Point and Distributed LoadingTip deflection errors and sensitivity results.

Point Loading CasesF1F2Sensor 1Sensor 2x1x2aTip DeflectionbCase 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 < 0, F2 > abs(F1).

Deflection Errors at Tip

Case 1F1 = 2e-3*9.81F2 = 5e-3*9.81Case 2F1 = 5e-3*9.81F2 = 2e-3*9.81Case 3F1 = -2e-3*9.81F2 = 5e-3*9.81

Sensitivity of Deflection Errors with Respect to x1

Case 1F1 = 2e-3*9.81F2 = 5e-3*9.81Case 2F1 = 5e-3*9.81F2 = 2e-3*9.81Case 3F1 = -2e-3*9.81F2 = 5e-3*9.81

Sensitivity of Deflection Errors with Respect to x2

Case 1F1 = 2e-3*9.81F2 = 5e-3*9.81Case 2F1 = 5e-3*9.81F2 = 2e-3*9.81Case 3F1 = -2e-3*9.81F2 = 5e-3*9.81

Deflection and Curvatures when x1 = 25, x2 = 82.

Case 1F1 = 2e-3*9.81F2 = 5e-3*9.81Case 2F1 = 5e-3*9.81F2 = 2e-3*9.81Case 3F1 = -2e-3*9.81F2 = 5e-3*9.81

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.

Distributed Loading CasesSensor 1x1x2aTip DeflectionbqSensor 2Case 1: a = L/2, b = L - a

Case 1: q = 10e-3*9.81/75Deflection Error at Tip Sensitivity w/Respect to x1 Sensitivity w/Respect to x2When x1 = 25, x2 = 82:

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).

Two Distributed Loadsx1x2aTip Deflectionbq2Sensor 2Sensor 1q1

One Distributed Load and End Point Loadx1x2aTip DeflectionbSensor 2Sensor 1qF

Three Loading TypesGiven 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.

Deflection Error at TipF1 = 2 grams, F2 = 5 grams

Sensitivity in x1 positionF1 = 2 grams, F2 = 5 grams

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

Deflection and Curvature when x1 = 20, x2 = 80. F1 = 2 grams, F2 = 5 grams

Error at tip = -0.0279mm

Error at tip = -0.0088mm

Error at tip = -0.0099mm

More Extreme CaseF1 and F2 are in opposite directions, and

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

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

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

Deflection and Curvature when x1 = 20, x2 = 80. F1 = -5 grams, F2 = 2 grams

Error at tip = 0.0697mm

Error at tip = -0.0088mm

Error at tip = -0.0099mm