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Point and Distributed Loading Tip deflection errors and sensitivity results.

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Point Loading Cases F1F1 F2F2 Sensor 1 Sensor 2 x1x1 x2x2 a Tip Deflection b Case 1: a = L/2, b = L. F1 > 0, F2 > F1. Case 2: a = L/2, b = L. F1 > 0, F1 > F2. Case 3: a = L/2, b = L. F1 abs(F1).

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Deflection Errors at Tip Case 1 F1 = 2e-3*9.81 F2 = 5e-3*9.81 Case 2 F1 = 5e-3*9.81 F2 = 2e-3*9.81 Case 3 F1 = -2e-3*9.81 F2 = 5e-3*9.81

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Sensitivity of Deflection Errors with Respect to x1 Case 1 F1 = 2e-3*9.81 F2 = 5e-3*9.81 Case 2 F1 = 5e-3*9.81 F2 = 2e-3*9.81 Case 3 F1 = -2e-3*9.81 F2 = 5e-3*9.81

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Sensitivity of Deflection Errors with Respect to x2 Case 1 F1 = 2e-3*9.81 F2 = 5e-3*9.81 Case 2 F1 = 5e-3*9.81 F2 = 2e-3*9.81 Case 3 F1 = -2e-3*9.81 F2 = 5e-3*9.81

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Deflection and Curvatures when x1 = 25, x2 = 82. Case 1 F1 = 2e-3*9.81 F2 = 5e-3*9.81 Case 2 F1 = 5e-3*9.81 F2 = 2e-3*9.81 Case 3 F1 = -2e-3*9.81 F2 = 5e-3*9.81

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Conclusions: Opposing loads lead to the same results as loads in the same direction (if magnitude is the same.) If the load at the inflection point is larger than the load at the end, the range of deflection error and sensitivities are greater. For all cases when a = L/2, the lowest deflection error at the tip is when x1 = 45 and x2 = 125.

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Distributed Loading Cases Sensor 1 x1x1 x2x2 a Tip Deflection b q Sensor 2 Case 1: a = L/2, b = L - a

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Case 1: q = 10e-3*9.81/75 Deflection Error at Tip Sensitivity w/Respect to x1 Sensitivity w/Respect to x2 When x1 = 25, x2 = 82:

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Conclusions: For Case 1, deflection error at tip when x1 = 25, x2= 82 is 0.0574. The best region for sensor placement seems to be when x1 = 28, x2 is in the range [122:138]. However, this is sensitive to x1. If one sensor is at L/2 (start of loading), the position of the second sensor is not important in improving the accuracy of the curvature readings (hence the displacement error).

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Two Distributed Loads x1x1 x2x2 a Tip Deflection b q2q2 Sensor 2 Sensor 1 q1q1

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One Distributed Load and End Point Load x1x1 x2x2 a Tip Deflection b Sensor 2Sensor 1 qF

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Three Loading Types Given two point loads, distributed loads were found such that the curvature at x = 0 and x = L/2 was the same. The distributed loading caused regions of similar deflection errors.

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Deflection Error at Tip F1 = 2 grams, F2 = 5 grams

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Sensitivity in x1 position F1 = 2 grams, F2 = 5 grams

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Sensitivity in x2 position F1 = 2 grams, F2 = 5 grams

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Deflection and Curvature when x1 = 20, x2 = 80. F1 = 2 grams, F2 = 5 grams Error at tip = -0.0279mmError at tip = -0.0088mmError at tip = -0.0099mm

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More Extreme Case F1 and F2 are in opposite directions, and

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Deflection Error at Tip F1 = -5 grams, F2 = 2 grams

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Sensitivity in x1 position F1 = -5 grams, F2 = 2 grams

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Sensitivity in x2 position F1 = -5 grams, F2 = 2 grams

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Deflection and Curvature when x1 = 20, x2 = 80. F1 = -5 grams, F2 = 2 grams Error at tip = 0.0697mmError at tip = -0.0088mmError at tip = -0.0099mm

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