Figure 2: Recovery of dynamic stress-strain from full-field displacement data MODEL REDUCTION Model reduction is necessary in order to develop expansion approaches for modal data for the unmeasured translational DOF as well as for rotational DOF. For this work, the expansion is needed for augmenting the limited set real-time operating data to provide a full field displacement solution. 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 [5], Dynamic Condensation [6], SEREP [7], and a Hybrid method [8]. 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 {X }=[T]{X }a @ (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) 189
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