξnω 2 n|≈0. (3.37 ) Investigation of Euclidian norms of the coefficient matrices and numerical analysis of the above matrix terms associated with ISPE experimental data and shell simulated data indicates that the Eq. 3.37 terms are in the following order of magnitude relationship: . |−ω 2 nϕR + ˜KϕR|+0(ζn ˜KϕR,ζn ˜KϕI) →|−ω 2 nϕR + ˜KϕR|≈0. (3.38 ) This completes the heuristic reliability proof of RTMA. That being said, it must be emphasized that the real mode approximation does not amount to exact replication of undamped system modes. 3.8 Recapitulation and Conclusions The recapitulation and conclusions to this “somewhat comprehensive” critique of experimental modal analysis are stated with respect to the seven key steps of the Integrated Test Analysis Process (ITAP) for structural dynamic systems. 3.8.1 System Dynamic Model (Covered in Previous Publications) An appropriate finite element (FEM) system dynamic model requires adherence to several practical, well-documented guidelines, specifically: (a) Strictly enforced consistency of the FEM with engineering drawings (CAD). (b) Selection of component sufficiently refined grid spacing to appropriately capture relevant frequency band dynamics. (c) Inclusion of quasi-static effects (differential stiffness) due to gravity or steady acceleration and hydrostatic pressure loads . (d) Provision for localized joint flexibilities to permit subsequent model updating (reconciliation). (e) Inclusion of nonlinear features, when significant, especially at component interfaces (early Space Shuttle experiences). 3.8.2 Modal Test Plan (Covered in Previous Publications) The commonly accepted practice for modal test planning focuses on development of a test-analysis model (TAM) employing Guyan reduction of a detailed FEM that is consistent with a selected set of test accelerometer degrees of freedom (DOF). Moreover, “residual kinetic energy” strategies for selection of appropriate selection of TAM degrees of freedom are widely accepted today. The most important products of the modal test planning step are the TAM mass matrix, which is used for validation of experimental modes through an orthogonality check, FEM natural frequencies, and (real) mode shapes (in particular, the TAM DOF order partition).
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