Dynamic Calibration of Force Transducers Using Sinusoidal Excitations

Transcription

1 SENSORDEVICES 011 : The Second Inernaonal Conerence on Sensor Devce Technologes and Applcaons Dynamc Calraon o Force Transducers Usng Snusodal Excaons Chrsan Schlegel, Garela Kekenap, Bernd Glöckner, Rol Kumme Physkalsch-Technsche Bundesansal, Bundesallee 100 D Braunschweg, Germany e-mal: Chrsan.Schlegel@p.de Garela.Kekenap@p.de Bernd.Gloeckner@p.de Rol.Kumme@p.de Asrac The prmary calraon o orce ransducers usng snusodal excaons wh elecrodynamc shaker sysems wll e descred. Frs a vew commen concernng he mporance o dynamc orce measuremens wll e gven. Tha wll e ollowed y a mahemacal descrpon o he ascs o dynamc measuremens ased on lnear derenal equaons. Some useul approxmaons are gven o average measured daa. The echncal equpmen wll e nroduced ogeher wh a dscusson concernng he racealy as well as he uncerany consderaon. Fnally an exemplary calraon perormed on a sran gage ransducer wll e presened. Keywords: dynamc orce, orce calraon, laser vromeer, acceleraon measuremen I. INTRODUCTION In he las ew decades very precse sac orce measuremens were developed and are now rounely used or calraon servces n many naonal merology nsues (NMI s) around he world. The orce scale whch s covered nowadays reaches rom µn-mn [1-]. Therey, relave measuremen unceranes down o 10-5 are oaned usng deadwegh machnes, whch are he es sandard o realze a raceale orce. The orce, F, s jus he produc o he SI ase un mass, m, and he gravaonal acceleraon, g, ollowng Newon s law, F=m a, wh he acceleraon, a=g. Besdes he precse realzaon o a orce n a sandard machne, here mus e seleced orce ransducer avalale whch can e used as a ranser sandard o gve he prmary calraon o he secondarly calraon laoraores and ndusry. The crucal ac s now ha oen hese sac calraed orce ransducers are used n dynamc applcaons. Tha s he reason why more and more NMI s have esalshed procedures or a dynamc calraon o orce ransducers and also oher sensors. Currenly n he European Merology Research Programme (EMRP) one promoed research opc s he: Traceale Dynamc Measuremen o Mechancal Quanes, whch ncludes, apar rom a work package aou dynamc orce, also work packages aou dynamc pressure, dynamc orque, he elecrcal characerzaon o measurng amplers and mahemacal and sascal mehods and modellng [3]. Smlar o he sac calraon phlosophy prmary calraons have o e provded whch guaranee racealy o he SI ase uns and also ranser ransducers (reerence sandards) o ranser hese calraons e.g. o an ndusral applcaon. Ths ranser urned ou o e he mos complcaed ask ecause o he crucal nluence o envronmenal condons presen n ceran applcaons. Mosly he ransducers are clamped rom oh sdes whch lead o sensvy losses due o he dynamcs o hese connecons whch are more or less no nnely s. On he oher hand he resonan requency oen shs down o lower requences whch can also drascally change he sensvy. The prolem can e solved o a ceran exen y modellng he whole consrucon ncludng all relevan parameers. For ha reason s also mporan o deermne he orce ransducer parameers lke sness and dampng whch can e oaned durng a dynamc calraon. Ths arcle descres one possly or a prmary dynamc orce calraon usng snusodal excaons. The whole procedure as well as mos o he se-ups where developed over wo decades and are exensvely descred n [4]. Oher mehods as well as analyss procedures or dynamc orce calraon are descred n [5-9]. Copyrgh (c) IARIA, 011. ISBN:

2 SENSORDEVICES 011 : The Second Inernaonal Conerence on Sensor Devce Technologes and Applcaons II. MATHEMATICAL DESCRIPTION To oan an analycal handle or he descrpon o he dynamc ehavour o a dynamcally exced orce ransducer, he well-known sprng-mass-damper model can e appled. In gure 1 one can see a smpled pcure o a orce ransducer whch s equpped wh a es mass, m. The connecon o ha mass o he ransducer s modelled y a ceran sness, k c, and a dampng consan, c. The ransducer sel consss o a oom mass, m, and a head mass, m. Boh masses are also conneced y a sprng wh sness, k, and a correspondng dumpng consan,. The coordnaes n space o all hree masses are hen gven y he vecor (x, x, x ), only a vercal movemen s consdered. A perodcal orce acs rom he oom on he mass, m, (see Fg.1). Ths orce s generaed y an elecrodynamc shaker sysem. The acceleraon o he op mass, & x&, he acceleraon on he shaker ale, & x&, and he orce ransducer elecrcal sgnal are measured durng he calraon procedure. Ths ransducer sgnal s drecly proporonal o he maeral enson/compresson and can e descred n he model y he derence o he sprng coordnaes x -x. The sysem depced n gure 1 can e nally modelled y he ollowng sysem o lnear derenal equaons wh consan coecens: m&& x = k m&& x = k m && x = k c c( x x ) ( x& c ) ( x x ) + ( x& c ) k ( x x ) ( x& ) ( x x ) + ( x& ) + F I should e noed ha he sysem can e smpled he couplng o he op mass has praccally no nluence on he dynamc process. Ths would correspond o he specal case, k c, c 0, and he op mass as well as he head mass o he ransducer can e summarzed as one mass ody. In he calraon process he dynamc sensvy, whch s he rao eween he measured orce ransducer sgnal and he acng dynamc orce, s measured as ollows. x x x m + = m k c µ µ 1 ω 1 + ω kc µ kc kc 1 + ω 1+ ω c k kc In equaon he reduced mass µ=(m m )/(m +m ) was nroduced. Ths equaon can e drascally smpled he op mass couplng s negleced and one apples a Taylor seres developmen o he second order or he requency ω: (1) () x x x m + m k ( 1 p ) 1 ω = p1 ω k Ths equaon s very convenen or ng purposes o approxmae he measured sensves jus y he wo parameers, p 1 and p. The measured sensvy s calculaed rom he rao o he ransducer sgnal, U, and he acng dynamc orce: (3) Fgure 1. Schemac model o a orce ransducer equpped wh a op mass. The ransducer sel can e seen as a sprng-mass-damper, whch consss o wo masses conneced y a sprng. The adapaon o he es mass s also modelled y a damped sprng, whose sness s mosly much sronger han ha o he ransducer. S = ( m U + m ) && x K corr S 0 (1 pω ) The parameer, K corr, akes no accoun he vercal acceleraon graden over he mass ody. Fne elemen smulaons have shown ha he ndvdual mass pons o he mass ody have slghly deren acceleraons n he vercal drecon [4]. Ths correcon acor can e negleced, que small masses are used (only a ew (4) Copyrgh (c) IARIA, 011. ISBN:

3 SENSORDEVICES 011 : The Second Inernaonal Conerence on Sensor Devce Technologes and Applcaons cenmeres n hegh). The acor S 0 s he sac sensvy oaned or he lmng case ω=0, wherey p=p 1. As one can see rom approxmaon (4), he sensvy drops down quadracally wh ncreasng he requency ω. Besdes he amplude o he sensvy accordng o equaon (4), also he phase sh eween he acceleraon x and he orce sgnal U can e derved y he model: ϕ( ω) = an ϕ( ω) an ω kc µ 1(1/,, ) ω k 4 ω kc µ ω kc µ 1 + (1/,, ) + 3(1/,, ) ω k k ω (5) Fgure. Schemac measuremen se-up or he snusodal orce calraon The que complcaed equaon (5) conans uncons 1-3 whch are all proporonal o 1/k c, so ha hese erms can e negleced or he lmng case o nne couplng sness o he op mass. In addon he arcus-angen uncon can e approxmaed y a Taylor seres o he rs order or he requency ω, whch leads o a lnear phase sh eween he acceleraon- and orce ransducer sgnal. III. MEASUREMENT SET-UP The essenal prerequses or a prmary snusodal orce calraon are seen n gure. Frs o all one needs an excer. A PTB we have hree elecromagnec shaker sysems, a small one or orces up o 100 N and 10 Hz unl khz, a medum one up o 800 N or 10 Hz ll 3 khz and a large shaker up o orces o 10 kn and requences o 10 Hz o khz. The shakers conss o wo pars, he vraon excer sel and a power ampler. The knd o excaon s deermned y he chosen sgnal creaed y a uncon generaor. Ths sgnal drecly modulaes he curren sgnal whch drves he col o he shaker armaure.the acceleraon o he op mass can e measured prncpally n wo deren ways, eher y a prmary mehod usng a laser Doppler vromeer or y acceleromeers. Usually he vromeer s used, whch consss o a laser head provdng a 63.8 nm red laser eam and a ceran conroller. There are wo knds o laser heads, one wh a xed eam and he oher one wh a scannng opporuny. The scannng vromeer s ale o scan suraces n an angle regon o ±5º n he x- and y- drecons. Ths oers he possly o nvesgae surace vraons. The modular conroller consss o deren dgal processng uns, wo velocy decoders, a dsplacemen decoder and a dgal quadraure decoder. In summary a requency range rom 0-.5 MHz wh a maxmum velocy o 10 m/s and a resoluon o 0.0 µm/s can e realzed. The sgnal processng nsde he decoders s ully dgal, he oupu s provded as an analogue sgnal. For precse calraons he dgal quadraure encoder s used n connecon wh ceran soware whch calculaes he dsplacemen accordng o he arcus-angen procedure. The analogue IQ oupu sgnals are undled ogeher wh he ransducer oupu sgnal and he acceleraon sgnal rom he shaker armaure n a juncon ox whch s hen caled o a 5 MHz PC samplng card. IV. TRACEABILITY AND UNCERTAINTY CONSIDERATION The snusodal calraon o orce ransducers s a prmary calraon mehod whch means ha all measured quanes are raceale o he SI ase uns and all measurng equpmen used s calraed usng ceran sandards, whch are well esalshed procedures. The calraon o he weghs used as op masses s done accordng o he nernaonal recommendaon OIML R [10]. Accordng o hs documen he op masses can e classed a leas as Class M 1, whch leads o a maxmum error or a 1 kg wegh o 50 mg or a relave sandard measuremen uncerany o Apar rom he mass deermnaon, he acceleraon measuremen s he mos mporan par o he calraon. In gure 3 he racealy chans are shown or deren ways Copyrgh (c) IARIA, 011. ISBN:

4 SENSORDEVICES 011 : The Second Inernaonal Conerence on Sensor Devce Technologes and Applcaons Fgure 3. The racealy chans o he acceleraon measuremen are shown or a prmary and secondary mehod. Le hand sde shows he prmary mehod usng laser nereromeers/vromeers, he rgh hand sde show he racealy usng acceleromeers wh ceran condonng amplers. o acceleraon measuremen. There are, n prncple, wo ways, he prmary mehod usng nereromeers/vromeers and he secondary mehod ased on a ceran elecrcal chan. The vromeer measuremen can der accordng o he nvolved overlap o ceran laser eams n he homodyne or heerodyne nereromeers. Boh nsrumens are ased on moded Mach-Zehnder nereromeers. In he heerodyne nereromeer, he measurng laser eam s sped wherey one par s addonally mxed wh a hgh requency usng a Bragg cell, usually 40 MHz, o provde he Doppler encodng. The homodyne nereromeers are used or larger dsplacemens whch can e deermned y counng he nererence maxma, whch s also known as he rnge counng mehod. The dsplacemen s, herey, only a uncon o he laser wave lengh and he numer o rnges. Frnge counng can e perormed wh very precse nsrumens, lke he hgh perormance couner Fluke PM6681. Accordng o he rnge counng equaon, see gure 3, he uncerany s very small. The relave counng error o he Fluke couner s a 1 khz, Couns/Couns and he relave wavelengh error s n he order o λ/λ The man error s made he dsplacemen comes n he order o λ/, ecause hs s he resoluon lm o he rnge counng. Keepng n mnd a lower lm or he dsplacemen o 400 µm, one can oan unceranes o 0.1% n he range rom 10 Hz o 1.5 khz. As menoned aove, he heerodyne echnque s ased on he arcus-angen calculaon o he quadraure sgnal. I one plos he IQ measurng pons n a dmensonal grd, as seen n gure 3, one oans a crcle n he mos perec case. The ully dgal quadraure encodng avods all errors made n ormer mes y analogue lers and mxng devces. By deaul he whole elecrcal chan o he vromeer conroller s calraed y he manuacurer hrough couplng o very precse known arcal dsplacemen sgnals drecly n he conroller. These measuremens resul n unceranes whch are elow 0.1% [11]. On he oher hand, he vromeer used or he snusodal orce calraon was calraed agans he naonal acceleraon sandard. Therey, he acceleraon values oaned y he measurng program had devaons rom he sandard se-up o %. Wh a clear conscence one can oan an uncerany a leas o 0.1% or he requency range o 10Hz-1.5 khz wh he heerodyne mehod. The rgh-hand sde o gure 3 shows he convenonal mehod o acceleraon measuremen usng acceleromeers n comnaon wh ceran condonng amplers. Normally a charge ampler s used whch can e calraed wh a very precse reerence capacy and a hgh accurae volmeer. The relave sandard measuremen uncerany o oh devces s a ew 10-4 % accordng o he calraon cercae whch was oaned y a sandard calraon procedure a PTB. For he use n a calraon se-up, one has o consder he whole measurng chan conssng o he acceleromeer and s condonng ampler. The measurng chan can e ncluded y a calraed sensvy acor, S qa, whch commonly has an uncerany around 0. %, as llusraed y example n gure 3. Accordng o he charge ampler calraon or he acceleromeer measurng chan also he orce ransducer can e handled, a pezoelecrc ransducer s used. For he case o ransducers ased on he sran gage echnque, a specal calraon devce, also called rdge sandard, was developed [4]. The rdge sandard smulaes, n prncple, a orce ransducer and s ased on a Wheasone rdge whose rdge volage s, as n he real case, provded y he ampler. In place o a orce ransducer he rdge sandard s conneced o he condonng ampler. The oupu o he ampler s a dynamc rdge deunng whch can e seered hrough an analogue npu sgnal rom an arrary uncon generaor wh volage ampludes. Insde he devce he npu sgnal s ransormed o a mv volage. The sgnal whch s seen rom he ampler can e measured n addon on an auxlary oupu channel. The oal (comned) measuremen uncerany, u c, o he snusodal calraon can hen e separaed no wo man pars, a se-up dependen par, u s and a par whch s oaned y he acual calraon measuremen, u m : c = us u. m u + The par, u s, s, n prncple, a consan gven or a ceran se up and relecs he smalles achevale measuremen uncerany. Ths par depends - accordng o equaon 4 on hree pars, he uncerany o he acceleraon measuremen, he uncerany o he condonng ampler calraon and he uncerany o he mass deermnaon. Noe ha he Copyrgh (c) IARIA, 011. ISBN:

5 SENSORDEVICES 011 : The Second Inernaonal Conerence on Sensor Devce Technologes and Applcaons Fgure 4. There s shown an example o he sandard uncerany evaluaon, u s, o he dynamc sensvy a a requency o 400 Hz. The calculaon was done accordng he Gude o he expresson o uncerany n measuremen, GUM [1]. uncerany o he head mass, m, o he ransducer s no ncluded n hs par, ecause hs mass has o e deermned wh he ad o he acual measuremen. Accordng o he numers or he ceran unceranes gven aove, hs par resuls n an uncerany conruon o %. Fgure 4 shows as an example he uncerany evaluaon or u s whch s oaned a 400 Hz. The par, u m, ncludes he uncerany o he nernal mass deermnaon and depends urher manly on he sandard devaons o he perormed measured sensvy pons. By usng he scannng vromeer or he acceleraon measuremen on he op mass one can measure up o 100 pons, dependng on he acual geomery o he wegh. Through hs opporuny specal dsurng nluences lke rockng modes or mechancal adapaon nluences o he ransducer can e aken no accoun. I should e noed ha hese nluences conrue more han oher errors made, y e.g., he sne approxmaon o he measured daa or he unceranes caused y specal ler echnques appled n he analyss procedure. Experence has shown ha he uncerany par, u m, s on average elow 1 khz eween 0.4-1% and aove 1 khz around 1-%. V. MEASUREMENT EXAMPLE Fgure 5 llusraes he oupu o a specal calraon. In hs case a 5 kn orce ransducer ased on he sran gage echnque was measured wh ve op masses as seen n he upper panel o gure 4. As seen rom he sensvy plo, all masses are n good agreemen elow a requency o 1 khz. Fgure 5: Dynamc sensvy, upper panel, o a 5 kn sran gage orce ransducer measured y ve op masses. The lower panel shows he unceranes o he ndvdual measurng pons. Ths s also conrmed y he comned relave sandard measuremen uncerany, gven n %, whch can e seen n he he lower panel o gure 5. The deren op masses cause deren resonance requences whch lay n he range o Hz. Measurng pons near he resonance and also eyond naurally have a gger uncerany. To acqure a gure o mer, all he sensvy curves oaned wh he deren op masses can e ed wh a uncon accordng o equaon 4 and he mean value or he oaned parameers can e calculaed. Includng he unceranes o he ndvdual pons n he procedure moreover, leads o a realsc error also or he parameers. The averaged resuls, ogeher wh he oaned uncerany range are shown n gure 6. The sensvy a requency, =0, was scaled o 100 % o llusrae he sensvy drop as a uncon o requency n an easy readale way. Thus, can mmedaely e seen, ha he ransducer shows only 96% o s sensvy a 1.6 khz. Copyrgh (c) IARIA, 011. ISBN:

6 SENSORDEVICES 011 : The Second Inernaonal Conerence on Sensor Devce Technologes and Applcaons sde resonances o he ransducer, he correspondng daa wll e no urher consdered. In addon specal adapers can e developed o suppress hese eecs. REFERENCES Fgure 6: Averaged sensvy accordng o he descred procedure normalzed a an ndcaon o 100% gven y he ransducer a =0 Hz. VI. CONCLUSION The raceale snusodal calraon o orce ransducers was demonsraed. The calraon manly depends on he acceleraon measuremen and he calraon o he elecrcal chan o he condonng amplers used. For he mos accurae acceleraon measuremen, laser Doppler vromeers can e used whch are raced ack o he laser wavelengh. In he case o pezoelecrc orce ransducers he charge amplers can e very precsely calraed usng a reerence capacy and a prmary calraed mulmeer. For sran gage ransducers, a specal calraon rdge sandard was developed o dynamcally calrae volage raos. In chaper IV was shown ha one can perorm he calraon dependng on he nvolved op masses wh relave sandard measuremen unceranes o %. The man uncerany conruons are no caused y he se-up u raher y he mechancal nluences lke adapaons and he rockng modes o he ransducer. These dsurng nluences can e deeced durng a calraon measuremen y applyng addonal sensors lke, e.g., raxal acceleromeers. I a ceran hreshold o ransverse acceleraon s exceeded, e.g., caused y rockng modes or [1] F. Tegmeer, R. Kumme, M. Sedel, Improvemen o he realzaon o orces eween MN and 5 MN a PTB he new 5 MN orce sandard machne, XIX IMEKO World Congress, pp , Lson, 009. [] Ch. Schlegel, O. Slanna, G. Haucke, R. Kumme, Consrucon o a Sandard Force Machne or he range o 100 µn 00 mn, IMEKO 010, TC3, pp , Paaya, 010. [3] European Merology Research Programme (EMRP), hp:// las access [4] R. Kumme, Invesgaon o a prmary mehod or a dynamc calraon o orce measurng nsrumens: a conruon o reduce he measurng uncerany, docoral hess (n German), PTB, [5] S. Echsäd, C. Elser, T. J. Esward and J. P. Hesslng, Deconvoluon lers or he analyss o dynamc measuremen processes: a uoral, 010, Merologa, 47, [6] G. Wegener and Th. Bruns, Tracealy o orque ransducers under roang and dynamc operang condons, 009, Measuremen, 4, [7] M. Kousch, Th. Bruns and E. Franke, Challenges n Praccal Dynamc Calraon, 009, Advanced mahemacal and compuaonal ools n merology and esng: AMCTM VIII (008 Pars France), eded y Franco Pavese, Seres on Advances n Mahemacs or Appled Scences - Vol. 78, pp. 04-1, World Scenc Pulshng, Sngapore. [8] C. Elser and A. Lnk, Uncerany evaluaon or dynamc measuremens modelled y a lnear me-nvaran sysem, 008, Merologa, 45, [9] M. Kousch, The 50 kn prmary shock orce calraon devce a PTB, IMEKO 010, Thaland, Paaya, Novemer 010. [10] Inernaonal Recommendaon, OIML R 111-1, Inernaonal Organzaon o Legal Merology, 004. [11] G. Segmund, Sources o Measuremen Error n Laser Doppler Vromeers and Proposal or Uned Speccaons, 8h. In. Con. on Vraon Measuremens y Laser Technques, Proc. o SPIE Vol Y, 008. [1] ISO/IEC Gude 98-3:008: Uncerany o measuremen Par3: Gude o he expresson o uncerany n measuremen. ISO, Gen 008, ISBN Copyrgh (c) IARIA, 011. ISBN: