Application of local mean decomposition to the surveillance and diagnostics of low-speed helical gearbox

Mechanism and Machine Theory 47 (2012) 62–73

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Application of local mean decomposition to the surveillance and diagnostics of low-speed helical gearbox Yanxue Wang a, b,⁎, Zhengjia He b, Jiawei Xiang a, Yanyang Zi b a b

School of Mechanical Engineering, Guilin University of Electronic Technology, Guilin, 541004, PR China School of Mechanical Engineering, Xi'an Jiaotong University, Xi'an 710049, PR China

a r t i c l e

i n f o

Article history: Received 1 December 2010 received in revised form 25 June 2011 accepted 14 August 2011 Available online 10 September 2011 Keywords: Local mean decomposition Instantaneous time-frequency spectrum Gearbox Fault diagnosis

a b s t r a c t Gears are common power transmission elements and are frequently responsible for transmission failures. Instantaneous time-frequency spectrum (ITFS) resulted from local mean decomposition is applied to the surveillance and early fault diagnosis of a finishing rolling mill in this paper. Results of practical signals demonstrate that ITFS is effective and reliable for the early detection of gear local fault. In addition, a new parameter to evaluate the damage severity of the gearbox is also developed based on the marginal spectrum derived from ITFS. The utility of the new gear fault symptom has been investigated using practical vibration signals. Results show that the new parameter is only sensitive to the changes caused by the deterioration of a monitored unit and insensitive to the influence of the variable non-deterioration factors such as varying speed and loads. This new index may thus find its wide applications for machine prognostics in the near future. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Gear mechanisms are widely used in rotating machinery and their operational safety has become the significant subject of the intensive investigation and research with the increasing demand for high performance, safety and lower maintenance costs of the machinery. Vibration monitoring as a part of preventive/predictive maintenance programs is the main goal and has proven to be highly effective. Based on the vibration measurement and analysis, the major effort in the gearbox diagnosis is to develop the reliable methods or parameters. Some of those techniques have been shown successfully detect damage in gearbox under certain conditions. However, if a gearbox often works at nonstationary operating conditions, those current techniques may fail to provide sufficient time between warning and failure in order to implement the safety procedures. On the other hand, inaccurate interpretation of operational conditions may also result in false alarms and unnecessary repairs downtime. The process of defect formation in gears can lead both to intensification of nonlinear phenomena as well as to occurrence of non-stationary effects even if in its early stages. So far, many techniques have been developed in the surveillance and diagnosis of gear. A simple feature for monitoring the condition of gearboxes in nonstationary operating conditions is proposed [1]. This diagnosis technique is based on the gearbox in bad condition is more susceptible to load than the gearbox in good condition, and the measure of the susceptibility is considered as a feature of gearbox condition. Time-frequency analysis technique can well describe the nonstationary and nonlinear nature of the measured vibration signals. Fault diagnosis of gear drives by means of time-frequency analysis is proposed in the past decades [2]. A demodulation and residual technique has been proposed to early detect the local gear damage [3]. Wavelet transform is used to detect the location of tooth defects in a gear system [4]. Newland investigated the harmonic wavelet transform to analyze the

⁎ Corresponding author at: School of Mechanical Engineering, Xi'an Jiaotong University, Xi'an 710049, PR China. Tel.: + 86 29 82667963; fax: + 86 29 82663689. E-mail address: [email protected] (Y. Wang). 0094-114X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2011.08.007

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63

transient vibration excited by defects of rolling bearings [5]. Multiple fault signatures detecting technique for rotating machinery are developed based on the dual-tree complex wavelet transform [6]. Mean frequency variation of a scalogram is used for the advancement monitoring of distributed pitting damage in gears. The level of the mean frequency of the gradually decreases when the fault severity is increased [7]. Instantaneous energy density is proposed as a feature for gear fault detection. The instantaneous energy density is shown to obtain high values when defected teeth are engaged [8]. However, the current time-frequency analysis techniques need to know a lot about the signal before it can be processed effectively. Also, if the signal analyzed is composed of several different components, the best predefined parameters such as window function and mother wavelet, differ for each component. Adaptive time-frequency decomposition techniques without the above limitations are used, for example linear chirp modulated Gaussian function is proposed to describe transient vibration, which provides a precise interpretation of complex signals in the time-frequency plane [9]. Frequency family separation method based on empirical mode decomposition (EMD) and Hilbert energy spectrum method is developed for gear fault diagnosis [10]. Local mean decomposition (LMD) is a data-driven method and recently developed by Smith [11]. The corresponding boundary processing and the determining step size of moving average used in the LMD method have been done by the authors of this paper. Moreover, the differences between the two adaptive methods LMD and EMD are given in [12] where it showed LMD is better than EMD in four aspects. LMD technique is further investigated in this work to the surveillance and diagnostics of an industrial gearbox. At the present time, severity assessment for rotating machine vibration levels is still most heavily governed by experience. Most industrial rotating machines are not mass produced like consumer products. A new gear diagnosis feature is developed to reliably quantify the degradation scenarios of the local damage. This work is organized as follows. Algorithm of LMD is briefly introduced in Section 2. Some criteria for condition monitoring and assessment are proposed in Section 3 based on LMD. In Section 4, the condition monitoring and diagnosis of an industrial gearbox are conducted via the proposed parameters. Conclusions are given in Section 5. 2. The local mean decomposition algorithm LMD was originally developed to decompose modulated signals into a small set of product functions (PFs), each of which is the product of an amplitude envelope signal and a frequency modulated (FM) signal. LMD scheme essentially involves progressively separating a FM signal from an amplitude envelope signal. This separation algorithm can be briefly described as follows. Given a signal x(t), the procedure begins by letting r0(t) = x(t) and setting i = 0. 1) Let c0(t) = ri(t), set j = 0. 2) Identify the extrema of cj(t), compute the local mean value and magnitude via the two successive extrema using mi =(cj(ti)+ cj(ti +1))/2, ai =|cj(ti)−cj(ti + 1)|/2. 3) Smooth local mean and local magnitude via moving averaging and get mj(t) and aj(t). 4) Compute the assessment function cj + 1(t) = (cj(t) − mj(t))/aj(t). a) If the aj(t) meet the condition j→∞ lim aj ðt Þ¼1 , set i = i + 1, let

 d arccos cj ðt Þ and PFi ðtÞ ¼ Ai ðtÞ⋅cj ðtÞ Ai ðt Þ ¼ ∏ aj ðt ÞFi ðt Þ ¼ 2π ⋅dt j ri(t) = ri − 1(t) − PFi(t). Go to step 1). b) Else set j = j + 1, go to step 2). 5) This procedure continues until no more PF can be extracted, for example ri(t) is a constant. Interested readers should refer to [11,13] for details. 3. New criteria for condition assessment based on LMD Similar to the Hilbert–Huang spectrum [14], time-frequency distribution can be constructed based on the results of LMD. Three energy-based features: time-frequency concentration, NP4 and energy dispersion ratio are computed via the ITFS. 3.1. The instantaneous time-frequency spectrum and its sharpness measurement While LMD decompose a signal x(t) into several PFs, the relating instantaneous amplitude and frequency components are also derived. As such, instantaneous time-frequency spectrum (ITFS) Cx(t, f) can be written as follows:

 J J Cx ðt; f Þ ¼ ∑ Aj t; Fj ðt Þ ¼ ∑ Aj ðt Þδ f −Fj ðt Þ j¼1

ð1Þ

j¼1

where Aj(⋅) is the instantaneous amplitude, Fj(⋅) is the instantaneous frequency, δ(⋅) is the Dirac function and J is the total of instantaneous amplitude or frequency. It can be seen Cx(t,f) is indeed a weighted non-normalized joint amplitude-frequency-time distribution and the weight assigned to each time-frequency cell is the local amplitude. In general, a gear transmission vibration signature consists of three significant components: a sinusoidal component due to time varying loading, a broad-band impulsive component due to impact, and random noise [15]. For an undamaged gear transmission, the sinusoidal components such as meshing frequency and its harmonics dominate. However, as damage propagates through the

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Y. Wang et al. / Mechanism and Machine Theory 47 (2012) 62–73

Amplitude

(a) 2 0 -2

600

(d)

(c)

(b)

Frequen cy /Hz

500 400 300 200 100

0.4

0.2

0

0

0

0.2

0.4

Magnitude

0.6

0.8

1

Time /s

0

0.25

0.5

Amplitude

Magnitude

(e) 0.03 0.02 0.01

Fig. 1. (a) The simulated temporal signal, (b) Fourier spectrum of the temporal signal, (c) ITFS of the simulated signal, (d) marginal spectrum in frequency domain, and (e) marginal spectrum in time domain.

system, the sinusoidal components reduction in amplitude. In addition, both the broad-band impulsive components and the random noise have become more prevalent [16]. As is known to all, local gear faults such as spalling on a limited number of gear teeth produce sharp transients in the vibration signal of a transmission. Based on the gear meshing vibration model given in [17], a simulated gear signal in time domain with local defect is shown in Fig. 1(a) where impacts cannot be clearly observed due to its weak energy. Fig. 1(b) shows the corresponding Fourier spectrum while Fig. 1(c) displays the ITFS obtained by LMD. Sinusoidal components can be detected using ITFS and more importantly impulsive components are also successfully detected in the time-frequency domain. In addition, it should be noted that ITFS may contain different information compared with the classical Fourier spectrum. The components shown in ITFS may not be necessarily identical to those in Fourier spectrum, which can be found in Fig. 1(c) and (b). Based on the Cx(t, f) derived from the LMD, the sharpness or concentration of the time-frequency distribution can be measured in terms of [18] T

fs =2

4

K ⋅∫ ∫ jCx ðτ; υÞj dτdυ Mx ¼  0 0 2 T fs =2 ∫∫ jCx ðτ; υÞj2 dτdυ

ð2Þ

0 0

0

f1

f2 Fig. 2. The nominal bounds for computing EDR.

f3

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Fig. 3. Setup of the industrial finishing rolling mill.

where K is only a scale coefficient, T and fs are the sampling time and frequency of the signal. Generally, the concentration of the time-frequency distribution itself is very small. It had better be enlarged to a value greater than one and thus be easily used to monitoring its value. For a kind of signal, K should be fixed, for example, K is set to 10 6 for the real gear vibration signals used in this work. When K = 10 6 the Mx of the simulated signal in Fig. 1(a) is 24.98. 3.2. Time-frequency marginal spectrum The relating marginal spectrum can be also defined from ITFS. Marginal spectrum in frequency domain can be written as T

Hf ðυÞ ¼ ∫ Cx ðt; υÞdt

ð3Þ

0

where T is the sampling time of the signal. This marginal spectrum offers a measure of the total energy contribution from each frequency value. Fig. 1(d) shows the marginal spectrum in frequency domain of the simulated signal. A new diagnostic symptom proposed in this paper is based on this marginal spectrum which will be described in the next section. Likewise, the marginal spectrum can be also defined in time domain which denotes the instantaneous energy density of the signal f =2

Ht ðτÞ ¼ ∫ s Cx ðτ; f Þdf

ð4Þ

0

where fs is the sampling frequency of the signal. Fig. 1(e) shows the instantaneous energy of the simulated signal. It can be found that the instantaneous energy still cannot effectively detect the impulsive signatures due to the weak impulsive energy (or resulting from incipient defect). As the degradation of the defect, instantaneous energy density can be used as a feature for gear fault detection [8]. Kurtosis for the instantaneous signal energy is a parameter NP4 which is developed to the gear diagnosis [19]. NP4 is non-dimensional and depends only on the shape of the power distribution, and it can be written as
4 P N N∑i¼1 Ht ðτi Þ− Ht ðτÞ NP4x ¼ h   i2 P ∑Ni¼1 Ht ðτi Þ− Ht ðτÞ 2

ð5Þ

where i is the sample index, Ht(τi) is the instantaneous energy at sample index i, H t ðτÞ is the mean of Ht(τ) and N is the total number of temporal signal. Initial research on the NP4 parameter is based on Wigner–Ville distribution, while NP4 is computed based on the ITFS derived from the LMD technique in this work. The NP4 of the simulated signal shown in Fig. 1(a) is 3.140. This parameter is used to the surveillance of the gearbox. Table 1 Data sheet of the gearbox in a finishing mill. Parameters

Values

Central distance Module of teeth Number of teeth Pressure angle Face width

1350 mm 30 22/65 β = 10°40′ 560 mm

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Y. Wang et al. / Mechanism and Machine Theory 47 (2012) 62–73

(a)

(b)

Fig. 4. (a) The picture of the local scuffing damage in pinion and (b) shows.

3.3. The proposed energy dispersion ratio ITFS derived by LMD introduces new type of information that may be very useful in the machine condition monitoring and fault diagnosis. Total signal energy via LMD time-frequency distribution can be written as T

f =2

E ¼ ∫ ∫ s Cx ðt; f Þdtdf

ð6Þ

0 0

where T and fs are the sampling time and frequency of the signal, respectively. Those energy-based features often can be applied by researchers for the fault diagnosis or classification [20]. However, the working condition may be changed in practice and thus some features such as local instantaneous energy defined in the [8] may be sensitive to the varying operations and cannot well reflect the development of the defect. Energy dispersion ratio (EDR) is developed in this work for the quantification of the energy distribution in the marginal spectrum derived from ITFS. EDR is defined as the ratio of the energy in the frequency band of interests to the total energy which can be written as f

ρ0;f1 ¼

∫ 1 Hf ðυÞdυ

ð7aÞ

0 f =2 ∫0s Hf ðυÞdυ

f

ρf1 ;f2 ¼

∫ 2 Hf ðυÞdυ

f1 ∫fs =2 Hf ðυÞdυ 0

ð7bÞ

where f1 and f2 are the frequency boundaries, ρ is the EDR and Hf(υ) is the marginal spectrum derived by Eq. (3). The chosen of the bounds in Eqs. (7a) and (7b) is critical in the applications. As mentioned above marginal spectrum describes the energy distribution. Vibration signal energy is concentrated in the meshing frequency when gear collected from a normal gear at low rotating speed, while it is concentrated in the harmonics of meshing frequencies in the case of high-speed rotating. As such, there

Table 2 EDR of the vibration signal received before the mill breakdown. Time (day)

MF (Hz)

8 7 6 5 4 3 2 1

89.38 101.3 60.00 73.13 95.00 82.50 90.00 93.75

EDR ρI

ρII

ρIII

0.1386 0.1637 0.1818 0.2013 0.2490 0.2742 0.3249 0.3465

0.6203 0.6037 0.6364 0.5789 0.5240 0.5008 0.4068 0.4292

0.2389 0.2066 0.1805 0.2171 0.2239 0.2243 0.2667 0.2235

NP4

Mx

4.032 3.647 4.593 5.677 4.292 3.474 6.956 12.35

11.06 7.737 9.796 9.183 7.870 8.421 9.578 11.38

Y. Wang et al. / Mechanism and Machine Theory 47 (2012) 62–73

67

Table 3 EDR of the vibration signal received after the maintenance action. Time (day)

MF (Hz)

EDR

1 2 3 4 5 6 7 8

70.00 80.00 76.25 86.25 53.13 72.50 76.25 83.13

ρI

ρII

ρIII

0.1098 0.1280 0.0923 0.1125 0.1441 0.1192 0.0660 0.0716

0.6140 0.5632 0.6349 0.5468 0.5655 0.5293 0.6999 0.6021

0.2711 0.3028 0.2649 0.3268 0.2883 0.3492 0.2194 0.3143

NP4

Mx

3.566 3.178 2.961 3.547 2.885 3.299 3.541 3.372

7.085 9.017 9.260 4.952 7.224 5.213 5.352 4.051

exists a minimum between the low component (mainly relating to the shaft rating) and the main MF component or between different harmonics of meshing frequency as is shown in Fig. 2. This minimum is found and considered as a boundary used in Eqs. (7a) and (7b). Boundary frequencies f2 and f3 can be chosen according to the practical signals, because these two bounds mainly indicate the high frequency and its harmonics. EDR is a ratio, thus it can be insensitive to the influence of the variable non-deterioration factors such as varying speed and loads that are very useful for the real applications. 4. Industrial gearbox applications Gearbox is one of the key units in the finishing rolling mill. Thus its surveillance, diagnosis and condition evaluation are very important for keeping its high performance, safety and reducing the maintenance costs. 4.1. The setup of the industrial gearbox Gearbox used in this paper is the key equipment in the hot strip finishing rolling mill and it can directly influence the quality products and the long-term safe operation. The main driven transmission comprises a one-stage helical gearbox with a reduction of 2.9545 and it usually works in a low rotating speed (the high speed shaft frequency is only about 2–5 Hz) and variable load conditions. Thus, traditional index such as rms and kurtosis may not well reflect the status due to the variable working conditions. The sketch of the finishing rolling mill in a steelworks is shown in Fig. 3. The power is provided by two parallel direct current electromotors. The corresponding specifications of the gearbox used in the drive system are given in Table 1. Vibration signals are collected using portable data acquisition (DAQ) unit (Telesen8823 produced by BaoSteel Inspection). This DAQ unit can simultaneously record 8 channels data and 7 kinds of vibration parameters and it can also be used for long-period condition monitoring. In addition, vibration velocity transducers with magnetic bases are used in the investigation. One day, the abnormality of the main gearbox was found and then the mill group was stopped quickly. Local scuffing on the pinion gear teeth has been observed in the following maintenance actions and then a new pinion gear was replaced as soon as possible. Fig. 4 displays the damaged pinion. All the velocity sensors were attached to the outer casing of the gearbox. Signals in vertical direction received from the first sensor shown in Fig. 3 are used in this work. The sampling frequency is set to 2560 and the number of sampling is 4096. In practice, industrial communities would rarely allow their assets to run to failure and operational

12 Kurtosis Peak-to-Peak RMS

10

Score

8 Maintenance 6 4 2 0 8

7

6

5

4

3

Before breakdown

2

1 0 1 Time /day

2

3

4

5

6

7

8

After maintenance

Fig. 5. The evaluation of kurtosis, peak-to-peak and rms of the history data recorded 8 days before and after the machine shutdown.

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Y. Wang et al. / Mechanism and Machine Theory 47 (2012) 62–73

15 NP4 Mx Maintenance

Score

10

5

0

8

6

7

5

4

3

2

1 0 1 Time /day

2

3

Before breakdown

4

5

6

7

8

After maintenance

Fig. 6. The evaluation of NP4 and Mx index of the history data recorded 8 days before and after the machine shutdown.

data are also not always recorded. Due to the above limitations, historical event data are not abundant. As such, only data recorded on 8 days before and after the shutdown of the rolling mill are used in this work, while data on the day for maintenance were not recorded. Those 16 signals are analyzed and used to identify the progressing gear defect using the proposed method in the following subsections. The vibration signals were acquired every day in order not to give an effect on the normal product. Since signals were randomly recorded in each day, the working conditions of the mill such as rotating speed and load may be different. Therefore, the meshing frequency (MF) (=fr Zp, fr is high speed shaft frequency and Zp is number of pinion teeth) is also varying. The MFs of all the 16 vibration signals are given in Tables 2 and 3. Not only the shaft frequency but also the loads vary during the 16 days. Those widely varying shaft frequencies and loads on the rolling mill may result in a vibration response that may be completely different from one operating condition to the next, which brings great challenge for the current parameters and techniques.

4.2. Surveillance based on statistical analysis The traditional techniques for vibration-based transmission damage detection are typically based on the some statistical measurement of the vibration signal. The mostly used signatures in time domain for the surveillance are the kurtosis, rms and peak-to-peak values. Kurtosis is the fourth normalized moment of a given signal x and provides a measure of the peakedness of the signal, i.e. the number and amplitude of peaks present in the signal [15]. It is given by

P

N∑Ni¼1 ðxi −x Þ4 Kurtosisx ¼  2 ∑Ni¼1 ðxi −xP Þ2

ð8Þ

(a)

(b) 1

0

600

MF=93.75Hz

I III

II

500

0.8

400 300

1

0.2347s

200 1.5 100 0

0

1

0.5

Time /s

1.5

2

Magnitude

Frequency /Hz

0.5 0.6

0.4

0.2

0

0

100

200

300

400

500

600

Frequency /Hz

Fig. 7. (a) ITFS of the vibration signal recorded on the 1st day before the mill shutdown where dashed line denotes the MF; (b) the relating marginal spectrum (dashed line is MF and dash dot line is the frequency limits).

Y. Wang et al. / Mechanism and Machine Theory 47 (2012) 62–73

(a)

69

(b) 600

1.5

0

II

I

MF=95.00Hz

III

500

300

1 0.2316s

200

Magnitude

Frequency /Hz

0.5 400

1

0.5

1.5 100 0

0

0.5

1

1.5

0

2

0

100

200

Times /s

300

400

500

600

Frequency /Hz

Fig. 8. (a) ITFS of the vibration signal recorded on the 4th day before the mill shutdown where dashed line denotes the MF; (b) the relating marginal spectrum (dashed line is MF and dash dot line is the frequency limits).

where xP is the mean of the signal x and N is the total number of data points. It is positive for a distribution consisting of a sharp single peak and rises with an increase in peakedness of a distribution. The root mean squared (rms) is defined as the square root of the average of the sum of the squares of the signal samples

RMSx ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  N 2 ∑i¼1 xi N

ð9Þ

where x is the mean of the signal and N is the total number of data points. Peak-to-peak value is the change between peak (highest amplitude value) and trough (lowest amplitude value, which can be negative). Peak-to-peak is a straightforward measurement on an oscilloscope. Fig. 5 shows the scores of kurtosis and rms derived using Eqs. (8) and (9) as well as peak-to-peak amplitude of the vibration signals. As it can be seen kurtosis values are more sensitive than the other two parameters and all the three parameters are nonmonotonic and have strong fluctuations with the progressing failure. In addition, it is much difficult to define threshold for the condition evaluation, because the kurtosis, rms and peak-to-peak values of the signals received after the machine maintenance action are close to those values before the machine shutdown. The reason is that these parameters may be sensitive to the operation conditions and cannot directly reflect the degradation of the damage. Parameters Mx and NP4 defined in Eqs. (2) and (5) respectively, are applied in the condition monitoring of the rolling mill. Scale coefficient K used in Eq. (2) is also set to 10 6. All the computed NP4 and Mx are given in Tables 2 and 3. Fig. 6 shows the NP4 and Mx parameters over time. It can be found that NP4 better demonstrates the increase than the Mx with the damage progressing. However, NP4 still doesn't have the consistence and thus it is also difficult to define the reliable threshold for monitoring.

(b)

(a)

2

0

600

MF=89.38Hz II

I

III

1.5

0.5 400 1

300 0.2461s

200

1.5

Magnitude

Frequency /Hz

500

1

0.5

100 0

2 0

0.5

1

Times /s

1.5

0

0

100

200

300

400

500

600

Frequency /Hz

Fig. 9. (a) ITFS of the vibration signal recorded on the 8th day before the mill shutdown where dashed line denotes the MF; (b) the relating marginal spectrum (dashed line is MF and dash dot line is the frequency limits).

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Y. Wang et al. / Mechanism and Machine Theory 47 (2012) 62–73

(a)

(b) 600

1

0

I

500

0.8

Magnitude

Frequency /Hz

0.5 400 1

300 200

MF=80.00Hz

III

II

0.6

0.4

1.5 0.2

100 0

0.5

0

1

1.5

2

0 0

100

200

Times /s

300

400

500

600

Frequency /Hz

Fig. 10. (a) ITFS of the vibration signal acquired on the 2nd day after the mill maintenance where dashed line denotes the MF; (b) the relating marginal spectrum (dashed line is MF and dash dot line is the frequency limits).

4.3. Fault diagnosis via the ITFS derived by LMD The above statistical analysis can give a simple alarming 1 day in advance of the shutdown, but statistical analysis has limit ability to detect the early damage. In addition, the details of the damage such as the location or type cannot be given using the statistical analysis. As such, more efficient technique should be used for the early fault detection and identification. As mentioned above the local faults give impulse perturbations in the signal whose frequencies are equivalent to gear shaft rotations. Thus, the proposed ITFS of the LMD is applied to detect the weak impulsive components in time-frequency domain. Figs. 7(a), 8(a) and 9(a) show the ITFS of the history data recorded on the 1st day, the 4th day and 8th day before the machine shutdown, respectively. Strong impulsive components (marked by arrows) can be clearly observed. In the meanwhile, while the frequency of the impulsive component is equal to the rotating frequency of the pinion as is shown in ITFS figures. Those impacts demonstrate that local damage was developed in the high speed gear, as is shown in Fig. 4. Depending on the severity of the disturbances in the form of a transient shock impulse result from a defect, the high signal energy exhibits at all scales coinciding in time instances. The corresponding marginal spectra are shown in Figs. 7(b), 8(b) and 9(b). The energy over the frequency gradually decreases as can be seen from the visual perspective due to the progressing failure. Fig. 10 shows the ITFS of the vibration signal collected 2 days after the maintenance action. The regular impulsive components cannot be observed from Fig. 10(a), moreover the signal energy is more concentrated in the area of MF from Fig. 10(b). 1

(a)

0.8

(b)

MF=70.80Hz

MF=80.00Hz

(c)

MF=72.50Hz

(g) MF=76.25Hz

(d) MF=86.25Hz

MF=76.25Hz

0.6

Magnitude

0.4 0.2 0

(e) MF=53.13Hz (f)

0.8

(h) MF=83.13Hz

0.6 0.4 0.2 0 0

200

500

0

200

500

0

200

500

0

200

500

Frequency /Hz Fig. 11. Marginal spectra of the history data acquired after the mill maintenance (dashed dot lines denote the boundary), (a) to (h) denote the 1st to 8th days, respectively.

Y. Wang et al. / Mechanism and Machine Theory 47 (2012) 62–73

(a)

71

(b)

600

1

0

MF=47.50Hz

I

500

II

III

0.8

400 300

1

200

Magnitude

Frequency /Hz

0.5

1.5 100 0

0

0.5

1

1.5

0.6

0.4

0.2

0 0

2

100

200

300

400

500

600

Frequency /Hz

Times /s

Fig. 12. (a) ITFS of the vibration signal in the normal condition of mill where dashed line denotes the MF; (b) the relating marginal spectrum (dashed line is MF and dash dot line is the frequency limits).

Fig. 11 shows the marginal spectra of the vibration signals recorded after maintenance where it can be found that most of the energy is concentrated around the meshing frequency. To further demonstrate its effectiveness, ITFS and the relating marginal spectrum of another signal collected from the same location under normal machine condition are given in Fig. 12(a) and (b), respectively. Similarly, signal energy also focuses around the MF and even the total energy is lower than those shown in Figs. 10(b) and 11. These above analyses demonstrate the ITFS can effectively and reliably detect the local damage in advance of 8 days, which is very important for the corporation in avoiding a catastrophic failure from occurring in the practical applications. 4.4. Damage severity assessment via the proposed new criteria ITFS can effectively identify the local damage of the pinion, but it cannot give the degradation scenarios of the damage quantitatively. Depending on the severity of the disturbances, the intensity of signal energy and distribution change. The more severe the fault is, the

(a)

1 Early warning

Warning

Magnitude

0.8 II

0.6 0.50 0.4 I

0.2 0

(b)

0.15 1

2

3

4

5

6

7

8

Time /day 1 Healthy

Magnitude

0.8 II

0.6 0.50

0.4 I

0.2 0

0.15 1

2

3

4

5

6

7

8

Time /day Fig. 13. Severity evaluation of the rolling mill over time, (a) the 1st to 8th days before machine shutdown, and (b) the 1st to 8th days after maintenance.

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higher the value of signal energy gets and the wider the stretch of the energy distribution ranging through a number of scales becomes. A better way to quantify this energy distribution variation is to use the developed EDR parameters. Marginal spectra of the vibration signal received on the 1st to 8th days after the maintenance actions are displayed in Fig. 11. Frequency boundary f1 used in Eq. (7a) is set according to the technique mentioned in Section 3.3, while f1 and f2 are defined as 200 Hz and 500 Hz respectively. EDR parameters of history data recorded before machine shutdown are given in Table 2. As can be seen the ρI and ρII EDR shows the consistent increase over time and well denotes the developing defect. However, ρIII always fluctuates around the 0.2 and it cannot indicate good tendency over time. Thus ρI and ρII are used as diagnostic symptoms to assess the severity of the damage in this work. The EDR parameters of history data recorded after maintenance actions are given in Table 3. The corresponding thresholds can be suitably set in practical applications. According to the derived values of the EDR parameter, the thresholds of the early warning and failure are set to 0.15 and 0.5 in this work, respectively. As such, the normal and early warning status can be defined as ρI ≤0.15∪ρII ≥0.5 and ρI ≥0.15 ∪ρII ≥0.5 respectively. The sever failure status is denoted by ρI ≥0.25 ∪ρII b 0.5, which shows the machine may have serious risk of functional loss and possible severe consequential damage and action is required as soon as possible. Fig. 13(a) and (b) shows the evaluation results. It can be observed in Fig. 13(a) that the machine status of early warning and the alarm can be correctly denoted and distinguished with the progressing of the natural damage. Fig. 13(b) shows the tendency of the ρI and ρII values of the vibration signals received after machine maintenance where all the values are subject to the above rules. As such, the rolling machine was operated in a normal condition. The ρI and ρII of other signals shown in Fig. 12(a) are 0.1109 and 0.5534 which also satisfy the rules and indicate the good operation condition of the machine. Therefore, it is demonstrated that the EDR parameter is only sensitive to the changes caused by the deterioration of a monitored unit and insensitive to the variable influence of the non-deterioration factors such as varying speed and loads. This two EDR indexes and their monotonicity used for assessment may increase detection confidence and reduce false alarms. 5. Conclusions Based on the ITFS constructed by LMD, a fault detection technique is developed for the industrial gearbox applications. ITFS can early and reliably detect the local gear tooth damage using practical vibration signals. A new gear diagnosis parameter EDR is also presented and investigated in the paper. It is demonstrated that the proposed new parameter EDR has better consistence over time than the traditional kurtosis, rms and peak-to-peak values as well as energy-based features NP4 and Mx derived from ITFS. Thus EDR as severity factor can well denote the degradation scenarios and quantitatively evaluate the damage if the relating threshold is set suitably. It is also verified that varying machine operating conditions have little effect on the severity evaluation when EDR is adopted. Two-parameter evaluation could give more confidence in the observation of deterioration, which may be valuable in managing maintenance programs. Surveillance, diagnostics and severity assessment of the industrial gearbox are all investigated in this work. To simplify the application of the proposed technique for the end users, condition monitoring software with artificial intelligence can be developed. The prognostics are another important aspect of a condition-based maintenance. Remaining life prognosis using the new parameter EDR may be conducted in the future research. Acknowledgments The financial sponsorship from the projects of National Natural Science Foundation of China (51105085,51175097,51035007) and Guangxi key Technologies R&D Program of China (1099022-1, 10123005-12) is gratefully acknowledged. We also thank H. Liu in Shanghai Baosteel Group Corporation for providing the data. References [1] W. Bartelmus, R. Zimroz, A new feature for monitoring the condition of gearboxes in non-stationary operating conditions, Mechanical Systems and Signal Processing 23 (5) (2009) 1528–1534. [2] G. Meltzer, Y.Y. Ivanov, Fault detection in gear drives with non-stationary rotational speed-part I: the time–frequency approach, Mechanical Systems and Signal Processing 17 (5) (2003) 1033–1047. [3] F. Combet, L. Gelman, P. Anuzis, R. 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