Gearbox is a widely used component for power transmission and speed change. Time synchronous average (TSA) is one of the most effective methods for vibration monitoring and diagnosis of gearboxes. Traditional TSA technique requires key-phase signal and constant operating speed. So the application of TSA is difficult in many situations such as in the case of gearboxes used in wind power generators and automobiles. A new method to implement TSA without key-phase signal for variable speed condition gearbox is proposed in the paper. The reported method is based on the estimation of instantaneous speed with time-frequency domain filtering and equal angular interval re-sampling of vibration signal. Experimental investigation performed in a variable speed gearbox test rig indicates that the proposed method can eliminate the influence of large speed fluctuation of gearboxes and provide satisfactory TSA results

Guangrui Wen

Tingting Wu

Xining Zhang

English

Journal of Vibroengineering

  VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2012. VOLUME 14, ISSUE 4. ISSN 1392-8716 
1766 
901. A new time synchronous average method for 
variable speed operating condition gearbox 
 
Xining Zhang
1
, Guangrui Wen
2
, Tingting Wu
3
 
 
Department of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, China 
2Corresponding author 
E-mail: 1zhangxining@mail.xjtu.edu.cn, 2grwen@mail.xjtu.edu.cn, 3wtingting@mailst.xjtu.edu.cn  
 
(Received 17 September 2012; accepted 4 December 2012) 
 
Abstract. Gearbox is a widely used component for power transmission and speed change. Time 
synchronous average (TSA) is one of the most effective methods for vibration monitoring and 
diagnosis of gearboxes. Traditional TSA technique requires key-phase signal and constant 
operating speed. So the application of TSA is difficult in many situations such as in the case of 
gearboxes used in wind power generators and automobiles. A new method to implement TSA 
without key-phase signal for variable speed condition gearbox is proposed in the paper. The 
reported method is based on the estimation of instantaneous speed with time-frequency domain 
filtering and equal angular interval re-sampling of vibration signal. Experimental investigation 
performed in a variable speed gearbox test rig indicates that the proposed method can eliminate 
the influence of large speed fluctuation of gearboxes and provide satisfactory TSA results. 
 
Keywords: time synchronous average, variable speed condition gearbox, time-frequency 
filtering, equal angular interval re-sampling. 
 
1. Introduction 
 
Being one of the most important parts of power transmission and speed changing, a gearbox 
is widely used in many industry fields. Due to its easiness to measure and analyze, vibration 
signals have been commonly employed in condition monitoring and diagnosis of gearboxes. 
The time synchronous average (TSA) technique is an effective vibration monitoring method for 
gearbox [1-3]. TSA can effectively attenuate the uncorrelated vibration components with the 
selected revolve frequency on spectrum. In the mean time, the TSA waveform intuitively 
reveals the condition of the monitoring gears on the selected shaft [4-7].  
The key problem of TSA technique is to obtain the vibration samples against revolving 
angle. Commonly, there are two ways to achieve the above purpose. One is to employ the 
phase-lock technique with key-phase signal, while the other is to use a re-sampling method. The 
first method can track the speed fluctuation and directly collect the vibration samples against 
revolving angle. But it requires key-phase sensor and complex phase-lock circuit, which fails to 
achieve good tracking performance in large speed fluctuation. In many circumstances, it is 
difficult to install the key-phase sensor on the gearbox especially on the intermediate shaft. 
Gearboxes in wind power generator, machine tool, helicopter, mining excavator and 
automobile, is operating under large speed fluctuation condition [8-9]. The re-sampling method 
is based on instantaneous speed estimation and can process large speed fluctuation [10]. 
TSA based on instantaneous speed estimation has attracted great interest recently.               
F. Bonnardot gave an angular re-sampling method without tachometer, which is based on the 
estimation of meshing frequency or it's harmonics by using a band-pass filter [11]. Sideband 
inside the filter band, which is produced by the slow speed fluctuation, is used to estimate the 
instantaneous speed. The method works well only under small speed fluctuation, which is less 
than the minimum speed of two meshing gears. On the basis of above work, F. Combet and     
L. Gelman extended automatic angular re-sampling algorithm for the estimation of the TSA 
signal of a particular shaft in a gearbox [12]. The innovation of the scheme is mainly 
concentrated on the automatic selection of the mesh harmonic, which is used for the shaft speed 
estimation. F. Combet and R. Zimroz proposed a method to estimate the instantaneous speed 
 901. A NEW TIME SYNCHRONOUS AVERAGE METHOD FOR VARIABLE SPEED OPERATING CONDITION GEARBOX. 
XINING ZHANG, GUANGRUI WEN, TINGTING WU 
 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2012. VOLUME 14, ISSUE 4. ISSN 1392-8716 
1767 
relative fluctuation with short-time scale transform of vibration signal in 2009 [13]. The method 
still suffers from the limitation of small speed fluctuation.  
In fact, the speed fluctuation of one shaft will result in the fluctuation of relative shaft 
rotation frequency and meshing frequencies. The instantaneous speed fluctuation can be well 
illustrated in joint time-frequency diagrams. The instantaneous speed can be extracted with 
time-frequency domain filtering. Based on the above thought, the paper gives a new time 
synchronous average method for variable speed gearbox without key-phase signal, and the 
method can be applied to the variable speed gearbox with large speed fluctuation. Variable 
speed gearbox test rig was built, and measurements were taken with accelerometers. The 
performance of the proposed method is verified by the processing of experimental data. 
 
2. Estimation of instantaneous speed 
 
Joint time-frequency analysis provides a tool for us to observe the signal both in time and 
frequency domain [14, 15]. Short Time Fourier Transform (STFT) is the most widely used 
method for non-stationary signal. So, it provides us a tool to view and track the instantaneous 
speed variations on the time-frequency diagram. 
The basic principle of STFT is as follow: break up the signal into small time segments with 
filter, and then analyze each time segment with Fourier transform to get the spectrum. The 
totality of such spectrum indicates how the spectrum is varying in time. The transform formula 
of STFT and ISTFT are: 
dtethtxfSTFT ftjx
πττ 2)()(),( −
+∞
∞−
−= ∫  (1) 
ωτω
π
τ ωτ dtdethtSTFTx jx )(),(
2
1
)( −= ∫ ∫
+∞
∞−
+∞
∞−
 (2) 
where )( τ−th  is the moving window around τ , ),( fSTFTx τ  is the time-frequency distribution.  
Suppose we have a frequency modulated signal: 
10sin(2 200 8sin(2 10 )) 10sin(2 400 8sin(2 10 )),x t t t tπ π π π= × + × + × + ×  
digitizing x  with the equal time interval sampling we can get the corresponding digital signal 
,  1,  2,  3,  ...,  .ix i N=  Fig. 1 provides the time-frequency diagram of signal x. The time-
frequency diagram directly illustrates the frequency varying of a component against the time. 
The variations of the instantaneous frequencies of each component are clearly observed on the 
diagram.  
 
 
Fig. 1. Time-frequency diagram  
 901. A NEW TIME SYNCHRONOUS AVERAGE METHOD FOR VARIABLE SPEED OPERATING CONDITION GEARBOX. 
XINING ZHANG, GUANGRUI WEN, TINGTING WU 
 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2012. VOLUME 14, ISSUE 4. ISSN 1392-8716 
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In order to extract the varying instantaneous frequency component around 400 Hz, the 
coordinates of each maximum amplitude points of the above selected component is searched 
along the time axis. The model of instantaneous frequency against time is built with the 
coordinates. With the model we can obtain the value of instantaneous frequency at any moment 
of time axial.  
 
3. Equal angular interval re-sampling of vibration signal 
 
Time synchronous average requires angular based sampling time series. For an equal time 
interval sampling vibration signal at varying operating speed condition, it is essential to perform 
equal angular interval re-sampling [16]. In order to determine time moments of each re-
sampling points, time-frequency domain filtering technique and Hilbert Transform are 
introduced. 
A. Time frequency filtering 
Suppose the model of the varying operating speed component is built with the above method. 
Band-pass filter curves in frequency domain are superimposed on the diagram at the point of 
maximum amplitude along the frequency axis of each moment on the time-frequency diagram. 
Fig. 2 indicates filter curve superimposed at point P(k, f). Fig. 3(a) gives the filtered time-
frequency diagram. With inverse short time Fourier transforms, the filtered signal (Fig. 3(b)) 
can be easily calculated. Obviously, the filtered signal is a mono-frequency signal at any 
moment.  
 
 
Fig. 2. Principle of time-frequency filtering 
 
B. Phase information extraction with Hilbert transform 
For a time series of )(tx , the definition of Hilbert transform is as follows: 
τ
τ
τ
π
d
t
x
ty ∫
+∞
∞− −
=
)(1
)(  (3) 
where )(ty  is the result of Hilbert transform. The analytic signal of signal )(tx  can be 
constructed as: 
)()()()()( tjetatjytxtz Φ=+=                                                                       (4) 
where )(ta  is the amplitude function, 2 2( ) ( ) ( ) ,a t x t y t= +  )(tΦ  is the phase function, 
 901. A NEW TIME SYNCHRONOUS AVERAGE METHOD FOR VARIABLE SPEED OPERATING CONDITION GEARBOX. 
XINING ZHANG, GUANGRUI WEN, TINGTING WU 
 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2012. VOLUME 14, ISSUE 4. ISSN 1392-8716 
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)(
)(
arctan)(
tx
ty
t =Φ . Fig. 4 gives the unwrapped phase curve of the filtered signal. 
 
 
(a) Filtered time-frequency diagram 
 
(b) Filtered signal 
Fig. 3. Result of time-frequency filtering 
 
 
Fig. 4. Phase curve against time 
 901. A NEW TIME SYNCHRONOUS AVERAGE METHOD FOR VARIABLE SPEED OPERATING CONDITION GEARBOX. 
XINING ZHANG, GUANGRUI WEN, TINGTING WU 
 
 
 
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C. Equal angular interval re-sampling 
The aim of re-sampling is to transform the original equal time interval signal to equal 
angular interval signal. Suppose the original signal and re-sampling signal time moments are 
( ),ix t 0,  1,  2,  ...,  1i N= − (∆T is an original sampling interval) and ( ),ky t  
0,  1,  2,  ...,  1.k M= − The corresponding re-sampling time moments are tk, 
0,  1,  2,  ...,  1.k M= −  The relation between )( kty  and )( itx  can be described as: 
1
0
sin[( / )( )]
( ) ( ) ,    0,  1,  2,  ...,  
( / )( )
N
k
k i
ki
T t i T
y t x t k M
T t i T
π
π
−
=
∆ − ∆
= =
∆ − ∆∑  (5) 
Practically, the above equation suffers from big computational volume, so it can be 
substituted with interpolation operation under the permission of computational precision. 
By far, the key of equal angular interval based re-sampling is to determine the right re-
sampling time moments ,  0,  1,  2,  ...,  1.kt k M= −  In fact, the re-sampling moments kt  can be 
easily obtained with the following steps (Fig. 5). Firstly, evenly divide the phase axial with 
appropriate angular interval .a∆  For example a∆  can be determined by 2 / ,a mπ∆ =  where 
m  indicates the number of points per circle in re-sample signal. Then the horizontal ordinates 
of each cross-point of parallel lines and the unwrapped phase curve are the re-sampling 
moments .kt  
 
 
Fig. 5. Determination of the re-sampling interval 
 
4. Implementation of time synchronous average 
 
Based on the above work, equal angular interval based re-sampling signal is obtained. 
Suppose re-sampled signal ( ) ( ) ( ),k k ky t f t n t= +  where ( ),  ( )k kf t n t  are period signal and 
noise respectively, and it is truncated into P sections with m points in each section. Then the 
time synchronous average can be described as:  
1 1 1
0 0 0
1 1 1 1
( ) ( ) ( ) ( ) ( ) ( )
P P P
k k k k k k
i i i
y t y t f t n t f t n t
P P P P
− − −
= = =
= = + = +∑ ∑ ∑ , 0,  1,  2,  ...,  .k m=  (6) 
Steps of implementation of time synchronous average for speed varying signal can be 
summarized bellow. 
1) Vibration signal collecting with equal time interval mode.  
2)  Performing time-frequency analysis to obtain time-frequency diagram. 
3)  Instantaneous speed abstracting and modeling. 
4) Performing time-frequency filtering and Hilbert transform to get phase curve. 
 901. A NEW TIME SYNCHRONOUS AVERAGE METHOD FOR VARIABLE SPEED OPERATING CONDITION GEARBOX. 
XINING ZHANG, GUANGRUI WEN, TINGTING WU 
 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2012. VOLUME 14, ISSUE 4. ISSN 1392-8716 
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5) Signal re-sampling with equal angular interval. 
6) Performing time synchronous average. 
 
5. Experiment and verification 
 
In order to verify the performance of the proposed method, experiment is conducted on the 
gearbox test rig (Fig. 6 and Fig. 7). Key phase signal is collected from the input axis with 
photoelectric sensor. Vibration signals are collected from left and right bearing cases of gear 
box. The number of teeth of each gear in the experiment is illustrated in Fig. 6. Sampling 
frequency is set to be 12000 Hz in the experiment. The range of speed fluctuation of the input 
shaft is 800 rpm ~1500 rpm  during the experiment. 
 
 
Fig. 6. Test rig for gearbox experiment 
 
 
Fig. 7. Structure of gearbox test rig 
 
Fig. 8 provides zoomed time-frequency diagram of the vibration signal. The curve of the 
instantaneous meshing frequency of gears in the input and intermediate shaft is clearly 
illustrated. The meshing frequency curve can be adopted to build the model and perform 
filtering, because there is a definite relation between meshing frequency and the input speed.  
Fig. 9 presents the comparison of the spectrums between the original signal and the re-
sampled signal. The average meshing frequencies of 495 Hz and 310 Hz, as well as modulation 
frequency of shaft rotation are clearly illustrated. Fig. 10 gives the waveform and spectrum of 
signal after time synchronous average. Waveform in Fig. 10(a) illustrates the vibration of gear 
 901. A NEW TIME SYNCHRONOUS AVERAGE METHOD FOR VARIABLE SPEED OPERATING CONDITION GEARBOX. 
XINING ZHANG, GUANGRUI WEN, TINGTING WU 
 
 
 
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on input shaft during one round rotation, and variation of vibration amplitude in one round 
rotation is revealed. 26 vibration cycles which are corresponding to 26 gear teeth respectively 
can be easily determined. The same result can also be observed in Fig. 10(c) for the gear on the 
intermediate shaft.  
 
Fig. 8. Zoomed time-frequency diagram  
 
 
Fig. 9. Spectrums before and after filtering 
 
6. Conclusion 
 
A new method to implement TSA without key-phase signal for variable speed condition 
gearbox is proposed in the paper. The presented method is based on the estimation of 
instantaneous speed with time-frequency domain filtering and the equal angular interval re-
sampling of vibration signal. From the above study, some main results can be summarized:  
(1) The method of instantaneous frequency estimation with time-frequency analysis and 
time-frequency filtering is feasible and practical, especially in the case of wide range of speed 
fluctuation of input shaft. 
 901. A NEW TIME SYNCHRONOUS AVERAGE METHOD FOR VARIABLE SPEED OPERATING CONDITION GEARBOX. 
XINING ZHANG, GUANGRUI WEN, TINGTING WU 
 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2012. VOLUME 14, ISSUE 4. ISSN 1392-8716 
1773 
 
 
Fig. 10. Waveform and spectrum after time synchronous averaging: 
(a) signal waveforms on input shaft, (b) FFT spectrum of gears on input shaft,  
(c) signal waveforms on intermediate shaft, (d) FFT spectrum of gears on intermediate shaft 
 
(2) Hilbert transform based equal angular interval re-sampling method can be employed to 
implement the re-sampling for vibration signal under varying speed condition. And in the mean 
time, equal angular interval re-sampling can eliminate the influence of speed variation to 
subsequent analysis and diagnosis. 
(3) The proposed time synchronous average for vibration signal under variable speed 
operating condition is effective, and it demonstrates wide perspective and applicability in the 
 901. A NEW TIME SYNCHRONOUS AVERAGE METHOD FOR VARIABLE SPEED OPERATING CONDITION GEARBOX. 
XINING ZHANG, GUANGRUI WEN, TINGTING WU 
 
 
 
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monitoring and diagnosis of wind power generators, automobiles and other applications using 
variable speed gear boxes. 
 
Acknowledgements 
 
This work was accomplished thanks to the Funds of National Natural Science Foundation of 
China (No. 51075323) and the National High Technology Research and Development Program 
of China (863 Program No. 2012AA040913). 
 
References 
 
[1] G. Dalpiaz, A. Rivola, R. Rubini Gear fault monitoring: comparison of vibration analysis 
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[2] S. Braun The synchronous (time domain) average revisited. Review Article. Mechanical Systems 
and Signal Processing, Vol. 25, Issue 4, 2011, p. 1087-1102. 
[3] Jian Song, Qing Wu, Changyong Pan, Zhixing Yang, Haitao Liu, Bingzhen Zhao, Xiao Li 
Two-way digital video transmission over medium-voltage power-lines using time-domain 
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[4] T. Heyns, P. S. Heyns, J. P. De Villiers Combining synchronous averaging with a Gaussian 
mixture model novelty detection scheme for vibration-based condition monitoring of a gearbox. 
Mechanical Systems and Signal Processing, Vol. 32, 2012, p. 200-215. 
[5] Enayet B. Halim, M. A. A. Shoukat Choudhury, Sirish L. Shah, Ming J. Zuo Time domain 
averaging across all scales: A novel method for detection of gearbox faults. Mechanical Systems and 
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[7] Ming Yang, Viliam Makis ARX model-based gearbox fault detection and localization under 
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[8] Zhipeng Feng, Ming Liang, Yi Zhang, Shumin Hou Fault diagnosis for wind turbine planetary 
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separation. Renewable Energy, Vol. 47, 2012, p. 112-126. 
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[10] P. D. McFadden A revised model for the extraction of periodic waveforms by time-domain 
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[11] F. Bonnardot, M. El Badaoui, R. B. Randall, J. Daniere, F. Guillet Use of the acceleration signal 
of a gearbox in order to perform angular resampling (with limited speed fluctuation). Mechanical 
Systems and Signal Processing, Vol. 19, 2005, p. 766-785. 
[12] F. Combet, L. Gelman An automated methodology for performing time synchronous averaging of a 
gearbox signal without speed sensor. Mechanical Systems and Signal Processing, Vol. 21, 2007,      
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[13] Francois Combet, Radoslaw Zimroz A new method for the estimation of the instantaneous speed 
relative fluctuation in a vibration signal based on the short time scale transform. Mechanical Systems 
and Signal Processing, Vol. 23, 2009, p. 1382-1397. 
[14] L. Cohen Time-Frequency Analysis. Prentice Hall, Englewood Cliffs, NJ, 1995. 
[15] M. Gröchenig Foundations of Time-Frequency Analysis. Birkhäuser, Boston, 2001. 
[16] Qiansheng Cheng Mathematical Theory of Signal Digital Processing. Petroleum Industry 
Publishing House, Beijing, 1979. 


A new time synchronous average method for variable speed operating condition gearbox

Zhang, Xining

Wen, Guangrui

Wu, Tingting

Journal of Mechatronics and Artificial Intelligence in Engineering

  VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2012. VOLUME 14, ISSUE 4. ISSN 1392-8716 1766 901. A new time synchronous average method for variable speed operating condition gearbox  Xining Zhang1, Guangrui Wen2, Tingting Wu3  Department of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, China 2Corresponding author E-mail: 1zhangxining@mail.xjtu.edu.cn, 2grwen@mail.xjtu.edu.cn, 3wtingting@mailst.xjtu.edu.cn   (Received 17 September 2012; accepted 4 December 2012)  Abstract. Gearbox is a widely used component for power transmission and speed change. Time synchronous average (TSA) is one of the most effective methods for vibration monitoring and diagnosis of gearboxes. Traditional TSA technique requires key-phase signal and constant operating speed. So the application of TSA is difficult in many situations such as in the case of gearboxes used in wind power generators and automobiles. A new method to implement TSA without key-phase signal for variable speed condition gearbox is proposed in the paper. The reported method is based on the estimation of instantaneous speed with time-frequency domain filtering and equal angular interval re-sampling of vibration signal. Experimental investigation performed in a variable speed gearbox test rig indicates that the proposed method can eliminate the influence of large speed fluctuation of gearboxes and provide satisfactory TSA results.  Keywords: time synchronous average, variable speed condition gearbox, time-frequency filtering, equal angular interval re-sampling.  1. Introduction  Being one of the most important parts of power transmission and speed changing, a gearbox is widely used in many industry fields. Due to its easiness to measure and analyze, vibration signals have been commonly employed in condition monitoring and diagnosis of gearboxes. The time synchronous average (TSA) technique is an effective vibration monitoring method for gearbox [1-3]. TSA can effectively attenuate the uncorrelated vibration components with the selected revolve frequency on spectrum. In the mean time, the TSA waveform intuitively reveals the condition of the monitoring gears on the selected shaft [4-7].  The key problem of TSA technique is to obtain the vibration samples against revolving angle. Commonly, there are two ways to achieve the above purpose. One is to employ the phase-lock technique with key-phase signal, while the other is to use a re-sampling method. The first method can track the speed fluctuation and directly collect the vibration samples against revolving angle. But it requires key-phase sensor and complex phase-lock circuit, which fails to achieve good tracking performance in large speed fluctuation. In many circumstances, it is difficult to install the key-phase sensor on the gearbox especially on the intermediate shaft. Gearboxes in wind power generator, machine tool, helicopter, mining excavator and automobile, is operating under large speed fluctuation condition [8-9]. The re-sampling method is based on instantaneous speed estimation and can process large speed fluctuation [10]. TSA based on instantaneous speed estimation has attracted great interest recently.               F. Bonnardot gave an angular re-sampling method without tachometer, which is based on the estimation of meshing frequency or it's harmonics by using a band-pass filter [11]. Sideband inside the filter band, which is produced by the slow speed fluctuation, is used to estimate the instantaneous speed. The method works well only under small speed fluctuation, which is less than the minimum speed of two meshing gears. On the basis of above work, F. Combet and     L. Gelman extended automatic angular re-sampling algorithm for the estimation of the TSA signal of a particular shaft in a gearbox [12]. The innovation of the scheme is mainly concentrated on the automatic selection of the mesh harmonic, which is used for the shaft speed estimation. F. Combet and R. Zimroz proposed a method to estimate the instantaneous speed  901. A NEW TIME SYNCHRONOUS AVERAGE METHOD FOR VARIABLE SPEED OPERATING CONDITION GEARBOX. XINING ZHANG, GUANGRUI WEN, TINGTING WU     VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2012. VOLUME 14, ISSUE 4. ISSN 1392-8716 1767 relative fluctuation with short-time scale transform of vibration signal in 2009 [13]. The method still suffers from the limitation of small speed fluctuation.  In fact, the speed fluctuation of one shaft will result in the fluctuation of relative shaft rotation frequency and meshing frequencies. The instantaneous speed fluctuation can be well illustrated in joint time-frequency diagrams. The instantaneous speed can be extracted with time-frequency domain filtering. Based on the above thought, the paper gives a new time synchronous average method for variable speed gearbox without key-phase signal, and the method can be applied to the variable speed gearbox with large speed fluctuation. Variable speed gearbox test rig was built, and measurements were taken with accelerometers. The performance of the proposed method is verified by the processing of experimental data.  2. Estimation of instantaneous speed  Joint time-frequency analysis provides a tool for us to observe the signal both in time and frequency domain [14, 15]. Short Time Fourier Transform (STFT) is the most widely used method for non-stationary signal. So, it provides us a tool to view and track the instantaneous speed variations on the time-frequency diagram. The basic principle of STFT is as follow: break up the signal into small time segments with filter, and then analyze each time segment with Fourier transform to get the spectrum. The totality of such spectrum indicates how the spectrum is varying in time. The transform formula of STFT and ISTFT are: dtethtxfSTFT ftjxπττ 2)()(),( −+∞∞−−= ∫  (1) ωτωπτ ωτ dtdethtSTFTx jx )(),(21)( −= ∫ ∫+∞∞−+∞∞− (2) where )( τ−th  is the moving window around τ , ),( fSTFTx τ  is the time-frequency distribution.  Suppose we have a frequency modulated signal: 10sin(2 200 8sin(2 10 )) 10sin(2 400 8sin(2 10 )),x t t t tπ π π π= × + × + × + ×  digitizing x  with the equal time interval sampling we can get the corresponding digital signal ,  1,  2,  3,  ...,  .ix i N=  Fig. 1 provides the time-frequency diagram of signal x. The time-frequency diagram directly illustrates the frequency varying of a component against the time. The variations of the instantaneous frequencies of each component are clearly observed on the diagram.    Fig. 1. Time-frequency diagram   901. A NEW TIME SYNCHRONOUS AVERAGE METHOD FOR VARIABLE SPEED OPERATING CONDITION GEARBOX. XINING ZHANG, GUANGRUI WEN, TINGTING WU     VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2012. VOLUME 14, ISSUE 4. ISSN 1392-8716 1768 In order to extract the varying instantaneous frequency component around 400 Hz, the coordinates of each maximum amplitude points of the above selected component is searched along the time axis. The model of instantaneous frequency against time is built with the coordinates. With the model we can obtain the value of instantaneous frequency at any moment of time axial.   3. Equal angular interval re-sampling of vibration signal  Time synchronous average requires angular based sampling time series. For an equal time interval sampling vibration signal at varying operating speed condition, it is essential to perform equal angular interval re-sampling [16]. In order to determine time moments of each re-sampling points, time-frequency domain filtering technique and Hilbert Transform are introduced. A. Time frequency filtering Suppose the model of the varying operating speed component is built with the above method. Band-pass filter curves in frequency domain are superimposed on the diagram at the point of maximum amplitude along the frequency axis of each moment on the time-frequency diagram. Fig. 2 indicates filter curve superimposed at point P(k, f). Fig. 3(a) gives the filtered time-frequency diagram. With inverse short time Fourier transforms, the filtered signal (Fig. 3(b)) can be easily calculated. Obviously, the filtered signal is a mono-frequency signal at any moment.    Fig. 2. Principle of time-frequency filtering  B. Phase information extraction with Hilbert transform For a time series of )(tx , the definition of Hilbert transform is as follows: τττπdtxty ∫+∞∞− −=)(1)(  (3) where )(ty  is the result of Hilbert transform. The analytic signal of signal )(tx  can be constructed as: )()()()()( tjetatjytxtz Φ=+=                                                                       (4) where )(ta  is the amplitude function, 2 2( ) ( ) ( ) ,a t x t y t= +  )(tΦ  is the phase function,  901. A NEW TIME SYNCHRONOUS AVERAGE METHOD FOR VARIABLE SPEED OPERATING CONDITION GEARBOX. XINING ZHANG, GUANGRUI WEN, TINGTING WU     VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2012. VOLUME 14, ISSUE 4. ISSN 1392-8716 1769 )()(arctan)(txtyt =Φ . Fig. 4 gives the unwrapped phase curve of the filtered signal.   (a) Filtered time-frequency diagram  (b) Filtered signal Fig. 3. Result of time-frequency filtering   Fig. 4. Phase curve against time  901. A NEW TIME SYNCHRONOUS AVERAGE METHOD FOR VARIABLE SPEED OPERATING CONDITION GEARBOX. XINING ZHANG, GUANGRUI WEN, TINGTING WU     VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2012. VOLUME 14, ISSUE 4. ISSN 1392-8716 1770 C. Equal angular interval re-sampling The aim of re-sampling is to transform the original equal time interval signal to equal angular interval signal. Suppose the original signal and re-sampling signal time moments are ( ),ix t 0,  1,  2,  ...,  1i N= − (∆T is an original sampling interval) and ( ),ky t  0,  1,  2,  ...,  1.k M= − The corresponding re-sampling time moments are tk, 0,  1,  2,  ...,  1.k M= −  The relation between )( kty  and )( itx  can be described as: 10sin[( / )( )]( ) ( ) ,    0,  1,  2,  ...,  ( / )( )Nkk ikiT t i Ty t x t k MT t i Tππ−=∆ − ∆= =∆ − ∆∑  (5) Practically, the above equation suffers from big computational volume, so it can be substituted with interpolation operation under the permission of computational precision. By far, the key of equal angular interval based re-sampling is to determine the right re-sampling time moments ,  0,  1,  2,  ...,  1.kt k M= −  In fact, the re-sampling moments kt  can be easily obtained with the following steps (Fig. 5). Firstly, evenly divide the phase axial with appropriate angular interval .a∆  For example a∆  can be determined by 2 / ,a mπ∆ =  where m  indicates the number of points per circle in re-sample signal. Then the horizontal ordinates of each cross-point of parallel lines and the unwrapped phase curve are the re-sampling moments .kt    Fig. 5. Determination of the re-sampling interval  4. Implementation of time synchronous average  Based on the above work, equal angular interval based re-sampling signal is obtained. Suppose re-sampled signal ( ) ( ) ( ),k k ky t f t n t= +  where ( ),  ( )k kf t n t  are period signal and noise respectively, and it is truncated into P sections with m points in each section. Then the time synchronous average can be described as:  1 1 10 0 01 1 1 1( ) ( ) ( ) ( ) ( ) ( )P P Pk k k k k ki i iy t y t f t n t f t n tP P P P− − −= = == = + = +∑ ∑ ∑ , 0,  1,  2,  ...,  .k m=  (6) Steps of implementation of time synchronous average for speed varying signal can be summarized bellow. 1) Vibration signal collecting with equal time interval mode.  2)  Performing time-frequency analysis to obtain time-frequency diagram. 3)  Instantaneous speed abstracting and modeling. 4) Performing time-frequency filtering and Hilbert transform to get phase curve.  901. A NEW TIME SYNCHRONOUS AVERAGE METHOD FOR VARIABLE SPEED OPERATING CONDITION GEARBOX. XINING ZHANG, GUANGRUI WEN, TINGTING WU     VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2012. VOLUME 14, ISSUE 4. ISSN 1392-8716 1771 5) Signal re-sampling with equal angular interval. 6) Performing time synchronous average.  5. Experiment and verification  In order to verify the performance of the proposed method, experiment is conducted on the gearbox test rig (Fig. 6 and Fig. 7). Key phase signal is collected from the input axis with photoelectric sensor. Vibration signals are collected from left and right bearing cases of gear box. The number of teeth of each gear in the experiment is illustrated in Fig. 6. Sampling frequency is set to be 12000 Hz in the experiment. The range of speed fluctuation of the input shaft is 800 rpm ~1500 rpm  during the experiment.   Fig. 6. Test rig for gearbox experiment   Fig. 7. Structure of gearbox test rig  Fig. 8 provides zoomed time-frequency diagram of the vibration signal. The curve of the instantaneous meshing frequency of gears in the input and intermediate shaft is clearly illustrated. The meshing frequency curve can be adopted to build the model and perform filtering, because there is a definite relation between meshing frequency and the input speed.  Fig. 9 presents the comparison of the spectrums between the original signal and the re-sampled signal. The average meshing frequencies of 495 Hz and 310 Hz, as well as modulation frequency of shaft rotation are clearly illustrated. Fig. 10 gives the waveform and spectrum of signal after time synchronous average. Waveform in Fig. 10(a) illustrates the vibration of gear  901. A NEW TIME SYNCHRONOUS AVERAGE METHOD FOR VARIABLE SPEED OPERATING CONDITION GEARBOX. XINING ZHANG, GUANGRUI WEN, TINGTING WU     VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2012. VOLUME 14, ISSUE 4. ISSN 1392-8716 1772 on input shaft during one round rotation, and variation of vibration amplitude in one round rotation is revealed. 26 vibration cycles which are corresponding to 26 gear teeth respectively can be easily determined. The same result can also be observed in Fig. 10(c) for the gear on the intermediate shaft.   Fig. 8. Zoomed time-frequency diagram    Fig. 9. Spectrums before and after filtering  6. Conclusion  A new method to implement TSA without key-phase signal for variable speed condition gearbox is proposed in the paper. The presented method is based on the estimation of instantaneous speed with time-frequency domain filtering and the equal angular interval re-sampling of vibration signal. From the above study, some main results can be summarized:  (1) The method of instantaneous frequency estimation with time-frequency analysis and time-frequency filtering is feasible and practical, especially in the case of wide range of speed fluctuation of input shaft.  901. A NEW TIME SYNCHRONOUS AVERAGE METHOD FOR VARIABLE SPEED OPERATING CONDITION GEARBOX. XINING ZHANG, GUANGRUI WEN, TINGTING WU     VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2012. VOLUME 14, ISSUE 4. ISSN 1392-8716 1773   Fig. 10. Waveform and spectrum after time synchronous averaging: (a) signal waveforms on input shaft, (b) FFT spectrum of gears on input shaft,  (c) signal waveforms on intermediate shaft, (d) FFT spectrum of gears on intermediate shaft  (2) Hilbert transform based equal angular interval re-sampling method can be employed to implement the re-sampling for vibration signal under varying speed condition. And in the mean time, equal angular interval re-sampling can eliminate the influence of speed variation to subsequent analysis and diagnosis. (3) The proposed time synchronous average for vibration signal under variable speed operating condition is effective, and it demonstrates wide perspective and applicability in the  901. A NEW TIME SYNCHRONOUS AVERAGE METHOD FOR VARIABLE SPEED OPERATING CONDITION GEARBOX. XINING ZHANG, GUANGRUI WEN, TINGTING WU     VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2012. VOLUME 14, ISSUE 4. ISSN 1392-8716 1774 monitoring and diagnosis of wind power generators, automobiles and other applications using variable speed gear boxes.  Acknowledgements  This work was accomplished thanks to the Funds of National Natural Science Foundation of China (No. 51075323) and the National High Technology Research and Development Program of China (863 Program No. 2012AA040913).  References  [1] G. Dalpiaz, A. Rivola, R. Rubini Gear fault monitoring: comparison of vibration analysis techniques. Proceedings of the Third International Conference on Acoustical and Vibratory Surveillance Methods and Diagnostic Techniques, Senlis, France, 2006, p. 623-637. [2] S. Braun The synchronous (time domain) average revisited. Review Article. Mechanical Systems and Signal Processing, Vol. 25, Issue 4, 2011, p. 1087-1102. 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A new time synchronous average method for variable speed operating condition gearbox

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