Adaptive modelling of transient vibration signals

ARTICLE IN PRESS Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 20 (2006) 825–842 www.elsevier.com/locate/jnlabr/ymssp Adaptive modelling of transient vibration signals Fenglin Wang, Chris K. Mechefske Department of Mechanical and Materials Engineering, Queen’s University, Kingston, Ont., Canada K7L 3N6 Received 31 May 2004; received in revised form 24 November 2004; accepted 20 December 2004 Available online 5 February 2005 Abstract Conventional vibration signal processing techniques are most suitable for stationary processes. However, most mechanical faults in machinery reveal themselves through transient events in vibration signals. Timeseries modelling, including autoregressive moving average (ARMA) modelling and autoregressive (AR) modelling, is an efficient approach for transient signal analysis. Based on the adaptive prediction technique, this paper applies the principle of the adaptive line enhancer (ALE) to the modelling of transient vibration signals. The time-series models, adaptive algorithms and the rational time–frequency transfer function are investigated in the paper. Simulation and experimental studies with different time–frequency–amplitude distributions and transient vibration responses are described. The results show that the adaptive modelling method can trace the time–frequency signal and extract dynamic features such as time–frequency distributions and time–amplitude distributions from sample signals. Given the simple programming and potentially easy implementation in on-line applications, this method should have application in machine monitoring and fault diagnosis. r 2005 Elsevier Ltd. All rights reserved. Keywords: Time-series modelling; Adaptive line enhancer; Time–frequency analysis; Transient vibration signals 1. Introduction Transient vibration processes are typical in machinery operation during machine run-up, shutdown, or changing speed or load conditions in a short time. These operation processes will Corresponding author. Tel.: +1 613 533 3148; fax: +1 613 533 6489. E-mail addresses: [email protected] (F. Wang), [email protected] (C.K. Mechefske). 0888-3270/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2004.12.004 ARTICLE IN PRESS 826 F. Wang, C.K. Mechefske / Mechanical Systems and Signal Processing 20 (2006) 825–842 excite the dynamic properties of the mechanical system and are useful when acquiring information such as machine resonant frequencies, vibration modes or even initial machine fault features. This important information, hidden in the transient vibration process, is usually in the form of chirp signals with both frequency and amplitude continuously varying with time. Conventional signal analysis techniques cannot be used to extract the useful information in the form of the time–frequency–amplitude characteristics. There are several available signal processing methods including parametric and non-parametric approaches, for example, time-series modelling, wavelet transform (WT) and short-time Fourier transform (STFT), which have been studied and used to analyse the transient signal in machine condition monitoring [1–6], structure vibration analysis [7] and machine condition monitoring [8]. By assuming stationarity of the signals within short intervals, the STFT is used to analyse the transient signal and achieve good time–frequency feature in some situation [9]. However, the STFT suffers from its inherent limitation of the spectrum versus narrowing the time intervals, and from the averaging effect of signal spectrum when the sudden frequency components occur in the transient process. Decomposing the transient signal into its orthogonal complementary parts with low-pass filter and high-pass filter, and iterating the decomposition in time scale, WT [10] has been increasingly applied to transient vibration analysis in wide research area. Although WT possesses the multi-resolution analysis of the transient signal, it also has shortcomings in analysis of the continuous transient signal such as the frequency overlapping and sensitive to the signal noise [8]. Comparatively, parametric signal processing approaches or time-varying coefficient models are efficient in tracking and modelling the chirp signals [11,12]. Identification of the time-varying parameters is crucial for the modelling process. From the viewpoint of the linear prediction filter, the identification of the modelling parameters is the process of solving time-varying Toeplitz matrix equations by using the Prony method [1–6] or the Levinson–Durbin algorithm [13]. On the other hand, the adaptive line enhancer (ALE) [14], adaptive time-series modelling, has long been applied to on-line machine diagnosis and fault detection [15,16]. The structure of an ALE is illustrated in Fig. 1 and its application in the enhancement of signal-to-noise ratio (SNR) is shown in Fig. 2. After digitalising with an A/D converter, the input signal x(n), composed of a vibration signal s(n) with background noise v(n), goes into two channels. One is treated as the expected + X(n) e(n) − Z-m y(n) Z-1 Z-1 W1n Z-1 W2n + Z-1 W3n + WLn + Adaptive filter algorithms Fig. 1. The adaptive line enhancer with FIR filter structure. ARTICLE IN PRESS F. Wang, C.K. Mechefske / Mechanical Systems and Signal Processing 20 (2006) 825–842 827 The ALE prediction + - Low pass filter e(n) X(n) X(t) LPF A/D -m Y(n) Adaptive filter Fig. 2. Application of the ALE in digital signal prediction. signal to an adaptive filter. The other goes through a decorrelation delay m and then acts as a reference input signal to the adaptive filter model. The adaptive algorithm will continue to adjust the time-varying filter model coefficients according to the principle of least mean square (LMS) error between its output and expected signal. After optimum convergence process of the adaptive iteration, the adaptive time-varying parametric model will be obtained. The output y(n) of the model will approach the predicted signal x(n), and the background noise is then suppressed. This is called ‘‘the ALE’’. Focusing on adaptive parametric time-varying modelling, this paper applies the adaptive filtering algorithms and ALE structure to modelling the chirp signals, swept sinusoidal excitation signals and transient vibration responses. First, the chirp excitation signal, transient response and the wavelet transformation are introduced and briefly discussed. Time-series models, adaptive modelling algorithms and its time–frequency-tracking ability are then explored. Multipleharmonic stationary signals, frequency- and amplitude-varying chirp signals and transient vibration responses are modelled with adaptive model process. Experimental studies on modelling of a machinery run-up process and transient vibration responses are engaged in a machinery faulttesting device. The results show the time–frequency–amplitude characteristics extracted from the transient vibration process, and also indicate that the adaptive modelling method could be applied to other transient signal analyses as well. 2. Chirp signals, vibration response and wavelet time–frequency analysis Chirp signals with continuous frequency and amplitude varying are common non-stationary signals widely encountered in machine transient process, sonar object detection or even bats’ object shooting. Among various chirp signals, linear chirp signals with linearly increasing frequency with respect to time give the best SNR excitation and become one of the typical signals used in structure modal testing. The linear chirp signal is often measured when changing machine speed such as a rapid acceleration or deceleration process. The linear chirp signal can be expressed as 
 
 1 2 f1
f0 2 (1) t ; xðtÞ ¼ x0 ðtÞ sin y ¼ x0 ðtÞ sin o0 t þ bt ¼ x0 ðtÞ sin 2pf 0 t þ p tS 2 ARTICLE IN PRESS 828 F. Wang, C.K. Mechefske / Mechanical Systems and Signal Processing 20 (2006) 825–842 where x0 ðtÞ is the varying amplitude; y is the total angular rotation; o0 is the starting angular velocity with starting frequencyf 0 ; f 1 is the finishing sweep frequency; tS is the sweep time interval; and b is the constant value of angular acceleration. With conventional FFT techniques, the sweep process must be slowly enough to calculate the frequency response function (FRF). The sweep frequency band must cover the rotation speed of the machinery to get the machine resonant frequencies and corresponding vibration peaks. For transient machinery operation, the linear chirp signal can be used to simulate machine excitation signal. According to the modal expansion concept, a multiple-degree-of-freedom (mdof) system with zero initial conditions can be used to calculate the transient vibration process of the mechanical system. The system response of the linear chirp excitation can be expressed by using the Duhamel integral.   Z t qffiffiffiffiffiffiffiffiffiffiffiffiffi I X ~ ark  ~ ask 2
zonk ðt
tÞ qffiffiffiffiffiffiffiffiffiffiffiffiffi xðtÞe sin onk 1
zk ðt
tÞ dt; (2) yrs ðtÞ ¼ k¼1 onk 1
z2k 0 where yrs is the response of the mass of order r under the excitation acting on the mass of order s; ~ ark and ~ ask are eigenvectors of M
1 K; K is the stiffness matrix of the first-order mdof system; M is the mass matrix of the first-order mdof system; onk is the k order natural frequency of the mdof system; and zk is the k order damping ratio of the system. In view of signal filtering, WT uses a series of orthogonal filters to sample the transient signal. The orthogonal high- and low-pass filters are used to extract one sample from two and decompose the transient signal into a series of constituent parts in time domain. Due to its orthogonal property, WT can extract time–frequency features effectively and keep the signal information ‘‘unaffected’’ during the decomposition process. So, the WT is suitable for the analysis of non-stationary signals, especially for the signals from machine fault monitoring where abnormal condition causes the sudden change of signal frequency components. However, after anytime of wavelet decomposition, the sampling frequency decreases by one-half, which makes frequency aliasing exist in the high-frequency decomposition coefficients. Every time frequency aliasing will influence the consecutive decomposition and will cause the more serious frequency aliasing as described in [17]. For example, the WT of a constant liner chirp signal xðtÞ ¼ sinð2p  125t2 Þ by using wavelet packet algorithm [18] is shown in Fig. 3. From the figure it can bee seen that many interference terms exist in the time–frequency distribution. The occurrence of the time–frequency component shift or overlapping in WT is inevitable due to its basic convolution definition [8]. In some cases this will mislead the analysis of signal. By the frequencyshifting processing, the corrected WT transform of the chirp signal is shown in Fig. 4. This example indicates that WT may not be quite suitable in the analysis of continuous frequencyvarying chirp signals. 3. Adaptive signal modelling and its time–frequency-tracking ability Generally transient signals can be modelled with time-dependent parameter autoregressive (AR) model and autoregressive moving average (ARMA) model. The relations of input series x(n) and the output series y(n) of AR and ARMA models can be expressed in the following ARTICLE IN PRESS Amplitude (lin) F. Wang, C.K. Mechefske / Mechanical Systems and Signal Processing 20 (2006) 825–842 1 0 -1 16 14 12 10 Frequency (*64 Hz) 8 6 4 1 2 0.2 0.4 0.6 0.8 Time (s) 0 Amplitude (lin) Fig. 3. The chirp signal WT decomposition of using Wickerhauser wavelet packet. 1 0 -1 16 14 12 10 Frequency (*64 Hz) 8 6 4 1 0.6 2 0 0.2 0.8 0.4 Time (s) Fig. 4. The corrected WT of the chirp signal by rearranging constituent parts. 829 ARTICLE IN PRESS 830 F. Wang, C.K. Mechefske / Mechanical Systems and Signal Processing 20 (2006) 825–842 time-varying difference equation: xðnÞ ¼ M X ai ðnÞxðn
iÞ þ gs ; M X bi ðnÞxðn
iÞ þ (3) i¼1 yðnÞ ¼ i¼0 N X aj ðnÞyðn
jÞ þ gs ; (4) j¼1 where gs is Gaussian noise, and ai ðnÞ and bj ðnÞ are time-varying parameters related to the poles and zeros of the rational transfer function of the system. Comparatively, there are the finite impulsive response (FIR) filter structure and the infinite impulsive response (IIR) filter structure in adaptive filtering. For FIR ALE structure, the relation between the output and the input of digital signals is yðnÞ ¼ L X wk ðnÞxðn
m
kÞ: (5) k¼0 The input X(n) and the time-varying weighting factors W(n) can be defined in the following vectors: X ðn
mÞ ¼ ½xðn
mÞ; xðn
m
1Þ; . . . ; xðn
m
LÞT ; W ðnÞ ¼ ½w0 ðnÞ; w1 ðnÞ; . . . ; wL ðnÞT : When applying the LMS algorithm in [14], the adaptive FIR modelling process can be expressed as yðnÞ ¼ X ðn
mÞT W ðnÞ; eðnÞ ¼ xðnÞ
X ðnÞT W ðnÞ; W ðn þ 1Þ ¼ W ðnÞ þ 2m  eðnÞX ðnÞ; ð6Þ where m is the convergence factor of the algorithm and L is the length of the adaptive filter. After the adaptive process converges to its optimum, the transfer function of the FIR filter model can be written as L Y ðzÞ X HðzÞ ¼ ¼ wk ðnÞz
m
k : X ðzÞ k¼0 (7) For adaptive IIR filter structure, the relation between the output and the input of digital signals is yðnÞ ¼ M0 X i¼0 b^i ðnÞxðn
m
iÞ þ N0 X a^ j ðnÞyðn
jÞ; (8) j¼1 where a^ j ðnÞ is the time-varying AR coefficient estimation of the IIR filter; b^j ðnÞ is the time-varying moving average coefficient estimation of the IIR filter; and M0 and N0 are the lengths of the adaptive forward filter and recursive filter separately. ARTICLE IN PRESS F. Wang, C.K. Mechefske / Mechanical Systems and Signal Processing 20 (2006) 825–842 831 In vector notation, the input digital signals and the time-varying weight factors are defined as Uðn
mÞ ¼ ½xðn
mÞ; xðn
m
1Þ; . . . ; xðn
m
MÞ; yðn
1Þ; yðn
2Þ; . . . ; xðn
NÞT : V ðnÞ ¼ ½b^0 ðnÞ; b^1 ðnÞ; . . . ; b^M ðnÞ; a^ 1 ðnÞ; a^ 2 ðnÞ; . . . ; a^ N ðnÞT By utilising the RLMS algorithm proposed by Feintuch [19], the adaptive IIR modelling algorithm can be expressed as yðnÞ ¼ Uðn
mÞT V ðnÞ; eðnÞ ¼ xðnÞ
UðnÞT V ðnÞ; V ðn þ 1Þ ¼ V ðnÞ þ 2w  eðnÞUðnÞ; ð9Þ where w is the convergence factor of the algorithm. After the adaptive process converges to its optimum, the transfer function of the IIR filter model can be written as PM 0 ^
m
i Y ðzÞ BðzÞ i¼0 bi ðnÞz : (10) ¼ ¼ HðzÞ ¼ PN 0 X ðzÞ 1
AðzÞ 1
j¼1 a^ j ðkÞz
j When the input signal is Gaussian noise, the adaptive modelling process of Eqs. (5) and (8) will approach to the AR modelling of Eq. (3) and the ARMA modelling of Eq. (4). When the adaptive iteration reaches its optimum, the transfer function estimations of Eqs. (7) and (10) will tend to be the time-series models of AR modelling and ARMA modelling. As predicted by B. Widrow et al. in [14], ‘‘the ALE has capabilities that may exceed those of conventional spectral analysers when the unknown sine wave has finite bandwidth or is frequency modulated’’. Eqs. (7) and (10) indicate that the adaptive transfer functions of FIR and IIR structure will track and model the transient signals by varying their time-varying parameters. Actually, the time–frequency-tracking ability of the ALE system was discussed in [20]. The theoretical transfer function of the ALE with FIR structure after the adaptive process converges to its optimum was written as  
m SNR z 1
½z=zðtÞ
L : (11) HðzÞ ¼ 1 þ L SNR zðtÞ 1
½z=zðtÞ
1 Its time-varying FRF is: H½o; oðtÞ ¼ SNR sin½L=2½o
oðtÞT e
j½o
oðtÞðmþðL
1Þ=2ÞT ; 1 þ L SNR sin½1=2½o
oðtÞT (12) where zðtÞ is the z transform of the input time-varying frequency o(t) and T is the sampling frequency of the signal. Eqs. (11) and (12) indicate that the adaptive FIR model functions as a frequency-tracking filter at the center of the input time-varying frequency oðtÞ: Theoretical FRF with time-shift frequency inputs are shown in Fig. 5. By freezing the time scale, the ALE of FIR structure with the length of 128 weights was compared with the DFT transform of 256 points in [14]. The results show that the ALE has the same frequency resolution when using the same amount of input data. Digital filter ARTICLE IN PRESS 832 F. Wang, C.K. Mechefske / Mechanical Systems and Signal Processing 20 (2006) 825–842 Fig. 5. The time–frequency function changing with the input shift frequencies. theory also indicates that theoretically the FIR filter modelling with the filter length of N+1 will possess the efficient bandwidth of 1/(N+1). 4. Simulation results Based on the adaptive algorithms and modelling structure analysis, simulation studies on the adaptive modelling of harmonic signals, linear chirp signals, and transient vibration responses were carried out. 4.1. Modelling of multiple-harmonic stationary signals First, we begin a tracking simulation of a multiple-harmonic signal with arbitrary initial phases given by uðtÞ ¼ 0:5 sinð25ptÞ þ 0:4 sinð50pt þ 0:1pÞ þ 0:3 sinð75pt þ 0:2pÞ þ 0:2 sinð100pt þ 0:3pÞ þ 0:1 sinð125pt þ 0:4pÞ þ 0:05 sinð150pt þ 0:5pÞ þ 0:025 sinð175pt þ 0:6pÞ: ð13Þ There are seven frequency components unchanged with time in this signal. The first five amplitudes are higher than the last two. By using the adaptive modelling method, the time–frequency distributions and tracking filter features of the signal are shown in Fig. 6. From this figure, it can be seen that a tracking comb-like narrowband filter centered at the five harmonic ARTICLE IN PRESS F. Wang, C.K. Mechefske / Mechanical Systems and Signal Processing 20 (2006) 825–842 833 Fig. 6. The tracking ability of the ALE with the input of a multiple-harmonic signal. frequencies is first formed, and then the same filter function with the last two harmonic components are adaptively formed as time progresses. These results further confirm that to the harmonic stationary signal the adaptive modelling method functions as a group of comb-like narrowband tracking filters widely used in on-line weak signal detection and machinery condition monitoring. 4.2. Modelling of linear chirp signals The accelerating process of a machine can be simulated by a constant linear chirp signal, which is given by uðtÞ ¼ 0:5 sinðp  25t2 Þ: (14) The time–frequency distribution calculated by the adaptive modelling approach is shown in Fig. 7a. It can be seen that, except for some variable wave amplitude and several interrupted time–frequency distributions at the zero-status start of the adaptive iteration process, the time–frequency features of the signal are fully revealed. The time–frequency–amplitude distribution of the signal is in agreement with the theoretical analysis of the linear time–frequency distribution (Fig. 5). Further, a transient chirp signal composed of two different frequency rates was also simulated. The signal is given by uðtÞ ¼ 0:5 sinðp  28t2 Þ þ 0:5 sinð2p  70 þ pt2 Þ: (15) ARTICLE IN PRESS 834 F. Wang, C.K. Mechefske / Mechanical Systems and Signal Processing 20 (2006) 825–842 Fig. 7. Time–frequency distribution of swept sinusoidal signals. (a) One frequency-varying component and (b) two frequency-varying component. The first frequency component is linearly increasing with time and the second frequency component is varying slowly with time. The time–frequency–amplitude distributions calculated by the adaptive modelling approach are shown in Fig. 7b. It can be seen that the adaptive modelling method can track the two combined frequency components quite well, and the resolution of the time and frequency are reasonably good. The next transient signal to be simulated is a time–frequency and time–amplitude-varying signal and it is expressed as uðtÞ ¼ 0:5 sinðp  28t2 Þ þ 0:5½1 þ 0:2 sinð2p  6tÞ sinð2p  88tÞ: (16) Two frequency components exist in the signal: one has the frequency changing quickly with time and the other has a constant frequency but the amplitude changes periodically. The time–frequency–amplitude distributions are shown in Fig. 8. From the figure it can be seen that the signal characteristics with both varying time–frequency and varying time–amplitude distributions are extracted by the adaptive modelling approach. 4.3. Modelling of transient vibration responses A general analytical solution for the transient vibration response of the constant linear chirp excitation is not possible and numerical techniques must be used. The transient vibration response of the chirp signal through a 2 dof system and a typical error of the adaptive modelling are shown in Fig. 9. From the figure, it can be seen that the transient vibration responses are frequency- and amplitude-varying chirp signals. The adaptive modelling process converges quickly and can track the transient signals. ARTICLE IN PRESS F. Wang, C.K. Mechefske / Mechanical Systems and Signal Processing 20 (2006) 825–842 835 Fig. 8. The distribution of the signal with varying time–frequency–amplitude. Fig. 9. The transient response of a 2 dof system and the error of the adaptive iteration. The time–frequency distributions of transient vibration responses to a 1 dof system and a 2 dof system are shown in Fig. 10. It can be seen that the vibration modes and transient working conditions of the vibration system can be modelled and revealed by the adaptive modelling ARTICLE IN PRESS 836 F. Wang, C.K. Mechefske / Mechanical Systems and Signal Processing 20 (2006) 825–842 Fig. 10. The frequency–time distribution of sdof and 2 dof vibration responses. (a) 1 dof system and (b) 2 dof system. method. The time–frequency–amplitude distributions of the transient vibration responses have the following characteristics: (1) Compared to continuous time–frequency distribution of the linear chirp signal, the distribution of the transient vibration responses are discontinuous and interrupted by the system resonances. (2) The amplitude distributions of transient vibration process are often condense or intensified in the resonant peak area with some gaps along the time–frequency axis . (3) After passing through a resonant peak, there are some residual amplitude responses with decaying magnitude with respect to time. (4) There is a time- or frequency-delay between the maximum response amplitude and its theoretical resonant peak for the run-up transient vibration response a s discussed in [18]. Fig. 11 shows the time–frequency distribution of a transient vibration response through a 4 dof system under the constant linear chirp excitation. Its natural frequencies are 14, 28, 38, and 44 Hz, respectively, and all the features discussed above are further evidenced. 5. Experimental results Experimental studies on the adaptive modelling of the machine transient operating process and transient vibration responses were engaged in a machinery fault-testing device as shown in Fig. 12. The rotor device is composed of a changeable electrical motor, changeable mechanical transmissions, and mechanical loading. Our focus is on the transient processes and resonant responses of the device. The highest rotor speed is 3500 rpm (58.3 Hz) and the transient frequency ARTICLE IN PRESS F. Wang, C.K. Mechefske / Mechanical Systems and Signal Processing 20 (2006) 825–842 837 Fig. 11. The distribution of transient vibration responses from a 4 dof system. Low pass filter Signal amplifier Data acquisition & computer system Signal amplifier Electrical motor Mechanical transmission Loading Fig. 12. Schematic of the machinery fault-testing device. band was set as 0–55 Hz. The sampling frequency is 1000 Hz and the signals were sampled and processed by using adaptive modelling process. First, an electrical motor with an electrical current sensor was installed and connected with a large rotational inertia gearbox in free loading condition. The time–frequency distribution of the run-up current signal is shown in Fig. 13. It can be seen that the run-up process follows a linear-frequency-increasing period and then changes to a constant frequency phase in which the rotor is in the steady-state or constant speed condition (see lower right-hand corner of Fig. 13). Second, the resonance case with quick running up of the testing device was devised for modelling of transient vibration processes, in which a motor with a small rotational inertia rotor system was installed. There are two close resonant peaks located between 30 and 45 Hz in the rotor resonant system. During the experiment both the acceleration transient process and the deceleration transient process were operated and measured. All the A/D sampling time was set a ARTICLE IN PRESS 838 F. Wang, C.K. Mechefske / Mechanical Systems and Signal Processing 20 (2006) 825–842 Fig. 13. The time–frequency distribution of the run-up current signal. Fig. 14. The frequency–time distributions of machinery run-up or close-down. (a) Acceleration process of 0–53 Hz and (b) deceleration process of 55–22 Hz. little earlier than the start of machinery run-up and after slow-down in order to capture the entire transient response and to avoid the adaptive transient process and high-peak response affecting the modelling of the experimental run-up process. The transient response data were then processed by the adaptive modelling approach off-line. The time–frequency distributions of the vibration responses are shown in Fig. 14. Considering the fact that mechanical noise was caused by the power transmission elements such as several pairs of rolling bearings, the time–frequency distribution is not as good as the simulation results. So, it is recommended that a tracking filter should be used to acquire a better SNR transient vibration response in later experimental ARTICLE IN PRESS F. Wang, C.K. Mechefske / Mechanical Systems and Signal Processing 20 (2006) 825–842 839 Fig. 15. The frequency–time–amplitude distributions of motors under free loading. (a) Healthy run-up process and (b) faulty run-up process. research. Even though the noise exists in the sampling signal, it can be seen that the amplitude spectral distributions are concentrated between 30 and 50 Hz in the figure. In the acceleration case, the first resonant peak response occurs at about 38 Hz with some residual amplitude responses. The second resonance peak appears at about 45 Hz with some spectral distribution gaps. About 6 s later the running process changed into the steady-state working condition at the constant speed frequency of 53 Hz. In the deceleration case, the slow-down process runs from the constant frequency at 55 Hz and then reduces to the speed at 22 Hz. The high resonant peak occurs at about 43 Hz with spectral concentration and the low resonance peak appears at about 35 Hz with some residual amplitude responses. A third experiment was conducted to analyse the run-up process of the electrical motor current. Comparison measurements were made between a healthy motor and a faulty one with a broken rotor bar. The healthy motor and the faulty motor were installed and run in the fault-testing device under free loading. The transient current signals of the run-up process were sampled and the time–frequency–amplitude properties were analysed and are shown in Fig. 15. From the figure it can be seen that both the healthy motor and the faulty one have starting current peaks at the beginning and end of the transient period before entering steady-state speed. However, the starting current peak of faulty motor is higher than that of healthy motor. The two motors were subsequently tested under mechanically loaded conditions (70 lb-in). The time–frequency–amplitude properties were analysed and are shown in Fig. 16. From the figure it can be seen that the amplitude difference between the healthy motor and the faulty one is significant. These results indicate that the amplitude of the transient process might be regarded as one criterion for judging the fault condition in an electric motor. 6. Conclusions The transient vibration process is difficult to analyse and model due to its continuous frequency- and amplitude-varying characteristics. Non-parametric approaches such as short-time ARTICLE IN PRESS 840 F. Wang, C.K. Mechefske / Mechanical Systems and Signal Processing 20 (2006) 825–842 Fig. 16. The frequency–time–amplitude distributions of motors under loading. (a) Healthy run-up process and (b) faulty run-up process. Fourier transform (STFT) and wavelet transform (WT) have their inherent limitations of spectrum versus the time intervals or ‘‘frequency aliasing’’ problems when being used to analyse the time–frequency–amplitude chirp signals. Parametric approaches of time-series AR modelling and ARMA modelling have been previously proven to be efficient tools in modelling the transient vibration signal and revealing its dynamic properties. This paper, from the viewpoint of an adaptive prediction filter, puts forward an innovative approach based on the principle of the adaptive liner enhancer (ALE) to tracking and modelling of linear chirp vibration signals. Simulation and experimental studies indicate that the new modelling approach is capable of tracking and modelling the transient vibration signals. The ALE procedure also has the characteristic of potentially easy on-line application as follows: (1) The adaptive prediction filter principle and adaptive algorithms can be applied to time-series modelling and the parameter estimation process. The application of the ALE system to the transient signal analysis was shown as a successful example. (2) When the adaptive iteration reaches its optimum, the transfer function estimations of FIR filter and IIR filter will converge to the time-series models of AR modelling and ARMA modelling. These adaptive time-varying modelling approache s possess the capacity of tracking time–frequency–amplitude chirp signals and modelling the transient vibration process. The model parameters can be estimated by using the least mean square (LMS) algorithm and the recursive least mean square (RLMS) adaptive algorithms. (3) To the stationary harmonic signal, the adaptive modelling approach functions as a group of comb-like narrowband tracking filters, which is widely used in weak signal detection and machinery condition monitoring. The adaptive modelling approach can track and model singles with constant amplitude swept sinusoidal excitation , compound swept sinusoidal excitation, and varying time–frequency and time– amplitude sinusoidal excitation. (4) Due to the superior ability of suppressing the wide band noise and enhancing the SNR of vibration signals, the adaptive ALE modelling approach not only can extract the time–frequency–amplitude spectral distributions from transient linear chirp signals and the ARTICLE IN PRESS F. Wang, C.K. Mechefske / Mechanical Systems and Signal Processing 20 (2006) 825–842 841 vibration responses in simulation condition but can also work in machine transient operating processes with background noise condition. Besides, the adaptive modelling method has its clear physical meanings in time–frequency–amplitude spectrum compared to the results of wavelet transformation. (5) Experimental studies of tracking the non-stationary run-up and slow-down process indicate that the adaptive modelling method can model the transient machine operating process with clear physical meanings. When passing through the natural frequency of the system, the time– frequency–amplitude spectrums are interrupted and intensified by the resonant peak responses along the time–frequency distribution. After passing through a resonant peak, there are occasional residual amplitude responses with magnitude decay with respect to time. (6) The adaptive modelling method was successfully applied to monitoring the transient runningup process of a healthy motor and a faulty one with a broken rotor bar. The results indicate that it can be applied to monitoring other transient vibration processes. Acknowledgements The authors would like to express many thanks to Mr. Wei Shao and Mr. Colin Zhou for the helpful discussions in the numerical vibration analysis of multiple dof vibration system and to Dr. Weidong Li for his help carrying out the experimental portion of this work. This work was funded by the Natural Sciences and Engineering Research Council of Canada. References [1] C.K. Mechefske, Z. 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