1 Practical Techniques for Scaling of Optically Measured Operating Deflection Shapes 3 1.2 Theoretical Background Within the current paper, two significant techniques are utilized, drive point scaling and mass sensitivity scaling. Both techniques are described in the following sections. 1.2.1 Drive Point Scaling Technique The operating deflections shapes tend to be highly correlated with the corresponding mode shape of the system. This correlation can be expressed as fugk D’kf‰gk (1.1) where ’0k is the scale factor between the operating deflection shapes (f‰gk) and the scaled mode shapes (fugk) at the kth mode of the structure. The scaled mode shapes and operating deflection shapes can be expressed as 8ˆ ˆ< ˆˆ: u1k u2k u3k : : : 9> >= >>; Dfugk; 8ˆ ˆ< ˆˆ: ods1k ods2k ods3k : : : 9> >= >>; Df‰gk (1.2) The drive point scaling technique relies on the use of a single drive point measurement to properly scale operating deflection shapes. The FRF obtained from the single drive point measurement can be shown as ŒH.s/ s Dj¨ DŒH.j¨/ D mX kD1 ŒAk .j¨ pk/ C A k j¨ p k (1.3) For a particular mode k, Eq. (1.3) can be expressed as ŒH.s/ s Dpk Df ukg qk s pkf ukg T (1.4) with the residue matrix calculated using ŒA.s/ k Dqk fukgfukg T (1.5) Equation (1.5) can be further simplified for a single reference point as 8ˆ ˆ< ˆˆ: a1k a2k a3k : : : 9> >= >>; Dqkurk 8ˆ ˆ< ˆˆ: u1k u2k u3k : : : 9> >= >>; !aijk Dqkuikujk ; qk D 1 2j¨k (1.6) where uik andujk are the kth mode of the structure at point i and j respectively. The residues are directly related to the system mode shapes and are scaled by qk, the scaling constant for the k th mode. From Eq. (1.6), it becomes evident that by taking a measurement at the drive point location, urk, a value for modal scaling can be determined. The drive point measurement provides a scaled mode shape for the drive point DOF as arrk Dqkurkurk (1.7) With this drive point scaling, the scaled mode shapes, for the corresponding drive point measurement location, can be used to scale the operating mode shapes
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