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Application of EMD method to friction signal processing
ARTICLE IN PRESS Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 22 (2008) 248–259 www.elsevier.com/locate/jnlabr/ymssp
Application of EMD method to friction signal processing Kejian Guo, Xingang Zhang, Hongguang Li, Guang Meng State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, PR China Received 17 April 2007; received in revised form 29 June 2007; accepted 4 July 2007 Available online 20 July 2007
Abstract Due to measurement limitation, the measured friction signal often contains noise and other force components such as elastic forces. Traditional Fourier-based analysis methods are not suitable to process nonlinear and non-stationary signal. In this paper, the combination of median filter and empirical mode decomposition (EMD) method is used to analyze the measured friction signal. Median filter is a nonlinear process useful in reducing random noise, while EMD method has offered a powerful method for nonlinear and non-stationary data processing. The background noise and the noise arising from the measurement system in the measured friction signal are removed using median filter first. Then the other force components except the real friction force can be extracted from the measured friction signal using the EMD method. The residue after extracting can be taken as a relatively clean and real friction force. This method is compared with the traditional Fourier-based methods and wavelet decomposition method. The comparison results both in time domain and in Hilbert spectrum can show the superiority of the EMD method in dealing with the problem of friction signal processing. r 2007 Elsevier Ltd. All rights reserved. Keywords: EMD method; Median filter; Nonlinear and non-stationary; Traditional Fourier-based analysis; Wavelet decomposition; Hilbert spectrum
1. Introduction In the system with friction contact, the measured friction signal often contains noise and other force components. With friction model becoming more and more complex, the identification of friction parameters also becomes more difficult [1,2]. And most of these identification work is done in the hypothesis of clean data or data with relatively little noise [3,4]. With the test data contaminated by noise and other force components badly, the parameter identification work even can be corrupted. Thus, it is very much necessary to develop a method to extract real friction force from the measured signal. Many de-noising theory and methods have been developed to eliminate these noise [5,6]. Among them, median filter is often used to remove noise. Median filtering is a nonlinear process useful in reducing random noise. In many different kinds of digital signal processing, median filtering follows this basic prescription. It consists of sliding a window of an odd number of elements along the signal, replacing the center sample by the median of the samples in the window. In particular, the median is hardly affected by one or two discrepant Corresponding author. Tel.: +86 21 34200664 322.
E-mail address: [email protected] (G. Meng). 0888-3270/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2007.07.002
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values among the signals in the window. Consequently, median filtering is very effective at removing various kinds of noise by ignoring it. However, in friction measurement other force components, such as elastic forces, often are included into the friction force data. These components often cannot be removed from measured
Fig. 1. Test apparatus.
Fig. 2. Measured time-domain friction signal (shake table moved in sinusoidal wave. The amplitude is 20 mm, and the frequency is 0.5 Hz).
Fig. 3. The spectrum of measured friction signal (18.438 and 56.823 Hz are detected).
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data through de-noising process. And traditional Fourier-based analyses simply cannot be applied to nonlinear and non-stationary time series. In friction experimental research field, there are few articles to discuss this problem. The purpose of this paper is to focus on friction signal processing. Empirical mode decomposition (EMD) method is employed to analyze measured friction signal. The EMD method had been pioneered by Huang et al. [7], with which any complicated data set can be decomposed into a finite and often small number of intrinsic mode functions (IMFs). This method has offered a powerful method for nonlinear and nonstationary data process. Especially, the IMF components produced by the EMD method usually have physical meanings [8]. The processed results through EMD method maintain the nonlinear and non-stationary characteristics of original signals. The IMFs produced by EMD method have the character of intra-wave modulation, and can concentrate the information of the same component which can only be delivered by Fourier frequency into one component. Recently, the EMD method has been used successfully to identify the dynamic characteristics of linear MDOF structures with proportional and non-proportional damping [9–11]. In Ref. [12], the EMD method is employed for the identification of a 2-story shear-beam building model with nonlinear stiffness.
Fig. 4. Measured time-domain acceleration signal at frame by accelerometer 2.
Fig. 5. The spectrum of acceleration signal at frame (the local natural frequency 18.438 and natural frequency 56.823 are detected).
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In Section 2, the experimental system is described and the forces composing the measured friction signal are studied. The combination of media filter and EMD method is used to extract the real friction force from measured signal in Section 3. In this section, the comparisons of the results processed using the method proposed in this paper with other methods (including traditional Fourier-based methods and wavelet decomposition method) are given. The conclusion is finally presented in Section 4. 2. Experimental setup The measurement of friction force is described in this section. The test apparatus is illustrated in Fig. 1. In this system, the specimen 1 is fixed to a shake table, and is carried to make reciprocating movement. The force sensor is fixed to a frame. A universal joint connected the force sensor and another friction segment. The joint can eliminate forces generated by bend moment and ensure that the force sensor only measures the horizontal friction force. However, due to the constant variation of friction force direction, the vibration of the fixing frame will be irritated. Thus, the force measured by the force sensor will contain other components coming from the frame vibration except friction force. It is described by the following formula: f m ¼ f friction þ f inertia þ f elastic þ f noise þ f others ,
(1)
where fm represents the measured force, ffriction is the real friction force, finertia is the inertia force, felastic is the elastic force produced mainly by the vibration of frame, and fnoise represents the background noise and the noise produced by the measurement system. With the increase of the frequency and amplitude of shake table movement, felastic will become prominent and cannot be ignored. That is to say, the measured force sometimes has large difference with the real friction force. This will lead to some corruption of analysis work such as friction model identification, or the identified model cannot describe the friction force accurately. Thus, the other components, especially the components with large proportion, must be removed from the measured force. 3. Results and discussion In this section, the combination of median filter and EMD method is applied to the measured friction signal processing. The sampling period here is 1.92E4 s. The background noise is around 860 Hz due to the cooling system of vibration table. The frame has a local natural vibration frequency at the location of connecting with the force sensor, which is about 18 Hz. Another natural frequency of the frame is about 57 Hz. The vibration
Fig. 6. The first IMF extracted from measured friction signal.
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table moved in sinusoidal wave, the amplitude is 20 mm, and the frequency is 0.5 Hz. The measured timedomain friction signal is illustrated in Fig. 2. Fig. 3 shows its Fourier spectrum, in which the frequencies of 18.438 and 56.823 Hz are detected. The acceleration signal at the connecting location is measured by accelerometer 2, which is shown in Fig. 4. Fig. 5 gives its Fourier spectrum. From Fig. 5, it can be seen that the frequencies of the acceleration signal are mainly 18.438 and 56.823 Hz. From Figs. 2–5, it can be found the measured friction forces contain some elastic components, which are produced by the vibration of the frame. The frequencies of the elastic force components are mainly 18.438 and 56.823 Hz. To get the real friction force from the measured signal, the noise, the inertia force, and the elastic force must be eliminated from the measured data (shown in Eq. (1)). At the same time, the information of the real friction force must be preserved completely. First, the median filter is employed to remove the noise. It included background noise and the noise arising from the measuring system. The inertia force can be extracted through calculation by the following formula: f inertia ¼ ma,
(2)
where the acceleration a is measured by the accelerometer 1. After the two parts are removed, the remainder forces are mainly friction force and elastic forces produced by the vibration of fixing frame.
Fig. 7. The Fourier spectrum of the first IMF (the main frequency is 56.823 Hz).
Fig. 8. The second IMF extracted from measured friction signal.
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Fig. 9. The Fourier spectrum of the second IMF (the main frequency is 18.438 Hz).
Fig. 10. The residue signal after extracting.
Fig. 11. The Fourier spectrum of the residue signal.
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Here, the EMD method is applied to the remainder signal to extract the elastic force signals. Fig. 6 gives the first IMF, the Fourier spectrum of which is shown in Fig. 7. It can be seen that the frequency of the first IMF is around 56.823 Hz, which coincides with the natural frequency of frame. Figs. 8 and 9 illustrate the second IMF and its Fourier spectrum, respectively. The frequency (18.438 Hz) (shown in Fig. 9) is the same with the local natural frequency of the frame. These two frequencies can also be found in the Fourier spectrum of the frame acceleration signal (in Fig. 5). Thus, it can be confirmed that these two force components in the measured friction signal are mainly elastic forces coming from fixing frame vibration. The residue signal and its Fourier spectrum are shown in Figs. 10 and 11, respectively. The residue signal can be taken as a relatively clean and real friction force. To demonstrate the superiority of the proposed method in this paper, the traditional band block filter and low-pass filter were compared to the proposed method. The result processed using 50–70 and 12–25 Hz band
Fig. 12. The result processed using 50–70 and 12–25 Hz band block filter.
Fig. 13. The result processed using 12 Hz low-pass filter.
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Fig. 14. The result processed using 10 Hz low-pass filter.
Fig. 15. The result processed using 8 Hz low-pass filter.
block filter is exhibited in Fig. 12. Figs. 13–15 give the result processed using 12, 10, and 8 Hz low-pass filter, respectively. It can be shown that when the 12–25 and 50–70 Hz band block filter or 12 Hz low-pass filter is used (Figs. 12 and 13), the result approximates, but obviously is not as smooth as the result processed by EMD method (in Fig. 10). There are still fluctuations in the processed friction force. However, as the threshold frequency of low-pass filter gets lower, the result becomes even worse (Figs. 14 and 15). The invalidation of the Fourier-based filtering methods is mainly because the friction signal is not the simple superposition of some harmonic signals, but nonlinear, while the traditional Fourier methods are applicable to linear signals. Then completely removing the part of some certain frequencies, perhaps, can gain the end of getting rid of the unwanted components, at the same time will lead to the lost of the information of the real friction force.
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Fig. 16. The reconstructed approximation signal at level 8 by wavelet decomposition method (the ‘db5’ wavelet is used).
Fig. 17. The reconstructed approximation signal at level 7 by wavelet decomposition method (the ‘db5’ wavelet is used).
Wavelet decomposition method is also applied to identify real friction force from measured signal, and the result is compared with the result using the proposed method in this paper. Through analysis and result comparison using different kinds of wavelet to process original signal, the ‘db5’ wavelet is chosen finally. Figs. 16–18 present the reconstructed approximation signal at levels 8, 9, and 7, respectively. From these figures, it can be seen the 8-level approximation signal is the most ideal result (in Fig. 16). This result is compared with the processed result using EMD method in Fig. 19. From the comparison, it can be found that the result derived from wavelet decomposition method is in accord with the result using EMD method in most sliding phase. However, at the beginning of sliding phase (around velocity reversal point), the friction signal processed using wavelet decomposition method has some fluctuations (the solid blue line in Fig. 19). The result processed using the EMD method is smooth and can show the real friction force consecutively (the solid red line).
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Fig. 18. The reconstructed approximation signal at level 9 by wavelet decomposition method (the ‘db5’ wavelet is used).
Fig. 19. The comparison of result processed by the method in this paper and the result processed by wavelet decomposition method (the dot line is measured signal, the red line is the result processed by the method in this paper, the blue line is the reconstructed signal at level 8 with ‘db5’ wavelet decomposition.).
The Fourier spectra of the results processed using EMD method, wavelet decomposition method, low-pass filter, and band block filter are exhibited in Fig. 20. From this figure, it is difficult to distinguish which method is more applicable. The reason for the invalidation is that Fourier spectral analysis works well only for data from linear process, while the friction signal is from nonlinear process. Hilbert spectrum, which claims to deal with both nonlinear and non-stationary time series, is introduced to analyze the processed friction signals. Fig. 21 gives the Hilbert spectra of the results processed using the four methods. For the results derived from EMD method, the instantaneous frequency keeps a nearly constant value when friction is in sliding phase (the protuberant part corresponds to the sticking phase of friction around velocity reversal point) (shown in
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Fig. 20. The Fourier spectrum for processed friction.
Fig. 21. The Hilbert spectrum for processed friction signal.
Fig. 21a). Whereas the instantaneous frequency of the signal processed using wavelet decomposition method has some fluctuations at the beginning of sliding (shown in Fig. 21b). And there are fluctuations during the whole sliding phase in the Hilbert spectrum of the result derived from low-pass filter (shown in Fig. 21c). As for the Hilbert spectrum of the friction signal processed using band block filter, it is confusing to find something (shown in Fig. 21d). Except the spectrum of the processed signal from EMD method, negative values of instantaneous frequency appear in the other three spectra. This confirms that the processed signals contain not only the component of friction force, but also others in those three cases. Thus, from the comparisons of Hilbert spectra, it can be concluded that the EMD method can get better result than the other three methods.
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4. Conclusions Besides real friction force, measured friction signal usually includes noise and other force components, such as elastic forces. And traditional Fourier-based analysis method is not applicable to this kind of signal processing. The EMD method has offered a powerful method for nonlinear and non-stationary data processing. It can decompose any complicated data set into a finite and often small number of IMF. And especially, these IMF components usually have physical meanings. In this paper, the combination of median filter and EMD is used to get the real friction force from measured signal. First the background and measurement noise are removed by median filter. Then the EMD method is employed to extract the elastic force signals. There are two kinds of signals being extracted from the measured friction signal. They correspond to the two vibration modes of fixing frame. Thus, it can be confirmed that these signals belong to elastic forces produced by the vibration of the frame. The residue after extracting is taken as a clean and real friction force. The results derived from band block filter, low-pass filter, and wavelet decomposition method are also compared with the result processed using EMD method both in time domain and in Hilbert spectrum. From these comparisons, it can be shown that the EMD method not only can analyze measured signal and extract different components according to real conditions, but also can get relatively more reasonable results. Thus, the method in this paper has superiority in processing measured friction signal. Acknowledgment This work was supported by the National Natural Science Foundation of China (NSFC) under the Grant no.10502032. References [1] E.J. Berger, Friction modeling for dynamic system simulation, Applied Mechanics Review 55 (6) (2002) 535–577. [2] J. Swevers, F. Al-Bender, C.G. Ganseman, T. Prajogo, An integrated friction model structure with improved presliding behavior for accurate friction compensation, IEEE Transactions on Automatic Control 45 (4) (2000) 675–686. [3] J.H. Kim, H.K. Chae, J.Y. Jeon, S.W. Lee, Identification and control of systems with friction using accelerated evolutionary programming, IEEE Control Systems Magazine 16 (4) (1996) 38–47. [4] D.D. Rizos, S.D. Fassois, Presliding friction identification based upon the Maxwell slip model structure, Chaos 14 (2) (2004) 431–445. [5] A. Papoulis, Probability, Random Variables, and Stochastic Process, third ed, McGraw-Hill, New York, 1991. [6] S. Hoyos, B. Li, J. Bacca, et al., Weighted median filters admitting complex-valued weights and their optimization, IEEE Transactions on Signal Processing 52 (10) (2004) 2776–2787. [7] N.E. Huang, Z. Shen, S.R. Long, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proceedings of the Royal Society of London A 454 (1998) 903–995. [8] N.E. Huang, M.L. Wu, S.R. Long, A confidence limit for the empirical mode decomposition and the Hilbert spectral analysis, Proceedings of the Royal Society of London A 459 (2003) 2317–2345. [9] J.N. Yang, Y. Lei, Identification of natural frequencies and damping ratios of linear structures via Hilbert transform and empirical mode decomposition, in: M.H. Hamza (Ed.), Proceedings of IASTED International Conference on Intelligent Systems and Control, Acta Press, Santa Barbara, 1999, pp. 310–315. [10] J.N. Yang, Y. Lei, Identification of natural civil structures with nonproportional damping, Proceedings of the SPIE: Smart Structures and Materials, Smart Systems for Bridges, Structures, and Highways 3988 (2000) 284–294. [11] J.N. Yang, Y. Lei, System identification of linear structures using Hilbert transform and empirical mode decomposition, in: Proceedings of the 18th International Conference on Modal Analysis, San Antonio, vol. 1, 2000, pp. 213–219. [12] C.W. Poon, C.C. Chang, Identification of nonlinear normal modes of structures using the empirical mode decomposition method, in: M. Tomizuka (Ed.), Smart Structures and Materials 2005: Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems, Proceedings of the SPIE – The International Society for Optical Engineering 5765 (2005) 881–891.
Application of EMD method to friction signal processing Kejian Guo, Xingang Zhang, Hongguang Li, Guang Meng State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, PR China Received 17 April 2007; received in revised form 29 June 2007; accepted 4 July 2007 Available online 20 July 2007
Abstract Due to measurement limitation, the measured friction signal often contains noise and other force components such as elastic forces. Traditional Fourier-based analysis methods are not suitable to process nonlinear and non-stationary signal. In this paper, the combination of median filter and empirical mode decomposition (EMD) method is used to analyze the measured friction signal. Median filter is a nonlinear process useful in reducing random noise, while EMD method has offered a powerful method for nonlinear and non-stationary data processing. The background noise and the noise arising from the measurement system in the measured friction signal are removed using median filter first. Then the other force components except the real friction force can be extracted from the measured friction signal using the EMD method. The residue after extracting can be taken as a relatively clean and real friction force. This method is compared with the traditional Fourier-based methods and wavelet decomposition method. The comparison results both in time domain and in Hilbert spectrum can show the superiority of the EMD method in dealing with the problem of friction signal processing. r 2007 Elsevier Ltd. All rights reserved. Keywords: EMD method; Median filter; Nonlinear and non-stationary; Traditional Fourier-based analysis; Wavelet decomposition; Hilbert spectrum
1. Introduction In the system with friction contact, the measured friction signal often contains noise and other force components. With friction model becoming more and more complex, the identification of friction parameters also becomes more difficult [1,2]. And most of these identification work is done in the hypothesis of clean data or data with relatively little noise [3,4]. With the test data contaminated by noise and other force components badly, the parameter identification work even can be corrupted. Thus, it is very much necessary to develop a method to extract real friction force from the measured signal. Many de-noising theory and methods have been developed to eliminate these noise [5,6]. Among them, median filter is often used to remove noise. Median filtering is a nonlinear process useful in reducing random noise. In many different kinds of digital signal processing, median filtering follows this basic prescription. It consists of sliding a window of an odd number of elements along the signal, replacing the center sample by the median of the samples in the window. In particular, the median is hardly affected by one or two discrepant Corresponding author. Tel.: +86 21 34200664 322.
E-mail address: [email protected] (G. Meng). 0888-3270/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2007.07.002
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values among the signals in the window. Consequently, median filtering is very effective at removing various kinds of noise by ignoring it. However, in friction measurement other force components, such as elastic forces, often are included into the friction force data. These components often cannot be removed from measured
Fig. 1. Test apparatus.
Fig. 2. Measured time-domain friction signal (shake table moved in sinusoidal wave. The amplitude is 20 mm, and the frequency is 0.5 Hz).
Fig. 3. The spectrum of measured friction signal (18.438 and 56.823 Hz are detected).
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data through de-noising process. And traditional Fourier-based analyses simply cannot be applied to nonlinear and non-stationary time series. In friction experimental research field, there are few articles to discuss this problem. The purpose of this paper is to focus on friction signal processing. Empirical mode decomposition (EMD) method is employed to analyze measured friction signal. The EMD method had been pioneered by Huang et al. [7], with which any complicated data set can be decomposed into a finite and often small number of intrinsic mode functions (IMFs). This method has offered a powerful method for nonlinear and nonstationary data process. Especially, the IMF components produced by the EMD method usually have physical meanings [8]. The processed results through EMD method maintain the nonlinear and non-stationary characteristics of original signals. The IMFs produced by EMD method have the character of intra-wave modulation, and can concentrate the information of the same component which can only be delivered by Fourier frequency into one component. Recently, the EMD method has been used successfully to identify the dynamic characteristics of linear MDOF structures with proportional and non-proportional damping [9–11]. In Ref. [12], the EMD method is employed for the identification of a 2-story shear-beam building model with nonlinear stiffness.
Fig. 4. Measured time-domain acceleration signal at frame by accelerometer 2.
Fig. 5. The spectrum of acceleration signal at frame (the local natural frequency 18.438 and natural frequency 56.823 are detected).
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In Section 2, the experimental system is described and the forces composing the measured friction signal are studied. The combination of media filter and EMD method is used to extract the real friction force from measured signal in Section 3. In this section, the comparisons of the results processed using the method proposed in this paper with other methods (including traditional Fourier-based methods and wavelet decomposition method) are given. The conclusion is finally presented in Section 4. 2. Experimental setup The measurement of friction force is described in this section. The test apparatus is illustrated in Fig. 1. In this system, the specimen 1 is fixed to a shake table, and is carried to make reciprocating movement. The force sensor is fixed to a frame. A universal joint connected the force sensor and another friction segment. The joint can eliminate forces generated by bend moment and ensure that the force sensor only measures the horizontal friction force. However, due to the constant variation of friction force direction, the vibration of the fixing frame will be irritated. Thus, the force measured by the force sensor will contain other components coming from the frame vibration except friction force. It is described by the following formula: f m ¼ f friction þ f inertia þ f elastic þ f noise þ f others ,
(1)
where fm represents the measured force, ffriction is the real friction force, finertia is the inertia force, felastic is the elastic force produced mainly by the vibration of frame, and fnoise represents the background noise and the noise produced by the measurement system. With the increase of the frequency and amplitude of shake table movement, felastic will become prominent and cannot be ignored. That is to say, the measured force sometimes has large difference with the real friction force. This will lead to some corruption of analysis work such as friction model identification, or the identified model cannot describe the friction force accurately. Thus, the other components, especially the components with large proportion, must be removed from the measured force. 3. Results and discussion In this section, the combination of median filter and EMD method is applied to the measured friction signal processing. The sampling period here is 1.92E4 s. The background noise is around 860 Hz due to the cooling system of vibration table. The frame has a local natural vibration frequency at the location of connecting with the force sensor, which is about 18 Hz. Another natural frequency of the frame is about 57 Hz. The vibration
Fig. 6. The first IMF extracted from measured friction signal.
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table moved in sinusoidal wave, the amplitude is 20 mm, and the frequency is 0.5 Hz. The measured timedomain friction signal is illustrated in Fig. 2. Fig. 3 shows its Fourier spectrum, in which the frequencies of 18.438 and 56.823 Hz are detected. The acceleration signal at the connecting location is measured by accelerometer 2, which is shown in Fig. 4. Fig. 5 gives its Fourier spectrum. From Fig. 5, it can be seen that the frequencies of the acceleration signal are mainly 18.438 and 56.823 Hz. From Figs. 2–5, it can be found the measured friction forces contain some elastic components, which are produced by the vibration of the frame. The frequencies of the elastic force components are mainly 18.438 and 56.823 Hz. To get the real friction force from the measured signal, the noise, the inertia force, and the elastic force must be eliminated from the measured data (shown in Eq. (1)). At the same time, the information of the real friction force must be preserved completely. First, the median filter is employed to remove the noise. It included background noise and the noise arising from the measuring system. The inertia force can be extracted through calculation by the following formula: f inertia ¼ ma,
(2)
where the acceleration a is measured by the accelerometer 1. After the two parts are removed, the remainder forces are mainly friction force and elastic forces produced by the vibration of fixing frame.
Fig. 7. The Fourier spectrum of the first IMF (the main frequency is 56.823 Hz).
Fig. 8. The second IMF extracted from measured friction signal.
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Fig. 9. The Fourier spectrum of the second IMF (the main frequency is 18.438 Hz).
Fig. 10. The residue signal after extracting.
Fig. 11. The Fourier spectrum of the residue signal.
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Here, the EMD method is applied to the remainder signal to extract the elastic force signals. Fig. 6 gives the first IMF, the Fourier spectrum of which is shown in Fig. 7. It can be seen that the frequency of the first IMF is around 56.823 Hz, which coincides with the natural frequency of frame. Figs. 8 and 9 illustrate the second IMF and its Fourier spectrum, respectively. The frequency (18.438 Hz) (shown in Fig. 9) is the same with the local natural frequency of the frame. These two frequencies can also be found in the Fourier spectrum of the frame acceleration signal (in Fig. 5). Thus, it can be confirmed that these two force components in the measured friction signal are mainly elastic forces coming from fixing frame vibration. The residue signal and its Fourier spectrum are shown in Figs. 10 and 11, respectively. The residue signal can be taken as a relatively clean and real friction force. To demonstrate the superiority of the proposed method in this paper, the traditional band block filter and low-pass filter were compared to the proposed method. The result processed using 50–70 and 12–25 Hz band
Fig. 12. The result processed using 50–70 and 12–25 Hz band block filter.
Fig. 13. The result processed using 12 Hz low-pass filter.
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Fig. 14. The result processed using 10 Hz low-pass filter.
Fig. 15. The result processed using 8 Hz low-pass filter.
block filter is exhibited in Fig. 12. Figs. 13–15 give the result processed using 12, 10, and 8 Hz low-pass filter, respectively. It can be shown that when the 12–25 and 50–70 Hz band block filter or 12 Hz low-pass filter is used (Figs. 12 and 13), the result approximates, but obviously is not as smooth as the result processed by EMD method (in Fig. 10). There are still fluctuations in the processed friction force. However, as the threshold frequency of low-pass filter gets lower, the result becomes even worse (Figs. 14 and 15). The invalidation of the Fourier-based filtering methods is mainly because the friction signal is not the simple superposition of some harmonic signals, but nonlinear, while the traditional Fourier methods are applicable to linear signals. Then completely removing the part of some certain frequencies, perhaps, can gain the end of getting rid of the unwanted components, at the same time will lead to the lost of the information of the real friction force.
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Fig. 16. The reconstructed approximation signal at level 8 by wavelet decomposition method (the ‘db5’ wavelet is used).
Fig. 17. The reconstructed approximation signal at level 7 by wavelet decomposition method (the ‘db5’ wavelet is used).
Wavelet decomposition method is also applied to identify real friction force from measured signal, and the result is compared with the result using the proposed method in this paper. Through analysis and result comparison using different kinds of wavelet to process original signal, the ‘db5’ wavelet is chosen finally. Figs. 16–18 present the reconstructed approximation signal at levels 8, 9, and 7, respectively. From these figures, it can be seen the 8-level approximation signal is the most ideal result (in Fig. 16). This result is compared with the processed result using EMD method in Fig. 19. From the comparison, it can be found that the result derived from wavelet decomposition method is in accord with the result using EMD method in most sliding phase. However, at the beginning of sliding phase (around velocity reversal point), the friction signal processed using wavelet decomposition method has some fluctuations (the solid blue line in Fig. 19). The result processed using the EMD method is smooth and can show the real friction force consecutively (the solid red line).
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Fig. 18. The reconstructed approximation signal at level 9 by wavelet decomposition method (the ‘db5’ wavelet is used).
Fig. 19. The comparison of result processed by the method in this paper and the result processed by wavelet decomposition method (the dot line is measured signal, the red line is the result processed by the method in this paper, the blue line is the reconstructed signal at level 8 with ‘db5’ wavelet decomposition.).
The Fourier spectra of the results processed using EMD method, wavelet decomposition method, low-pass filter, and band block filter are exhibited in Fig. 20. From this figure, it is difficult to distinguish which method is more applicable. The reason for the invalidation is that Fourier spectral analysis works well only for data from linear process, while the friction signal is from nonlinear process. Hilbert spectrum, which claims to deal with both nonlinear and non-stationary time series, is introduced to analyze the processed friction signals. Fig. 21 gives the Hilbert spectra of the results processed using the four methods. For the results derived from EMD method, the instantaneous frequency keeps a nearly constant value when friction is in sliding phase (the protuberant part corresponds to the sticking phase of friction around velocity reversal point) (shown in
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Fig. 20. The Fourier spectrum for processed friction.
Fig. 21. The Hilbert spectrum for processed friction signal.
Fig. 21a). Whereas the instantaneous frequency of the signal processed using wavelet decomposition method has some fluctuations at the beginning of sliding (shown in Fig. 21b). And there are fluctuations during the whole sliding phase in the Hilbert spectrum of the result derived from low-pass filter (shown in Fig. 21c). As for the Hilbert spectrum of the friction signal processed using band block filter, it is confusing to find something (shown in Fig. 21d). Except the spectrum of the processed signal from EMD method, negative values of instantaneous frequency appear in the other three spectra. This confirms that the processed signals contain not only the component of friction force, but also others in those three cases. Thus, from the comparisons of Hilbert spectra, it can be concluded that the EMD method can get better result than the other three methods.
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4. Conclusions Besides real friction force, measured friction signal usually includes noise and other force components, such as elastic forces. And traditional Fourier-based analysis method is not applicable to this kind of signal processing. The EMD method has offered a powerful method for nonlinear and non-stationary data processing. It can decompose any complicated data set into a finite and often small number of IMF. And especially, these IMF components usually have physical meanings. In this paper, the combination of median filter and EMD is used to get the real friction force from measured signal. First the background and measurement noise are removed by median filter. Then the EMD method is employed to extract the elastic force signals. There are two kinds of signals being extracted from the measured friction signal. They correspond to the two vibration modes of fixing frame. Thus, it can be confirmed that these signals belong to elastic forces produced by the vibration of the frame. The residue after extracting is taken as a clean and real friction force. The results derived from band block filter, low-pass filter, and wavelet decomposition method are also compared with the result processed using EMD method both in time domain and in Hilbert spectrum. From these comparisons, it can be shown that the EMD method not only can analyze measured signal and extract different components according to real conditions, but also can get relatively more reasonable results. Thus, the method in this paper has superiority in processing measured friction signal. Acknowledgment This work was supported by the National Natural Science Foundation of China (NSFC) under the Grant no.10502032. References [1] E.J. Berger, Friction modeling for dynamic system simulation, Applied Mechanics Review 55 (6) (2002) 535–577. [2] J. Swevers, F. Al-Bender, C.G. Ganseman, T. Prajogo, An integrated friction model structure with improved presliding behavior for accurate friction compensation, IEEE Transactions on Automatic Control 45 (4) (2000) 675–686. [3] J.H. Kim, H.K. Chae, J.Y. Jeon, S.W. Lee, Identification and control of systems with friction using accelerated evolutionary programming, IEEE Control Systems Magazine 16 (4) (1996) 38–47. [4] D.D. Rizos, S.D. Fassois, Presliding friction identification based upon the Maxwell slip model structure, Chaos 14 (2) (2004) 431–445. [5] A. Papoulis, Probability, Random Variables, and Stochastic Process, third ed, McGraw-Hill, New York, 1991. [6] S. Hoyos, B. Li, J. Bacca, et al., Weighted median filters admitting complex-valued weights and their optimization, IEEE Transactions on Signal Processing 52 (10) (2004) 2776–2787. [7] N.E. Huang, Z. Shen, S.R. Long, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proceedings of the Royal Society of London A 454 (1998) 903–995. [8] N.E. Huang, M.L. Wu, S.R. Long, A confidence limit for the empirical mode decomposition and the Hilbert spectral analysis, Proceedings of the Royal Society of London A 459 (2003) 2317–2345. [9] J.N. Yang, Y. Lei, Identification of natural frequencies and damping ratios of linear structures via Hilbert transform and empirical mode decomposition, in: M.H. Hamza (Ed.), Proceedings of IASTED International Conference on Intelligent Systems and Control, Acta Press, Santa Barbara, 1999, pp. 310–315. [10] J.N. Yang, Y. Lei, Identification of natural civil structures with nonproportional damping, Proceedings of the SPIE: Smart Structures and Materials, Smart Systems for Bridges, Structures, and Highways 3988 (2000) 284–294. [11] J.N. Yang, Y. Lei, System identification of linear structures using Hilbert transform and empirical mode decomposition, in: Proceedings of the 18th International Conference on Modal Analysis, San Antonio, vol. 1, 2000, pp. 213–219. [12] C.W. Poon, C.C. Chang, Identification of nonlinear normal modes of structures using the empirical mode decomposition method, in: M. Tomizuka (Ed.), Smart Structures and Materials 2005: Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems, Proceedings of the SPIE – The International Society for Optical Engineering 5765 (2005) 881–891.