TOWARD HELICOPTER GEARBOX DIAGNOSTICS FROM A SMALL NUMBER OF EXAMPLES

Mechanical Systems and Signal Processing (2000) 00(0), 000}000doi:10.1006/mssp.2000.1297, available online at http://www.idealibrary.com on

TOWARD HELICOPTER GEARBOX DIAGNOSTICSFROM A SMALL NUMBER OF EXAMPLES

M. ZACKSENHOUSE, S. BRAUN AND M FELDMAN

Faculty of Mechanical Engineering, Technion}Israel Institute of Technology,Haifa, Israel

AND

M. SIDAHMED

University de Technologie de Compiegne, France

(Received 30 November 1999; accepted 11 February 2000)

The United States Navy funded Westland Helicopters LTD to carry out a series of testson the CH-46E helicopter transmission for the purpose of generating a database that can beused to evaluate diagnostic tools in general, and neural networks in particular. The databaseis extensive in the number of accelerometers, load conditions and diversity of faults.However, only 1}4 instances (levels) of each fault were included. Thus, the number ofexamples that can be extracted to train any classi"cation scheme is limited, and well belowthe minimum required. We argue that instead the database is best used to develop speci"csignal generation models for the CH-46E helicopter gearbox. Such models can then be usedto suggest meaningful features and derive appropriate signal-processing tools. Speci"cally,we demonstrate that the meshing vibrations induced by the large collector gear located onthe quill shaft are signi"cant and may interact with the vibrations induced by other elementsattached to the same shaft. An appropriate model is developed and the e!ect, termedcross-gear-pair interaction, is studied using di!erent signal-processing tools.

( 2000 Academic Press

1. INTRODUCTION

Helicopter gearboxes are critical systems, where in-#ight malfunctions can lead to cata-strophic results. Many investigations, theoretical and experimental have been conducted inthis area for decades. Recently, the main power transmission of a US Navy CH-46Ehelicopter was tested by Westland Helicopters LTD on their test rig under tightly control-led conditions. Some details of the test rig, controlled test parameters, instrumentation andvibration data were made public domain [1], enabling interested researchers to test varioussignal-processing techniques. A special workshop was even dedicated to some initial resultsobtained by independent research groups [2].

The available database includes simultaneously recorded vibration signals from eightaccelerometers, and a tachometer signal. A multitude of test conditions was conductedunder di!erent loads with arti"cially induced (typical) defects. Some tests involved twodi!erent instances of the same type of fault at di!erent levels of severity. Other tests wereconducted during the initiation and development of the fault. Seven type of faults wereincluded, one at a time, along with a base test with a fault-free transmission. It is indeed rarethat such a huge database is made available to the community engaged in vibration signalprocessing.

MSSP=20001297 Susan KGU BG

0888}3270/00/050789#21 $35.00/0 ( 2000 Academic Press

The Westland test was conducted to evaluate di!erent classi"cation schemes and espe-cially the power of neural network for monitoring the health of the helicopter transmission.However, while the test is indeed extensive in the number of accelerometers and thediversity of faults, it does not provide a su$cient number of examples for training a neuralnetwork. This issue is addressed in Section 2. In Section 2.1 we review relevant bounds onthe number of examples required to train a classi"er and conclude that at least 40 examplesare needed and that this number increases with the number of features available to theclassi"er. In Section 2.2, we enumerate the number of training examples that can be derivedfrom the Westland database and conclude that at most 20 examples are available for eachload level. We conclude this section with a speci"c classi"cation example demonstratingthat with a relatively simple 16-dimensional feature space the 10 examples, derived from thedata available on CD-ROM, are easily separated, (Section 2.3). We reason that theWestland database is better used toward re"ning signal-generation models that can then beused to identify physically relevant features or cues for each defect.

Simple signal-generation models, reviewed in Section 3.1, are tailored for a single gearpair and assume that each vibratory source is small. In contrast, the CH-46E transmissionincludes multiple gear pairs, which may interact with each other. We consider a simpli"edversion of the transmission and extend the simple models to include possible cross-gear-pairinteraction (Section 3.2). In Section 3.3, the resulting model is applied to the CH-46Etransmission to predict the location of signi"cant spectral lines in the induced vibrations,including those associated with cross-gear-pair interaction.

The diagnostic performance of a classi"er depends strongly on the type of featuresavailable as input. Di!erent signal-processing methods have been developed to extractsigni"cant features of gearbox-induced vibrations. Spectral methods, reviewed inSection 4.1, result in multiple spectral lines, whose origin may be revealed with referenceto the relevant signal-generation models. Many faults, like those associated with toothwear, induce vibrations that are synchronised with the shaft rotation. Time-domainaveraging reviewed in Section 4.2 enhances deterministic components of such vibrationsand are therefore powerful in extracting deterministic vibration sources that are syn-chronised with the rotation of a speci"c shaft. Averaging in the auto-correlation domaininstead of the time domain, and computing the resulting spectral correlation density(Section 4.3) may reveal stochastic vibration sources that are synchronised with the rotationof the shaft.

In this paper, we focus on the analysis of the vibrations induced by Fault 4 (Spiral BevelInput Pinion Gear Tooth Spalling, level 1 or 2) as compared with the analysis of thevibrations induced by the fault-free transmission. Each of the three methods describedabove is strongly a!ected by the cross-gear-pair interaction as demonstrated in Sections 5.1,5.2 and 5.3, respectively. Finally, the e!ect of torque level on the magnitude of selectedspectral peaks is studied in Section 5.4.

2. CLASSIFICATION AND GENERALIZATION

A classi"cation scheme is based on a set of features establishing the input to the classi"erand a set of classes establishing its output. The Westland database establishes eight outputclasses, one for each of the seven faults included in the study and one for the fault-free case.A full diagnostic system would thus include eight output classes. It is also possible to reducethe number of output classes by considering a system for partial classi"cation. A binarydiagnostic system, for example, may be designed to indicate abnormal conditions or todetect a speci"c fault. We therefore restrict our discussion to binary classi"cation.

2 M. ZACKSENHOUSE ET AL.

The input to the classi"er is a vector of features extracted formed the measurements. Eachtraining example derived from a speci"c test conducted by Westland is described as a pointin the feature space and assigned the appropriate output class corresponding to the faultcondition. As the number of independent features increases it is expected that pointsassociated with di!erent faults would become better separated. In such a space, it becomeseasier to de"ne decision regions that assign each example to the proper class (see example inSection 2.3). However, given the small number of examples that can be derived from thedatabase (see Section 2.2), the generalisation power of such a classi"er is questionable, asdiscussed below (Section 2.2).

2.2. A GENERALISATION POWER

A binary classi"er c(x; w) classi"es vector x as belonging to one of two classes dependingon its vector of parameters w. The classi"er may be implemented, for example, by a neuralnetwork in which case w denotes the weights of the connections. Speci"cally, we mayconsider the family of classi"ers implemented by a Perceptron neural network [3] having 20inputs. Given the considered family of classi"ers, the corresponding parameters are adjustedto obtain a high rate of correct self-classi"cation, i.e. classi"cation of the available trainingexamples. Given m training examples, the achieved rate of correct self-classi"cation isdenoted by g

m(c) [4].

The utility of the classi"er is determined by the probability that a randomly selected newexample is correctly classi"ed. This unknown probability, denoted by g (c), de"nes thegeneralisation performance of the classi"er. The generalisation performance g (c) is upperbounded by g

m(c) since the classi"er was tuned to the particular set of training examples. In

the extreme, the classi"er may be based on a very large feature space so that the set ofavailable examples is perfectly classi"ed [g

m(c)"1] while the predictive power is very poor

so g (c) is very small [4].Given that g

m(c) is a biased over-estimator of g (c), it is critical to determine conditions for

assuring that the bias is small. Speci"cally, we are interested in assuring that with probabil-ity of at least 1!d, the bias is less than e, i.e.

Prob(maxc

Dgm(c)!g(c) D'e))d (1)

where the maximum is over all classi"ers of the considered family. The parameter e is thedesired accuracy, while the parameter d is the desired con"dence. For example, we mayrequire (as in the rest of the paper) that with 99% con"dence (d"0.01) the probability ofcorrect classi"cation is within 10% (e"0.1) of the rate of correct self-classi"cation. Theconditions for satisfying equation (1) depend on the diversity of the considered family ofclassi"ers, and on the number of training examples available. In general, as the consideredfamily of classi"ers is more diverse, more examples are needed to restrict the selection of theappropriate classi"er.

It has been shown that for "nite capacity classi"ers (having "nite number of parameters)the probability that g

m(c) overestimates g (c) by more than e can be made arbitrary small by

increasing the number of examples [3, 4]. This stems from a theorem due to Vapnik andChervonenkis [5], which states that

Prob(maxc

D gm(c)!g (c)D'e))4D(2m)exp(!e2m/8) (2)

where D (m) is the growth function and will be further discussed next. At this stage we notethat for "nite classi"ers, the growth function D (m) increases at the most as a polynomialfunction of m. Thus, by increasing the number of training examples m, the probability thatthe bias is more than e can be made smaller than any desired con"dence requirement d.

3HELICOPTER GEARBOX DIAGNOSTICS

The growth function D (m) denotes the number of distinct binary classi"cations(dichotomies), which can be generated by the considered family of classi"ers on a set of mtraining examples. Note that the number of potential di!erent classi"cations on m examplesis 2m, and distinguish between two cases:

D (m)"2m (3a)

The considered family of classi"ers can generate all of the possible classi"cations, so inthis case, the maximum bias given by the right-hand side of equation (2) cannot be madesmaller by increasing m.

(b) The considered family of classi"ers has a "nite capacity, so there is a maximumnumber of examples, denoted by the VC dimension d

VC, above which it cannot generate all

the possible classi"cations. In this case, the growth function is upper bounded by [3, 4]

D (m)(mdVC#1 (3b)

and the maximum bias given by the right-hand side of equation (2) can be made arbitrarysmall by increasing m.

Consider, for example, the family of linear classi"ers (such as those implemented by thePerceptron neural network [3]), which includes all the classi"ers whose decision regions arehalf-spaces. The VC dimension is equal to p#1 where p is the dimension of the featurespace (i.e. the number of inputs to the network) [6]. Thus, the upper limit for the growthfunction [equation (3b)] depends on the number of examples m and the number of inputs p.The right-hand side of equation (2) can be made smaller than d, by limiting the number ofinputs p to

p(e2m!8 log(4/d)

8 log(2m). (4)

This bound is plotted as a solid curve in Fig. 1(a) for e"0.1 and d"0.01. Point A, forexample, implies that according to equation (4), 105 training examples are needed to traina Perceptron neural network with 10 inputs. This would assure that with 99% con"dencethe true classi"cation rate does not di!er by more than 10% from the rate of self-classi"cation.

As the complexity of the family of classi"ers increases, the VC dimension increases and sodoes the requirement on the number of training examples. Consider, for example, the familyof non-linear classi"ers implemented by feedforward neural nets with p inputs and a singlehidden layer with k threshold neurons. For binary classi"ers, the output includes a singlethreshold neuron, so the total number of threshold neurons is M"k#1, and the numberof weights is ="k (p#1). The VC dimension has been shown [7] to be bounded by

2[k/2]p)dVC

)2= log2(eM)"2k(p#1)log

2(e(k#1)) (5)

where the square brackets indicates the largest integer smaller than the argument, ande denotes the natural log base. For a speci"c family of feedforward neural nets with k"10threshold neurons, the VC dimension d

VCdimension d

VCis bounded by

10p"2[k/2]p)dVC

)2k(p#1)log2(e (k#1))"98(p#1). (6)

Using equation (2), the number of examples needed to train such a network for goodgeneralisation can be determined. However, since the required number of examples is verylarge it is outside the range of Figure 1(b).

These bounds, derived from equation (2), are in general conservative, requiring a largenumber of training examples. For consistent classi"ers, i.e. classi"ers that correctly classify


Figure 1. Bounds on the number of examples m required for training a z z z inputs. (a) A linear classi"er de"nedby the Perceptron neural network. (b) A non-linear classi"er de"ned by a two-layer feedforward neural networkwith 10 hidden threshold units. Upper bounds derived based on the VC theorem (*) are most demanding. Upperbounds for consistent classi"ers (} } }) follow and lower bounds () ) )) are least demanding. The intermediate curvesre#ect the rule-of-thumb recommendations (} )} ) }).

the training examples, a better bound on the number of required training examples mucan

be derived [6]:

mu"max C

4

elog

2

2

d,8d

VCe

log2

13

e D . (7)

Thus, a classi"er that correctly classi"es all the m'mutraining examples, is acceptable

in the sense that with probability of at least 1!d, new examples are misclassi"ed witha probability of not more than e. The resulting bounds for a Perceptron and for a feed-forward neural network with 10 hidden units are plotted in Fig. 1(a) and 1(b), respectively(dashed line).

The above limits upper bounds on the number of required training examples in the sensethat a larger number of examples guarantees, regardless of the underling probability ofevents, that the resulting classi"er is acceptable. Lower bounds can be derived as theminimum number of examples below which there is de"nitely an underlying probability ofevents for which the classi"er is not acceptable.

For a con"dence requirement of d(0.01 and accuracy requirement of e(0.125 thelower bound m

l, derived in [6}8], is given by

ml"max C

1!ee

ln1

d,dVC

!1

32e D . (8)

Speci"cally, a consistent classi"er on less than mlexamples may, with probability of at least

1!d, misclassify over a fraction e of the cases. For small classi"ers (in terms of dVC

) the "rstterm (1!e)ln(1/d)/e is dominant, and so the bound is independent of the structure of theclassi"er. Requiring a con"dence parameter of 1% (d"0.01) and accuracy of 10% (e"0.1),the lower bound is m

l:42. Thus, if fewer than 42 examples are used, there is a 1% chance

that the resulting classi"er may perfectly classify all the training examples and yet misclas-sify over 10% of the cases, independent of the type of the classi"er and the learningalgorithm. For large classi"ers (large d

VC), the bound is determined by the second term and

depends on the structure of the classi"er, as described by the VC dimension. The resultingbounds are shown as dotted curves in Fig. 1.


In practice, a general rule of thumb can be used, whereby an intermediate number ofexamples of the order of=/e is recommended for training neural networks with= weights.The corresponding relations are shown as dashed-dotted curves in Fig. 1.

2.2. GENERALISATION POWER AND THE WESTLAND DATABASE

The Westland database [1] includes three types of tests: (1) a fault-free test (test 9), (2)static tests (tests 2}5) and (3) dynamic tests (tests 6}8). During each static test (including thefault free test) the state of the fault did not vary signi"cantly, so the test represents onlya single example for the classi"er. While each test was conducted for an extended period oftime (few hours), the relevant statistical features are not expected to change since there is nochange in the state of the system (the gearbox) nor in the defect. Thus, the static tests provideeight examples, one from the faultless test, one from test 2 and two from each of the othertwo-level tests. During each dynamic test the severity of the fault had been developing andso di!erent examples may be extracted. Test number 7, for example, involved a propagatingcrack on the Collector gear. Four defect sizes were determined by inspection and so this testmay be divided into four corresponding sections. Assuming that four examples can beextracted from each dynamic test, we get a total of 20 training examples.

Each test was conducted at di!erent load levels and so there are di!erent trainingexamples at di!erent load levels. However, the preferred strategy is to design multipleload-dependent classi"ers rather than a single load-independent classi"er. This strategy ispreferred since the load level heavily in#uences the produced vibrations as demonstrated inSection 5.4. Since this parameter can be measured during normal operation it should beused to improve classi"cation. Training each load-dependent classi"er can be based only onthe relevant training examples, which are limited to 20 at best. It is further noted that withinthe initial publication of the Westland Data Files on a CD-ROM, there are only 10examples (two from tests 3 and 4, and one from all the other tests).

The generalisation bounds reviewed above (Section 2.1, with the accuracy and con"dencerequirements stated there) indicate that the number of examples should be at least 42 andshould grow with the number of features used. Assuming, for example, that 80 features(10 from each of the eight accelerometers) are used as inputs to a two-layer feedforwardneural network of threshold units with 10 hidden units, the number of weights is="800.The recommended number of examples is of the order of =/e. Thus, for 10% accuracy(e"0.1), few thousands of examples are required.

A common approach to the problem of high dimensionality is to select a small number offeatures. While feature extraction can be accomplished automatically, by neural networksfor example, this approach requires a large number of examples. An alternative approach isto select the relevant features based on physical models of the speci"c system. This canrelate speci"c characteristics of the measured vibrations to various mechanical faults. Inprinciple, this approach would necessitate a smaller training set than the number dictatedby the general approach discussed above. (In the extreme, hypothetical case, where a one-to-one correspondence between a physically based feature and a fault exists, no trainingwould even be necessary). Conversely, discarding (or not using) such existing knowledge,would be contrary to the most basic philosophy of utilising all possible information forpowerful decision making.

We thus conclude that the Westland Database should "rst be utilised toward re"ning andvalidating gearbox vibration-generation models for the speci"c transmission of the CH-46EHelicopter. The data can be used to check the assumptions underlying such models,determine which simpli"cations are appropriate and which mechanisms are dominant.Once a signal-generation model is tailored to the CH-46E helicopter gearbox, it can be used


Figure 2. Classi"cation example for Westland test data based on the rms of the vibrations (horizontal axis ofeach panel) and the rms of the time-averaged vibrations (vertical axis of each panel) across the eight accelerometers.Letters denote the fault type.

to suggest which features are most relevant for detecting speci"c faulty conditions andwhich signal-processing methods are appropriate for extracting these feature. This is therational for presenting the material covered in the other sections of the paper.

2.2. FAULT DIAGNOSTIC EXAMPLE

This section demonstrates that the Westland database can be classi"ed using, forexample, a 16-dimensional feature space. The selected feature space is limited to (only) twofeatures from each of the eight accelerometer: (1) the root mean square (rms) of the originalsignal, and (2) the rms of the time-averaged signal (time-domain averaging is reviewed inSection 4.2). Time averaging is based on the rotation of the central quill shaft, so vibrationsinduced by faulty elements that rotate with the quill shaft (namely Faults 3, 4, 7 and 8) areenhanced. Thus, the rms of the time-averaged vibrations produced by each of these defects isexpected to be only weakly attenuated while the rms of the time-averaged vibrationsproduced by each of the other defects is expected to be heavily attenuated.

The 10 examples provided in the Westland database (the portion available on a CD-ROM) are shown in Fig. 2 across the 16-dimensional feature space de"ned above. All theexamples are taken while the torque level is at its maximum (100%) level. The 16-dimensional space is described by eight two-dimensional sub-spaces, each corresponding tothe two features extracted from one of the eight accelerometers. The indicated letterscorrespond to the fault type, i.e. b}h for Faults 2}8, and i for the fault-free case.


The straight lines overlaid on the sub-spaces are handcrafted decision boundaries thatseparate the 10 examples into the corresponding eight classes. Faults 2, 4, 5, 6, 7 and 8,corresponding to classes b, d, e, f, g and h, are easily distinguished based on measurementsfrom accelerometer 6, 4, 2, 3, 1 and 8, respectively. The remaining cases can be classi"ed aseither corresponding to a fault-free transmission or to Fault 3 by using the right decisionline on the feature plane de"ned by the measurements taken from the second accelerometer.This example demonstrates that by using a large feature space, it is indeed easy to separatethe 10 examples into the corresponding eight classes. However, as noted before thegeneralisation power of this classi"cation scheme, and hence its usefulness, are extremelydoubtful.

It is noted that most faults are linearly separated from the other cases based on only oneor two features. However, this is highly expected given that there are only 10 examples andthat 16 features have been evaluated. Indeed, ranking m training examples according to anyone feature, there is a probability of 1/m that any particular example be ranked "rst. Thus,the probability that out of p independent features, a particular example is ranked highestby at least one feature is 1!((m!1)p. For the 10 examples in the 16-dimensioned spaceconsidered here, this probability is higher than 80%. However, unless the speci"c featureused to distinguish a particular defect is physically relevant, the resulting classi"cationalong that feature is questionable.

3. SIGNAL GENERATION MODELS

3.1. BASIC MODELS

Vibration signals are heavily a!ected by development of wear, spalling, pitting,cracks and even more by tooth breakage. A catastrophic failure can follow many ofthese defects, hence the high interest in developing and improving diagnostic capabilitiesbased on the vibration signals. Helicopter gears operate at a constant speed, so it is notpossible to design diagnostic procedures that involve changing the operating speedto di!erentiate between structural transfer function and rotational e!ects. Therotational e!ects, including unbalanced shaft rotation, gear meshing and possiblebearing e!ects, are treated as signal generators providing the inputs that excite thestructural modes of the system. The Helicopter transmission is composed of multiple gearpairs and thus the resulting spectra are dominated by multiple spectral lines. Signalgeneration models of varying complexity have been developed and some classical works arefound in [9}11].

Relatively simple models can be based on elementary systems composed of two gears,such as pinion/gear, planetary, etc. Taking as an example the system depicted in Fig. 3(a),the basic model predicts a periodic vibration y (t) given by

y(t)"+k

xksin(2nk f

mt#/

k) (9)

with a fundamental meshing frequency of fm"f

r1n1"f

r2n2.

Distributed e!ects like eccentricities as well as localised defects like spalling, cracks andobviously breakage introduce modulation e!ects. These e!ects are due to the multiplicativee!ect induced by time-varying torsion loads. The simplest case would be the e!ect ofeccentricity, generating a once per revolution torque variation. Most of the modulatingfunctions include a strong deterministic component, but some (like wear-induced slipping)are predominantly stochastic processes.


Figure 3. Simpli"ed gear trains. (a) A pair of spur gears. (b) A gear train composed of two pairs of spur gearsconnected by a common shaft. (c) A simpli"ed version of the Westland Helicopter transmission, including the spurpinion and collector gear, the common quill shaft, and the spiral bevel pinion/gear.

The amplitude, phase or combined modulation mechanisms have been noticed, resultingin

y (t)"+ xk[1#a

k(t)] sin(2nk f

mt#/

k#b

k(t)) (10)

where a(t) and b (t) are modulating functions. For defects generating deterministic modula-tion e!ects, these are basically of a periodic character described by

ak(t)"+

p

akp

sin(2np fr1

t#akp

)#+q

akq

sin(2nq fr2

t#bkq

) (11a)

bk(t)"+

p

bkp

sin(2np fr1

t#akp

)#+q

bkq

sin(2nq fr2

t#bkq

). (11b)


For deterministic e!ects, synchronised with the rotating frequencies, the model thuspredicts frequencies of

f (k, p, q)"k fm$p f

r1$q f

r2, k"1, 2,2, p, q"0, 1, 2,2 (12)

The combined modulation terms at frequencies with both p and q positive may besigni"cant when the modulations by both shafts are strong, but are negligible otherwise.

More elaborate models have been developed and veri"ed. In this "rst preliminary workwe will base our analysis on the above model.

3.2. MULTIPLE-STAGE GEAR SYSTEMS

Many transmission systems consist of multiple stages, where many sub-elementsare driven at di!erent rotational speeds. A simple example is given in Fig. 3(b). Itis somewhat surprising to note the scarcity of models analysing such a relatively simplesystem.

One simple approach is to extend the single gear-pair model described by equation (10).Accordingly, the vibrations generated by one gear-pair induce torque variations, whichmodulates the vibrations induced by the other gear pair. Each gear pair generates vibra-tions and induces torque variations ¹, as per the basic single gear-pair model [equation(12)]:

¹1Jy

1(t) for "rst pair

¹2Jy

2(t) for second pair. (13)

The vibrations generated by each gear pair are modulated by the torque variationsinduced by the other gear pair. Assuming just amplitude modulation, the resulting vibra-tions in the "rst gear pair, for example, are

y"y1(1#Ky

2). (14)

The predicted frequencies would now be the following:

f (k1, p

1, q

1, k

2, p

2, q

2)"k

1fm1

$p1

fr11

$q1

fr12

$p2

fr2$p

2fr21

$q2

fr22

k1, k

2"1, 2,2, p

1, p

2, q

1, q

2"0, 1, 2,2 . (15)

At expected, the meshing frequencies associated with each gear pair and their respectfulside bands are predicted. However, we also note the prediction of cross-gear-pair terms atfrequencies like

f (1, 0, 0, 1, 0, 0)"fm1

$fm2

f (1, 1, 0, 1, 0, 0)"fm1

$fm2

$fr11

(16)

3.3. THE CH46 HELICOPTER TRANSMISSION (SIMPLIFIED)

The Westland CH-46E gearbox, rig test, test description and some of the acquiredvibration data are available on the Internet [1]. The gearbox is relatively complex, drivingboth the main shaft and many auxiliary devices. The simpli"ed main section of theWestland CH-46E Helicopter is shown in Fig. 3(c) including the input, quill and outputshafts, the spur pinion/collector gear pair and the spiral bevel pinion/gear pair. Fault 4,spiral bevel pinion tooth spalling, studied here, is expected to induce meshing vibrations atfm1

. Thus, we are interested in the meshing frequency fm1

of that gear, its harmonics and thee!ect of modulation by mechanisms associated with the rotation of the quill and outputshafts as well as the previous gear stage.


TABLE 1

Frequencies of interest

il Function f (i) (Hz)*

f (k1, p

1, q

1)

1 f (1, 0, 0) 1108.92 f (2, 0, 0) 2217.83 f (3, 0, 0) 3326.74; 5 f (1, 1, 0) 1066.2; 1151.66; 7 f (2, 1, 0) 2175.2; 2260.58; 9 f (3, 1, 0) 3284.0; 3369.410; 11 f (1, 0, 1) (1093); (1126.5)12; 13 f (2, 0, 1) 2200.2; 2235.414; 15 f (3, 0, 1) (3309.1); (3344.3)

f (k2, p

2, q

2)

16 f (1, 0, 0) 3155.7517; 18 f (1, 1, 0) 3029; 5; (3282.0)s19; 20 f (1, 0, 1) 3113.1; 3198.421; 22 f (1, 2, 0) 2903.3; 3408.223; 24 f (1, 0, 2) 3070.5; 3241.125; 26 f (1, 1, 1) 2986.9; (3324.6)t27; 28 (3072.2)u; (3239.3)B

f (k1, p

1, q

1, k

2, p

2, q

2)

29; (30) f (1, 0, 0, 1, 0, 0) 2046.9; (4264.7)31; 32 f (1, 1, 0, 1, 0, 0) 2004.2; 2089.5

f (1, 0, 0, 1, 0, 1)

Note: 1Number in parentheses indicates a frequency that is out of the range shown inFig. 4.

*Frequency in parentheses indicates that the corresponding spectra lines are notmarked in Fig. 4.

sIndistinguishable from f (8).tIndustinguishable from f (3).uIndistinguishable from f (23).BIndistinguishable from f (24).

The following parameters of the spiral bevel pinion/gear pair are speci"ed:

n1"26, n

2"63

fr11

"42.65, fr12

"17.60. (17)

The meshing frequency and its "rst harmonics are speci"ed in Table 1 and marked asf (i), i"1}3. The modulations of these frequencies by mechanisms associated with therotation of the quill and output shafts according to equation (15) or (12) are marked asf (i), i"i"4}9, and as f (i), i"10}15, respectively.

The parameters of the spur pinion/collector gear pair are

n3"74, n

4"25

fr21

"42.65, fr22

"126.23. (18)

The meshing frequency is speci"ed in Table 1 and marked as f (i), i"16. Its modulationsby mechanisms associated with the rotation of the input and quill shafts according toequation (15) or (12) are marked as f (i), i"17, 18, 21, 22, and as f (i), i"19, 20, 23, 24,respectively, and as f (i), i"25}28, for the combined e!ects.


The spur gear pair and the spiral bevel gear pair share the quill shaft so fr11

"fr21

. Thetwo gear pairs may a!ect each other through modulation of the angular frequency of thequill shaft or the transmitted load, as suggested by equation (14). The resulting vibrationsin the frequencies speci"ed by equation (16) re#ect the cross-gear-pair interaction and aremarked in Table 1 as f (i), i"29}32.

4. FEATURE EXTRACTION AND SIGNAL-PROCESSING METHODS

Signal-processing tools are applied to identify the signi"cant components of the signal-generation model and to investigate how they are a!ected by the defects. In particular, thesigni"cance of the cross-gear-pair interaction and it e!ect on di!erent signal-processingtools are demonstrated. We focus on extracting features relevant for detecting Fault 4,Spiral Bevel Pinion Tooth Spalling, and compare accleration measurements recorded withfaultless transmission with those recorded with two defected spiral bevel pinions (Fault 4,levels 1 and 2). Accelerometers 3, 4 and 7 (coded d04, d08 and d40) are situated close tothe spiral bevel gear and are most relevant or detecting this fault. We demonstate theanalysis on the data recorded from accelerometer 4 (coded d08).

4.1. SPECTRAL LINES

The power spectrum density (PSD) is estimated by averaging 12 periodograms of non-overlapping sections of the data using the Matlab [12] function PSD. Each periodogram iscomputed from a block of N"32 768 points using the fast Fourier transform (FFT). Theaccelerometer measurements were sampled at a sampling frequency of f

s"103 116Hz, so

the resulting frequency resolution is 3.15 Hz. A Hanning window is applied to enhance theanalysis of closely spaced spectral components.

The estimated PSD of a mixed process, which includes sinusoidal processes superim-posed on a second-order process, is composed of peaks at the frequencies of the sinusoidalcomponents superimposed on the spectrum of the second-order process. The estimatedPSD is customary used to (1) estimate the frequencies of the sinusoidal components fromthe location of the peaks of the estimated PSD, and (2) estimate the amplitude of thesinusoidal components from the magnitude of the peaks [13]. In our analysis, we either(1) check whether there are peaks at the frequencies of interest, or (2) compare themagnitude of the same peaks over di!erent conditions.

The peak amplitude of the spectrum at a prescribed frequency of interest is determined asthe maximum value of the spectrum among the three closest spectrum frequencies. Givena prescribed frequency of interest, i.e. one of the frequencies identi"ed above, the peakamplitude at that frequency is detected in the di!erent spectrums computed from thedi!erent vibration signals taken under di!erent loads. By plotting the peak amplitude vs theload level, the e!ect of the load on that spectra line is determined.

4.2. TIME-DOMAIN AVERAGING

Time-domain averaging is applied to enhance deterministic e!ects that are synchronisedwith the rotation of the relevant shaft [11]. The output from an optical tachometer, whichprovides 256 pulse per revolution, is available to determine the period of the relevant shaftgiven the known gear ratio. The relevant shaft for this study is the quill shaft, so thetachometer periods are scaled according to its rotation frequency to give one pulse per quillshaft revolution. The acceleration signals are sectioned according to the resulting pulses andaveraged. The period of the quill shaft includes, on average, 2400 samples with a standarddeviation of 2 samples. So the variability in the shaft rotation is less than 0.1%, and the


resulting e!ect on the time averaging is negligible. The envelope of the averaged signal isgenerated using the Hilbert transform (Matlab's Hilbert function), whose magnitude pro-vides the instantaneous amplitude [14].

The time-domain-averaged signal reveals the vibration induced by the meshing of eachtooth on the relevant gear. To detect abnormal meshing of an individual tooth it wassuggested [11] to further divide the averaged signal into a regular and a residual part. Theregular part is obtained by passing the averaged signal through a multi-band-pass "lter,with pass-bands centred at the meshing frequency and its harmonics. It is essentially thetime-domain average of the vibrations induced by a single tooth. The residual part is thenobtained by subtracting the regular part.

We modify this method to deal with cross-gear-pair interaction e!ects. Ideally, theregular e!ect of the spiral bevel pinion tooth meshing would be extracted by passing onlycomponents at the fundamental and harmonics of this meshing frequency. However, asnoted in Table 1, the third harmonic is very close to the meshing frequency of the collectorgear. Furthermore, the second and fourth harmonics are very close to the cross-gear-pairinterference frequencies.

To attenuate the vibrations induced by mechanisms associated with the cross-gearinterference, e.g. to attenuate f (29) while passing f (2), the width of the pass-bands is keptnarrow (200 Hz). The centre frequencies are located at the meshing frequency of the spiralbevel pinion and its second, fourth and "fth harmonics. The third harmonics is omitted as itis too close to the meshing frequency of the collector gear. The attenuation in the stop bandsis limited to one order of magnitude given the narrow width of the "lters. The multi-band-pass "lters are implemented using Matlab's Remez function.

4.3. CYCLO-STATIONARY ANALYSIS

Averaging methods are based on the assumption that the defect induces vibrations thatare synchronised with the rotation of the relevant shaft. A critical issue concerns thestochastic nature of the induced vibrations, speci"cally whether the mean-induced signal(i.e. the deterministic component) is signi"cant. A strong mean signal could be detectedusing time-domain averaging either directly or upon further "ltering into regular andresidual components. However, when the induced vibrations are due to a purely randomprocess their average would tend to zero, and the time-domain averaging would fail tocapture the e!ect.

Assuming that the induced vibrations are band-limited random processes, their auto-correlation function is periodic at the period ¹ of the relevant shaft:

R(t, q)"R(t#¹, q) (19)

where q is the time lag [15]. By averaging the auto-correlation function instead of theoriginal time series, second-order moments synchronised with the rotation of the relevantshaft are also enhanced. This method for analysing second-order cyclo-stationaryprocesses have been developed by Gardner [15, 16] and results in the spectral correlationdensity (SCD) S ( f, a). The SCD is a bivariate function of the classical spectra frequency f,and the cyclic frequency a"1/¹, and reduces to the classical power spectral density whena"0.

For "nite a, the SCD is the FFT of the averaged auto-correlation, where the average isbased on the period ¹. The SCD can be interpreted as the statistical link between thecomponents at frequency f

1"f!a/2 and f

2"f#a/2. The correlation between a speci"c

component at f1"f

cand its side band at f

2"f

c#f

ddue to modulation by a signal with


frequency fd, is determined by the magnitude of the spectral correlation density at

a"f2!f

1"f

d

f"( f1#f

2)/2"f

c#f

d/2. (20)

5. RESULTS

5.1. SPECTRAL LINES

The spectral lines predicted by the model are listed in Table 1 and marked on the PSDsshown in Fig. 4. The PSDs are computed from measurements recorded by the 4thaccelerometer (code number d08) with a faultless transmission (top panels) and witha faulty spiral bevel pinion (Fault 4, level 2) (bottom panels). Most of the predicted spectrallines are signi"cant as detailed in [17]. We note that some of the predicted spectral lines arevery close to each other and cannot be di!erentiated given the 3.15 Hz resolution of thePSDs. For example, the spectral line at f (8) is indistinguishable from the spectral line atf (18). The "rst is due to the modulation of the third harmonic of the spiral bevel gearmeshing by the rotation of the quill shaft, while the second is due to the modulation of thespur gear meshing by the rotation of the input shaft.

As another example consider the mixed side bands resulting from simultaneous modula-tion of the spur gear meshing vibrations by mechanisms associated with both the rotation ofthe input shaft and the rotation of the quill shaft. The four resulting frequencies areidenti"ed in Table 1 as f (25), f (26), f (27) and f (28). However, the last three frequencies areindistinguishable from the previously identi"ed frequencies f (3), f (23) and f (24), respectively.The peak corresponding to the "rst side-band frequency f (25) is clearly identi"ed in Fig. 4.However, given that the spur gear ratio is close to three (74/25&3), this peak may alsore#ect the modulation due to mechanisms associated with the fourth harmonic of the quillshaft.

The most dominant spectral peak in the range 0}3500 Hz is associated with the meshingfrequency f (16) of the spur gear pair, which includes the large collector gear. The meshingvibrations induced by the collector gear are transmitted through the quill shaft to the spiralbevel gear as modelled in Section 3.2, and may give rise to cross-gear-pair interaction. Theinteraction frequencies are speci"ed in Table 1 as f (29) and f (30). The "rst frequency f (29)

appears on the left in the middle panels of Fig. 4. It results from the modulation of the toothmeshing vibrations of one pair of gears by mechanisms associated with the meshingvibrations induced by the other gear pair. Either vibration source might also be modulatedby the rotation of the quill shaft connecting the two gear pairs, resulting in the pair ofside-bands at f (31), f (32), which are also apparent in the PSDs.

5.2. TIME-DOMAIN AVERAGING

The modulation of the tooth meshing vibrations induced by one gear pair by mechanismsassociated with the tooth meshing of the other gear pair can be demonstrated in the timedomain using synchronised averaging. During one revolution of the quill shaft the averagedacceleration (Fig. 5) includes exactly 74 clear cycles, corresponding to the 74 teeth on thelarge collector gear. In the faultless transmission, these strong and almost identical cyclesseem to mask any other e!ect due to the meshing of the spiral bevel pinion connected tothat same shaft (Fig. 5, top panel). However, with the faulty spiral bevel pinion, the cyclesdue to tooth meshing of the collector gear are amplitude modulated with 26 modulationcycles per one revolution of the quill shaft, corresponding to the 26 teeth on the spiral bevelpinion. The modulation is best seen at the amplitude envelope marked in bold.


Figure 4. Frequencies of interest. Spectrums of vibration signals picked by accelerometer d08 with fault-freecomponents (top row) and fault spiral bevel pinion (bottom row; Fault 4, level 2, spiral bevel tooth spalling). Thespectrums are sectioned to focus on frequencies of interest, which are marked and numbered. The actualfrequencies are speci"ed in Table 1.

To extract the e!ect of the spiral bevel gear tooth meshing, we band-pass "lter theaveraged signal, passing only the frequencies associated with that meshing frequency and itsharmonics. As mentioned above, the third harmonic is excluded as it is too close to thestrong peak at the meshing frequency of the collector gear. The results are shown inFig. 6 where the e!ect of the collector tooth meshing is suppressed. The e!ect of the spiralbevel pinion tooth meshing is indeed more pronounced; however the e!ect of the collectorgear tooth meshing is still apparent.

5.3. CYCLO-STATIONARY CORRELATION

The cross-gear-pair interaction is revealed using cyclo-stationary analysis by lookingfor the correlation between the meshing vibrations induced by the collector gear atfm2

"3155.75 Hz, and its modulation by the meshing vibrations induced by the spiral bevelgear. Thus, with f

c"f

m2"3155.75 Hz, and f

d"f

m1"1108.9, equation (20) indicates

that this correlation is measured by the SCD at a"fd"f

m1"1108.9 Hz and

f"fc#f

d/2"f

m2#f

m1/2"3710.2 Hz. The SCD of the vibrations recorded by acceler-

ometer d08 are shown in Fig. 7 for a"1108.9 Hz as a function of f. The strong peak atf"3710 indicates that the correlation between the component of the vibrations at thecollector gear meshing frequency and its side band due to the modulation by the meshing


Figure 5. Cross-gear pair modulation. Averaged time signals derived from vibration signals picked by acceler-ometer d08 with fault-free components (top panel) and faulty spiral bevel pinion (bottom panel). The envelopesignals are the instantaneous amplitude signals computed by the Hilbert transform, revealing cross-gear-pairmodulation in the vibrations evoked with the faulty pinion in place.

vibrations induced by the spiral bevel gear, is signi"cant. This correlation is a directindication of the cross-gear-pair interaction.

The SCD's peak is especially strong in the vibrations induced by the severely damagedspiral bevel pinion (Fault level 2) but is also apparent in the vibrations induced by theless-damaged spiral bevel pinion (Fault level 1). In contrast, the peak is very small in thevibrations induced by a fault-free pinion. These observations are independent of the torquelevel, and are apparent in the vibrations induced at the lowest torque level [27%, Fig. 7(a)],and at the highest torque level [100%, Fig. 7(b)].

5.4. SPIRAL BEVEL TOOTH SPALLING EFFECTS

The e!ect of spiral bevel pinion tooth spalling, both levels 1 and 2, on the magnitude ofthe peaks in the power spectrum is summarised in Fig. 8. Each panel describes the e!ect ofspiral bevel tooth spalling on the magnitude of one spectra peak as a function of torquelevel. The three graphs in each panel were derived from measurements recorded with nofault (solid), Fault 4, level 1 (dashed) and Fault 4, level 2 (dash-dotted).

The e!ect of the fault on the spiral bevel gear tooth meshing frequency f (1) and its sidebands f (4) and f (5) is depicted in the upper row. The fault does not a!ect any of these spectralines in a consistent way. Similarly, the e!ect on the harmonics of the meshing frequencyand their side-bands f (1), i"2, 3, 6}9, is inconsistent as shown in [17].


Figure 6. Demodulating the cross-gear-pair interaction. The time-averaged signals of Fig. 5 are "ltered withpass-bands at the fundamental, second, fourth and "fth harmonics of the bevel pinion meshing frequency. Themeshing vibrations induced by the collector gear are partially attenuated so that the e!ect of the meshingvibrations induced by the bevel gear is more signi"cant.

The e!ect of the fault on the spectra peaks associated with the meshing frequency of thecollector gear f (16) and its side bands at f (19), f (20) are shown in the second row. The peak atthe meshing frequency of the collector gear is consistently higher when a faulty spiral bevelpinion is installed. However, in general such an e!ect may result from other sources, such asincreased wear on that gear.

The last row depicts the e!ect of the fault on the cross-gear-pair interaction and itsmodulation due to the rotation of the quill shaft. The spectra lines associated with the


Figure 7. Cyclo-stationary analysis. The spectral correlation density (SCD) of the vibrations recorded byaccelerometer d08 at the lowest torque level (27%, a) and at the highest torque level (100%, b) with fault-freecomponents (bottom panels), and damaged spiral bevel pinion (Fault 4, levels 1 and 2, middle and top panels,respectively). The SCD is shown for a"f

ml"1108.9 Hz, the tooth meshing frequency of the spiral bevel gear, as

a function of f. The SCD at (a, f ) indicates the correlation between components at fl"f!a/2 and f

2"f#a or

between a component at fc"f!a/2 and its side band at f

c#f

d"f#a/2 due to amplitude modulation by a signal

at frequency fd. The indicated peak is at f"f

m2#f

m1/2 where f

m2"3155.75 Hz is the tooth meshing frequency of

the collector gear.

modulation of the cross-gear-pair interference, i.e. on the peaks at f (31) and f (32), areconsistently higher when the fault is present. These vibrations are evoked by the combinedmodulation of the collector tooth meshing vibrations by the tooth meshing of the spiralbevel pinion and by the quill shaft rotation.


Figure 8. E!ects of spiral bevel pinion tooth spalling. The magnitude of the spectrums at selected frequencies ofinterest as a function of torque level. The analysed vibrations z z z with fault-free components (*) and two di!erentfaulty pinions: Fault 4, level 1 (} } }) and Fault 4, level 2 (} ) } )}). Frequencies of interest associated with the toothmeshing frequency of the spiral bevel gear (top centre), tooth meshing frequency of the collector gear (middlecentre) and the cross-gear-pair interaction (bottom-centre), and their modulation due to the rotation of the quillshaft (left and right columns, respectively).

6. DISCUSSION

The Westland tests are limited in the number of examples included from each fault whileextensive in the number of accelerometers recorded and the diversity of faults. Thevibrations recorded by each accelerometer are characterised in the frequency domain bymultitude of spectral lines. It is thus possible to extract a large number of features to


characterise the vibrations induced by the Helicopter transmission during normal andabnormal operations. However, the limited number of examples prohibits any meaningfullearning of the underlying classes of vibration signatures. We support this claim withgeneral bounds derived in the literature on the number of examples needed for learningprobability-free classi"cation. We also note that apart from the examples needed to trainthe classi"er, additional examples are necessary for testing and cross-validating the result-ing classi"er.

We conclude that the Westland data base can be best used toward re"ning and tuningsignal-generation models for the speci"c case of the CH-46E Helicopter transmission. Suchmodels can then be used to suggest signi"cant features and appropriate signal-processingalgorithms for diagnostic purposes. We demonstrate this approach by noting the signi"cantin#uence of the meshing vibrations induced by the large collector gear. Spiral bevel toothspalling results in modulating these strong vibrations, a phenomenon we term cross-gear-pair interaction. This e!ect, which is usually ignored based on the assumption that eachvibratory source is small, is thus identi"ed as very signi"cant for the CH46 transmission andmotivates the development of an appropriate signal-generation model.

The e!ect of the cross-gear-pair interaction is demonstrated using di!erentsignal-processing tools. It is argued that feature extraction methods targeted atdetecting faults on the quill shaft and attached gears should be modi"ed to account forthis large e!ect. One possibility is to "lter this e!ect; however it may not be possiblewithout attenuating other potentially informative components. Another possibility isto identify how the investigate fault modulates the meshing vibrations induced bythe collector gear, as shown in Fig. 5. In particular, we note that the e!ect of Fault 4 onthe modulation of the cross-gear-pair vibrations by the quill shaft rotation increaseswith the fault level, at all torque levels. Furthermore, the correlation of thecross-gear-pair interaction with the collector gear meshing vibrations also increases withthe fault level.

Some of the faults, like the spiral bevel tooth spalling fault studied here, were tested withtwo di!erent faulty parts in place. The e!ect of these two instances of the same type of faultis inconsistent on many of the spectra lines. This important observation has been de-monstrated for some of the spectra lines in the PSDs of the vibrations recorded by the 4thaccelerometer (code number d08), but similar phenomena are also evident in the vibrationsrecorded by other accelerometers. The inconsistent e!ect of the fault level is also apparent inFig. 2 with respect to the two instances of Fault 3 (class c in Fig. 2). These may raiseadditional doubts regarding the utility of the experiments for deriving powerful discrimi-nates. The inconsistent e!ect may be attributed to disassembly and re-assembly of thesystem prior to each test. Accounting for the potentially signi"cant role of this procedure, itmay be better to induce the faults of interest on the installed parts instead of replacing thefaultless parts with faulty ones. This strategy would facilitate direct comparison of thevibratory signature and assure that the fault-induced e!ects are not masked by assembly-induced modi"cations in the system.

The e!ect of the load level is signi"cant. In general, it seems that the vibration leveldecreases with the load level, but some spectra lines demonstrate the opposite e!ect. Sincethe load level may be measured independently it is advised to develop multiple load-dependent classi"ers instead of a single load-dependent classi"er.

ACKNOWLEDGEMENT

This research was supported by the fund for the promotion of research at the Technion.


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AUTHOR QUERY FORM

JOURNAL TITLE:MSSP DATE: 25/02/2000ARTICLE NO. : 20001297

Queries and/or remarks

Manuscript Details required Author’s responsePage/line

p. 22 provide authors name in Ref. [19]p. 24 Figure caption 1 is not clearp. 31 Figure caption 8 is not clear


TOWARD HELICOPTER GEARBOX DIAGNOSTICS FROM A SMALL NUMBER OF EXAMPLES

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