Gearbox Health Monitoring Experiment Using Vibration Signals

Abu-Mahfouz, Issam A

Abstract

Machine deterioration prognosis is one of the most important considerations to reduce the cost of maintenance. This experiment demonstrates the processing of vibration signals for gear fault detection. An artificial defect was introduced to the gearbox as a full tooth breakage (missing tooth). Vibration signals were collected for run-up operation and at some fixed speeds. The kurtosis statistical parameter was used as a measure of impact severity in the time domain of the vibration signal. The fast Fourier transform (FFT) and the wavelet analysis (two widely applied techniques for machine health monitoring) were also used for spur gear fault detection. The kurtosis was found to be higher for the case of faulty gears. Also, the experiment demonstrated that both FFT and wavelet techniques were very sensitive to gear faults and imperfections.

(ProQuest-CSA LLC: … denotes formulae omitted.)

I. Introduction

Gears are primarily used to transmit power or torque between shafts of rotating machinery components while maintaining a certain angular velocity at high efficiency. However, gear tooth and assembly faults might cause the performance of the geared system to deteriorate reducing their efficiency and in some cases leading to machine failure. It is for this reason that gearbox condition monitoring is of significant importance to assure adequate and continuous machine operation. Gear damage can result from faults due to localized gear defects, such as tooth fatigue fracture, tooth flank wear (spalling), pitting fatigue, and backlash. This results in energy waste, noise, impact, and wear. The detection of potential gear failure modes, and the fault diagnosis in gears without disassembly, has become one of the most important research areas. If these defects can be detected and tracked early as they develop, an appropriate maintenance decision can be taken before catastrophic failure occurs.

Vibration based gear fault detection and diagnosis has been the most popular technique because of its rich content of patterns sensitive to gear fault symptoms. Many researchers have investigated the use of time and frequency based techniques for gear damage monitoring. Randall1 discussed the applicability of vibration signal spectrum, synchronous signal averaging, and cepstrum analysis to the monitoring and diagnosis of gearbox faults. Wilson, et al.,2 used the beta kurtosis and the continuous wavelet transform for assessing different gear faults.They concluded that diagnosis, based only on dominant meshing frequency residual, should not be used independently for gear health condition monitoring. Nairn and Andrew3 presented a case history in which they compared the merits of acoustic and vibration signals in detecting growing gear cracks using the smoothed pseudo-Wigner-Ville distribution. Wang and McFadden4 demonstrated the application of the time-frequency based energy distribution for early detection of gear failure. Lin and McFadden5 used the Bspline wavelet-based linear wavelet to monitor the development of a gear crack. Wang and McFadden,6 and Williams and Zalubas,7 used the wavelets and time-frequency, respectively, to monitor vibration signals from a helicopter gearbox. A thorough discussion on gear noise and vibration is presented in Smith.8 This list of studies is not intended as a comprehensive survey for literature in the field of gear health monitoring. This paper presents the application of selected time and frequency based vibration analysis techniques to demonstrate the diagnosis of localized gear faults.

II. Experimental Sot-up

The test rig, shown in Figure 1, consists of a singlestage gearbox with two identical, commercially available spur gears.Table 1 presents design data for the gears. One gear was driven by a 0.25 hp permanent magnet DC variable speed electric motor, while the other gear was connected to an air blower. The gears and bearings were just slightly lubricated to minimize the damping effect of the lubricant and to allow for almost direct contact between the mating teeth faces.A local defect was introduced into the output gear by completely machining out one tooth simulating a broken tooth (missing tooth fault) as shown in Figure 2. The gear with the missing tooth was positioned in such a way that the defected tooth came into gear mesh at an approximately 260° rotation angle with respect to the one-per-revolution proximity sensor position reference signal. The gearbox was tested through a run-up speed range from 175 rpm to 800 rpm, with 25 increments of 25 rpm each. This speed range was adequate for this experiment as it passes through two natural mode frequencies of the test rig, i.e., 95 Hz and 400Hz.

The vibration signal generated by the gears was measured by a PCB accelerometer (mod-el 607A61, with sensitivity of 95 mV/g at 100 Hz) mounted on the bearing housing nearest to the faulty gear (Figure 2). The accelerometer and the position reference signals were sampled at 10 Ksps and then recorded via a National Instruments DAQPAD-6070E connected via FireWire OEEEl 394) PC card to a Pentium IV computer (1.2 GHz with 256 MB RAM). The raw vibration data was averaged over 100 gear rotations. This process is known as synchronous averaging and it filters out vibrations that were not synchronized with gear rotation. The analysis in the following section was performed on these time-averaged segments using the MATLAB environment.9

III. Signal Analysis

Often interactions of vibration-based fault conditions are highly nonlinear and their temporal trends behave as a time-varying series.Time- and frequency-based methods, which can lead to clear identification of the nature of faults, are widely used to describe machine conditions. The following three analysis methods were selected for identifying the missing tooth fault.

Kurtosis (K) is a measure of peakedness, but can also be used to describe how outlier-prone a distribution is. The kurtosis of the normal distribution is zero. Distributions that are more outlier-prone than the normal distribution have kurtosis greater than three. Distributions that are less outlier-prone have kurtosis less than three.The kurtosis of a distribution is the fourth statistical moment and is defined as:

… (2)

where … is the average vibration amplitude or mean;

…

x^sub i^ is the instantaneous amplitude of the vibration signal and N is the number of vibration data points within one full cycle of gear rotation. The variance σ^sup 2^ of the data segment is the mean square value about the mean defined as follows:

… (3)

The kurtosis K is rather sensitive to the occurrence of spikes or impulses in the time domain of the vibration signal.

Fast Fourier Transform (FFT): Simply stated, the Fourier method is still the most powerful technique for signal analysis. It transforms the signal from the time domain to the frequency domain usually referred to as the spectral domain. The Fourier transform of the function x(t) is defined via the integral

… (4)

The MATLAB function fft was used to implement the FFT algorithm on the gearbox vibration signals.

Continuous Wavelet Transform (CWT):This is one of many time-frequency domain analysis techniques currently available for the detection of gear faults. Wavelets10 are mathematical functions, which are well suited to the expansion of non-stationary signals. For sake of completeness, only a brief discussion of the theory is presented here.The continuous wavelet transform of a signal x(t) is defined as the convolution integral of xft) with a translated and dilated versions of a mother wavelet function ψ. Any waveform, which satisfies some conditions,11 may be used for the wavelet ψ. After experimenting with several wavelets, it has been found that the Mexican hat wavelet gave acceptable results. The Mexican hat function is proportional to the second derivative function of the Gaussian probability density function and is defined as

… (5)

The CWT coefficients represent how closely correlated the wavelet is with the original signal x(t).The higher the value of the coefficient, the more the similarity. The CWT of a signal x(t) is the family of Coefficients C(a,b), which depends on the wavelet position b and its scale a.

… (6)

Small scaling parameter values result in narrow windows and serve for precisely localized registration of high frequency phenomena. On the other hand, large scaling parameter values result in wide windows and serve for the registration of slow phenomena.

IV. Results

Extensive experimentation and signal analysis were carried out to study the vibration patterns of the gearbox with both healthy and faulty gears. The results of implementing the kurtosis measure, the FFT, and the CWT are presented and discussed as follows.

The kurtosis statistical moment was calculated using Equation (1) for the vibration signal during run-up speed variation from 175 rpm to 800 rpm in 25 equal speed increments. For each speed the vibration signal was collected for the duration of 200 gear rotations and the kurtosis statistical moment was calculated for the complete segment. The results were presented in Figure 3 for two cases of gear loading, (a) gearbox driving the blower, and (b) gearbox running free. It can be clearly seen that the kurtosis of the faulty gear was higher than that of the healthy gear for most of the speed range. The missing tooth caused higher impacts at the fault location. These impacts generated pulsating transients that propagated within the transmission system resulting in more disturbances. The impacts and their transients were indicated by the higher values of the K moment.

The FFT plots were delineated in Figure 4 as waterfall diagrams.These plots show the FFT for each speed during run-up operation. Two traces of the meshing frequency (IX) and its second multiple (2X) can be clearly observed. The meshing frequency was calculated as the number of teeth (38) multiplied by the running speed in Hz. For example, spikes appear close to frequencies ranging from 110 Hz to 500 Hz corresponding to speeds increasing from 175 rpm to 800 rpm. It was also clearly noticed that, even for the case of healthy gear, many spikes and sidebands, which were not necessarily multiples of the meshing frequency, were excited. Two of such resonant peaks were fixed close to 100 Hz and 400 Hz, and were related to the test rig resonant modes. A fixed peak just below 300Hz was more distinct in the case of the missing tooth. Other frequency sidebands might be due to a combination of tooth surface imperfections, gear run-out, and gear roundness or misalignment errors. Nevertheless, the main observation was that the faulty gear has higher amplitude harmonics and excite, due to periodic impacts at the missing tooth, broader range of spectra compared to the healthy gearbox.The FFT amplitudes increased noticeably as the speed increased. The contribution of the first meshing frequency (IX) grew significantly, while the second meshing frequency (2X) increased rather slowly.

The continuous wavelet transform (CWT) provided an alternative signal representation with a flexible time-frequency resolution. Contour plots of the CWT results utilizing the Mexican hat were shown in Figure 5.The wavelet scalograms at two speeds (625 rpm and 800 rpm) were presented for 128 scales. Although the wavelets were presented here as a timescale domain, they can be interpreted as a frequency-time domain using the relationship presented in Blatter.10 The time axis is presented in terms of sampling steps instead of seconds. The correlation between gear rotation angle and the number of time steps can be readily deduced from the sampling rate of 10,000 samples/second and the gear speed in rpm. For example, at 625 rpm one gear cycle corresponds to 960 time steps, and similarly, at 800 rpm one gear cycle corresponds to 750 time steps.The wavelet images clearly presented the fractal nature (self similarity) of the gearbox vibration signals through the repetitive bands of coefficients at different instants in time.This similarity was a result of repeated impacts and meshing imperfections of a similar nature on the machined surface roughness of the tooth face. Both rapidly changing features (small scale) and slowly changing features (large scale) of the vibration signals were influenced by gear condition and running speed. High-frequency (low scale) regions represented the impacts of subsequent meshing teeth. Overall, the faulty gear (Figure 5(b)) showed a higher energy content (denser contour lines and darker regions) than the healthy gear. For the healthy gear signal, the vibrational energy was uniformly distributed.Whereas for the case of a missing tooth, some energy concentrations around the 260° gear rotation position (where the missing tooth comes into mesh) were clearly revealed by the presence of abrupt energy changes.

V. laboratory Experiment

The work explained in this paper is implemented as part of a more comprehensive laboratory in the ME458 Noise Control in Machinery course at PSU Harrisburg. The experiment duration is two hours including set up, data acquisition, signal processing and result collection as shown in the Figures 3, 4, and 5. The students are then asked to write a formal report and discuss their observations. The experiment enabled the students to relate machine parameters such as operating speed, meshing frequency, and fault location to other monitoring system parameters such as sampling rate and extracted vibration features in both the time and frequency domains. The experiment is designed to accomplish the following objective:

1. Investigate the use of various signal analysis techniques in fault detection and diagnosis.

2. Effective implantation of computer based data acquisition system to extract essential vibration features using available hardware and software.

3. Use of instrumentation and transducers in a practical example on non-destructive testing.

Future expansion will include the use of these techniques to detect other faults, such as localized pitting, misalignment, and eccentricity.

VI. Conclusion

In this experiment, gearbox vibration signals were analyzed to obtain diagnostic information for identifying the missing tooth fault. The statistical kurtosis measure successfully showed higher values for the faulty gear during a run-up test. Both the FFT and the CWT were able to predict distinctive patterns for gear fault and were very sensitive to gearbox speed. The FFT waterfall diagrams revealed that impacts due to the missing tooth excited broader range of spectral energy at higher speeds.The CWT provided a rich visual tool by displaying symptoms of the missing tooth as abrupt amplitude variations at the missing tooth location.This experimental setup, and the adopted analysis techniques, proved to be very useful in demonstrating current trends in machine health monitoring for maintenance technology for engineering and engineering technology students.

References

1. Randall, R. B., 1982, “A New Method of Modeling Gear Faults.” ASME Journal of Mechanical Design, April 1982, Vol. 104, pp. 259-267.

2. Wilson, Q. W., Fathy, I., and M. Farid, G., 2001, “Assessment of Gear Damage Monitoring Techniques Using Vibration Measurements.” Mechanical Systems and Signal Processing, Vol. 15(5), pp. 905-922.

3. Nairn, B., and Andrew, B., 2001, “A Comparative study of Acoustic and Vibration Signals in Detection of Gear Failure Using Wigner-Ville Distribution.” Mechanical Systems and Signal Processtng, Vol. 15 (6), pp. 1091-1107

4. Wang, W. J., and McFadden, P. D., 1993, “Early Detection of Gear Failure By Vibration Analysis-1. Calculation of The Time-Frequency Distribution.” Mechanical Systems and Signal Processing, Vol. 7(3), pp. 193-203.

5. Lin, S. T, and McFadden, P. D., 1997, “Gear Vibration Analysis by B-Spline Wavelet-Based Linear Wavelet Transform.” Mechanical Systems and Signal Processing, Vol. 11(4), pp. 603-609.

6. Wang, W. J., and McFadden, P. D., 1996, “Application of Wavelets to Gearbox Vibration Signals for Fault Detection.” Journal of Sound and Vibration, Vol. 192(5), pp. 927-939.

7. Williams, W. J., and Zalubas, E. J., 2000, “Helicopter Transmission Fault Detection Via Time-Frequency, Scale, and Spectral Methods.” Mechanical Systems and Signal Processing, Vol. 14(4), pp. 545-559.

8. Smith, J. D., 1999, Gear Noise and Vibration. Marcel Dekker, Inc. New York, NY.

9. MATLAB, The Math Works, Inc., Natick, MA.

10. Blatter, C., 1998, “Wavelets: A Primer.” A K Peters, Ltd., Natick, Massachusetts.

11. CH. K. Chui, 1992, “An Introduction to Wavelets.” In series: Wavelet Analysis and its applications, Vol. I. Boston, MA: Academic Press.

Issam A. Abu-Mahfouz

Issam A. Abu-Mahfouz obtained his B.E. and M.Sc. in Mechanical Engineering from Kuwait University in 1986 and 1989, respectively. His Ph.D. degree in the field of Machinery Vibrations and Chaos was received from case Western Reserve University, Cleveland, Ohio, USA in 1993· He joined the Cleveland Trencher Company as a project engineer until 1995, and then with Allied Machine and Engineering Corp., Dover, Ohio, as a research and development engineer for the next four years. During this period Abu-Mahfouz was also an Adjunct Professor in Mechanical Engineering Technology at Kent State University, NewPhiladelphia, Ohio. Presently he is Associate Professor of Mechanical Engineering at Penn State University, Harrisburg, PA. His interests include Chaotic Vibrations, metal cutting, neural networks and fuzzy logic for machine health monitoring and manufacturing process diagnostics.

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