Study on Vibration Signal Denoising of Electric Spindle Based on Wavelet Transform

Gao Rong Department of Mechanical Engineering

Huai Yin Institute of Technology Huaian ，China

ggrr7012@163.com

Abstract：An testing and processing method of electric spindle

status is presented in this article, It is based on wavelet

transform. Wavelet transform is selected as its processing

platform ,it is very important. The theory of wavelet transform

and Mallat method are studied deeply.The main point

discussed in this paper is how to handle the noises, especially

about acceleration. Finally,according to signals and noises

presenting different characteristics after wavelet transform,

the inverse wavelet transform can be used to reconstruct

signal，The simulating results show that signal denoising and

comeback will be realized.It is significant that can present

actual characteristics,it is helpful to science study and decision

by CNC.

Key words：Electric spindle; Wavelet transform; Vibration；

Denoising

I. INTRODUCTION High speed electric spindle is the key functional

component of high precision CNC machine tool, in which the main spindle electric is mounted to the spindle and become a complete built-in spindle unit. With the developments of CNC machine tool and high speed cutting technologies, Electric spindles are to have high speed, high power, high accuracy, high rigidity, accurate pointing (accurate stop) and ultra high speed. Before electronic spindles can be widely applied for ultra high speed machining, they must be compact in structure, light in weight, with a small inertia and excellent dynamic performance, and vibration, pollution and noise should also be avoided. Therefore, we must be able to control electric spindles by collecting and processing various spindle signals such as rotation speed, vibration and temperature rise. However, due to complexity of working environments of NC machines, as well as influence of cutting parameters and the interference of strong external electromagnetism, signals reflecting the running status of electric spindles are

often overwhelmed by noise signals. So we must extract status signals out from noise signals, in order to control the CNC system[1-3]. II. PROCESSING METHODE OF ELECTRIC SPINDLE VIBRATION SIGNAL Being a new field of mathematics, wavelet analysis is developing fast in recent years which can realize localized transform of time and frequency and effective extraction of information from signals, just as Fourier transform and windowed Fourier transform. Fourier transform proves less effective for detection of HF signals and research of LF signals; however, wavelet analysis has a flexible time-frequency window which enables automatic narrowing when at “center frequency” to increase the calculation accuracy. Besides, wavelet analysis can handle problems when Fourier transform proves helpless by making multiscale analysis of functions or signals through dilation and shift operations, and can make the position of source signals and their singularities accurately reappear. Therefore, wavelet transform has been extolled as “Mathematical Microscope”, representing a significant milestone progress in the development of harmonic analysis. Wavelet analysis has already obtained significant achievements which are of scientific and application value in many subjects of mathematics, such as signal processing, image processing, quantum mechanics, electronic countermeasure, computer recognition, data compression, CT imaging, seismic data processing, edge detection, artificial synthesis of music and voice, mechanical fault diagnosis, analysis of atmosphere and ocean wave, fractal mechanics, flow, turbulence and celestial mechanics. Except for solving of differential equations, wavelet analysis is in principle applicable wherever Fourier analysis can be applied, and better results can be obtained. Analysis and materials show that wavelet transform can effectively extract characteristics of the spindle’s status signals (acceleration signal, for instance), and predict the development trend of status signals of the spindle[4-6]. III. WAVELET THOERY OF ELECTRIC SPINDLE VIBRATION SIGNAL

2009 International Conference on Intelligent Human-Machine Systems and Cybernetics

978-0-7695-3752-8/09 $25.00 © 2009 IEEE

DOI 10.1109/IHMSC.2009.24

62

Suppose ( ) ( )RLt 2∈ψ , wherein ( )tψ stands

for the square integrable function and ( )2L R stands for

the square integrable space, and the Fourier transform of

( )tψ is ( )ωψ∧

; when ( )ωψ∧

meets the admissible

condition of ( )

∞<=

∧

∫ ωω

ωψψ dC R

2

, ( )tψ is

regarded as a basis wavelet or a mother wavelet. After

dilation and shift of generating function ( )tψ , we can get a

wavelet sequence. In case of succession, the wavelet

sequence is ( ) ⎟⎠⎞

⎜⎝⎛ −=

abt

atba ψψ 1

, 0;, ≠∈ aRba ,

wherein a stands for the dilation factor and b stands for the shift factor.

The continuous wavelet transform for randomly

continuous function ( ) ( )RLtf 2∈ :

( ) ( ) dta

bttfafbaW Rbaf ⎟⎠⎞

⎜⎝⎛ −== ∫

− ψψ 21

,,, <1>

And the reconstruction formula (inverse transform) is:

( ) ( ) dadba

btbaWaC

tf fRR ⎟⎠⎞

⎜⎝⎛ −= ∫∫ + ψ

ψ

,112

<2>

In application, what is normally used is binary wavelet transform. During continuous wavelet transform as defined in formula <1> mentioned above, discrete sampling will be

carried out for scale factor a and shift factor b: 2 ja = ,

2 jb k= ( Zkj ∈, ). Now, continuous wavelet transform

becomes binary discrete sampling, and the transform formula is:

21( ) ( ) ( )2 2

jj j

x tW f x f t dtψ+∞

−∞

−= ∫ <3>

If wavelet ( )tψ meets the condition of

( ) BAzj

j ≤≤∑∈

−∧ 2

2 ωψ , wherein A and B are constants,

then there certainly is a reconstruction wavelet ( )f t that

enables ( ) ( )2 2*j j

j Zf t w f x t

∈

=∑ [7-8].

IV. WAVELET DENOISING OF ELECTRIC SPINDLE VIBRATION SIGNAL A. Mallat Algorithms

Suppose { }Znknk ∈,,ψ is the orthogonal wavelet base

of2L , then ( )xf can be expanded for any

2Lf ∈ as

below:

( ) ( )xxf knZn,k

knd ψ∑∈

=

Znkfd knkn ∈∈ ,,ψ

<4>

Because Nj

N

jVWV −

−

−=⊕⊕=

10, we get

∑=

−− +=N

knk fgf

10

, wherein Nnkk VfWg −−−− ∈∈ , ,

and such decomposition is unique. Again, 00 Vf ∈ ,

therefore { } 20 lC Znn ∈∈ , enabling n

ZnnCf 00

0 ϕ∑∈

=,

wherein nn fC 00 ,Φ=

; and now we get:

⎪⎪⎩

⎪⎪⎨

⎧

+=+=

+=+=+=+=

+−+−−+−−+−−

−−−−−−−

−−−−

)1()1()1()1(

2212121

1101010

kkkkkkk fgfQfPf

fgfQfPffgfQfPf

<5>

Wherein jj QP ,is the orthogonal projection operator

of2L towards jV

and jW,

and )1( −−−− = kkk fPP, )1( −−−− = kkk fQQ

.

As represented in:

63

⎪⎩

⎪⎨

⎧

=

=

−∈

−

−∈

−

∑

∑

nkZn

knk

nkZn

knk

dg

Cf

,

,

ψ

ϕ

<6>

Then we get:

⎪⎪⎩

⎪⎪⎨

⎧

==

==

−∈

−−

−∈

−

∑

∑

njZj

jnn

njZj

jnn

gCfQd

hCfC

20

,111

20

,11

21,

21,

ψ

ϕ

<7>

And generally:

⎪⎪⎩

⎪⎪⎨

⎧

=

=

−∈

−

−∈

−

∑

∑

njZj

kj

kn

njZj

kj

kn

gCd

hCC

21

21

21

21

, ,3,2,1=k <8>

The process of 0C decomposition into

Nddd ,,, 21and

Nc is referred as finite orthogonal wavelet decomposition

which proves especially useful for signal processing. B. DECOMPOSITION AND RECONSTRUCTION OF ELECTRIC SPINDLE VIBRATION SIGNAL

Perform Mallat's two-dimensional discrete wavelet transform of status signals of the electric spindle, and the decomposition process is shown in Fig. 2.

( )(0 nH and )(0 nG respectively stand for the low-pass

filter and the high-pass filter). Mallat’s algorithm uses a set of decomposition filters, namely H (low-pass filter LPF) and G (high-pass filter HPF), to filter source signals and proceeds to carry out alternate sampling (taking every other) of output results to realize wavelet decomposition after which we get two halves: one is the smooth portion of the source signal created by the LPF, and the other is the detail portion of the source signal created by the HPF.

Fig.2. Map of Two Dimensions Wavelet Decomposition

The reconstruction process is shown in Fig.3 where

)(1 nH and )(1 nG respectively stand for the LPF and

the HPF. A set of h and g composite filters are used to filter the results of wavelet decomposition, and signal reconstruction is realized through alternate sampling (zero padding between two adjacent points). Multi-level wavelet decomposition is realized through concatenation, and wavelet transform of each level is carried out on the LF component created by decomposition of the previous level; therefore, reconstruction is the inverse operation of decomposition. Information is comparatively rich with concentrated energy on LF components, while HF components have rich detail information of less energy.

Fig.3. Map of Two Dimensions Wavelet Reconstruction V. EXPERIMENT

After wavelet transform, signals obtain excellent multi-resolution representation of space-frequency; not only status characteristics of the source signal can be maintained, but also HF information of signals can be perfectly extracted, with favorable frequency characteristics at low frequency and favorable space selectivity at high frequency. After wavelet transform, noises and signals show different characteristics: noises are singular almost everywhere, with

an average amplitude inverse scale factor 2 j , and the number of average modular maximum values are also

inverse 2 j ; that is to say, the energy of noises drops sharply as the scale increases. Under most circumstances, however, signals are of better smoothness, and their amplitudes basically remain unchanged as the scale varies, if wavelet transform is performed with several small scales. So we can set certain threshold values for different scales, and wavelet coefficient modular maximum values below the threshold values can be regarded as results of noise effect; then we can perform quantitative treatment of threshold values by applying methods such as maximum likelihood estimate to

64

eliminate most of the noises without distorting reconstructed signals.

Wavelet decomposition and reconstruction algorithm can be used for treatment of any non-stationary signal. However, if acceleration signal is regarded as the treatment object, characteristics of acceleration signals can hardly be found out and the development trend of signals cannot be predicted, due to serious influence of noise signals, and therefore we may fail to correctly judge vibration situations of the electric spindle. Shown in Fig. 4 is wavelet noise elimination from acceleration signals, and we can see that wavelet signal reconstruction (i.e. noise signal elimination) can distinctly eliminate noises and make the source signal reappear in a relatively accurate manner; therefore, reconstructed signals can well get rid of the influence of noises while retaining major characteristic information of source signals.

Fig.4 Wavelet Denoising Result of Acceleration Signal

VI. CONCLUSIONS (1) Wavelet transform can be used to process vibration status signals, and simulated processing of acceleration signals of the electric spindle shows that wavelet transform can effectively extract characteristics of the spindle’s status signals, and predict the development trend of status signals of the spindle. (2) Our case study shows that wavelet theory has great application potentials in noise elimination. Monitoring of status signals of the electric spindle can help us handle technical problems with the design and operation of electric

spindles and accurately extract important parameters such as vibration signals, which eventually will strongly guarantee spindle control and decision-making for CNC system.

ACKNOWLEDGMENT

The author like to thank the Natural Science Foundation of Jiangsu Higher Education ,China (No. 07KJD460017).

REFERENCES [1]Yang Suzi，Wu Bo，Li Bin．Future Discussion on Trends in the Development of Advanced Manufacturing Technology[J] ． Chinese Journal of Mechanical Engineering，2006，42(1)：1-5. [2]Jiang Ke-rong Wang Zhisen. A research on automobile anti-braking system wheel speed signal denoising based on wavelet transform[J],Journal of Agricultural Engineering, 2007，38（11）：1-3. [3] Hu Chang2hua , ZHANG J un2bo , XIA J un , et al . System analysis and design based on MATLAB2Wavelet transform[M] . Xi’an :Xidian University Press ,1999. [4] TANG H P，WANG D Y，ZHONG J. Investigation into the electromechanical coupling unstability of a rolling mill[J]. Journal of Materials Processing Technology，2002，(12)9：294-298． [5] PHAM T H ， WENDLING P F ， SALON S J ， et al.Transient finite element analysis of an induction electric with external circuit connections and electromechanical coupling[J]. IEEE Transactions on Energy Conversion，1999，14(4)：1 407-1 412． [6] Burly S，Darnell M。Robust impulse noise suppression using adaptive wavelet denoising [C]，Munich, Germany：IEEE International Conference on Acoustics, Speech and Signal Processing，1997：3417-3420. [7] Zhao Guoliang ,Yang Junchun ,Sun Shen.Noise rejection method of wavelets for ECG signal[J]. Journal of Harbin Engineering University,2004，25(5)：631-634. [8] Pan Quan.Application and Methods of wavelets[J],

Journal of Electronics and Information，2007，29（1）：236-242.

65