Study of Time-Frequency Order Tracking of Vibration Signals of Rotating Machinery in Changing State

Xiaoping Zhao Institute of Vibration Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing

210016，xiaopingzhao@yahoo.cn Qingpeng Kong

Collage of electronics and information, Hangzhou Dianzi university, xiasha higher

education district, Hangzhou. 310018，jonah119@163.com

Qintao Guo College of Mechanical and Electrical Engineering, Nanjing University of

Aeronauticsand Astronautics, Nanjing 210016

Abstract

Order tracking is an important method for condition monitoring and fault diagnosis, dynamic analysis and design in rotating machinery. Especially, the hidden fault is usually easier to discover at engine run up/down. Order tracking is implemented by band pass filter in time-frequency domain and time signal of desired order is reconstructed. Compared to other existing order analysis methods, this technique is not only more intuitive and more powerful, and can also be applied where rotational speed information is not available. The experimental results show that the method is suitable for vibration response character extraction of rotating machinery. 1. Introduction

Vibration analysis of rotating machinery is an important part of industrial predictive maintenance programmes, so that wear and defects in moving parts can be discovered and repaired before the machine breaks down, thus reducing operating and maintenance costs. One method of vibration analysis is known as order tracking. The aim of the order tracking is to extract selected orders from the vibration signal [1].

The order-tracking (OT) techniques assigning a specific shaft speed as a basic order have been applied in the analysis of dynamic signals measured from rotary machinery for more than two decades [2]. Comparing the GOT with other OT schemes, only the GOT technique links two families of signal processing tools together,

namely, the TF representation (TFR) that handles transient and nonstationary signals and the OT techniques that characterise and reconstruct target order/spectral components embedded in a processed signal.

In this paper，the GOT scheme is applied in a engine without tacho information, and investigating dynamic characteristics of the machine. As results, while the Gabor coefficients characterized in an rev/minF (or rev/ min-order) plane are partially selected, the waveform of an embedded single or multiple order/spectral components be reconstructed. This is the basic idea of the GOT technique that cannot be accomplished through other OT [2].

2. Discrete Gabor expansions

In 1946, Dennis Gabor, a Hungarian-born British physicist, suggested expanding a signal into a set of functions that are concentrated in both the time and frequency domains [4]. Then, use the coefficient as the description of the signal’s local property. The resulting representation is now known as the Gabor expansion.

Applying the sampling theorem and the Poisson-sum formulate [5] [6], Wexler and Raz obtained a discrete Gabor expansion for the finite and periodic sequence ( )s i� with a period L as follows:

1 1

, ,0 0

( ) ( )M N

m n m nm n

s i C h i− −

= =

= ∑∑ �� (1)

, ( ) ( ) n Nim n Lh i h i m M W Δ= − Δ� � (2)

2008 International Symposiums on Information Processing

978-0-7695-3151-9/08 $25.00 © 2008 IEEEDOI 10.1109/ISIP.2008.116

561

2008 International Symposiums on Information Processing

978-0-7695-3151-9/08 $25.00 © 2008 IEEEDOI 10.1109/ISIP.2008.116

559

2 /n Ni n Ni LLW e πΔ Δ= (3)

1*

, ,0

( ) ( ) ( )L

m n m ni

C i s i r i−

=

=∑ � � (4)

Where MΔ and NΔ are time and frequency sampling intervals, respectively. M and N are the numbers of sampling points in time and frequency domains.

Qian [4] introduced a more efficient method developed recently is an iterative algorithm in which we continuously apply the same mask and reconstruct the time waveform, reference the papers[1][2][7].

3. Gabor Order tracking

Gabor transform suggested that the coefficient ,m nC

reflected a signal’s behavior in vicinity of[ , ]m M n NΔ Δ , in this method, lengths of the analysis and synthesis widow functions are the same, while perfect reconstruction is guaranteed.

3.1. The main steps of GOT

Step 1: Dynamic signals are acquired from an operational machine in a changing state;

Step 2: Set synthesis window function ( )h i and analysis window function ( )iγ ;

Step 3: Compute the Gabor coefficient ,m nC ; Step 4: By time-frequency analysis, the instantaneous

frequency 1( )if t can be acquired through peak search based adaptive short time Fourier transform (ASTFT);

Step 5: By using the Gabor coefficient, the 2D map and 3D map can be displayed.

Step 6: Time-varying Filtering matrix Order tracking is implemented by band pass filter in

time-frequency domain and time signal of desired order is reconstructed. After signal component ( pth ) is selected,

instantaneous frequency (IF) ( )pif t will be obtain

through 1( ) ( )pi if t p f t= × . Compute the

coordinates ( , )m n of the selected order component which in the time-frequency plane, (where m and n are time and frequency coordinates, respectively.). And search the IF coordinate kn at the sampling point km in the Gabor TF grid.

( ) / 0, , 1pk i kn INT f m N k M⎡ ⎤= Δ = −⎣ ⎦ � (14)

Where INT is integrate， NΔ is frequency step。 Step 7: Reconstruction Take the Gabor coefficient of p th component into equ

(1), make the inverse TF transform and realized the reconstruction of the selected component.

Fig 1. Processing diagram of GOT technique

3.2. Influences of dual functions on Gabor

expansions

The key issue of implementing the Gabor expansion is the selection of synthesis window function ( )h i and analysis window function ( )iγ . In order to have localized Gabor coefficient, the particularly interesting choice of

( )iγ is one that is most similar to the given ( )h i since such constraint will lead to localized and perfect reconstruct.

Select Gaussian window function, such as：

[ ]22 0.252

0.5( 1)( ) ( ) exp

2i L

h i πδδ

−⎧ ⎫− −⎪ ⎪= −⎨ ⎬⎪ ⎪⎩ ⎭

(15)

Where 0 i L≤ < 。

2

2M LN

δπ

Δ=Δ

Lδ � （16）

Acquire vibration data

Gabor coefficient,m nC

ASTFT acquire the instantaneous frequency of base axis

Select order p and Time-varying Filtering matrix

Display 2D,3D

Time-Frequency filter

Gabor expansion and reconstruction

Synthesis window function ( )h i Analysis window function ( )iγ

562560

During extensive tests found that the error Γ decreases

as the over sampling ratio,NMΔ

, increases[4], Fig2

demonstrates ( )h i and corresponding ( )opt iγ . Note that

the difference between ( )h i and ( )opt iγ is inversely proportional to the over sampling rate. For critical sampling, the solution of ( )iγ is unique. As the over

sampling rate NMΔ

increases, ( )opt iγ becomes closer

to ( )h i . It is shown that for the GOT there tradeoffs between the length of element function, over-sampling rate, computation efficiency, and TF resolution while used in tracking target spectral/order components. From Fig.2, we can see that the error Γ decrease apparently and satisfy the request of application when over sampling is 4.

Fig 2. ( )h i (solid line) and ( )opt iγ (dotted line)(as the

oversampling rate increates, ( )opt iγ becomes more and

more close to ( )h i 4. Tests 4.1. Numerical simulation Table 1. Spectrum components composed in synthetic signal

Order components

1 2 3

Amplitude Linearly increasing From 0 to 0.8

1.5 (fixed) Linearly increasing From 0 to 0.5

The simulation signal is designed to validate the implemented GOT schemes, the reference speed linearly discrete from 0 rpm to 8000 rpm in 5s. The synthetic signal comprises three order components including order 1, 2, and 3. The amplitudes of those orders are illustrated in Table1. A sampling frequency 51.2 kHz is employed in this signal.

(a) Gabor coefficient of simulation signal

(b) Gabor coefficient of the second order signal

(c) Second order signal

563561

(d) Reconstructed the second order signal

Fig 3. GOT of Simulate signal OT is applied to the simulation signal and the

corresponding spectrograms are shown as Fig3 (a), the order components of signal were clear exposed. The Gabor coefficient of the second component of signal was shown as Fig3 (b). Compare Fig3 (d) with Fig3 (c), it is clearly that the amplitude of the front the end of reconstruct signal have some different because of the Gabor transform window and zero pad. But the center part of the reconstructed signal is the same as the simulation. So GOT is feasible.

4.2. Engineering applications

(a) Gabor spectrum

In the section, a rotation machine was done to demonstrate the validity of Gabor based order tracking. The analysis function is a 1024-point Hamming window. The number of frequency bins N is equal to the window length wL .The over sampling rate is four (see Fig2). Consequently, the difference between the analysis and synthesis windows is negligible.

The order components of signal were clear exposed (see Fig4 (a)). The horizontal axis represents time t in

seconds while vertical axis denotes frequency f scaled by Hz.Fig4(b)is the first order component and Fig4 (c) is the Second order component through time-frequency filter. The reconstruction of the first order is shown Fig4 (d).

(b) The spectrum of the first order

(c) The spectrum of the second order

(d) The actual signal and Reconstruction

Fig 4. GOT of actual signal 5. Conclusions

In this paper, discrete Gabor expansion for order tracking-a popular application is introduced in the engine machine. Simulations and actual tests indicate the

564562

applying condition is satisfied, the order tracking based Gabor is feasible in engineering applications.

This method is an effective supplement to traditional methods on order tracking and is more attractive on the condition where the keyphasor equipment’s installation is inconvenient. Comparing the GOT with other OT schemes, only the GOT technique links two families of signal processing tools together, namely, the TF representation (TFR) that handles transient and nonstationary signals and the OT techniques that characterise and reconstruct target order/spectral components embedded in a processed signal. *Supported by National Natural Science Foundation of China（50675105） 6. References [1] K.R.Yfe, and E.D.S.Munck, ¡°Analysis of computed

order tracking.¡± Mechanical Systems and Signal Processing (1997) 11(2), 187-205.

[2] Hui Shao, Wei Jin, and Shie Qian, Order Tracking by Discrete Gabor Expansion. IEEE Transactions on Instrumention and Measurement , Vol. 52, No.3,June 2003.

[3] Characteristics of the Vold-Kalman order tracking filter. H.Herlufsen. S.Gade, H.Konstantin-Hansen, B&k, Denmark H.Vold, Vold solution Inc, USA.

[4] D.Gabor, ¡°Theory of communication.¡± J.IEEE, vol.93, no. Ⅲ,pp. 429-457, November 1946.

[5] Shie Qian. Discrete Gabor Transform. IEEE transactions on signal processing. Vol.41.No.7.July.1963.

[6] S.Qian and D.Chen. Joint Time-Frequency Analysis. Englcwood Cliffs, NJ: Prentice-Hall, 1996.

[7] Xiang-Gen xia, Shie Qian. An Iterative Algorithm for Time-Variant Filtering in the Discrete Gabor Transform Domain. IEEE 1997.

[8] kong Qing-peng, Song Kai-chen. Study of time-frequency order tracking of vibration signals in engine speed changing stage. Journal of Vibration Engineering. Vol.18 No.4.Dec.2005.

[9] Xiang-Gen xia, Shie Qian. Convergence of an Iterative Time-Variant Filtering Based on Discrete Gabor Transform. IEEE Transations on Signal Processing, Vol. 47, No.10, October 1999.

[10] Y Gao, Y Gao. Order Tracking Based on Robust Peak Search Instantaneous Frequency Estimation. Journal of Physics:Conference series 48 (2006) 479-484.

[11] Kong Qing-peng, Song Kai-chen. Study on least-quare adaptive filter order tracking in rotating machinery. Journal of Zhejiang University (Engineering Science). Vol.40 No.9. Sep.2006.

565563