Application of the Continuous Wavelet Transform to Bearing Defect Diagnosis

Contributed by the Tribology Division for publication in the ASME JOURNAL OF TRIBOLOGY. Manuscript received by the Tribology Division July 2, 2002 revised manuscript received March 25, 2003. Associate Editor: L. Chang. Among the techniques used for rolling element bearing defect diagnosis, vibration monitoring via envelope analysis is widely used and perhaps the most popular approach for detecting bearing incipient failures. Acoustic emission techniques also show some promising results in detecting bearing defects before they appear on the material surface 1. Envelope analysis provides a mechanism for extracting the amplitude modulation of the structural resonance, which could be excited by the impact due to the contact of bearing defect with other components in the bearing 2. However, this technique suffers from the difficulty of determining the best frequency band for enveloping due to the dynamic nature of the signal pattern. The impulsive patterns are prominent in the onset of the localized defects but become embedded in the wide band signals when defects spread to a larger area of bearing elements. Enveloping using narrow band filters needs constant re-tuning as defects develop into more severe stage, not mentioning the disturbance from the environmental noise. Since recently, the wavelet transform has been explored in the applications involving transient and non-stationary signals 3. In this paper, continuous wavelet transform using the modulated Gaussian wavelet is shown to be effective in highlighting the characteristics pertinent to bearing defect-induced signal patterns. We present two case studies on the experimental data from freight car bearings. Wavelet transform uses basis functions representing small waves with compact support, called wavelets 4. In particular, all basis functions are obtained from their mother wavelet ψt by scaling, that is, ψs=1/sψt/s, where s∈R+ is called the scale factor. For a signal ft∈L2R, wavelet transform of the signal is performed through the convolution of the signal with the scaled and translated mother wavelet: (1)Wfs,t=ft*ψst=∫−∞+∞fuψst−udu.Values of Wfs,t indicate the degree of similarity between the signal and ψst at particular values of s,t. As s decreases, we are looking into finer details of the signal. A large number of members such as Haar, Daubechies, Symlets, Morlet and Gaussian wavelets exist in the wavelet family. They differ from each other in their properties such as orthogonality, vanishing moments, support length, regularity, symmetry etc. For this study, the modulated Gaussian function is chosen as the mother wavelet, that is, (2)ψt=e−σ2t2ej2πf0t.In the above, σ denotes the width of the wavelet and f0 represents the frequency of the damped oscillation of the wavelet. When a localized bearing defect such as spalling or brinelling occurs on the working surface of bearing components, impact is generated each time when the defect is rolled over, resulting in a damped impulse response of the vibration signal. The modulated Gaussian wavelet defined in Eq. (2) resembles closely the shape of a damped impulse response. Therefore, we expect a compact and efficient matching between the wavelet and the defect-induced impulse patterns in the vibration signal. Scaled version of Eq. (2) can be regarded as a series of wavelet filters with different center frequencies and bandwidths. Scale factors for rolling element bearing defect diagnosis are selected with the consideration that bearing defect frequencies are covered by one of the wavelet filters. Adjacent wavelets should overlap to certain degrees so that there is no leakage at the bearing defect frequencies. For the examples studied in this work, all bearing characteristic frequencies are below 250 Hz. Large number of case studies showed that the defect induced signal patterns are most sensitive to scale factors ranging from 0.025 to 1. Therefore, we suggest the following three-step heuristic strategy for choosing the scale factors: decompose the subject signal on scales ranging from s=0.1 to s=0.9 with the increment Δs=0.1, that is, s=0.1,0.2,…,0.9. Search in the decomposed signals to see if there is any match of the dominant frequency component with bearing defect frequencies. If so, conclude on the type of bearing defect based on the bearing defect frequency. Otherwise, go to step 2. further decompose the subject signal on twice finer scales by using the following additional scales: s=0.05,0.15,…,0.95. Search in the additional decomposed signals for a close match of components with defect frequencies. Conclude on the defect type if a match is found. Otherwise, go to step 3. perform one more decomposition of the signal on further finer scales. The additional scales are therefore: s=0.025,0.075,…,0.975. Search again for the defect frequency components. Conclude on the defect type if a match is found. Otherwise, we can conclude that the subject bearing is in good condition. We’ll show in the following section that this strategy works well with the studied examples. In our study, double row tapered rolling element bearings used in freight cars were investigated. Vibration signals were collected via accelerometers mounted on the bearing housing of the test rig in the Transportation Technology Center in Colorado. Experiments were performed using bearings with “seeded” defects and under different speed and load conditions. Bearing outer race ring is fixed stationary on the bearing housing while inner race ring is tightly fit with the rotating shaft of the wheel. Bearing characteristic frequencies include the inner race defect frequency fbpfi, outer race defect frequency fbpfo, cage rotational frequency fc, and the roller rotational frequency around its own axis fbsf. Kinematics model of a ball bearing based on bearing geometry and speed under ideal operating conditions is used to estimate these frequencies 2: (3)fbpfi=Zfo−fi1+dD cos α2(4)fbpfo=Zfo−fi1−dD cos α2(5)fc=fi1−dD cos α2+fo1+dD cos α2(6)fbsf=fo−fi2Dd 1−dD cos2 αIn the above, fi and fo represent the inner race and outer race rotating frequencies respectively; d is the ball diameter and D is the diameter of the center line; α is the operating contact angle and Z is the number of balls. Bearing characteristic frequencies can be used to determine the location of bearing defects. For instance, if a defect has developed on the outer raceway, impact will be generated each time when a roller runs over the damaged area. The period between adjacent impacts is 1/fbpfo. If a defect has developed on the inner raceway, we wish to detect the impulsive patterns with the frequency fbpi and modulated with the inner race rotating frequency fi. For bearings used in this study, relevant geometric data are d=2.1396 cm 0.84235″,D=20.296 cm 7.99057″,Z=23, α=10°. We present the following two cases: Figure 1 shows the picture of bearing with condemnable single spall on the outer raceway. Vibration signal is given in Fig. 2 measured from the bearing at the wheel speed of 35.7632 m/s (80 mph) and under 146.7913 kN (33 klb) axial load. The sampling frequency is 13 kHz. Some periodicity of impulse clusters can be observed from the signal but it is difficult to determine which impulse marks the onset of the damped response from the defect-induced impact. Using the modulated Gaussian wavelet with the scales chosen by the heuristic strategy as described earlier, the signal was decomposed following step 1 and 2. Figure 3 shows the decomposed signal on scales ranging from 0.075 to 0.6. In particular, the decomposed signal on scale s=0.15 was singled out and shown in Fig. 4 since equally spaced impulses can be clearly measured. Period between the two adjacent impulses can be identified by either directly measuring the time period on the decomposed signal or performing a spectrum calculation. Impulse frequency is estimated as 127.5 Hz, which is very close to the roller passing outer race frequency 128.7 Hz calculated by Eq. (4). Discrepancy between the two numbers is due to two factors: (1) the kinematics model used for Eqs. (3–6) is for ball bearings, which is an approximation for the tapered roller bearings used in our case; and (2) bearing operating conditions are never ideal without slip and imperfection. From our practical experience, we allow within 3% difference between the theoretical value and the measurement to make affirmative conclusions. Figure 5 shows a bearing with condemnable multiple cone spalls. Multiple spalling indicates a more severely damaged bearing. Figure 6 shows the vibration signal measured from the bearing at wheel speed of 13.4112 m/s (30 mph) with 146.7913 kN (33 klb) axial load. The sampling frequency was decreased to 1.3 kHz in this case. From the time domain signal, repetitive impacts corresponding to roller passing inner race frequency and the modulation with shaft rotating frequency can be observed. Figure 7 shows the decomposed signal at scale s=0.075. Two dominant components at the frequencies of 57.8 Hz and 4.8 Hz are detected from the decomposed signal. These two numbers are identified to correspond to the theoretical inner race defect frequency 59.4 Hz and inner race rotational frequency 4.7 Hz at the wheel speed. Comparing the decomposed signal (Fig. 7) with the original one (Fig. 6), they both provide the right information for this case but the decomposed signal provides cleaner signal patterns that is most relevant to our interest. It shows that wavelet analysis can denoise a signal without appreciable degradation. In this paper, the continuous wavelet transform is used to treat the bearing vibration signals for defect detection and diagnosis. The modulated Gaussian function was used as the mother wavelet to resemble the pattern of the damped impulse response. Experimental results showed that the continuous wavelet transform can be used to identify the time locations of the impulse clusters generated by the contact of the defect with the mating surfaces in the bearing. Subsequently, diagnostic conclusions can be reached by comparing the time period of the impulse clusters with the bearing defect characteristic frequencies. It also showed that the method is robust to the variation of bearing operating conditions and the damage severity. It can be used for rolling element bearing defect diagnosis. The authors would like to thank the National Research Council of Canada for a previous contract through which the experimental data were provided.