Point and Distributed Loading

1
Point and Distributed Loading

• Tip deflection errors and sensitivity results.

2
Point Loading Cases
b
a
F2
F1
Sensor 2
Sensor 1
Tip Deflection
x1
x2
Case 1 a L/2, b L. F1 gt 0, F2 gt F1. Case 2
a L/2, b L. F1 gt 0, F1 gt F2. Case 3 a L/2,
b L. F1 lt 0, F2 gt abs(F1).
3
Deflection Errors at Tip
Case 1 F1 2e-39.81 F2 5e-39.81 Case 2 F1 5e-39.81 F2 2e-39.81 Case 3 F1 -2e-39.81 F2 5e-39.81

4
Sensitivity of Deflection Errors with Respect to
x1
Case 1 F1 2e-39.81 F2 5e-39.81 Case 2 F1 5e-39.81 F2 2e-39.81 Case 3 F1 -2e-39.81 F2 5e-39.81

5
Sensitivity of Deflection Errors with Respect to
x2
Case 1 F1 2e-39.81 F2 5e-39.81 Case 2 F1 5e-39.81 F2 2e-39.81 Case 3 F1 -2e-39.81 F2 5e-39.81

6
Deflection and Curvatures when x1 25, x2 82.
Case 1 F1 2e-39.81 F2 5e-39.81 Case 2 F1 5e-39.81 F2 2e-39.81 Case 3 F1 -2e-39.81 F2 5e-39.81

7
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.

8
Distributed Loading Cases
b
a
q
Sensor 1
Sensor 2
Tip Deflection
x1
x2
Case 1 a L/2, b L - a
9
Case 1 q 10e-39.81/75
Deflection Error at Tip Sensitivity
w/Respect to x1 Sensitivity w/Respect
to x2
When x1 25, x2 82
10
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 122138.
  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).

11
Two Distributed Loads
b
a
q2
q1
Sensor 2
Sensor 1
Tip Deflection
x1
x2
12
One Distributed Load and End Point Load
b
a
q
F
Sensor 2
Sensor 1
Tip Deflection
x1
x2
13
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.

14
Deflection Error at TipF1 2 grams, F2 5 grams


15
Sensitivity in x1 positionF1 2 grams, F2 5
grams


16
Sensitivity in x2 positionF1 2 grams, F2 5
grams


17
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

18
More Extreme Case

• F1 and F2 are in opposite directions, and

19
Deflection Error at TipF1 -5 grams, F2 2
grams


20
Sensitivity in x1 positionF1 -5 grams, F2 2
grams


21
Sensitivity in x2 positionF1 -5 grams, F2 2
grams


22
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