Abstract Cyclostationarity matches the rhythmic feature of gearbox vibration and thus provides an effective alternative for gearbox condition monitoring. This paper concerns with the applications and the effectiveness of cyclostationarity from the first order to the third order in gearbox condition monitoring. To this objective, basic theories on cyclostationarity analysis for signal feature representation are reviewed. A fatigue test is carried out on a gearbox to obtain the gearbox signals of running-in stage, normal wearing stage and ultimate wearing stage. Based on the first-order cyclostationarity, the double-time synchronous averaging and the first-order cyclostationary spectrum is proposed for feature representation; the results of these approaches demonstrate their effectiveness in showing the gearbox condition degradation and even the gear structural fault. The spectrum of the second-order cyclic cumulant shows the degradation of the gearbox condition through the interactions between cyclic frequency and frequency. The cyclic bispectrum based on third-order cyclostationarity represents the gearbox condition degradation effectively through the interactions between frequencies. All the conclusions verify cyclostationarity analysis as an effective tool for gearbox condition monitoring. r 2004 Published by Elsevier Ltd.
ARTICLE IN PRESS Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 19 (2005) 467-482 www.elsevier.com/locate/jnlabr/ymssp Cyclostationarity analysis for gearbox condition monitoring: Approaches and effectiveness Z.K. Zhu*, Z.H. Feng, F.R. Kong Department of Precision Machinery and Instrumentation, University of Science and Technology of China, Hefei, Anhui 230027, People's Republic of China Received 31 March 2003; received in revised form 19 November 2003; accepted 29 February 2004 Abstract Cyclostationarity matches the rhythmic feature of gearbox vibration and thus provides an effective alternative for gearbox condition monitoring. This paper concerns with the applications and the effectiveness of cyclostationarity from the first order to the third order in gearbox condition monitoring. To this objective, basic theories on cyclostationarity analysis for signal feature representation are reviewed. A fatigue test is carried out on a gearbox to obtain the gearbox signals of running-in stage, normal wearing stage and ultimate wearing stage. Based on the first- order cyclostationarity, the double-time synchronous averaging and the first-order cyclostationary spectrum is proposed for feature representation; the results of these approaches demonstrate their effectiveness in showing the gearbox condition degradation and even the gear structural fault. The spectrum of the second-order cyclic cumulant shows the degradation of the gearbox condition through the interactions between cyclic frequency and frequency. The cyclic bispectrum based on third-order cyclostationarity represents the gearbox condition degradation effectively through the interactions between frequencies. All the conclusions verify cyclostationarity analysis as an effective tool for gearbox condition monitoring. r 2004 Published by Elsevier Ltd. 1. Introduction The detection and understanding of condition degradation in the gearbox is important for gearbox design and maintenance. For gearbox condition monitoring, the processing and analysis of the measured signal is the common way of extracting reliable feature representative of the gearbox condition. Cyclostationarity analysis is a relatively new area of signal processing [1-3]. *Corresponding author. E-mail address: zzkustc@yahoo.com.cn (Z.K. Zhu). 0888-3270/$ - see front matter r 2004 Published by Elsevier Ltd. doi:10.1016/j.ymssp.2004.02.007 ARTICLE IN PRESS 468 Z.K. Zhu et al. / Mechanical Systems and Signal Processing 19 (2005) 467-482 Because of its attractive properties, cyclostationarity matches the key feature of the gearbox vibration and shows its feature in a natural way as the gearbox condition goes. The gearbox vibrations arisen by the teeth meshing, the transient fault and the gearbox resonance are all characterised by rhythmic variations. These characteristics are the justifications for adopting the cyclostationarity analysis for gearbox condition monitoring. Recent research on cyclostationarity for condition and fault diagnosis mainly focuses on the second- and the third-order cyclostationarity [4-10]; however, the first-order cyclostationarity has always been restricted to the synchronous averaging, which is rather rigid in feature representation. This paper explores the cyclostationarity of the gearbox vibration and the applications of the cyclostationarity through the third order in gearbox feature representation and condition monitoring, as well as the effectiveness and ?exibility of these approaches. The outline of this paper is arranged as follows. Section 1, the introduction; Section 2, the basic theories on cyclostationarity and their applications in feature representation; Section 3, the test and the demonstration of the cyclostationarity of the gearbox vibration; Section 4, the proposed first-order approaches for gearbox condition monitoring; Section 5, two approaches based on the second-order cyclostationarity for gearbox condition monitoring; Section 6, the cyclic bispectrum of cumulant for gearbox condition monitoring; Section 7, the conclusions. 2. Cyclostationarity and its applications in feature representation The theories of cyclostationarity analysis were introduced in [11], and extensively studied in [1,2], and the applications of this theory include communication, sonar, radar, electrocardiograms and telemetry and so on. The applications of cyclostationarity in mechanical feature representation are proposed in [12-14], and are intensively studied for mechanical fault diagnosis and condition monitoring. Cyclostationarity analysis is a forceful tool for condition monitoring because it matches the characteristic of the rotating machinery, which is characterised by seasonal and rhythmic variations when working. 2.1. Basic parameters about cyclostationary process Consider a cyclostationary signal xðtÞ; its basic parameters are its cyclic moment M a ðtÞ and its kx cyclic cumulant C a ðtÞ; where t ¼ ðt 1 ; y; t n À 1 Þ: For the signal xðtÞ for ÀNotoN; its kth-order kx cyclic moment M a ðtÞ is defined to be the sine wave extracting operation of the kth-order kx lag-product over the t: M a ðt 1 ; t 2 ; y; t k À 1 Þ ¼ /xðtÞxðt þ t 1 Þ?xðt þ t k À 1 Þe À j2pa t S t ; kx where, a is known as the cyclic frequency. ð1 Þ The demonstration of the cyclostationarity of the signal is based on its cyclic moment. If there exists one aa0 that satisfies M a ðt 1 ; t 2 ; y; t k À 1 Þa0; this signal is kth-order cyclostationary. kx From Eq. (1), it is obvious that the cyclic moment through the third order can be written as M a x ¼ /xðtÞe À j2pa t S t ; 1 ð2 Þ ARTICLE IN PRESS Z.K. Zhu et al. / Mechanical Systems and Signal Processing 19 (2005) 467-482 M a xðtÞ ¼ /xðtÞxðt þ tÞe À j2pa t S t ; 2 469 ð3 Þ M ð 4 T he cy cli c cu m ula nt is co m pu ted thr ou gh the cy cli c m o me nt ac co rdi ng to the fol lo wing f o r m u l a : X " X Y a p q # C a ðt 1 ; y; t k À 1 Þ ¼ S kx q p¼ 1 I p ¼ I ðÀ1Þ q ðq À 1Þ! a 1 þ?þa q ¼a p¼1 M n p ; x ðt I pÞ : ð 5 Þ From the above formula, we can see that the cumulant through the third order can be written based on the lower-order cyclic moments: C a x ¼ M a x ; ð 1 1 C a xðtÞ ¼ M a xðtÞ À M a xM a xðtÞ; ð7Þ 2 2 1 1 C a xðt 1 ; t 2 Þ ¼ M a xðt 1 ; t 2 Þ À M a x½M a xðt 1 Þ þ M a xðt 2 Þ þ M a xðt 2 À t 1 ÞŠ þ 2ðM a xÞ 2 : ð8Þ 3 3 1 2 1 1 1 2.2. Potential applications of cyclostationary features in condition monitoring 2.2.1. The first-order cyclostationarity based feature detection The application of the first order is actually the synchronous averaging operation for cyclic period T ¼ 1=a; which is equivalent to mean function for cyclic frequency a: 1 N À1 xðt þ kTÞ: X m Þ ¼ N N k ð9Þ In the computing of synchronous averaging, the cyclic period T of the signal should be known in advance. However, this is not easy when the signal feature is not clear. What is more, only one certain cyclic frequency can be analysed one time, so this operation is not capable of detecting the signal feature. To detect the cyclic frequency, we propose a novel operation called double-time synchronous averaging (DTSA) defined to be 1 N À1 xðt þ kTÞ: X DTSAðt; TÞ ¼ N lim N -N k ¼ ð 1 0 T he D TS A is the ge ne ral isa tio n of sy nc hr on ou s av era gi ng be ca us e T be co me s a va ria ble he re. This generalisation makes the DTSA capable of detecting the cyclic period of the signal because the DTSA represents the signal first-order cyclic moment with respect to all the cyclic time. To represent the signal first-order feature for cyclic frequency, we defined the first-order cyclic spectra (FCS) as the average operation of DTSA over delay time t: FCSðTÞ ¼ /DTSAðt; TÞS t : ð11Þ Through the DTSA and the FCS, even though the angular position of the signal is not known, the first-order cyclostationarity feature in the signal can be detected. ARTICLE IN PRESS 470 Z.K. Zhu et al. / Mechanical Systems and Signal Processing 19 (2005) 467-482 The advantage of the DTSA over the conventional synchronous averaging is that the DTSA needs no information about the angular position of the signal, which simplifies the test hardware greatly. Moreover, the DTSA is capable of detecting the cyclostationary feature of the signal, while the synchronous averaging can only show the cyclostationary feature for a certain cyclic frequency. 2.2.2. The second-order cyclostationary characteristics for feature representation The cyclic moment M a xðtÞ is capable of illustrating the feature of the signal. The spectrum 2 correlation density function (SCDF), the Fourier transform of M a ðtÞ is also effective in detecting 2 the feature of the signal. The SCDF measures the correlation between frequency lines centered on f and separated by shift of 7a: There are other functions for estimating the cyclostationarity feature based on the second-order cyclic moment. The most effective one is the degree of cyclostationarity (DCS) proposed in [12]. The DCS is defined to be Zfi DCS a ¼ fiM a xðtÞfifi 2 dt 2 Z fi fiM 0 xðtÞfifi 2 dt: 2 ð12 Þ Both the second-order cyclic moment, the SCDF and the DCS have been intensively studied for signal feature representation [5,12,13]. The other category of feature representation of the second-order cyclostationarity is based on the second- order cumulant, which includes the second-order cyclic cumulant and its spectrum. The spectrum of the second-order cyclic cumulant is defined to be the Fourier transform of C a xðtÞ 2 with respect to t: ZN C x a ð f Þ ¼ ÀN C a xðtÞe À j2p ft dt: 2 ð13 Þ Analogous to the SCDF, the spectrum of the second-order cyclic cumulant measures the second-order cumulant in frequency domain. In this paper, because the signal of interest is cyclostationary of both the first order and the second order, to purify the nth-order impure sine wave, we must extract all the impure terms of lower order. The second-order cumulant is obligatory to represent the signal feature of the second order. It is obvious that the second-order cyclic cumulant can be obtained by subtracting the first- order moment from the second-order cyclic moment. 2.2.3. The third-order cyclostationary characteristics for feature representation The cyclic bispectrum of moments and the cyclic bispectrum of cumulants are the practical application of the third-order cyclostationarity. The cyclic bispectrum of moment is defined to be the two-dimension Fourier transform of the third-order cyclic moment of the signal: M a xðo 1 ; o 2 Þ ¼ 3 XN XN M a xðt 1 ; t 2 Þe À jðt 1 o 1 þt 2 o 2 Þ : 3 ð14 Þ t 1 ¼ÀN t 1 ¼ÀN The cyclic bispectrum of cumulant is defined to be the two-dimensional Fourier transform of the third-order cyclic cumulant of the signal: C a xðo 1 ; o 2 Þ ¼ 3 XN XN C a xðt 1 ; t 2 Þe À jðt 1 o 1 þt 2 o 2 Þ : 3 ð15 Þ t 1 ¼ÀN t 1 ¼ÀN ARTICLE IN PRESS Z.K. Zhu et al. / Mechanical Systems and Signal Processing 19 (2005) 467-482 471 For the third-order cyclic features of noise component equal to zero, the cyclic features based on the third- order cyclostationarity has a natural tolerance to the stationary and non-stationary Gaussian noise that may be corrupting the signal of interest [1,2]. The third-order features exhibit the property of signal selectivity, which means that they are capable of estimating the feature of specific signal components, especially for the signal corrupted by noise. 3. The test on the gearbox and the cyclostationarity demonstration 3.1. The test rig To study the condition development in the gearbox lively, our experiment concerns with a fatigue test of an automobile transmission gearbox, which have 5 forward speeds and one backward speed. The structure of the gearbox is shown in Fig. 1. The vibration signal was acquired by an accelerometer mounted on the outer case of the gearbox, when it was loaded in the third speed. Fig. 2 shows the location of the accelerometer and the photo of the gearbox. We recorded the vibration signal from the beginning to the end of the test until the beginning of the cycle 7, when a tooth of the driving gear is broken. No distinct fault occurred until the tooth is broken, so it is reasonable that cycle 1 through cycle 6 correspond to a complete wearing process. Cycles 1 and 6 correspond to the running-in stage and the ultimate running stage; cycles 2- 5 correspond to the normal wearing stage. Table 1 gives the relation between test time and gearbox condition. For a gear transmission, the meshing frequency f m is calculated by f m ¼ nz=ð60iÞ; where, z is the number of gear teeth, n the rotating speed of the input shaft and i the transmission ratio. In this test, z ¼ 27; n ¼ 1600 rpm and i ¼ 1:44: It follows from the equation that the meshing frequency of the third speed gears is 500 Hz: The sampling frequency is set to be 1500 Hz: At this frequency, the working parameters of the third speed are shown in Table 2. input shaft third speed first speed fifth speed output shaft counter shaft forth speed second speed reverse speed Fig. 1. Schematic diagram of the structure of the gearbox. ARTICLE IN PRESS 472 Z.K. Zhu et al. / Mechanical Systems and Signal Processing 19 (2005) 467-482 Fig. 2. The gearbox and the sensor locations. Table 1 Relations between the gearbox condition and experiment cycle Cycle no. Cycle 1 Cycles 2-5 Cycle 6 Wearing stage Running-in stage Normal wearing stage Ultimate wearing stage Meshing times (thousand) 0-700 700-3500 3500-4200 Table 2 Working parameters of the third speed gears The third speed gears Constant meshing gears Driving gear Driven gear Driving gear Driven gear Number of teeth 25 27 24 32 Rotating period (s) 0.05 0.054 0.04 0.03 Rotating frequency (Hz) 20 18.5 25 33.3 Sampling points per cycle 75 81 60 45 Meshing frequency 500 640 Meshing period (s) 0.002 0.0156 Sampling points per meshing 3 2.34 The typical vibration signals of each stage are shown in Fig. 3. The time domain signal fails to demonstrate other characteristic feature of the gearbox vibration except the increase of the signal amplitude. 3.2. Theoretical analysis on the cyclostationarity of the gearbox vibration The vibrations of the gearbox are mainly produced by the shock between the teeth of the two meshing gears, and this vibration is filtered by the structure of the machine between the source of the vibrations and the measurement point. This filtration is supposed to be linear so that the model of our measured signal can be regarded as the model of meshing signal. The transmission . xðtÞ; ð7Þ 2 2 1 1 C a xðt 1 ; t 2 Þ ¼ M a xðt 1 ; t 2 Þ À M a x½M a xðt 1 Þ þ M a xðt 2 Þ þ M a xðt 2 À t 1 ÞŠ þ 2 M a xÞ 2 : ð8Þ 3 3 1 2 1 1 1 2. 2. Potential. i XN n¼ÀN ð2pnf i Þ 2 a n ; i e 2 pjnf i t þjf i 22 Þ and one random component of zero mean a i ; r ðtÞ ¼ ÀC i XN n¼ÀN ð2pnf i Þ 2 a n ; i e 2 pjnf i tþjf