Applicaion of Speed Transform o he diagnosis of a roller bearing in variable speed Julien Roussel 1, Michel Hariopoulos 1, Edgard Sekko 1, Cécile Capdessus 1 and Jérôme Anoni 1 PRISME laboraory 1 rue de Loigny la Baaille, 8 Charres, France {Michel.Hariopoulos, Julien.Roussel, Cecile.Capdessus, Edgard.Sekko}@univ-orleans.fr Laboraory of Vibraion and Acousics Universiy of Lyon (INSA), F-6961 Villeurbanne CEDEX, France {Jerome.Anoni}@insa-lyon.fr Absrac In consan speed operaion, vibraion signals of roaing machinery and heir saisics usually exhibi periodiciies. These periodiciies can be brough ou hrough Fourier analysis, by applying eiher specral analysis or cyclosaionary analysis. In variable speed operaion, all he componens whose frequencies are ied o he speed of he machine follow he speed variaions. Looking for consan frequency oscillaory funcions wihin he signal is no as relevan as in consan speed operaion. Since many diagnosis ools have been developed based on Fourier ransform, i is worh looking for equivalen ools in variable speed. The usual echniques consis eiher in segmening he signal ino slices shor enough for he speed o be consan, or in re-sampling he signal so ha he samples correspond o fixed angular posiions raher han being emporally equi-spaced. The firs mehod s drawback is ha he shorer he slices, he poorer he resoluion of frequency analysis, while he second mehod s drawback is ha by resampling he signal, one alers he resonance phenomena, ha are no ied o he speed of he shaf. Here, we propose o adap Fourier ransform o he vibraion signal, raher han alering he signal for i o be adaped o Fourier analysis. The new ool ha we propose, called Speed Transform, consiss in decomposing he signal over a basis of elemenary oscillaory funcions whose frequencies follow he speed variaions. I has been shown in [1] o have equivalen properies o Fourier ransform, provided ha he speed is linearly varying. I can be applied o he signal wihou any prior resampling or segmenaion, which allows exending o variable speed operaion mos of he classical ools based on Fourier analysis for he consan speed operaion. In his paper, Speed Transform is applied o he vibraion signal of a roller bearing. We show how i allows compuing under variable speed operaion classical parameers such as envelope specrum and specral correlaion, which hen become envelope Speed Transform and Speed correlaion. 1. Inroducion During many years, vibraion analysis echniques have been applied in consan speed operaion, in order o ensure ha he vibraion signals were saionary or cyclosaionary. Bu i is no always possible o ge some records performed in such peculiar condiions. Indeed, he machinery roaion speed canno be se according o he surveillance needs. Furhermore, some damages can be revealed in non saionary operaion, for example by exciing some resonance during a speed-up. For hese reasons some echniques have been developed laely for he analysis of vibraion signals in non saionary operaion. In saionary operaion he Fourier ransform is widely used o pu in evidence some repeiive phenomena ied o he presence of a damage on a roaing par. In non saionary operaion, racking sricly periodic componens is no relevan any more. Among he echniques adaped o variable roaion speed, some are based on filering, bu ohers sill rely on Fourier ransform, hough requiring some aleraion eiher of he signal or of he analysis ool. One can spli he signal ino slices and suppose he roaion speed is consan over he duraion of a slice, or resample he signal for i o be expressed versus roaion angle raher han ime. The drawback of he firs 1
echnique is a worsened frequency resoluion, while resonance phenomena are alered by he laer. Some new echniques consising in a modified ransform have been proposed. The Velociy Synchronous Discree Fourier Transform [] performs angular resampling and Fourier ransforming a he same ime, while he Time Varian Discree Fourier Transform [3] is a modified (non-orhogonal) Fourier ransform whose kernel follows he speed variaions. The ool applied here consiss in calculaing a Fourier ransform whose kernel follows speed variaions over a duraion sufficien o ensure orhonormaliy, provided ha he speed variaions are linear. In secion we firs sudy wha happens o roller bearing vibraions in variable roaion speed hrough calculaions performed over a simplified model. We deduce from hese calculaions which componens of he auocorrelaion funcion follow he speed variaions and propose o exend classical analysis echniques such as envelope specrum and specral correlaion hrough he use of Speed Transform insead of Fourier Transform. Two new analysis echniques are derived, called Envelope Speed Transform and Speed Correlaion, and applied o simulaed signals fiing o he simplified model. In secion 3 hese new ools are applied firs o a simulaed roller bearing vibraion signal and hen o a real-life one.. Simplified model of he bearing vibraion signal in variable speed operaion.1 Signal model in consan speed operaion In consan speed operaion, he vibraion signal of a damaged roller bearing is mainly produced by shocks ha excie he mechanical srucure. These shocks occur a periodic inervals, he periodiciy being alered by some jier due o he movemens of he rollers wihin he cage. Such a model can be described as follows [4]. s = m δ Td h (1) m is an ampliude modulaion, periodic a he roaion speed frequency, T d is he period of he shocks, δ Td = + n= δ nt d is a pulse rain sanding for he exciing shocks. Some random jier can affec he pulse imes nt d, h is he response of he mechanical srucure. This can be viewed as a sum of complex exponenials wih complex random ampliudes. The frequencies of hese periodic waves are all he possible combinaions of he damage shocks frequency (f d = 1 T d ) harmonics and he roaion frequency (f r ) harmonics f k1,k = k 1 f d + k f r wih k 1, k Z due o he modulaion m. The vibraion signal can hus be wrien as follows. s = + k + 1 = k = () a k1,k e πj f k1,k The ampliudes a k1,k are complex and random. They ake ino accoun: he resonance of he srucure, he phases of he oscillaory funcions, he jier on he shocks period.
These ampliudes are usually correlaed, due o he fac ha all hese oscillaory componens are produced by he same physical phenomenon. As a consequence, boh envelope specrum and specral correlaion exhibi specral lines when applied o such a signal [5]. They hus are classically used as analysis ools in order o deec he characerisic shocks produced by any damage and diagnose he roller bearing.. Exension of a simplified model o variable speed In order o sudy from a heoreical poin of view wha happens in ime-varying operaion, he auocorrelaion of he signal should be calculaed. Indeed, he specral correlaion is obained from he auocorrelaion funcion by applying Fourier ransform over boh ime and ime-lag, while he envelope specrum is obained by applying Fourier ransform over ime for zero ime-lag. A simplified model given in Eq (3) will be sudied here in order o avoid heavy calculaions. s = a 1 e πj f 1 + a e πj f (3) The respecive complex random ampliudes a 1 and a of he wo componens can be correlaed o each oher or no. Their respecive frequencies are ime-varying, For insance, hey can follow he roaion frequency variaions. The auocorrelaion funcion is defined by R s, τ = E s s τ where E[...] sands for ensemble averaging and * for complex conjugae. The calculaion of he auocorrelaion of he signal defined by Eq. (3) is given in appendix. In wha follows, he frequencies of he wo componens are supposed o be linearly varying and equal o f 1 = α 1 + β 1 and f = α + β. In he peculiar case of zero ime-lag (τ = ) i is equal o: R s, = E s = E a 1 + E a + Re a 1 a cos π α α 1 + β β 1 (4) The auocorrelaion funcion depends only on ime and follows he variaions of α = f f 1. In consan speed operaion, i would be periodic a a fixed frequency α = f f 1 and he signal would be cyclosaionary a frequency α. The envelope specrum would exhibi a specral line a cyclic frequency α. We propose o exend his echnique o he variable speed case by applying o he signal an Envelope Speed Transform (EST). Provided ha boh f 1 and f are proporional o he roaion frequency variaions f r, he EST should exhibi an order line a he order K such ha α = Kf r. In he mos general case he auocorrelaion can be decomposed ino wo auo-erms R 1, τ and R, τ and wo cross-erms R,1, τ and R 1,, τ, whose expressions are given below. R 1, τ = E a 1 e πj α 1τ+β 1 τ α 1 τ R, τ = E a e πj α τ+β τ α τ (5) R,1, τ = E a a 1 e πj α α 1 R 1,, τ = E a 1 a e πj α 1 α + β β 1 e πj α 1 τ +β 1 τ e πjα 1τ + β 1 β e πj α τ +β τ e πj α τ (6) The wo auo-erms now depend on τ insead of τ alone in he consan speed case. The wo cross-erms are each a produc of hree erms: The firs one depends only on ime and follows he variaions of α = f 1 f, 3
The second one depends only on τ, The hird one depends on he produc τ, as in he auo-erms. I should be possible o deec he firs erm by applying a Speed correlaion, hough, due o he hird erm, i should no exhibi such a sharp order line as he cyclic specral line observed in he saionary case..3 Speed Transform based diagnosis ools Speed ransform was firs inroduced in [1]. I consiss in decomposing a signal over a basis of oscillaory funcions b n whose frequencies follow he speed variaions. I consiss in calculaing 1 T all he basis funcions: T s b d for πj o b o = e f r (7) Where f r denoes he roaion frequency and o he order. More deails abou he accuracy of he ransform and is asympoic properies can be found in [1]. This basis was proved o be asympoically orhonormal in he case of linear speed variaions. Speed ransform exhibis speed lines a all orders corresponding o oscillaory componens whose frequencies are proporional o f r. I is an efficien ool for he esimaion of he ampliude of componens whose frequencies follow he speed variaions. We propose o exend wo signal processing ools classically used for roller bearing diagnosis in consan speed o he case of linear speed variaions by replacing Fourier Transform (FT) by Speed Transform (ST). By replacing FT by ST in he envelope specrum, we obain an Envelope Speed Transform (EST), calculaed hrough he following seps: s() Hilber ransform Speed Transform S(o) Figure 1: Synopic of he Envelope Speed Transform calculaion, where o sands for he order relaively o he roaion frequency Specral correlaion funcion is a Fourier ransform of he auocorrelaion funcion boh over ime and ime-lag. Fourier ransform over ime gives cyclic frequency while Fourier ransform over ime-lag gives specral frequency. When applied o cyclosaionary signals, specral correlaion exhibis specral lines in cyclic frequency. We hus replace Fourier Transform by Speed Transform over ime, while we sill apply Fourier ransform over ime-lag. The funcion obained, ha will be called in wha follows Speed Correlaion, hus depends on specral frequency and order insead of specral frequency and cyclic frequency. In he presence of componens whose frequencies follow he speed variaions, i should exhibi order lines versus order..4 Applicaion of he proposed ools o he simplified signal The simulaed signal generaed here can be described by Eq. 3 wih he following parameers: Sampling frequency: f e = 1 khz Number of samples:, which corresponds o. seconds Roaion frequency: f r() = 5 + 5 * Frequency of he firs componen: f 1() = * f r() 4
Order Frequency of he second componen: f () = 3 * f r() The ampliudes a 1=a are complex, random, Gaussian and oally correlaed. The achomeer signal is a cosine wave a he roaion frequency. Speed envelope specrum is applied o he whole lengh of he signal, wih Hamming windowing, and inerpolaed by. The EST of he simplified signal is displayed on Fig.. As expeced from he heoreical sudy, an order line appears a order K such ha f 1 f = K f r (), i.e. K=1. Envelope Speed Transform 1.8 1.6 1.4 1. 1.8.6.4. 1 3 4 5 6 7 Order Figure : Envelope Speed Transform of he simplified signal The Speed Correlaion of he same signal was calculaed for an order range going from.95 o 1.5 by seps of.1. I was esimaed by averaged periodogram over 1 slices wih /3 overlap. As can be observed on he resul displayed on Fig. 3, an order line appears a order one, in spie of he weighing erm ha depends on τ. 1.5 1.4.7 1.3 1. 1.1 1.99.98.97.96.6.5.4.3..1.95 1 3 4 5 Specral frequency (Hz) x 1 4 Figure 3: Speed correlaion of he simplified signal 5
3. Applicaion of he Speed Transform o roller bearing signals 3.1 Applicaion o a simulaed roller bearing vibraion signal Once validaed on he simplified signal he wo proposed echniques were applied o a simulaed roller bearing signal following Eq. 1. The parameers of he model are he following ones: Mean diameer: D m = 39mm Ball diameer: d = 7.5mm Number of rolling elemens: Z = 13 Conac angle: α = Resonance frequency: f res = 8 khz Roaion frequency: f r = 5 + 3.33 Ampliude modulaion: m = 1 + cos π f r Ouer ring damage of frequency: f d = 5.5 f r 7% random jier on he damage period Signal duraion: seconds Sampling frequency: 1 khz Addiive Gaussian random noise. The Signal o Noise Raio is SNR = 7dB The envelope Speed Transform of his simulaed vibraion signal is given in Fig. 4. I exhibis a classical envelope specrum paern, wih a speed line a order 5.5 and sidebands due o he modulaion a he roaion frequency..16.14 Envelope Speed Transform Harmonics of order 1, and 3 of f d.1.1 f r.8.6.4. 1 3 4 5 6 7 8 9 1 11 1 13 14 15 16 17 18 19 1 3 4 5 6 7 8 9 3 Order Figure 4: Envelope Speed Transform of he simulaed roller bearing vibraion signal The speed correlaion was compued on he same simulaed vibraion signal for orders ranging from 4 o 6.5 by sep of.1. I was esimaed by averaged periodogram over 1 slices wih /3 overlap. I is normalized by he energy of he specrum, so ha he funcion displayed in Fig. 5 corresponds o Speed coherence (equivalen o specral coherence if compued wih Fourier ransform). I exhibis a paern ha is characerisic of he chosen damage, wih a peak of energy a order 5.5 and sidebands due o he roaion frequency. These peaks are wider han acual lines, which is probably due o he facor varying in τ. 6
Order 6.5.8 6.7.6 5.5.5 5.4.3 4.5..1 4 1 3 4 5 Specral frequency (Hz) x 1 4 Figure 5: Speed coherence of he simulaed roller bearing vibraion signal 3. Applicaion o a real-life roller bearing vibraion signal The signal was recorded wih an acceleromeer on a Speraques es bench. The roller bearing characerisics are he following ones: Number of balls: Z = 8 Ball diameer: d =.315 inches Mean diameer: D m = 1.319 inches The sampling frequency is f s = 51. khz Damage on he ouer ring From hese characerisic, we can deduce ha he damage frequency should be equal o f d = 3.5 * f r wih f r he roaion frequency, so ha he envelope Speed Transform should exhibi lines a order o d = 3.5 and is harmonic orders. The roaion speed, esimaed from a achomeer signal, is ploed on Fig. 6. 7
6 Variaions of he roaion speed in Hz 5 4 3 1 5 1 15 5 3 Time (s) Figure 6: Variaions of he roaion frequency Boh envelope specrum and envelope Speed Transform were compued over a par of he signal aken beween 8s and 16.5s and beforehand high pass filered a f s / 4. The wo ransforms are ploed on Fig. 7. On he chosen ime inerval, he mean roaion frequency is 3.5 Hz. The envelope specrum is displayed from Hz o 13 Hz in order o ake ino accoun he harmonics of ha mean frequency up o he 4 h. Almos nohing can be deeced from he classical envelope specrum, whereas he envelope Speed Transform exhibis order lines a order O d = 3.5, which is very close o he heoreical order o d = 3.5 and is harmonic orders..4 Envelope specrum.3..1 4 6 8 1 1 Frequency (Hz) 1.5 1 F d =3.5 Envelope Speed Transform.5 5 1 15 5 3 35 4 Order Figure 7: Envelope specrum and envelope Speed Transform of he real-life vibraion signal. 8
4. Conclusion Here, some classical ools of roller bearing vibraions analysis have been exended o ime varying operaion condiion by replacing Fourier ransform by Speed ransform. The relevance of such echniques have been shown by sudying he effec of speed variaions on a simplified model of hese vibraions. We proposed wo new analysis ools, called envelope speed ransform and speed correlaion. Envelope Speed Transform seems o be a very promising diagnosis ools boh because i is very well fied o he ime varying model of he vibraions, and because i is applied over he whole lengh of he signal, which ensures he orhonormaliy of speed ransform. Speed correlaion also gave ineresing resuls on he simulaed signals, hough i does no appear from he heoreical sudy as well fied o he model as envelope Speed Transform, and he applicaion o shor slices of he signal can deeriorae he orhonormaliy. The heoreical aspecs of Speed Transform and is exension o he case of non linear variaions are sill under sudy. References [1] C. Capdessus, E. Sekko, J. Anoni, Speed Transform, a New Time-Varying Frequency Analysis Technique, CMMNO 13, Ferrara, Ialy, 8-1 May 13. [] P. Borghesani, P. Pennacchi, S. Chaeron, R. Ricci, The velociy synchronous discree Fourier ransform for order racking in he field of roaing machinery, MSSP, available online a hp://dx.doi.org/1.116/j.ymssp.13.3.6. [3] J. R. Blough, D. L. Brown, H. Vold, The Time Varian Discree Fourier Transform as a Tracking Order Mehod, SAE Paper Number 976, 1997. [4] P. D. McFadden, J. D. Smih, Model for he vibraion produced by a single poin defec in a rolling elemen bearing, Journal of Sound and Vibraion, 1984, 96(1), pp. 69-8. [5] R. B. Randall, J. Anoni, S. Chobsaard, The relaionship beween specral correlaion and envelope analysis in he diagnosis of bearing fauls and oher cyclosaionary machine signals, Mechanical Sysems and Signal Processing, (1), 15(5), p.p. 945-96. A Auocorrelaion funcion of he simplified model in variable speed operaion s = a 1 e πj f 1 + a e πj f Ampliudes a 1 and a are complex, random and can be eiher correlaed or no. A.1 General case R s, τ = E s s τ R s, τ = E a 1 e πj f 1 + a e πj f a 1 e πj f 1 + a e πj f τ τ R s, τ = R 1, τ + R, τ + R,1, τ + R 1,, τ R 1, τ = E a 1 e πj f 1 f 1 = E a 1 e πj τ f 1 R, τ = E a e πj f f = E a e πj τ f τ τ R,1, τ = E a a 1 e πj R 1,, τ = E a 1 a e πj f τ f 1 f τ 1 f 9
A. Peculiar case of consan frequencies R 1, τ = E a 1 e πj f 1τ R, τ = E a e πj f τ R,1, τ = E a a 1 e πj f f 1 +f 1 τ R 1,, τ = E a 1 a e πj f 1 f +f τ The wo componens R 1, τ and R, τ depend only on τ. The wo cross erms depend boh on and τ. These erms are periodic versus ime wih a frequency f 1 f, so ha s is cyclosaionary a ha frequency. A.3 Peculiar case of uncorrelaed ampliudes In his case he cross-erms R,1, τ and R 1,, τ are equal o zero. The auocorrelaion funcion hus does no depend on ime and s is saionary. A.4 Case of he ime varying frequencies Le us suppose ha he frequencies f 1 and f are linearly varying. f 1 = α 1 + β 1 f = α + β R 1, τ = E a 1 e πj α 1τ +β 1 τ α 1 τ R, τ = E a e πj α τ +β τ α τ R,1, τ = E a a 1 e πj α α 1 R 1,, τ = E a 1 a e πj α 1 α + β β 1 e πj α 1 τ +β 1 τ e πj α 1τ + β 1 β e πj α τ +β τ e πj α τ A.5 Peculiar case τ = R 1, = E a 1 R, = E a R,1, τ = E a a 1 e πj α α 1 R 1,, τ = E a 1 a e πj α 1 α + β β 1 + β 1 β The auocorrelaion funcion hen becomes: R s, = E s = E a 1 + E a + Re a 1 a cos π α α 1 I hus depends on and follows he variaions of f 1 f. + β β 1 1