Removal of Order Domain Content in Rotating Equipment Signals by Double Resampling

Transcription

1 Removal of Order Domain Conen in Roaing Equipmen Signals by Double Resampling By: Charles L. Groover Marin W. Trehewey Deparmen of Mechanical and Nuclear Engineering Penn Sae Universiy Universiy Park, PA 680 USA Kenneh P. Maynard Michell S. Lebold Applied Research Laboraory Penn Sae Universiy Universiy Park, PA 680 USA Submied o: Mechanical Sysems and Signal Processing Corresponding Auhor: Marin W. Trehewey 39 Reber Building Penn Sae Universiy Universiy Park, PA 680 USA FAX: Submied: June 003 Revised: Sepember 003

2 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling Absrac Fixed frequency conen (i.e., componen or srucural resonances) in specra obained from roaing equipmen can be masked by he srong sources a harmonics (order) of he shaf running speed. The raio of he fixed frequency componens o he order componens can be greaer han 60 db making inerpreaion of resonances in he specra difficul. Hence he order componens are view as a corruping phenomenon. An approach o remove he order componens from he specra, wihou affecing he remaining frequency domain informaion is presened in his work. The echnique uilizes a sequence of daa sampling and ransformaions, beween he ime, order and frequency domains as follows:. Vibraion daa is sampled using a consan ime basis ( ).. The imes corresponding o a consan angular basis ( θ) are deermined. 3. The vibraion daa is inerpolaed o a consan angular basis ( θ). 4. The consan angle sampled daa is ransformed via he FFT o he order domain. 5. The high ampliude order componens are now exacly bin cenered and can be removed from he specra. 6. An inverse FFT is applied o reurn o a consan angular incremen sampled ( θ) array, sans order conen. 7. The consan angular incremen sampled ( θ) array is inerpolaed o an array sampled wih consan ime basis ( ). 8. A FFT is applied and hen sandard specral esimaion procedures are applied o compue he vibraion specra wih he high level orders removed. The heoreical and implemenaion deails of he double resampling approach are discussed. The approach is applied o experimenal orsional vibraion daa acquired from a laboraory es rig designed o simulae a urbine roor. The es resuls show ha he mehod can recover fixed frequency componens (i.e., urbine blade naural frequencies) in he presence of order componens 50 db higher.

3 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 3. Inroducion Forced vibraion in roaing equipmen can generally be separaed ino wo caegories; ) variable frequency componens relaed o he machine s roaion rae; and ) fixed frequency relaed vibraion. Specral analysis is commonly used o separae and idenify he frequencies of imporance. For synchronous machines all he frequency componens remain fixed regardless if hey are caused by he machine s inheren roaion (e.g., imbalance) or a fixed frequency componen (e.g., srucural resonance). However, for asynchronous machines he running speed changes, eiher drifing or experiencing rapid speed changes depending on he applicaion. In his case, all he vibraion relaed o a machine s roaion (i.e., imbalance) change proporionaely in frequency o he running speed variaions while he fixed frequency componens remain consan. To rack he frequency variaions in relaion o changes in he equipmen running speed, order analysis is used. Frequency and order analysis are similar, bu wih differen independen argumens. Frequency analysis applies a Fourier ransform o a vibraion signal digiized wih a uniform sampling ime inerval ( ). The resuling specrum s independen variable is in Hz. Order analysis applies a Fourier ransform o a vibraion signal sampled on he basis of a uniform angular shaf roaion incremen ( θ). The independen specral variable is orders, or muliples of shaf running speed. Hence, any vibraion direcly caused by he machine s roaional speed will remain a a fixed order regardless of running speed. A significan body of work exiss boh in order processing algorihms [,,3] and applicaions [4,5]. Order analysis has proven o be effecive o rack vibraion componens in relaion o variaion in running speeds. However, an arifac of he ransformaion from he

4 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 4 frequency domain o he order domain is ha all fixed frequency componens (i.e., resonance vibraion) will change orders as he running speed varies. Therefore, in applicaions where fixed frequency componens are imporan, frequency based analysis is appropriae. Whereas, when roaional relaed componens are imporan, order analysis is appropriae. In some applicaions boh are imporan and i is necessary o be able o separae he componens. A very demanding applicaion requiring accurae frequency idenificaion in he presence of roaional dependen componens is found in he healh monioring of urbine blades [6] in urbomachinery. The monioring mehod uses he urbine blade bending naural frequencies as a srucural healh diagnosic feaure. The blade naural frequencies are deeced by analysis of he roaing shaf orsional vibraion signaure. As a crack develops in a blade is naural frequency drops. Hence, he measuremen of he blade frequencies can be ulimaely used as a diagnosic meric o monior he srucural inegriy of he blade. The deecion of he blade frequencies in he orsional domain requires ha blade modes couple wih shaf orsional modes. Because of he physical scale of he blades o he shafing, he blade bending o shaf orsion coupling produces very small orsional vibraion levels. The relaively large ampliude vibraion caused by roaional relaed componens furher compounds he deecion of he blade frequencies. The high level roaion componens can poenially make he harvesing he very small signals associaed wih blade vibraion in he orsional domain very difficul, if no infeasible. Oher examples of problems of his ype may be found when i is necessary o separae fixed order from fixed frequency componens. For example, gear ooh healh

5 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 5 diagnosics in drive rains and gearboxes [7]. Again, he bending frequencies of he gear eeh are moniored and used as a healh diagnosic feaure and can difficul o disinguish from he srong order componens. Impac esing can be used o idenify he naural frequencies in roaing equipmen while operaing [8]. When he vibraion response is acquired from an impac, i will inherenly include he fixed frequency resonances and he roaing relaed componens. The fixed order componens need o be idenified and separaed o enable idenificaion of he naural frequencies. The raio of he fixed frequency componens o he order componens can be greaer han as 60 db making inerpreaion of resonances in he specra difficul. Hence, he order componens are viewed as a corruping phenomenon. An approach o remove he order componens from he specra, wihou affecing he remaining frequency domain informaion is presened in his work. The echnique uilizes a sequence of daa sampling and ransformaions, beween he ime, order and frequency domains as follows:. Vibraion daa is sampled using a consan ime basis ( ).. The imes corresponding o a consan angular basis ( θ) are deermined. 3. The signal ampliudes corresponding o he consan angular basis ( θ) are deermined via a compued order resampling inerpolaion. 4. The consan angle sampled daa block is ransformed via he FFT o he order domain. 5. The high ampliude order componens are now exacly bin cenered and can be removed from he specra, wihou affecing daa in adjacen bins.

6 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 6 6. The order domain specral daa block, sans order conen, is inverse ransformed back o an array sampled wih a consan angular basis ( θ). 7. The consan angle inerval array is inerpolaed back o a conac ime inerval array ( ). 8. An FFT and sandard specral esimaion procedures are hen applied o esimae he frequency based vibraion specra wih he high level orders removed. In he following secions he heoreical basis of he order removal mehod will firs be presened. The processing echnique will hen be demonsraed wih orsional vibraion daa acquired from a laboraory es sand. The paper concludes wih an assessmen of he mehod s capabiliies.. Double Resampling for Order Conen Removal Consider a simulaed analog daa signal and he synchronous keyphasor signal from a hypoheical piece of roaing equipmen. The simulaed signal could represen a number of possible physical measuremens including, an acceleromeer, microphone, load cell, srain gage, ec. and will use generic unis of volage in his discussion. The signal and keyphasor are shown in Fig. (). Fig. (A) shows a fixed frequency 0 Hz signal superimposed on a higher ampliude signal which is increasing in frequency. The keyphasor signal, Fig. (B), shows a sharp pulse occurring once every shaf revoluion. I is obvious from he keyphasor ha he shaf speed is acceleraing since he ime beween he keyphasor pulses is decreasing. The frequency specrum of he signal is shown in Fig. (). The 0 Hz fixed frequency componen is clearly idenifiable along wih broader low frequency conen around -3 Hz associae wih he changing shaf speed. As he shaf

7 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 7 speed increases he fixed roaion componen would evenually dominae he 0 Hz fixed frequency componen when he frequencies coincide. I is his possible corruping face of he roaional relaed componens ha arise in some applicaions and is he moivaion for his effor. To simplify he idenificaion of low level fixed frequency specral feaures in he presence of high level shaf speed harmonics is presened in his secion. The mehod uilizes a series of resampling inerpolaions along wih forward and inverse Fourier ransforms as follows: Sep : Fixed Time Sample Daa Acquisiion Firs, a sufficienly long discree daa array from an analog ransducer signal, x(), on he roaing equipmen operaing a he desired condiions is acquired. ( ) x( r ) x r = () where r is an ineger index and is he ime based sampling inerval. The sample rae should be appropriae for he desired analysis frequency range and ani-alias proeced by he use of filers. A once per revoluion key phasor signal synchronized o he acquired dynamic signal should also be recorded. The key-phasor achomeer signal should use a significanly higher sample rae as o provide a very accurae esimae of he shaf speed. Fig. (3) depics he analog signal wih he discree array superimposed when digiized wih a fixed sampling frequency. Noe, ha here are varying numbers of discreized daa poins during each respecive cycle of he shaf s roaion. This is direcly due o using a fixed frequency sampling rae applied o an angularly acceleraing shaf.

8 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 8 Sep : Fixed Angular Sample Time Idenificaion A compued order racking algorihm is now uilized o change he independen variable o he shaf roaion angle (θ) from ime (). The discree angular sampling process will produce a discree array wih an idenical number of discreized daa poins per shaf revoluion, regardless of he speed. The angular based sampling inerval is defined as: 360 o θ = () O where O represens an ineger number of angular incremens around he shaf. The angular discreizaion produces he array: ( θ ) = x( r θ) x r (3) where r is an index. The compued order inerpolaion mehod o obain he array in Eqn. (3) was firs proposed in [] and has proven o be effecive and robus. The firs sep is he deerminaion of he imes ha correspond o he fixed angular sampling inervals, (r θ). Calculaion of hese new sampling imes requires an accurae reference of he shaf roaion. Accuracy of he fixed angle sampling imes is highly dependen upon correcly deecing he edge of he achomeer signal. Afer acquiring accurae keyphasor imes, he desired imes corresponding o r θ can be deermined. The assumpion ha he reference shaf experiences a consan acceleraion is made [] in order o creae he relaionship beween shaf angular posiion and ime: () = b + b + b θ (4) 0

9 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 9 The coefficiens of Eqn. (4) can be deermined by solving hree independen equaions using he imes (,, 3 ) recorded for hree coniguous keyphasor pulses, as shown in Eqn. (5). = b b b 0 (5) The soluion of Eqn. (5) for he coefficiens yields []: ( ) ( ) [ ] ( )( )( ) b + = (6) ( )( )( ) b + = (7) ( )( )( ) b = (8) Once he coefficiens have been deermined, he desired sampling imes can be calculaed by solving quadraic Eqn. (4) for ime as a funcion of angular shaf posiion (r θ). Two soluions exis, bu only one yields realisic resuls (posiive values of ime) as shown in Eqn. (9): ( ) ( ) 0 r r b b b 4b b θ + = θ (9) where: r is an ineger and θ r = r θ. New polynomial coefficiens from Eqns. (6), (7) and (8) are deermined for every new keyphasor pulse. The desired sampling imes are calculaed over he range of 0.5 θ r.5 for each polynomial in order o avoid he overlap beween he consecuive

10 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 0 coefficien soluions. The only requiremen is ha he record be boh an ineger number of shaf revoluions and an ineger muliple of θ []. The approximaion used o creae Eqn. (4) enables his mehod o correcly rack a sysem wih consan acceleraion. More complex models of he shaf roaion are possible, bu he increased compuaion ime involved and lack of increased accuracy have no warraned heir applicaion. Errors due o a physical sysem no fully saisfying he approximaion are generally small, since he coefficiens are updaed for each achomeer pulse. Sep 3: Compued Fixed Angular Inerval Sample Inerpolaion Now ha he imes corresponding o he fixed angular inerval sample have been calculaed ((θ r )), he signal ampliude values a hese respecive imes mus be deermined. The discree ampliudes of original roaing equipmen signal, x(), are known only a values corresponding o r. Since he imes (θ r ) will in general be differen, an inerpolaion is necessary. Various inerpolaion schemes may be used o obain he fixed angle sampled signal ampliudes, x(θ r ), from he fixed ime sampled daa array x( r ) [9]. The rouine used in [], enails oversampling he signal by expanding he original ime vecor and applying a finie impulse response (FIR) filer. The FIR filer allows he original daa o pass hrough and inerpolaes beween he values so ha he mean squared errors are minimized. The mehod in [] uses an oversampling facor of along wih a 0-poin FIR filer designed o give a passband flaness of ±0.08% and a sopband rejecion greaer han 04 db []. To improve he compuaional speed he acual filer used in his algorihm is implemened as a lookup able sored in he memory. This leads o some round off error yielding a dynamic range closer o 80 db [0].

11 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling An alernaive inerpolaion was implemened in his work. Afer he desired fixed angle sampling imes have been deermined, (θ r ), he nex sep is o up-sample he daa in order o enable he use of a simple inerpolaion mehod. The discree ime sampled daa array, x( r ), was increased by an up-sampling facor of 3 producing a well defined waveform. A cubic spline inerpolaion was used o calculae he signal ampliudes corresponding o he desired consan angle sampling imes, (θ r ) from he up-sampled array. Cubic spline inerpolaion was shown o have he leas amoun of error when used in his ype of applicaion [,]. The process produced a new consan angle sampled daa array x(θ r ). The acual coding used he Malab inerp command []. A graphical depicion of he consan angle inerpolaion, x(θ r ), from he fixed ime sep sampled daa, x( r ), is shown in Fig. (4). The original analog signal is graphed as a solid line and a blown up ime segmen of he daa in Fig. (3) is shown. The consan ime sampled daa is marked by he aserisk (*) symbol. This daa is he discree array ha would be available from he analog-o-digial converer. The diamond ( ) marked poins represen he discree waveform ampliude when sampled wih respec o a consan angle of roaion ( θ). The ime values corresponding o he angular incremens are deermined by he procedure discussed in sep and he inerpolaion process described herein esimaes signal ampliudes. Sep 4: Order Domain Specral Esimaion A FFT algorihm is now applied o he consan angle sampled daa array, x(θ r ) by; N ( o ) = x ( θ ) r = 0 π n r i N X e (0) n r

12 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling The order specrum in unis of (v rms) can be calculaed from Eqn. (0) and is shown in Fig. (5). I clearly shows he high ampliude componen relaed o he firs order, or he shaf roaional running speed. Unlike he frequency based specrum in Fig. (), his componen does no smear as he running speed changes. Whereas he fixed frequency componen a 0 Hz in Fig. () becomes smeared as he shaf roaional speed increases. Sep 5: Order Removal The corruping high magniude order conen can now be removed from he signal. The key o he removal is he fac ha all he order componens are exacly bin cenered due o angular based sampling, as seen in Fig. (5). The index k ha corresponds o ineger orders in he specrum compued from he FFT in Eqn. (0) can be deermined from Eqn. (), producing Eqn (). ( θ O) k = j () where j is an ineger and j N θ O. The upper limi of j can be adjused o a lower value or o only roublesome orders if desired. I is a hese values in he order specrum ha i is desirable o remove heir effecs from he signal. Furhermore, i is imporan o reain fideliy in boh he real and imaginary values of he specrum since an inverse Fourier ransform will be applied laer. To accomplish he order conen removal several alernaive mehods were evaluaed. A direc zeroing of he order conen was unrealisic, since i did no compensae for he signal levels from oher sources. This was paricularly problemaic when an order and a resonance were in he same region. An alernaive mehod uses a linear inerpolaion of he bins complex values direcly surrounding each ineger order.

13 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 3 [ X( o )] + Re[ X( o )] Im[ X( o )] Im[ X( o )] Re X ~ (o ) k k + i k + k + k = + () where he index k is defined in Eqn. (). Noe, he symbol is used o denoe a quaniy in which he order conen has been removed. The mehod expressed in Eqn. () was found o work reasonably well, bu occasionally encounered difficulies. When processing signals from a sysem ha have well-defined phase shifs [3], such as a synchronous moor drive, he removal was ineffecive. An alernaive mehod based on he local complex minima surrounding an ineger order is shown in Eqn. (3): ( o ) min[ X( o ),X( o ) ] X ~ k k k + = (3) This mehod was found o be more robus wih acual experimenal daa under various operaional condiions [3]. Hence, i is used subsequenly in his work. Fig.(6) depics he order domain specrum wih he firs order conen removed. I is apparen ha he high ampliude firs order conen is absen while he remaining par of he specrum in unaffeced. Sep 6 Fixed Angular Inerval Array Wihou Order Conen An inverse FFT is now applied o he order domain array, X ~ ( o n ). N r = 0 π n r i N x~ ( θ ) = X ~ ( o ) e (4) r This produces a consan angle sampled array in which he ineger relaed order conen has been removed. n

14 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 4 Sep 7 Inerpolaion o Fixed Time Sample Array Wihou Order Conen A reverse operaion o Sep 3 is now performed whereby he fixed angular sampled array is inerpolaed o a fixed ime sample array. Recall ha he imes corresponding o he angular sample basis (θ r ) are equal o (θ r ). The signal x~ ( θ r ) is inerpolaed o obain he signal ampliudes ha occur on a consan ime base sample, r. This resuls in a fixed ime sampled array, x~ ( r ), wih he order conen removed. Similar o he coding in Sep 3, he implemenaion uses he Malab inerp command wih a cubic spline []. The resuls of Seps 6 and 7 when applied o he daa shown in Fig. (6) is shown in Fig. (7). The resuling ime waveform is a sine wave wihou he high ampliude order relaed componens seen in he original waveform in Fig. (). Sep 8 Frequency Based Specral Esimaion Wihou Order Conen The FFT algorihm is now applied o he discree ime sampled array, x~ ( r ), o compue a frequency based ransform. N r = 0 π n r i N X ~ ( f ) = x~ ( ) e (5) n Daa windows and ensemble averaging mehods are applied in conjuncion wih Eqn. (5) o improve he qualiy of he specral esimaes. The resuling specrum is shown in Fig.(8). Comparison o Fig. () shows ha he order conen is effecively removed leaving only he fixed frequency componens in he specrum. r 3. Capabiliies of Fixed Order Conen Removal Procedure The capabiliies of he order removal process will be examined using acual experimenal vibraion daa from a roaing equipmen es rig. Consider a commonly

15 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 5 used sysem o measure orsional vibraion of a roaing shafing sysem [4,5] as shown in Fig (9). Signal deecion involves four main aspecs; shaf encoding; ransducion; analog demodulaion; and daa discreizaion. The shaf encoding used a arge zebra srip wih alernaing black and whie sripes prined on paper or polyeser and glued o he shaf. Alernae mehods include he use of a iming gear or opical encoder. The ransducer was an infrared inensiy based reflecive fiber opic sensor. An analog demodulaor was used o produce a volage signal proporional o he shaf orsional vibraion from he carrier signal generaed by passage of zebra ape on he roaing shaf. The volage from he demodulaor is hen discreized for furher processing, such as specral analysis. The implemenaion and uilizaion of his hardware configuraion for orsional vibraion measuremen was previously presened in [5]. Fig. (0) shows a picure of a laboraory es rig developed o sudy orsional vibraion of a shaf wih a simulaed bladed disk assembly. The shaf is suspended by oil impregnaed flanged brass bearings, and is driven by a /7 h hp moor, wih a maximum speed of 0,000 rpm, using a DC power supply. A he opposie end of shaf from he moor was a simulaed bladed disk assembly. Eigh sainless seel hreaded rods are used o represen he blades. Two lock nus are posiioned a he end of each rod o produce a movable mass. The movable nus allow changes o be made in each rod s naural frequencies by changing heir respecive radial locaion. The insrumenaion followed he signal deecion schemaic diagram in Fig (9). The encoding zebra ape consised of 60 black and whie lines mouned ono an aluminum wheel nex o he drive moor. A single black sripe on he side of he

16 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 6 aluminum wheel creaed a one per revoluion keyphasor. Two fiber opic probes were posiioned o sense he zebra ape and he keyphasor as shown in Fig. (). Torsional vibraion daa was colleced while he sysem roaed a a consan speed of approximaely 3400 rpm. The daa was acquired wih a fixed sampling frequency and processed wih an Agilen (HP) E433B card in a VXI mainframe. The specrum was calculaed via a FFT algorihm using 30 ensemble averages and is shown in Fig. (). The specrum shows high ampliude conen a ineger muliples of he nominal 57 Hz running speed, or a fixed orders. The coupled blade bending-orsional mode occurs around 0 Hz and is barely apparen in he specrum. The shaf orsional mode is a approximaely 60 Hz and is almos unnoiceable. The high ampliude order conen obscures he low level orsional naural frequency signals. The skirs on he fixed order conen is caused by leakage due o he nonperiodic capure of he signal. The corruping high ampliude order conen resuls from he zebra ape encoding approach. Ideally he zebra ape has idenical secions of alernaing black and whie sripes. However several pragmaic issues relaed o prining and insallaion inroduce errors caused by:. Priner resoluion. The priner may be incapable of exacly reproducing he desired zebra srip widh due o he dpi (dos-per-inch) resoluion. This inroduces a bias error whereby srips are produced wih varying numbers of prined raser lines.. End effecs. When insalling he ape around he shaf circumference here may be an overlap or void when aemping o mach he firs and las sripe

17 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 7 on he shaf. This produces a disincly differen response ha he evenly spaced sripes and occurs a a once per revoluion cycle. 3. Tape insallaion errors. In heory he ape sripes should be insalled parallel o shaf axis. Pracically his is difficul o accomplish. Also, he ape may srech when placing i around he shaf. These errors produce a sysemaic paern for every revoluion of he shaf ha deviaes from he ideal regularly spaced condiion. These errors manifes hemselves as specral conen frequencies ha are ineger muliples of he shaf running speed. The daa in Fig. () was acquired wih he shaf roaing a a consan speed. The corruping naure of he order componen would furher exacerbae he idenificaion of he fixed frequency componens if he running speed were o vary. As he running speed changes he specral frequencies would change proporionaely. Hence, even broader high ampliude frequency regions would resul having he poenial o dominae and compleely obscure he idenificaion of a naural frequency. This example was seleced because i clearly demonsraes he corruping naure ha fixed order relaed componens can have on fixed frequency componens. Furhermore, i represens a raher severe se of condiions and hence serves an excellen case o illusrae he proposed mehod o eliminae he order conen from he specrum. Daa is again colleced from he es sand, bu his ime he processing discussed in Secion are applied. The discree ime daa array and keyphasor ime samps were acquired by he Agilen VXI sysem. The daa was processed using a rouine programmed in Malab. For he sake of breviy, only criical graphics associaed wih he performance of he algorihm o he order removal will be presened. Fig. (3) shows one record afer

18 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 8 Seps -4 have been applied. For visual clariy, a symbol marks each bin individually. The high ampliude order conen is a exac muliples of he shaf running speed. Furhermore, is bin-cenered naure is visible by he fac ha only one daa poin possesses he high ampliude wih he adjacen bins being considerably lower ampliude. This characer is in conras o he skir behavior of he frequency specrum in Fig. () and is he key o he abiliy o remove i. The resuls afer he order removal process is described in Sep 5 is applied o he same daa sample is shown in Fig. (4). The corruping order conen is now gone. Nex, seps 6-8 are hen compleed. The process is repeaed o obain a oal of 30 ensemble frequency records ha are averaged ogeher. The final specrum, wih he order conen removed, is shown in Fig. (5). Noe, ha he orsional naural frequencies of he shafing sysem ha were almos non apparen in he original specrum in Fig. () can be readily idenified. Furhermore, comparing he resuls in Fig. () o Fig. (5) demonsraes ha a dramaic improvemen in he effecive dynamic range is realized. 4. Concluding Remarks This work has discussed an algorihm by which order relaed specral conen can be separaed and removed from fixed frequency conen on signals acquired on roaing equipmen. The processing enhancemen is produced by a sequence of digial resampling o faciliae ransformaion beween he ime, order and frequency domains. Processing in he respecive domains allows he masking effec of he high level order componens o be eliminaed. The required inerpolaions and Fourier ransforms have been coded in Malab for evaluaion of he mehod. The algorihm has been shown no only o be

19 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 9 effecive in removing he fixed order conen bu also improves he effecive dynamic range by removing he order relaed skirs in he frequency domain. Laboraory ess clearly demonsraed he proposed mehod s abiliy o improve orsional vibraion signals allowing he deecion of low ampliude signals. The approach has been applied in several oher demanding siuaions wih excellen resuls. A laboraory es was performed where a fixed frequency orsional mode coincided wih an ineger muliple of he running speed [3]. Iniially he low ampliude fixed frequency componen was compleely masked by he order componen. Afer applicaion of he order removal algorihm, presenly herein, he orsional componen was readily idenifiable. A field es was also performed on a hree-megawa hydroelecric power uni o demonsrae he poenial o apply his daa processing algorihms in an indusrial environmen [3, 5]. This es showed ha applicaion of he proposed mehod produced a more useful orsional vibraion signaure han he originally acquired daa. Furhermore, i demonsrae is abiliy o handle issues ha ofen arise in he field ha may no be apparen in he more conrolled laboraory seing. More invesigaion is necessary o more fully characerize and quanify he algorihm s capabiliies. The processing relies on a sequence of inerpolaions, which has poenial sources of error. Alernaive inerpolaion schemes should be ried. Because mos inerpolaion mehods are based upon polynomials, which require a well-defined signal o be accurae, his mehod was iniially used o faciliae boh resampling seps, ime-o-order-resampling and order-o-ime resampling. If a Fourier series inerpolaion mehod could be applied o boh seps hen he required oversampling could be reduced while sill yielding accurae resuls [6]. This inerpolaion scheme could, herefore

20 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 0 reduce he compuaional and memory demands. A horough error analysis of he various inerpolaion schemes is needed o evaluae how hese inheren errors will manifes hemselves in he respecive specra. Experience wih he algorihm o dae, has yielded excellen resuls in several demanding orsional vibraion laboraory and field measuremen siuaions. This bodes well for he algorihm s poenial in applicaions where i is desired o separae fixed frequency from order componens.

21 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling References. Poer, R., and Gribler, M. 989 SAE Paper 893. Compued Order Tracking Obsolees Older Mehods.. Fyfe, K. R., and Munck, E. D. S., 997 Mechanical Sysems and Signal Processing, (). Analysis of Compued Order Tracking. 3. Vold, H. and Leuridan, J., 993 SAE Paper High Resoluion Order Tracking a Exreme Slew Raes Using Kalman Tracking Filer. 4. Vold, H., Herlufsen, H., Marins M., and Corwin-Renner, D., 997 Sound and Vibraion, 3(5). Muli Axle Order Tracking wih Vold-Kalman Tracking Filer. 5. Agilen Technologies, Palo Alo, CA, USA. Effecive Machinery Measuremens Using Dynamic Signal Analyzers, Applicaion Noe AN Maynard, K. P., and Trehewey, M. W., 999 Proceedings of he 53 rd Meeing of he Sociey for Machinery Failure Prevenion Technology, Virginia Beach, Virginia, USA April 9-. On The Feasibiliy of Blade Crack Deecion Through Torsional Vibraion Measuremens. 7. McFadden, P.D., 987 Mechanical Sysems and Signal Processing (). Examinaion of a Technique for he Early Deecion of Failure in Gears by Signal Processing of he Time Domain Average of he Meshing Vibraion. 8. Marscher, W.D., 999 Proceedings of he 7 h Inernaional Modal Analysis Conference, Sociey for Experimenal Mechanics. The Deerminaion of Roor Criical Speeds while Machinery Remains Operaing hrough Use of Impac Tesing. 9. IEEE Programs for Digial Signal Processing, 979 IEEE press, New York, John Wiley & Sons. 0. McDonald, D., and Gribler, M, 99 Proceedings of he 9 h Inernaional Modal Analysis Conference, Sociey for Experimenal Mechanics. Digial Resampling: A Viable Alernaive for Order Domain Measuremens of Roaing Machinery.. Munck, E.D.S., 994 MS Thesis, Universiy of Albera, Canada. Compued Order Tracking Applied o Vibraion Analysis of Roaing Machinery.. Malab, The MahWorks, Inc. Naick, MA, USA. 3. Groover, C.L., 000 MS Thesis, Deparmen of Mechanical and Nuclear Engineering, Penn Sae Universiy. Signal Componen Removal Applied o he Order Conen in Roaing Machinery. 4. Vance, J. M., 988 Roordyamics of Turbomachinery, John Wiley & Sons, New York. 5. Maynard, K.P, Trehewey, M.W. and Groover, C. L., rd Meeing of he Sociey for Machinery Failure Prevenion Technology, Virginia Beach, VA, USA. Applicaion of Torsional Vibraion Measuremen o Shaf Crack Monioring in Power Plans. 6. Schanze, T., 995 IEEE Transacions on Signal Processing, 43(6). Sinc Inerpolaion of Discree Periodic Signals.

22 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling Lis of Symbols FFT Fas Fourier Transform f frequency Im Imaginary par of a complex values quaniy i = k index correspond o an ineger order min selecs he minimum of complex valued argumens on a norm basis N FFT block size n array index o order O number of angular sampling incremens per shaf revoluion Re real par of a complex valued quaniy r array index coninuous ime variable r discree ime array in ineger muliples of (θ r ) ime array corresponding he fixed angular sample a θ r X(o n ) order domain discree Fourier ransform X ~ ( o n ) order domain discree Fourier ransform wih ineger order conen removed X ~ ( f n ) frequency domain discree Fourier ransform wih ineger order conen removed x() coninuous funcion of ime, or an analog signal x( r ) discree array sampled wih a fixed ime incremenal basis, x~ ( r ) discree array sampled wih a fixed ime incremenal basis,, wih ineger order conen removed x(θ r ) discree array sampled wih a fixed angular incremenal basis, θ x~ ( θ r ) discree array sampled wih a fixed angular incremenal basis, θ, wih ineger order conen removed ime based sampling inerval θ angle based sampling inerval

23 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 3 a. Signal (v) 0 - b Time.5 Keyphasor (v) Time Figure. Signal from a simulaed piece of roaing equipmen. a. Analog dynamic signal b. Once per revoluion keyphasor

24 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling v rms Frequency (Hz) Figure. Specrum of dynamic signal in Figure A sampled wih a consan ime inerval ( ).

25 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling vols Time Figure 3. Analog signal from a simulaed piece of roaing equipmen and he corresponding discree array sampled wih a consan ime inerval ( ).

26 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 6. θ θ θ. θ analog signal vols Time Figure 4. Analog signal from a simulaed piece of roaing equipmen wih discree arrays; * consan ime sampling inerval ( ); consan angle sampling inerval ( θ).

27 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling v rms Orders Figure 5. Order specrum of signal from a simulaed roaing equipmen in Figure wih consan angle inerval samples ( θ) via compued order sampling mehod.

28 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling v rms Orders Figure 6. Order specrum of signal from a simulaed roaing equipmen in Figure wih order conen removed.

29 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 9 Figure 7. Double resampled ime waveform in Figure wih order conen removed.

30 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 30 Figure 8. Double re-sampled specrum for ime waveform in Figure wih order conen removed.

31 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 3 Phoosensiive opic ransducer Fiber opic cable Angsrom resolver A-D converer Equally spaced black and whie sripe code 9. Analog demodulaor Shaf experiencing orsional vibraion Figure 9. Schemaic of orsional vibraion measuremen sysem on a roaing shaf.

32 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 3 Figure 0. Torsional vibraion es sand.

33 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling 33 Figure. Close up of orsional vibraion zebra ape insallaion and fiber opic ransducer placemen.

34 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling db Blade Bending Mode Region Shaf Torsional Mode Region Frequency (Hz) Figure. Torsional vibraion specrum from experimenal es sand in Fig. (0) wih a running speed of 340 rpm (57 Hz), wih 30 ensemble averages.

35 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling db Orders (Muliples of Shaf Speed) Figure 3. Order domain specrum for a single record.

36 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling db Orders (Muliples of Shaf Speed) Figure 4. Order domain specrum wih order conen removed for one ensemble sample.

37 Groover, Trehewey, Maynard & Lebold - Order Removal by Double Resampling Blade Bending Naural Frequencies Firs Shaf Torsional Naural Frequency db Frequency (Hz) Figure 5. Torsional vibraion specrum in Fig. () afer he order removal processing has been applied wih 30 ensemble averages.