J Intell Manuf (2012) 23:213–220DOI 10.1007/s10845-009-0346-y

Quantification of condition indicator performanceon a split torque gearbox

Eric Bechhoefer · Ruoyu Li · David He

Received: 27 August 2009 / Accepted: 16 October 2009 / Published online: 13 November 2009© Springer Science+Business Media, LLC 2009

Abstract The requirement for higher energy density trans-missions (lower weight) in helicopters has led to the devel-opment of the split torque gearbox (STG) to replace thetraditional planetary gearbox by the drive train designer. Thismay pose a challenge for the current gear analysis methodsused in health and usage monitoring systems (HUMS). Gearanalysis uses time synchronous averages to separates in fre-quency gears that are physically close to a sensor. The effectof a large number of synchronous components (gears or bear-ing) in close proximity may significantly reduce the faultsignal (reduce signal to noise ratio) and therefore reducethe effectiveness of current gear analysis algorithms. In thispaper, quantification of condition indicator performance on asplit torque gearbox is reported. The vibration signatures areprocessed through a number of gear analysis algorithms toquantify the gear fault performance. The performance metricis separability.

Keywords Split torque gearbox · Fault diagnosis ·Condition indicators

Introduction

Helicopter performance, economy and development cost area function of weight. The split torque gear (STG) box (White

E. BechhoeferGoodrich Sensors and Integrated Systems, 100 Panton Rd,Vergennes, VT, USAe-mail: Eric.Bechhoefer@Goodrich.com

R. Li · D. He (B)Department of Mechanical and Industrial Engineering,University of Illinois-Chicago, Chicago, IL, USAe-mail: davidhe@uic.edu

R. Lie-mail: rli8@uic.edu

1982) is an alternative to traditional planetary gearbox design.Potentially, the STG saves weight, can be more reliable, havereduce transmission noise, and improved efficiency. Thesebenefits have driven the helicopter manufacturing communityto develop products using the STG. For example, the Coman-che helicopter was designed with a STG, and the new Siko-sky CH-53K will incorporate the STG design to transmit over18,000 shp to the rotor blades. It is likely that STG will beincorporated into more designs in the future (Gmirya 2008).

Because of the limited experience in building helicopterwith STG, there is no condition based monitoring data on thistype of gear box. Studies have been conducted to model andanalyze vibration dynamics of the STG (Krantz 1995), andanalysis on gear loading has been conducted (Krantz 1996).Yet, these studies do not give insight into fault detection ofgears on this type of design. Gear diagnostics use time syn-chronous averages to separates in frequency gears that arephysically close. The effect of a large number of synchro-nous components (gears or bearing) in close proximity maysignificantly reduce the fault signal (increase signal to noise)and therefore reduce the effectiveness of current gear analysisalgorithms

In order to gain experience in performing HUMS types ofanalysis on STG, Goodrich working with the University ofIllinois at Chicago (UIC) have build a test gearbox for the pur-pose of testing condition indicators (CI) used in HUMS andcondition based maintenance practices. The primary designconsiderations where emulation of synchronous gear signalsthat would be found in a STG (see Figs. 1, 2).

The input spur gear is a 40-tooth gear, driving three, inputspur idler gears of 72 teeth. The idler shafts drives three, 48-tooth output spur idler which drives a single 64-tooth outputspur gear. Accelerometers where mounted on the input drivepinion and on each output drive idler. The fault was charac-terized as removal of 20% of the gear tooth on one of the

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Fig. 1 Input drive side of STG test rig

Fig. 2 Output drive side of the STG test rig

output drive idler. The location of each accelerometer wasplaced such that: there was optimal fault detection, and thatthere would be some potential for sensor fusion (Fig. 5).

The STG test rig is representative of an operational STG inthe generation of synchronous tones test CBM/HUMS anal-ysis gear analysis algorithm. The cost, size and power wheredriving considerations in the design of this STG, which wasultimately developed by SpecrtaQuest, Inc. A more repre-sentative gearbox design, such at seen in Krantz (1996) fromthe Comanche STG is given in Fig. 3.

Among the traditional technique for detection of thelocalized gear faults, the time domain synchronous average(TSA) (Combet and Gelman 2007) is the most popular tech-nique. It is powerful in removing the noise and other non-synchronous components from the original vibration signal.

Fig. 3 Comanche STG showing compact size and high gear reductionratio

However, TSA requires an extra reference signal. Recently,the time-frequency analysis methods, such as, short-timeFourier transform (SFT) (Kaewkongka et al. 2003), Wig-ner-Ville distribution (Kim et al. 2007), wavelet analysis,and Hilbert-Huang transform (HHT), are well developed andproven to be effective in gear fault detection. Among thesemethods, HHT is the most recently developed method andit outperforms other methods in gear fault detection (Liuet al. 2006). Since in a STG type gearbox, there are severalidentical gears meshing simultaneously. This unique behav-ior of the STG type gearbox may significantly reduce thefault signal and therefore reduce the effectiveness of cur-rent gear analysis algorithms. No fault detection methodshave been applied to the vibration signals on the STG typegearbox.

In this paper, quantification of condition indicator perfor-mance on a split torque gearbox is reported. The vibrationsignatures are processed through a number of gear analysisalgorithms to quantify the gear fault performance. The per-formance metric is separability. The remainder of the paper isorganized as follows. In section Gear analysis, the gear anal-ysis background is introduced. The theoretical details on thegear analysis are presented in section “Gear analysis descrip-tion”. In section “Advance signal processing techniques”,three advanced signal processing methods are investigated.Finally, section “Conclusions” concludes the paper.

Gear analysis

Gears are complicated dynamic structures with a number offailure modes. Any CBM/HUMS type of analysis needs tocapture these damage modes to ensure appropriate mainte-nance can be conducted prior to failure. For example, geardamage can be characterized by:

• Ageing induced defects such as pitting, spalling andincreased surface roughness.

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• Assembly and/or wear resulting in misalignment,improper backlash or looseness.

• Manufacturing error resulting in an eccentric gear or non-uniform gear tooth spacing.

• Mesh impact/soft tooth due to gear tooth chipping, crack-ing of the gear tooth base.

Each fault could induce one or more CI to indicate an anoma-lous condition. While studies have been conducted to modelgear fault dynamics (Wemhoff et al. 2007; Decker 2003), noone CI is seen to be sensitive to all gear fault modes. Addi-tionally, it is unlikely that, in the near future, any CI willdenote an absolute level of damage.

Implicit in absolute levels of damage is a probability ofgear continuing to transmit torque over some period of time,e.g. a CI value of xx indicates that the gear will transmittorque for 100 h with a probability of .95. However, both teststand and real world experience have shown that a number ofCI values change relative to a level of damage. This suggeststhat a stochastic or hypothesis testing methodology can beused to indicate impending gear failure. In fact, most HUMSuse statistical limits (e.g. mean plus 3 standard deviation, forexample) to indicate when maintenance is appropriate.

Given the literature of gear CI, in this study 20 gear/shaftbased analysis and 38 gear analysis were conducted on nom-inal and faulted gear box, and a measure of seperability (abil-ity to discern a statistically significant change) between thenominal and faulted gear was measured.

Finally, non-traditional gear analysis was attempted usingadvance signal processing techniques. A formal developmen-tal description and results will be given after the traditionalgear analysis results.

Gear analysis description

The 58 gear analysis were all based on processing timedomain vibration data into a synchronous average (SA). SAis a technique where gear and shaft vibration are resampled toremove variation is shaft RPM speed. This has two benefits:

• Each bin in the results Fourier transform (FFT) repre-sents one shaft revolution. All the energy associated witha shaft order or a gear mesh is then just the FFT at theappropriate bin index + 1 (bin 1 is the DC component,and with an AC coupled system, should be zero. The firstharmonic for a 40 tooth gear is bin 41).

• Any spurious noise is average. In general, the noise reduc-tion is 1/sqrt(number of revolutions). Each revolution isestimated by the product of the time between tachometerzero crossings (usually rising edge) and the shaft ratiofrom the shaft on which the tachometer signal is taken.

A number of SA techniques have been developed (Combetand Gelman 2007). In this study, the optimal interpolationmethod was chosen. In this method, a linear-phase FIR fil-ter is constructed that minimizes the weighted, integratedsquared error between an ideal piecewise linear function andthe magnitude response of the filter over a set of desired fre-quency bands. The generalized filter is Eq. 1:

h(t)=sin (π t/T )/(π t/T )×cos (Rt/T )/(

1 − 4R2t2/T 2)

(1)

where R is the roll off factor and T is the symbol period.Essentially, the filter h(t) interpolates n samples into N sam-ples, where

N = 2ceiling(log2(n)) (2)

for radix 2 FFTs.The pseudo code for the SA is:

BT: average elapse time between tachometer impulses.N : number of points in the FFTPhi0: Initial shaft anglePhi: shaft angleRev: number of revolutions in SARat: shaft ratio relative the tach signalZct: zero crossing timeIncr: shaft increment =1/NIsamp: interpolated sampleSamp: time domain signalSR: sample rate;SA: synchronous average

For r = 0 to r < Rev,Phi0 = Rat + r ;

For k = 0; k < NPhi = (Phi0 + k*Incr)/Rat;Dt = conv(h(t),zct(t))Isample = conv(h(t), Samp(Dt ∗ S R))SA(k) = SA(K ) + isamp;

EndEnd

SA = SA/Rev

The CI are features of the SA. There are typically three typesof CI features that are derived from the SA:

• Shaft algorithms which detect fault on any gear associ-ated with the shaft (e.g. if a gear is attached to either endof the shaft, it is not possible to identify which gear isfaulted). These typically include statistics on the SA, orratio of statistics on the SA (Table 1).

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Table 1 Shaft based algorithm results

Condition indicator T Condition indicator T

SO1 amplitude .66 Gear dist. fault 2.34

SO2 amplitude 1.14 Energy operator CF 0.04

SO3 amplitude .84 EO Pk 2 Pk 2.4

SA Pk 2 Pk 4.0 EO skewness 0.04

SA crest fact. 2.55 EO kurtosis .005

SA skewness 1.4 EO RMS 3.5

SA kurtosis 3.4 Residual skewness 1.0

SA RMS 3.0 Residual kurtosis 0.65

SA phase kurtosis .04 Residual Pk 2 Pk 2.75

Residual RMS 2.8 Residual crest fact. 1.24

• Gear algorithms based on the ratio of gear mesh ampli-tudes or the ratio of gear mesh amplitude to shaft fun-damental amplitude. These algorithms can identify gearsby the number of teeth (Table 2).

• Gear algorithms based on statistics/features of the SAband passed filtered around the gear tooth mesh rate (e.g.narrow band signal analysis, Table 3).

For a full description of the gear algorithms used see(Wemhoff et al. 2007).

The measure of seperability was calculated using thepooled sample standard deviation. The sample size was 20acquisitions per sample set, where the populations for thenull set came from the nominal gear (no damage) and thealternative set came from the damage gear population. Thetest statistics is then:

T = E [Y1] − E [Y2]

Sp√

2/n(3)

where

Sp =√

(n − 1) S21 + (n − 1) S2

2

2n − 2(4)

Table 2 Gear algorithm based on mesh tones

Condition indicator T Condition indicator T

Mesh rate 2.6 Sideband mod 1 4.3

Mesh 2nd harmonic 3.1 Sideband mod 2 4.2

Mesh 3rd harmonic 0.55 Sideband mod 3 2.5

G2 rate 4.3 Gear mesh 1 1.44

G2 2nd harmonic 1.4 Gear mesh 2 2.8

G2 3rd harmonic 1.88 Gear mesh 3 3.8

SA RMS 3.0

Table 3 Narrow band signal analysis

Condition indicator T Condition indicator T

Narrow band 6th M 4.1 Derivative FM RMS 3.0

NB 5th moment 1.3 DFM kurtosis 1.2

NB PK 2 PK 0.1 AM RMS 1.12

NB kurtosis 4.4 AM CF 7.3

NB skewness 0.04 AM skewness 0.9

NB CF 7.1 AM kurtosis 0.6

NB RMS 1.1 DAM RMS 1.92

FM RMS 0.22 DAM CF 0.37

FM CF 2.8 DAM skewness 0.38

FM skewness 0.3 DAM kurtosis .19

FM kurtosis 1.5

A test statistic T greater than 3.58 is considered signifi-cant and would indicate that the CI could detect the fault(Wackerly et al. 1996).

The large statistical difference (T of 4) seen in the SAPeak to Peak suggests that the mesh tones are increased inthe damage gear.

There was a number of gear CIs that would indicate gearfault.

This should not be considered an easy case to performanalysis. A number of observations come out of the study.Data was acquired at a number of torque (e.g. brake) levels.Essentially, no detection was possible at lower torque set-tings. This can be attributed to the nature of the STG, wherethe torque paths, and therefore loads, are split. Additionally,the gears relative to the driving loads, are large: the drivemotor can supply only 3 HP. In other words, the gearbox wasover designed for the loads present. What this shows is thatthose algorithms with a T > 3.58 are remarkable sensitiveto gear fault.

Advance signal processing techniques

Test stands provide good opportunities to test new algo-rithms and techniques. Given this, three signal processingtechniques where evaluated as potential new methodologyfor gear fault detection. The first technique, beam form-ing, would take advantage to n number of sensor to rejectunwanted noise by “steering” the sensor array sensitivity tothe desired signal source. The second techniques, linear pre-diction coding, models the periodic signals (e.g. shaft andgear mesh tones) and dynamically predicts the signal d stepsin the future, removing them. This, in theory leaves onlyfault signatures or random noise. This technique is callednarrowband interference cancellation. The third technique,empirical mode decomposition (EMD) would decompose the

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Fig. 4 Example: array of two accelerometers

vibration signal into several intrinsic mode functions (IMF)and then the interested IMF component would be analyzedto reveal the gear fault.

Beam former technology

An accelerometer, hard mounted to the gearbox, will gen-erally receive a number of undesired signals in addition tothe signal of the component under analysis. These extrane-ous signals are of random phase, and tend to increase thenoise of the system, masking signals of interest. This can bea more serious matter in STG where there are synchronoussignals that differ only in phase and amplitude when reachinga sensor.

It would be useful if a beam former could be used as aspatial filter that linearly combined the output of n acceler-ometers so that signals arriving from some particular direc-tions are amplified, while signals from other directions areattenuated. Consider a sensor array consisting of two accel-erometers, A and B (see Fig. 4).

The signal s(n) = A cos(ωn) and v(n) = B *cos(ωn),arriving at angles 0 and �, with respect to the line perpen-dicular to the line combining A and B, which are the inputsto the array filter which consists of a phase-shifter and sub-tractor. The signal s(n) arrives at elements A and B at thesame time, where the arrival times of signal v(n) at A and Bare different. Thus,

sA (n) = sB(n) = α cos (ωn)

vB (n) = β cos (ωn)

vA (n) = β cos (ωn − φ)

(5)

where A and B are used to denote the signals picked up byaccelerometers A and B, respectively, and phi is the phase-shift arising from the time delay of arrival of v(n) at elementA with respect to its arrival at B.

It is assumed that s(n) is the desired signal and v (n) is aninterference, then by inspection we can find a phase shifterequal to f with which we can cancel out v(n). The desiredsignal reaches the filter output as A cos(ωn)−cos(ωn −�),which is non-zero and still holding the information containedin a. The algorithm to do this is based on the least-mean-square (LMS) algorithm developed by Widron and Hoff(Manolakis et al. 2000).

Fig. 5 Accelerometer location

Using two sensors from the STG, a beam-former to nullssignals that are not coming perpendicularly into the sensorwas tested. The accelerometers where mounted parallel toeach shaft, such that signatures that arrive off axis would befrom a gear that are not of interest. Theoretically, this wouldimprove the signal to noise of the gear under analysis, whichhas the benefit of increasing the discriminate capability ofthe gear algorithms.

Testing of the beam former showed no improvement gearfault discrimination. This was disappointing and, on furtherinvestigation, not surprising. In RADAR for example, eachantenna element is a dipole, with sensitivity not a function ofarrival angle. This is not the case of shear style accelerome-ter, which has sensitivity of, at most, 10◦ on axis. Due to theplacement of the accelerometers (see Fig. 5) there was verylittle signal interference from the adjacent gears.

Due to this limitation, additional accelerometer mountingbosses where machined into the gearbox shaft supports. Thiswill allow future analysis of beam former techniques in thefuture.

Narrowband interferance canceller

Most gear algorithms used in HUMS are based on measuringfrequencies associated with gear mesh tones. For example,Mesh Analysis is the ratio of gear mesh harmonics to the basemesh tone. Similarly, G2 analysis is the ratio of the signalaverage peak to peak value and the gear mesh tone. Someanalysis, such as demodulation analysis, measures change ingear mesh tone amplitude or frequency. However, none ofthese analyses measure the fundamental artifact of a soft or

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cracked gear tooth: a high frequency impulse due to the localfault.

This type of fault is difficult to detect with current Fouriertransforms (FFT) in that this transform measures sinusoidstones and is relatively insensitive to periodic impulse trains.Additionally, in a gear box, there are a number of narrowband tones and broad band noise which mask the desiredimpulse train. The system can be modeled as:

x (n) = s (n) + y (n) + y (n)

where

s(n) is the signal of interesty(n) is the signal associated with gear mesh, shafts, e.g.

interference.v(n) is random noise.

The interference signal is typically large compared to thesignal of interest. It is necessary to remove the interferingsignal y(n) from x(n) while preserving the signal of inter-est s(n). Since the measured signal x(n) and the interferencey(n) are correlated, one can estimate the interference usingan optimal linear estimator:

yh (n) = ct x (n − D)

Rc = d

R = E[x (n − D) xt (x − D)

]

d = [x (n − D) y (n)] (6)

where D is the delay operator. If yh(n) = y(n), the outputof the filter is x(n) − yh(n) = s(n) + v(n). This means wecan completely remove the interference and only the desiresignal and noise remains.

In practice, signal y(n) is not available. One can use aminimum means square error D-step linear predictor, suchthat:

e(n) = x(n) + at x(n − D)and Ra = −r,

where

r = E [x (n − D) x (n)] .

For this modeling assumptions to hold true:

• The signal of interest s(n) and the noise signal v(n) areuncorrelated

• The noise signal v(n) is white• The signal of interest s(n) is wideband and has a short

correlation length (e.g. its impulsive), r(l) = 0 for l > D.• The interference signal y(n) has a long correlation length:

its autocorrelation length takes significant values over therange 0 <= l <= M , for M > D

Fig. 6 Gearbox spectrum and LPC derived filter response

The first modeling assumption suggests:

E[x (n − k) yt (n)

] = E[y (n − k) yt (n)

] = ry (k)

where

rx (l) = rs (l) + ry (l) + rv (l)

Assuming that the second and third modeling assumptionshold true:

rx (l) = ry (l) , for l! = 0, 1, . . . D − 1

This means that the linear predictor at time n is based on datadelayed by d lags (e.g. the vector x(n−D)). This suggest that,under these conditions, d = r, and c = a. This gives the resultthat the optimal interference filter coefficients c is equal to theD stop linear predictor of a, which can be determined fromthe input signal x(n). The signal with interference removedis then:

x (n) − yh (n) = x (n) + at x (n − D) = e (n) ;Note that a is and FIR filter based on delayed data. See Fig. 6,which shows the filter response and the spectrum of the gear-box. For a full description of the analysis, see (Manolakiset al. 2000).

As an example, analysis was conducted on the good andfaulted gear. The fault, as noted, was a chipped tooth, thatshould result in a 1/revolution impact. The sample rate is102 KHz, with 2 seconds of time domain data. The lag (D)

was 1, and the filter size (M) was 400. The ratio of the errorpeak to peak to standard deviation was 81.4 for the faultedgear, and 37.5 for the nominal case. Figure 7 clearly showsthe period impulse associated with the gear fault.

Periodic impulses in the time domain generate multipleharmonics in the frequency domain. The real Cepstrum ofthese types of signal can be used to identify impulsive sig-nals that cannot be seen in FFTs (Fig. 8).

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Fig. 7 Detection of gear 1/rev gear impulse

Fig. 8 Cepstrum of error signal

Figure 8 shows artifacts in the Cepstrum that is indica-tive of periodic impluses with a time period of 1/Rev. Thesesignals are not present in the good data.

Empirical mode decomposition

EMD was proposed by (Huang et al. 1998). It is a newmethod for adaptive analysis of non-linear and non-station-ary signals. The EMD technique decomposes a signal intoa finite sum of components called intrinsic mode functions(IMF). Hilbert transform is then applied to IMFs to obtainboth the envelope signal and the instantaneous frequency.The detailed procedure of EMD technique can be found inHuang et al. (1998)

In general, the Hilbert transform H [x(t)]for a given signalx(t) is defined as

H [x(t)] = y(t) = 1

π

∫ +∞

−∞x(τ )

τ − tdτ (7)

Fig. 9 Fourier spectrum of health gear data

When y(t) is obtained, the analytic signal z(t) can be definedas:

z(t) = x(t) + j y(t) = E(t)e jϕ(t) (8)

where E(t) = |x(t) + y(t)| and ϕ(t) = tan−1 y(t)x(t)

The envelope signal is defined as

E(t) =√

x2(t) + y2(t) (9)

The EMD is applied to the vibration signal of both healthyand damaged gearbox. Among the IMF components the 3rdIMF component was found to be associated with the gearmeshing frequency of 1600 Hz. The Fourier spectrum of theenvelope signal of the IMF 3 is calculated and shown inFigs. 9 and 10. From Fig. 10, it is easy to find the high peakvalue at the gear fault frequency, which is 33.33 Hz. Since weare interested in the gear fault frequency of 33.33 Hz, the kur-tosis value of the envelope spectrum between 30 and 40 Hzcan reveal the fault of the gearbox. The kurtosis is a fourth-order statistic normalized by the square of the variance, inthe discrete domain it can be calculated as following,

K urtosis = 1

N

N∑i=1

(x (ti ) − μ)4

σ 4 (10)

Forty groups of vibration data, twenty for healthy gearboxand twenty for damaged gearbox, are processed by EMDand the envelope spectrum of the 3rd IMF components arecalculate and the kurtosis values in the frequency range of30 to 40 Hz are obtained. The mean value of the spectrumkurtosis for the damaged gear is 4.4968 and the standarddeviation is 2.2180. The mean value of the spectrum kurtosisfor the healthy gear is 2.6733 and the standard deviation is0.6986.

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Fig. 10 Fourier spectrum of damaged gear

Conclusions

This paper evaluated a number of gear fault detection algo-rithms when applied to a split torque gearbox design. Manytraditional Condition Indicators (CI) where shown to beeffective in detecting a chiped gear tooth. Access to nom-inal and faulted data allowed evaluation of other advancedsignal processing techniques. An array processing techniquefor beam forming did not perform well. However, a narrow-band interferance canceller may potentially lead to betterdetection of chiped tooth on gears. Success was also foundin using empirical mode decomposition.

We anticipate additional work on this gearbox. Newaccelerometer mounting points will be machined to allowarray processing techniques to be developed. AdditionalCI development will be conducted using both NarrowbandInterference Canceller and Empirical Mode Decompositiontechniques. These new CI performance will be quantifiedagainst the current CI performance. Eventually, we wish tobroaden the scope of the faults used to include gear toothmachining error and gear misalignment.

References

Combet, F., & Gelman, L. (2007). An automated methodology forperforming time synchronous averaging of a gearbox signalwithout speed sensor. Mechanical systems and signal processing,21(6), 2590–2606.

Decker, H. J. (2003). Crack detection for aerospace quality spur gears.NASA TM-2003021220 REV 1.

Gmirya, Y. (2008). Split torque gearbox for rotor wing aircraft withtranslational thrust system. USPTO, 7,413,142, August 19.

Huang, N. E., Steven, S. Z., Long, R., & Manlic, W. (1998). Theempirical mode decomposition and the Hilbert spectrum fornonlinear and non-stationary time series analysis. ProceedingRoyal Society of London A, 454(1971), 903–995.

Kaewkongka T., Au Y. H. J., Rakowski R. T., & Jones B. E. (2003).A comparative study of short time fourier transform for bear-ing condition monitoring. International Journal of Comadem, 6,41–48.

Kim B. S., Lee S. H., Lee M. G., Ni J., Song J. Y., & Lee C. W.(2007). A comparative study on damage detection in speed-upand coast-down process of grinding spindle-typed rotor-bearingsystem. Journal of Materials Processing Technology, 187, 30–36.

Krantz, T. (1995). Vibration analysis of a split path gearbox. NASAtechnical memorandum 106875, AIAA-95-3048.

Krantz, T. (1996). Code to optimize load sharing of split-torque trans-missions applied to the comanche helicopter. grc.nasa.gov/WWW/RT/RT1995/2000/2730k.

Liu, B., Riemenschneider, S., & Xu, Y. (2006). Gearbox fault diagno-sis using empirical mode decomposition and Hilbert spectrum.Mechanical Systems and Signals Processing, 20, 718–734.

Manolakis, D., Ingle, V., & Kogon, S. (2000). Statistical and adaptivesignal processing, New York, NY: McGraw-Hill.

Wackerly, D., Mendenhall, W., & Scheaffer, R. (1996). Mathematicalstatistics with applications (p. 441). Belmont, CA: WadsworthPublishing Co.

Wemhoff, E., Chin, H., & Begin, M. (2007). Gearbox diagnos-tics development using dynamic modeling. American HelicopterSociety 63th Annual Forum.

White, G. (1982). Split torque transmission. USPTO, 4,489,625,November 23.

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