Segmentation and Scattering of Fatigue Time Series Data by Kurtosis and Root Mean Square

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1 Segmentation and Scattering of Fatigue Time Series Data by Kurtosis and Root Mean Square Z. M. NOPIAH 1, M. I. KHAIRIR AND S. ABDULLAH Department of Mechanical and Materials Engineering Universiti Kebangsaan Malaysia 436 UKM Bangi, Selangor MALAYSIA Abstract: - This paper presents the method of classifying and scattering of fatigue data by time series segmentation and segment-by-segment analysis of fatigue damage based on its relation with segmental kurtosis and root mean square (r.m.s.) values. The time series was segmented using piecewise linear representation (PLR) based segmentation algorithms. Statistical analysis and fatigue damage calculation was made on each segment of the time series and patterns of data scatteringwere identified based on the plots of relationship between segmental damage and its corresponding kurtosis and/or root mean square. The information gained from the data scattering could then be made useful for fatigue data scattering and editing. Key-Words: - Time series, segmentation, data scattering, kurtosis, root mean square, fatigue damage. 1 Introduction It has been established over the years that proper evaluation of statistical properties will give reasonable diagnostic indication of damage in critical automotive components [1]. Although there are a large number of such statistical attributes such as root mean square value, crest factor, skewness, kurtosis, and so on, kurtosis has emerged as one of the good indicators of damage of automotive components such as gears. In this study, the fatigue data was obtained from field tests conducted on a lower suspension arm of a car. This component has been selected because it was defined as one of the critical components in automotive parts [2]. This paper discusses on the segmentation of fatigue data (represented as time series), the statistical analysis of each segment of the data, and the scattering of data which would help in accelerating fatigue testing by means of fatigue data editing. 2 Literature Background For the purpose of this study, a time series segmentation algorithm that inputs a time series and returns a Piecewise Linear Representation (PLR) was used to segmentise the time series data. Based on the studies by Keogh et al. [3], three most common methods of time series segmentation algorithms are as follows: Sliding Windows (online): A segment is grown until it exceeds some error bound. The process repeats with the next data point not included in the newly approximated segment. Top-Down (batch): Time series is recursively partitioned until some stopping criterion is met. Bottom-Up (batch): Starting from the finest possible approximation, segments are merged until some stopping criterion is met. Tests performed by Keogh et al. [3] showed that a segmentation algorithm that has a global perspective of the data produces the best PLR with the least amount of error. Such algorithms are called batch algorithms, and of the two segmentation methods that fall under this category, Bottom-up segmentation algorithm has proven to be the best at performing batch segmentation with the least amount of error. This is based on the survey conducted by Keogh et al. [3], whose results are shown in Figure 1. By definition, a PLR refers to the approximation of a time series T, of length n, with K straight lines [3]. The Bottom-up algorithm first creates the finest approximation of the data, which contains at most n/2 segments. Then it recursively calculates the cost of merging each pair of adjacent segments and proceeds to merge the segments beginning with the lowest cost pair. The number of segments in the PLR will gradually be reduced until a stopping criterion is met. 64 ISSN:

2 Fig. 1: Comparison of the three segmentation algorithms on ten diverse datasets by Keogh et al. In real applications, mechanical signals can be classified as having a stationary or a non-stationary behaviour. Stationary signals exhibit the statistical properties remain unchanged with the changes in time and the statistics of non-stationary signal is dependent on the time of measurement [4]. The most commonly used statistical parameters are the mean value, the root-mean-square (r.m.s.) value and the kurtosis [5]. The r.m.s. value, which is the 2 nd statistical moment, is used to quantify the overall energy content of the signal and is defined by the following equation: r. m. s n 1 = x n j = 1 2 j 1 2 The kurtosis, which is the signal s 4 th statistical moment, is a global signal statistic which is highly sensitive to the spikiness of the data. K = 1 n ( r. m. s ) n 4 ( x j x ) 4 j = 1 For a Gaussian distribution the kurtosis value is approximately 3.. In some definitions of kurtosis, a deduction of 3 is added to the definition in order to maintain the kurtosis of a Gaussian distribution to be equal to zero. For clarity and convenience, in this study the former definition of kurtosis (where the Gaussian distribution has a kurtosis value of 3) was used since the kurtosis function in MATLAB uses this definition. Therefore kurtosis values which are higher than 3. indicate the presence of more extreme values than should be found in a Gaussian distribution. Kurtosis is used in engineering for detection of fault symptoms because of its sensitivity to high amplitude events [6]. 3 Methodologies The fatigue data for this study was obtained from field tests conducted on the lower suspension arm of a mid-sized sedan car. The material for the lower suspension arm is SAE145 steel, and this material s specifications were used in all fatigue damage calculations. The road load conditions were from a stretch of highway road to represent consistent load features and an in-campus road to represent load features that might include braking, rough road surfaces and speed bumps. Because the Bottom-Up segmentation method produces the best PLR with the least amount of error, for the purpose of this study, the Bottom-Up segmentation algorithm which was developed by Keogh et al. [3] was used to segmentise the time series signals. As the algorithm was run, the number of segments in the PLR will gradually be reduced until a stopping criterion is met. The stopping 65 ISSN:

3 criterion for the algorithm was set to be the number of segments in the resulting PLR, which for the purpose of simplicity and statistical acceptability, was decided to be 3 segments. The segmented data was then analysed using the GlyphWorks software package, where the fatigue damage for each segment of the time series was calculated. The segmented data was also run through a MATLAB algorithm that calculates the kurtosis and r.m.s. values of each segment. Another MATLAB algorithm generates comparison scatter plots of fatigue damage against kurtosis and r.m.s. values. Based on these scatter plots, patterns of data scattering, if any, were identified and noted. 4 Results and Discussions 4.1 Segmentation Segmentation on the time series data was done by implementing a segmentation algorithm, which was defined as an algorithm that inputs a time series and produces a piecewise linear representation (PLR) of the time series. A MATLAB routine developed by Keogh et al. [3] was used for the purpose of segmenting the time series into 3 segments. As evident from Figure 2, these segments were not uniform in size; their lengths vary from one segment to the other. This is because the Bottom-up algorithm segmented the time series so that each segment and its corresponding linear representation would have the least amount of error. The PLR-based segmentation was used to ensure that like features in the time series data would be isolated and grouped into the same segments, and that further analyses of each segment would help us determine which parts of the data signal made significant contributions to the overall fatigue damage calculations from the multiaxial strains the lower suspension arm of the car was subjected under for each set of different road conditions. The two statistical parameters chosen for the segmentby-segment analysis of the segmented time series data were the kurtosis and the root mean square (r.m.s). 2 Highway Pt 1 Time Series data segmented into 3 segments - Highway Pt 1 Piecewise Linear Representation with 3 segments UKM1 Pt1 Time Series data segmented into 3 segments UKM1 Pt1 Piecewise Linear Representation with 3 segments Fig. 2: Segmentation of two time series and their Piecewise Linear Representations, Highway data and in-campus road data 66 ISSN:

4 4.2 Kurtosis Kurtosis shows the presence of significantly high amplitudes or peaks in each segment, which supposedly translates into a higher fatigue damage value for the particular segment. Scatter plots of kurtosis versus fatigue damage for both sets of data are shown in Figure 3. the in-campus road because of the mostly consistent surface conditions of the highway road, whereas the in-campus road includes instances of speed bumps and braking conditions as well as both smooth and rough road surfaces. The maximum kurtosis value for the highway load data is close to 3, whereas for the in-campus road the maximum kurtosis is closer to 13. This means that under in-campus road load conditions, the lower arm suspension of the car is subjected to significantly higher strains than under highway road conditions. 4.3 Root mean square (r.m.s.) Fig. 3: Scatter plots of kurtosis against fatigue damage for two time series data, highway data and in-campus road data From Figure 3 we can see some patterns of scattering, where small damage values correspond to small kurtosis values and vice versa. The variations in kurtosis values are due to the randomness of the data and the variety in size of each segment. Shorter segments with higher amplitudes usually result in higher kurtosis values whereas longer segments with lower amplitudes would result in lower kurtosis values. The kurtosis versus damage scatter plot for the highway are not as widely distributed as the one for Fig. 4: Scatter plots of r.m.s. against fatigue damage for two time series data, highway data and in-campus road data The root mean square value shows the overall energy content of the segment of data; therefore for higher damage values the r.m.s. values are 67 ISSN:

5 theoretically higher. Scatter plots of r.m.s. versus fatigue damage for both sets of data are shown in Figure 4. For this statistical parameter we can see some scattering of data, although the points are more widely distributed, also due to the randomness of the data and the variety in size of each segment. We can observe from Figure 4 that the r.m.s. versus damage scatter plot for the in-campus road has a wider distribution compared to the highway road. The maximum r.m.s. value is also higher, around 96, compared to highway road s which is around 615. This is due to the variety in frequency and magnitude of strain the lower suspension arm was subjected under when the car was driven on the in-campus road. Energy levels would vary each time the brakes were applied, or when the car goes over a rough surface or a speed bump. The case is less evident in the energy levels of each segment in the highway road load data, as most of the time when the car was driven in the highway the lower suspension arm would be subject to comparatively lower and more consistent loads than when the car was driven on the in-campus road. 5 Conclusions The study has demonstrated the use of linear segmentation of time series data for fatigue analysis. Combining time series segmentation with statistical analysis has produced reliable results. By analysing the data this way, we may identify trends and patterns of data scattering based on critical statistical parameters. From the scattering of data we may acknowledge which parts of the data made significant contribution and which did not. Finally based on our findings we may eliminate or exclude certain parts of the data in order to make further study and analysis of the signal much faster and more efficient without significant loss of data. In our case we can clearly see that a scatter of the kurtosis produced better and more evident data scattering than that of root mean square. Based on these results it is suggested that the kurtosis method is the more preferable one of the two if one were to use similar methods as presented in this paper when studying time series and data scattering. Finally, after identifying the scattering of data in the signal, fatigue data editing through the elimination of certain non-contributory or insignificant segments of the signal may help in reducing the length and complexity of the data and may thus speed up the process of fatigue testing of metal components of mechanical systems or any similar application. 6 Acknowledgements 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 , for supporting these research activities. References: [1] Rao, V. B., 1999, Kurtosis as a Metric in the Assessment of Gear Damage: The Shock and Vibration Digest, Vol. 31, No. 6, pp [2] Nadota, Y. and Denier, V., 24, Fatigue failure of suspension arm: experimental analysis and multiaxial criterion, Engineering Failure Analysis, Vol. 11, pp [3] Keogh, E., S. Chu, D. Hart and M. Pazzani, 21. An Online Algorithm for Segmenting Time Series: Data Mining. ICDM 21, Proceedings IEEE International Conference on 29 Nov - 2 Dec 21, pp [4] Bendat, J. S. and Piersol, A. G., 1986, Random Data: Analysis and Measurement Procedures, 2nd Edition, Wiley-Interscience, New York. [5] Hinton, P. R., 1995, Statistics Explained: A Guide for Social Science Students, Routledge, London. [6] Qu, L. and He, Z., 1986, Mechanical Diagnostics, Shanghai Science and Technology Press, Shanghai, P. R. China. 68 ISSN: