Identification of harmonics and sidebands in a finite set of spectral components

Transcription

1 The Tenth Internatonal Conference on Condton Montorng and Machnery Falure Preventon Technologes Identfcaton of harmoncs and sdebands n a fnte set of spectral components Tmothée Gerber and Nadne Martn GIPSA-lab Grenoble Campus, 11 rue des Mathématques, Sant Martn D Hères, France +33 (0) Tmothee.Gerber@gpsa-lab.grenoble-np.fr, Nadne.Martn@gpsa-lab.grenoble-np.fr Cornne Malhes IRIT-TéSA Unversté de Toulouse, 2 rue Charles Camchel, Toulouse, France Abstract Spectral analyss along wth the detecton of harmoncs and modulaton sdebands are key elements n condton montorng systems. Several spectral analyss tools are already able to detect spectral components present n a sgnal. The challenge s therefore to complete ths spectral analyss wth a method able to dentfy harmonc seres and modulaton sdebands. Compared to the state of the art, the method proposed takes the uncertanty of the frequency estmaton nto account. The dentfcaton s automatcally done wthout any a pror, the search of harmoncs s exhaustve and moreover the dentfcaton of all the modulaton sdebands of each harmonc s done regardless of ther energy level. The dentfed seres are characterzed by crtera whch reflect ther relevance and whch allow the assocaton of seres n famles, characterstc of a same physcal process. Ths method s appled on real-world current and vbraton data, more or less rch n ther spectral content. The dentfcaton of sdebands s a strong ndcator of falures n mechancal systems. The detecton and trackng of these modulatons from a very low energy level s an asset for earler detecton of the falure. The proposed method s valdated by comparson wth expert dagnoss n the concerned felds. 1. Introducton System montorng s a key element n a predctve mantenance strategy (1). Vbraton analyss s one of the oldest and most used technques. It conssts n computng the spectral densty of vbraton sgnals, recorded at senstve ponts of the system (e.g., the bearngs or gearboxes). The presence of harmonc or modulaton seres s then used as ndcators of wear or damage of one or more mechancal parts of the system (2). Few studes have focused on the problem of dentfyng harmoncs at the output of a spectral analyss: (3) presented a method based on correlaton but does not take the uncertanty due to the estmaton of the frequences nto account, whle (4) proposed a method based on statstcal tests wth the a pror hypothess that the power of an harmonc seres s a decsve crteron. A method developed n (5) and appled to the dagnoss of helcopter engnes assocates the detected peaks to known peaks from an underlyng model, thus nducng an a pror model.

2 The Tenth Internatonal Conference on Condton Montorng and Machnery Falure Preventon Technolog The man dea of ths paper s to automatcally dentfy the harmonc seres and sdebands takng the uncertanty n frequency estmaton nto account and wthout ntroducng any a pror on the sgnal. If the number of system parts to be montored s large, the number of sgnals to be analysed becomes szeable. Therefore, there s a need to automatcally perform spectral analyss and after that the readng of the acheved spectra. Many spectral analyss tools are already able to detect all spectral components of an analysed sgnal. Each detected component s usually characterzed by some parameters, dependng on the tool used. In general, these parameters nclude at least the central frequency of the detected peak and the estmaton error of the central frequency, estmaton strongly related to the spectral resoluton. Assumng the knowledge of these two parameters, we propose a method based on spectral nterval ntersectons, n order to dentfy the harmonc and modulaton seres from a fnte set of spectral components. 2. Context The context of our work takes place at the output of a spectral analyss tool whch provdes a set S of spectral components. Wthn ths set, each component C s characterzed by at least ts central frequency ν, the uncertanty ν drectly lnked to the spectral resoluton and ts ampltude A { 1( ν1 ν1 1) 2( ν2 ν2 2) F ( νf νf F) } S = C,, A, C,, A,, C,, A, (1) where F s the total number of spectral components detected. In the present paper, we use the automatc spectrum analyser AStron (6)(7). Thanks to ts method of detecton and automatc dentfcaton of nose, AStron detects only the relevant components (snusods or narrowband). Moreover, a method mplemented n AStron allows the estmaton of the central frequency of components wth a better precson than the spectral resoluton (8). The purpose of ths study conssts then n dentfyng the harmonc seres and sdebands n the set S of detected spectral components. 3. Harmonc seres and modulaton sdeband dentfcaton After the defntons of a harmonc seres and a harmonc famly, a method s proposed to dentfy the harmonc seres. The same method s then extended around the detected harmoncs to dentfy the modulaton sdebands. 3.1 Defnton of a harmonc seres and of a harmonc famly Mathematcally, a harmonc seres s characterzed by a fundamental frequency ν and * defned as a set of spectral components of frequences r ν,r N representng the harmonc order. Two seres of fundamental frequences ν and ν j belong to the same famly f ther rato s a ratonal number, that s to say, 2

3 The Tenth Internatonal Conference on Condton Montorng and Machnery Falure Preventon Technolog ν p p,q N N =. (2) q * ( ) * such as νj In ths case, the famly s defned by all the components of both seres and s characterzed by a fundamental frequency equal to ν 0 = ν j / p = ν / q, even f ths frequency s not detected and may be not present n the spectrum. Ths defnton mples that f two harmonc seres of fundamental frequency ν and ν j have a harmonc ν k n common, then the two seres are part of the same famly. Otherwse, f ν and ν j are ncommensurable, each seres belongs to a dstnct harmonc famly. It s worth notng that a sgnal contanng more than one harmonc famly cannot be perodc. From a system mantenance pont of vew, t s nterestng to dentfy all the harmonc seres snce each one may be assocated wth a dfferent part of the system. Groupng harmonc seres n famly s an addtonal ndcator to dentfy nterrelated components. 3.2 Harmonc seres detecton Harmonc seres dentfcaton from estmated components s a nontrval problem because of estmaton errors. In fact, estmaton errors do not preserve the accuracy of the relaton between an estmated frequency and ts harmoncs. So, n order to fnd the harmonc frequency of order r of an estmated frequency ν, lookng for a detected component at a frequency exactly equal to r ν wll not be suffcent. In ths paper, we propose to use the uncertanty ν of each detected component to bypass the drawbacks of the non-exact frequency estmaton. Each estmated frequency ν s thus represented by a confdence nterval of wdth ν centred on ν. The harmonc detecton s then completed by ntersecton of these ntervals, as detaled n what follows. A component C (ν, ν, A ) of S s the fundamental of a harmonc seres, referred to as H, f C j (ν j, ν j, A j ) S exsts such that ν j ν j ν ν a j;b jν = j ;ν + j r ν ;r ν , (3) * wth (a j, b j ) R², a j b j, and r N { 1}. The harmonc order r ncreases sequentally, startng at 2 and stoppng when the end of the spectrum s reached. The search for harmonc components s then performed n a sequental manner (r=2,3, ) lookng for the components C j whch are harmoncs of C. However, ths can rase a problem of harmonc dentfcaton when several detected peaks satsfy (3), for the same order r. Fg. 1 presents the case of two components of frequences ν j and ν j+1 satsfyng (3), the component ν beng consdered as a potental fundamental frequency. Therefore a crteron has to be added to (2) n order to dentfy the successve harmoncs of a seres. We propose to use a crteron of mnmum dstance to select ν (r) the harmonc of order r of the fundamental frequency ν. 3

4 The Tenth Internatonal Conference on Condton Montorng and Machnery Falure Preventon Technolog For a gven order of harmoncs, ν (r) has to satsfy the followng complete crteron ν = ν / mn ν r ν. (4) ( r) j ν j satsfyng ( 2) j In the case of Fg. 1, based on ths second crteron, the component wth frequency ν j s chosen as the harmonc of order r. Fgure 1. Harmonc search based on nterval ntersecton: ν s consdered as a potental fundamental of a gven seres and ν j and ν j+1 are two canddates for the harmonc of order r. Based on the dstance crteron, ν j wll fnally be retaned. Ths method rases a second problem: the search nterval for harmoncs grows lnearly wth the harmonc order r. Consderng a fundamental ν wth ts estmaton error ν, the uncertanty of ts r order harmonc frequency s equal to r. ν. As a consequence, for hgh order harmoncs the probablty of gettng multple canddates and selectng a wrong one ncreases. To prevent the search nterval growng, each tme a component s dentfed as a harmonc of C (satsfyng (2) and (3)), the parameters ν and ν are updated as a j+bj bj aj ν=, ν=. (5) 2r r Ths strategy s llustrated on Fg. 2. If r s the order of the last harmonc added n the seres and no harmoncs have been dentfed for the orders r+1, r+2,, r+k, the search nterval wll not be updated and wll contnue to grow. To prevent the search nterval to become large compare to spectral resoluton, the search for harmoncs n ths seres stops when no harmoncs have been detected for k consecutve order. In our mplementaton, we choose k = 8. Fgure 2. Parameter updates: ν and ν are updated to avod the search nterval growng. In grey, the prevous values of ν and ν. Each tme a harmonc s detected, updated values (n blue) are consdered. 4

5 The Tenth Internatonal Conference on Condton Montorng and Machnery Falure Preventon Technolog In the proposed algorthm, the search of harmonc seres s exhaustve. One by one, each detected spectral component s consdered as a potental fundamental of a harmonc seres. Further processng descrbed n Secton 3.5 has to be done to determne the fnal lst of dentfed harmonc seres. Moreover, the fundamental frequency can be mssng from a sgnal or t could have not been detected by the prevous spectral analyss. To avod the non-detecton of the harmonc seres because of the non-presence of the fundamental, we create an artfcal component Ĉ for each real component C n S wth frequency ˆ ν = ν /2 and uncertanty ˆ ν = ν /2. If the seres detected from Ĉ and C are dentcal, we merge them and consder only the component C as a fundamental of a harmonc seres. 3.4 Modulaton sdeband detecton Sdebands are usually the result of an ampltude or frequency modulaton process. In the spectrum, they take the form of spectral components equally spaced on both sdes of the carrer frequency, symmetrcally. For computatonal tme reason, each component of the spectrum s not consdered as a potental carrer frequency. The search for sdebands s only made around the components belongng to the harmonc seres H prevously dentfed. Assumng that ν s the fundamental frequency of the harmonc seres H, for each component C j of order r n H, the search for modulaton seres s made n 3 steps, llustrated n Fg. 3: A - Frst, we look for sdebands above C j, n the search nterval [ν j ; ν j + ν ] = [r ν ; (r+1) ν ]. To proceed, we dentfy all the harmonc seres M k Cj+ present n ths nterval, consderng C j as the new frequency reference, k representng the seres ndex. In the example of Fg. 3, two seres are dentfed, n orange (wth fundamental ν 0 ) and purple (wth fundamental ν 1 ). B - Then, the same process s appled below C j, n the search nterval [ν j - ν ; ν j ] = [(r 1) ν ; r ν ] to dentfy the harmonc seres M k Cj-. In the example of Fg. 3, two seres are extracted from the set of detected frequences, n red (same fundamental ν 0 ) and n green (wth fundamental ν 2 ). C - Fnally, we compare the fundamental frequences from the M k Cj+ seres to the fundamental frequences from M k Cj-. If two seres have the same fundamental frequency (wth a possble error of maxmum ν ), both seres are merged and are now consdered as a modulaton seres. Thus, n the example of Fg. 3, the modulaton seres of fundamental ν 0 s selected as a symmetrcal seres around frequency ν j = r ν j and two non-symmetrcal seres are kept on both sdes of ν j, of fundamentals ν 1 and ν 2. 5

6 The Tenth Internatonal Conference on Condton Montorng and Machnery Falure Preventon Technolog Fgure 3. Modulaton sdebands detecton: (A) Two harmonc seres (n orange and n purple) dentfed above the carrer frequency rν. (B) Two harmonc seres (n red and n green) dentfed below the carrer frequency. (C) Search for symmetry and fuson: one modulaton seres (n orange) fnally detected. A modulaton seres s not always symmetrc. There can be more components above the carrer frequency than below, and vce versa. The proposed method allows the dentfcaton of such non-exactly symmetrc sdebands. An example s gven n Fg. 4.C wth 3 sdebands below the carrer frequency, and only 2 sdebands above. 3.5 Charactersaton crtera The proposed method s exhaustve and dentfes every harmonc and modulaton seres present n the spectrum. As a consequence, the number of seres detected s large and some of them are not always relevant. Nevertheless, n the lterature, there s no precse defnton of a harmonc seres (apart from a mathematcal pont of vew). Moreover, the relevancy of a seres depends on the applcaton and the physcal context of the studed sgnals. That s the reason why keepng all the seres detected s necessary. Rather than elmnatng the false seres, the proposed method classfes the detected seres thanks to the followng three charactersaton crtera. 6

7 The Tenth Internatonal Conference on Condton Montorng and Machnery Falure Preventon Technolog These crtera have been chosen as a comparson of each detected seres to the correspondng perfect one. A perfect seres s defned as a spectral comb gong tll the end of the spectrum wth no harmonc mssng. The frst crteron denoted D, hghlghts the densty of the seres, n order to dfferentate seres wth several harmoncs mssng from seres wth almost all harmoncs present card D= r ( H ) max, (6) wth r max the rank of the last harmonc n the seres H. A seres n whch lots of harmoncs are mssng wll have a small densty whereas a full seres ncludng all harmonc orders wll have a densty equal to one. The second crteron s based on N max whch s the maxmum sze of the seres based on the frequency of ts fundamental ν and of the hghest frequency ν F n the set S. Ths has to be compared to r max to defne the second crteron, the rchness R of the seres max r max ν R = wth N max = Nν F, (7) wth provdng the nteger part. Ths wll help to consder n a dfferent way two seres wth the same cardnal and the same harmoncs orders. For example a seres of fundamental frequency ν = 510 Hz ncludng only harmoncs of orders 2 and 3 for a sgnal n whch the maxmum detected frequency s ν F = 2000 Hz carres more weght than a seres ncludng also the same harmonc orders but wth a fundamental frequency ν j = 23 Hz. The frst seres wll have a crteron equal to 1, whch s the maxmum possble whereas the second seres wll only get a 0.035, whch s a very low value. Classcally used, the thrd crteron s the Total Harmonc Dstorton THD (9) THD = 2 2 A+A A N 1 A (7) Ths crteron wll be helpful n applcatons where ampltude behavour n harmonc seres s known a pror and awated. For modulaton sdebands, these crtera are computed on the seres below and above the carrer frequency. The combnaton of the followng crtera allows classfyng the harmonc seres and modulaton sdebands by relevancy. In addton, the seres are grouped n famly as defned. 7

8 The Tenth Internatonal Conference on Condton Montorng and Machnery Falure Preventon Technolog 4. Results The method has been tested on synthetc and real-world sgnals. The results for a current sgnal of a fan are shown n Fg. 4, below the spectral analyss result, as a schematc representaton of the detected seres. The method dentfes a seres of 28 consecutve harmoncs at the fundamental frequency Hz and two modulaton sdebands seres around ths carrer frequency wth cardnal 3 and 7 and ther respectve frequences Hz and Hz. All the detected seres have hgh densty, that s to say D = 1. Harmonc seres near the fundamental 50 Hz was expected. Its THD s very low (1.19 %) and s under the maxmum 2 % guaranteed by the energy suppler. Fgure 4. Seres dentfcaton on the spectral component set from the current sgnal of a fan: (A) The 50 Hz harmonc seres (B) A zoom on the two modulaton seres around the 50 Hz. (C) A second zoom on the Hz modulaton seres. 8

9 The Tenth Internatonal Conference on Condton Montorng and Machnery Falure Preventon Technolog The presence of two seres of sdebands s characterstc of two defects on the fan, dentfed by an expert n mantenance. The seres of fundamental frequency Hz s generated by a msalgnment of the belt. Its rchness s very low (R = 0.03), but ts densty s maxmal (D = 1) and the THD s hgh (8.74 %). The second modulaton seres wth fundamental frequency Hz, s due to a broken shaft. Its crtera are very hgh: maxmal densty D = 1, maxmal rchness R = 1 and a very hgh TDH = 146 %. These two seres of modulaton are also present around the harmoncs of 50 Hz,.e. around 100 Hz, 150 Hz, 200 Hz, etc. 5. Conclusons The method proposed n ths artcle dentfes harmonc seres and modulaton sdebands n a fnte set of spectral components, and wthout any a pror on these seres. On vbraton sgnals, rch n spectral components, even the low-energy harmonc seres are dentfed. The dentfcaton of modulaton sdebands around these harmoncs s an excellent ndcator for the early detecton of faults n condton montorng systems. References 1. P F G Marquez et al, Condton montorng of wnd turbnes: Technques and methods, Renewable Energy 46 (2012), p A Boulenger et C Pachaud, Analyse vbratore en mantenance : survellance et dagnostc des machnes, Dunod, B Redorter et B Laget, Détecton des harmonques dans un spectre de vbratons par des méthodes de tratement d mages, TS (1989). 4. M Zeytnoglu et K M Wong, Detecton of harmonc sets, IEEE Transactons on Sgnal Processng (1995), p L M Gelman et al, Condton Montorng Dagnoss Methods of Helcopter Unts, Mechancal Systems and Sgnal Processng 14.4 (jul. 2000). 6. N Martn et al, Vers une carte d dentté spectrale, GRETSI, C Malhes et al, A spectral dentty card, EUropean SIgnal Processng COnference (EUSIPCO), 2006, p M Durnern, Une stratége pour l nterprétaton en analyse spectrale. Détecton et caractérsaton des composantes d un spectre, Thèse INPG. 1999, Url: 9. G. R. Slone, The audophle's project sourcebook, McGraw-Hll/TAB Electroncs (2001), p. 10, ISBN