XVIII IMEKO WORLD CONGRESS Metrology for a Sustainable Development September, 17 22, 2006, Rio de Janeiro, Brazil

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1 XVIII IEKO WORLD ONGRESS eology fo a Susanable Developen Sepebe, 17, 006, Ro de Janeo, Bazl INNOVATIVE DEVELOPENTS AND INTEGRATION OF SINE-APPROXIATION AND TIE INTERVAL ANALYSIS ETHODS FOR PRIARY VIBRATION ALIBRATION BY HETERODYNE INTERFEROETRY Qao Sun Naonal Insue of eology, Bejng, hna, sunq@n.ac.cn Absac: In hese pape, nnovave developens n sne appoxaon and e neval analyss ehods (SA and TIA fo pay vbaon calbaon by heeodyne nefeoey ae descbed n deal. The fequency deodulaon of novel TIA s based on esaon of e nevals beween neghboung peaks and valleys nsead of adonal posve o negave zeo-cossngs of Lase Dopple Inefeoee (LDI sgnal, whle he splfed SA uses dgal efeence sgnal o geneae quadaue sgnals and pefo phase deodulaon and dffeenaon o esoe e hsoy of velocy. Because of splfed algoh and low equeens on hadwae, hese developens lead o a successful negaon of SA and TIA, boh effcen and cos effecve, fo accuae esul and elable copason n a sngle easueen, and o a novel pleenaon of naonal edu fequency vbaon sandad (10 Hz o 10 khz n Naonal Insue of eology (NI, hna. The valdy and accuacy of he novel ehods s poved by sulaon and wll be fuhe vefed by a blaeal copason undeway. Keywods: pay vbaon calbaon, Sne appoxaon ehod, e neval analyss ehod, heeodyne nefeoey. 1. INTRODUTION Heeodyne lase nefeoey echnque poves s value n vbaon calbaon, especally when vbaon dsplaceen s n nanoee ange a hgh fequency wh odeae acceleaon level appled [1-3]. Two ehods, SA and TIA, have been esablshed, well nvesgaed and pleened n vaous vesons fo calbaon of coplex sensvy of acceleoees based on phase deodulaon and fequency deodulaon especvely [4-8]. The cuen heeodyne pleenaon of SA and TIA ae ndependen n hadwae and sofwae wh espec o sgnal acquson and pocessng. Hgh-speed daa acquson cads (50 S/s and hgh-esoluon e neval analyses (50 ps have poved o be possble hadwae choces fo good easueen esuls [4]. Unfounaely, s alos unavodable n ehe case o soe and pocess a lage sze of daa saples. Recen sudy has shown ha lowe saplng ae can speed up he calculaon pocedue of TIA o soe degee, bu esul n noleably naccuae easueen esuls [9]. Howeve, s no necessaly o be so wh newly developed algoh fo TIA, fo whch fequency deodulaon s based on esaon of e nevals beween neghboung peaks and valleys of LDI sgnal nsead of successve posve o negave zeo-cossngs. I akes possble o apply a elavely slow saplng ae (5 S/s o dgse LDI sgnal ha can be used fo boh SA and TIA. On he ohe hand, he splfed SA of heeodyne veson uses dgal efeence sgnal o geneae quadaue sgnals and pefo phase deodulaon and dffeenaon o esoe e hsoy of velocy. All hese lead o he possbly of sngle channel saplng of LDI sgnal wh noal speed daa acquson cad nsead of dual channel hgh-speed one o expensve hgh-esoluon e neval analyse, and o a novel pleenaon and negaon of SA and TIA, boh effcen and cos effecve, fo accuae esuls and elable copason n a sngle easueen. The nnovave negaon and successful pleenaon of splfed SA and novel TIA fo he naonal edu fequency vbaon sandad povdes NI, hna wh bee pay vbaon calbaon capables ove a wde fequency ange fo 10 Hz o 10 khz. Based on suffcen heoecal nvesgaon and expeenal vefcaon, hese wo ehods ae vald and effcen enough o be adoped, aong ohe oe convenonal pay vbaon calbaon ehods, fo a blaeal copason n vbaon easueen undeway.. THEORETIAL BAKGROUND AND PRA- TIAL ONSIDERATIONS The auho s pevous nvesgaon of pay vbaon calbaon by hoodyne nefeoey has evealed he sulaon of slow elave oons shown as df beween he devce unde es and he lase nefeoee a hgh fequency when odeae acceleaon level s appled [10]. Two an oon dsubances ae caused by hu of powe aplfe and esonance of vbaon solao. These oon dsubances physcally exs dung he opeaon of evey vbaon calbaon syse even wh he bes powe aplfe and vbaon solaon syse avalable eployed. The

2 phenoenon s oe obvous a hgh vbaon fequency, whch s shown n Fg. 1(a. The slow df can be easly ecognzed fo dsplaceen e hsoy ecoveed by phase deodulaon, wh a saplng ae of 5 S/s fo he dgsaon of LDI sgnal whose cae fequency s 1.5 Hz fo a saplng peod of 1 s a 10 khz. The nfluence fo slow oon dsubances can be effecvely suppessed by nuecal dffeenaon of dsplaceen e hsoy. The esulng velocy e hsoy s deonsaed n Fg. 1(b wh hgh fequency nosy conen aplfed o soe exen. Fg. 1. Dsplaceen (a and velocy (b e hsoes of vbaon oon a 10 khz supeposed anly by oon dsubances (a and nose (b. In fac, he fequences and velocy apludes of vaous elave oons ae of he os neesng aspec fo fequency deodulaon, whch ae calculaed by fequency doan analyss ehod and shown n Table 1. In hs case, only wo an oon dsubances ae aken no accoun: df caused by hu of powe aplfe (abou 50 Hz and vbaon solao (abou 4 Hz. Even lowe fequency dsubance caused by esonan effec of vbaon solao does ex, bu has neglgble nfluence on acceleaon easueen esuls a vbaon fequency lowe han 10 khz. Table 1. Fequences and velocy apludes of vbaon oon and oon dsubances a hgh vbaon fequences. Vbaon fequency (khz Vbaon velocy (/s Fequency of dsubance1 (Hz Velocy of dsubance1 (/s Fequency of dsubance (Hz Velocy of dsubance1 (/s I should be noed ha he acceleaon aplude unde easueen s que sable because he long-e df has no sgnfcan nfluence on he copaably vey hgh acceleaon level easued, noally abou 100 /s. Theefoe, hgh easueen accuacy fo volage aplude and nal phase of acceleoee oupu sgnal can be acheved snce s sensve eleen coesponds o he change of he acceleaon. The specfc pocedue fo calculaon of hs volage sgnal can be efeed o [1]. of oon dsubances should be coveed fo sgnal pocessng and fuhe calculaon based on he peodcal chaacescs; (3 Fo suppesson of df effec, TIA based on fequency deodulaon of LDI sgnal s pefeed fo acceleaon easueen. As fo SA, nuecal devave s ecoended o be used fo dsplaceen e hsoy ecoveed fo phase deodulaon befoe leas squae appoxaon s appled fo he calculaon of velocy aplude and nal phase. By akng he eal suaon nfluence quanes no accoun, hee poan conclusons can be dawn whch lay a sold bass fo he feasbly and easueen accuacy of nnovave pleenaon of SA and TIA: (1 Though e neval analyse can easue e nevals of a LDI sgnal wh a esoluon of 50 ps, hs easueen accuacy ensued by hadwae can no assue he age low unceany of he whole calbaon syse because he elave oons obscue he deal e nevals odulaed only by vbaon oon unde easueen beween successve zeo-cossngs; ( To educe he unfavouable nfluence fo elave oons, nege cycles 3. SIPLIFIED SA The SA of heeodyne veson has been descbed a lo n pevous publcaons [,4,7-9]. Alos n each case, a efeence sgnal geneaed by xng of wo sgnals fo nenal quaz oscllao of Bagg cell and exenal synhess geneao s equed fo he geneaon of quadaue sgnals, a bass of phase deodulaon. I s poposed n [] ha he efeence sgnal can be geneaed by dgal sgnal pocessng pocedue on he pecondon ha he exac value of cae fequency can be acqued accuaely enough. Ths s no easy because he dve fequency of

3 Bagg cell ay df fo e o e, whch akes dffcul o ack he acual nsan fequency value even wh exa nsuen fo fequency easueen. On he ohe hand, s unnecessay o do so snce he nfoaon of nees s he Dopple odulaon conen caed n a naow bandwdh ahe han he nsan cae fequency value self. Fuhe oe, he unfavouable effec on fequency o phase odulaon caused by fequency nsably of he cae sgnal (noally n he ode of 10-6 s no copaed wh ha caused by elave oons. Afe he down xng of ognal LDI sgnal, he odulaed easueen sgnal can be expessed as: u = uˆ cos[ ω + ϕ ( + ϕ ] ( 0 (1 wh a odulaon e whose aplude 4πsˆ ˆ ϕ =. s λ ϕ ˆ ϕ cos( ω+ ϕ ( s = ( ϕˆ s popoonal o he dsplaceen, ϕ sands fo dsplaceen nal phase angle and ω fo vbaon angula fequency. The nal phase angle of he odulaed easueen sgnal s denoed by ϕ and he ansfoed cae angula fequency by ω. 0 The dgal efeence sgnal wh angula fequency can be geneaed pecsely by a pocedue as: ω u ( = uˆ cosω (3 Afe dgal xng and fowad-and-evese low pass fleng, he expeced quadaue sgnals ae geneaed n he fo of: 1 u ˆ ˆ 1( = u u cos[( ω ω + ϕ ( + ϕ0 ] (4 1 u ˆ ˆ ( = u u sn[( ω ω + ϕ ( + ϕ0 ] (5 whee he odulaon phase s ϕ = ( ω ω + ϕ ( + ϕ od 0 (6 I should be noed ha û s he aplude of elecc odulaed easueen sgnal and û s he aplude of dgal efeence sgnal, and ha s no necessay o ee he elaonshp uˆ = uˆ. The adonal goal s o oban quadaue sgnals n base band sla o hoodyne nefeoee quadaue oupu sgnals bu wh bee qualy by keepng ω ω as pecsely as possble. onsequenly, an elecc pah fo geneaon of a efeence sgnal o a devce fo easung nsan dve fequency of Bagg cell s needed. Afe he phase deodulaon of he sae LDI sgnal daa as fo Fg. 1(a bu whn a longe e peod of 0 s, dsplaceen e hsoy s deonsaed n Fg. (a, wh 50 Hz elave oon clealy seen. Fg.. Dsplaceen e hsoes of vbaon oon a 10 khz based on phase deodulaon of quadaue sgnals wh (a (b ω ω. ω ω and When hee s a consan dffeence beween ansfoed cae and efeence fequences, a speedly vayng phase equal o ( ω ω s added o ognal odulaon phase coponen epesenng he easuand. If hs fequency dffeence s hgh enough o nclude he acual bandwdh of he LDI sgnal odulaed anly by vbaon oon, he consequen phase deodulaon wll eveal a dsplaceen sees alos lnealy nceasng wh e as shown n Fg. (b. In hs case, 50 khz fequency dffeence s used fo he geneaon of quadaue sgnals and seves as a new cae fequency odulaed by vbaon oon and ohe oon dsubances. The lnealy nceasng phase coponen appeas as a velocy consan n Fg. 3, whch pesens he esulng e hsoy afe he dgal dffeenaon of he dsplaceen sees shown n Fg. (b. Fg. 3. Velocy e hsoy afe dgal dffeenaon of dsplaceen sees n Fg. (b. The pocessng of hs velocy sees by convenonal sne-appoxaon leads o he aplude and nal phase of he easued velocy, and fnally o hose of acceleaon unde easueen. 4. INNOVATIVE TIA The TIA of heeodyne veson has been pleened wh e neval analyse o hgh speed AD cad fo daa

4 acquson. Though he coon goal s o acque he e nevals beween successve zeo-cossngs, he foe hadwae choce can povde hs nfoaon decly and he lae akes possble o locae he nsan e sees of successve zeo-cossngs by hd ode polynoal fng [4]. Howeve, he pevous nvesgaon on TIA algoh fo he lae case has ndcaed s wo dsadvanages as boh e consung and naccuae [9]. The nnovave TIA has no specal equeens on daa acquson facly because s algoh focuses on he e nevals beween neghboung peaks and valleys nsead of zeo-cossngs. Even wh a sall nube of saples pe peod of he LDI sgnal ha s close o he Nyqus-l, he e locaons of all s peaks and valleys can be obaned faly accuaely wh hs new algoh whle polynoal fng wll ceanly fal o fnd exac e locaons of zeo-cossngs. The key les n he well-known fac ha LDI sgnal s sable and affeced lle by vaous elecc dsubances copaed wh hoodyne nefeoey sgnal. The ognally sapled LDI sgnal ay consss of only a few saples pe peod as shown n Fg. 4(a whee 5S/s saplng ae s appled fo he cae fequency of 1.5 Hz. Fou saples pe peod as an aveage s defnely no suffcen fo dec deecon of ehe zeo-cossngs o peaks and valleys. Theefoe, sofwae-based esaplng ha copues new values based on exsng sgnal saples s a us fo new TIA algoh. A cubc splne nepolaon algoh o copue he esapled values s appled o ognally sapled sgnal, yeldng sooh ansons beween he saples, shown n Fg. 4(b. The splne nepolaon eques he nepolan o pass hough all he sapled daa fo he copuaon of he nepolaon and uses pecewse-connuous cubc polynoals fo nepolaon of he daa se avodng pobles assocaed wh hghe degee, sngle polynoal nepolaon. Fo he peaks and valleys deecon, a pogessve quadac f on a wndow conssng of a cean nube of daa pons s appled eavely wh he slde of he wndow [11]. Fo he f, fnds he second devave, and checks o see f he slope changes, o deene f hee s a peak o a valley. The quadac f algoh euns he peak locaons as floang pon nubes, no as nege ndex values. Ths feaue s an advanage of he algoh because effecvely nepolaes beween he daa pons whle fndng peaks and valleys. The funcon can heefoe easue a peak o valley ha has geae aplude han any daa pons nea he peak o valley. Ths nepolaon povdes a good ndcaon of he ue value of he peak o valley n he ognal sgnal. Based on he e locaons of all he peaks and valleys of he odulaed easueen LDI sgnal, a sees of e nevals beween neghboung peaks and valleys can be calculaed by = +, = 0,1,..., N 1. A 1 non-equdsan e sees of nsan fequences can be obaned by he elaonshp: = + * 1 f ( * ( * + 1 whee, 0,1,..., 1 = = (7 N. The acual velocy sees s gven by v( * λ * = f ( (8 4 I needs o be enoned ha a hghe deodulaon esoluon fo velocy calculaon can be acqued based on e nevals beween neghboung peaks and valleys, whch s wce as ha of convenonal TIA algoh. The aplude and nal phase of velocy can be obaned hen by he calculaon of sne-appoxaon, whch fnally leads o hose of acceleaon. 5. SIULATIONS 5.1. Sulaon odel The heoecal odel fo ansfoed LDI sgnal s expessed n equaon 1. Howeve, he eal easueen condon s oe coplcaed affeced by nfluence fo nose and dsubance fo elave oons, ec. The LDI sgnal hen can be wen as: Fg. 4. Plos of ognally sapled LDI sgnal (a and esapled sgnal wh splne nepolaon (b. u ( = ( uˆ v + 1 uˆ n 1 cos[ ω + ϕ + ϕ + ˆ ϕ cos( ω + ϕ ] (9 0 p =0 s

5 whee û v s volage aplude devaon affeced by ando nose and denoed as pecen of deal sgnal volage aplude û. ϕ denoes he phase dsubance n degee. p The las e consss of vaous odulaon es of vbaon and oon dsubances wh dffeen angula fequences and nal phase angles because n ealy, low fequency oon dsubances caused by hu and esonance also appea as odulaon coponens n LDI sgnal. To acheve easonable sulaon esuls, all he nfluence quanes enoned above have o be well known fo eal ess of a cean syse. In ou sulaon ess, he daa of elave oons n able 1 ae used. The ando nose level of he sgnal s 3% n elaon o s volage aplude, and he phase dsubance 0.. Ohe condons concenng he sulaed sgnals geneaed ae: acceleaon level equal o value obaned fo eal expeens (fo abou 75 o 90 /s, vbaon fequency fo 4 o 10 khz, saplng ae 10 S/s, saplng peod 0 s, and effecve esoluon 8 bs. Snce he nal phase angles of cae fequency, vbaon oon and oon dsubances have sgnfcan nfluences on easueen accuacy, hey ae andoly geneaed whn he angle ange fo 0 o 180 fo each Vbaon fequency (khz ean (% sulaon es. On accoun of hs nfluence, one hunded sulaon ess ae pefoed unde all he sae es condons wh he only excepon of fou dffeen andogeneaed nal dsplaceen phase angles. Based on sulaed LDI dgal sgnal wh led pecson descbed above, he ahec ean and expeenal sandad devaon of he calculaed devaons beween nonal acceleaon aplude and nal phase values and hose calculaed by SA and TIA can be esaed... Sulaon esuls In Table, soe of he esuls fo he nuecal sulaon ess ove fequency ange fo 4 khz o 10 khz ae pesened o llusae he effecveness of SA and TIA fo good easueen esuls of acceleaon aplude and nal phase. Ths hgh fequency ange s oublesoe and of pacula nees because long e df wh lage dsplaceen aplude s sulaed and also odulaed he LDI sgnal. Fo aplude and nal phase easueen, he ean and sandad devaon values fo boh ehods ae below 0.1% and 0.1 wh confdence level of 95% hough hey ncease gadually wh hghe fequences. Table. opason esuls of sulaon ess fo SA and TIA fo 4 o 10 khz. SA Aplude Inal phase Aplude Inal phase Sandad devaon (% ean ( Sandad devaon ( ean (% Sandad devaon (% TIA ean ( Sandad devaon ( Obvously, he esuls fo he sulaon ess conf he expeced good pefoance and ageeen of he wo ehods, due o: (1 nege cycle of elave oons s used as saplng peod and he nfluence s nzed by sneappoxaon; ( non-lnea effecs of he phoo deeco as well as of all sgnal pe-pocessng sages do no affec he negy of he Dopple odulaon conen as only hgh fequency A sgnals ae ansed; (3 ando nfluences such as a cuen dsubance, nefeoee esdual effec and fequency nsably of oscllao n Bagg cell, have no been aken no accoun. 6. ONLUSION A successful pleenaon of novel SA and TIA (heeodyne veson has been descbed. Insead of usng hgh esoluon e neval analyse o hgh speed daa acquson cad o esae e nevals beween successve zeo cossngs of LDI sgnal, he novel TIA s based on he esaon of e nevals beween neghboung peaks and valleys of he sgnal, whch s dgsed by a saplng ae of 5S/s. On he ohe hand, he splfed SA akes use of he sae LDI sgnal as fo TIA and dgal efeence sgnal povded by sgnal pocessng pocedue o geneae equed quadaue sgnals fo phase deodulaon. Ths nnovave developen allows, aong ohe hngs, a possble negaon of SA and TIA based on he sae daa acquson hadwae of one AD cad whn one vbaon calbaon syse.

6 The esaplng pocedue and peak/valley deecon algoh fo TIA s effcen and poweful, even when he saplng ae of AD cad s close o he Nyqus-l, whch would be defnely possble f convenonal TIA based on deecon of e locaons of successve zeo cossngs wee appled. To educe he sgnfcan nfluence on fequency/phase odulaon fo elave oons a hgh fequency, he splfed SA pefos phase deodulaon and dffeenaon o esoe e hsoy of velocy befoe he aplude and nal phase of acceleaon unde easueen ae calculaed. The novel algoh fo TIA and SA esuls n no only lowe and less equeens on hadwae fo daa acquson and efeence sgnal geneaon bu also a elable copason beween fequency and phase deodulaon ehods n a sngle easueen, whch s fuhe nvesgaed by sulaon expeens akng no accoun vaous nfluences obseved unde eal suaon. Based on he eseach esuls of SA and TIA enoned above, he naonal edu fequency vbaon sandad of NI has been fuhe developed and eployed, aong ohe naonal vbaon sandads, n a blaeal copason fo 10 Hz o 10 khz unde way. [8] L Zhang, Rolf Kue, Invesgaon of a hoodyne and a heeodyne lase nefeoee fo dynac foce easueen, Poc. SPIE 5503, pp , 004 [9] Thoas Buns, chael Kobusch, Daa acquson and pocessng fo PTB s pac foce sandad achne, Poc. Of he 19h IEKO T3 onfeence, 005 [10] Qao Sun, W. Wabnsk and T. Buns, Invesgaons of pay hgh fequency vbaon calbaon usng Sne-appoxaon ehod: pobles and soluons, eas. Sc. Technol. no publshed, 006 [11] Naonal Insuens, Sgnal Pocessng Toolse Use anual, 00 [1] H. J. von aens, E. leens, Fnal epo on key copason AUV.V-K1, eologa 40 Tech Suppl., 003 AKNOWLEDGENTS The auho would lke o hank D. Thoas Buns and ohe vbaon eologss of PTB fo he suppos n blaeal copason and ohe scenfc eseach elaed. REFERENES [1] ISO , ehods fo he calbaon of vbaon and shock ansduces. Pa 11: Pay vbaon calbaon by lase nefeoey, ISO 1999 [] A. Lnk, H. J. von aens, Aplude and phase easueen of snusodal vbaon n he nanoee ange usng lase nefeoey, easueen 4, pp.55-67, 1998 [3] H. J. von aens, A. Taubne, and W. Wabnsk, A. Lnk, H. J. Schlaak, Taceably of vbaon and shock easueens by lase nefeoey, easueen 8, pp.3-0, 000 [4] W. Wabnsk, H. J. von aens, Te neval analyss of nefeoee sgnals fo easung aplude and phase of vbaons, Poc. SPIE 868, pp , 1996 [5]. S. Veldan, H. J. von aens, Phase calbaon of laboaoy sandad acceleoees usng lase nefeoey, Poc. SPIE 5503, pp , 004 [6] Holge Ncklch, Uwe Buehn, Paccal expeences n pay vbaon calbaon usng lase vboey easueen unceanes n wde fequency ange applcaons, Poc. SPIE 5503, pp , 004 [7] L Zhang, Jun Peng, Pay acceleaon calbaon by heeodyne lase nefeoee and PXI nsuen, Poc. SPIE 5503, pp , 004