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附錄一 英文參考文獻(xiàn)
Application of slice spectral correlation density to gear defect detection
G Bi, J Chen, F C Zhou, and J He
The State Key Laboratory of Vibration, Sound, and Noise,Shanghai Jiaotong University, Shanghai,People’s Republic of China
The manuscript was received on 16 October 2005 and was accepted after revision for publication on 3 May 2006.DOI: 10.1243/0954406JMES206
Abstract: The most direct reflection of gear defect is the change in the amplitude and phase modulations of vibration. The slice spectral correlation density (SSCD)method presented in this paper can be used to extract modulation information from the gear vibration signal; amplitude and phase modulation information can be extracted either individually or in combination.
This method can detect slight defects with comparatively evident phase modulation as well as serious defects with strong amplitude modulation. Experimental vibration signals presenting gear defects of different levels of severity verify to its character identification capability and indicate that the SSCD is an effective method, especially to detect defects at an early stage of development.
Keywords: slice spectral correlation density, gear, defect detection, modulation
1 INTRODUCTION
A gear vibration signal is a typical periodic modulation signal. Modulation phenomena are more serious with the deterioration of gear defects. Accordingly, the modulation sidebands in the spectrum get incremented in number and amplitude.Therefore, extracting modulation information from these sidebands is the direct way to detect gear defects. A conventional envelope technique is one of the methods for this purpose. It is sensitive to modulation phenomena in amplitude, but not in phase. A slight gear defect often produces little change in vibration amplitude, but it is always accompanied by evident phasemodulation. Employing the envelope technique for an incipient slight defect does not produce satisfactory results.
In recent years, the theory of cyclic statistics has been used for rotating machine vibration signal and shows good potential for use in condition monitoring and diagnosis [1–3]. In this article, spectral correlation density (SCD) function in the second-order cyclostationarity is verified to be a redundant information provider for gear defect detection. It simultaneously exhibits amplitude and phase modulation during gear vibration, which is especially valuable for detecting slight defects and monitoring their evolution.The SCD function maps signals into a two-dimensional function in a cyclic frequency (CF) versus general frequency plane (a–f). Considering its information redundancy [4] and huge computation,the slice of the SCD where CF equals the shaft rotation frequency is individually computed for defect detection,which is named slice spectral correlation density (SSCD). The SSCD is demonstrated to possess the same identification capability as the SCD function. It can be computed directly from a time-varying autocorrelation with less computation and, at the same time, has clear representation when compared with a three-dimensional form of the SCD.
2 SECOND-ORDER CYCLIC STATISTICS
A random process generally has a time-varying autocorrelation[5]
Where is the mathematic expectation operator and t is the time lag. If the
autocorrelation is periodic with a period T0, the ensemble average can be estimated with time average
The autocorrelation can also be written in the Fourier series because of its periodicity
WhereCombining with equation(2), its Fourier coefficients can be given as [5]
Where is the time averaging operation, is referred to as the cyclic autocorrelation (CA),and a is the CF. SCD can be obtained by applying Fourier transform of the CA with respect to the time lag t
The SCD exhibits the characteristics of the signal in a–f bi-frequency plane. All non-zero CFs characterize the cyclostationary (CS) characters of the signal.
3 THE GEAR MODEL
The most important component in gearbox vibration is the tooth meshing vibration, which is due to the deviations from the ideal tooth profile. Sources of such deviations are the tooth deformation under load or original profile errors made in the machining process. Generally, modulation phenomena occur when a local defect goes through the mesh and generates periodic alteration to the tooth meshing vibration in amplitude and phase. To a normal gear, the fluctuation in the shaft rotation frequency and the load or the tiny difference in the teeth space also permits slight amplitude modulation(AM) or phase modulation (PM). Therefore, the general gear model can be written as [6, 7]
where fx is the tooth meshing frequency and fs is the shaft rotation frequency. am(t) and bm(t) denote AM and PM functions, respectively. The predominant component of the modulation stems from the shaft rotation frequency and its harmonics; other minute modulation components can be neglected.AM and PM, either individually or in combination,cause the presence of sidebands within the spectrum of the signal. Band-pass filtering around one of the harmonics of the tooth meshing frequency is the classical signal processing for the detailed observation of the sidebands. The filtered gear vibration signal can be expressed as follows
where fh denotes one of the harmonics of the tooth meshing frequency. The subscript m is ignored for simplification in this equation and in the following discussion. The study emphasis of this paper is the filtered gear vibration signal model in equation (7),and its carrier is a single cosine waveform and modulated parts are period functions.
4 CS ANALYSIS OF THE GEAR MODEL
According to the analysis mentioned earlier, the gear vibration signal can be simplified as a periodic signal modulated in amplitude and phase. The modulation condition reflects the severity extent of potential defect in gear. In this section, AM and PM cases are studied individually, and the CS analysis of the gear model is developed on the basis of their results.
4.1 AM case
The model of AM signal is derived from equation (7)
The analytic form of x(t) in equation (8) can be written as
Substitution of ^x(t) into equation (4) can deduce the CA of x?(t)
Where is the envelope of
is equal to as a provider of modulation information.It is the Fourier transform of according to equation (11). In addition, the Fourier transform of with respect to the time lag is the corresponding SCD .thus can be computed using twice Fourier transform of with respect to time t and time lag t,respectively
According to integral transform, becomes
where H(v) is the Fourier transform of a(t)
After substituting H(v) into equation (13) and uncoupling f and a using the properties of d function, the final expression of an be obtained
has a totally symmetrical structure in four quadrants. Equation (15) is just a part of it in the first quadrant, and others are ignored for simplification. According to equation (15), is composed of some discrete peaks. In addition, these peaks regularly distribute on the a–f plane. Despite the comparatively complex expression, the geometrical description of is simple. These peaks nicely superpose the intersections of the cluster of lines. Then, these lines can also be considered as the character lines of .
4.2 PM case
PM signal derived from equation (7) is
The CA of its analytic form can be represented as
The CA in the PM case also has the envelope–carrier form, as in the AM case. Therefore, the envelope of the CA is used to extract modulation information from the signal. Its corresponding SCD is also denoted as .The PM part, b(t), comprises finite Fourier series.The CS analysis of the PM case starts with the sinusoidal waveform .Bessel formulais employed in the computation. The final result of this simple case can be expressed as
The geometrical expression of equation (18) is also related to lines,and is nonzero only at their intersections. The number of the lines does not depend on the number of harmonics in the modulation part, but is infinite in theory even for a single sinusoidal PM signal. In fact, Bessel coefficients limit discrete peaks in a range centring around the zero point of a–f. The amplitude of other theoretical character peaks out of the range is close to zero with the distance far away from the zero point.When the PM function comprises several sinusoidal waveforms as shown in equation (16), components of it can be expressed as bi(t), where i is Application of SSCD to gear defect detection 1387 from 0 to I. The envelope of CA can be written as
Where equals unity. According to the two-dimensional convolution principle, the corresponding SCD ofcan be represented by
where the sign means the two-dimensional convolution on the bi-frequency plane. The expression of is shown in equation (18) with fs replaced byifs and B by Bi and b by bi. Despite more complex expression of the SCD in the multiple sinusoidal modulation case, the result of the two-dimensional convolution between has the same geometrical distribution, as it does in the single sinusoidal modulation case. The distance between the character lines of along the general frequency axis is the fundamental frequency fs. Therefore, convolution does not create new character peaks, but changes their amplitude. Equation (18) also represents the SCD of the signal in equation (16), although the coefficients Cln are changed by the two-dimensional convolution.
4.3 CS analysis of the gear vibration signal
The second-order CS analysis of the general gear model in equation (7) is developed on the basis of the AM and PM cases. The CA of the analytic signal also has the envelope–carrier form, and the envelope of the CA is expressed as follows
Two parts in the sign { .} in equation (21) are relatedto AM and PM functions, respectively. Therefore, the corresponding SCD of has the form of two dimensional convolution of two components issued from AM and PM functions
The expressions of and are given in equations (15) and (18). The two-dimensional convolution between and just causes the superposition of the character peaks in and , as it does in the PM case. Owing to the same geometrical characters, the convolution can not change the distribution, but involves change in
the number and amplitude of the effective character peaks (whose amplitude is larger than zero). Therefore,the CS characters of the gear model are also represented by lines , as it does in the AM and PM cases.
4.4 SSCD analysis of the gear vibration signal and its realization
Three modulation cases have a uniform CS character, according to the above analysis. Lines f = on the bi-frequency plane are their common character lines.Figure 1 shows its distribution.Only the part in the first quadrant is displayed because of the identical symmetry of in four quadrants. The number of these discrete points and the amplitude of the spectrum peaks reflect the modulation extent of the signal.The SCD provides redundant information for gear modulation information identification. In fact, some slices of it are sufficient for the purpose. For the AM case, the slice of , where CF is (in the first quadrant), can be derived from equation (15)
The slice contains equidistant character frequencies,and the distance between them is fs. The PM case and the combination modulation case have the similar result, which can also be expressed by equation (23), whereas the coefficients Cl have different expressions. Therefore, , where is composed of discrete peaks All these character spectrum peaks correspond toodd multiples of the half shaft rotation frequency.The number and amplitude of the peaks reflect the modulation extent, thereby reflecting the severity extent of the potential defect in the gear.Similar situations will be encountered when analysing other Fig. 1 Diagram of CS character distribution slices of the SCD where CF equals the integer multiples of the shaft rotation frequency.The information redundancy of the SCD function always becomes an obstacle to its practical application in the gear defect detection. The sampling frequency must be high enough to satisfy the sampling theorem. Simultaneously, identifying modulation character relies on the fine frequency resolution.Long data series are needed because of these two factors.Therefore, huge matrix operations bring heavy burden to the computation.Moreover, sometimes it is hard to find a clear representation for the redundant information in the three-dimensional space.Therefore, the SSCD, as shown in the above analysis,is presented as a competent substitute for the SCD in detecting gear defects. In this article, the SSCD is specialized to the slice of the SCD where CF equals a certain character frequency. The SSCD can be acquired directly from the time-varying autocorrelation without computing the CA matrix and other subsequent matrix operations. Its realization is detailed as follows:
(a) use the Hilbert transform to get the analytic signal ^x(t);
(b) compute the time-varying autocorrelation of the analytic signal as described in equation (2);
(c) select the CF a0, which equals a certain prescient character frequency, and then compute
the slice of the CA (a0 equals fs for gear defect detection);
(d) compute the envelope of the slice CA . It cannot be attained directly from the slice CA,therefore, a technique is involved for another form of Utilizing the equation, arrive at the squared modulus of ;
(e) apply the Fourier transform of with respect to the time lag t and obtain the final result of the SSCD.The SSCD can be computed according to the steps listed above. Nevertheless, the manipulation of replacing the envelope slice CA by the squared modulus of it will change the spectrumstructure. Original half character frequencies are converted into integer form (lfs) together with the appearance of some inessential high frequency components.These changes do not impact the character identification capability of the SSCD, on the contrary,it gives more obvious representation.
5 SIMULATION
Two modulated signals are used to identify the capability of the SCD and the SSCD in modulation character identification. All modulation functions of these signals are finite Fourier series. Figure 2 shows the AM case simulated according to equation(8). The AM function a(t) comprises three cosine waveforms, representing 10 Hz and its double and triple harmonics and amplitude of 1, 0.7, and 0.3 units, respectively. All initial phases in the model are randomly decided by the computer. The carrier frequency is 100 Hz, sampling frequency 2048 Hz,and the data length 16 384. Figure 3 shows the case of the combination of AM and PM simulated according to equation (7). The PM function b(t) comprises two sinusoidal waveforms with the frequency of 10 and 20 Hz and amplitude of 3 and 1 units,respectively. Other parameters are identical to the AM case.Figures 2(a) to (c) show the time waveform, the contour of its SCD analysis, and the SSCD where CF is equal to 10 Hz, respectively. Only the results of the SCD in the first quadrant are given because of its symmetry. All character points in the contour of the SCD are at the intersections of the lines f =. Their distribution is regular in the AM case. The
Fig. 2 One simulated AM signal: (a) the time waveform, (b) the contour of its SCD, and (c) the SSCD at 10 Hz
SSCD in Fig. 2(c) comprises Fig. 2 One simulated AM signal:
(a) the time waveform, (b) the contour of its SCD, and (c)the SSCD at 10 Hzand its integer multiples and reflects themodulation condition in this signal as the SCD.
Fig. 3 Another simulated modulated signal with modulation phenomena in amplitude and phase: (a) the time waveform, (b) the contour of its SCD, and (c) the SSCD at 10 Hz
Figure 3 shows the case of the combination of AM and PM.All character points in the contour of the SCD are also at the intersections of the character lines 10 Hz. In addition, the SSCD also comprises 10 Hz and its several integer multiples.When PM is involved, the results from the PM part interact with those from the AM part by the two dimensional convolution. The number of the character peaks manifestly increases when compared with the original AM case in the contour of the SCD. The number of character peaks in the SSCD also augments.Therefore, according to the SCD or the SSCD, the same conclusion can be drawn: the second simulated signal is strongly modulated when compared with the first.Simulation results indicate that either the SCD or the SSCD has the capability of identifying the present and the extent of the modulation, disregarding its existence in amplitude or phase. The SSCD possesses the virtues of less computation and clear representation.These two factors seem to be indifferent for simulated signals, but are valuable when encountering very long data series in practice.
6 EXPERIMENTAL RESULTS
Three experimental vibration signals employed in this section came from 37/41 helical gears. They represented healthy, slight wear (wear on addendum of one tooth of 41 teeth gear), and moderate wear status (wear on addendum of one tooth profile of 41 teeth gear and two successive tooth profiles of 37 teeth gear), respectively. The shaft rotation frequency of the 37 teeth gear minutely fluctuates 16.6 Hz. Signals were sampled at 15 400 Hz under the same load. The data length was 37 888. Before the SSCD analysis, all experimental signals were band-pass filtered around four-fold harmonics of the tooth meshin frequency in order to identify the change in themodulation sidebands in different defect status.These filtered signals are analysed by a conventional envelope technique and the SSCD. The comparison between their results dedicates the effect of theSSCD.Figure 4 shows the case of the healthy status.Figures 4(a) to (c) are the time waveform of the experimental signal, its envelope spectrum, and its SSCD analysis at the shaft rotation frequency of the 37 teeth gear, respectively. The envelope spectrum and the SSCD have the similar spectrum structure Fig. 3 Another simulated modulated signal with modulation phenomena in amplitude and phase: (a) the time waveform, (b) the contourof its SCD, and (c) the SSCD at 10 H
Fig. 4 First experimental gear signal: (a) the time waveform, (b) the envelope spectrum, and (c)the SSCD
comprising the rotation frequency and several negligible harmonics. Demodulated sidebands in these two spectra are few and low because there are some modulation phenomena during the gear’s normal operation. The fluctuation in the load, the minute rotational variation, and the circular pitch error in the machining process are the possible sources of the slight modulation. There is no comparability between numeric values of the envelope spectrum and the SSCD because of different computing procedures.
The slight wear case is shown in Fig. 5. Wear on one tooth profile of one of the helical meshing gears does not result in significant deviation from its normal running. Therefore, there is a little increment in amplitude in the time waveform plot. In the envelope spectrum, compared with the normal case, the amplitude of these demodulated sidebands augments a little, and the extent seems to enlarge. The increment in number and amplitude of the sidebands is attributed to the modulation condition of the signal. However, the alteration is too slight to provide enough proof for the existence of some defect in the gear. In fact, a slight defect evidently always modulates the phase of the gear vibration signal and produces little change in the amplitude.Therefore, the envelope spectrum is not sensitiveto a slight gear defect due to its fail to the PMphenomena.Figure 5(c) shows the SSCD analysis of the slightlywearing gear. Mor