Key Engineering Materials Vols. 462-463 (2011) pp 1115-1120 © (2011) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/KEM.462-463.1115
Variable Amplitude Loading Strains Data Distribution using Probability Density Function and Power Spectral Density A. Lenniea, Z. M. Nopiahb, S. Abdullah, M. N. Baharin, M.Z. Nuawi and A. Arifin Faculty of Engineering and Built Environment, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia a
[email protected],
[email protected]
Keywords: Energy, Variable amplitude loadings, Probability distribution function, Power spectral density
Abstract. This paper presents a relative study on variable amplitude (VA) strain data distribution using the approach of probability density function (PDF) and power spectral density (PSD). PDF is a technique to identify the probability of the value falling within a particular interval, and a PSD is to measure the power of a signal by converting it from the time domain to the frequency domain. The objective of this study is to observe the applicability of both techniques in detecting the pattern behaviour in terms of energy and probability distribution. For this reason, a set of case study data consist of nonstationary VA pattern with a random behaviour was used. This kind of data was measured by fixing a strain gauge that connected to the strain data acquisition on the lower suspension arm of a mid-sized sedan car. The data was measured for 60 seconds at the sampling rate of 500 Hz, which gave 30,000 discrete points. The distribution of collected data was then calculated and analysed in the form of both PDF and PSD, and they were then compared for further analysis. The findings from this study are expected for determining the pattern behaviour that exists in VA strain signals. Introduction In statistics and signal processing approach, a time series is a sequence of data points, typically measured at successive times spaced at uniform time intervals [1]. Methods for time series analysis may be divided into three classes, and they are time domain methods, frequency domain methods and time-frequency domain methods. The time domain method is suitable for simple structures where only a few hot spots need to be investigated [2]. In the frequency domain, the power spectral density has been calculated for determining the related stress response. The frequency domain method includes the spectral analysis that also leads to the development of time-frequency analysis. A signal is a series of number that come from measurement, typically obtained using some recording method as a function of time. In the case of fatigue research, the signal consists of a measurement of a cyclic load, such as force, strain and stress against time [3]. In actual applications, mechanical signals can be classified to have a stationary or non-stationary behaviour. Stationary signal behaviour showed that the statistical property values remained unchanged with the changes in time. For non-stationary that is common for the fatigue analysis cases, however, the statistical property values of a signal are dependent to the time of measurement [4]. Each component experienced different stress value on different point. Thus, the probability density function (PDF) and power spectral density (PSD) were performed to study the signal pattern between the maximum stress point and the minimum stress point. This paper discusses some related issues to the analysis on VA strain data behaviour of lower suspension arm using PDF and PSD. In addition, signal statistical properties such as the mean, the root-mean-square (RMS) and the kurtosis values also considered for the pattern behaviour study. The goal of this study is to observe the
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applicability of both PDF and PSD techniques for detecting the strain signal pattern in terms of energy and probability distribution. Literature Background In mathematics, a probability density function (PDF) represents a probability distribution in terms of integrals. The term of probability distribution function has also been used to denote the probability density function. In amplitude probability analysis [5], the time at level analysis is useful as it is more common to represent the data as amplitude PDF. Time at level analysis is useful for determining the statistical amplitude content of a signal and can be used to detect anomalies such as signal drift, spikes, clipping, etc. Frequency analysis data is typically presented in graphical form as a power spectral density (PSD). Essentially a PSD displays the amplitude of each sinusoidal wave of a particular frequency. The mean squared amplitude of a sinusoidal wave at any frequency can be determined by finding the area under the PSD over that frequency range. PSDs are useful for detecting resonance in components, aliasing in the data, frequency interference, etc. Another objective of time series analysis is to determine the statistical characteristics of the original function by manipulating the series of discrete numbers. In a normal practice, the global signal statistical values are frequently used to classify random signals [6]. The most commonly used statistical parameters are the mean, the root-mean-square (RMS) and the kurtosis values. For a signal with a number n of data points, the mean value of x is given by
x=
1 n ∑ xj n j =1
(1)
The RMS value, which is the 2nd statistical moment, is used to quantify the overall energy content of the signal [7]. For a zero-mean signal the RMS value is equal to the standard deviation (SD) value. For a discrete data set the RMS value is defined as, RMS =
1 n 2 ∑xj n j =1
(2)
Kurtosis is mathematically defined in Eq. 3, which is the signal 4th statistical moment that highly sensitive to the spikiness of the data. Higher kurtosis values 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.
Kurtosis =
1 n( SD) 4
n
∑ (x
j
− x) 4
(3)
j =1
Methodology A variable amplitude (VA) strain loading was used in this study as an input signal for the lower suspension arm of a mid-sized sedan car. The fatigue data acquisition system, i.e. SoMat eDAQTM Data Acquisition, was used for the strain data measurements. The strain loading was then measured using two strain gauges of 2-mm in size which were placed at the maximum stress area and the minimum stress area, and they were notated for channel 1 and channel 2, respectively. The signal was recorded on the right lower suspension arm of a car travelling over a pavé road at the average speed of 40 km/h and a highway route at the average speed of 70 km/h. The strain data was collected twice of each road in order to provide a variety of data set pattern for channel 1 and
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channel 2. Fig. 1 shows a model of a lower arm suspension which indicates the position of strain gauges during the tests. Maximum stress point (Channel 1)
Minimum stress point (Channel 2)
Fig. 1: Lower suspension arm component. [8] The data was collected in the unit of microstrain (µε) for 60 seconds at the sampling rate of 500 Hz, which gave 30,000 discrete points. The strain signal was then analysed based on the amplitude in order to obtain the PDF and PSD distribution plots, as well as the global signal statistics for the data. The distributions function and signal statistical properties are then compared for both pavé road and highway route. Results and Discussion The data in Fig. 2 was collected on a straight pavé road at Putrajaya and a highway route of Sistem Lingkaran-Lebuhraya Kajang (SILK) of Selangor. These strain signals with a VA pattern in the strain format, were measured on the front left lower suspension arm of a mid-sized sedan car. The road conditions were a stretch of highway road to represent mostly consistent load features, and a stretch of brick-pavéd road to represent a noisy with a consistent load features.
(a) Straight pavé road (Run 1)
(b) Straight pavé road (Run 2)
(c) Highway route (Run 1)
(d) Highway route (Run 2)
Fig. 2: Time history plot for a straight pavé and a highway route surfaces
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The captured of these random strain loadings were analyzed using the time domain approach. It showed that the high strain amplitudes for the pavé road were recorded at 295 µε and 75 µε for channel 1 and channel 2, respectively. For highway route, high strain amplitudes were recorded at 321 µε for channel 1 and 81 µε for channel 2. The peak in strain signal occurs when vehicle was in braking or driven condition on an uneven road. Fig. 3 illustrates the amplitude of PDF diagram for both straight pavé and highway route. The distribution for both types of roads seems to have the shape of a classic symmetrical bell curve, and it is a type of Gaussian Normal distribution. This kind of shape is expected especially for any measured random vibration signals.
(a) Straight pavé road (Run 1)
(b) Straight pavé road (Run 2)
(c) Highway route (Run 1)
(d) Highway route (Run 2)
Fig. 3: Graph of PDF distributions for both straight pavé and highway route surfaces This PDF diagram of channel 1 has a wide range of bell shape compared to channel 2. It can be related to the amplitude range of its signal, which the significant range is wider than the similar trend of channel 2. In addition, the signal of channel 1 is seems to be skewed to right when referring from the origin point, but the peak of channel 2 is around origin point. It means that the distribution of channel 1 is found to be a non-Gaussian pattern. For channel 2, however the distribution is closed to the Gaussian pattern. The PSD diagram of channel 1 shows a broad band signal while channel 2 shows a little bit narrow band pattern. A broad band signal covers a wide range of frequencies, and it might consist of a single, wide spike or a number of distinct spikes as shown in the PSD diagram. A narrow band pattern only covers a narrow range of frequencies as all of these patterns can be seen in Fig. 4.
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PSD graph of pave road for run 1
PSD graph of pave road for run 2 140
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(a) Straight pavé road (Run 1)
(b) Straight pavé road (Run 2)
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PSD graph of highway for run 2 250
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(d) Highway route (Run 2)
Fig. 4: Graph of PSD diagram for both straight pavé and highway route surfaces This PSD pattern of Fig. 4 represents the texture of road surface that is more noisy. In addition, the PSD is also used to determine the power of a time series or time domain signal. It also tells where the average power is distributed as a function of frequency. In this case, power distribution of channel 1 is more in the range of energy distribution compared to channel 2. This is related that channel 1 that contains more vibration compared to channel 2 during the vehicle movement. The statistical analysis is more concerned with reducing a long time signal into a few numerical values that can describe its characteristics. The global signal statistical properties values were then calculated and the results were tabulated in Table 1. The mean value is the average value or the centre of area of the PDF about the x-axis. In this table, mean value of channel 1 is higher than channel 2. The mean value represents the highest peak point in PDF diagram and when the mean value is not equal to zero, it give the meaning of the distribution is non-Gaussian. As exhibited in Fig. 3, the PDF diagram on channel 1 shows a non-Gaussian distribution, however, it is not for the channel 2 signal. RMS can be said as the indicator for a vibration signal energy in a time series and the kurtosis value represented the amplitude range in a time series. The RMS value of channel 1 is found to be higher than in channel 2. It meant that the maximum stress area create more vibration potential compared to the minimum stress area. With referring to the PSD diagram in Fig. 4, it was found that the power distribution on channel 1 is wider than channel 2. The kurtosis value of channel 1 was found to be lower than channel 2, as it is because kurtosis was highly sensitive to the spikiness of the data. The differences of kurtosis value on pave road was higher than the highway route because the pavé road surface represented more noisy with a consistent load features.
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Table 1: Global signal statistical properties for the collected data Kurtosis Mean (µε) RMS (µε) Signal Ch. 1 Ch. 2 Ch. 1 Ch. 2 Ch. 1 Ch. 2 Pavé (Run 1) Pavé (Run 2) Highway (Run 1) Highway (Run 2)
45.05 43.60 63.22 66.86
-1.49 -3.36 6.79 -7.23
56.48 54.77 72.36 75.51
8.36 8.591 12.26 12.27
3.88 3.59 4.23 3.66
5.18 4.26 4.48 3.48
Diff. (%) 25.11 15.63 5.71 -5.02
Conclusion This study discussed the capability of these two techniques for detecting the pattern behaviour in terms of energy and probability distribution that exist in a strain signal. This study presented an analysis on VA loading strains data distribution using both PDF and PSD techniques. A set of case study data consist random pattern of VA strain loadings were used. The amplitude of PDF diagram illustrated a symmetrical bell curve following the Gaussian Normal distribution and this curve is well expected for a random vibration signal. The result can be concluded that the non-Gaussian distribution is related to a broad band signal, while for the Gaussian distribution is for a narrow band signal. Using the statistical analysis, it was found that higher kurtosis values indicate the presence of more extreme values than should be found in the Gaussian distribution. For a nonstationary signal, the engineering-based signal analysis is important to explore the characteristics and behaviour of the signal. Thus, the outcomes of this data analysis can then be used in the related fault detection. The findings from this study are expected can be used to determine the pattern behaviour that exist in any VA signals, not only specifically for strain loadings. Acknowledgement The authors would like to express their gratitude to Universiti Kebangsaan Malaysia through the fund of UKM-KK-02-FRGS0129-2009 for supporting this research. References [1]
D.S.G. Pollock: A handbook of time-series analysis, signal processing and dynamics (Academic Press, Cambridge 1999).
[2]
X. Wang and J.Q. Sun: Journal of Sound and Vibration Vol. 280 (2005), p. 455-465.
[3]
Y. Meyer: Wavelets: Algorithm and Applications (SIAM, Philadelphia USA, 1993).
[4]
J.S. Bendat and A.G. Piersol: Random data: Analysis and Measurement Procedures (WileyInterscience, New York 1986).
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S. Abdullah, M.D. Ibrahim, Z.M. Nopiah and A. Zaharim: J. Applied Sciences Vol. 8 (2008), p. 1590–1593.
[7]
J. Draper: Modern metal fatigue analysis (Safe Technology Ltd., United Kingdom 1999).
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N.A. Al-Asady, S. Abdullah, A.K. Ariffin, and S.M. Beden: International Journal of Mechanical and Materials Engineering Vol. 4 (2009), p. 141-146.
