Experimental Correlation of an N-Dimensional Load Transducer by Finite Element Analysis Tim Hunter, Ph.D., P.E. Wolf Star Technologies, LLC - or - Getting analysis to match test
What are the loads?
Linear Systems and Loads In other words F = K x εc = F
Back to the Blackboard
A closer look [m x n] [n x m] [m x m] [ε1,1 ε1,2 ε1,3 ε1,4 1,1 C 1,2 C 1,3 C ε 2,1 ε 2,2 ε 2,3 ε 2,1 C 2,2 C 2,3 2,4 C ε 3,1 ε 3,2 ε 3,3 ε 3,1 C 3,2 C 3,3 3,4][C C 4,1 C 4,2 C 4,3]= [F 1 0 0 0 F 2 0 0 0 F 3] Strain locations gauges m = # of load cases (e.g. 3) n = # of gauges (e.g. 4) n m
A closer look Strain signature due to load case 1 turned on [ε1,1 ε1,2 ε1,3 ε1,4 1,1 C 1,2 C 1,3 C ε 2,1 ε 2,2 ε 2,3 ε 2,1 C 2,2 C 2,3 2,4 C ε 3,1 ε 3,2 ε 3,3 ε 3,1 C 3,2 C 3,3 3,4][C C 4,1 C 4,2 C 4,3]= [F 1 0 0 0 F 2 0 0 0 F 3] Strain locations gauges
A closer look Strain signature due to load case 2 turned on [ε1,1 ε1,2 ε1,3 ε1,4 1,1 C 1,2 C 1,3 C ε 2,1 ε 2,2 ε 2,3 ε 2,1 C 2,2 C 2,3 2,4 C ε 3,1 ε 3,2 ε 3,3 ε 3,1 C 3,2 C 3,3 3,4][C C 4,1 C 4,2 C 4,3]= [F 1 0 0 0 F 2 0 0 0 F 3] Strain locations gauges
A closer look Strain signature due to load case 3 turned on [ε1,1 ε1,2 ε1,3 ε1,4 1,1 C 1,2 C 1,3 C ε 2,1 ε 2,2 ε 2,3 ε 2,1 C 2,2 C 2,3 2,4 C ε 3,1 ε 3,2 ε 3,3 ε 3,1 C 3,2 C 3,3 3,4][C C 4,1 C 4,2 C 4,3]= [F 1 0 0 0 F 2 0 0 0 F 3] Strain locations gauges
The Big Unknown From εc= F How do we find C? Let F = I, then εc= I Perform a psuedo-inverse εc= I ε T εc= ε T I [ε T ε] 1 [ε T ε]c= [ε T ε] 1 ε T I C= [ε T ε] 1 ε T
The Big Unknown From εc= F How do we find C? Let F = I, then εc= I Perform a psuedo-inverse εc= I ε T εc= ε T I [ε T ε] 1 [ε T ε]c= [ε T ε] 1 ε T I C= [ε T ε] 1 ε T
The Big Unknown From εc= F How do we find C? Let F = I, then εc= I Perform a psuedo-inverse εc= I ε T εc= ε T I [ε T ε] 1 [ε T ε]c= [ε T ε] 1 ε T I C= [ε T ε] 1 ε T
The Big Unknown From εc= F How do we find C? Let F = I, then εc= I Perform a psuedo-inverse εc= I ε T εc= ε T I [ε T ε] 1 [ε T ε]c= [ε T ε] 1 ε T I C= [ε T ε] 1 ε T
The Big Unknown From εc= F How do we find C? Let F = I, then εc= I Perform a psuedo-inverse εc= I ε T εc= ε T I [ε T ε] 1 [ε T ε]c= [ε T ε] 1 ε T I C= [ε T ε] 1 ε T Can be derived from FEA model
Seeing [C] An extremely large number of strain locations and orientations exist. Which ones are best? C= [ε T ε] 1 ε T The optimal strains locations are found when ε T ε max Search is done via a Galil-Kiefer D-Optimal search algorithm
Hangman Structure
Static Test Setup 16
Procedure 1. Clamp fixture in vise in varying orientations (6) 2. Load fixture with various static load magnitudes (4) and collect strain data (increase load in steps: 5 10.2 15.4 25.2 lbs) 3. Perform preliminary static load strain correlation between test and analysis at all gauge locations 4. Manipulate strain data to reduce point count 5. Use strain data in WST/True Load Post-Test to predict loads 6. Compare predicted loads to known loads 7. Compare replicated strain time histories to measured time histories 17
Static Load Application Load was applied with shot bags weighing 5, 5.2, 5.2, 9.8 lbs (in that order) Load was applied in 6 directions (see below) Lateral Longitudinal Vertical 18
Static Load Application cont. 45 o 60 o 45 o Lat/Long 45 o Lat/Vert 60 o Long/Vert 45 o 19
Data Manipulation 0 lbs 5 lbs 10.2 lbs 15.4 lbs 25.2 lbs 0 lbs Data Extraction & Decimation (using Glyhworks) Reduction from 60,000 to 150 datapoints 20
Post-Test GUI Load Pre-Test Setup Load strain data Create report & load scales 21
Lateral Static Load Load Factors should be: Lateral = 1 Longitudinal = 0 Vertical = 0 22
Lateral Static Load G1: 2.8% Error G2: 0.3% Error G3: 5.2% Error G4: 3.7% Error G5: 0.2% Error G6: 5.6% Error Note: very noisy, < 20 23
Longitudinal Static Load Load Factors should be: Lateral = 0 Longitudinal = 1 Vertical = 0 24
Longitudinal Static Load G2: 15.7% Error G3: 2.1% Error G1: 9.6% Error G4: 0.3% Error G5: 7.2% Error G6: 0.1% Error Note: very noisy, < 20 25
Vertical Static Load Load Factors should be: Lateral = 0 Longitudinal = 0 Vertical = 1 26
Vertical Static Load G3: 11.2% Error G1: 0.0% Error G2: 0.3% Error G4: 1.6% Error G6: 2.4% Error G5: 0.3% Error Note: very noisy, < 20 27
Lat/Long 45 degrees Load Factors should be: Lateral = 0.707 Longitudinal = 0.707 Vertical = 0 28
Lat/Long 45 degrees G1: 4.7% Error G2: 0.3% Error G3: 0.8% Error G4: 0.0% Error G5: 0.2% Error G6: 0.1% Error Note: very noisy, < 20 29
Lat/Vert 60 degrees Load Factors should be: Lateral = 0.866 Longitudinal = 0 Vertical = -0.5 30
Lat/Vert 60 degrees G1: 0.4% Error G2: 0.3% Error G3: 15.8% Error G4: 3.8% Error G5: 0.2% Error G6: 11.2% Error Note: very noisy, < 20 31
Long/Vert 45 degrees Load Factors should be: Lateral = 0 Longitudinal = 0.707 Vertical = -0.707 32
Long/Vert 45 degrees G1: 0.3% Error G2: 0.9% Error G3: 2.6% Error G4: 0.3% Error G5: 0.7% Error G6: 0.1% Error Note: good strain replication on all gauges great correlation 33
Static Load Summary Predicted Theoretical Lateral Long. Vert. 25.2 lb Load Application Direction Lat/Long Lat/Vert Long/Vert 45 o 60 o 45 o Lateral 1.068 1 0.073 0 0.006 0 0.809 0.707 0.937 0.866 0.047 0 Load Scale Factor Long. Vertical 0.004-0.006 0 0 1.019 0.040 1 0 0.026 1.039 0 1 0.714 0.024 0.707 0-0.526 0.005-0.5 0 0.751-0.717 0.707-0.707 Magnitude 1.068 1 1.022 1 1.039 1 1.079 1 1.075 1 1.039 1 Angle b/w ideal and predicted 0.39 o 4.67 o 1.47 o 3.79 o 0.74 o 2.91 o Angle calculated as: 1 AB cos AB 34
Static Load % Error < 5% 5% - 10% > 10% Lateral Long. Vert. Load Application Direction Lat/Long Lat/Vert Long/Vert 45 o 60 o 45 o Lateral 6.8 7.3 0.6 14.4 9.0 4.7 Load Scale Factor Error (%) Long. 0.4 1.9 2.6 1.0 5.2 6.2 Vertical 0.6 0.4 3.9 2.4 0.5 1.4 Magnitude 6.8 2.2 3.9 7.9 7.5 3.9 Mag. Angle Offset (degrees) 0.39 o 4.67 o 1.47 o 3.79 o 0.74 o 2.91 o 35
Proving Ground Loads
Proving Ground Vert Long Lat
Force Correlation Measured Force F= m ẍ Test strains From FEA Force from strain gauges
Strain Correlation
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