68 W. Fladung and K. Napolitano Fig . 7. 1 Idealized representation of two orthogonal shapes and an imperfectly extracted shape that is a mass-weighted linear combination of the two exact shapes 0 0.5 1 1.5 2 2.5 3 3.5 Modal Extraction Error (Percent) 0 5 10 15 20 25 Self Orthogonality Off Diagonal (Percent) Modal Extraction Error Effect on Off-Diagonal Terms Fig . 7. 2 Off-diagonals of test self-orthogonality matrix are very sensitive to small errors in modal extraction Test self-orthogonality is defined as .Oij =ФT i MФj, where Mis the mass matrix and the magnitudes of the shapes are defined such that .ФT i MФi =1.In an idealized case with a perfect mass matrix, .ФT i MФj =1 for i = j, and . ФT i MФj =0 for i /=j. Thus, orthogonality can be thought of as a mass-weighted dot product used to calculate the projection of one shape onto another. In the ideal case, as shown in Fig. 7.1, two “exact” shapes, ФE1 andФE2, can be represented on a unit circle as being perpendicular to each other. Assume that an “imperfect” shape, ФT1,can be represented as being rotated from the exact shape ФE1 and that ФT2 is a perfect modal extraction for ФE2. The orthogonality between ФT2 and ФT1, OT1− T2, is the projection of ФT2 ontoФT1. The orthogonality betweenФT1 andФE1 is OT1− E1, which would be 1 if the modal extraction of ФT1 had been perfect, but is less than 1 by the amount ΔOT1 − E1. The error for the first shape (ΔOT1− E1) versus the orthogonality for the second shape (OT1−T2) is depicted graphically as 1 − cos 1 and cos 2 in Fig. 7.2, which shows that small percentage errors in modal extraction can lead to large off-diagonals in orthogonality. For example, a 1% error in a test mode (ФT1) can lead to an off-diagonal term of 14%.
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