Vibration-based fault detection of meshing shafts

9th 9th IFAC IFAC Symposium Symposium on on Fault Fault Detection, Detection, Supervision Supervision and and 9th IFAC on Fault Detection, Supervision and Safety of Symposium Technical Processes Processes 9th IFAC Symposium on Fault Detection, Supervision Safety of Technical 9th IFAC Symposium on Fault Detection, Supervision and and Safety of Technical Processes Available online at www.sciencedirect.com September 2-4, 2015. Arts et Métiers ParisTech, Paris, Safety of Technical Processes September 2-4, 2015. Arts et Métiers ParisTech, Paris, France France Safety of Technical Processes September 2-4, 2-4, 2015. Arts Arts et Métiers Métiers ParisTech, Paris, Paris, France September September 2-4, 2015. 2015. Arts et et Métiers ParisTech, ParisTech, Paris, France France

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Vibration-based Vibration-based Vibration-based Vibration-based

fault detection fault detection fault detection fault detection shafts shafts shafts shafts

of of of of

meshing meshing meshing meshing

∗,∗∗ ∗ ∗∗ ∗,∗∗ Herve Morel ∗ Jean-Philippe Cassar ∗∗ Girondin Girondin Morel Cassar ∗ ∗∗ ∗,∗∗ Herve ∗ Jean-Philippe ∗∗ ∗∗ Girondin ∗,∗∗ Herve Morel Jean-Philippe Cassar ∗,∗∗ ∗ Girondin Herve Morel Jean-Philippe Cassar ∗∗ Midzodzi Girondin Komi Herve Morel Pekpe Jean-Philippe Cassar ∗∗ Komi Midzodzi Pekpe ∗∗ ∗∗ Komi Midzodzi Midzodzi Pekpe Pekpe ∗∗ Komi Komi Midzodzi Pekpe ∗ ∗ AIRBUS HELICOPTERS F-13725 Marignane Cedex, France AIRBUS HELICOPTERS F-13725 Marignane Cedex, France ∗ ∗∗ ∗ HELICOPTERS F-13725 Cedex, France ∗ AIRBUS AIRBUS HELICOPTERS F-13725 Marignane Cedex, France ∗∗ LAGIS -- UMR CNRS 8219 Universit´ eeMarignane Lille 1 Boulevard Langevin AIRBUS HELICOPTERS F-13725 Marignane Cedex, France LAGIS UMR CNRS 8219 Universit´ Lille 1 Boulevard Langevin ∗∗ ∗∗ LAGIS - UMR CNRS 8219 Universit´ e Lille 1 Boulevard Langevin ∗∗ LAGIS - UMR CNRS 8219 Universit´ e Lille 1 Boulevard 59655 Villeneuve d’Ascq LAGIS - UMR CNRS 8219 Universit´ e Lille 1 Boulevard Langevin Langevin 59655 Villeneuve d’Ascq 59655 Villeneuve Villeneuve d’Ascq d’Ascq 59655 59655 Villeneuve d’Ascq Abstract: In this article, vibration-based monitoring of cracked shafts is addressed in the Abstract: In In this this article, article, vibration-based vibration-based monitoring monitoring of of cracked cracked shafts shafts is is addressed addressed in in the the Abstract: Abstract: In vibration-based monitoring of cracked shafts in context of mechanical transmission of helicopters. The is based on Abstract: In this this article, article, vibration-based monitoring ofapproach cracked developed shafts is is addressed addressed in the context of mechanical transmission of helicopters. The approach developed is based on the context of mechanical transmission of helicopters. The approach developed is based on the context transmission of The is on detection cyclostationary components produced by crack. The cyclostationary theory is context ofof mechanical transmission of helicopters. helicopters. Theaa approach approach developed is based based on the the detectionof ofmechanical cyclostationary components produced by by crack. The Thedeveloped cyclostationary theory is detection of cyclostationary components produced aa crack. cyclostationary theory is detection of cyclostationary components produced by crack. The cyclostationary theory is first presented on a mechanical point of view. Then a signal processing indicator, the spectral detection of cyclostationary components produced by a crack. The cyclostationary theory is first presented on a mechanical point of view. Then a signal processing indicator, the spectral first presented on a a mechanical point of view. view. Then signalcyclostationary processing indicator, indicator, the spectral first presented on of aaa order signal processing coherence, is introduced in order to quantify the second content of signal. first presented on a mechanical mechanical point of view. Then signalcyclostationary processing indicator, the spectral coherence, is introduced introduced in order order point to quantify quantify theThen second order contentthe of aaaspectral signal. coherence, is in to the second order cyclostationary content of signal. coherence, is introduced in order to quantify the second order cyclostationary content of a The characteristics of the vibration produced in the faulty case are explained from a mechanical coherence, is introduced invibration order to quantify the second order cyclostationary content of a signal. signal. The characteristics of the produced in the faulty case are explained from a mechanical The characteristics of the vibration produced in the faulty case are explained from a mechanical The characteristics the in faulty case explained from mechanical model of cracked shaft. It is then shown that small variations the structural The characteristics of the vibration vibration produced in the the faulty case are are in explained from aa parameters mechanical model of aa a cracked cracked of shaft. It is is then thenproduced shown that that small variations in the structural structural parameters model of shaft. It shown small variations in the parameters model of a cracked shaft. It is then shown that small variations in the structural parameters of the crack creates cyclostationary modulations around the meshing frequencies. This justifies model of a cracked shaft. It is then shown that small variations in the structural parameters of the crack creates cyclostationary modulations around the meshing frequencies. This justifies of the crack creates creates cyclostationary modulations around thethe meshing frequencies. This justifies of crack cyclostationary the meshing justifies thethe spectral coherence as a a relevant relevantmodulations indicator to toaround monitor crack. frequencies. The spectal spectal This coherence is of the crack creates cyclostationary modulations around thethe meshing frequencies. This justifies the spectral coherence as indicator monitor crack. The coherence is the spectral coherence as a a relevant relevant indicatora to to monitor the crack. crack. The spectal coherence is the spectral coherence as indicator monitor the The spectal coherence is successful applied on bench reproducing crack propagation on an actual helicopter shaft. the spectral coherence as a data relevant indicatora to monitor the crack. The spectal coherence is successful applied on bench data reproducing crack propagation on an actual helicopter shaft. successful applied on bench data reproducing aa crack propagation on actual helicopter shaft. successful applied on data propagation on an an helicopter shaft. successful applied on bench bench data reproducing reproducing a crack crack propagation an actual actual helicopter shaft. © 2015, IFAC (International Federation of Automatic Control) Hosting by on Elsevier Ltd. All rights reserved. Keywords: vibration, vibration, helicopter, helicopter, health health monitoring, monitoring, shaft, shaft, HUMS HUMS Keywords: Keywords: vibration, helicopter, health monitoring, shaft, HUMS Keywords: Keywords: vibration, vibration, helicopter, helicopter, health health monitoring, monitoring, shaft, shaft, HUMS HUMS 1. INTRODUCTION 1. INTRODUCTION INTRODUCTION Influence 1. Influence of of the the microstructure microstructure and and load load factor factor 1. Influence of the microstructure and load factor 1. INTRODUCTION INTRODUCTION Influence of the microstructure and load Influence of the microstructure and load factor factor da Early fault detection is a crucial problem in helicopter log da (meter by cycle) dN Early fault detection is a crucial problem in helicopter log Propagation rate da (meter by cycle) da dN Early fault detection detection isActually, aa crucial crucial problem problem in helicopter helicopter Propagation rate at at 50Hz 50Hz (meter by by cycle) log dN Early fault in log da (meter maintenance strategy. miss-detection of Propagation rate rate at at 50Hz 50Hz Early fault detection isActually, a crucialone problem in helicopter (meter by cycle) cycle) log dN Propagation maintenance strategy. is one miss-detection of aaa dN Propagation rate at 50Hz maintenance strategy. Actually, one miss-detection of maintenance strategy. Actually, one miss-detection of a small fault could lead to its propagation and to system maintenance strategy. Actually, one miss-detection of a small fault could lead to its propagation and to system small fault and couldeventually lead to to its propagation and to system system small fault could lead to breakdown to an accident. Health moni10−5 small fault and couldeventually lead to its its propagation and to system breakdown to propagation an accident. accident.and Health moniStrong 10−5 1 mm/minute Low −5 breakdown and eventually to an Health moniStrong 10−5 breakdown and eventually to an accident. Health moni1 mm/minute Low toring methods are effective ways of performing Condition Strong 10 Strong breakdown and eventually to an accident. Health monitoring methods are effective ways of performing Condition 1 Strong Strong 10−5 −7 1 mm/minute mm/minute Low 10 Strong Low toring methods are helicopter effective ways ways of the performing Condition Strong 1 mm/minute Low toring methods are effective of performing Condition 10−7 Strong Monitoring. In the field, Health function is −7 toring methods are helicopter effective ways of the performing Condition 1 mm/hour Strong Monitoring. In the field, Health function is 10 −7 10−7 1 mm/hour Monitoring. In the helicopter field, the Health function is I 10−9 Monitoring. In helicopter Health is 1 part of the HUMS and Usage Monitoring Systems) II II 10−9 III 1 mm/hour mm/hour Monitoring. In the the(Health helicopter field, the Health function function is part of of the the HUMS HUMS (Health andfield, Usagethe Monitoring Systems) 1 mm/day II 10−9 III 1 1 mm/hour mm/day part (Health and Usage Monitoring Systems) II 10−9 III part of the HUMS (Health and Usage Monitoring Systems) I 10 and is specified by the ‘CAP753” Group [2012] and the 1 −9 part of the HUMSby(Health and UsageGroup Monitoring Systems) mm/day 1 mm/day mm/week −11 II 10 III and is specified the ‘CAP753” [2012] and the 10 1 mm/day mm/week and is [2014]. specified by the the ‘CAP753” ‘CAP753” Group [2012] and and the 10−11 and is specified by Group [2012] the 1 −11 EASA Among Methods, vibration analysis 10 1 mm/week mm/week and is [2014]. specified by theHealth ‘CAP753” Group [2012] and the EASA Among Health Methods, vibration analysis 10−11 1 mm/week −11 10 EASA [2014]. Among Among Healthmatches Methods, vibration analysis log∆K EASA [2014]. Health Methods, vibration analysis using non-intrusive sensors well with helicopter log∆K ∆K EASA [2014]. Among Healthmatches Methods, vibration analysis threshold using non-intrusive sensors well with helicopter ∆Kthreshold log∆K using non-intrusive sensors matches matches well suitable with helicopter helicopter log∆K using non-intrusive sensors well with ∆Kthreshold requirements. Accelerometers are the most sensors log∆K ∆K using non-intrusive sensors matches well suitable with helicopter requirements. Accelerometers are the the most most sensors ∆Kthreshold threshold requirements. Accelerometers are suitable sensors requirements. Accelerometers are the most suitable sensors for non-intrusive health monitoring of helicopters in terms requirements. Accelerometers are the most suitable sensors for non-intrusive non-intrusive health health monitoring monitoring of of helicopters helicopters in in terms terms Fig. 1. Log/log diagram of a crack’s length as a function of for for non-intrusive monitoring of 1. Log/log diagram of aa crack’s length as aa function of of space, space, cost and andhealth qualification Randall [2011]. in for non-intrusive health monitoring of helicopters helicopters in terms terms Fig. Fig. 1. Log/log diagram of length as of of cost qualification Randall [2011]. Fig. 1. Log/log diagram of aa crack’s crack’s length as aa function function of the applied stress intensity factor. The first part of the of space,helicopter cost and and qualification qualification Randall [2011]. Fig. 1. Log/log diagram of crack’s length as function of of space, cost Randall [2011]. the applied stress intensity factor. The first part of the Among elements, the health of shafts bearings of space,helicopter cost and qualification Randall [2011]. the applied stress intensity factor. The first part of the Among elements, the health of shafts bearings the applied stress intensity factor. The first part of the curve corresponds to the initiation and the last part of Among helicopter elements, the health of shafts bearings the applied stress intensity factor. The first part of the Among helicopter elements, the health of shafts bearings curve corresponds to the initiation and the last part of plays a key role for the power transmission chain integrity. Among helicopter health of shafts bearings curve corresponds to the the initiation and the the last last part of plays aa key key role for for elements, the power powerthe transmission chain integrity. integrity. curve corresponds to initiation and part of the curve corresponds to to the destruction the shaft plays role the transmission chain curve corresponds to the initiation and the last part of plays a key role for the power transmission chain integrity. the curve corresponds to to the destruction the shaft Its monitoring is therefore worth to be carried out with plays a key role for the power transmission chain integrity. the curve corresponds to to the destruction the shaft Its monitoring is therefore worth to be carried out with the curve corresponds to to the destruction the shaft Clavel and Bombard [2009]. ∆K is the stress intensity Its monitoring is therefore worth to be carried out with the curve corresponds to to the destruction the shaft Its monitoring is therefore worth to be carried out with Clavel and Bombard [2009]. ∆K is the stress intensity the monitoring most effective effective methods. worth One of oftothe the most catastrophic Its is therefore bemost carried out with Clavel and Bombard [2009]. ∆K is the stress intensity the most methods. One catastrophic Clavel and factor. the most effective methods. One of the the most catastrophic Clavel and Bombard Bombard [2009]. [2009]. ∆K ∆K is is the the stress stress intensity intensity the most effective methods. One of catastrophic factor. failure modes of shafts is the transverse crack leading to the most effective methods. One of the most most catastrophic factor. failure modes of shafts is the transverse crack leading to factor. failure modesfailure of shafts shafts is the the transverse crack leading leading to factor. failure modes of is transverse crack to the complete of the power transmission. A crack is failure modesfailure of shafts is the transverse crack leading the complete complete of the the power transmission. A crack crack to is the failure of power transmission. A is the complete failure of the transmission. A crack is in general caused by in heavily loaded parts and the complete failure offatigue the power power transmission. A crack is in general caused by fatigue in heavily loaded parts and in general caused by fatigue fatigue in heavily heavily loaded parts and in general by in and where local notch may occur (corrosion, fretting, matting in general caused by fatigue in heavily loaded loaded parts and where localcaused notch may may occur (corrosion, (corrosion, fretting,parts matting where local notch occur fretting, matting this diagram and is in general hard to guess since it where local notch may occur (corrosion, fretting, matting coups. . . ). Crack propagation can be decomposed into this diagram and in hard guess since where local notch may occur (corrosion, fretting, matting coups. . . ). Crack propagation can be decomposed into this diagram andonis isintrinsic in general general hard to toyoung guessmodulus) since it it coups. . .. ). ). Crack Crack propagation can1 (Clavel be decomposed decomposed into this diagram and is in general hard to guess since it depends strongly (geometry, coups. . propagation can be into three stages presented in diagram and Bombard this diagram and is in general hard to guess since it depends strongly on intrinsic (geometry, young modulus) coups. . . ). Crack propagation can1 (Clavel be decomposed into depends strongly on intrinsic (geometry, young modulus) three stages stages presented in diagram diagram and Bombard Bombard three presented in 1 (Clavel and depends strongly on intrinsic (geometry, young modulus) and extrinsic factors (cycles, temperatures, load). The three stages presented in diagram 1 (Clavel and Bombard [2009]). depends strongly on intrinsic (geometry, young modulus) and extrinsic factors (cycles, temperatures, load). The three stages presented in diagram 1 (Clavel and Bombard [2009]). and extrinsic factors (cycles, temperatures, load). The [2009]). and extrinsic factors (cycles, temperatures, load). The crack can also open and close dynamically due to the [2009]). and extrinsic factors (cycles, temperatures, load). The crack can also open and close dynamically due to the [2009]). crack can also open and close dynamically due to the This diagram represents the crack growth rate against the crack can also open and close dynamically due to the deflation of the shaft itself. This phenomenon is called This diagram represents the crack growth rate against the crack can also open and close dynamically due to the deflation of the shaft itself. This phenomenon is called 1 This diagram represents the crack growth rate against the This diagram represents the crack growth rate against the deflation of the shaft itself. This phenomenon is called 1 (Bachschmid and variation of the stress intensity factor deflation of the shaft itself. This phenomenon is called This diagram represents the crack growth rate against the ”‘breathing”’. variation of the stress intensity factor 1 (Bachschmid and deflation of the shaft itself. This phenomenon is called 1 ”‘breathing”’. (Bachschmid and variation of of the the stress stress intensity5.2.3.1). factor 1 The variation intensity factor (Bachschmid and ”‘breathing”’. Pennacchi paragraph first stage is ”‘breathing”’. (Bachschmid and variation of[2008], the stress intensity5.2.3.1). factor The Pennacchi [2008], paragraph first stage is ”‘breathing”’. presents in Papadopoulos [2008] an overview Pennacchi [2008], paragraph 5.2.3.1). The first stage is Pennacchi [2008], paragraph 5.2.3.1). The first stage is the initiation and is very fast. The second stage is the Papadopoulos presents Papadopoulos [2008] overview Pennacchi [2008], paragraph 5.2.3.1). The first stage is Papadopoulos the initiation and is very fast. The second stage is the Papadopoulos presents in inmethods Papadopoulos [2008] an an monitoroverview the initiation and is very fast. The second stage is the Papadopoulos presents Papadopoulos [2008] overview the crack modeling for vibration the initiation and is very fast. The second is the propagation the propagate steadily Papadopoulos presents in inmethods Papadopoulos [2008] an an monitoroverview of the crack modeling for vibration the initiationand and iscrack very tends fast. to The second stage stage is and the of propagation and the crack tends to propagate steadily and of the crack modeling methods for vibration monitorpropagation and the crack tends to propagate steadily and of the crack modeling methods for vibration monitoring. Cracks have many consequences on the dynamics of propagation and the crack tends to propagate steadily and linearly. The last stage is the failure and is, as initiation, of the crack modeling methods for vibration monitoring. Cracks have many consequences on the dynamics of propagation and the crack tends to propagate steadily and linearly. The last stage is and as initiation, ing. Cracks have many consequences on the dynamics of linearly. The lastcrack stage trajectory is the the failure failure and is, is, asfound initiation, ing. Cracks have many consequences on the dynamics of shafts: change or appearance of coupling between modes, linearly. The last stage is the failure and is, as initiation, very fast. The cannot be with ing. Cracks have many consequences on the dynamics of shafts: change or appearance of coupling between modes, linearly. The lastcrack stage trajectory is the failure and is, asfound initiation, very fast. The cannot be with shafts: change or appearance of coupling between modes, very fast. The crack trajectory cannot be found with shafts: change or appearance of coupling between modes, new vibration components, new resonances during runvery fast. The crack trajectory cannot be found with shafts: change or appearance of coupling between modes, new vibration components, new resonances during runvery fast. The crack trajectory cannot be found with new .vibration vibration components, new resonances during run1 This parameter represents the strain field around the crack. new components, new resonances during runups. .. The authors classify the methods into two groups: 1 This parameter represents the strain field around the crack. new components, new resonances during runups. ..vibration authors classify the methods into two groups: 1 This parameter represents the strain field around the crack. 1 ups. .. The The authors classify the methods into two groups: ups. . The authors classify the methods into two groups: parameter represents the strain field around the crack. 1 This ups. . . The authors classify the methods into two groups: This parameter represents the strain field around the crack.

Victor Victor Victor Victor Victor

Copyright 2015 IFAC IFAC 560 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015, 2015 560 Copyright ©under 2015 responsibility IFAC 560Control. Copyright 560 Peer review© of International Federation of Automatic Copyright © 2015 2015 IFAC IFAC 560 10.1016/j.ifacol.2015.09.585

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→ signal processing: detection of new resonance, modes identification using the free response function, active controlling. . . → model-based: Kalman filter, adaptive filters. . .

In Carneiro [2000], the author presents a model of cracked rotor (breathing or non-breathing) and deduce the vibration signatures for known quasi-periodic excitations. Then he proposed to detect a crack using a temporal distance between the recorded data and reference signals. The author argues that frequential monitoring is not feasible with this approach because of the non-linear effects. However no real theoretical or practical clues are presented. In general the temporal methods require a high signal-to-noise ratio. In the helicopter field the signal-to-noise ratio is very low and potentially variable, moreover parasitical components are numerous. The performances of temporal methods in this context are very poor since they require a controlled excitation and a high signal-to-noise ratio. In the thesis Mani [2006], the author proposes LavalJeffcott and Euler-Bernoulli models. The crack is monitored using the response of the system subject to resonances. The author utilizes a periodic excitation to maximize the signal-to-noise ratio and concludes with wavelet analysis to monitor the pattern of the fault. These excitations may not be easily reproduced on helicopters due to the additional workload on operations. In Bigret and F´eron [1994], a Laval model allows deducing that superharmonics of shaft’ rotation speed occur when the speed is constant. Bench tests carried in Bigret and F´eron [1994] prove that the angular position of the shaft with respect to the unbalance modifies highly the fault pattern. Classic condition indicators are presented to monitor the shaft such as RMS, crest-to-crest, shaft harmonics. . . . In the same reference, the author supposes that these fault patterns are uniquely caused by a crack. However, high bending or rotor-stator contact may cause the same issue. Fault identification is then compromised. For example, even harmonics of shaft’ rotation are used as indicator of shaft misalignment Bigret and F´eron [1994] Pennacchi et al. [2012] Randall and Antoni [2011] Wiig [2006] or for spline monitoring Bechhoefer and Bernhard [2006]. The authors of Bachschmid and Pennacchi [2008] et Bigret and F´eron [1994] suggest to rely on trend monitoring to allow fault isolation without further explanations. These references rely strongly on a pre-existent historic and lack a joint vibration and signal processing approach that is of vital importance to design indicators used for monitoring. The next paragraphs will introduce cyclostationary theory and how it connects with the vibrations of cracked shafts. 2. CYCLOSTATIONARY ANALYSIS 2.1 Mechanical sources of cyclostationarity Standard models of mechanical signals recorded on rotating transmission assume that the recorded signals can be decomposed into a periodic and a random part (background noise independant from the periodic part). This model implies that the dynamic response of the system, generating the first part, is purely deterministic, or in other words that the statistical properties of the system are con561

561

stant. However, for rotating machines like helicopters, the structural parameters of the system are actually periodic. These features can be linked to some excitations of the structural parameters. The author of Antoni [2000] classify excitations into 3 groups: periodic excitations (inertia, meshing), random excitation with possible amplitude or frequency modulation and localized excitations created by repetitive impacts (shocks, explosion...). The last two classes produce periodic uncertainties in the vibrations due to the phenomena causing these signals. The transfer function between the excitations and the accelerometer may also vary periodically, causing periodic amplitude or frequency modulation. The statistical properties of the signal change periodically with the periods attached with the different sub-systems. These periodic variations reflect some mechanical uncertainties like the instantaneous load, the surfaces in contact, and small differences in the geometry. These signals are called cyclostationary. 2.2 Definitions The mathematical expectation is noted E, the Dirac distribution is δ, x is a stochastic process as defined in Gardner and Spooner [1994] and x∗ is its conjugate. Cyclostationary is now presented for the first two orders using the concept of temporal and spectral cumulants Gardner and Spooner [1994] Spooner and Gardner [1994]. It is possible to define cyclostationary for higher orders but its application is more difficult and not necessarily more fruitful according to Antoni [2000]. Yet the use of the first two orders is enough to extract the vibration patterns useful for monitoring. (1) The process {x(t)} is first order cyclostationary (CS1) with respect to the period T = 0 when: E {x(t)} is T periodic (1) The quantity E {x(t)} is called the first order temporal cumulant of x(t). (2) The process {x(t)} is second order cyclostationary (CS2) with respect to the period T = 0 when:     τ τ  E x t+ −E x t+ 2   2 τ τ  ∗ − E x∗ t − × x t− 2 2 is T periodic with respect to the variable t (2) This quantity is noted the second order temporal cumulant of x(t). The formula is quite close to the autocovariance function. The main difference is that the two signals in the expectation are centered on t for the cumulant and on t − τ2 for the autocovariance function. The variable τ has a role of temporal shift. A typical CS1 signal is a additive mixture of a periodic signal and a noise. A typical CS2 signal is a multiplicative mixture of a periodic signal and a noise. In Antoni et al. [2004], the authors propose a cyclostationary classification of gearbox vibrations: meshing and inertia vibrations are mainly CS1 and bearing vibrations are mainly CS2. 2.3 Spectral representation of cyclostationarity Fourier transform It is possible to rewrite x(t) with the first order temporal cumulant defined in paragraph 2.2:

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x(t) = E {x(t)} + (x(t) − E {x (t)}) (3) The first component in the right side of the equation is the first order temporal cumulant. The second component is the residu and contains the second order temporal cumulant. Since periodicity is a core concept of cyclostationary analysis, it is more practical to used Fourier representation:  x(t) = e2πif t dX(f ) (4)

dX(f ) is called the spectral increment of x of the Fourier decomposition. Applying this decomposition in equation 3, it is possible to find a similar relationship for dX(f ): dX(f ) = E {dX(f )} + (dX(f ) − E {dX(f )}) (5) The process E {x(t)} is then a countable sum of sinusoids definied by the frequencies (fn )n∈N . It is possible to write dX(f ) as a series:  E {dX(f )} = (6) dX(fn )δfn (f ) with fn ∈ R

|E {dY (f − α) dX ∗ (f + α)}| λY X (f ; α) =     1/2 2 2 E |dY (f − α)| E |dX (f + α)|

∈[0; 1] (11) here dY is called the spectral increment of the signal y(t) according in Cramer’s decomposition. The main advantage of such an indicator is that its values for a healthy (λ = 0) or cracked (λ = 1) shaft can be defined before carrying any specific tests. This point makes sense when it is required to make seeded fault tests on systems with complex architectures. 3. VIBRATION OF A CRACKED SHAFT 3.1 Model for a healthy shaft The system under study is a meshing shaft rotating at Ω as presented in Lalanne and Ferraris [1998]. Z

n∈N

Consequently x(t) − E {x(t)} does not contain any sinusoidal term. In the right part of equation 3, the first component E {x(t)} is “singular” and the second component x ˜(t) = x(t) − E {x(t)} is absolutely continuous.

First order According to the definition given in paragraph 2.2 and Bonnardot [2004], only the singular component is useful to define first order cyclostationarity. We can then rewritte E {dX(f )} using equation 6 supposing first order cyclostationary:  E {dX(f )} = dX(n/T )δn/T (f ) (7) n∈N

The quantity E {dX(f )} is the first order spectral cumulant and is noted C1x (f ). C1x (f ) = E {dX(f )}

(8)

Second order The definition 2 of second order of cyclostationarity can be applied in the same way to the Fourier domain and as explained in Antoni and Randall [2004], it is possible to find the spectral expression of the second order spectral cumulant :     α τ ˜ ∗ (f − α ) ˜ + α )dX (9) e−2πi[t 2 + 2 f ] E dX(f 2 2 f α

Based on the paragraph 2.2 decomposition, it appears that the second order analysis uses the autocorrelation function of the absolute continue component of dX(f ). The second order spectral cumulant is defined by:      ˜∗ f − α ˜ f + α dX E dX 2 2 Cum2x (α, f ) = (10) dαdf Where E is the expectation. x is CS2 if its second order spectral cumulant is non-zero. The CS2 information is contained in the cyclic frequencies. Spectral Coherence The second order spectral cumulant can be interpreted as a scalar product between two frequency channels. Using basic linear algebra, it is possible to define a ”normalized” indicator that measures the alignment between these two frequencies. Such an indicator is called the ”spectral coherence” and is defined by: 562

Bearing

Y

O X

Unbalance

ke (t) Gear

Fig. 2. Model of the studied shaft. The equation of motion of transverse displacements is: (12) Mx ¨ + (k + ke (t)) x = F With F = mdΩ2 cos 2πΩt the unbalance force. Where the reduced parameters are : → M : reduced mass → k: reduced stiffness (including the bearing) → ke : meshing stiffness (ZΩ periodic functions, where Z is the number of teeth of the gear) → m: mass of unbalance → d: distance to unbalance

This second order differential equation presents a resonance frequency fres :  1 k (13) fres = 2π M It is now possible to compute shaft vibrations for several configurations. In equation 12 the sinusoidal excitation is created by an unbalance. In practice, this mass cannot be totally canceled API684 [2013]: there is then always an Ω periodic vibration component. The accelerations defined by equation 12 have then a sinusoidal component at frequency Ω and the amplitude increases then Ω gets closer to fres . Other excitations can contribute to the global excitations, such as the ones caused by the main rotor aerodynamic load. 3.2 Cyclostationarity for a cracked shaft Let us consider a shaft with a transversal crack where the following hypotheses apply: (1) A crack induces a contact loss between two sections of the shaft

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(2) The surface of the loss of contact is variable for one rotation (breathing phenomenon) (3) The considered shaft contains a gear The hypothesis 1 implies that the inertia moment is modified. The hypothesis 2 expresses that the shaft stiffness is changing over one rotation. The influence of a constant force applied radially on the shaft is described for two configurations in figure 3: (1) the crack is facing the force. The opening works then in traction and tends to open. The stiffness loss is then the highest. (2) the crack is opposite to the force. The opening works then in compression and tends to close. The stiffness loss is then the lowest.

Closed

Open

x

number of teeth on the shaft. The Fourier decomposition of k − ∆kx0 (t) + ke (t) is:  k − ∆kx0 + ke (t) = Ke (ne )e2πine ZΩt (16) ne ∈Z

With Ke (ne ) the coefficients of the meshing stiffness Fourier series. Since ke (t) has harmonics multiple of ZΩ, then x needs also to have the same feature to be a solution. The product between these harmonics and kc (t) in equation 14 creates modulations at frequency Ω around these harmonics. For that reason, the solution q1 can be written as a Fourier series with this pattern:  X(pZ + n)e2πi(pZ+n)Ωt (17) x(t) = p∈N n∈Z

The natural p is indexing the harmonic of ZΩ generated by the meshing and n is indexing modulations. It is possible to substitute this decomposition in equation 14 in order to study how X (pZ + n) relates to kc . The equation of motion of 14 for x at frequency (pZ + n) Ω becomes: 2

− M (2π (pZ + n) Ω) X (pZ + n)  Ke (ne ) [X ((p + ne ) Z + n) + X ((p − ne ) Z + n)] +

z

Z

563

z

z Crack

ne ∈Z

x

+

x

X



nc ∈Z

Ωt

Fig. 3. Breathing phenomenon. Three transversal views (z, x) of the shaft are presented: the left one is the shaft in a random position and the last two ones show the crack in compression and in traction. Lateral views are shown on the top of the last two shafts (y, z). The hypothesis 3 implies that there will be a dynamical coupling between the crack and the meshing. For helicopter shafts that rotate slowly, the gyroscopic effect and the damping can be neglected. We want to monitor the influence of the crack on the meshing. Gears are high frequency phenomena in comparison with the unbalance force F that is small and low frequency. For these reason, when studying the displacements related to the meshing the force F can be neglected. The calculation of the Lagrangian leads to the new equations of motion: M x + λx˙ + (k − ∆kx0 − ∆kx (t) + ke ) x = F (14) A complete analysis of this kind of equation is presented in Richards [1983] and stability analysis is presented in Mani [2006]. The reduced parameters M , k have been described in paragraph 3. The coefficients that model the crack are: → ∆kx0 : is quantifying the permanent loss of stiffness. → ∆kx (t): is quantifying the dynamic loss of stiffness over one rotation. The latest stiffness models the breathing of the crack. For example if the crack opens and close rapidly, then ∆kx (φ) is more like a rectangle function. In the general case, it is an Ω periodic function that can be decomposed into a Fourier series:  ∆kx (t) = Kc (nc )e2πinc Ωt (15) nc ∈Z

With Kc (nc ) the coefficients of the Fourier series of kx . It is reminded that the meshing frequency ke is ZΩ, with Z the

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Kc (nc ) [X(pZ + n + nc ) + X(pZ + n − nc )] = 0

(18) It is possible to compute numerically the solutions by truncating the Fourier series of X. The Ω modulations around the meshing harmonics is a crack symptom. There is actually a direct relationship between the smoothness of the crack breathing and the amount and level of modulations. Moreover some factors may modify the dynamics of the shaft : → variations in the transmitted load due to flight conditions → geometrical errors caused by the crack

These factors contribute to the randomization of the loss of stiffness. ∆kx is then a random variable modulated by its mean value, which can be estimated from quasistatic analysis. As described in Antoni [2000] and Richards [1983], this randomization creates new second order cyclostationarity. 3.3 Spectral coherence substantiation

One typical case of random excitation with possible amplitude or frequency modulation was presented in paragraph 2. The application to our case implies that the modulation patterns around the meshing harmonics show cyclostationary content that relates to the randomized stiffness. Based on equation 18, the first sidebands around the meshing fundamental (X(Z ± 1)) are linear combination of the harmonics of Kc. since these harmonics are stochastic, then so are the sidebands. Moreover these sidebands are generated by the same stochastic phenomenon, and are then correlated. Based on that conclusion, it is proposed to use |λ(ZΩ; Ω)| to check the spectral correlation between the right 1Ω sidepeak and the left 1Ω sidepeak around the fundamental of the meshing frequency ZΩ. In our applications |λ(ZΩ; Ω)| is 0 for a perfectly safe system and is 1 for a completely failed system. Unlike the methods

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mechanical transmissions. It has been demonstrated that the cyclostationary theory and specifically the spectral coherence are suitable ways to model and detect random harmonic patterns. The mechanical model of a shaft has been introduced for the healthy and cracked case. It permitted to justify the spectral coherence as a monitoring indicator. Ultimately a representative bench test has permitted to show that this indicator can indicate crack’s propagation. REFERENCES

Fig. 4. Spectral coherence measured of the cracked shaft with propagation. The general shape of the curve is in accordance with figure 1. The horizantal axis has been rescaled. presented in the introduction, this indicator does not require historic data and is justified by a joint mechanical and signal processing approach. This is advantageous to aeronautic, where a theoretical substantiation is needed and a few fault cases are available. 4. BENCH TEST 4.1 Materials The bench test consists of a helicopter gear shaft mounted in a configuration that allows representative torque and speed. An accelerometer has been mounted on the bearing of the shaft next to the gear. This accelerometer is ICPlike and has its first resonance frequency around 50 kHz. A notch has artificially been created in the shaft to initiate propagation. The shaft is subjected to successive cycles of load until shaft rupture. A cycle consist to a short phase of over torque that is approximately 30 % higher (overtorque) than its nominal value and a longer phase of nominal torque. The crack propagated until the complete failure of the shaft. The paragraph 3.3 justifies the use of the spectral coherence λ(Z, 1) between the sidebands X(Z − 1) and X(Z + 1) to detect the crack. An increase in lambda implies that the crack is propagating. The spectral coherence is implemented using Welch method and 50 hanning windows. 4.2 Results The spectral coherence is presented in figure 4 against time. The overtorque phases are hidden since they produce abnormal values of spectral coherence that are not representative of standard helicopter flights. The constant increase of this indicator indicates the crack propagation over time. Moreover the scattering of the spectral coherence remains lower than 0.03 and decreases as the cracks grows. It means that the risk of non detection when a crack is present is reduced. 5. CONCLUSION In this article, vibration-based monitoring of cracked shafts has been presented in the context of helicopter 564

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