72 Y.F. Xu and W.D. Zhu 9.2.4 Approximation of MSs of an Undamaged Plate While the CDIs proposed above can be used to identify damage in a plate, they require use of MSs of an undamaged plate, which are usually unavailable in practice. Assuming that existence of relatively small damage in a plate does not cause prominent changes in its MSs in the neighborhood of the damage, one can approximate MSs of the associated undamaged plate using polynomials that fit the corresponding MSs of the damaged plate, provided that the undamaged plate is geometrically smooth and made of materials that have no stiffness and mass discontinuities. The modal assurance criterion (MAC) value in percent between Zd;30 and Zu;30 is 99:85%, which indicates that they are almost identical to each other and validates the assumption on the existence of relatively small damage. A similar technique has been proposed in [19] to approximate MSs of an undamaged beam using polynomials that fit the corresponding MSs of the damaged one. MSs of an undamaged plate corresponding to those of a damaged one are not measured in this work, and it is proposed that a MS of the undamaged plate be obtained from a polynomial of a properly determined order that fits the corresponding MS of the damaged plate: zp .x; y/ D n X kD0 k X iD0 ai;k ix iyk i (9.18) where n is the order of the polynomial, which controls the level of approximation of the polynomial fit to the MS of the damaged plate, .x; y/ are coordinates of a point on an undeformed plate, andai;k i are coefficients of the polynomial that can be obtained by solving a linear equation Va Dz (9.19) inwhichVis theN nC1 PpD1 p ! -dimensional bivariate Vandermonde matrix withNbeing the dimension of z, the Vandermonde matrix can be expressed by VD 2 6 6 6 4 1 x1 y1 : : : x n 1 : : : x i 1y n i 1 : : : y n 1 1 x2 y2 : : : x n 2 : : : x i 2y n i 2 : : : y n 2 : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 xN yN : : : x n N : : : x i Ny n i N : : : y n N 3 7 7 7 5 (9.20) a D a0;0 a1;0 a0;1 : : : an;0 : : : ai;n i : : : a0;n T is the nC1 PpD1 p ! -dimensional coefficient vector, and z is the MS vector of the damaged plate to be fit. Solving Eq. (9.19) for the coefficient vector is equivalent to solving an unconstrained least-squares problem min 1 2 k Va zk2 for an optimum minimizer a [20], which is usually an over-determined problem, i.e., N > nC1 PpD1 p. A solution can be obtained using the singular-value decomposition (SVD) of V[20], which gives VDU S 0 WT (9.21) where UandWare N Nand nC1 PpD1 p ! nC1 PpD1 p ! orthogonal matrices, respectively, and Sis a nC1 PpD1 p ! nC1 PpD1 p ! diagonal matrix. An optimum minimizer a based on the SVD of Vcan be obtained by a DWS 1UT 1z (9.22) where U1 is the first nC1 PpD1 p columns of U. When n in Eq. (9.18) becomes a large value, S can be ill-conditioned, which can result in a low level of approximation of the associated polynomial fit. To avoid ill-conditioning of S, it is proposed that x
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