statistical properties component response

F A T I G U E 2 0 0 7

ON THE STATISTICAL PROPERTIES OF COMPONENT RESPONSE TO ROAD INPUTS

Dr. Colin J Dodds1 and Mr. Chandrakant Awate2

Often complete vehicle, aggregate and component fatigue testing in the laboratory is conducted using one specific track recording of vehicle component response data. These data are usually edited using some damage criteria and become the targets which are reproduced in the laboratory. The fatigue failures that are obtained in the laboratory test could be biased due to the limited data set. A study was carried out where data were collected using multiple drivers and multiple repeats from the same test tracks. Multiple sets of component and aggregate response data were analyzed in terms of damage and other fatigue related properties. In practical engineering situations it is not possible to collect such large volumes of data. The paper discusses how the statistical properties of the fatigue parameters obtained from this study can be applied to the situation where only one data set is available to estimate the durability performance of the component or aggregate.

INTRODUCTION

In 1988 Rivolo and Vianello [1] from Fiat Auto s.p.A presented a paper in which they defined the challenge of verifying the reliability objectives in vehicles and their components (table 1).

TABLE 1 - “Why the challenge” after Rivolo & Vianello.

Necessary Available Long test periods Increasingly shorter test times Very strict objectives to be evaluated with good statistical significance

Limited sampling

Very wide spread of usage related environment conditions

Specific test conditions at test facilities

Measurements made by customers Test drivers and trained engineers

1 Dodds and Associates, Edinburgh, Scotland 2 Tata Motors, Pune, India

F A T I G U E 2 0 0 7

Although all OEMs recognize this challenge and are aware that a large quantity of data is a prerequisite for achieving the high reliability objectives for the vehicle and its components, the reduction in vehicle development time over the last 18 years has, in turn, reduced the time available to collect these data. Vehicle development times less than two years are becoming the norm! This paper describes the results of a study at Tata Motors in Pune, to investigate if the statistics from a short durability environment defined by a single pass of the proving ground and then transcribed into the laboratory test can be applied to estimate the durability environment of the proving ground test as a whole.

It is worthwhile at this stage to comment on the probability required in the estimation of an extreme load in a component. The value is optional, but should be within 90% to 99.9%. It is important to notice that the reliability of wheel suspension and its components should be >99.9% during customer usage. A danger in statistical analysis is that insufficient data are available causing wide scatter and a low confidence in the distribution function. Care was taken throughout the analysis to estimate the accuracy of each distribution function and the AD statistic and corresponding p-value are given for each distribution (see later).

The test program was implemented using a chassis based LCV with front independent suspension and a rigid rear axle. The major components and aggregates of the vehicle were instrumented and response data collected on a customer-correlated proving ground. Multiple response measurements using three different drivers were made to investigate the scatter of the damage and the other statistical properties of the recorded data on the proving ground tracks.

In line with industry practice, an accelerated durability simulation test was developed using a damage based editing procedure on one pre-selected test schedule. The vehicle responses in this accelerated test schedule, which had been reduced to 33% of the time in the original schedule, were used as the targets to derive the excitation time series for the 4-post simulator [2]. The simulation test derived by this methodology was implemented on the same vehicle that was used to record the data. The excitation signals (the actuator drives derived from the targets) were repeated a requisite number of times to simulate the laboratory test equivalent of 12,800km on the proving ground or 160,000km of customer service. Although 86% of the incidents that occurred on this vehicle in the laboratory test correlated well with known incidents on similar vehicles on proving ground tests, there was concern regarding the incidents that were observed on the proving ground and not in the laboratory test and vice versa. In addition, there was concern regarding the relative timing of similar incidents on the proving ground and laboratory tests.

In an attempt to resolve these issues and to better understand the properties of one such set of recorded data as a representation of the overall proving ground test the multiple data sets were analyzed.

COMPONENTS UNDER STUDY

Although 115 channels of data were collected for this study it is not practical to review all these data in this paper and five component responses (table 2) were chosen for presentation. In addition, other channels were added when necessary to augment the discussion. The results presented were typical of all the strain channels and the authors suggest that the conclusions on the statistical properties of response to rough road inputs (proving ground) are generally valid.

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TABLE 2 Components under study

Component Transducer Number Description

LH front coil spring 10 Strain ~ vertical loadRH chassis frame mid location 18 Top of frame bending strain LH chassis frame mid location 45 Top of frame bending strain LH cab ‘A’ pillar 47 ‘A’ pillar axial/bending strain Anti-roll bar 27 Torsion in bar

In each case the strain was measured close to a critical location or located to measure a specific load. Each component response was initially verified as being sensitive to vertical input. This was necessary to ensure that the response on the 4-post rig would adequately simulate the component response. The correlation was obtained by calculating the joint probability distribution (JPDF) between each component response and the wheel-to-body (WTB) displacement transducer closest to the component (figure 1)3. Perfect correlation is indicated by all points lying on the diagonal as exhibited by the front coil spring (figure 1a); that is large excursions in the vertical displacement cause large strain excursions in the component. This condition is not a prerequisite for the subsequent statistical analysis in this paper, but only for the accuracy of the component response in the laboratory test on a vertical input simulator. Additionally, the implication of this correlation is that if we were to increase the amplitude of the WTB displacement we would increase the strain in the component pro-rata, but always subject to any non-linearity in dynamic response. Obviously not all transducer locations exhibit this characteristic and the JPDF of the lateral force in a ball joint is shown in figure 1f for comparison. In this case large excursions in the vertical displacement cause small strains in the ball joint as expected.

COMPONENT RESPONSE STATISTICS OF PROVING GROUND RUNS

Initially it seemed reasonable to check the repeatability between the three drivers and between each run by one driver. Boxplots are a diagrammatic form of presenting this type of data. They are based on a summary of the data which can be written down in terms of five numbers: the median, the upper and lower hinges and the upper and lower extremes. However, in our case, as each driver completed only three runs, the boxplot shows the median value and the upper and lower hinges represent the other two values on either side of the median.

Three statistical quantities were calculated for each run: the strain range, RMS and kurtosis. Whilst the first two will be familiar to the reader the kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. The kurtosis, M4, or as it is sometimes called the fourth moment of the data is defined as:

( )( ) 4

1

4

4 1 sNYY

MN

i i

−

−= ∑ =

3 Figures are located at the end of the text.

F A T I G U E 2 0 0 7

In the above equation, Y is the mean value and s , the standard deviation. Data sets with high kurtosis tend to have a distinct peak near the mean and decline rather rapidly into a long tail. Data sets with low kurtosis tend to have a flat top near the mean and have heavy tails. A uniform distribution would be the extreme case. The kurtosis for a normal distribution is three. Time series amplitude distributions for different values of kurtosis are shown in table 3.

TABLE 3 Amplitude distribution v kurtosis

Up-road s05

-200

0

200

400

600

800

1000

1200

0 20 40 60 80 100 120

bin

Freq

uenc

y

Pave s05

-1000

100200300400500600700800900

0 50 100 150 200 250bin

Freq

uenc

y

Twist s05

050

100150200250300350400450

0 20 40 60 80 100 120

bin

Freq

uenc

y

Kurtosis = 8 Kurtosis = 3 Kurtosis = 2

The reason that the kurtosis was chosen to compare the data sets was that a low kurtosis value indicates a signal where there will be a significant number of peak-valley ranges at higher amplitudes than predicted from a normal distribution and hence a higher induced damage and consistent between short runs. However, as it happens, all the kurtosis values in these data were high and in some cases very high!

The boxplots for four different components are shown in figure 2. Based on the measured RMS values of the data, driver one, D1 produces consistent results while the other two are inconsistent. However, the repeatability from each driver in terms of the strain range and kurtosis is dependent on the location of the gauge. No distinct conclusion can be drawn from these data except that each run over the track is a unique event. The strain range and kurtosis are influenced, in these short samples, by individual bumps or holes which the driver may or may not drive over. If we assume that each of these runs is part of a stationary random process4 then we can compare the statistics for each gauge location across each run. This assumption is reasonable especially as each track was run at constant speed. Figure 3 shows that the strain range and RMS values for each run are governed by a normal distribution where the values for the nine recorded events lie within two standard deviations of the mean. This was true of all gauges, even those on the suspension bump stops (figure 4) where the response is highly non-linear. The normal distribution is a valid model since the Anderson-Darling, AD, statistic is small, <1 in all cases and the associated P-value is greater than a level of significance of 0.1. The lower the AD statistic the better fit the distribution model to the data.

The kurtosis values are high and, with the exception of some suspension components, are indicative of a very spiked amplitude distribution with a long thin tail. The majority of the data lies close to zero and there are few data points at the extremes. This has a significant implication for the estimation of fatigue damage. The damage on the proving ground tracks will be caused by a very

4 A stationary process will exhibit the same statistical properties over time. It does not imply that the statistical properties of different runs are the same.

F A T I G U E 2 0 0 7

few high cycles and the number and level of these few cycles will be important. This phenomenon will be reflected in the rainflow histograms where the majority of damage is accumulated by very few high cycles transposed into the damage histogram (figure 5). It can be seen that the higher the kurtosis the fewer rainflow cycles contribute to the damage estimation. This problem is well known when we try to obtain a good estimate of the rainflow histogram from a small data sample. In proving ground tests there is no way round this situation unless we run each track for a large number of times and concatenate the data into one long sequence. However, this defeats the purpose of collecting small samples.

Frequency content

The frequency content for each measurement location is the same independent of the measurement location, run or driver. This is something that would be expected as the frequency response characteristics are dependent on the dynamic properties of the vehicle unless the road surface has an inherent sinusoidal content. Road profiles in general do not exhibit this characteristic [3]. The area under each power spectra shown in figure 6 is equal to the mean square of the signal.

∫∞

=0

2 )( dffGs

It is obvious that this varies between run and driver as indicated earlier in the distribution of RMS values and the variation is illustrated by the different local mean square energy at the resonant peaks in the spectra.

Damage

The damage estimate for each run was calculated using the same material properties for all locations. Bearing in mind that the damage estimate is highly influenced by the short data sample it seemed prudent to normalize the data in each channel over all runs by dividing by the mean damage. Again, a normal probability distribution fitted these data. In figure 7 the probability modal of normalized pp-strain and damage is estimated for a combination of eight channels. They reveal that 99% of the scatter in the damage estimation is within a factor of 2 on the mean while the pp-strain is within a factor of 20% on the mean. Figure 8 shows the correlation between pp-strain and damage for these same data points with what the authors term the “simulation target chart” limits of a factor of two in damage and 20% in pp-strain. Although there is significant scatter in this cross-plot, the regression line almost equally divides the damage around the mean line. Figure 9 presents the “achieved simulation target chart” for the laboratory simulation. The damage in selected channels is estimated from the proving ground run (used to derive the drive signals for the laboratory test) is divided by the damage in the same channels recorded on the test vehicle on the rig. The authors’ experience over 20 years is that if the ratio of the road/rig damage in correlated components in a full vehicle simulation falls within a factor of 2 then the simulation is very acceptable.

F A T I G U E 2 0 0 7

DISCUSSION AND CONCLUSION

Based on all the observed data and subsequent analyses the authors believe that more information can be extracted for a single proving ground run which will help improve the reliability of the simulation test. Larsson [4] and Pedersen [5] from Volvo and Blackmore [6] from Jaguar showed that peak strains could be estimated from recorded data using an extreme value distribution. A number of extreme value models are available [7] and the one used here is the Type 1 or the Gumbel distribution given by

]exp[]Pr[ /)( βμ−−=< xexX

That is, the probability that X is less than an extreme value x is given by the double exponential with parameters μ and β. The parameter, μ is the location parameter or mode, in this case i.e. the most likely value of x; the parameter, β is the intensity factor or scale parameter, i.e. the slope of the line when plotted logarithmically. For any given component one run was selected at random and the rainflow histogram estimated from the proving ground data. A threshold rainflow range was selected and all the ranges above the threshold were extracted. The extreme value distribution was fitted to these data. The results from two channels are presented in figure 10 along with the normal distribution of the maximum pp-strains recorded for the nine runs. The 99.9 percentile pp-strains predicted form these two distributions are in reasonable agreement (around 10%). The exercise was repeated and in this case three runs were selected on one channel and compared with the maximum pp-strains from all nine runs (figure 11). The resultant 99.9 percentile pp-strains are again acceptable. Although the accuracy depended on the run that was selected these results give some confidence in that we can achieve a better estimate of the peak loads using this methodology.

The question now arises – can we use this information to improve the reliability of the laboratory simulation? We would like to increase our confidence in the test vehicle experiencing similar high loads that would occur during the complete 12,800km proving ground bogie.

Johannesson, [8] has just published a paper in which he describes how the higher peaks that were not recorded in the data set can be extrapolated using the extreme value statistics and he has applied this in work to a single channel fatigue test data where frequency in Hz is not important. He also demonstrated how an augmented rainflow matrix could be developed from the measured matrix and the extreme value statistics. Hay [9, 10] and Sherratt et al [11] have proposed models for generating multi-channel test data from frequency spectra with given rainflow distributions. The authors believe that based on the results of this study, it would be a worthwhile exercise to combine and expand the work of Johannesson, Hay and Sherratt to provide a new simulation methodology whereby the target channels for the multi-axes simulation contain the correct frequency content and extrapolated rainflow from a single recorded data set from the proving ground. These new targets could be used to derive the actuator drive signals for the simulation test where the extreme values in the drive signals are randomly estimated in each repeat of the drive signal.

The laboratory test so derived would better simulate the complete proving ground test where the multiple passes required to achieve the mileage required are sufficient to ensure that the same extreme values would be achieved.

F A T I G U E 2 0 0 7

ACKNOWLEDGEMENTS

The authors of this paper would like to express heartfelt thanks to the following individuals: Mr. R. R. Akarte, Consultant adviser, Mr. S. R Ravishankar, Sr. General Manager, Mr. G R Nagbhushan, Deputy General Manager and S. M. Panse, Assistant General Manager, all ERC, TATA Motors, for their valuable guidance and infrastructural support throughout this project, without which its successful completion would not have been possible.

The authors also gratefully acknowledge the help they received from team members in the ERC instrumentation, data acquisition and laboratory simulation teams who spent many hours on this project.

REFERENCE LIST

(1) Rivolo P.F. and Vianello M., “Vehicle test planning for reliability targets verification”, Proceedings of international conference on “Metodologie e mezzi innovative per la sperimentazione nel settore automotoristico” Firenze, November 1988.

(2) Panse S., Awate C. and Dodds C. J., “Validation of an accelerated test on a 4-post road simulator”, Proceedings of SIAT symposium, Pune India January 2007.

(3) Dodds C. J. and Robson J. D., “The description of road surface roughness”, J Sound Vibration, 31(2) pp175-183, 1975

(4) Larsen M., “Validation of design spectra for passenger wheel suspensions” Proceedings of durability performance testing and analysis symposium, Troy Mi. 1995

(5) Pedersen A., “Basic aspects of durability life evaluation of commercial vehicle components” Proceedings of durability performance testing and analysis symposium, Troy Mi. 1995

(6) Blackmore P. A., “Living with doubt and uncertainty…Statistics and engineering durability”, EIS symposium on advances in computer modeling techniques and laboratory tests, Gaydon, October 1999.

(7) Kotz S. and Nadarajah S., “Extreme value distributions” Imperial College Press, England, 2000 – ISBN 1-86094-224-5.

(8) Johannesson P., “Extrapolation of load histories and spectra” Fatigue and fracture of engineering materials and structures, 29, pp 210-207, 2006

(9) Hay N. C., “The simulation of random environments for structural dynamics testing” PhD thesis, #624.1771, Napier U. Edinburgh, 1989

(10) Hay N.C., “Matching frequency content when regenerating from a rainflow count” EIS symposium on fatigue caused by vibration, May 1997.

(11) Sherratt F., Bishop N. W. N. and Dirlik T., “Predicting fatigue life from frequency domain data” J EIS, 18, September 2005.

F A T I G U E 2 0 0 7

Figure 1 - Joint probability distribution (JPDF) between

LH wheel-to-body transducer (#6) and each response channel. Figure 1a – Coil spring Figure 1b – RH frame bending

Figure 1c – LH frame bending Figure 1d – ‘A’ pillar

Figure 1e – torsion bar Figure 1f – ball joint lateral force

F A T I G U E 2 0 0 7

Figure 2 – Box plots comparing driver variability and consistency.

8

7

6

5

4

D3D2D1

140

120

100

D3D2D1

960

940

920

900

880

Kurtosis RMS

Range

Boxplot of Kurtosis, RMS, Range, channel 10 for each driver

12

11

10

9

D3D2D1

80

70

60

50

D3D2D1

1100

1000

900

800

700

Kurtosis RMS

Range


18

16

14

12

10

D3D2D1

21.0

19.5

18.0

16.5

15.0

D3D2D1

320

300

280

260

240

Kurtosis RMS

Range


8

7

6

D3D2D1

80

75

70

65

60

D3D2D1

840

810

780

750

720

Kurtosis RMS

Range


Each driver is denoted by D1, D2 and D3 and the boxes show the median value of each of the three runs per driver and the other two values at the hinges (i.e. the top and bottom of the box)

F A T I G U E 2 0 0 7

Figure 3 – Probability model for RMS and Range – all nine runs

Channels 10, 45, 47 and 25

Standard deviation

Perc

ent

420-2-4

99

95

90

80

706050403020

10

5

1

420-2-4

99

95

90

80

706050403020

10

5

1

RMS RangeMean 1.233581E-17StDev 1N 9AD 0.266P-Value 0.596

RMS

Mean -4.29286E-15StDev 1N 9AD 0.304P-Value 0.504

Range

Normal - 95% CIProbability Plot of RMS, Range - Channel 10

Standard deviation

Perc

ent

420-2-4

99

95

90

80

706050403020

10

5

1

420-2-4

99

95

90

80

706050403020

10

5

1

RMS RangeMean -9.86865E-17StDev 1N 9AD 0.190P-Value 0.858

RMS


Range


Standard deviation

Perc

ent

420-2-4

99

95

90

80

706050403020

10

5

1

420-2-4

99

95

90

80

706050403020

10

5

1


RMS

Mean 1.853456E-15StDev 1N 9AD 0.367P-Value 0.348

Range


Standard deviation

Perc

ent

420-2-4

99

95

90

80

706050403020

10

5

1

420-2-4

99

95

90

80

706050403020

10

5

1


RMS


Range

Normal - 95% CIProbability Plot of RMS, Range - channel 25

Figure 4 – JPDF WTB displacement to strain, channel 16 (bump stop) and probability model. This is a highly non-linear response parameter

RMS (left) - Range (right) in micro-strain

Perc

ent

403020100

99

95

90

80

7060504030

20

10

5

1

8006004002000

99

95

90

80

7060504030

20

10

5

1

RMS RangeMean 20.54StDev 5.810N 9AD 0.326P-Value 0.444

RMS

Mean 425.0StDev 114.8N 9AD 0.225P-Value 0.743

Range

Normal - 95% CIProbability Plot of RMS, Range - channel 16 LH bump stop

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Figure 5a – Rainflow (left) and damage (right) histograms from channel 10

Kurtosis = 5

Figure 5b – Rainflow (left) and damage (right) histograms from channel 18

Kurtosis = 11

Figure 5c – Rainflow (left) and damage (right) histograms from channel 16

Kurtosis = 98

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Figure 6 – Power spectra channel 10, 45 and 47

Figure 6a – Channel 10 – left. 3-runs driver-1; right, 1-run each from the 3-drivers

Figure6b – Channel 45 – left, 3-runs driver-2; right, 1-run each from the 3-drivers

Figure 6c – Channel 47 – left, 3-runs driver-3; right, 1-run each from the 3-drivers

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Figure 7 – Probability model for damage scatter and pp-strain

Damage as a ratio over the mean damage in 9 runs per channel

Perc

ent

2.52.01.51.00.50.0-0.5

99.9

99

9590

80706050403020

10

5

1

0.1


Probability Plot of %damageNormal - 95% CI

pp-strain as a ratio over the mean strain in 9 runs per channel

Perc

ent

1.41.31.21.11.00.90.80.70.6

99.9

99

9590

80706050403020

10

5

1

0.1


Probability Plot of %StrainNormal - 95% CI

Figure 8 – pp-strain v damage 8 locations

%Strain

%da

mag

e

1.31.21.11.00.90.8

2.0

1.5

1.0

0.5

0.0

1

1

0.8 1.2

2

0.5

Scatterplot of %damage vs %Strain

%Strain

%da

mag

e

1.31.21.11.00.90.8

2.0

1.5

1.0

0.5

0.0

1845464749585960

Channel

Scatterplot of %damage vs %Strain

Left - regression fit for all data and right - the individual channels are identified. The area bounded by the dotted lines represents for these data what the authors call the simulation target. That is the damage at the same locations in the laboratory test should line within the same boundary.

Figure 9 – Target chart used in verifying the accuracy of the simulation test

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Figure 10 – Estimation of extreme pp-strain from one run

micro-strain

Perc

ent

110010501000950900850800

99.9

99

95

90

80

7060504030

20

10

5

1

99.9

1017

.5


Normal - 95% CIProbability maximum pp-strain per run - all drivers

micro-strain

Perc

ent

150012501000750500

99.99

99

98

95

80

50

205

99.9

1143

Loc 752.3Scale 56.55N 12AD 0.333P-Value >0.250

Largest Extreme Value - 95% CIExtreme value estimation - channel 10 rainflow ranges from 1 run

Channel 10

Range (pp-strain) 99.9% from normal distribution of maximum ranges each run = 1018μE (95% confidence)

Range (pp-strain) 99.9% from largest extreme value distribution using rainflow ranges from one run that are greater than 60% of the maximum value in that run = 1143μE (95% confidence)

micro-strain

Perc

ent

400350300250200

99.9

99

95

90

80

7060504030

20

10

5

1

99.9

354.

0


Normal - 95% CIProbability maximum pp-strain per run - all drivers

micro-strain

Perc

ent

500400300200100

99.99

99

98

95

80

50205

99.9

321.

4

Loc 177.6Scale 20.82N 9AD 0.513P-Value 0.183

Largest Extreme Value - 95% CIExtreme value estimation - channel 47 rainflow ranges from 1 run

Channel 47

Range (pp-strain) 99.9% from normal distribution of maximum ranges each run = 354μE (95% confidence)

Range (pp-strain) 99.9% from largest extreme value distribution using rainflow ranges from one run that are greater than 60% of the maximum value in that run = 321μE (95% confidence)

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Figure 11 – Estimation of extreme pp-strains

Range

Perc

ent

150012501000750500

99.9

99

95

90

80

7060504030

20

10

5

1

99.9

1291


Normal - 95% CIProbability Plot of max pp-strain range, channel 18 - all drivers

pp-strain range

Perc

ent

1750150012501000750500

99.99

99

98

95

80

50205

99.9

1262

1464

1043

546.5 103.6 9 0.304 >0.250564.7 130.2 7 0.303 >0.250517.9 76.02 9 0.337 >0.250

Loc Scale N AD P

D1AD2BD1C

Variable

Largest Extreme Value Probability Plot - channel 18 - D1A, D2B, D1C