Direct Least-Squares Formulation of a Stiffness Adjustment Method

Abstract

A direct least-squares formulation of the stiffness matrix adjustment method known as the KMA method (Kabe, A. M., Stiffness Matrix Adjustment Using Mode Data, AIAA Journal, Vol. 23, No. 9, 1985, pp. 1431-1436) is presented. The KMA method belongs to a class of procedures that refine dynamic models using test-measured modes and structural connectivity. Viewed as a constrained minimization problem, most of these methods have applied Lagrange multiplier techniques in their development. It is shown that these two approaches result in linear systems of equations that are algebraically equivalent and therefore have identical solutions. By virtue of this equivalence, the direct least-squares formulation of the KMA method is shown to be equivalent to its original Lagrange multiplier formulation. The direct least-squares versions of other model refinement methods that preserve structural connectivity are also discussed. In this respect, this approach provides a rigorous framework for unifying these optimal update procedures and indicates how their solutions should compare. Numerical results are presented that illustrate the equivalence between the two formulations and also the conditions under which the various methods yield identical solutions. The results also indicate that because of improved numerical conditioning, the direct least-squares approach yields more accurate computational solutions and generally requires less computer storage than methods that were developed using Lagrange multipliers.