Small Deflections Model

1
Small Deflections Model

• Determining Sensor Placement

2
Beam_Param_Op.m

• Follows math explained in Bending in 2
  Planes.ppt
• Models done of
• No Inflection Point (Only end-load)
• One Inflection (and end-load)
• Load in Middle of Needle
• Small Deflections vs. Large Deflections
• Varying Load Concentration Optimization

3
No Inflection

• Sensor position does not affect deflection error
  (see Sensitivity plots).

4
No Inflection
Sensitivity of Error w/respect to x1.
Force at tip 2e-39.81 N Below Deflection
Error at Tip at various positions for 2 sensors.
Sensitivity of Error w/respect to x2.
5
One Inflection

• A force was applied mid-way along the needle and
  at the tip.
• Deflection in error can be differentiated by
  sensor position.
• Largest error is still small (about 0.6mm).
• Sensitivity plot shows that sensor can be placed
  almost anywhere from the base to mid-length along
  the needle.

6
One Inflection
Sensitivity of Error w/respect to x1.
Force at 75mm 2e-39.81 N Force at tip
5e-39.81 N Below Deflection Error at Tip at
various positions for 2 sensors.
Sensitivity of Error w/respect to x2.
7
One Inflection
Deflection error at tip in this case is 0.0605
mm. As can be seen in the curvature plot, the
actual and estimated graphs intersect at x1 and x2
Force at 75mm 2e-39.81 N Force at tip
5e-39.81 N Based on Above Results, Sensor x1
placed at 30mm. Sensor x2 placed at 110mm.
8
Sensor Placement Around Inflection Point.
In both cases, the inflection was at 75mm. When
the second sensor was closer to the inflection
point, the deflection error at the tip became
0.002mm

• X1 30 X2 110

• X1 33 X2 80

9
Load in Middle of Needle

• One load only applied at center of needle.
• Error in curvature proportional to distance
  between inflection point and second sensor
  position.
• Deflection error is sensitive to sensor location
  (unlike end-loaded only case).

10
One Load
Sensitivity of Error w/respect to x1.
Force at 75mm 2e-39.81 N Force at tip 0
N Below Deflection Error at Tip at various
positions for 2 sensors.
Sensitivity of Error w/respect to x2.
11
One Load
Deflection error at tip in this case is 0.0020
mm. As can be seen in the curvature plot, the
actual and estimated graphs intersect at x1 and
x2. The error in the curvature is relative to the
distance between the actual applied load and the
second sensor position.
Force at 75mm 2e-39.81 N Force at tip 0
N Sensor x1 placed at 33mm. Sensor x2 placed at
80mm.
12
Larger Deflection

• X1 33, X2 80

• F1 5e-39.81 N
• F210e-39.81 N

Sensitivities similar as in small deflection case
above. Plot of Deflection Error at Tip
Tip Deflection Error at given X1, X2 0.0050mm.
13
Varying Load Concentration

• Suppose the applied load is concentrated at some
  location along the needle.
• Vary location of applied load (0-150mm)
• Optimize best location for two sensors.