9 Non-Model-Based Damage Identification of Plates Using Curvature Mode Shapes 69 r a b c p p q q1 q6 q5 q4 q3 q2 s qi ai qi bi p Fig. 9.3 (a) One-ring neighborhood of a triangulated mesh associated withp, (b) an acute triangle4pqrand(c) a hexagon one-ring neighborhood with a side length of s Fig. 9.4 Pseudo-code for calculating the new surface area Ap associated withp and GZ;p D 2 Pi2N1.p/ i AP (9.10) where N1.p/ is the number of points that are connected with p in the one-ring neighborhood, and i is the angle associated withqi, as shown in Fig. 9.3a. With calculated HZ;p and GZ;p, principal curvatures at pcan be calculated by Max Z;p DHZ;p CpGZ;p HZ;p (9.11) and Min Z;p DHZ;p pGZ;p HZ;p (9.12) Similar to calculating curvatures of a one-dimensional structure, that of a MS of a plate can be jeopardized by measurement noise. To alleviate adverse effects of measurement noise on the operators introduced above, a hexagon onering neighborhood that has a side length of s associated with each measurement point is constructed for the operators. The hexagon one-ring neighborhood associated with pprojected onto the undeformed plate is equilateral, as shown in Fig. 9.3c. Coordinates of qi of the hexagon one-ring neighborhood, where i D1;2; : : : ;6, are obtained from interpolation based onZ. For a measured MS with an unknown noise level, one needs to progressively test different values of s from smaller to larger ones. A proper value of s is the one with which the resulting CMS becomes smooth and has a clear global trend. To illustrate adverse effects of measurement noise and the effectiveness of the scheme, white noise is added toZd;30 witha signal-to-noise ratio (SNR) of 60db to simulate measurement noise. Maximum CMSs associated withZd;30 from the scheme with s D0:002, 0:005 and 0:015 are shown in Fig. 9.5a through c, respectively; the maximum CMS associated with noisefree Zd;30 is shown in Fig. 9.5d. It can be seen that measurement noise is amplified and becomes dominant in the resulting maximum CMS from the scheme with s D0:002, since differences between the value of a noise-free Zd;30 at a point and those in the hexagon one-ring neighborhood with a side length of s are small compared with those associated with a noisy Zd;30. Figures 9.5b and c show that maximum CMSs can be obtained with a lower noise level with a larger value of s. Though
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