THEORY The approach used in the merging of the different sets of data finds its roots in model reduction and model expansion with some extension of those approaches to address the problem identified. The specific theory of the approach can be found in the reference papers identified; only summarizing equations will be presented herein and details can be found in the references. Model Reduction Model reduction is necessary in order to develop expansion approaches for modal data for the various sets of data to be expanded. For this work, the reduction transformation matrices are used for the expansion process. The key is that the reduced set of ADOF will be different for all the various sets of data to be extracted. The expansion can be performed to the full space of the finite element model or to an intermediate space which is reduced from the full size of the finite element model but larger than the largest of the different measurement systems/scenarios utilized. The reduction techniques are the basis of the expansion discussed in this work. These techniques have been presented in earlier work cited in the references; only summarizing equations are presented below. Several model reduction methods have commonly been used for expansion of measured data. Four common methods are Guyan [4], Dynamic Condensation [5], SEREP [6], and a Hybrid method [7]. In these methods, the relationship between the full set of degrees of freedom and a reduced set of degrees of freedom can be written as n a {X }=[T]{X } (1) All of these methods require the formation of a transformation matrix that can project the full mass and stiffness matrices to a smaller size. The reduced matrices can be formulated as > @ > @ > @> @ M T M T n T a (2) > @ > @ > @> @ K T K T n T a (3) For the specific work in this paper, only the SEREP method has been used for the expansion of mode shapes. The System Equivalent Reduction Expansion Process (SEREP) produces reduced matrices for mass and stiffness that yield the exact frequencies and mode shapes as those obtained from the eigensolution of the full size matrix. The SEREP transformation is formed as > @ > @> @g a n UT U U (4) Implementation of Reduction/Expansion For the work presented in this paper, each of the different types of data acquired from each of the different types of measuring systems needs to be merged together to form one hybrid/combined set of data. The schematic of Figure 2 shows this process schematically. The bottom of the schematic shows the full space finite element model representation of the structure; this may possibly be reduced to a relatively large model with fewer DOFs than the full finite element model but more than any of the test ADOF sets considered. The data set for the rigid body modes will have a set of measurement points at ADOF(RB) that are used in conjunction with the U(RB) modes to form a Tu(RB) expansion matrix. Next the data set for the low frequency flexible modes will have a set of measurement points at ADOF(LF) that are used in conjunction with the U(LF) modes to form a Tu(LF) expansion matrix. And finally, the data set for the high frequency modes will have a set of measurement points at ADOF(HF) that are used in conjunction with the U(HF) modes to form a Tu(HF) expansion matrix. It is very important to note that all of the individual ADOF sets can be totally different from each other and do not need to contain any similar or corresponding points. The “jelling mechanism” comes from the transformation matrix for each of the techniques which expands all the different data sets to a common set of points; this can be the full set of finite element DOFs or the larger reduced order ADOF model which is an umbrella for all the individual ADOF sets from the different test scenarios. 167
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