Jeff Sundermeyer Engineering Specialist Advanced Virtual Product Development Caterpillar Inc.

Transcription

1 Validation of Physics-Based Computer Simulations of Non-Stationary Random Processes via Hypothesis Testing in the Time Domain Jeff Sundermeyer Engineering Specialist Advanced Virtual Product Development Caterpillar Inc. CAT, CATERPILLAR, their respective logos, Caterpillar Yellow and the POWER EDGE trade dress, as well as corporate and product identity used herein, are trademarks of Caterpillar and may not be used without permission Caterpillar All Rights Reserved

2 A Broad Range of Products and Industries

3 The Next Great Chapter in Mining

4 Agenda Variability in the Test Data Correlation of Simulation with Test Data Taking into Account the Test Data Variability Hypothesis Testing of Time Histories A Mini-Monte Carlo Exercise Introducing Variability to the Simulation Hypothesis Testing of Fatigue Damage Rates Testing, Simulation, & Just Making the Loads Up (Random Generation) Constraining Randomly Generated Loads to Obey Newton s Laws Value Summary

5 Enormous variability in the test data each time the machine crossed the ditch. No one test crossing is truly representative. Different angle of attack, ground properties, speed, ditch geometry, etc.

6 Example simulation load response for vertical equalizer bar pin force on chassis: MODEL >> Note the variability in the test data: Duration of the event Magnitudes, phases, random fluctuations, etc. TEST >> How can we deal with the variability in the test data?

7 Time History Hypothesis Testing Output No. of Model Curves = 1 No. of Test Curves = 9 Draw Time Step (delta_t) = Significance for Rejection (alpha) = 0.05 Denominator For Standardizing Model Curve = Constant Sigma N: 9 *************************************************************************** Hypothesis Tests for Time History Equivalence Test of Signal Mean Equivalence: Mean of Standard Normal Distribution: 2.5% limit mean_t 97.5% limit * Test of Signal Mean Equivalence: Standard Deviation of Standard Normal Distribution: 2.5% limit sigma_t 97.5% limit * ***************************************************************************

8 Not rejected Equalizer bar pin FY Left sprocket FX Left sprocket FY Left pivot shaft FX Left pivot shaft FY Left trunnion FX Left trunnion FY Lift cylinder FX Lift cylinder FY EE pin FX 2 EE pin FY 2 DD pin FX 2 DD pin FY 2 AA pin FX 2 AA pin FY 2 Left equal. bar contact FY (2 nd Impact) Left equal. bar contact FY (1 st Impact) Rejected None

9 Thoughts on Fatigue Damage Comparison Each crossing of the ditch in the test world will have a radically different damage rate. That is why they have to cross the ditch so many times in the loads test. It is not reasonable to expect the damage rate from our one simulated crossing to match the average damage rate from all of the test crossings at all locations. The best we can reasonably expect is to have a damage rate that is a believable member of the observed test distribution of damage rates.

10 Damage Rate from Simulation = E-07

11 Damage Rate from Simulation = E-07

12 Damage Rate from Simulation = 3.26E-08

13 Damage Rate from Simulation = E-07

14 Damage Rate from Simulation = E-08

15 Damage Rate from Simulation = E-07

16 Damage Rate from Simulation = E-07

17 Damage Rate from Simulation = E-07

18 What if I run the model 10 times varying the key parameters randomly, resulting in a larger sample of virtual damage rates? I am hoping that the average damage rate from this larger sample will be closer to the test average damage rate (i.e. reciprocal of the fatigue life). Key random parameters (assumed uniform distribution) Angle of attack [+/- 25%] Ground stiffness [+/- 80%] Height of first windrow [+/- 20%] Height of second windrow [+/- 20%] Depth of ditch [+/- 20%] Angle between ditch walls [+/- 7%]

19 Graphical representation of the two samples. TEST Nominal Run SIMULATION

20 How about equivalence of mean? Let s use the two sample T-test in Minitab. Two-Sample T-Test and CI: R4406X1e7, flex model R4406 X 1E7 Two-sample T for R4406X1e7 vs flex model R4406 X 1E7 N Mean StDev SE Mean R4406X1e flex model R4406 X 1E Difference = mu (R4406X1e7) - mu (flex model R4406 X 1E7) Estimate for difference: % CI for difference: (-0.493, 0.677) T-Test of difference = 0 (vs not =): T-Value = 0.33 P-Value = DF = 16 We cannot reject the null hypothesis of equivalence.

21 How about equivalence of variance? Let s use the test for equal variances in Minitab. We cannot reject the null hypothesis of equivalence.

22 An Interesting Progression in the Creation of Load Time Histories Measuring them physically at the test facility during actual events Simulating these events with computational models Making up the load histories via a random statisticalbased generation!

23 High Level Flowchart for the Statistical Generation of Time Histories Seed set X s (t) Alignmen t points Discrete Fourier Transform Map into normalized time R x Normalize d time Seed Set X s (t*) Perform ensemble autocorrelation x si * t X si * t * St Standardized Seed Set x si (t*) X * t * St * X t Rx (Power Spectrum) Rx Perturb Both via Chi-Square & Sampling Error of Mean Inverse Fourier Transform with Uniform Random Phase * r i t * S p t * * * * X git X p t ri t S p t * X p t Alignmen t points Map back into real time X gi t

24 No. of Model Curves = 9 No. of Test Curves = 9 Draw Time Step (delta_t) = Significance for Rejection (alpha) = 0.05 Denominator For Standardizing Model Curve = Instantaneous Sigma N: 12 ********************************************************************** ********** Hypothesis Tests for Time History Equivalence Test of Signal Mean Equivalence: Mean of Student-t Distribution: 2.5% limit mean_t 97.5% limit * Test of Signal Mean Equivalence: Standard Deviation of Student-t Distribution: 2.5% limit sigma_t 97.5% limit * Test of Signal Variance Equivalence: Mean of F-Distribution: 2.5% limit mean_f 97.5% limit * Test of Signal Variance Equivalence: Standard Deviation of F- Distribution: 2.5% limit sigma_t 97.5% limit * The null hypothesis of equivalence cannot be rejected at this level of significance (0.05) I can run the strong hypothesis test to see if the generated curves are believably from the same population as the seed set, as a confirmation of the process. We did not reject the null hypothesis.

25 How can we force statistically generated loads (or measured loads for that matter) to obey Newton s laws? Let s look at a simple example of a bar in tension. f 1 (t) f 2 (t) f 1 (t) - f 2 (t) = 0 f 2. (measured or generated point) or f f Let s blow this up. 2. f 2 f 1. Most probable point in load space ( f 1, f2) f 1 1 T f T dt f 1 f1 f 2 1 T f T dt f 1 (applicable to measured loads)

26 What happens if I perturb all of my statistically generated loads via this scheme? I know my new perturbed loads will be in equilibrium, but will my new equalizer bar pin vertical force, for example, pass the hypothesis test against the seed set?

27 No. of Model Curves = 10 No. of Test Curves = 10 Draw Time Step (delta_t) = Significance for Rejection (alpha) = 0.05 Denominator For Standardizing Model Curve = Instantaneous Sigma N: 17 ************************************************************* Hypothesis Tests for Time History Equivalence Test of Signal Mean Equivalence: Mean of Student-t Distribution: 2.5% limit mean_t 97.5% limit * Test of Signal Mean Equivalence: Standard Deviation of Student-t Distribution: 2.5% limit sigma_t 97.5% limit * Test of Signal Variance Equivalence: Mean of F-Distribution: 2.5% limit mean_f 97.5% limit * Test of Signal Variance Equivalence: Standard Deviation of F-Distribution: 2.5% limit sigma_t 97.5% limit * We do not reject the null hypothesis!

28 Possible uses for load perturbation: Forcing statistically generated loads to obey equilibrium. Forcing measured machine test loads to obey dynamic equilibrium. Quantifying the uncertainty of machine test loads. Propagate the uncertainty in the loads through the FEA. Express calculated strain as a probability density function sweeping forward in time. Correlate measured strain with calculated strain via the hypothesis testing of time histories. Load measurement engineers can use the standard deviation of uncertainty as a quality control metric, and also as a metric to compare various calibration methods, devices, etc. The lower the sigma, the better!

29 Hypothesis Testing of Finite Element Models f( t) f i ( t), fi Output of load uncertainty characterization s( t) Af ( t) FEM transforms mean load to mean strain n si A 2 ij 2 fi j1 Load uncertainty can be propagated to calculated strain uncertainty via FEM s meas i s si calc i x i Transform the measured strains to hypothesized drawings from the standard normal distribution. Statistical summary on the transformed measured strains should yield 95% confidence intervals that contain 0 and 1 for the mean and standard deviation, respectively.

30 Here is what some sample output might look like for FEM hypothesis testing. This was a preliminary effort on the D10T track roller frame. CHANNEL N_PTS MEAN_PASS SIGMA_PASS TEST_MEAN_OF_STD_NORMAL TEST_STDDEV_OF_STD_NORMAL R1201T_IR ( * ) ( * ) R1201F_IR ( * ) ( * ) R1201L_IR ( * ) ( * ) R1203T_IR ( *) ( *--- ) R1203F_IR (* ) ( *----- ) R1203L_IR ( * ) ( * ) R1204T_IR ( * ) ( * ) R1204F_IR ( * ) ( * ) R1204L_IR ( * ) ( * ) R1206T_IR ( * ) ( * ) R1206F_IR ( * ) ( *) R1206L_IR ( * ) ( *) R1210T_IR ( * ) ( * ) R1210F_IR ( * ) ( * ) R1210L_IR (* ) ( *) R1211T_IR ( * ) ( * ) R1211F_IR ( *---- ) ( *) R1211L_IR ( * ) ( * ) R1212T_IR ( * ) ( * ) R1212F_IR (* ) ( *- ) R1212L_IR ( * ) ( * ) R1215T_IR ( * ) ( * ) R1215F_IR ( * ) ( * ) R1215L_IR ( * ) ( * ) R1217T_IR ( *) ( *----- ) R1217F_IR ( *- ) ( * ) R1217L_IR ( * ) ( * ) R1222T_IR ( * ) ( * )

31 Why do hypothesis testing of FEM? If you pass the test, then any disagreement between calculated and measured strain is explainable by the load measurement uncertainty. There is no point in continuing to try to improve the model (i.e. the A matrix) If you fail the test, then the disagreement between calculated and measured strain cannot be blamed on the load measurement uncertainty. The problem really does lie in the A matrix (the finite element model). The value is realized by not wasting effort chasing after the wrong cause.

32 Summary Attempting to achieve the average damage rate of a loads test event at all rosette locations with one event simulation is a statistical impossibility. We should give up trying to do impossible things. We need to introduce variability to our structural simulation activities. Variability & load measurement uncertainty in machine level test results should be recognized in the machine level simulation correlation activities and in the finite element model correlations.

33 Questions?