Time Series Behaviour of Lower Arm Suspension Fatigue Data using Classical Decomposition Method

Z. M. NOPIAH, M.N.BAHARIN, S. ABDULLAH, M. I. KHAIRIR Department of Mechanical and Materials Engineering

Universiti Kebangsaan Malaysia 43600 UKM Bangi, Selangor

Malaysia zmn@vlsi.eng.ukm.my

Abstract— The study of time series behaviour refers to the analysis of certain unique attributes that exist in the time series data. The presence of these attributes in the data series may influence the decision making process. These attributes are generally grouped into four main component types which are trend, cyclical, seasonal and irregular components. In this study, fatigue signal data with three different road factors from a lower arm suspension for a mid-sized car were used as the case study. “Classical decomposition” time series method was used to segregate and to analyse the existence components in a systematic manner. Although fatigue data is a time series signal, not all components were considered. This is due to the nature of fatigue behaviour itself which is different from a normal time series data. From the study, it was found that only trend, cyclical and irregular component existed in the fatigue data signal. The study also revealed the additive effect that existed between these three types components as the absolute sizes of the seasonal variation are independent of each other. Keywords-component; time series behaviour, trend, cyclical, seasonal, irregular, fatigue, additive

I. INTRODUCTION In fatigue data analysis, data editing plays an important role in calculating the damage caused by the stress loading. The function of fatigue data editing is to remove the small amplitude cycles for reducing the test time and cost. By using this approach, large amplitude cycles that cause the majority of the damage are retained and thus only shortened loading consists of large amplitude cycles is produced [1]. Although data editing is important, the study on time series behavior for fatigue data should be done in getting a better understanding for the data behaviour. This paper discusses on the variation of time series that exist in fatigue time series data. A study by using three types of fatigue time series data were used as a case study. The main objective is to observe the time series component that exist in the fatigue data.

II. LITERATURE BACKGROUND

A. Time Series Time series data can be described as a set of data

collected or arranged in a sequence of order over a successive equal increment of time [2]. Whilst, a signal is one type of time series data that involve with a series of number that come from a measurement. Typically it is obtained by using some recording method as a function of time [3].

B. Fatigue Fatigue failure is a process that involves with crack

initiation and propagation of a component under repeated loading. The fatigue behaviour of mechanical components under service loading and its evaluation are usually affected by numerous uncertainties and characterized by several random variables such as materials properties, structural properties and load variation [4].

In the case of fatigue research, the data was measured based on signal that consists of a measurement of cyclic loads, i.e. force, strain and stress against time [3].

C. Global Statistic In normal practice, the global signal statistical values are frequently used to classify random signals. In this study, mean, root mean square (r.m.s.) and kurtosis were used [3]. For a signal with n data points, the mean value of x is given by

(1)

On the other hand, root mean square (r.m.s) value, which is the 2nd statistical moment, is used to quantify the overall energy content of the signal and is defined by the following

equation: (2)

∑=

=n

jjx

nx

1

1

2/1

1

21..⎭⎬⎫

⎩⎨⎧= ∑

=

n

ijx

nsmr

2009 International Conference on Signal Processing Systems

978-0-7695-3654-5/09 $25.00 © 2009 IEEE

DOI 10.1109/ICSPS.2009.180

984

where xj is the jth data and n is the number of data in the signal. The kurtosis, which is the signal 4th statistical moment, is a global signal statistic which is highly sensitive to the spikeness of the data. It is defined by the following equation:

(3)

where r.m.s is the root mean square as calculated in Equation 4 and x is the mean value of the signal data. Kurtosis is used in engineering for detection of fault symptoms because of its sensitivity to high amplitude events [4].

D. Classical Decomposition The identification of fatigue data behavior is based on

the existence of time series component which involves with the identification of trend (Tt), cyclical (Ct), seasonal (St) and irregular (It) component in time period t [2].These components were also recognized as variation that exist in the time series data [5]. The method that will be used in the identification of this component is called “classical decomposition” of time series. This process is used to segregate and to analyse the existence components in a systematic manner.

The trend component represents the long-run growth or decline over time. On the other hand, the cyclical component refers to the rises and falls of the series over unspecified period of time. The seasonal component, also known as seasonal variation, refers to the characterization of regular fluctuations occurring within a specific period of time. The irregular or random error is defined as the difference between the actual observation and the underlying pattern [2].

There are several methods used in identifying the trend. The simplest way in identifying trend in time series is the familiar “linear trend + noise”, for which the observation at time t is a random variable Xt given by

tttX εβα ++= (4) where α, β are constant and εt denotes a random error term with zero mean. The trend in (4) is a deterministic function of time and is sometimes called a global linear trend [5]. In time expansion, usually the detection of cyclic movement are based on the fluctuation of data upper on the upper side trend line and alternatively during period of contraction on the lower side of the trend line [2]. In this study, the identification of seasonal variation are based on the autocorrelation of time series[5]. The identification of random error is based on the patterns that

exist between the actual observation and the underlying pattern. Although these components are individually identified, they are also related to each other in a certain mathematical functional form. The type of relationship that these components have is divided into two: multiplicative effect and additive effect. The function of multiplicative effect can be explained as follows:

ttttt ISCTy ***= for Tt ,...,3,2,1= (5) It means that the components are interacted to each other such that the sizes of the seasonal variation increase in accordance with in the level of data. On the other part, additive effect can be explained by the function as follows:

ttttt ISCTy +++= for Tt ,...,3,2,1= (6) It involve with the assumption that the components of the series are interacted in additive manner [2]. Although signal data is one type of time series data, there exist a difference in terms of behavior between these signal data and other types of time series such as weather or economical. The differences of this behavior are dependent on the application of signal that was measured.

III. METHODOLOGY

A. Data Collection The data that was used in this case study is variable

fatigue strain loading data. It was collected from an automobile component during vehicle road testing. It was obtained from a fatigue data acquisition experiment using strain gauges and data logging instrumentation.

The collected fatigue data were measured on the car’s front lower arm suspension as it was subjected to the road load service. All the data that were measured from this experiment are recorded as strain time histories. In order to collect a variety of data for this study, the car was driven on three different road surfaces which are explained in Fig.1. All data were taken from different road conditions: - pave, highway and campus route

Figure 1. Main Factor for Fatigue Road Condition

4j

n

1j4 )xx(

)s.m.r(n1K −= ∑

=

Road Condition

Pave’ Highway Campus

985

Experimentally, the data was collected for 200 seconds at a sampling rate of 500 Hz, which gave 100,000 discrete data points. This frequency was selected for the road test because this value does not cause the essential components of the signal to be lost during measurement. The road load conditions were from a stretch of highway road to represent mostly consistent load features, a stretch of brick-paved road to represent noisy but mostly consistent load features, and an in-campus road to represent load features that might include turning and braking, rough road surfaces and speed bumps.

B. Analysis The analysis of this study involve with 3 main stages. It

can be explained by the following figure.

Figure 2. Flowchart in understanding the fatigue time series behaviour

The first stages involve with display the global input statistics and followed by identification of trend component. The identification of cyclical and irregularities component followed after the identification of trend existence in the fatigue data. Fig.3 shows the example of scatter diagram with linear trend that has been used in detecting the trend component.

Figure 3. Fatigue trend time series plot over index data

Fig.4 shows the example of residual plot that has been used in the identification of irregularities in fatigue time series data. The plot below is based on the fitted values from the linear equation from trend analysis. The whole analysis is explained in the results and discussion section.

Figure 4. Fatigue time series residual plot over index data

Figure 5. Fatigue time series for Pave road condition

Fig.5, Fig.6 and Fig.7 shows the time series plot for fatigue signal data that has been used in this study.

0 20 40 60 80 100 120 140 160 180 200-150

-100

-50

0

50

100

150

200

Figure 6. Fatigue time series for Highway road condition

Display global statistic

Identification of trend component

Identification of cyclical and irregularities component

Conclusion of relationship between components

Input time series

0 20 40 60 80 100 120 140 160 180 200-150

-100

-50

0

50

100

150

200

250

Dam

age

Time[seconds]

986

Figure 7. Fatigue time series for Campus Route road condition

IV. RESULT AND DISCUSSION From Fig.5, 6 and 7, it shows that there exists random

shock behaviour especially in campus route data. Random shock refers to changes in mean and later returns to the normal level. The occurrence of this behaviour can affect the trend identification for fatigue data.

Referring to Table 1, the pave gives the highest value of mean which are 68.08. On the other hand, campus route shows the highest value of RMS which is 88.22. Kurtosis results show that all of the data are non-Gaussian distribution since all the kurtosis value exceeds 3.

TABLE I. THE GLOBAL STATISTICS

Data Mean RMS Kurtosis

Pave 68.08 78.05 3.29

Highway 52.71 58.45 3.49

CampusRoute 64.74 88.22 4.55

Table 2 explained all the behaviour for all component that exist in the fatigue data except for seasonal component. In this analysis behavior of time series, seasonality component was not considered as important such as the event that occur in the fatigue data is not the regular fluctuations occurring within a specific period of time. The existence of regular fluctuations which involve with high amplitude events in fatigue data are independent each other. These high amplitude events are recognized as the fatigue damaging events. From the analysis by using classical decomposition method, it was found that all the data have positive trend, cyclical and random effect.

TABLE II. FATIGUE TIME SERIES COMPONENT TABLE

By referring to the equation (7), (8) and (9), it shows the linear equation for pave, highway and campus route data respectively. From the linear equation, the slope for campus route data has the highest value which is 0.00045.

t*00014.014.61yt += (7)

t*00037.034.34yt += (8)

t*00045.045.42yt += (9)

From the analysis, it was found that all the components that exist for the case study data has additive effect since all the variation that exist are random and independent to each other.

V. CONCLUSION From this study, it was found that campus route has the

highest value for RMS and slope in the data. The high value in RMS and slope that exist in the campus route data may due to the existence of several regular irregularities behaviour for trend identification which is short term memory random shock. Although the slope changes over time for the whole case study data are really low, the study reveals that the mean value exist in the data are not constant.

VI. SUGGESTION Further studies on the existence of random shock

behaviour should be performed. The purpose of this research is to investigate the effect of random shock behaviour to the fatigue damage that exist in the data.

ACKNOWLEDGMENT The authors would like to express their gratitude to

Universiti Kebangsaan Malaysia and Ministry of Science, Technology and Innovation, through the fund of UKM-GUP-BTT-07-25-152, for supporting these research activitie

Component

Data Trend Cyclical Seasonal Irregular Pave Positive Yes

Not Applicable Random

Highway Positive Yes

Not Applicable Random

Campus Positive Yes Not Applicable Random

0 20 40 60 80 100 120 140 160 180 200-200

-100

0

100

200

300

400

500

Dam

age

Time[seconds]

987

REFERENCES

[1] S. Abdullah, J.C. Choi, J.A. Giacomin, and J.R. Yates, “Bump Extraction Algorithm for Variable Amplitude Loading”, International Journal of Fatigue, Vol 28, 2005, pp. 675-691.

[2] M.A.Lazim, “Introductory Business Forecasting, a practical approach”, 2nd edition, 2007, University Publication Centre (UPENA), Uitm 2007

[3] S. Abdullah, M.D. Ibrahim, Z.M.,Nopiah, and A. Zaharim, “Analysis of a variable amplitude fatigue loading based on the quality statistical approach”, Journal of Applied Sciences, Vol 8, 2008,1590-1593

[4] S.Abdullah, S.N. Sahadan and M.Z.Nuawi, “On the need of the 4th order of Daubechies Wavelet Transforms to Denoise a Nonstationary Fatigue Loading”, Proceedings of the 7th WSEAS on International Conference on Signal Processing, Robotics and Automation

[5] C.Chatfield, “The Analysis of Time Series, An Introduction”, 5th edition, 1996, Chapman & Hall/CRC 1996

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