24 C.-H. Wang et al. Fig. 4.7 The depths of an aluminum block evaluated by using laser triangulation methods at different x-positions. (a) Typical images obtained from Left- and Right cameras used for depth evaluation by laser triangulation method. (b) The depths observed from left- and right- cameras as the object is moved along Cx-direction to different positions height were enlarged as the in-plane displacement increased. Meanwhile, the step-height between front- and middle- surface determined from left- camera is slightly decreased as the in-plane displacement increased, but the height determined by the right-camera is converse. As for the determined height between middle- and rear- surfaces, the determined heights from both left- and right- cameras at different locations along x-axis are departure from each other, however the offsets are almost the same in this study. In this study the out-of-plane runoff caused by moving the object along the linear stage should be ignored. Considering the determined out-of-plane displacement is monotonously increased as the in-plane displacement increased which means there is a small angle introduced by the precision stage and the structure used to support the camera-pair. However, from laser triangulation measurement results, the height of middle- and rear-surface remain the same value as the object move from 0 to 20 mm away from its origin; the heights of front- and middle- surface were different at different locations along x-axis and the maximum difference is about 1.36 mm which is small than the corresponding out-of-plane displacement determined by 3DDIC. 4.3.4 Correcting the In-Plane Displacement According to the measurement results presented previously, the evaluated in-plane displacement is 9% above the nominal in-plan displacement and the out-of-plane displacement is available with about 1/4 in-plane displacement in magnitude. In addition, according to laser triangulation measurement, the aluminum block was moved by a precision stage should be moved toward x-axis parallel to the camera-pair baseline; that means there is no significant out-plane displacement will be introduced as the measurement is executed. Since the angle defined by in-plane and out-of-plane displacement is almost a constant all over the displacement range, therefore, the deviation of in-plane displacement and the associated unforeseen out-of-plane displacement are considered to be introduced by the cameras layout. To correct the deviation, a single pinholecamera model with half-baseline distance and the heading angle with respect to the object is 12.5686ı which is introduced by u w to simplify the camera-pair implemented for 3D DIC imaging system. Then, in this study, the in-plane displacement can be corrected by the following equation, u bCu cos. C'/ cos. C' C / b (4.1) Where is the angle defined by u w, ® is angle spand by half baseline of the camera-pair with angle vertex at pinhole, is the angle spand by in-plane displacement along x-axis with its vertex locates at pinhole, b is the half baseline of camera-pair, b is defined to be distance of baseline b projecting with angle ®along the direction of u. Assuming b and distance between pinhole and object are known, then the angle ®and b can be determined. As shown in Fig. 4.8, the blue line is the in-plane displacement deviation with respect to the nominal one and the red line is the in-plane displacement deviation which is corrected by Eq. (4.1). The results show that the maximum difference in magnitude between the nominal displacement and evaluated in-plane displacement before and after correcting both occur when the block aluminum is moved
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