FEM Sensitivity Method for Uncertainty and Reconciliation Analyses Robert N. Coppolino, CTO Measurement Analysis Corporation 23850 Madison Street Torrance, California 90505 ABSTRACT Parametric variation of large order finite element models is required for evaluation of uncertainty and test-analysis reconciliation analyses. Established procedures for computation of modal frequency and mode shape derivatives are widely used in such studies. While modal derivatives accurately describe sensitivities for small parametric changes, they may be ineffective when modal frequencies are closely spaced or repeated. Over the past decade, an alternative modal sensitivity procedure has been employed for modal test-analysis reconciliation, without rigorous proof of its validity. This procedure, based on definition of residual shape functions that augment baseline system mode shapes, produces reduced mass and stiffness (sensitivity) matrices. The resultant sensitivity formulation is extremely effective for computation of altered system modes associated with large parametric variations, regardless of the presence of closely-spaced or repeated modal frequencies. This paper provides a rigorous validity proof of the alternative modal sensitivity formulation. Residual shape functions are similar to well-known quasi-static residual vectors for both localized and highly distributed model changes. A simple illustrative example is provided to demonstrate effectiveness of the technique. 1. INTRODUCTION Efficient computation of structural dynamic modal frequency and mode shape sensitivities associated with variation of physical stiffness and mass parameters is essential for (1) practical design sensitivity and uncertainty studies and (2) reconciliation of finite element models with modal test data. Sensitivity analysis procedures fall in two distinct categories, namely (a) modal derivatives for small parametric variation and (b) altered system modes associated with “large” parametric variation. The latter category is generally applicable to modal testing, which often requires significant local parameter changes at joints to effect FEM-test reconciliation[1]. However, many investigators and commercial software packages employ estimated modal derivatives in optimization strategies, which address FEM-test reconciliation objectives. Since the 1960’s, methods for computation of modal frequency and mode shape derivatives have evolved. Fox and Kapoor[2] introduced an exact derivative formulation that required knowledge of all modes of the original system; application of the procedure when a truncated set of modes was employed produced compromised derivatives. In response to this difficulty, Nelson[3] derived an exact formulation for computation of mode shape derivatives for truncated mode sets. Efforts to refine and extend application of mode shape derivatives for finite parameter change sensitivity computations have been pursued by many investigators (including the present author). However, the need for modal frequency and mode shape sensitivities that map over very large ranges for multiple parameters suggests alternative Rayleigh-Ritz strategies. The Rayleigh-Ritz method, actually introduced by Ritz[4], is one of the most significant developments in analytical mechanics of the past century. This method provides a logical energy formulation for consistent reduction of mass and stiffness matrices employing a set of trial vectors as a reduction transformation. Effectiveness and accuracy of the reduction process depends on selection of an appropriate trial vector set. When a truncated set of baseline system mode shapes is used as the trial vector set, the Rayleigh-Ritz method often produces poor estimates for the altered system[5]. Augmentation of the truncated baseline system mode shapes with appropriately defined additional vectors, however, has been found to produce extremely accurate altered system modal frequencies and mode shapes. Quasi-static residual vectors[6], appended to a truncated set of mode shapes, were found to produce extremely accurate modes for offshore oil platform models subjected to localized alterations[7]. This paper introduces a new procedure for selection of augmented trial vectors that is appropriate for structures that are subjected to highly distributed, as well as localized, alterations. T. Proulx (ed.), Linking Models and Experiments, Volume 2, Conference Proceedings of the Society for Experimental Mechanics Series 5, 375 DOI 10.1007/978-1-4419-9305-2_27, © The Society for Experimental Mechanics, Inc. 2011
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