LASER DOPPLER VIBROMETER AND IMPULSE SIGNAL PHASE DEMODULATION IN ROTATION UNIFORMITY MEASUREMENTS

Transcription

1 15 th Iteratioal Cogress o Soud ad Vibratio 6-1 July 28, Daejeo, Korea LASER DOPPLER VIBROMETER AND IMPULSE SIGNAL PHASE DEMODULATION IN ROTATION UNIFORMITY MEASUREMENTS Jiri Tuma 1 1 Faculty of Mechaical Egieerig, VSB - Techical Uiversity of Ostrava, 17. listopadu 15, Ostrava, CZ 78 33, Czech Republic jiri.tuma@vsb.cz Abstract The paper deals with two methods for measuremets of agular vibratio durig rotatio, which is a source of machie vibratio ad cosequetly emittig oise. Characteristics of the agular vibratio allow studyig oise ad vibratio problems at its very source. The first metioed method for agular vibratio measuremets is based o usig a two-beam laser Doppler vibrometer ad the secod oe o employig the phase demodulatio of impulse sigals produced by icremetal rotary ecoders (IRC) or by the teeth of a toothed wheel. The phase demodulatio requires usig some post-processig to get a result i the form of the phase variatios. The laser output sigal is proportioal to the istataeous agular velocity while the phase sigal has to be coverted i agular frequecy by the phase differetiatio with respect to time. To obtai the agular acceleratio, which is proportioal to the drivig momet or brakig torque, the velocity sigal has to be coverted to the acceleratio sigal usig the agular frequecy differetiatio with respect to time agai. The accuracy of the phase demodulatio is depedig o the errors i the IRC impulse gratig period. Ecoder error aalysis is a importat part of the paper. 1. INTRODUCTION The mai sources of rotatig machie vibratios are rotor ubalace, misaligmet of shafts ad o-uiform drivig torque. All these excitatios result i a dyamic force actig the bearig supports. Bearig vibratio excites vibratio of the machie housig, which icreases the oise level. This paper is focused o the shaft agular vibratio as a cosequece of the o-uiformity of a drivig torque. Rotatioal speed is measured i terms of the umber of revolutios per miute (RPM) while the agular vibratio is measured i terms of the agle, agular velocity or agular acceleratio. The uiform rotatioal speed at the costat RPM correspods to a icrease i the shaft rotatio agle proportioally to the elapsed time. The agle time history, havig the form of the sum of a term that is depedig liearly o time ad a term that is radomly or regularly varyig i time aroud zero, results from the agular vibratio durig rotatio. The agular velocity is obtaied as the first derivative of the agle while the agular acceleratio is computed as the secod derivative of the agle. There are may possible approaches to measurig agular vibratio durig rotatio 1

2 ICSV July 28 Daejeo Korea Tagetially mouted accelerometers Laser torsioal vibratio meters based o the Doppler effect Icremetal rotary ecoders (several hudreds of pulses per revolutio). I practice, measuremets based o the use of ecoders domiate. Istataeous agular velocity is proportioal to the reciprocal value of the time iterval, which elapses betwee cosecutive impulses. The measuremet methods for the legth of the time iterval are as follows: Sample umber & Iterpolatio High frequecy oscillator (1 MHz) & Impulse couter Phase demodulatio. The simplest method for evaluatio of the istataeous rotatioal speed is the reciprocal value of the time iterval betwee two cosecutive pulses. If the impulse sigal is sampled the the time iterval betwee the adjacet impulses is determied by the cout of samples ad iterpolatio of some values, which results i 5 times more accuracy of the time iterval legth tha is idicated by the actual samplig iterval. The accuracy is satisfyig for the RPM measuremet based o oly oe pulse per shaft rotatio. This method is ot suitable if a large umber of pulses per revolutio is geerated, which results i a few samples i betwee successive impulses ad the time iterval legth is impossible to estimate at a satisfyig accuracy. If the strigs of ecoder impulses, as a aalogue sigal, cotrols a gate for the high frequecy clock sigal (up to 1 GHz) that is a iput of a impulse couter, the this method works properly. This priciple is implemeted i the sigal aalysers produced by Rotec. The primary output of these aalysers is agular velocity. This paper is dealig with the agular vibratio measuremets based o impulse sigal phase demodulatio usig the theory of complex aalytic sigals, a imagiary part of which is the Hilbert trasform of its real part. The agular vibratio measuremet usig the phase demodulatio is compared with this measuremet usig the laser Doppler vibrometer. 2. PRINCIPLE OF THE PHASE DEMODULATION The phase demodulatio carried out the Bűel & Kjær sigal aalyser of the 355 type, which has bee produced sice It seems that this sigal processig method is margialized ow. I view of the paper s author, there are two mai approaches to the phase demodulatio, amely the aalytic sigal, which is created usig the Hilbert trasform [1], ad quadrature mixig. This paper is focused o the aalytic sigals. The sigal cotaiig iformatio o the istataeous agular velocity is a impulse sigal produced by a ecoder attached to the free ed of the shaft. The ecoders produce a trai of impulses rather the a siusoid. As the spectrum of the impulse sigal cosists of several harmoics of the basic impulse frequecy, the first step i the phase demodulatio procedure is to separate the frequecy bad cotaiig a carrier compoet icludig sidebad compoets by usig a bad-pass filter. The secod step i the process of demodulatio is creatig a aalytic sigal of the phase-modulated harmoic sigal. The aalytic sigal is obtaied from the origial sigal by extedig this real sigal by the imagiary part, which is the Hilbert trasform of the real part. 2

3 ICSV July 28 Daejeo Korea There are two methods how to evaluate the Hilbert trasform for sampled sigals. The first oe is based o usig the Fourier trasform [1] ad the secod oe employs digital filters approximatig the Hilbert trasform as a o-causal digital filter [2,3]. The Fourier trasform (FT) of real sigals is composed from complex cojugate pairs of compoets with a positive ad egative frequecy (greater tha the Nyquist frequecy). The Hilbert trasform i the frequecy domai ca be expressed as a frequecy trasfer fuctio shiftig the phase of each metioed compoet by π 2 or + π 2, which meas multiplyig them by either j or + j. The frequecy rage of the sampled real sigals is limited to half the samplig frequecy, f S 2 (Nyquist frequecy). Therefore, the argumet of the trasfer π, +π fuctio is limited to the iterval ( ) j, + π > ω G HT ( exp( jω) ) =. (1) j, π < ω < Let x be a sample sequece, which is a real part of the aalytic sigal. The first step of the aalytic sigal imagiary part computatio cosists i shiftig the phase of the FT compoets X k, which are correspodig to the sample sequece x, to become the FT compoets Y k, which are correspodig to a sample sequece y. The iverse Fourier trasform of the FT compoets Y k results i the Hilbert trasform of the sample sequece x, which is correspodig to the aalytic sigal imagiary part y. The aalytic sigal is a sequece of the complex quatities x + j y, where the idex =,1,..., N 1 desigates the order of the m samples i the iput vector for the FT ( N = 2 ). The digital filter performig the Hilbert trasform is called the Hilbert trasformer. The ideal impulse respose of this digital filter is as follows +π 1, = 2k g HT ( ) = ( ( ω) ) ( ω ) ω = π GHT exp j exp j d. (2) 2 2 π, = 2k + 1 π The Hilbert trasformer is a low pass, high-pass, or bad-pass filter of the FIR (Fiite Impulse Respose) or IIR (Ifiite Impulse Respose) filter types. The order of the FIR filter is limited ad turs to the causal filter while the filter liear phase is preserved. The phase-modulated harmoic sigal x is put ito the Hilbert trasformer to obtai the Hilbert trasform y givig the imagiary part of the aalytic sigal x + j y. The group delay of the Hilbert trasformer, which is created usig the FIR filter, is equal to half the digital filter order; therefore the real part of the aalytic sigal x is delayed by the correspodig umber of samples. The agle ϕ = ata( y x ) for the complex values x + j y rages from π to + π ad cotais jumps at π or at + π. The true phase ϕ of the aalytical sigal as the time fuctio has to be uwrapped. The uwrappig algorithm is based o the fact that the absolute value of the phase differece Δϕ = ϕ ϕ 1 betwee two cosecutive samples of uwrapped agle sequece is less tha + π Δϕ < π ϕ + 2 π ϕ, Δϕ > +π ϕ 2π ϕ. (3) The uwrapped phase chage per oe impulse rotatio is equal to 2 π radias. The phase 3

4 ICSV July 28 Daejeo Korea chage, which is correspodig to oe complete ecoder revolutio, is equal to the product of 2 π ad the umber K of the impulses per ecoder revolutio. To establish the depedece of the shaft rotatio agle o the omial revolutio assumig the steady-state rotatio, the phase ormalizatio has to be computed is such a way as follows ϕ K ϕ, =,1,..., N 1. (4) The ormalised phase as a fuctio of the omial rotatio is composed from a term, which is idetical with the omial rotatio, ad the phase variatio determiig the phase-modulatio sigal. After removig the liear term, the phase-modulatio sigal is obtaied. The additioal oise i the measured sigals ca be reduced by sychroized averagig (filtratio or sigal ehacemet). There are two possible ways how to do it. It is possible to average either the impulse sigal or the phase-modulatio sigal. Both the solutios require to resample a impulse sigal accordig to the rotatioal frequecy i such a way that records, correspodig to the complete revolutio, cotaiig the same umber of samples (usually power of two). After resamplig sigals the time uit (secod) is replaced by the agle uit (omial revolutio) ad the frequecy i Hz is replaced by the frequecy i order, which is a multiple of the base frequecy, usually the shaft rotatioal frequecy. The omial rotatio, which is equal to uity, correspods to oe complete revolutio. The effect of the averagig order will be demostrated i the example. 3. INCREMENTAL ROTARY ENCODER ACCURACY Usig the Fourier trasform to compute the phase-modulatio sigal will be demostrated by ecoder accuracy testig. The ecoders uder test, show i Figure 1, are of HEIDENHAIN origi, the ERN 46-5 type. To compute error i pulse distributio agaist the agle of rotatio, both the ecoders were mouted o a shaft what esured their same rotatioal speed (see Figure 2). Accuracy was assessed at the rotatioal speed of 14 RPM. The pulse strig geerated by the ecoders was sampled at the frequecy of Hz. E 2 E 1 Figure 1. Heidehai ecoders of the ERN 46-5 type (5 pulses per revolutio) Figure 2. Arragemet of ecoders to be tested RMS db/ref Order [-] Resamplig (Time : Real (Ecoder 1)) Resamplig (Time : Real (Ecoder 2)) Figure 3. Autospectrum of phase modulated sigal geerated by the ecoders E1 ad E2 4

5 ICSV July 28 Daejeo Korea As the ruig was ot perfectly uiform, both the impulse sigals were uder the ifluece of the same phase modulatio. The ecoder speed variatio results i the phase modulatio of the impulse sigal base frequecy. As oticed above the phase-modulated sigal cotais sidebad compoets aroud the carryig compoet. The frequecy of the carryig compoet is equal to 5 orders as it is show i Figure 3. Usig the method described above, the phase differece betwee modulatio sigals of both the ecoders gives the error i the distributio of impulses. The idividual phase differeces versus the omial revolutio are show i Figure 4. The sychroously averaged phase differece of the 16 metioed phase differeces is show as the blue lie i Figure 5. The phase differece computed for the sychroous average of 16 resampled impulse sigal records, which are correspodig to 16 successive revolutios, is show i Figure 5 as the red lie.,3,3,2,2,1,1 deg, deg, -,1 -,1 -,2 -,2 -,3 -,3,,5 1,,,5 1, Nomial revolutio [-] Nomial revolutio [-] Figure 4. Idividual phase differece for 16 complete revolutios vs. omial revolutio Figure 5. Average of 16 phase differeces (blue lie) ad phase differece for average of the impulse sigal (red lie) 1E-1 Phase differece 1E-1 Phase differece 1E-2 1E-2 RMS deg 1E-3 1E-4 1E-5 RMS deg 1E-3 1E-4 1E-5 1E E Averaged spectrum Order [-] Order [-] Averaged time sigal Figure 6. Averaged spectrum of phase differeces (blue lie) ad spectrum of averaged phase differeces (red lie) Figure 7. Spectrum of the averaged impulse sigals The order spectra of the phase differeces ca be computed either for the sigals show i 5

6 ICSV July 28 Daejeo Korea Figure 4 or for the sigals show i Figure 5. It meas that sychroous averagig is carried out i either the frequecy or agle domai. The averaged spectrum i the frequecy domai for the 16 idividual phase differeces is show i Figure 6 (blue lie). I the same figure the spectrum for the averaged phase differeces i the agle domai is show as well. The spectrum of the phase differeces based o the averaged impulse sigals is show i Figure 7. The order spectra i Figure 6 ad 7 have the frequecy axis i orders. The quatity order determies a part of a circle related to the error level. For istace, the order 2 meas the circle arc of the legth, which is the twetieth part of the circle circumferece. The correspodig spectrum value determies the RMS of the impulse-pair distace error. For icremetal rotary ecoders with lie couts up to 5, the maximum directioal deviatio at 2 C ambiet temperature ad slow speed (scaig frequecy betwee 1 khz ad 2 khz), lies withi ± 1/2 gratig period. The ecoder producig 5 impulses works with the maximum error i impulses distributio of.36 degree. The distace error for a pair of impulses at the twetieth part of the circle circumferece is a value less tha.1 degree. It is evidet that the error level i Figure 6 i relatio to the error level i Figure 7 is a little bit less. It ca be recommeded to prefer the computatio of the phase-modulatio sigal for each rotatio to the computatio of the average impulse sigal before the phase demodulatio. 3. COMPARISON OF THE LASER TORSIONAL VIBRATION METER AND ENCODER INCREMENTAL ROTARY ENCODER MEASUREMENT The laser torsioal vibratio meter of a type 2523 of the BK origi cosists of a meter ad a dual-beam laser trasducer. Without cotactig a rotatig compoet, this sesor primary determies the istataeous chages i agular velocity from the frequecy differece of the retroreflected, Doppler-shifted beams. Measuremets are idepedet of the target cross-sectio. The heart of the system is a low power (less tha 1.5 mw) Ga-Al- As laser diode producig 78 m ivisible light. The laser beam is split ito two equal-itesity parallel beams separated by a distace of 1 mm. The agular velocity measuremet rage is ragig from.3 to 7 degrees per secod (RMS) ad the frequecy rage from.3 to 1 Hz. The electric had drill was employed as a source of the agular vibratio durig rotatio. The measuremets were realized while this tool was ruig at a idle speed of approximately 4 RPM. The ecoder of the type, which is producig 124 impulses per revolutio, was attached to the ed of the spidle while the other ed was fixed by the drill chuck. The pla of the laser beams was perpedicular to the spidle axis ad the beams were focused o the cylider of the 15 mm diameter as it is show i Figure 8. V ,,3,5,8 Time [s] Figure 8. Arragemet of ecoders to be tested Figure 9. Impulse sigal 6

7 ICSV July 28 Daejeo Korea The laser ad ecoder output sigals are sampled at the rate of Hz. It meas about te samples per a impulse as it is show i Figure 9. The impulse sigal cotais may harmoics of the impulse base frequecy, which is equal to 6656 Hz. As the phase demodulatio works oly for the phase-modulated harmoic sigal, the impulse sigal spectrum is limited to the frequecy bad 6656 ± 3 Hz. This way of the passbad filtratio i the frequecy domai, usig the poit FFT ad IFFT, results i the purely phase-modulated sigal. The impulse sigal is trasformed ito the aalytical sigal usig the 16-order FIR filter actig as the Hilbert trasformer. The real ad imagiary part of the aalytic sigal gives the sigal phase limited to a iterval from π to + π radias. After uwrappig the phase ad the phase ormalisatio to the spidle rotatio (divisio by the umber of the ecoder impulses per revolutio), the rotatio agle as a fuctio of time is obtaied. To compare the laser output sigal, measurig the agular vibratio i uit of degrees per secod, with the result of the ecoder sigal phase demodulatio, the phase, as the rotatio agle, has to be differetiated with respect to the time to obtai the same uit of the phase demodulatio output sigal as the laser output sigal. Takig ito accout that the spectrum of the laser output sigal is cut dow to a frequecy of 1 Hz, the 147-order FIR filter actig as a differetiator is combied with the low pass filter. The impulse ad frequecy respose is show i Figure 1 ad 11. Filter coefficiets ,,1,2 Time [s] Magitude i db Frequecy [Hz] Figure 1 Differetiator impulse respose Figure 11 Differetiator frequecy respose Two FIR filters, which are used i successive steps, isert i the sigal processig the delay, which is give by half the order sum of both the FIR filters. For the metioed sample frequecy, this delay of the agular vibratio sigal i relatio to the laser output sigal is equal to ms. To reduce the sigal backgroud oise, the sychroous averagig of the records, correspodig to the spidle complete revolutio, is computed. The averaged agular velocity i degrees per secod is show i Figure 12 as a fuctio of a omial revolutio, which is assumig the uiform rotatio of the spidle. As it is evidet, the laser sigal is delayed i relatio to the ecoder sigal by ms. It ca be explaied by the iteral filters employed i the laser measurig chai. It ca be cocluded that the laser total delay is about ms. The order spectrum, which is show i Figure 13, cofirms the almost idetical results of measuremet usig the differet measuremet methods. All the described algorithms are implemeted i Sigal Aalyzer, the idoor software for supportig sigal processig educatio, traiig ad research. 7

8 ICSV July 28 Daejeo Korea deg/s ,,2,4,6,8 Nomial revolutio [-] Laser Ecoder Figure 12. Laser sigal after differetiatio ad ecoder sigals after phase demodulatio vs. omial revolutio RMS db/ref 1 deg/s Order [-] Ecoder Laser Figure 13. Frequecy spectrum of agular vibratio i db scale (ref. 1 deg/s) CONCLUSIONS The paper describes two methods for agular vibratio measuremet. The first oe is based o the phase demodulatio of impulse sigals ad the secod oe employs a two-beam Doppler laser vibrometer. The metioed measuremet method, which is based o usig the phase demodulatio, was demostrated o the icremetal rotary ecoder accuracy testig ad had drill rotatio uiformity. The secod preseted measuremet demostrates employig the laser vibrometer for agular vibratio measuremet ad compares both the described measuremet methods at the same time. The compariso of the sychroous averagig i the agle ad frequecy domai results i recommedatio accordig that it is better to average the modulatio sigal to average the iput impulse sigal. The matchig of the ecoder ad laser measuremets allows determiatio of the laser delay. REFERENCES [1] J. Tuma, Phase Demodulatio of Impulse Sigals i Machie Shaft Agular Vibratio Measuremets. Proceedigs of Teth iteratioal cogress o soud ad vibratio (ICSV1). 7-1 July, 23, Stockholm, Swede, pp [2] J. Tuma, Simple gear set trasmissio error measuremets. Thirteeth iteratioal cogress o soud ad vibratio (ICSV13), 2-6 July, 26, Viea, Austria, 8 p., ISBN [3] J. Tuma, Dyamic trasmissio error measuremet. Egieerig Mechaics, Vol 13, No 2, pp , ISSN This research has bee doe at the VSB-Techical Uiversity of Ostrava as a part of the project No. 11/7/1345 ad has bee supported by the Czech Grat Agecy. 8