Combined Expansion and Orthogonalization of Experimental Modeshapes

The paper describes a method of combined expansion and orthogonalization (CEO) of experimental modeshapes. Most model updating and error localization methods require a set of full length, orthogonal with respect to the mass matrix, eigenvectors. In practically every modal experiment, the number of measurements is less than the order of the model, and hence modeshape expansion, i.e., adding the unmeasured degrees of freedom, is required. This step is then followed by orthogonalization with respect to the mass matrix. Most current methods use two separate steps for expansion and orthogonalization, each one optimal by itself, but their combination is not optimal. The suggested method combines the two steps into one optimization problem for both steps, and minimizes a quadratic criterion. In the case of an equal number of analytical and experimental modeshapes, the problem coincides with the Procrustes problem and has a closed form solution. Otherwise the solution involves nonlinear equations. Several examples show the advantage of CEO, especially in cases where the measurements are limited either in number or in space, i.e., not spanned through the entire structure.