PRINCIPLE AND SOFTWARE TOOLS FOR MACHINE-SHAFT ANGULAR-VIBRATION MEASUREMENTS

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1 Colloquium DYNAMICS OF MACHINES 5 Prague, February 8 9, 5 C C E M S PRINCIPLE AND SOFTWARE TOOLS FOR MACHINE-SHAFT ANGULAR-VIBRATION MEASUREMENTS Jiří Tůma Abstract: (I Eglish) The paper deals with the priciple ad software tools for machie-shaft agular-vibratio measuremets i terms of rotatio agle, agular velocity or agular acceleratio. Icremetal rotary ecoders producig a strig of impulses or aother impulse geerator, which frequecy is proportioal to the istataeous rotatioal frequecy, is the source of iformatio about agular vibratios. The basic approach to the sigal processig employs the Hilbert trasform usig the Fourier trasform ad the digital filters. The theory is demostrated o gear trasmissio-error measuremets. Key words: mai Eglish key words (accordig to their importace) 1. INTRODUCTION The mai sources of rotatig machie vibratios are ubalace of rotors, misaligmet of shafts ad o-uiform drivig torque. All these excitatios result i dyamic force affectig bearig supports. Bearig vibratio excites vibratio of the machie housig, which icrease the oise level. This paper is focused at the shaft agular vibratio as a cosequece of the o-uiformity of 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 growig up the shaft 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 agular vibratio durig rotatio. The agular velocity is obtaied as the first derivative of the agle while the agular acceleratio is evaluated as the secod derivative of the agle. There are may possible approaches to measurig agular vibratio durig rotatio o Tagetially mouted accelerometers o Laser torsioal vibratio meter based o the Doppler effect o 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 is elapsed betwee cosecutive impulses. The measuremet methods for the legth of the time iterval measuremets are as follows: Prof. Ig. Jiří Tůma, CSc., Faculty of Mechaical Egieerig, VSB Techical Uiversity of Ostrava, 17. listopadu, CZ 78 33, Czech Republic, jiri.tuma@vsb.cz

2 o Sample umber & Iterpolatio o High frequecy oscillator (1 MHz) & Impulse couter o 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 iterpolatio of some values 5 times more accurately tha 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 the large umber of pulses per revolutio is geerated, which results i a few samples betwee impulses ad the time iterval legth is impossible to estimate at satisfyig accuracy. If the strig of ecoder impulses as a aalogue sigal cotrols a gate for the high frequecy clock sigal (1 MHz or 1 GHz) that is a iput of a impulse couter the this method works properly. This priciple is implemeted i the sigal aalyzers produced by Rotex. The primary output of these aalyzers is agular velocity. This paper is dealig with agular vibratio measuremets based o the impulse sigal phase demodulatio usig the Hilbert trasform.. HILBERT TRANSFORM The real harmoic sigal X ca be modeled i the complex plae as a sum of two rotatig vectors, X P ad X N, at the same agular frequecy but opposite directio. The vector X is parallel to the real axis if both these vectors have to be complex cojugate; it meas that they are of the same magitude ad iitial phase, see Figure 1. The aalytical sigal, correspodig to the vector X, is defied as the doubled vector X P. To evaluate the aalytic sigal, the vector X N is to be removed. Usig the Hilbert trasform (HT) of the vector X, desigated by Y, ca do this. The vector Y is a sum of two rotatig vectors Y P ad Y N at opposite directio ad the same agular frequecy as well. The priciple of evaluatig the aalytic sigal Z as a sum of X+jY, where j is a imagiary uity, is show i Figure. The vector X N ca be vaished by the vector jy N resultig from double rotatio of the vector X N by. The first rotatio by of the vector X N i positive directio results i the vector Y N. The secod rotatio correspods to multiplyig this vector by the imagiary uity j. I cotrast to the positive rotatio of the vector X N by, the vector X P rotates i the egative directio by ad results i the vector Y P. The vector Y P is a product of multiplyig the vector X P by the imagiary uity. Therefore, the vector Z is the same as the redoubled vector X P. X = X P + X N Z = X P X N Y = N Y N j X N j Y = N X P Y P = j X P j j X N = X N Figure 1. Harmoic sigal Figure. Hilbert Trasform The magitude of the aalytical sigal is referred to as a sigal evelope while the vector rotatio agle is the aalytical sigal phase. The amplitude of the aalytical sigal correspodig to a harmoic sigal (cosie fuctio with arbitrary iitial phase shift) has its evelope i the form of a costat fuctio while the harmoic fuctio phase is a liear fuctio of time. The liear fuctio correspods to the uiform rotatio, which is rotatio at the costat rotatioal speed. The o-uiformity i rotatio results i the additioal term to the metioed liear term i the phase fuctio of time.

3 The Fourier trasform of real sigals is composed from complex cojugate pairs of compoets with positive ad egative frequecy. The Hilbert trasform i the frequecy domai ca be expressed as a frequecy trasfer fuctio shiftig the phase of each metioed compoet by i the opposite directio, which meas multiplyig them by -j or +j j, ω > G HT ( jω) = (1) j, ω < The impulse respose correspodig to the trasfer fuctio (1) is its Fourier trasform 1 () + 1 g HT t = ( ω) exp( ω ) ω =, GHT j j t d t () t Note that the impulse respose is ot causal for its o-zero values for the egative time. The frequecy rage of the sampled real sigals is limited to the half of the samplig frequecy f S. Therefore, the argumet of the frequecy trasfer fuctio is, + limited to the iterval ( ) o o j, + > ω G HT ( exp( j ω) ) = (3) j, < ω < + 1, = k g HT ( ) = GHT ( exp( jω) ) exp( jω) dω = (4) ( ), = k + 1 There are two approaches to evaluate the Hilbert trasform of a sampled sigal Fast Fourier Trasform (FFT) Digital filters Figure 3 demostrates the priciple of usig FFT while usig the digital filter is show i Figure 4. x(t) real part x(t) Hilbert Trasformer y(t) imag part Figure 3. HT usig FFT Figure 4. HT usig digital filters The digital filter performig the Hilbert trasform is called the Hilbert trasformer. It is a high-pass or bad-pass filter of the FIR (Fiite Impulse Respose) or IIR (Ifiite Impulse Respose) type. 3. PRINCIPLE OF THE PHASE DEMODULATION The agular vibratio ca be measured by usig shaft ecoders givig usually a trai of pulses, rather the a siusoid. As the impulse sigal cosists of several harmoics of the basic impulse frequecy the first step i phase demodulatio procedure is to separate the frequecy bad cotaiig a carrier compoet ad with sidebad compoets by usig a bad-pass filter. A example of the phase-modulated sigal is show i Figure 5.

4 1-1,1,,3,4,5,6,7,8,9 1 Revolutio Figure 5. Phase-modulated sigal Let the sampled sigal be desigated by x k, k =, 1,..., N 1. The relatioship betwee the compoets X ad Y, which are correspodig to the Fourier trasform of the sampled sigal x k ad its Hilbert trasform y k respectively, is give by the followig formula Y = j sig N i X. (5) ( ) The agle of the complex values rages from to + ad cotais jumps at or at + (see Figure 6). The true phase ϕ of the aalytical sigal as the time fuctio must be uwrapped. The uwrappig algorithm is based o the fact that the absolute value of the differece betwee two cosecutive phase samples is less tha ϕ < ϕ + ϕ, ϕ > + ϕ ϕ. (6) rad,1,,3,4,5,6,7,8,9 1 Revolutio Figure 6. Phase of aalytical sigal ragig from to + The result of a phase uwrappig of a harmoic sigal is show o the left diagram i Figure 7. The phase o this diagram is ot a liear fuctio of time, because addig a harmoic sigal to the liear fuctio iflueces it. The uwrapped phase chage per oe complete rotatio is equal to the multiplied by the umber N of the impulses per rotatio. To evaluate uiformity of rotatio, the phase ormalizatio has to be performed ϕ k N ϕ k. (7) The phase ormalisatio results i the diagram phase scale that correspods to a agle of the complete revolutio. The correspodig agle of rotatio is a fuctio composed from a liear term ad a phase modulatio sigal. The liear term correspods to the steady-state rotatioal speed. After removig the liear term the phase modulatio sigal is obtaied ad it is see o the right diagram i Figure rad 3 1,,4,6,8 1 Revolutio,15,1,5 rad -,5 -,1 -,15,,4,6,8 1 Revolutio Figure 7. Uwrapped ad removed liear tred i a phase of aalytical sigal

5 4. EXAMPLES OF ANGULAR VIBRATION MEASUREMENTS Noise ad vibratio problems i gearig are maily cocered with the smoothess of the drive. The parameter that is employed to measure smoothess is a Trasmissio Error (T.E.). This parameter ca be expressed as a liear displacemet at a base circle radius defied by the differece of the output gear s positio from where it would be if the gear teeth were perfect ad ifiitely stiff. May refereces have attested to the fact that a major goal i reducig gear oise is to reduce the trasmissio error of a gear set. The basic equatio for T.E. of a simple gear set is give as TE( m) = Θ Θ1 r (8) 1 where 1, are teeth umbers of piio ad wheel respectively, Θ 1, Θ are agles of rotatio of the metioed gears ad r is a wheel radius. T.E. results ot oly from maufacturig iaccuracies such as profile errors, tooth pitch errors ad ru-out, but from a bad desig. The pure tooth ivolute deflects uder load due to the fiite mesh stiffess caused by tooth deflectio. A gearcase ad shaft system deflects due to load as well. While ruig uder load oe of very importat parameters, tooth cotact stiffess, is varyig what excites the parametric vibratio ad cosequetly oise. There are may possible approaches to measurig T.E., but, as Derek Smith poits out [1], i practice, measuremets based o the use of ecoders domiates. The sketch of the gear set cosistig of the 1- ad 44- tooth gears uder test ad attached icremetal rotary ecoders, desigated by E1 ad E is show i Figure 8. Both the ecoders are of Heidehai origi, the ERN 46-5 type. A perfectly uiform rotatio of gear produces a ecoder sigal havig i its frequecy spectrum a sigle compoet at the frequecy that is a multiple of the gear rotatioal frequecy. As both the ecoders geerate 5 impulses per ecoder rotatio, the frequecy of the sigle compoets i orders (a multiple of the ecoder rotatioal frequecy) is equal to the same umber as the umber of the impulses. Θ 1 Θ E 1 E 1 =1T =44T piio wheel Figure 8. Measuremet arragemet The gear speed variatio results i the phase modulatio of the impulse sigal base frequecy. As oted above the phase-modulated sigal cotais sidebad compoets aroud the carryig compoet. The distace of the domiatig sidebad compoets from the carryig compoets equals to the iteger multiple of the tooth umber as it is show i Figure 9. The frequecy scale of both the frequecy spectra is i order; it meas the multiples of the gear rotatioal frequecy. The frequecy of the carryig compoet is equal to 5 orders while the sidebad compoet associated with the correspodig gear is at the distace of ± 1k or ± 44k, where k = 1,,..., order uits from the metioed carryig compoet frequecy. Take otice of the fact that the domiatig compoets i both the sidebads exceed the backgroud oise level at least 1 times or eve more. Both the spectra were evaluated from time sigals that are a result of sychroized averagig of 1 revolutios of gears uder test. The phase modulatio sigal i degrees durig the piio revolutio is show i Figure 1. The ehaced sigal cotais five harmoics of the toothmeshig frequecy, each of them with 3 pairs of sidebads that cause the amplitude modulatio of agle variatio. Whe all these sidebads are removed a purely periodic sigal is obtaied. Therefore, oe of these periods correspodig to the gear tooth pitch rotatio ca be take as a represetative to characterize agular vibratio i average.

6 RMS db/ref 1 V Ehaced Spectrum, 1-Tooth Gear Order [-] RMS db/ref 1 V Ehaced Spectrum, 44-Tooth Gear Order [-] Figure 9. Frequecy spectrum of phase modulated sigal geerated by the E1 ad E ecoders deg,6,4,, -, -,4 -,6 -,8 Time History : Piio 1T : Ehaced Time(Impulsy5),,1,,3,4,5,6,7,8,9 Revolutio [-] Figure 1. Phase modulatio sigal i degrees durig the piio revolutio The agular vibratio i agle ca be trasformed ito the arc legth. The arc legth differece is ot a fial step for evaluatio of T.E.. As both the ecoder sigals are recorded separately the true phase delay betwee these sigals is to detect. This problem ca be solved thaks to the fact that the average toothmesh resposes, for example i acceleratio of some poit o the gear case, to dyamic forces actig betwee meshig teeth are theoretically the same. Therefore, both the ecoder pulse sigals are sampled together with the acceleratio sigal. Two-stage averagig of the twice-measured acceleratio sigal gives average toothmesh resposes that are delayed agaist each other. The lag for the maximum correlatio gives the value of relative delay. T.E. is give as the differece betwee the agular vibratio sigals i the arc legth produced by the meshig gears. T.E. [micro] Tooth pitch rotatio 5 RPM, +4 Nm 5 RPM, +8 Nm Figure 11. Trasmissio error agaist rotatio agle i tooth pitch rotatio The result for 3 times repeatig tooth pitch rotatios ad levels of loadig at the iput shaft is show i Figure SOFTWARE TOOLS There is a lack of software tools supportig the phase demodulatio o the market with sigal aalyzers. At the begiig of the 9 s the BK Compay offered the sigal aalyzer supportig the Hilbert trasform usig FFT while the ewest model, LabShop PULSE, performs oly the evelope aalysis.

7 Figure 1. Automatio software for LabShop PULSE Figure 13. Sigal Aalyzer, the software for sigal processig icludig phase demodulatio

8 As a result of a cotract with idustry the Techical Uiversity of Ostrava developed the automatio software called Agular Vibratio ad based o Object Likig ad Embeddig., which exted the PULSE LabShop software by fuctioality for agular vibratio evaluatio (see Figure 1). The secod stage of developig the tools for agular vibratio results i special software, called Sigal Aalyzer, supportig teachig the sigal processig [3]. Both of the metioed approaches to the evaluatio of the Hilbert trasform are implemeted. The software cotais features for sigle/double differetiatio/itegratio i the limited frequecy bad. The approach based o the digital filters is out of the BK sigal aalyzer scope. The iput data for the Sigal Aalyzer software ca be take from various type of sources, for istace from LabShop PULSE. The mai widow of Sigal Aalyzer is show i Figure 13. Sigal Aalyzer cotais the special istrumets for trasmissio error evaluatio. 6. CONCLUSION The paper reviews the field of diagostics based o the measuremet of agular vibratio, which gives useful iformatio about the operatioal coditio of machies. The example, illustratig the theory of aalytical sigals, is focused o the problem of the gearbox trasmissio error measuremet. The istataeous value of trasmissio error results from variatio of the gear agle revolutio from a liear term depedig o steady state rotatio. The source of iformatio about agular vibratios is a strig of impulses with the frequecy proportioal to the rotatioal speed. The measuremet method is based o the phase demodulatio of impulse sigals usig the theory of the aalytical sigals. 7. REFERENCES [1] Derek Smith J. Gear Noise ad Vibratio, 1st ed. New York Basel : Marcel Dekker Ic., ISBN: [] TŮMA, J. Sigal Phase demodulatio of impulse sigals i machie shift agular vibratio measuremets. I Proceedigs of Teth iteratioal cogress o soud ad vibratio. Stockholm: IIAV, , P 59. [3] TŮMA, J. Sigal Aalyse, the software support for educatio of sigal processig. I Proceedigs of 4 rd Iteratioal Carpathia Cotrol Coferece. Košice : TU Košice, , s ISBN ACKNOWLEDGEMENT This research has bee coducted at the Departmet of Cotrol Systems ad Istrumetatio as a part of the research project No. 11/4/153 ad has bee supported by the Czech Grat Agecy. The author beefits of the research work doe for the TATRA ad Skoda-Auto Compaies