Analysis on Singularity of Fault Signals of High ... - Semantic Scholar
JOURNAL OF COMPUTERS, VOL. 6, NO. 4, APRIL 2011
755
Analysis on Singularity of Fault Signals of High Spindle Based on Hermitian Wavelet Gao Rong Faculty of Mechanical Engineering, Huai Yin Institute of Technology, Huaian 223003, China Email: ggrr7012@163. com
Abstract—The signal in the nature ofsingularity is always caused by mechanical fault of CNC machine tool. It is important to recognize the singularity correctly for mechanical fault diagnosis. This paper deals with the wavelet t ransform and the relation between the modulus maxima and the singularity detection, and based on the excellent property of the Hermitian wavelet, poses the conception that applies it in fault diagnosis of the high speed spindle. On the base of analyzing singularity characters and wavelet detection mechanism of the singular signals, the Hermitian continuous wavelet transform is applied in extraction and analysis of the fault signals. Fault detection and positioning of high speed spindle is completed by the modulus maxima point distribution in transform. Results of simulation verify the effectiveness of the method. Index Terms—Hermitian, Singularity, Fault, Wavelet transform
Spindle,
I. INTRODUCTION As the electronics and automation technology development, the numerical control technology application is used more and more widely. Numerically-controlled machine tool's high-speed spindle technology is also very important, But also tend to malfunction, in order to display the use efficiency of NC machine, CNC machine is fast, high precision and compounding direction, it will be efficient, high precision and high flexible sets a body, so the comprehensive performance of machine put forward more and more high demand, high-speed machining technology also more and more get the attention. Superhigh speed NC machine tools is to realize the ultra-high speed processing of the material base, high-speed spindle is extra-high speed machine tool "core" part, its performance directly determine the tools of ultra-high speed processing performance, it not only require high speed precision and require continuous output high torque ability and a wide range of constant power operation. Process monitoring system can forecast the spindle component that can work long time, namely the axis of life and how long, Manuscript received December1, 2010; revised Jan. 1, 2011; accepted Jan. 11, 2011. Natural Science Foundation of Jiangsu provience and Jiangsu Higher Education, China (No. 10KJA460004 and NO. BK2009662), and Natural Science Foundation of Key Lab of Jiangsu provience Digital Manufacturing, China (NO. HGDML-0902 ).
© 2011 ACADEMY PUBLISHER doi:10.4304/jcp.6.4.755-760
otherwise continue to work of words, will cause spindle, machine tool, cutting tool and workpiece damage. High-speed spindle fault signal is different from the machine working normally signal, ex-ceptionally fault signal has a singularity and contain abundant, rich in fault state characteristic information is failure causes, the location of a kind of indirect reflect and characterization. In signal singularity detection Morlet, due to the wavelet in support of interval is repeatedly shocks (concussion times greater than 5 times), commonly according to the Nyquist sampling theorem, need more data points to express Morlet wavelet and wavelet transform is actually the signal convolution filtering points more filter will inevitably smoothed out part of signal singularity. From this point, not Morlet wavelet singularity detection for the ideal of wavelet and wavelet filter based on Hermitian not smooth signal transformation, this is exactly the singularity singularity detection need. II. ELECTRIC SPINDLE TECHNOLOGY The use of high speed spindle of the numerically controlled machine tool is becoming more and more widespread, characterized by the corresponding high speed, accuracy, speed precision and efficiency. The high speed spindle is advantageous in compact structure, light weight, low inertia, and good dynamic characteristic. It is able to improve the dynamic balance of machine tools and is free of vibration, pollution and noise. In the industrial developed countries, like USA, Germany, Japan, Switzerland and Italy, the high speed electric spindle structure [1-4] is widely used on the high speed numerically controlled machine tool. However, the high speed spindles are always found to have noise or other mechanical failures during running (such as breakdown of bearings and sealing devices or running collision of rotors and casing). Especially, the impact of initial stage faults is often smaller than the vibration and hardly to be found, which will be a great challenge for the safe operation of the high speed spindle. The electric spindle is a set of components consisting of a spindle and its accessories: an electric spindle, an HF converting device, am oil mist lubricator, a cooling device, a built-in encoder and a tool changer. Its main characteristic is that the motor is positioned inside the spindle to drive it, realizing the integration of motor and spindle. Supplying spindle of Siemens is shown in fig. 1.
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JOURNAL OF COMPUTERS, VOL. 6, NO. 4, APRIL 2011
showing discontinuity, which is the discontinuity point of Class 2. In signal processing, the term pulse has the following meanings: A rapid, transient change in the amplitude of a signal from a baseline value to a higher or lower value, followed by a rapid return to the baseline value. A rapid change in some characteristic of a signal, e. g., phase or frequency, from a baseline value to a higher or lower value, followed by a rapid return to the baseline value. In telephony pulse dialing is a way of dialing a telephone number using interrupted electrical pulses. Step Function has the following meanings: A step function is a special type of function whose graph is a series of line segments. The graph of a step function looks like a series of small steps. Slope function is used widely, in a chip, the time it takes for a signal to switch from 0 to 1 or 1 to 0. Although extremely fast, it is not instantaneous and can be measured in picoseconds (ps) and nanoseconds (ns).
Fig. 1 Supplying the Spindle
The electric spindle includes some features: A. By positioning the motor directly inside the spindle, we can save intermediate transmission devices, ensuring a compact structure, high mechanical efficiency, low noise, small vibration, high accuracy, etc. B. The application of AC frequency control and vector control enables large output power, wide adjustment range and excellent torque characteristics. C. Simple mechanical structure and small rotation inertia help realize high speed and high acceleration, as well as fast and accurate stop at a given angle. D. Such a design makes it easier for the electric spindle to attain high-speed rotation, with better dynamic accuracy and stability. E. Without the influence of intermediate transmission devices, the spindle can run more stably, and therefore bearings of the spindle can be used for a longer period of time. III.
FAULT SIGNALS AND SINGULARITY OF HIGH SPEED SPINDLE
Funcations called singularity are useful as signals models, In general, the singularity signals generally happen in two conditions (as fig. 2): one is that the amplitude of signals appears a sudden change at a moment leading to the discontinuity of signals and the sudden change of amplitude is the discontinuity point of Class 1; and the other is that the signals is smooth in appearance without sudden changes of amplitude, yet, the first order differential equation appears a sudden change,
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Fig. 2. Modes of High Speed Spindle Fault Singals
Under normal conditions, the high speed spindle signals often appear as the impulse, step and ramp functions [5]. IV. HERMITIAN WAVELET THEORY Let ψ (t ) ∈ L 2 (R ) , where ψ ( t ) is the square 2
integrable function, and L
( R ) is the square integrable ∧
space, then the Fourier transform of ψ ( t ) isψ (ω ) . If ∧
ψ (ω )
meets
the ∧
condition C ψ =
∫
ψ (ω ) R
ω
admissibility
2
d ω < ∞ , then ψ (t ) is
called a Base Wavelet or Mother Wavelet. Let the generating function ψ (t ) be flexible and translated, then a wavelet sequence is obtained. For the continuous conditions, the wavelet sequence is 1 ⎛t −b⎞ a, b ∈ R; a ≠ 0 where a ψ a ,b (t ) = ψ⎜ ⎟ ⎝ a ⎠ a is an extension factor, b is a shift factor. For any continuous function f (t ) ∈ L2 (R ) , the wavelet transform is −1 ⎛ t − b ⎞ (1) W f (a , b ) = f ,ψ a ,b = a 2 ∫ R f (t )ψ ⎜ ⎟dt ⎝ a ⎠ Where the reconstruction formula (inverse transform) is
JOURNAL OF COMPUTERS, VOL. 6, NO. 4, APRIL 2011
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Then signal f (x) of reconfiguration by finding the 1 f (t ) = Cψ
∫
R+
∫
1 W a2
R
f
(a , b )ψ ⎛⎜ t − b ⎝ a
⎞ ⎟ dadb ⎠
~
(2)
function f ( x ) to realize: In making that meet the scales on 2j,
The first order and second order derivatives of
1
Gaussian function g ( t ) =
ψ
(1 )
dg ( t ) = − dt
(t ) =
1 2π
e
2π te
−
−
t2 2
t
~
W f (2 , x
2
2
are
e 2π Its real part Re( ψ ( t )) is ψ
imaginary part
ψ
(1 )
Im( ψ ( t ))
(1 − t 2 )e
−
t2 2
+i
1 2π
t 2e
−
t2 2
(2)
( t ) wavelet and
is the opposite number [6] of
( t ) wavelet.
V. DETECTION MECHANISM OF SINGULARITY OF HIGH SPEED SPINDLE FAULT SIGNALS A. Wavelet singularity and reconstructing signal Wavelet transform has good spatial localization properties. Research of Mallat indicated that if wavelet W (x) have n order disappear torque and n times continuously differentiable and tight supported, wavelet transform mode of local maxima can undertake function local singularity analysis. Hypothesisθ (x ) is a zero mean smooth functions. Its telescopic functionsθs (x ) for
θ
s
(x) =
1 s
θ ( xs )
ψ
ψ
(x ) =
dθ (x) dx
(x ) =
d 2θ ( x ) dx 2
1
2
1
The infinitely differentiable (derivable) function is called as smooth or non-singularity in mathematics. If a function is discontinuous at some part or a certain order derivative, then it is called as singularity at this point and this point is called as the singular point of signals. Meanwhile, the Lipschitzα Exponent of this extreme point may be used to measure the degree of singularity. Let n be a non-negative integer with n＜α ≤n＋1, f (x) here refers to Lipschitz α at point x0, if there is two constants A and h0＞0 and a polynomial of degree n, pn (h), such that for any h≤h0, then the formula holds α . f (x0 + h ) − p n (h ) ≤ A h
If the above formula is exact for all x0∈ (a, b) and x0 ＋h∈ (a, b), then f (x) is called as an uniform Lipschitzα at (a, b). The Lipschitz Exponent demonstrates the smooth degree if comparing f (x) with polynomial of degree n. The greater the Lipschitz Exponent is, the smoother the function is. And a smaller Lipschitz Exponent represents that the function is great in change at some point. If the function is continuous and differentiable at one point, then the Lipschitz Exponent of this point is 1; if the function is derivable at one point with a bounded but discontinuous derivative, then the Lipschitz Exponent is also 1. If there is a Lipschitz Exponent ＜1 for f (x) at x0, then x0 is called as the singular point[7] of f (x). C. Multi-resolution Analysis The fundamental idea of multi-scale analysis is to construct a series of successive linear space V j , where
ψ 1 (x) and ψ 2 (x) meet the singularity detection requirements, the corresponding wavelet transform for W
>,
j, p
B. Lipschitz exponent Analyses
t2 − 2
1
= (1 + it − t 2 )
>=< f ,ψ
j, p
great value [18].
t2
2π
) =< f ,ψ
~
− 1 d g (t ) (4) = − (1 − t 2 ) e 2 ψ ( 2 ) (t ) = 2 dt 2π HaroldSzu constructs the complex Hermitian wavelet, the expression is
1
j, p
f ( x ) various requirement,
and only in point x j , p , W f ( j , x ) has achieved 2
(3)
2
ψ (t ) = ψ ( 2 ) (t ) − iψ (t ) =
~
j
~
d ⎡ dθs ⎤ (x) = s f (θ s )( x ) f ( s , x ) = f ( x )ψ s1 ( x ) = f ( x ) ⎢ s dx ⎣ dx ⎥⎦
j∈ Z
is for different resolution and V j
is the
approximation of L 2 ( R ) at a resolution of 2 , the higher the resolution is, the higher the degree of approximation is. Based on the decomposition algorithm of wavelet: If j
()
⎡ d 2θs ⎤ d2 for a given a scaling function φ t there is a closed W 2 f ( s, x) = f ( x)ψ s2 ( x) = f ( x) ⎢ s 2 ( x) = s 2 2 f (θ s )( x) 2 ⎥ dx dx ⎣ ⎦ subspace V j of a linear spanning space L 2 ( R )
Asher ψ is binary wavelet, binary wavelet transform {Wf (2j, x) }j∈Z is signal f (x) of completely and stable said. In the various scales 2j, if Wf (2j, x) local extremely mould point located in{xj, p} and its value is
Wf ( 2 j , x j , p ) =< f ,ψ
ψ
j, p
(x) =
© 2011 ACADEMY PUBLISHER
j, p
⎡ x − x j, p 1 ψ ⎢ j 2 ⎢⎣ 2
> ⎤ ⎥ ⎥⎦
between the delay φ ( t ) and extension φ ( 2 − j t − k ) ; then for a wavelet function there is ψ ( t ) and a linear
spanning closed space W
(
j
between the delay
)
ψ (t )
and extension ψ 2 − j t − k , where W j is called as wavelet space and V j as scale space. And there is ∩ V j = {0 } 2 j , ∪ V j = L 2 ( R ) j∈ z
j∈ z
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JOURNAL OF COMPUTERS, VOL. 6, NO. 4, APRIL 2011
Detection Process of Singular Points of Fault Signals is shown in fig. 3.
⋅ ⋅ ⋅ ⊂ V − 1 ⊂ V 0 ⊂ V1 ⊂ ⋅ ⋅ ⋅ ⊂ V j ⊂ ⋅ ⋅ ⋅ , L2 ( R ) = ⋅ ⋅ ⋅ ⊕ W − 1 ⊕ W 0 ⊕ W 1 ⊕ ⋅ ⋅ ⋅ ,
V k +1 = V k ⊕ W k
Then, for any signal f ( x ) ∈ L
2
( R ) , there is a unique
decomposition:
f ( x ) = ⋅⋅⋅ + g −1 ( x ) + g 0 ( x ) + g1 ( x ) + ⋅⋅⋅ Where g j ∈ W j , k ∈ Z For a real signal x ( n ) , the following two formulas
may be used for decomposition a k( j ) = ∑ h ( n − 2 k )x n( j +1) , n
d k(
j)
=
∑ g ( n − 2 k )x (
j +1)
n
j ≥ 0 , j∈Z
, j ≥ 0 , j∈Z
n
( j)
Where, d k
( j)
and ak
are respectively the discrete
detail and approximation coefficients of signals at a
decomposition of j , and h ( n − 2k ) and g ( n − 2k ) are respectively the low and high pass filter coefficients. And there is g ( n ) = ( −1)
−n
h ( N − n ) where N is the
length of filter. The restructing algorithm of signals j j is xn( j +1) = h ( n − 2 k )xk( ) + g ( n − 2 k )d k( ) .
∑
∑
k
k
Any non-stationary signals may be handled by the above-mentioned wavelet decomposition and restructing algorithm. VI. DETECTION OF SINGULARITY OF HIGH SPEED SPINDLE FAULT SIGNALS
A. Wavelet Transform Decomposition Method A kind of symmetrical and compactly-supported interpolating wavelets may be constructed to determine the corresponding low and high pass digital filters {a n }
and {bn } , based on the Daubechies orthogonal scaling function and the corresponding wavelet function for the symmetry of wavelet function is of great significance for the accurate positioning of the moment of signal sudden change during the detection of high speed spindle fault signals. Though the interpolating wavelet is a non-translated orthogonal function and is not applicable to Mallat fast algorithm, the sampled data
{f
j
}
(n) ∈ v j on a known scale j, from the point of
filter, may be decomposed as the discrete data and
ω
j +1
f
j +1
(n)
(n) that reflect the low and high frequency
content, by the help of recurrence formulas
[8. 9]
{an } , {bn } ,
and then their
may be obtained.
B. Detection of Singular Points of Fault Signals
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Fig. 3. Detection Process of High Speed spindle Fault Singularity
It can be known from the definition and nature of the wavelet transformed modulus maximum that the modulus maximum of singular point of signals is of scaling transfer and the modulus maximum at adjacent scale is close in position and same in symbol. Therefore, among all scaling high frequency components, only the modulus of transferring to next scale (greater scale) and all modulus transferred from previous scale (smaller scale) are the singular points of signals. However, the wavelet transform of some normal signal points (zero crossing and peak points) may also have maximums. Under this condition, the multi-scale analysis[10。11] may be used by letting a certain smaller threshold value be ε and reserving maximums greater than ε . According to the wavelet transform decomposition algorithm, for the online treatment of singular points of high speed spindle fault signals, the computation of high and low frequency coefficient of scale j+1 may be made once getting the low frequency component of scale j, with the synchronized computation of all scaling high and low frequency components[11。12. 13]. VII. PROCESSING OF HIGH SPEED SPINDLE FAULT SIGNALS The original signals of the vibration acceleration may be obtained through the fault testing system of high speed spindle. See Fig. 4 for the original vibration acceleration signals of high speed spindle.
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759
signal feature points and accurate positioning feature point moments, and then judge high-speed spindle fault occurs or not and fault location.
Fig. 4. Initial Wave of Acceleration Signal
For large scales, the modulus maximum of wavelet transform of signals mainly belongs to the edges of the deterministic signals. In other words, the occurrence of mechanical noises corresponds to the edges of the deterministic signals [16. 17]. This theory may be used to determine the corresponding relationship between the location of the modulus maximum of wavelet transform, and the signal sudden change. However, for the multi-scale analysis, the processing methods for the separation of signals and noises vary with the different natures of the noises. For the deterministic noises, the characteristics of signals may be obtained through the filtering characteristic of wavelet transform for the pre-determined noise frequency range. For non-determined noises, the characteristics of singular points corresponding to the modulus maximum of wavelet transform could be used to determine the characteristic points of signals for the unpredicted frequency range of noises. Fig. 5 (a), (b), (c) and (d) are the transformed modular figures using db3 wavelet to decompose the original acceleration signals respectively. It is observed from the fig. 3 that, the modulus maximum from high frequency noises decreases gradually as the scale of wavelet transform increases from Scale 1 to Scale 4; and the rate of declining is greater than that of signals, which indicates that the Lipschitz Exponent of noise is smaller than that of signals. It can be seen that the high frequency noise is continuous and non-derivable at intervals as well as singular. Many modulus maximums of high frequency noises have changed, diminishing or disappearing; however, the modulus maximum points have increased. The characteristic points of signals can be clearly seen and determined for moments from fig. 4, making the further determination of faults and positions of high speed spindles. As shown from the Fig. 4, along with the wavelet transform scale from 1 to 4 scale scale degree of change, we find that high-frequency noise generated by modulus maxima is gradually attenuation, the attenuation rate than signal explain noise Lipschitz index than signal Lipschitz index is little, also can be seen everywhere, high-frequency noise is continuous't differentiable anywhere, i. e., is everywhere peculiar, high-frequency noise corresponding many modulus maxima points changed, the modulus maxima decrescent or disappear, the mode of the corresponding feature contrary extreme value point, increased, from figure 4 can be clearly seen
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Fig. 5. Scale Decomposition and Transformation of Initial Vibration Acceleration.
VIII. CONCLUSIONS A. There are two advantages to make singularity identification of Hermitian wavelet: one is that the signal transformation will cause no phase changes for the Fourier transformation of Hermitian wavelet is of real numbers; the other is that Hermitian wavelet has fewer number of oscillations in real and imaginary parts comparing with the Morlet wavelet, which makes it
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JOURNAL OF COMPUTERS, VOL. 6, NO. 4, APRIL 2011
possible to convolute signals with fewer data points in case of damaging the singularity of signals. B. When applying the singularity theory of Hermitian wavelet for the fault signal extraction of high speed spindle, the wavelet inverse transform signals may be reconstructed to extract the characteristics of fault signals of high speed spindle based on the different characteristics of signals and noises at scale spaces after wavelet transform. B. The testing emulation shows that using the singularity of Hermitian wavelet to process the fault signals of high speed spindle may realize the purposes of signal de-noising and restoring. Thus, we could conduct adjustment, displaying and alarming on the spindle movement of the numerically controlled machine tool and further improve the service life of the high speed electric spindle through forecasting the mechanical breakdown signals, controlling system and mastering the operation conditions of the spindle timely and accurately. ACKNOWLEDGMENT The author would like to thank the Natural Science Foundation of Jiangsu provience and Jiangsu Higher Education, China (No. 10KJA460004 and NO. BK2009662), and Natural Science Foundation of Key Lab of Jiangsu province Digital Manufacturing (NO. HGDML-0902). 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] Li Songsheng. Analysis of the Dynamatic Characteristics of the Ball Bearing-rotor System in High -speed Electric Spindles[J]. Journal of Mechanical Science and Technology, 2006, 25, (12): 1447-1450. [3] Meng Jie, Chen Xiaoan. Electromechanical coupled Dynamical Modeling of High-speed Motorized Spindle’s Motor-spindle Subsystem[J]. Chinese Journal of Mechanical Engineering, 2007, 43 (12): 160-165. [4] MA Bao. Singularity Detection Through the Hermitian Wavelet and Rotating Mechanical Fault Diagnosis[J]. Moden machine, 2006, 5: 58-59. [5] Ding Na. Analysis on Singularity of Fault Signals of Rolling Bearing Based on Morlet Wavelet[J]. Bearing, 2006, 35 (1): 29-32. [6] He Zhen –jia. RecognizationofSi gnalSin gularityBasedonHermitian WaveletTransform[J]. Journal of Engineering Mathematics, 2001, 18 (12): 82-93. [7] Mallat S, Hwang W L. Singularity detection and processing with wavelet [J]. IEEE Trans on Information Theory, 1992, 38 (2): 617-643. [8] Ding Na. Study on Morlet Wavelets Singular ity[J]. Technology of Power and Electronics, 2004, 38 (5): 83-85. [9] Rao Ying, Mutant Character Detection of Wavelet Transform in Power System Fault Signal, 2005, 24 (4): 17-19. [10] Pan Quan. Application and Methods of Wavelet, 2007, 29 (1): 236-242. [11] Kang P. Characterization of vibration signals using continuous wavelet t ransformer for condition assessment © 2011 ACADEMY PUBLISHER
of on2load tap2changers [ J ]. Me2chanical Systems and Signal Processing, 2003, 17 (3): 5612577. [12] Donoho D L.. Denoising by soft-thresholding[J]. IEEE Transaction on Information, 1995, 3: 613 - 627. [13] Mallat S, Zhong Sifen. Characterization of signal from mutiscale edges [J]. IEEE Trans Inf Theory, 1992, 14 (7): 710-732. [14] Zhang Xiaoping, Desai M D. Adaptive denoising based on Surerisk [C]. IEEE Signal Processing Letters, 1998, 11 (5): 265-267. [15]Tord Bengtsson, Mare Foata. Monitoring tap changer operations [C]. CIGRE. Paris, France, 2003, 12 (209): 125. [16] Peng Wei-shao, Li Li-zheng, Hu Yan-yu, Wavelet preprocessing of analog circuit fault signal[J]. J. Cent. South Univ. (Science and Technology), 2008, 39 (3): 1169-1173. [17] Hu Han-hui. Sintering fan faults diagnosis based on wavelet analysis[J], 2008, 39 (3): 1169-1173. [18] Tang Ying. Study on Vibration Signal Singularity Analysis by Wavelet for Rolling Element Bearing Detection. Journal of Vibration Engineering, 2002, 15 (1): 111-113.
Gao Rong, male, born in 1970, Ph. D, Associate professor in Faculty of Mechanical Engineering, Huaiyin Institute of Technology, China, is a postdoctoral in department of precision instruments & mechanology, TsinghuaUniversity, Beijing, China. majors in the CNC technology and mechanics, published more than 30 passages.
755
Analysis on Singularity of Fault Signals of High Spindle Based on Hermitian Wavelet Gao Rong Faculty of Mechanical Engineering, Huai Yin Institute of Technology, Huaian 223003, China Email: ggrr7012@163. com
Abstract—The signal in the nature ofsingularity is always caused by mechanical fault of CNC machine tool. It is important to recognize the singularity correctly for mechanical fault diagnosis. This paper deals with the wavelet t ransform and the relation between the modulus maxima and the singularity detection, and based on the excellent property of the Hermitian wavelet, poses the conception that applies it in fault diagnosis of the high speed spindle. On the base of analyzing singularity characters and wavelet detection mechanism of the singular signals, the Hermitian continuous wavelet transform is applied in extraction and analysis of the fault signals. Fault detection and positioning of high speed spindle is completed by the modulus maxima point distribution in transform. Results of simulation verify the effectiveness of the method. Index Terms—Hermitian, Singularity, Fault, Wavelet transform
Spindle,
I. INTRODUCTION As the electronics and automation technology development, the numerical control technology application is used more and more widely. Numerically-controlled machine tool's high-speed spindle technology is also very important, But also tend to malfunction, in order to display the use efficiency of NC machine, CNC machine is fast, high precision and compounding direction, it will be efficient, high precision and high flexible sets a body, so the comprehensive performance of machine put forward more and more high demand, high-speed machining technology also more and more get the attention. Superhigh speed NC machine tools is to realize the ultra-high speed processing of the material base, high-speed spindle is extra-high speed machine tool "core" part, its performance directly determine the tools of ultra-high speed processing performance, it not only require high speed precision and require continuous output high torque ability and a wide range of constant power operation. Process monitoring system can forecast the spindle component that can work long time, namely the axis of life and how long, Manuscript received December1, 2010; revised Jan. 1, 2011; accepted Jan. 11, 2011. Natural Science Foundation of Jiangsu provience and Jiangsu Higher Education, China (No. 10KJA460004 and NO. BK2009662), and Natural Science Foundation of Key Lab of Jiangsu provience Digital Manufacturing, China (NO. HGDML-0902 ).
© 2011 ACADEMY PUBLISHER doi:10.4304/jcp.6.4.755-760
otherwise continue to work of words, will cause spindle, machine tool, cutting tool and workpiece damage. High-speed spindle fault signal is different from the machine working normally signal, ex-ceptionally fault signal has a singularity and contain abundant, rich in fault state characteristic information is failure causes, the location of a kind of indirect reflect and characterization. In signal singularity detection Morlet, due to the wavelet in support of interval is repeatedly shocks (concussion times greater than 5 times), commonly according to the Nyquist sampling theorem, need more data points to express Morlet wavelet and wavelet transform is actually the signal convolution filtering points more filter will inevitably smoothed out part of signal singularity. From this point, not Morlet wavelet singularity detection for the ideal of wavelet and wavelet filter based on Hermitian not smooth signal transformation, this is exactly the singularity singularity detection need. II. ELECTRIC SPINDLE TECHNOLOGY The use of high speed spindle of the numerically controlled machine tool is becoming more and more widespread, characterized by the corresponding high speed, accuracy, speed precision and efficiency. The high speed spindle is advantageous in compact structure, light weight, low inertia, and good dynamic characteristic. It is able to improve the dynamic balance of machine tools and is free of vibration, pollution and noise. In the industrial developed countries, like USA, Germany, Japan, Switzerland and Italy, the high speed electric spindle structure [1-4] is widely used on the high speed numerically controlled machine tool. However, the high speed spindles are always found to have noise or other mechanical failures during running (such as breakdown of bearings and sealing devices or running collision of rotors and casing). Especially, the impact of initial stage faults is often smaller than the vibration and hardly to be found, which will be a great challenge for the safe operation of the high speed spindle. The electric spindle is a set of components consisting of a spindle and its accessories: an electric spindle, an HF converting device, am oil mist lubricator, a cooling device, a built-in encoder and a tool changer. Its main characteristic is that the motor is positioned inside the spindle to drive it, realizing the integration of motor and spindle. Supplying spindle of Siemens is shown in fig. 1.
756
JOURNAL OF COMPUTERS, VOL. 6, NO. 4, APRIL 2011
showing discontinuity, which is the discontinuity point of Class 2. In signal processing, the term pulse has the following meanings: A rapid, transient change in the amplitude of a signal from a baseline value to a higher or lower value, followed by a rapid return to the baseline value. A rapid change in some characteristic of a signal, e. g., phase or frequency, from a baseline value to a higher or lower value, followed by a rapid return to the baseline value. In telephony pulse dialing is a way of dialing a telephone number using interrupted electrical pulses. Step Function has the following meanings: A step function is a special type of function whose graph is a series of line segments. The graph of a step function looks like a series of small steps. Slope function is used widely, in a chip, the time it takes for a signal to switch from 0 to 1 or 1 to 0. Although extremely fast, it is not instantaneous and can be measured in picoseconds (ps) and nanoseconds (ns).
Fig. 1 Supplying the Spindle
The electric spindle includes some features: A. By positioning the motor directly inside the spindle, we can save intermediate transmission devices, ensuring a compact structure, high mechanical efficiency, low noise, small vibration, high accuracy, etc. B. The application of AC frequency control and vector control enables large output power, wide adjustment range and excellent torque characteristics. C. Simple mechanical structure and small rotation inertia help realize high speed and high acceleration, as well as fast and accurate stop at a given angle. D. Such a design makes it easier for the electric spindle to attain high-speed rotation, with better dynamic accuracy and stability. E. Without the influence of intermediate transmission devices, the spindle can run more stably, and therefore bearings of the spindle can be used for a longer period of time. III.
FAULT SIGNALS AND SINGULARITY OF HIGH SPEED SPINDLE
Funcations called singularity are useful as signals models, In general, the singularity signals generally happen in two conditions (as fig. 2): one is that the amplitude of signals appears a sudden change at a moment leading to the discontinuity of signals and the sudden change of amplitude is the discontinuity point of Class 1; and the other is that the signals is smooth in appearance without sudden changes of amplitude, yet, the first order differential equation appears a sudden change,
© 2011 ACADEMY PUBLISHER
Fig. 2. Modes of High Speed Spindle Fault Singals
Under normal conditions, the high speed spindle signals often appear as the impulse, step and ramp functions [5]. IV. HERMITIAN WAVELET THEORY Let ψ (t ) ∈ L 2 (R ) , where ψ ( t ) is the square 2
integrable function, and L
( R ) is the square integrable ∧
space, then the Fourier transform of ψ ( t ) isψ (ω ) . If ∧
ψ (ω )
meets
the ∧
condition C ψ =
∫
ψ (ω ) R
ω
admissibility
2
d ω < ∞ , then ψ (t ) is
called a Base Wavelet or Mother Wavelet. Let the generating function ψ (t ) be flexible and translated, then a wavelet sequence is obtained. For the continuous conditions, the wavelet sequence is 1 ⎛t −b⎞ a, b ∈ R; a ≠ 0 where a ψ a ,b (t ) = ψ⎜ ⎟ ⎝ a ⎠ a is an extension factor, b is a shift factor. For any continuous function f (t ) ∈ L2 (R ) , the wavelet transform is −1 ⎛ t − b ⎞ (1) W f (a , b ) = f ,ψ a ,b = a 2 ∫ R f (t )ψ ⎜ ⎟dt ⎝ a ⎠ Where the reconstruction formula (inverse transform) is
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Then signal f (x) of reconfiguration by finding the 1 f (t ) = Cψ
∫
R+
∫
1 W a2
R
f
(a , b )ψ ⎛⎜ t − b ⎝ a
⎞ ⎟ dadb ⎠
~
(2)
function f ( x ) to realize: In making that meet the scales on 2j,
The first order and second order derivatives of
1
Gaussian function g ( t ) =
ψ
(1 )
dg ( t ) = − dt
(t ) =
1 2π
e
2π te
−
−
t2 2
t
~
W f (2 , x
2
2
are
e 2π Its real part Re( ψ ( t )) is ψ
imaginary part
ψ
(1 )
Im( ψ ( t ))
(1 − t 2 )e
−
t2 2
+i
1 2π
t 2e
−
t2 2
(2)
( t ) wavelet and
is the opposite number [6] of
( t ) wavelet.
V. DETECTION MECHANISM OF SINGULARITY OF HIGH SPEED SPINDLE FAULT SIGNALS A. Wavelet singularity and reconstructing signal Wavelet transform has good spatial localization properties. Research of Mallat indicated that if wavelet W (x) have n order disappear torque and n times continuously differentiable and tight supported, wavelet transform mode of local maxima can undertake function local singularity analysis. Hypothesisθ (x ) is a zero mean smooth functions. Its telescopic functionsθs (x ) for
θ
s
(x) =
1 s
θ ( xs )
ψ
ψ
(x ) =
dθ (x) dx
(x ) =
d 2θ ( x ) dx 2
1
2
1
The infinitely differentiable (derivable) function is called as smooth or non-singularity in mathematics. If a function is discontinuous at some part or a certain order derivative, then it is called as singularity at this point and this point is called as the singular point of signals. Meanwhile, the Lipschitzα Exponent of this extreme point may be used to measure the degree of singularity. Let n be a non-negative integer with n＜α ≤n＋1, f (x) here refers to Lipschitz α at point x0, if there is two constants A and h0＞0 and a polynomial of degree n, pn (h), such that for any h≤h0, then the formula holds α . f (x0 + h ) − p n (h ) ≤ A h
If the above formula is exact for all x0∈ (a, b) and x0 ＋h∈ (a, b), then f (x) is called as an uniform Lipschitzα at (a, b). The Lipschitz Exponent demonstrates the smooth degree if comparing f (x) with polynomial of degree n. The greater the Lipschitz Exponent is, the smoother the function is. And a smaller Lipschitz Exponent represents that the function is great in change at some point. If the function is continuous and differentiable at one point, then the Lipschitz Exponent of this point is 1; if the function is derivable at one point with a bounded but discontinuous derivative, then the Lipschitz Exponent is also 1. If there is a Lipschitz Exponent ＜1 for f (x) at x0, then x0 is called as the singular point[7] of f (x). C. Multi-resolution Analysis The fundamental idea of multi-scale analysis is to construct a series of successive linear space V j , where
ψ 1 (x) and ψ 2 (x) meet the singularity detection requirements, the corresponding wavelet transform for W
>,
j, p
B. Lipschitz exponent Analyses
t2 − 2
1
= (1 + it − t 2 )
>=< f ,ψ
j, p
great value [18].
t2
2π
) =< f ,ψ
~
− 1 d g (t ) (4) = − (1 − t 2 ) e 2 ψ ( 2 ) (t ) = 2 dt 2π HaroldSzu constructs the complex Hermitian wavelet, the expression is
1
j, p
f ( x ) various requirement,
and only in point x j , p , W f ( j , x ) has achieved 2
(3)
2
ψ (t ) = ψ ( 2 ) (t ) − iψ (t ) =
~
j
~
d ⎡ dθs ⎤ (x) = s f (θ s )( x ) f ( s , x ) = f ( x )ψ s1 ( x ) = f ( x ) ⎢ s dx ⎣ dx ⎥⎦
j∈ Z
is for different resolution and V j
is the
approximation of L 2 ( R ) at a resolution of 2 , the higher the resolution is, the higher the degree of approximation is. Based on the decomposition algorithm of wavelet: If j
()
⎡ d 2θs ⎤ d2 for a given a scaling function φ t there is a closed W 2 f ( s, x) = f ( x)ψ s2 ( x) = f ( x) ⎢ s 2 ( x) = s 2 2 f (θ s )( x) 2 ⎥ dx dx ⎣ ⎦ subspace V j of a linear spanning space L 2 ( R )
Asher ψ is binary wavelet, binary wavelet transform {Wf (2j, x) }j∈Z is signal f (x) of completely and stable said. In the various scales 2j, if Wf (2j, x) local extremely mould point located in{xj, p} and its value is
Wf ( 2 j , x j , p ) =< f ,ψ
ψ
j, p
(x) =
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j, p
⎡ x − x j, p 1 ψ ⎢ j 2 ⎢⎣ 2
> ⎤ ⎥ ⎥⎦
between the delay φ ( t ) and extension φ ( 2 − j t − k ) ; then for a wavelet function there is ψ ( t ) and a linear
spanning closed space W
(
j
between the delay
)
ψ (t )
and extension ψ 2 − j t − k , where W j is called as wavelet space and V j as scale space. And there is ∩ V j = {0 } 2 j , ∪ V j = L 2 ( R ) j∈ z
j∈ z
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Detection Process of Singular Points of Fault Signals is shown in fig. 3.
⋅ ⋅ ⋅ ⊂ V − 1 ⊂ V 0 ⊂ V1 ⊂ ⋅ ⋅ ⋅ ⊂ V j ⊂ ⋅ ⋅ ⋅ , L2 ( R ) = ⋅ ⋅ ⋅ ⊕ W − 1 ⊕ W 0 ⊕ W 1 ⊕ ⋅ ⋅ ⋅ ,
V k +1 = V k ⊕ W k
Then, for any signal f ( x ) ∈ L
2
( R ) , there is a unique
decomposition:
f ( x ) = ⋅⋅⋅ + g −1 ( x ) + g 0 ( x ) + g1 ( x ) + ⋅⋅⋅ Where g j ∈ W j , k ∈ Z For a real signal x ( n ) , the following two formulas
may be used for decomposition a k( j ) = ∑ h ( n − 2 k )x n( j +1) , n
d k(
j)
=
∑ g ( n − 2 k )x (
j +1)
n
j ≥ 0 , j∈Z
, j ≥ 0 , j∈Z
n
( j)
Where, d k
( j)
and ak
are respectively the discrete
detail and approximation coefficients of signals at a
decomposition of j , and h ( n − 2k ) and g ( n − 2k ) are respectively the low and high pass filter coefficients. And there is g ( n ) = ( −1)
−n
h ( N − n ) where N is the
length of filter. The restructing algorithm of signals j j is xn( j +1) = h ( n − 2 k )xk( ) + g ( n − 2 k )d k( ) .
∑
∑
k
k
Any non-stationary signals may be handled by the above-mentioned wavelet decomposition and restructing algorithm. VI. DETECTION OF SINGULARITY OF HIGH SPEED SPINDLE FAULT SIGNALS
A. Wavelet Transform Decomposition Method A kind of symmetrical and compactly-supported interpolating wavelets may be constructed to determine the corresponding low and high pass digital filters {a n }
and {bn } , based on the Daubechies orthogonal scaling function and the corresponding wavelet function for the symmetry of wavelet function is of great significance for the accurate positioning of the moment of signal sudden change during the detection of high speed spindle fault signals. Though the interpolating wavelet is a non-translated orthogonal function and is not applicable to Mallat fast algorithm, the sampled data
{f
j
}
(n) ∈ v j on a known scale j, from the point of
filter, may be decomposed as the discrete data and
ω
j +1
f
j +1
(n)
(n) that reflect the low and high frequency
content, by the help of recurrence formulas
[8. 9]
{an } , {bn } ,
and then their
may be obtained.
B. Detection of Singular Points of Fault Signals
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Fig. 3. Detection Process of High Speed spindle Fault Singularity
It can be known from the definition and nature of the wavelet transformed modulus maximum that the modulus maximum of singular point of signals is of scaling transfer and the modulus maximum at adjacent scale is close in position and same in symbol. Therefore, among all scaling high frequency components, only the modulus of transferring to next scale (greater scale) and all modulus transferred from previous scale (smaller scale) are the singular points of signals. However, the wavelet transform of some normal signal points (zero crossing and peak points) may also have maximums. Under this condition, the multi-scale analysis[10。11] may be used by letting a certain smaller threshold value be ε and reserving maximums greater than ε . According to the wavelet transform decomposition algorithm, for the online treatment of singular points of high speed spindle fault signals, the computation of high and low frequency coefficient of scale j+1 may be made once getting the low frequency component of scale j, with the synchronized computation of all scaling high and low frequency components[11。12. 13]. VII. PROCESSING OF HIGH SPEED SPINDLE FAULT SIGNALS The original signals of the vibration acceleration may be obtained through the fault testing system of high speed spindle. See Fig. 4 for the original vibration acceleration signals of high speed spindle.
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signal feature points and accurate positioning feature point moments, and then judge high-speed spindle fault occurs or not and fault location.
Fig. 4. Initial Wave of Acceleration Signal
For large scales, the modulus maximum of wavelet transform of signals mainly belongs to the edges of the deterministic signals. In other words, the occurrence of mechanical noises corresponds to the edges of the deterministic signals [16. 17]. This theory may be used to determine the corresponding relationship between the location of the modulus maximum of wavelet transform, and the signal sudden change. However, for the multi-scale analysis, the processing methods for the separation of signals and noises vary with the different natures of the noises. For the deterministic noises, the characteristics of signals may be obtained through the filtering characteristic of wavelet transform for the pre-determined noise frequency range. For non-determined noises, the characteristics of singular points corresponding to the modulus maximum of wavelet transform could be used to determine the characteristic points of signals for the unpredicted frequency range of noises. Fig. 5 (a), (b), (c) and (d) are the transformed modular figures using db3 wavelet to decompose the original acceleration signals respectively. It is observed from the fig. 3 that, the modulus maximum from high frequency noises decreases gradually as the scale of wavelet transform increases from Scale 1 to Scale 4; and the rate of declining is greater than that of signals, which indicates that the Lipschitz Exponent of noise is smaller than that of signals. It can be seen that the high frequency noise is continuous and non-derivable at intervals as well as singular. Many modulus maximums of high frequency noises have changed, diminishing or disappearing; however, the modulus maximum points have increased. The characteristic points of signals can be clearly seen and determined for moments from fig. 4, making the further determination of faults and positions of high speed spindles. As shown from the Fig. 4, along with the wavelet transform scale from 1 to 4 scale scale degree of change, we find that high-frequency noise generated by modulus maxima is gradually attenuation, the attenuation rate than signal explain noise Lipschitz index than signal Lipschitz index is little, also can be seen everywhere, high-frequency noise is continuous't differentiable anywhere, i. e., is everywhere peculiar, high-frequency noise corresponding many modulus maxima points changed, the modulus maxima decrescent or disappear, the mode of the corresponding feature contrary extreme value point, increased, from figure 4 can be clearly seen
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Fig. 5. Scale Decomposition and Transformation of Initial Vibration Acceleration.
VIII. CONCLUSIONS A. There are two advantages to make singularity identification of Hermitian wavelet: one is that the signal transformation will cause no phase changes for the Fourier transformation of Hermitian wavelet is of real numbers; the other is that Hermitian wavelet has fewer number of oscillations in real and imaginary parts comparing with the Morlet wavelet, which makes it
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possible to convolute signals with fewer data points in case of damaging the singularity of signals. B. When applying the singularity theory of Hermitian wavelet for the fault signal extraction of high speed spindle, the wavelet inverse transform signals may be reconstructed to extract the characteristics of fault signals of high speed spindle based on the different characteristics of signals and noises at scale spaces after wavelet transform. B. The testing emulation shows that using the singularity of Hermitian wavelet to process the fault signals of high speed spindle may realize the purposes of signal de-noising and restoring. Thus, we could conduct adjustment, displaying and alarming on the spindle movement of the numerically controlled machine tool and further improve the service life of the high speed electric spindle through forecasting the mechanical breakdown signals, controlling system and mastering the operation conditions of the spindle timely and accurately. ACKNOWLEDGMENT The author would like to thank the Natural Science Foundation of Jiangsu provience and Jiangsu Higher Education, China (No. 10KJA460004 and NO. BK2009662), and Natural Science Foundation of Key Lab of Jiangsu province Digital Manufacturing (NO. HGDML-0902). 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] Li Songsheng. Analysis of the Dynamatic Characteristics of the Ball Bearing-rotor System in High -speed Electric Spindles[J]. Journal of Mechanical Science and Technology, 2006, 25, (12): 1447-1450. [3] Meng Jie, Chen Xiaoan. Electromechanical coupled Dynamical Modeling of High-speed Motorized Spindle’s Motor-spindle Subsystem[J]. Chinese Journal of Mechanical Engineering, 2007, 43 (12): 160-165. [4] MA Bao. Singularity Detection Through the Hermitian Wavelet and Rotating Mechanical Fault Diagnosis[J]. Moden machine, 2006, 5: 58-59. [5] Ding Na. Analysis on Singularity of Fault Signals of Rolling Bearing Based on Morlet Wavelet[J]. Bearing, 2006, 35 (1): 29-32. [6] He Zhen –jia. RecognizationofSi gnalSin gularityBasedonHermitian WaveletTransform[J]. Journal of Engineering Mathematics, 2001, 18 (12): 82-93. [7] Mallat S, Hwang W L. Singularity detection and processing with wavelet [J]. IEEE Trans on Information Theory, 1992, 38 (2): 617-643. [8] Ding Na. Study on Morlet Wavelets Singular ity[J]. Technology of Power and Electronics, 2004, 38 (5): 83-85. [9] Rao Ying, Mutant Character Detection of Wavelet Transform in Power System Fault Signal, 2005, 24 (4): 17-19. [10] Pan Quan. Application and Methods of Wavelet, 2007, 29 (1): 236-242. [11] Kang P. Characterization of vibration signals using continuous wavelet t ransformer for condition assessment © 2011 ACADEMY PUBLISHER
of on2load tap2changers [ J ]. Me2chanical Systems and Signal Processing, 2003, 17 (3): 5612577. [12] Donoho D L.. Denoising by soft-thresholding[J]. IEEE Transaction on Information, 1995, 3: 613 - 627. [13] Mallat S, Zhong Sifen. Characterization of signal from mutiscale edges [J]. IEEE Trans Inf Theory, 1992, 14 (7): 710-732. [14] Zhang Xiaoping, Desai M D. Adaptive denoising based on Surerisk [C]. IEEE Signal Processing Letters, 1998, 11 (5): 265-267. [15]Tord Bengtsson, Mare Foata. Monitoring tap changer operations [C]. CIGRE. Paris, France, 2003, 12 (209): 125. [16] Peng Wei-shao, Li Li-zheng, Hu Yan-yu, Wavelet preprocessing of analog circuit fault signal[J]. J. Cent. South Univ. (Science and Technology), 2008, 39 (3): 1169-1173. [17] Hu Han-hui. Sintering fan faults diagnosis based on wavelet analysis[J], 2008, 39 (3): 1169-1173. [18] Tang Ying. Study on Vibration Signal Singularity Analysis by Wavelet for Rolling Element Bearing Detection. Journal of Vibration Engineering, 2002, 15 (1): 111-113.
Gao Rong, male, born in 1970, Ph. D, Associate professor in Faculty of Mechanical Engineering, Huaiyin Institute of Technology, China, is a postdoctoral in department of precision instruments & mechanology, TsinghuaUniversity, Beijing, China. majors in the CNC technology and mechanics, published more than 30 passages.
