64 N. Spivey et al. Fig. 7.2 Spring-mass two-degrees-of-freedom system The FRF for traditional modal testing is calculated using the shaker forces applied to DOF 1 and DOF 2 as references to obtain the full system response. The results of using these traditional FRFs are referred to in this paper as the “uncorrected” results, as shown in Eq. (7.2): a1 = −ω 2 −ω 2 m2 +2k −ω2m2 +2k −ω2m1 +k −k 2 −ω 2 k −ω2m2 +2k −ω2m1 +k −k 2 f1 f2 (7.2) where a is the acceleration. When implementing the FBC, however, if the force at DOF 1 and the acceleration at DOF 2 are used as references, then the resulting FRFs are associated with a structural system with dynamics associated with DOF 2 fixed, as shown in Eq. (7.3): a1 = −ω 2 −ω2m1 +k k −ω2m1 +k f1 a2 (7.3) Furthermore, the FRF associated with the force applied at DOF 1 is equivalent to an FRF associated with DOF 2 being fixed. This property is exploited in the FBC method by using drive point accelerations, instead of the traditionally used shaker forces, on the test fixture as references when calculating the FRF. The key necessity of the FBC method is at least one independent excitation source, usually modal shakers, for each degree of freedom that is desired to be fixed. Therefore, FBC modal testing requires multiple shakers used on the test fixture in addition to the test article. Although not described in this paper, the FBC technique could also use constraint shapes as references when the number of independent sources is larger than the number of independent DOF of the test fixture [16]. The fundamental FBC strategy is to use shaker accelerations as references, rather than the traditional shaker forces, when calculating FRFs. Personnel at ATA Engineering, Inc. (San Diego, California) have implemented the FBC modal methodology into their IMAT™ (Interface between MATLAB®, Analysis and Test) software (MATLAB is a registered trademark of The MathWorks, Natick, Massachusetts). The fixed base corrected FRF can be calculated directly using shaker accelerations on the fixture and shaker forces on the test article as references [18], or by performing a partial inversion of the baseline FRFs that have been calculated using all shaker forces as references [6, 17]. In fact, the results are equivalent if measured forces are used as basis vectors when calculating the FRF directly [19]. One advantage of calculating the FRF directly is that doing so removes the requirement to mount load cells to the shakers on the test fixture. The advantage of performing a partial inversion of the FRF matrix is that boundary conditions can be changed quickly by changing which DOF are to be inverted. One potential disadvantage of the FBC method is that the measured damping values of the FBC modes have been observed to be slightly different from expected values; sometimes very lightly damped modes may even be calculated to have slightly negative damping. In these cases, it may be better to report the damping values of the mode from the uncorrected test data
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