GEAR CRACK DETECTION UNDER VARIABLE ROTATING SPEED CONDITIONS USING A SINGLE CHANNEL ACCELEROMETER

International Journal of Mechanical and Production Engineering Research and Development (IJMPERD) ISSN (P): 2249–6890; ISSN (E): 2249–8001 Vol. 10, Issue 3, Jun 2020, 127–136 © TJPR Pvt. Ltd.

GEAR CRACK DETECTION UNDER VARIABLE ROTATING SPEED CONDITIONS USING A SINGLE CHANNEL ACCELEROMETER NGUYEN PHONG DIEN1 & Dr. NGUYEN TRONG DU2 1Associate

Professor, School of Mechanical Engineering, Hanoi University of Science and Technology, Hanoi, Vietnam

2Reserach

Scholar, School of Mechanical Engineering, Hanoi University of Science and Technology, Hanoi, Vietnam

ABSTRACT The present study aims to detect gear cracks early at variable gear speeds from vibration signals measured by a single channel accelerometer. The analysis procedure combines the Order Tracking Method (OTM) and the Order Cepstrum Method (OCM), in which a key-phase reference signal is not required. A suitable approach for gear crack detection in gearboxes operating under non-stationary conditions is also proposed and experimentally verified.

KEYWORDS: Gearbox, Order Tracking, Cepstrum & Gear Crack

Received: Mar 13, 2020; Accepted: Apr 03, 2020; Published: Apr 27, 2020; Paper Id.: IJMPERDJUN202012

Gears are essential elements in mechanical power transmission systems. They are used to change the speed and torque.

The working efficiency of them will decrease when it is damaged. Early fault identification systems for the gear enable detecting and diagnosing initial faults or fault causes. That allows maintenance planning or limitations to the extent of damages to reduce the downtime for production equipment. The traditional vibration-based signal processing methods were almost applied to gearboxes operating at constant speeds. However, when the gearbox runs under varying speeds

Original Article

1. INTRODUCTION

or loads, its mechanical vibrations will become non-stationary, whose amplitude and frequency are time-varying. As a result, traditional analysis techniques can not be applied. In recent years, Time-Synchronous Averaging (TSA) [1] and order tracking [2] are two commonly used methods to overcome these drawbacks. The TSA method has been known as a useful analysis tool for gearbox condition monitoring.

The conventional TSA method is performed by dividing the signal into segments by means of a reference phase signal, resampling each signal segment with the same number of sampling points, and averaging resampled signal segments to get the TSA signal that remains periodic components and attenuates asynchronous, non-periodic components and noises [3]. However, averaging signal segments may lead to the loss of useful information in the original signal.

Order tracking is another approach to remove asynchronous components without losing essential information on the signal. This technique uses the instantaneous phase to resample the original signal at a fixed angular increment.

This technique also requires a reference phase signal in the form of a pulse sequence (tacho-pulse or tacho-signal) that is usually generated by an additional phase probe or a tachometer. In many application cases, the installation of a tachometer is difficult to implement or increases measurement costs.

Therefore, order tracking methods without phase signals, namely the tacho-less method, have been proposed to overcome the limitation mentioned above. Bonnardot and his colleagues [4] firstly proposed a called tacho-less www.tjprc.org

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method and applied it in the detection of gearbox faults under operation conditions with small gear speed fluctuation. Zhao et al. [5] subsequently extended this tacho-less method to large speed variation cases.

If the temporal variation of the instantaneous rotation frequency of the gear is determined, it is easy to create the corresponding tacho-signal [6; 7]. Time-Frequency Analysis (TFA) is one of the most effective methods for Instantaneous Frequency (IF) estimation, and therefore receives much attention from researchers. Urbanek [8] estimated IF using time-frequency distribution by spectrogram and extracting the maximum-energy time-frequency ridge from time-frequency distribution of signals. However, the method such as spectrogram or windowed Fourier transform is usually restricted by time-frequency resolution and therefore, may give inaccurate evaluation results [9]. In the last decade, a number of more sophisticated methods have been proposed to improve time-frequency resolution. The synchrosqueezing transform [10] is one of them.

For the vibration diagnosis of machines, the tacho-less order tracking method is just a preprocessing step. The detection of damage location depends on the later stage of vibration signal processing. Wang et al. [11] estimated the instantaneous frequency with TFA to create the order spectrum for the detection of gear faults in wind turbine gearbox. Zhao et al. [12] combined tacho-less order tracking with envelope spectrum method and Kurtogram diagram for bearing diagnosis.

Wang et al. [13] also detect bearing fault based on phase demodulation function estimation and envelope order spectrum.

This study proposes an improvement in the estimation of the instantaneous phase and frequency of rotating machines. In this proposal, the estimation of the instantaneous phase is extracted from time-frequency distributions that are achieved by wavelet synchrosqueezing transform. Afterward, the OCM method [14] is performed with the resampled signal in the angle domain to detect gear crack faults in non-stationary operating conditions.

After the introduction, a theoretical background of the proposed method is recommended in section 2. In Section 3, the proposed method is demonstrated and verified by an experimental test conducted at a gearbox.

2. THEORETICAL BACKGROUND 2.1 Order Tracking using the Computed Tacho-Pulse This section presents a procedure to recover tacho pulses from a measured vibration signal on gearbox hosing by an accelerometer. Hence, the order tracking technique should be performed using these pulses. A flow chart of the procedure is represented on figure 1, including the following steps: Step 1: Construct a Time-Frequency Distribution (TFD) from the acceleration signal There are several algorithms available for representation of signals in the time-frequency domain, such as the windowed Fourier or the wavelet transform. In this article, The wavelet-based synchrosqueezing transform (WSST) [15] is suggested due to its outstanding abilities compared with others. Because the basic wavelet function of WSST can be flexibly adjusted to the original signal, we can achieve a time-frequency representation having high resolution. Furthermore, this method is secure with measurement noise, and then it can generate accurate time-frequency distribution of signals, even when the signal-to-noise ratio is small. In this context, the kth GMF harmonic f mk will be extracted from the original signal. These harmonics can be filtered by bandpass filtering only in the case that there is small speed variation, and the adjacent frequency lines in the spectrum are separable, which is illustrated in figure 2a. When the rotating speed has large fluctuations, the

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frequency-bands will overlap in the spectrum, as shown in figure 2b. Thus, it cannot be filtered by bandpass filtering. In order to get over this drawback, generalized demodulation based on the generalized Fourier transform (GFT) will be introduced as follows.

Figure 1: Principles of Tacho Pulse Recovery from the Acceleration Signal.

Step 2: Recover the kth GMF harmonic For a signal x(t), the GFT is given by XG  f  



 x t  e

 i 2 [ft  s0  t  ]

(1)

dt



where s0(t) is a time function that specifies the evolutionary phase behavior of x(t).

By introducing

u t   x t  e

i 2 s0  t 

(2)

Eq. (1) can be written in the following form XG  f  



 u t  e

 i 2 ft

(3)

dt



Appling the conventional inverse Fourier transform, it follows that u t  



 X  f e

i 2 ft

G

(4)

df



Comparison of Eq. (2) with Eq. (4) leads to

x t   e

i 2 s0  t 



 X  f e

i 2 ft

G

df

(5)



According to the property of Fourier transformation, assuming that X G  f     f  f0  , where  denotes Dirac delta function, this yields

x t   e

i 2 s0  t 



   f  f e 0

i 2 ft

i 2 f t  s  t  df  e  0 0 

(6)



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From Eq. (6), the instantaneous frequency is given by

f t  

d  t  2 dt

 f 0  s0  t 

(7)

Eq. (7) shows that, if the parameter s0  t  is chosen to satisfy the condition (as shown figure 3a):

s0 t   f t   f0

(8)

f(t) is then mapped at constant frequency f0. Therefore, to straighten kth GMF harmonic (figure 3b), we use the following algorithm X G  f   FFT  x(t )e 

i 2 s0  t 

 

Conducting a bandpass filter around straightened frequency f(t) (as illustrated in figure3c) to generate X G  f 0  .

Hence, kth GMF harmonic is recovered in the following algorithm (as shown figure 3d): x  t   iFFT  X G  f 0  e 

i 2 s0  t 

 

(9)

Step 3: Calculate the rotating frequency of the shaft as follows:

f sh 

f mk k .Z

(10)

in which k denotes harmonic order, and Z is the number of gear teeth

Figure 2: Two Frequencies are Separated (a) and Overlapped (b).

Step 4: Instantaneous phase of the signal is obtained by t

sh (t )  2 . f sh  u .du

(11)

0

Step 5: Calculate a tacho pulse instants occurring once per revolution by inverse function: tP  n   sh1  2 n 

(12)

where N is the integer number of shaft revolutions.

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2.2 Computed Tacho-Less Order Cepstrum The power cepstrum of time-signal x(t) is defined by the following formula:

Cx     F 1 logS x  f 

(13)

where F 1   denotes the inverse Fourier transform, S x  f  is the power spectrum with frequency variable f.

Variable τ is called quefrency, which is a measure of time. It is known that the spectrum of a periodic vibration at a fundamental frequency shows harmonics that are periodic with this frequency, corresponding to only a peak in the power cepstrum. We can use this property of cepstrum to identify periodic components in a complex multi-component signal, such as gearbox signal at stationary operating conditions [14]. We note that the power cepstrum can only be applied effectively when the signal is stationary, i.e., the signal with constant frequencies.

Figure 3: An example to Illustrate the Recovery of kth GMF Harmonic.

In the case of signals with rapidly changing frequencies, such as gearbox signals at variable rotation speeds, these signals need to be processed by order tracking method. Commonly, a reference signal of tacho pulses is required to implement the conventional order tracking for the vibration signal. If there is only a single channel accelerometer for this purpose, we can apply the computed order tracking without a reference signal (the tacho-less order tracking). This technique uses an angular resampling procedure of a time-sampled vibration signal, in which the original signal is resampled in shaft angle domain with constant angle increments using a computed tacho-signal and numerical interpolation.

The cepstrum of an angular resampled signal is called the order cepstrum, which is depicted by a cepstral graph versus the corresponding shaft angle. In summary, we will perform the following steps: 

Step 1. Recovering a reference signal of tacho pulses from the acceleration signal according to the procedure presented in subsection 2.1

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

Step 2. Implementing the tacho-less order tracking for the signal using numerical interpolation.



Step 3. Transforming the resampled signal into the cepstral domain using Eq. (13).

3. EXPERIMENTAL RESULT In this section, a single-stage gearbox with gear crack fault operating under speed-up conditions is considered. An accelerometer was installed on gearbox housing to get the vibration transferred from the gear. As you known from the comments above, to accurately separate tacho pulses, a requirement for the rotating frequency of shaft must be recovered as well as possible. Therefore, a tachometer was also utilized to get reference rotating frequency for testing its reconstruction. The ratio of gearbox is 14/40, with the pinion was cracked root of a tooth. The input shaft of the gearbox is driven by a motor that is controlled by an AC inverter to change the rotating speed from 500 rpm to 1000 rpm according to the predefined laws. A sampling rate of vibration signals is 10 kHz. Figure 4a displays the measured vibration signal in the time domain and its corresponding power spectrum (figure 4b). It is seen that the frequency spectrum is significantly obscured by spectral smearing. Thus, this is a serious problem with conventional analysis methods, and it has less effective in processing the vibration signal of the gearbox is sampled in the time domain under variable rotating speed condition.

Figure 4: Time-Domain Signal (a) and Spectrum (b).

The measured acceleration signal is analyzed by wavelet -based synchrosqueezing transform to generate the time-frequency representation as figure 5a. From this figure, the order harmonics of meshing frequency can be identified clearly. We chose one order harmonic that is displayed most clearly on the time-frequency representation. This step is significant for the successful separation of gear meshing frequency.

Figure 5: Time-Frequency Distribution (a) and First Order GMF is Extracted from the Original Signal (b).

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Figure 3 illustrates the exacting procedure. At first, the kernel function s0(t) in Eq. (8) is estimated using numerical integration, where f(t) is selected from first meshing harmonic, and f0 is the desired straightening frequency component of meshing harmonic at 200 Hz, as shown in Figure 5a. By applying Eq. (2) the first harmonic is then mapped at constant frequency f0 of 200 Hz. A bandpass filter is then employed to extract this straightened first harmonic. Finally, we obtain the waveform of the first gear meshing harmonic, as shown in Figure 5b by taking the inverse GFT. The extracted GMF is the first order meshing harmonic, which means the k factor equal to 1 in Eq. (10). The shaft rotating frequency is calculated by dividing the GMF by the number of teeth according to Eq. (10).

Figure 6: The Extracted and Measured Shaft Frequency.

The accuracy of the reconstructed shaft rotating frequency is evaluated by comparing it with the other frequency measured by a tachometer, as shown in figure 6. It can be observed that the reconstructed shaft frequency is almost hardly different from the original shaft rotating frequency. The tacho information is recovered from this shaft frequency will be used as input for resampling the acceleration signal into the angle domain. Thus, the order tracking method has been performed for the original vibration signal without utilizing any tachometer. After this step, because the effect of variable speed is removed from the vibration signal, then the vibration of the gearbox comes back steady state.

(a) (b) Figure 7: Cepstrum of the Resampled Vibration Signal (a) and Original Vibration Signal (b).

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In the next step, we can apply the order tracking or the order cepstrum method to analyze the resampled signal.

However, as mentioned in this paper [14], the order tracking is less efficient than the order cepstrum method in variable speed conditions. Thus, the cepstrum transform was chosen in this case.

Figure 7a shows the order cepstrum of the resampled signals. It can be seen from Fig. 7a that the order cepstrum of the tooth surface corresponds with 360° and its harmonics. The fault in gear can be explained by the peak value according to the impact pulses through crack position after each revolution. As such, the order cepstrum method has eliminated undesirable spectral smearing and modulation effects to detect symptoms of the fault quickly. Otherwise, the cepstrum transform is applied to the initial acceleration signal, as shown in Fig. 7b. It can be seen that it is not showing any signs of the gear fault.

4. CONCLUSIONS In this study, the wavelet-based synchrosqueezing transform has been applied to extract the instantaneous frequency and phase from a signal measured by a single channel accelerometer mounted at a gearbox housing. The acceleration signal is then resampled according to the recovered phase information to general the angle domain signal. In addition, the resampled signal is transformed into a cepstral domain to obtain the order cepstrum. The experimental results from a test rig have demonstrated that the proposed method is applicable for the detection of gear cracks in a gearbox operating with variable speeds.

ACKNOWLEDGMENT This research is funded by the Hanoi University of Science and Technology (HUST) under project number T2018-PC-214.

REFERENCES 1.

S. Braun. (2011). The synchronous (time domain) average revisited. Mechanical Systems and Signal Processing, 25(4), 1087–1102.

2.

E. D. S. Munck, & K. R. Fyfe. (1997). Computed Order Tracking Applied to Vibration Analysis of Rotating Machinery.

Mechanical Systems and Signal Processing, 11(2), 187-205.

3.

L. Zhang, & N. Hu. (2019). Time Domain Synchronous Moving Average and its Application to Gear Fault Detection. IEEE Access, 7, 93035-93048.

4.

F. Bonnardot, M. E. Badaoui, R. B. Randall, J. Danière, & F. Guillet. (2005). Use of the acceleration signal of a gearbox in order to perform angular resampling (with limited speed fluctuation). Mechanical Systems and Signal Processing, 19(4), 766 -785.

5.

M. Zhao, J. Lin, X. Wang, Y. Lei, & J. Cao. (2013). A tacho-less order tracking technique for large speed variations. Mechanical Systems and Signal Processing, 40(1), 76-90. doi:10.1016/j.ymssp.2013.03.024

6.

B. Boashash. (1992). Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals. Proceedings of the IEEE, 80(4), 520–538.

7.

B. B. (1992). Estimating and interpreting the instantaneous frequency of a signal. II. Algorithms and applications. Proceedings of the IEEE, 80(4), 540–568.

8.

J. Urbanek, T. Barszcz, J. Antoni, B. Samanta, & K. R. Al-Balushi. (2013). A two-step procedure for estimation of instantaneous rotational speed with large fluctuations. Mechanical Systems and Signal Processing, 38, 96-102.

9.

Z. Feng, F. Chu, & M. J. Zuo. (2011). Time–frequency analysis of time-varying modulated signals based on improved energy

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by

iterative

generalized

demodulation.

Journal

135 of

Sound

and

Vibration,

330(6),

1225-1243.

doi:10.1016/j.jsv.2010.09.030 10. G. Thakur, & H.-T. Wu. (2011). Synchrosqueezing-based Recovery of Instantaneous Frequency from nonuniform Samples.

International Journal of Applied Mathematics, 43, 2078-2095.

11. Y. Wang, Y. Xie, G. Xu, S. Zhang, & C. Hou. (2017). Tacholess order-tracking approach for wind turbine gearbox fault detection. Frontiers of Mechanical Engineering, 12:427.

12. M. Zhao, J. Lin, X. Xu, & Y. Lei. (2013). Tacholess envelope order analysis and its application to fault detection of rolling element bearings with varying speeds. Sensors (Basel), 13(8), 10856-10875. doi:10.3390/s130810856 13. Y. Wang, G. Xu, A. Luo, L. Liang, & K. Jiang. (2016). An online tacholess order tracking technique based on generalized demodulation for rolling bearing fault detection. Journal of Sound and Vibration, 367, 233-249. doi:10.1016/j.jsv.2015.12.041 14. N. P. Dien, & N. T. Du. (2020). Fault Detection for Rotating Machines in Non-stationary Operations Using Order Tracking and Cepstrum. In Advances in Engineering Research and Application (pp. 349-356).

15. Daubechies, J. Lu, & H.-T. Wu. (2011). Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool.

Applied and Computational Harmonic Analysis, 30(2), 243-261.

16. Mashitah Abu Bakar & Jamali Md Sah, " Dynamic Response Analysis for Development of Flexible Lightweight Vehicle Chassis using CAE Tools ", International Journal of Mechanical Engineering (IJME), Vol. 7, Issue 3, pp. 9-20 17. Tanuja P. Mote, Meenal. R. Majge & Gouri. P. Brahmankar, " Mechatronics in Automobiles ", International Journal of Electrical and Electronics Engineering (IJEEE), Vol. 5, Issue 5, pp. 13-24 18. Y. Vijayanand, U. Sree Krishnakanth, Ch. Rajesh Kumar & K. Laksmi Ganesh, " New Multilevel Cascaded PWM Inverter Topology for Hybrid Electric Vehicle Drive ", International Journal of Electrical and Electronics Engineering (IJEEE), Vol. 2, Issue 2, pp. 27-40 19. K. K. Padghan, A. K. Pitale J. P. Modak & A. P. Narkhedkar, " Design and Analysis of CAD Based Human Powered Flywheel ", BEST: International Journal of Management, Information Technology and Engineering (BEST: IJMITE), Vol. 2, Issue 7, pp.

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AUTHORS PROFILES

Mr. Nguyen Phong Dien holds a Ph.D. in Engineering Mechanics in 2002 from Chemnitz University, Germany. He has been an Associate Professor of HUST since 2006. He specializes in Engineering Dynamics and Vibration, Technical Diagnostics.

He is a member of the International Federation for the Promotion of Mechanism and Machine Science (IFToMM).

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Mr. Nguyen Trong Du holds a Ph.D. in Engineering Mechanics in 2016 from Hanoi University of Science and Technology, Vietnam. He specializes in Engineering Dynamics and Vibration, Technical Diagnostics.

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