Point and Distributed Loading

Tip deflection errors and sensitivity results.

Point Loading Cases

F1 F2Sensor 1 Sensor 2

x1

x2

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 < 0, F2 > abs(F1).

Deflection Errors at Tip

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

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

Case 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*9.81

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

Case 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*9.81

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

Case 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*9.81

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

Case 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 Cases

Sensor 1

x1

x2

a

Tip Deflection

b

q

Sensor 2

Case 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 x2

When 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 Loads

x1

x2

a

Tip Deflection

b

q2

Sensor 2Sensor 1

q1

One Distributed Load and End Point Load

x1

x2

a

Tip Deflection

b

Sensor 2Sensor 1

q F

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.

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

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

Sensitivity in x2 positionF1 = 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 Case

• F1 and F2 are in opposite directions, and

21 FF

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

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

Sensitivity in x2 positionF1 = -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