J umal Kejuruteraan 17 (2005) 71-84

Fatigue Data Editing Algorithm for Automotive Applications

Shahrum Abdullah, John R. Yates and Joseph A. Giacomin

ABSTRACT

This paper presents a wavelet based algorithm to summarise a long record of fatigue signal by extracting the bumps (fatigue damaging events) to produce a bump signal. With this algorithm the input signal is decomposed using the orthogonal wavelet transform and the wavelet levels are then grouped into characteristic frequency bands. Bumps are extracted from each wavelet group at a specific trigger level, which is set automatically according to the global signal statistics comparison between the original and bump signals. The accuracy of the algorithm has been evaluated by application to two experimentally measured data sets containing tensile and compressive preloading conditions. For both data sets, the bump signals length were at minimum of 40% of their respective original signals, and almost 90% original fatigue damage was retained in the bump signals, as calculated using the strain-life models of Smith-Watson- Topper and Morrow. Based on the results, this algorithm was found to be a suitable approach to summarise a long fatigue signal for the automotive usage.

Keywords: Fatigue, wavelet transform, automotive, bumps, trigger levels

ABSTRAK

Kertas kerja ini membentangkan pembangunan algoritma suntingan isyarat lesu. Algoritma ini berkemampuan untuk mengekstrak data beramplitud tinggi (kejadian hentakan) yang mengesani terhadap pengalahan lesu. Menerusi algoritma ini, isyarat masukan dipecahkan kepada aras-aras anak-gelombang bagi pembentukan kumpulan anak-gelombang. Kejadian­kejadian hentakan ini akan dikenalpasti menerusi aras penentu bagi setiap kumpulan anak-gelombang. Ketepatan algoritma ini telah diuji menerusi dua jenis isyarat lesu yang diukur pada ampaian kereta dan mengandungi pembebanan awal tegangan dan mampatan. Untuk kedua-dua isyarat ini, tempoh isyarat hentakan sekurang-kurangnya 40% daripada isyarat masukan dan hampir 90% pengalahan lesu daripada isyarat asal masukan dapat dikekalkan. Pengalahan lesu ini dikira berasaskan kepada model hayat­terikan Smith-Watson- Topper dan Morrow. Berdasarkan keputusan ini, algoritma ini berkemampuan untuk meringkaskan isyarat-isyarat lesu yang diperoleh dalam bidang otomotif.

Kata kunci: Lesu, pindahan anak-gelombang, otomotif, hentakan, aras penentu

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INTRODUCTION

Fatigue damage analysis is one of the key stages in the design of vehicle structural components. One of the vital input variables in the fatigue assessment of consumer products is the load history. For ground vehicles, these can be an extremely wide range of uses and hence a representative road load time history is hard to quantify, which is in the pattern of variable amplitude (VA) loading (Yan et al. 2001). Loads that are predicted to do little or no damage can be eliminated and the large amplitude cycles that cause the majority of damage should be retained for producing the simulated road load for accelerated durability testing of a component (Con Ie et al. 1997). Method to retain the large amplitude cycle to produce simulated histories is known as fatigue data editing.

Some data editing algorithms were developed in various domains such as time, peak and valley, frequency, cycles, damage and histogram. Among these, the editing process in the time domain is the most popular approach used (EI-Ratal et al. 2002), such as: the application of local strain parameter (ConIe & Topper 1980), the damage window joining function (Austen & Gregory 1995; EI-Ratal et al. 2002), the range of Smith-Watson-Topper (SWT) parameter (Stephens et al. 1997) and the effect of overload and underload to the crack opening stress criteria (DuQuesnay et al. 1993). Fatigue damage can also be analysed in the frequency domain using Fourier transforms (Bishop & Sherratt 1989; Li et al. 2001). However, the fatigue data editing research was rarely implemented, as the obtained results were not accurate as in the time domain editing (Morrow & VoId 1997). In this domain, the loading was low pass filtered on the basis of high frequency cycles have small amplitudes which are not damaging. This filtering method does not shorten the time series as the number of points is similar. In addition to frequency domain editing, the time-frequency or wavelet approach has been applied to the problem of fatigue data editing through its use in spike removal and denoising (Oh 2001).

Based on these literatures, none of them associated to the use of time and frequency localisation by identifying large amplitudes in a frequency band. This has led to the development of an algorithm incorporated with identification and extraction of the fatigue damaging events. This algorithm is used to produce a shortened signal, or the mission signal, that replicates equivalent global signal statistical and fatigue damage of the input signal.

THEORETICAL BACKGROUND

GLOBAL SIGNAL STATISTICS

In this algorithm, the global signal statistic values are used to evaluate algorithm effectiveness. The signal root-mean-square (r.m.s.) value, which is the 2nd statistical moment, is used to quantify the overall energy content of the oscillatory signal. For discrete data sets the r.m.s. value is calculated as

1 N )~

r.m.s.= ~ LX; , ) =1

(1)

73

where x . is the instantaneous value and N is the number of values in the }

sampled sequence. Kurtosis, which is the signal 4th statistical moment, is a measure of nongaussianity since it is highly sensitive to outlying data among the instantaneous values. For discrete data sets the kurtosis value is defined as

N

Kurtosis = 1 4 ~)Xj -.xt N x (r.rn .s.) j= i

(2)

The crest factor is defined as the ratio between a maximum value in the time history and the calculated r.m.s. value:

FATIGUE DAMAGE PREDICTION

Current industrial practice for fatigue life prediction is to use the Palmgren­Miner linear damage rule. For strain-based fatigue life prediction, this rule is normally applied with strain-life fatigue damage models. The strain-life fatigue life behaviour considers plastic deformation that occurs at the localised region where fatigue cracks begin with the influence of a mean stress. This approach was developed in response to analyse shorter fatigue life problems. It is clear that the service loadings of many machines, vehicles and structures could be best evaluated using a strain-based approach (Dowling 1998).

Some of the realistic service situations involve non-zero mean stresses. Two mean stress effect models are used in the strain-life fatigue damage analysis, i.e. Morrow and SWT strain-life models . Mathematically, the Morrow's model is defined by

(3)

The SWT strain-life model is mathematically defined as

(4)

where E is the material modulus of elasticity, amax is a true maximum stress, Ea is a true strain amplitude, 2N

f is the number of reversals to failure, a 'f

is a fatigue strength coefficient, his a fatigue strength exponent, E'f is a fatigue ductility coefficient and c is a fatigue ductility exponent.

The Morrow's strain-life model is consistent with the observation that mean stress effects are significant at low values of plastic strain and that they have little effect at high plastic strains. For loading sequences that are predominantly tensile in nature the SWT approach is recommended. In the case where the loading is predominantly compressive the Morrow approach can be used to provide more realistic life estimates (Dowling 1998). The damage caused by each cycle of the repeated loading is calculated by

74

reference to the material strain-life curve. The Nf

value can be obtained from Equation (3) and Equation (4) for both models. Therefore, fatigue damage for one cycle is defined as

Fatigue Damage, D = _1_ Nf

FATIGUE DATA EDITING ALGORITHM

(5)

A flowchart describing this fatigue data editing is presented in Figure 1. Three main stages of the algorithm can be observed in the flowchart: the application of the wavelet decomposition process, the identification of fatigue damaging events and the production of a mission signal. In the first stage, the power spectral density (PSD) of the input signal is calculated in order to determine its vibrational energy distribution in the frequency domain. This PSD approach is then applied in the wavelet decomposition process of the input signal. The next stage is a wavelet decomposition and wavelet level grouping procedure. The wavelet transform is the functions in the time-scale domain and it is a significance tool for presenting local features of a signal. In this type of transform, the data is moved into a scale domain as a basic function in order to provide the localised features of the original signal. Thus, wavelet transform gives a separation of components of a signal that overlap in both time and frequency and it gives a more accurate local description of the signal characteristics.

( START ')

NO

" .ITOP )

FIGURE 1. A flowchart of the algorithm

75

In this algorithm, the 12th order of Daubechies ' wavelet which is the basis functions forming an orthogonal set was chosen due to its efficiency in providing a large number of vanishing statistical moments. The 12th order representation was adopted due to its successful application in several previous studies involving automotive road data (Giacomin et al. 2000, 2001; Abdullah et al. 2004). The number of discrete sampling points in the time history determines how many wavelet levels can be decomposed. When the number of sampling points N is equal to 2" (N = 2"), the number of levels obtainable from the wavelet decomposition is n + 1. A wavelet grouping stage in WBE permits the user to group wavelet levels into single regions of significant energy. Each wavelet group is defined by the user to cover frequency regions of specific interest, such as high energy peaks caused by a subsystem resonances. This subdividing of the original signal (see Figure 2) permits analysis to be performed for each frequency region independently, avoiding situations where small bumps in one region are concealed by the greater energy of other regions of the frequency spectrum.

(j)(J Road lime hlStoty l<'4lut

Origmal road Ime hsloty IS dec~"'d by means of an Orthogonal Wavelel Transform (OWT) inlJ wavelet leVels

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TII1lO --FIGURE 2. Wavelet decomposition and grouping procedure (Giacomin et al. 2000)

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The second stage consists of bump identification and extraction process (Figure 3). A bump is defined as an oscillatory transient which have a monotonic decay envelope either side of the peak value. Bump events are identified from a given wavelet group time history by means of an automatic trigger level. At program launch the user specifies the maximum acceptable percentage difference between r.m.s. and kurtosis of the original signal and the mission signal. The trigger level is then automatically determined to achieve the requested statistics for each wavelet group. The r.m.s. and kurtosis values of this mission signal are compared to those of the original signal. If the statistics exceed the required difference, the trigger levels are reduced incrementally by a step size that is specified by the user until both statistical values of the mission signal achieve the user-specified tolerance. Figure 3(a) presents a set of possible trigger levels for an individual wavelet group. Once a bump event exceeding the pre-set trigger level, the time extent of the event is defined. Bump identification (Figure 3(b)), is performed by means of a search which identifies the inversion points at which the signal envelope inverts from a decay behaviour.

4----------r--------3---------+~-------

2---T----~~--,_---

·1 '-'---+lI-+---"tl-ltll--I'-iHl---'+

~--------~~--+---~~ ~--------~--------­~-------------------

(a)

MrillJt~ .'

(b)

~aYv'G

~J~'INfft'l\\ItH-'; =.,

(c)

FIGURE 3. (a) possible trigger level values, (b) decay enveloping of a bump, (c) production of a mission signal

Finally, a method to seek the start and finish points of a bump from all wavelet groups has been defined in Figure 3(c). This approach is used to produce a bump segment, so as to produce a mission signal. If a bump event is found in any of the wavelet groups a block of data covering the time extent of the bump feature is taken from the original data set. This strategy retains the amplitude and phase relationships of the original signal.

77

ALGORITHM VALIDATION

An artificial test signal (Figure 4(a)) was defined for validating the algorithm. The zero-mean signal contains 16000 data points sampled at 400 Hz and consists of a combination of sinusoidal and random signals of various amplitude and duration. This 40.0s signal was used to test the effectiveness of the algorithm when selecting bump events. It was intentionally defined to be a mixture of both high amplitude bump events and low amplitude harmonic background. Using this algorithm the bumps were extracted from each wavelet group of the validation signal at ±75% statistical difference between the original and the bump signals. With a signal length at 12.5s (Figure 4(b)) the mission signal was 31 % of the original test signal length. The properties of the both signals are listed in Table 1.

6

4

cu 2 '"0

:e 0 Q.. E -2 c:(

-4

-6

20

T ime (s ) (a)

30 40

6 ,----.------ --- --

4

<I> 2 "0 . .@

0 a. E « -2

-4

.{)

20 . 30 40

-------------- ---' Time [5]

(b)

FIGURE 4. Visualisation of the test signal: (a) original signal, (b) bump signal

TABLE 1. Properties of the test signal

Signal Type No. of data Signal length r.m.s. Kurtosis points (s)

Original signal 16000 40.0 1.51 7.37 Mission signal 4997 12.5 2.62 2.53

The original and mission signal were used as input into nSoft® software package in order to predict the fatigue damage potential. Two commonly

78

used mean strain-life models were used, i.e. SWT and Morrow, for comparing the lifetime of both signal types. This was not intended to be absolute measure of the fatigue life, but as a check to ensure that the fatigue damage of the two histories was almost unchanged. Using these approaches, the damage ratios between the mission signal and the original signal were 98.4%. In the case of the artificial test signal a large (±7S%) difference in r.m.s. and kurtosis values between the bump and original signals occurred because approximately 70% of the original signal contained low amplitudes.

VEHICLE SUSPENSION ARM CASE STUDY: RESULTS AND DISCUSSIONS

The signals were measured on the lower suspension arm of a vehicle travelling at 34 kmlh over a pave test track as shown in Figure Sea). The signals were measured at the top of the component for the tensile preloading (mean = IS.0me) and side of the component for the compressive preloading (mean = -48.1me). The strain gauge locations are shown in Figure S(b). The signals were sampled at 500 Hz for a total of 23000 data points which produced a total record length of 46 seconds.

FIGURE 5. (a) Pave test track used for the road tests, (b) Schematic view of strain gauge locations on the component for: (i) tensile preloading,

(ii) compressive preloading

Both signals were decomposed into 12 wavelet levels and assembled into four wavelet groups. The fatigue damage values were calculated for both the original and the mission signals using the SWT and Morrow strain­life models as implemented in the nSoft® software. For the simulation purposes, the material used was SAE 4340 steel, a fatigue notch factor (K

j)

value was 2.0 and the component to have an average machined surface finish.

Figure 6 shows several plots of the analysed signals. The original signals are shown in Figure 6(a) for the tensile preloading and in Figure 6(e) for the compressive preloading. The bump segments of the tensile and compressive mean loadings are shown in Figure 6(b) and Figure 6(f), respectively. Those figures show the exact locations of the bump segments in the respective original signal. These bump segments were joined in order to produce the mission signals, as shown in Figure 6(c) for the tensile preloading and Figure 6(g) for the compressive preloading. Using this data

79

editing technique, the mission signal length was 41 % of the tensile preloading signal, and 55% for compressive preloading signal. These were obtained at ±1O% of the r.m.s. and kurtosis difference between the original and mission signals. This is an acceptable difference value in fatigue damage prediction when considering the presence of the low amplitudes background in the fatigue signal. Using this value, the vibrational energy distribution for the original and its mission signals looks similar, which are figured out as the PSD plots in Figure 6(d) and Figure 6(h).

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(g)

81

o.zo

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LO %0 30 .0 50 60

(h)

FIGURE 6. Plots for the tensile preloading signal: (a) the original signal, (b) locations of bump segments, (c) the mission signal, (d) PSD of the original

and mission signals. Plots for the compressive preloading signal: (e) the original signal, (f) locations of bump segments, (g) the bump signal,

(h) PSD of the original and bump signals

The post-processing to determine the fatigue damage potential of both the original signals and their equivalents output was performed by means of the nSoft® software package. For the tensile preloading, the ratio of fatigue damage between the mission and original signals was 96% and 99% as calculated using the SWT and Morrow strain-life models, respectively. In addition, the compressive preloading had the fatigue damage ratio between the mission and original signals at 97% and 90% for the SWT and Morrow strain-life models, respectively. An accuracy of ±1O% for the global signal statistics and ±ll % for the fatigue damage between the mission and the original signals was obtained in this analysis. The properties of the two original signals and their mission signals are presented in Table 2. Almost all of the original damage was retained in the mission signal , which is shown in the histogram of Figure 7. Finally, the results of the signal length compression and fatigue damage values indicate that this wavelet-based algorithm can be successfully applied to compress the original signal without changing the original and phase relationship.

TABLE 2. Properties of the experimentally measured fatigue signal

Signal No. of Signal r.m.s Kurtosis Fatigue damage data length / block

points (s) SWT Morrow model model

T1 Original signal 23000 46.0 7.1 3.3 1.36 x 10.5 6.35 X 10.6

Mission signal 9413 18.8 7.8 3.6 1.31 x 10-5 6.32 X 10.6

T2 Original signal 23000 46.0 19.4 3.7 6.22 x 10-4 7.73 X 10-4

Mission signal 12674 25.3 21.0 4.0 6.01 x 10-4 6.95 X 10.4

82

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(c)

(d)

FIGURE 7 _ Fatigue damage histogram using the SWT mean stress correction for tensile preloading: (a) original signal, (b) mission signal; and for compressive

preloading: (c) original signal, (d) mission signal

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CONCLUSIONS

The main purpose of this algorithm is for the fatigue mission synthesis for accelerated fatigue tests. It is an algorithm which is able to extract fatigue damaging events with an automatic procedure of trigger level determination. This study focused on the fatigue damage analysis of two different preloading (tensile and compressive) data sets measured from a lower suspension arm of a vehicle travelling over a pave test track. Using this data editing procedure, the available experimentally fatigue signals were compressed by up to 40% with the maximum of 10% lost in the fatigue damage. Finally, it is found that this algorithm is an efficient approach for summarising long records of fatigue data.

ACKNOWLEDGEMENT

The authors wish to express their gratitude to Universiti Kebangsaan Malaysia, The University of Sheffield and Leyland Technical Centre for their supports.

NOTATION

b Fatigue strength exponent c Fatigue ductility exponent D Fatigue damage for one cycle E Modulus of elasticity N Number of data points N

f Number of cycles to failure for a particular stress range and mean

r.m.s. Root-mean-square x. Instantaneous value of the sampled process ..J x Signal mean for a time history 2N

f Reversals to failure

ea Strain amplitude for fatigue time history e max Maximum strain for fatigue time history e 'f Fatigue ductility coefficient S max Maximum stress for fatigue time history s 'f Fatigue strength coefficient sm Mean stress for fatigue time history sa Alternating stress for fatigue time history

REFERENCE

Abdullah, S. , Giacomin, 1.A. and Yates, 1.R., 2004. A mission synthesis algorithm for fatigue damage analysis, Proceedings of the Institution of Mechanical Engineers, Part D, Journal of Automobile Engineering 218(D3): 243-258.

Austen, 1. and Gregory, R., 1995. Component test during duration prediction and acceleration by fatigue analysis and fatigue editing. VTI Symposium 3(157): 169-187.

Bishop, N.W.M. and Sherratt, F. , 1989. Fatigue life prediction from power spectral density data, Part 1: Traditional approaches. Enviromental Engineering 2 : 11-14.

Conle, A. and Topper, T.H., 1980. Overstrain effects during variable amplitude service history testing. International Journal of Fatigue 2(5): 130-136.

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Con Ie, A., Grenier, G., Johnson, H., Kemp, S., Kopp, G. and Morton, M., 1997. Service history determination. In SAE Fatigue Design Handbook AE-22, pp 115-144. Society Automotive Engineers Inc, Warrendale USA.

Dowling N.E., 1998. Mechanical Behaviour of Materials: Engineering Methods for Deformation, Fracture and Fatigue, Second Edition. Prentice Hall, New Jersey USA.

DuQuesnay, D.L. , Pompetzki, M.A. and Topper, T.H. , 1993. Fatigue life prediction for variable amplitude strain histories. SAE Transactions (SAE930400) 102(5) : 455-465.

EI-Ratal, w., Bennebach, M. , Lin, X. and Plaskitt, R., 2002. Fatigue life modelling and accelerated test for components under variable amplitude loads . Symposium on Fatigue Testing and Analysis Under Variable Amplitude Loading Conditions, Tenth International Spring Meeting of SF2M, Tours, France.

Giacomin, J., Steinwolf, A. and Staszewski, W.1. , 2000. An algorithm for Mildly Nonstationary Mission Synthesis (MNMS). Engineering Integrity 7: 44-56.

Giacomin, J., Steinwolf, A. and Staszewski, w.J., 2001. Application of Mildly Nonstationary Mission Synthesis (MNMS) to Automotive Road Data. ATA 7th Int. Con! on the Role of Experimentation in the Modern Automotive Product Development Process , Florence, Italy.

Li, Q., Minnetyan, L. and Charnis, C.C. , 2001. Computational simulation under PSD fatigue loading. In Proceedings of Structures Structural Dynamics & Materials Conference, pp 3147-3154. Seattle USA.

Morrow, D. and Void, H. , 1997. Compression of Time Histories Used for Component Fatigue Evaluation, SAE930403 in PT-67. Recent Developments in Fatigue Technology, edited by Chernenkoff, R.A. and Bonnen, J.1. Society of Automotive Engineers (SAE), USA.

Oh, C-S., 2001. Application of wavelet transform in fatigue history editing. International Journal of Fatigue 23 : 241-250.

Stephens, R.I. , Dindinger, P.M. and Gunger, J.E., 1997. Fatigue damage editing for accelerated durability testing using strain range and SWT parameter criteria. International Journal of Fatigue 19: 599-606.

Yan, H., Zheng, X.L. and Zhao, K. , 2001. Experimental investigation on the small­load-omitting criterion. International Journal of Fatigue 23(5): 403-415.

Shahrum Abdullah Jabatan Kejuruteraan Mekanik dan Bahan Fakulti Kejuruteraan Universiti Kebangsaan Malaysia 43600 Bangi, Selangor D.E Malaysia

John R. Yates and Joseph A. Giacomin Department of Mechanical Engineering University of Sheffield United Kingdom