Spindle Health Diagnosis Based on Analytic Wavelet Enveloping

1850 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 55, NO. 5, OCTOBER 2006

Spindle Health Diagnosis Based onAnalytic Wavelet Enveloping

Li Zhang, Robert X. Gao, Senior Member, IEEE, and Kang B. Lee, Fellow, IEEE

Abstract—A new diagnostic technique for identifying structuraldefects in spindles was developed based on the analytic wavelettransform. The new technique extracts defect-induced impulsesfrom the spindle vibration signal and constructs their envelopesin a single step, eliminating the need for intermediate operationsas traditionally required. Theoretical background of the analyticwavelet transform was first introduced, and numerical simulationwas then conducted on a synthetic signal. The result was subse-quently compared with vibration signals measured on a spindletest bed. It was confirmed that the developed technique is effectivein detecting defect-induced impulses buried in the spindle vibra-tion signals that otherwise were undetectable using the traditionalspectral techniques.

Index Terms—Analytic wavelet, enveloping, spindle healthdiagnosis.

I. INTRODUCTION

IN THE machine tool industry, unexpected failure of spindlescan lead to severe part damage and costly machine down-

time, affecting the overall production logistics and productivity.Vibration signals measured on a spindle contain rich physicalinformation about the operating conditions; thus, vibrationsignal analysis has long been regarded as a physics-based tech-nique for spindle defect detection and health diagnosis [1]. In arotating spindle, the main causes contributing to its vibration in-clude manufacturing imperfection, assembly- and installation-related problems, and tool–work interactions [2].

Rolling element bearings are the most critical and vulnerablecomponent in a machine tool spindle. As the bearings wear out,localized defects may develop on the raceways (inner or outer)or within the rolling elements (balls and rollers) themselves.Interactions between the rolling elements and the defects ex-press themselves as impulsive inputs to the bearing and spindlestructure, which excites high-frequency structural resonance inthe form of vibration impulses. Due to the rotating nature ofthe spindle, these impulses are repetitive and periodic in nature,and their strength depends on both the defect geometry andthe spindle operating condition (e.g., applied load and shaftrotational speed). Furthermore, the specific location of a defectis reflected in its characteristic frequency [3], [4], e.g., ball spinfrequency (BSF), ball pass frequency for inner raceway (BPFI),

Manuscript received June 6, 2005; revised May 25, 2006. This work wassupported by the Smart Machine Tools Program at the National Institute ofStandards and Technology (NIST).

L. Zhang and R. X. Gao are with the Department of Mechanical andIndustrial Engineering, University of Massachusetts, Amherst, MA 01003 USA(e-mail: [email protected]; [email protected]).

K. B. Lee is with the Manufacturing Metrology Division, National Insti-tute of Standards and Technology, Gaithersburg, MD 20899 USA (e-mail:[email protected]).

Digital Object Identifier 10.1109/TIM.2006.880261

Fig. 1. Envelope and pattern of defect-induced vibration in rolling elementbearings.

and ball pass frequency for outer raceway (BPFO), as illustratedin Fig. 1. Therefore, spectral techniques can be employedto analyze spindle vibration signals for defect detection andidentification.

Spectral analysis of the envelope of a vibration signal (enve-lope spectrum) has shown to be a more effective approach to thedetection and identification of structural defects than spectralanalysis of the raw vibration signal itself [5], [6]. However,given that defect-induced vibration impulse is generally weakin amplitude and short in duration at the incipient stage [7], theeffectiveness of enveloping generally suffers from a low signal-to-noise ratio (SNR). In recent years, wavelet transform hasreceived considerable attention from the research communitydue to its ability in extracting time-dependent transient featuresfrom vibration signals with strong background noise by meansof a combined time–frequency analysis [8]. Combining theadvantages of enveloping and wavelet transform, this paperpresents a new method for machine health diagnosis basedon a specific class of wavelets called the “analytic wavelet,”which served as the base wavelet for spindle vibration signaldecomposition and analysis. The rationale for investigating thisclass of wavelets is that they extract defect features and con-struct the signal’s envelope in a single step, thus eliminating theneed for additional operations such as the Hilbert transform [9]or low-pass filtering to extract signal envelope. In addition, theanalytic wavelet is inherently flexible in the selection of relativebandwidth, thus, it can be adaptively applied to the signal underinvestigation.

0018-9456/$20.00 © 2006 IEEE

ZHANG et al.: SPINDLE HEALTH DIAGNOSIS BASED ON ANALYTIC WAVELET ENVELOPING 1851

II. THEORETICAL BACKGROUND

Once a structural defect develops on the surface of a bear-ing’s raceway, rolling elements will impact on the defect eachtime they roll over it and generate a vibration impulse. Theterm defect characteristic frequency refers to the frequencyat which the defect-induced impulse signal repeats itself witheach spindle rotation. To detect such signal, the envelope ofthe signal can be first extracted and subsequently analyzed inthe frequency domain to determine the magnitudes (or energycontent) at each of the characteristic frequencies. Higher en-ergy concentration at the characteristic frequencies indicatesthe existence of a defect at a specific location. Traditionally,enveloping is accomplished by rectifying and subsequentlylow-pass-filtering raw vibration signals measured for the ma-chine system being monitored, e.g., a spindle. The drawbackof traditional enveloping is that it suffers from low SNRs whenthe defect is at its incipient stage. Since the wavelet transformhas shown to be effective in extracting features from transientsignals, preprocessing of vibration signals by means of waveletanalysis would enhance the effectiveness of signal envelopingfor identifying the existence of structural defects.

The continuous wavelet transform (CWT) [10], [11] of asignal x(t) is defined as

CWT(τ, s) = |s|− 12

∫x(t)ψ∗

(t− τ

s

)dt (1)

where ψ∗(t) is the complex conjugate of the wavelet functionψ(t), and CWT(τ, s) represents the wavelet coefficient. Sincescaling a wavelet by a factor of 1/s in the time domain isequivalent to changing the frequency by a factor of s, the timewindow automatically widens when analyzing low-frequencycomponents and narrows when analyzing high-frequency com-ponents. While real-valued wavelets have been widely studied,a special class of complex-valued wavelets, which is termedanalytic wavelet, has remained a subject of thorough investiga-tion. An analytic wavelet ψ(t) is defined as a complex-valuedwavelet given by [12]

ψ(t) = w(t) + jw̃(t) (2)

where the operator •̃(t) denotes the Hilbert transform of •(t).The imaginary part w̃(t) is the Hilbert transform [9] of the realpart w(t) in the range −∞ < t <∞, which is expressed as

w̃(t) =1π

∞∫−∞

w(t)u− t

dt (3)

where u is a time variable. In the frequency domain, therelationship between w(t) and its Hilbert transform w̃(t) canbe expressed as

W̃ (f) = (−jsgnf)W (f) ={−jW (f), for f > 0

jW (f), for f < 0 (4)

where W̃ (f) is the Fourier transform of the imaginary part ofthe wavelet w̃(t), and W (f) is the Fourier transform of the realpart w(t). According to the linearity property of the Fourier

transform, which states that the Fourier transform of the sum-mation of the two terms in (2) is equivalent to the summationof the Fourier transform of each of the two terms, the analyticwavelet ψ(t) can be expressed in the frequency domain as

Ψ(f) = W (f) + jW̃ (f) (5)

where Ψ(f) denotes the Fourier transform of ψ(t). Using therelationship shown in (4), when the frequencies are negative(f < 0), the Fourier transform of the real part of the analyticwavelet W (f) and its Hilbert transform W̃ (f) cancel eachother out, resulting in an analytic wavelet with a one-sidedspectrum as

Ψ(f) =

2W (f), for f > 0W (0), for f = 00, for f < 0.

(6)

From (6), it follows that forming an analytic wavelet ψ(t) inthe time domain is equivalent to assigning Ψ(f) twice the spec-trum of its real partW (f) and then setting the spectrum of Ψ(f)to zero in the negative frequency range. The same property alsoapplies to any real-valued signals in general, i.e., by settingthe spectrum of a signal x(t) to zero in the negative frequencyrange and double its spectrum in the positive frequency range,its analytic signal can be formed. This property can be usedto explain why analytic wavelet transform is more efficientin extracting the envelope information than the conventionalenveloping method. As shown in (1), the CWT is, by definition,the convolution between the signal and the wavelet function inthe time domain; therefore, it can be computed in the frequencydomain by direct multiplication of the Fourier transform of thesignal X(f) with the Fourier transform of the wavelet functionΨ(f) [13]. The wavelet coefficient is then the inverse Fouriertransform of the product of X(f) and Ψ(f), i.e.,

CWT(τ, s) =s

2π

∞∫−∞

X(f)Ψ∗(sf)ej2πftdf. (7)

Since the negative frequencies of Ψ(f) are all zeros, themultiplication also sets the negative frequencies of X(f) tozeros, resulting in an analytic wavelet coefficient CWT(τ, s),which can be rewritten as

CWT(τ, s) = T (τ, s) + jT̃ (τ, s) (8)

where T (τ, s) is the real part of the analytic wavelet coefficient,and the imaginary part T̃ (τ, s) is the Hilbert transform of thereal part. The wavelet coefficient is a two-dimensional matrixwith each row as the wavelet transform of the vibration signalusing the corresponding wavelet scale s. The defect-inducedimpulses will then be extracted to one of the scales for analysis.The enveloping followed by Fourier transform are subsequentlyapplied to each row of the wavelet coefficient matrix to revealthe repetition frequency of the defect-induced impulses, i.e.,the characteristic frequency associated with the defect. Beinganalytic in nature, the envelope a(τ, s) can be obtained through


Fig. 2. Process of obtaining the wavelet envelope spectrum of a time-domain signal x(t).

the wavelet coefficient CWT(τ, s) itself without additionalrectifying and low-pass filtering steps as

a(τ, s) = ‖CWT(τ, s)‖ =√T 2(τ, s) + T̃ 2(τ, s) (9)

where ‖ • ‖ denotes the modulus operation. The resultant en-velope for each scale is subsequently Fourier-transformed toobtain the envelope spectrum A(f, s), which is defined by

A(f, s) =

∞∫−∞

a(τ, s)e−j2πftdt. (10)

Therefore, the analytic wavelet-based envelope spectrumcombines the advantages of both enveloping and wavelet trans-form techniques by extracting defect features and constructs thesignal’s envelope in a single step. Fig. 2 illustrates the completeprocess for obtaining the wavelet envelope spectrum of a signalgraphically.

When analyzing realistic spindle vibration signals, the scales(center frequencies) of the wavelet transform can be determinedby observing the resonant frequencies of the spindle system.As the rolling elements roll over a structural defect within aspindle bearing, impulsive signals will be generated, which, inturn, excite the spindle system to vibrate at one or more ofthe resonance frequencies [2]. Such resonant frequencies areidentified by experimental modal analysis. With the frequencyrange in which the structural resonance takes place, appropriatewavelet scales can be selected to cover the frequency range.

A commonly used analytic wavelet is the complex Morletwavelet [14], [15], which is defined as

ψm(t) =1√πγe−(

t2γ

)ej2πfct (11)

where fc is the center frequency of the wavelet, γ is the varianceof the Gaussian window, and fb = (1/γ)1/2 is defined as thebandwidth of the complex Morlet wavelet. The complex Morletwavelet is essentially a Gaussian window-modulated complex-

valued sinusoidal wave, which can be shown by substitutingthe complex exponential term in (11) with sine and cosinefunctions, i.e.,

ψm(t) =1√πγe−(

t2γ

)[cos(2πfct) + j sin(2πfct)] . (12)

Given that the Hilbert transform of cos(2πfct) is sin(2πfct),the complex Morlet wavelet satisfies the criterion of (2),which confirms that the complex Morlet wavelet is an analyticwavelet. In the frequency domain, the Fourier transform of thecomplex Morlet wavelet is expressed as

ψ̂m(f) = e−π2γ(f−fc)2. (13)

It can be seen that ψ̂m(f) is also a Gaussian function centeredat fc, with a variance of 1/γ.

Apart from the analytic nature, another advantage of theclass of analytic wavelets is that their relative bandwidths(i.e., the ratio between the absolute bandwidth and the centerfrequency) can be changed by varying the center frequencyand bandwidth parameters fc and fb, making it possible tocustomize the wavelet functions. Given that the wavelet trans-form is a “matching” operation between the base wavelet (i.e.,the analytic wavelet in this paper) and the signal to be ana-lyzed, customizing the base wavelet such that it best matchesthe relative bandwidth of the defect-induced impulses wouldimprove the performance of the wavelet transform. In practice,the relative bandwidth of a defect-induced impulse vibration isdependent on the structural stiffness and damping, which canbe estimated by means of experimental modal analysis.

To illustrate the variable relative bandwidth, examples of thecomplex Morlet wavelet in the time and frequency domains areshown in Fig. 3. In Fig. 3(a), the real and imaginary parts ofthe complex Morlet wavelet and its envelope under a centerfrequency of fc = 1 MHz and a bandwidth of fb = 0.5 MHzare shown. The complex Morlet wavelet with the same centerfrequency of 1 MHz but a different bandwidth of 1 MHz isshown in Fig. 3(b). Therefore, for the same center frequency,the absolute bandwidth of the complex Morlet wavelet canbe adjusted according to (11) to provide variable relativebandwidths.


Fig. 3. Complex Morlet wavelets with variable relative bandwidth (a) fc =1 MHz and fb = 0.5 MHz and (b) fc = 1 MHz and fb = 1 MHz.

In Fig. 3, it is seen that the spectrums of the complex Morletwavelets are one sided, with the amplitude in the negativefrequency range being zero. This is consistent with (6) andconfirms that the complex Morlet wavelet is analytic.

III. NUMERICAL EVALUATION

To quantitatively evaluate the effectiveness of the wave-let envelope spectrum technique, a synthetic defect-inducedvibration signal consisting of a series of exponentially decayingimpulses buried in white noise was simulated and processed.The simulated impulses have a center frequency of 1000 Hzand a repetition frequency of 80 Hz. This type of signal is com-monly seen in vibration signals generated from a ball bearingwith a localized outer raceway defect. The 80-Hz repetitionfrequency corresponds to the BPFO, which characterizes pulsegeneration due to interactions between the rolling balls and thedefect. The 1000-Hz center frequency characterizes the excitedspindle structural resonance by the ball–defect interactions. Thewhite noise simulates vibrations from other sources, such as the

Fig. 4. Defect identification from a synthetic broadband vibration signal undera noisy environment using conventional enveloping technique followed byFourier transform.

time-varying contact forces between the various components ofthe spindle structure and vibration input from the drive system.

Several synthetic signals with SNRs vary from −10 to−20 dB were simulated using conventional and analyticwavelet enveloping techniques. It was found that, for all SNRs,the wavelet enveloping spectrum has shown better performancein detecting the impulses buried in the noise. An example of asynthetic signal with an SNR of −10.35 dB is given, with itswaveform in the time domain shown in the upper plot of Fig. 4.The envelope spectrum of this signal was first obtained us-ing conventional enveloping technique, followed by a discreteFourier transform. A relatively strong peak was found at 80 Hz,as shown in the lower plot of Fig. 4. While this correlates wellwith the BPFO described earlier, the closeness of its magnitudeto that of other frequency components contributed by noise,e.g., the peak around 75 Hz, makes it difficult to clearly definethe existence of a structural defect on the outer raceway ofthe bearing. This illustrates the limitation of the conventionalenveloping technique in extracting transients of low-energysignals from strong background noise. For the analytic waveletenveloping technique, the similar limitation was observed at anSNR of −20 dB. Therefore, the analytic wavelet envelopingtechnique was able to achieve a 10-dB lower limit than theconventional enveloping technique.

The same signal was then processed using the analyticwavelet-based enveloping technique. Equally spaced scalesfrom 3 to 7 with an increment of 0.25 were chosen for thewavelet transform. These scales correspond to wavelet cen-ter frequencies from 700 to 1700 Hz (sampling frequency is5000 Hz), which cover the entire frequency range of theimpulse sequence. In addition, various wavelet relative band-widths were also tested, and the result shows that a relativebandwidth of 1.4, which corresponds to a variance γ of 0.5,had the best performance.

The wavelet transform of the original signal resulted in awavelet coefficient matrix CWT(τ, s), with each of its rowscontaining the extracted vibration signal corresponding to awavelet scale. The envelope of each row was then obtained


Fig. 5. Envelope spectrum of the synthetic vibration signal obtained based ona complex Morlet wavelet transform.

Fig. 6. Extracted envelope and envelope spectrum for scale 5.

by using (9). The set of envelopes obtained were subse-quently Fourier-transformed to generate an envelope spectrum,as shown in Fig. 5, in the form of a waterfall plot. Highmagnitude peaks indicating strong energy concentration areseen to be located around 80 Hz, for most of the waveletscales, whereas the highest magnitude peak is produced byscale 5. Comparing with the conventional enveloping techniqueshown in Fig. 4, the wavelet-based enveloping is shown to besignificantly more effective in detecting and identifying theexistence of the characteristic defect frequency BPFO withinthe vibration signal.

To further illustrate the effectiveness of the developed tech-nique, the extracted signal envelope and its spectrum at scale5 are plotted in Fig. 6, where the frequency peak at 80 Hz isclearly distinguishable from the rest of the frequency compo-nents with an SNR of 14.20 dB. The SNR is defined as the ratioof the BPFO frequency magnitude to the average magnitudeacross the entire frequency range. Such a value is more than10 dB larger than the SNR obtained from using the conventionalenveloping technique, as shown in Fig. 4.

Fig. 7. Envelope spectrum for two unmatched relative bandwidths.(a) Relative bandwidth = 0.25. (b) Relative bandwidth = 3.

Another advantage of the analytic wavelet, as indicated in(9), is that its relative bandwidth can be varied to match that ofthe impulse signals under investigation. Therefore, a properlyselected relative bandwidth can also improve the performanceof the impulse detection. For example, the ratio of the band-width to the center frequency of each of the simulated impulsesin Fig. 4 was calculated to be 1.4. Accordingly, the relativebandwidth of the wavelet was chosen to be 1.4 such that theshape of the wavelet function would best match that of theimpulses to most effectively extract transient features withinthe impulses from the background noise. To illustrate the ad-vantage of such a “matched” bandwidth selection operation, theperformance of the same wavelet with an unmatched relativebandwidth of 0.25 and 3.0, respectively, was investigated. Asillustrated in Fig. 7, the detected BPFO frequency peaks haveshown an SNR of 9.65 and 7.64 dB, respectively, about 32%and 46% weaker than the 14.20 dB achieved under the matchedbandwidth of 1.4.

IV. EXPERIMENTAL RESULTS

To experimentally verify the developed wavelet envelopingtechnique, a test bed was designed and constructed. As shownin Fig. 8, a spindle is supported by four angular contact ballbearings of 42 mm outer diameter on its front and rear ends.Two air cylinders (static and dynamic load cylinder) applyconstant and impulsive loads to the spindle, simulating thespindle operation conditions under stable preload (e.g., whenmachining a workpiece under constant speed and feed) or shockinput (e.g., when impacted due to tool–workpiece interaction).Four accelerometers were placed at the front and rear endsof the spindle, within the loading and unloading zones of thebearings, to measure their vibrations.

Based on the bearing pitch diameter D, rolling elementdiameter d, and number of rolling elements n, defect-relatedcharacteristic frequencies of the spindle bearings can beanalytically determined as a ratio to the shaft rotational speed.


Fig. 8. Spindle test bed and its functional diagram for the experimentalevaluation of wavelet-based diagnostic technique.

TABLE IFORMULAS TO CALCULATE DEFECT-RELATED CHARACTERISTIC

FREQUENCIES OF THE SPINDLE BEARINGS

In Table I, the formulas used to calculate four major defect typesare shown.

In the first phase of the experimental study, vibration datawere collected by the four accelerometers from the spindleassembly under various rotational speeds (1200, 3600, 6000,and 8400 r/min) and static loads (70, 419, 839, and 1258 N),respectively. Dynamic loads were purposely not applied duringthis phase of the experiments to establish a reference baselinefor an undamaged “healthy” spindle. The data sampling fre-quency used for data acquisition was 20 kHz, and for eachexperiment under a certain load/speed combination, a total datalength of 1.64 s was collected.

To ensure complete frequency range coverage of the spindlevibration signals, proper scales for the analytic wavelet trans-form needs to be chosen. For this purpose, a modal analysis wasconducted on the spindle structure, which identified a numberof major resonant frequencies in the range of 2–10 kHz. Con-

Fig. 9. Envelope spectrum of vibration signals measured on undamaged“healthy” spindle (static load = 839 N and spindle speed = 8400 r/min).

sidering that structural defects may excite the spindle structureat any of these frequencies, equally spaced scales from 1 to 12with an increment of 0.25 were chosen for the wavelet trans-form, which cover a frequency range from 1.667 to 20 kHz.

Wavelet enveloping of the vibration signals was performedaccording to the procedure illustrated in Fig. 2. As an example,Fig. 9 illustrates the spectrum of the envelope signal obtainedfrom the analytic wavelet transform, under the static load of839 N and rotational speed of 8400 r/min. For a clear presen-tation of the frequency bands where defect-excited structuralresonance occurs, the wavelet center frequency instead of thewavelet scale was used to label one of the axes in the coordinatesystem. In Fig. 9, the only appreciable frequency peak with amagnitude of 170 W/Hz is shown at 140 Hz, which correspondsto the spindle rotational speed of 8400 r/min. No characteristicfrequencies that relate to structural defects within the spindlebearing system were found, indicating that the spindle bearingsare in good condition.

After establishing the reference baseline, dynamic loading inthe form of a 13 300-N impulse lasting about 25 ms was consec-utively applied to the spindle system under a constant rotationalspeed of 3600 r/min, for a total of 700 times. At the end of100, 400, and 700 impacts, the impact test was paused, andvibration signals were collected by the four accelerometers, un-der various combinations of static loads and rotational speeds.Spindle damage as the result of the impacts was evaluated byprocessing the spindle vibration signals with analytic wavelettransform and subsequent spectral analysis of the envelope. Therelative bandwidth of the analytic wavelet was chosen to be 1.0,based on previous experimental modal analysis result. After300 impacts, the wavelet envelope spectrum of vibration signalcollected by accelerometers placed near the front bearingsstarted to show frequency peaks within the range of 0–1500 Hz.The magnitude of these peaks increased as the impacts onthe spindle accumulated, and after 700 impacts, most of thepeaks have exceeded 100 W/Hz in magnitude, as can be readilyidentified visually. An example of wavelet envelope spectrumfor the spindle vibration signal collected under 839 N static loadand 8400 r/min rotating speed is shown in Fig. 10.


Fig. 10. Envelope spectrum of vibration signals measured on damaged spin-dle, after 700 impacts (static load = 839 N and spindle speed = 8400 r/min).

Fig. 11. Envelope spectrum of the spindle signal at scale 3.25 before and afterimpacts. (a) Wavelet envelope of the spindle signal before impact applications.(b) Wavelet envelope of the spindle signal after 700 impacts were applied.

Comparing to Fig. 9, where the spindle was not dynamicallyimpacted, several frequency components with large magnitudeappeared within the frequency range of 0–1500 Hz, whichindicate the effect of the impacts. To quantitatively evaluate theeffect, wavelet envelope spectrum at scale 3.25 (correspondingto the wavelet center frequency of 6154 Hz) is shown in Fig. 11,for the cases of before and after the impact experiments.

Comparing Fig. 11(a) and (b), the three peaks in the< 500-Hz range, which are associated with the spindle shaftrotating frequency fr = RPM/60 and its two harmonics 2fr

and 3fr have shown to have increased in magnitude as the resultof impact accumulation. Under the shaft rotational speed of8400 r/min, these peaks are located at 140, 280, and 420 Hz,respectively. The increase of these frequency components canbe related to the increase of impact-induced spindle unbalance,which is the offset between the center of mass of the rotatingspindle and its center of rotation. Structural unbalance can berepresented in effect as a radial force that excites a rotating shaftat the frequency fr.

Based on the equations shown in Table I, the frequency peakat 935 Hz can be identified as the BPFI. The existence ofsuch a frequency component in the spectrum indicates that one

of the spindle front bearings has developed a localized defecton its inner raceway as the result of dynamic impact loading.Theoretically, the BPFI frequency at 8400 r/min is calculatedas 922 Hz. The 1.4% difference between the theoretical andexperimental values can be traced back to the combined effectof rolling element slippage and the slight drift of spindle speedfrom the nominal input values to the spindle drive controller.

The spectrum also displayed several other frequency peaksat 1075, 1215, and 1355 Hz, respectively. These frequenciescan be specified as BPFI + k · (RPM), with k = 1, 2, . . . , n,and reflect upon the combined effect of spindle unbalance andinner raceway defect. The frequency peak at 885 Hz does notmatch any of the spindle bearing characteristic frequencies. Itis postulated that this frequency component is associated withthe dynamics of the external housing bearing, which was usedto transmit mechanical impact force from the impact cylinderto the spindle. Further analysis is being conducted to identifythe nature of this frequency.

In addition to the spindle speed of 8400 r/min, the waveletenvelope spectrums for 4800, 6000, and 7200 r/min were alsoanalyzed, as shown in Fig. 12. Specifications of the individualfrequency components are summarized in Table II. It is seenthat all defect-related characteristic frequencies for 8400 r/minare consistently present under other spindle speeds, with theaverage magnitude exceeding 100 W/Hz. This result verifiesthat the existence of the defects is a physical phenomenon thatis independent of the spindle rotational speed.

Analysis of the wavelet envelope spectrum further revealedthat the vibration modes and associated resonant frequenciesexcited by the spindle defect have varied, as the spindle rota-tional speeds varied. As shown in Fig. 12(a), for 4800 r/min,vibration frequencies are concentrated in two wavelet frequencybands, which are centered at 2963 and 6154 Hz, respectively,with comparable magnitudes. As the speed increased to 6000and 7200 r/min, the average magnitude of the defect-relatedcharacteristic frequencies at the wavelet center frequency of2963 Hz has decreased to about 30% and 5% of that under4800 r/min, as shown in Fig. 12(b) and (c), respectively. The2963-Hz center frequency became nearly negligible after thespindle speed has reached 8400 r/min, as shown in Fig. 10.Such variations provide insight into the excitation frequency-dependent dynamics of the spindle structure and are helpful inchoosing appropriate wavelet scales for the wavelet envelopingalgorithm to reduce computational load and achieve computa-tional efficiency.

V. CONCLUSION

A wavelet envelope spectrum technique was developed fordetecting and diagnosing structural defects from vibration sig-nals measured on a realistic spindle test bed based on theanalytic wavelet transform. Analysis has shown that the wavelettransform of a real-valued signal using an analytic waveletproduces a complex wavelet coefficient, which, in turn, is an-alytic. Consequently, the envelope of the signal can be obtaineddirectly from the wavelet coefficient, without additional opera-tions. Numerical simulation of the vibration signal of a bearingwith an outer raceway defect has verified that the developed


Fig. 12. Envelope spectrum of spindle vibration signals measured after 700 impacts under different speeds. (a) Spindle speed is 4800 r/min. (b) Spindle speed is6000 r/min. (c) Spindle speed is 7200 r/min.

TABLE IIIDENTIFIED DEFECT-RELATED CHARACTERISTIC FREQUENCIES OF THE SPINDLE BEARINGS

wavelet envelope spectrum technique can successfully detectthe defect frequency in a bearing, with an SNR that is 10 dBhigher than that obtained from using the conventional envelop-ing technique. The wavelet envelope spectrum technique wasthen applied to monitor and diagnose a spindle test bed, whichwas subject to dynamic impact forces. Several characteristicfrequencies associated with the spindle unbalance and innerraceway defect resulting from the impact were successfully de-tected. In addition, the envelope spectrum has revealed that thedefect has excited different structural resonances at different ro-tational speeds, which were not found using conventional meth-

ods. In addition to spindles and bearings, the developed waveletenveloping technique can be applied to other types of machinesfor condition monitoring and health diagnosis purposes.

ACKNOWLEDGMENT

The authors would like to thank S. Fick and M. Huff ofthe Manufacturing Metrology Division at NIST for the col-laboration during the experimental study and R. Browner andS. Crossman at Timken Corporation for the support and assis-tance on the design and construction of the spindle test bed.


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Li Zhang received the M.S. degree from the Univer-sity of Electronic Science and Technology of China,Chengdu, China, in 2000 and the Ph.D. degree fromthe University of Massachusetts (UMass), Amherst,in 2004.

He is currently a Postdoctoral Research Associatewith the Department of Mechanical and IndustrialEngineering, UMass, and a Guest Researcher withthe National Institute of Standards and Technology,Gaithersburg, MD, working on a collaborative re-search project on smart machine tools. His research

interests include electromechanical systems design, smart sensing systems, ma-chine condition monitoring and health diagnosis, and sensor signal processing.

Robert X. Gao (M’91–SM’00) received the M.S.and Ph.D. degrees in mechanical engineering fromthe Technical University of Berlin, Berlin, Germany,in 1985 and 1991, respectively.

He is currently a Professor with the Department ofMechanical and Industrial Engineering, Universityof Massachusetts, Amherst. His research and teach-ing interests include physics-based sensing method-ology, coordination-based and energy-efficient sen-sor networks, mechatronic systems design, medicalinstrumentation, and wavelet transforms for machine

health monitoring, diagnosis, and prognosis.Dr. Gao was a recipient of the National Science Foundation CAREER Award

in 1996 and the University of Massachusetts Outstanding Engineering JuniorFaculty Award in 1999 and was the faculty advisor and a corecipient of the in-augural Best Student Paper Award from the International Symposium on SmartStructures and Materials of the International Society for Optical Engineers in1997. He is a Coeditor of the book Condition Monitoring and Control forIntelligent Manufacturing (Berlin, Germany: Springer-Verlag, 2006), the GuestEditor for a Special Section on Built-In-Test for the IEEE TRANSACTIONS

ON INSTRUMENTATION AND MEASUREMENT, which was published in 2005,and the Guest Editor for a Special Section on Sensors for the Journal ofDynamic Systems, Measurement, and Control, which was published by theAmerican Society of Mechanical Engineers in 2004. He currently serves as anAssociate Editor for the IEEE TRANSACTIONS ON INSTRUMENTATION AND

MEASUREMENT, an Associate Editor for the Journal of Dynamic Systems,Measurement, and Control, and an Editorial Board Member for the Interna-tional Journal of Manufacturing Research. He is also a Cochair of the TechnicalCommittee on Built-in-Test and Self-Test of the IEEE Instrumentation andMeasurement Society.

Kang B. Lee (S’69–M’74–SM’00–F’03) receivedthe B.S. and M.S. degrees in electrical engineer-ing from the Johns Hopkins University, Baltimore,MD, and the University of Maryland, College Park,respectively.

He is the Leader of the Sensor Development andApplication Group, Manufacturing Engineering Lab-oratory, National Institute of Standards and Tech-nology (NIST), Gaithersburg, MD. Since he joinedNBS, now NIST, in 1973, he has worked in thefields of electronic instrumentation design, sensor-

based closed-loop machining, robotic manufacturing automation, smart sensornetworking, and Internet-based distributed measurement and control systems.

Mr. Lee serves as the Chair of I&M Society’s Technical Committee on SensorTechnology TC9 that defines the IEEE 1451 standards. He also serves as anI&M Society Delegate to the Sensors Council.

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Spindle Health Diagnosis Based on Analytic Wavelet Enveloping

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