FAULT DIAGNOSIS OF GEARBOX BASED ON MATCHING PURSUIT

ZHI-PENG FENG1, JIN ZHANG2, RU-JIANG HAO3, MING J. ZUO4, FU-LEI CHU2

1Institute of Vehicular Engineering, University of Science and Technology Beijing, Beijing 100083, China 2Department of Precision Instruments, Tsinghua University, Beijing 100084, China

3Department of Mechanical Engineering, Shijiazhuang Railway Institute, Shijiazhuang 050043, China 4Department of Mechanical Engineering, University of Alberta, Edmonton T6G 2G8, Alberta, Canada

E-MAIL: zhipeng.feng@yahoo.com.cn

Abstract: Matching pursuit is effective in matching the characteristic

structure of signals and extracting the time-frequency features directly. It is employed to analyze the vibration signals of a gearbox under healthy and faulty statuses. Based on a compound dictionary, the periodic impulses characterizing the vibration of localized damaged gears are extracted in joint time-frequency domain, and the localized gear damage is detected and located. The analysis validates the effectiveness of matching pursuit in detecting and locating localized gear damage.

Keywords: Matching pursuit; Time-frequency analysis; Wavelet; Gear;

Fault diagnosis

1. Introduction

Gears are widely used in many kinds of machinery to transmit power and rotation. The smooth operation and high efficiency of gears is necessary for the normal running of machinery. Therefore, fault diagnosis of gears is a main topic in the field of machinery condition monitoring and fault diagnosis.

Due to the intrinsic dynamic characteristics and complicated ambient excitations, the on-site measured vibration signals of gearboxes have complexity and non-stationary, for example: the amplitude modulation and frequency modulation phenomena inherent with gear pair transmission; the multiple signal components like gear rotating frequency, gear pair meshing frequency, and their harmonics; and especially the transient impulses induced by gear damage. How to effectively extract the interested constituting components from complicated signals is an important issue for gearbox fault diagnosis.

Matching pursuit [1, 2] has unique advantages in analyzing such complicated and non-stationary signals. With a specially designed waveform dictionary adapted to the local structure of signals, matching pursuit can extract different

features such as impulses, harmonic oscillations, and modulation phenomena, etc. So far, matching pursuit has been studied in the machine diagnosis community. Liu et al [3] employed matching pursuit with a time-frequency dictionary to detect the localized defects of rolling element bearings. Fan et al [4] improved matching pursuit, and extracted impulses from the vibration signals of rotating machines so as to detect fault. The above researches show that matching pursuit has potential in analyzing the complicated vibration signals of gearboxes.

A large percentage of gear faults are induced by localized gear damage like pits, chips, and cracks on gear tooth surface. Such damage usually generate periodic impulses in the vibration signals during the running of damaged gears, with the period of impulse train depending on the number of damaged teeth and their distribution over the gear. In a word, periodic impulses characterize the vibration of damaged gears, and provide an intuitively understandable indicator of localized damage. So how to extract impulses from vibration signals is of great importance for gear damage detection and location.

In this paper, matching pursuit is used to analyze the vibration signals and recognize the tooth damage of a gearbox. In section 2, the basics on dictionary and matching pursuit are briefly introduced. In section 3, the vibration signals of a gearbox under healthy and faulty statuses are analyzed. With a specially designed compound dictionary, the periodic impulses characteristic of localized gear damage in the vibration signals are extracted by means of matching pursuit. From the time-frequency plot, the gear tooth damage is easily detected and located. Finally, some conclusions are drawn in section 4.

2. Matching pursuit with dictionary

2.1. Signal representation in dictionary

Any signal s(t) can be represented as a superposition of

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Proceedings of the 2010 International Conference on Wavelet Analysis and Pattern Recognition, Qingdao, 11-14 July 2010

elementary waveforms g�(t) ( ) ( )s t A g tγ γ

γ∈Γ

=� , (1)

or in an approximation form

1

( ) ( ) ( )i i

m

mi

s t A g t r tγ γ=

= +� , (2)

where A is a weight coefficient, � is a parameter associated with elementary waveforms, � is a parameter collection of elementary waveforms, and rm(t) is a residue.

The dictionary is a collection of parametric waveforms (a library of functions). Usually, the dictionary used in matching pursuit is over complete and redundant, i.e. the number of waveforms in the dictionary is larger than the length of the signal to be analyzed, and some elements in the dictionary can be represented in terms of other ones. Any parametric waveform in the dictionary is called an atom.

Arbitrary signals can be reconstructed by superposing a series of atoms associated with physical meaning, and the characteristics of signals can be interpreted in terms of the properties of atoms. For example, in the time-frequency analysis of signals, the energy, location and variation of signal components is represented by the building block associated with the constructing atom.

In order to match the local structure of signals, the dictionary must be carefully designed adapting to the signal properties. The Dirac dictionary is a collection of Dirac functions, which is suitable to analyze impulses. The Fourier dictionary is a collection of sinusoidal waveforms with their frequencies sampled more finely, and is effective in analyzing harmonic oscillations. The wavelet dictionary is a collection of translations and dilations of a basic mother wavelet, together with translations of a father wavelet, and it is efficient in analyzing signals with constant proportional bandwidth. Wavelet packets is a time-frequency dictionary frequently used, which includes a standard orthogonal wavelet dictionary, Dirac dictionary, and a collection of waveforms spanning over a range of bandwidths and durations.

2.2. Matching pursuit

Matching pursuit [1, 2] is a stepwise greedy approximation algorithm. Starting from a null initial model, it iteratively builds up an approximation to a given signal s(t) by adjoining at each stage an atom gi(t) which best correlates with the current residue si(t)

10

( ) ( ) ( )m

i i ii

s t A g t s t+=

= +� , (3)

where the signal residue

1( ) ( ) ( )i i i is t s t A g t+ = − . (4) When iteration index i=0, s0(t)=s(t). The weight Ai is the inner product between the signal residue si(t) and the atom gi(t)

( ) ( )i i iA s t g t dt∞ ∗

−∞= � (5)

In each iteration, matching pursuit selects from the given dictionary the atom that best minimizes the signal residue

2 2

1( ) ( )min ( ) min ( ) ( )

i ii i i ig t g t

s t s t A g t+ = − . (6)

This minimization problem is equivalent to a maximization problem

2

2

( ) ( )max max ( ) ( )

i ii i ig t g t

A s t g t dt∞ ∗

−∞= � . (7)

Matching pursuit is essentially a nonlinear optimization problem without analytical solution. It can be solved by means of some optimization routines, such as zooming algorithm, Newton-Raphson, and genetic algorithm.

3. Gearbox vibration signal analysis

3.1. Specification of gearbox experiment

The gearbox fault experiments are conducted on a SpectraQuest test rig as shown in Figure 1. The motor drives the gearbox through the power train which is composed of shaft 1, a convey belt (transmission ratio 2.5), and shaft 2 in sequence. The parameters of the drive gear 1 and driven gear 2 are listed in Table 1.

TABLE 1. GEARBOX PARAMETERS

Gear 1 Gear 2 Number of gear teeth 18 27

Rotating frequency (Hz) 6.667 4.444 Meshing frequency (Hz) 120

Figure 1 Schematic diagram of gearbox test rig

An accelerometer is mounted on top of the gearbox casing. During the experiment, the motor runs at a stable

Motor Bearing BearingShaft 1

Shaft 2

Bearing

Belt

Gear 1

Gear 2

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Proceedings of the 2010 International Conference on Wavelet Analysis and Pattern Recognition, Qingdao, 11-14 July 2010

speed of 1000 rpm. The vibration signals are collected at a sampling frequency of 6400 Hz.

Two kind of statuses of the gearbox are simulated. Under the normal status, both gear 1 and gear 2 are perfect. While under the faulty status, one tooth of gear 1 is broken (see Figure 2), whereas gear 2 is perfect.

Figure 2 Tooth breakage of gear 1

3.2. Compound dictionary

In order to match the characteristic components in gearbox vibration signals using matching pursuit, a compound time-frequency dictionary is constructed. It consists of the Fourier dictionary, Dirac dictionary, standard orthogonal wavelets dictionary (based on Symmlets of vanishing moment 4), and a collection of the waveforms spanning over a range of bandwidths and durations, which are used to pursue the harmonic oscillations, impulses, and transient phenomena in gear vibration respectively. Figure 3 shows the time-frequency distribution of some typical atoms. In Figure 3 and the following plots, the signal waveform is shown on the top, its power spectrum density at left, and a grey bar denoting the time-frequency distribution amplitude at right.

Sig

nal

Power spectrum 100 200 300 400 5000

0.1

0.2

0.3

0.4

Fourier atom

Time [s]

Fre

quen

cy [H

z]

(a) Fourier atom

Sig

nal

Power spectrum 100 200 300 400 5000

0.1

0.2

0.3

0.4

Dirac atom

Time [s]

Fre

quen

cy [H

z]

(b) Dirac atom

Sig

nal

Power spectrum

Wavelet atom

Time [s]

Fre

quen

cy [H

z]

100 200 300 400 5000

0.1

0.2

0.3

0.4

(c) Wavelet atom

Figure 3. Time-frequency plots of typical atoms

3.3. Time-frequency analysis

The gearbox signals under the normal and faulty statuses are analyzed using matching pursuit based on the compound dictionary. Since transient phenomena often exist in the vibration signals of gearboxes, it is suitable to represent the matching pursuit results in joint time-frequency domain. Figures 4 and 5 show the time-frequency analysis results based on matching pursuit.

Matching pursuit extracts the meshing frequency and its harmonics, the impulses, and other transient components which constitute the signals. For the healthy gearbox, most of the signal energy is associated with the meshing frequency and its harmonics up to the fourth order. A few impulses also appear in the time-frequency distribution, but their energy concentrate in a relatively lower frequency band 0-2000 Hz,

One tooth missing

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Proceedings of the 2010 International Conference on Wavelet Analysis and Pattern Recognition, Qingdao, 11-14 July 2010

and the time interval between consecutive impulses is not regular. These irregular impulses do not indicate any gear damage. It is consistent with the actual setting in the experiment.

While for the faulty gearbox, in addition to meshing frequency and some transient components, periodic impulses appear in the time-frequency distribution. Their energy distributed over a relatively wider frequency band 0-3200 Hz, and repeat in a very regular interval of about 150 ms (for example, the time interval between the consecutive impulses A, B, C, and D). This feature indicates the gearbox has damage. Furthermore, according to the repeating period of these impulses which exactly equals the rotating period of gear 1, it can be inferred that the damage exists on gear 1. This finding is also consistent with the actual setting in the experiment.

Sig

nal

Power spectrumTime [ms]

Fre

quen

cy [k

Hz]

100 200 300 400 500 6000

0.5

1

1.5

2

2.5

3

Figure 4. Normal gearbox signal

Sig

nal

Power spectrumTime [ms]

Fre

quen

cy [k

Hz]

100 200 300 400 500 6000

0.5

1

1.5

2

2.5

3

A CB D

Figure 5. Faulty gearbox signal

In spectral analysis, periodic impulses characteristic of localized gear damage usually appear as complicated sidebands around the gear meshing frequency and/or its harmonics. It is relatively more difficult to interpret the physical meaning embedded in the sidebands, thus it is not easy to detect and locate the gear damage according to the spectral structure of signals in frequency domain.

In time-frequency domain, the energy of an impulse concentrates in a very narrow rectangular area (which tends to be a line) vertical to the time axis whereas parallel to the frequency axis. According to this property, impulses can be easily identified on time-frequency plane. Based on a proper dictionary, matching pursuit can extract impulses from complicated signals. Followed by time-frequency analysis, it enables the task of identifying periodic impulses to become very easy, thus providing an effective approach to detect and locate gear damage.

4. Conclusions

The characteristic components of gearbox vibration signals are extracted using matching pursuit based on compound dictionary. According to the regular time interval between periodic impulses, the gear damage is detected and located successfully. The analysis shows the potential of matching pursuit in gearbox fault diagnosis.

Acknowledgements

This work is supported by National Natural Science Foundation of China (50705007, 50975185), Scientific Research Foundation for Returned Overseas Chinese Scholars, Ministry of Education, Beijing Natural Science Foundation (3102022), and Natural Sciences and Engineering Research Council of Canada.

References

[1] S. Mallat, Z. Zhang, “Matching pursuit with time-frequency dictionaries”, IEEE Transactions on Signal Processing, 1993, Vol. 41: 3397-3415

[2] S. Qian, D. Chen, “Signal representation using adaptive normalized Gaussian functions”, Signal Processing, 1994, Vol. 36, No. 1: 1-11

[3] B. Liu, S. F. Ling, R. Gribonval, “Bearing failure detection using matching pursuit”, NDT&E International, 2002, Vol. 35: 255-262

[4] H. Fan, Q. Meng, Y. Zhang, Q. Gao, F. Wang, “Matching pursuit based on nonparametric waveform estimation”, Digital Signal Processing, 2009, Vol. 19: 583-595

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