Determining a Prony Series for a Viscoelastic Material From Time Varying Strain Data

Transcription

1 NASA/TM-- ARL-TR-6 Drmnng a Prony Srs for a Vscolasc Maral From Tm Varyng Sran Daa Tzkang Chn U.S. Army Rsarch Laboraory Vhcl Tchnology Drcora Langly Rsarch Cnr, Hamon,Vrgna May

2 Th NASA STI Program Offc... n Profl Snc s founng, NASA has bn ca o h avancmn of aronaucs an sac scnc. Th NASA Scnfc an Tchncal Informaon STI Program Offc lays a ky ar n hlng NASA manan hs moran rol. Th NASA STI Program Offc s ora by Langly Rsarch Cnr, h la cnr for NASAÕs scnfc an chncal nformaon. Th NASA STI Program Offc rovs accss o h NASA STI Daabas, h largs collcon of aronaucal an sac scnc STI n h worl. Th Program Offc s also NASAÕs nsuonal mchansm for ssmnang h rsuls of s rsarch an vlomn acvs. Ths rsuls ar ublsh by NASA n h NASA STI Ror Srs, whch nclus h followng ror ys: TCHNICAL PUBLICATION. Rors of coml rsarch or a major sgnfcan has of rsarch ha rsn h rsuls of NASA rograms an nclu xnsv aa or horcal analyss. Inclus comlaons of sgnfcan scnfc an chncal aa an nformaon m o b of connung rfrnc valu. NASA counrar of r-rvw formal rofssonal ars, bu havng lss srngn lmaons on manuscr lngh an xn of grahc rsnaons. TCHNICAL MMORANDUM. Scnfc an chncal fnngs ha ar rlmnary or of scalz nrs,.g., quck rlas rors, workng ars, an bblograhs ha conan mnmal annoaon. Dos no conan xnsv analyss. CONTRACTOR RPORT. Scnfc an chncal fnngs by NASA-sonsor conracors an grans. CONFRNC PUBLICATION. Collc ars from scnfc an chncal confrncs, symosa, smnars, or ohr mngs sonsor or co-sonsor by NASA. SPCIAL PUBLICATION. Scnfc, chncal, or hsorcal nformaon from NASA rograms, rojcs, an mssons, ofn concrn wh subjcs havng subsanal ublc nrs. TCHNICAL TRANSLATION. nglshlanguag ranslaons of forgn scnfc an chncal maral rnn o NASAÕs msson. Scalz srvcs ha comlmn h STI Program OffcÕs vrs offrngs nclu crang cusom hsaur, bulng cusomz aabass, organzng an ublshng rsarch rsuls... vn rovng vos. For mor nformaon abou h NASA STI Program Offc, s h followng: Accss h NASA STI Program Hom Pag a h:// -mal your quson va h Inrn o hl@s.nasa.gov Fax your quson o h NASA STI Hl Dsk a 6-4 Phon h NASA STI Hl Dsk a 6-9 Wr o: NASA STI Hl Dsk NASA Cnr for AroSac Informaon 7 Sanar Drv Hanovr, MD 76-

3 NASA/TM-- ARL-TR-6 Drmnng a Prony Srs for a Vscolasc Maral From Tm Varyng Sran Daa Tzkang Chn U.S. Army Rsarch Laboraory Vhcl Tchnology Drcora Langly Rsarch Cnr, Hamon, Vrgna Naonal Aronaucs an Sac Amnsraon Langly Rsarch Cnr Hamon, Vrgna May

4 Th us of ramarks or nams of manufacurrs n h ror s for accura rorng an os no consu an offcal norsmn, hr xrss or ml, of such roucs or manufacurrs by h Naonal Aronaucs an Sac Amnsraon or h U.S. Army. Avalabl from: NASA Cnr for AroSac Informaon CASI Naonal Tchncal Informaon Srvc NTIS 7 Sanar Drv 585 Por Royal Roa Hanovr, MD 76- Srngfl, VA

5 Drmnng a Prony Srs for a Vscolasc Maral from Tm Varyng Sran Daa ABSTRACT In hs suy a mho of rmnng h coffcns n a Prony srs rrsnaon of a vscolasc moulus from ra nn aa s rsn. Loa vrsus m s aa for a squnc of ffrn ra loang sgmns s las-squars f o a Prony srs hrary ngral mol of h maral s. A nonlnar las squars rgrsson algorhm s mloy. Th masur aa nclus ram loang, rlaxaon, an unloang srss-sran aa. Th rsulng Prony srs, whch caurs sran ra loang an unloang ffcs, roucs an xclln f o h comlx loang squnc. KY WORDS: hrary ngral, vscolascy, wgh nonlnar rgrsson, Prony srs, mull loang sgmns INTRODUCTION In orr o rmn h m nn srss - sran sa n a lnar vscolasc maral, unr an arbrary loang rocss, h formaon hsory mus b consr. Th m nn consuv quaons of a sol vscolasc maral nclu hs hsory ffcs. Th loa srss an slacmn sran hsory, h loang ra slacmn ra, an m of loa alcaon on h scmn ar all n o rmn h consans n h consuv quaons. A common form for hs consuv quaons N / mloys a Prony srs.., a srs of h form α. Cr an rlaxaon ss ar mos commonly us o rmn h vscolasc maral rors, s Fgur. In al rlaxaon an cr ss, h slacmns or loas ar al o h scmn nsanly. In h ral s, scally for a larg srucural comonn, lmaons of h sng qumn rsul n a rlavly low sran ra an long

6 ro of loang. Th rsons urng h ro of loang s ycally gnor, an only h aa oban urng h ro of consan slacmn or consan loa ar us o rmn h maral rors., Ignorng hs long loang ro an h sran ra ffcs n h aa rucon nroucs aonal rrors n h rmnaon of h maral rors. Thr ar numrous mhos for rmnng h Prony srs from rlaxaon an/or cr aa. An arly mho nvolv consrucng log-log los of rlaxaon aa n whch sragh ln aroxmaons for h aa on h log-log grah yl h m consans.. s from h slos, an h xonnal coffcns.. α s ar oban from h nrcs. Ohr mhos hav also bn mloy. For xaml, Johnson an Qugly 4 rmn a rlaxaon m consan for a nonlnar mol whch s smlar o a on-rm Prony srs mol. Thy mnmz a las-squars rror masur, of h ffrnc bwn h nonlnar mol an masur aa, by ravly ngrang numrcally an nrnal varabl quaon. Whn amng o rmn rlaxaon m consans for hghr fly nonlnar mols, Johnson, al. 5 mloy ral an rror rocurs, smlar o arly lnar mhos, u o h comlxy of h rsulng nonlnar lassquars roblm. Mor rcnly, a fw auhors 6-8 mloy nonlnar omzaon mhos o oban a hgh qualy Prony srs rrsnaon of rlaxaon aa wh a mnmum numbr of rms n h srs. Th vscolasc mol can also b formula n ffrnal form. Ths s bcomng oular rcnly 9- snc h ffrnal mols can b ffcvly ncorora no fn lmn algorhms. Whn usng hs nrnal varabl mhos, ach Prony srs rm s assoca wh a maral nrnal sa varabl. In h scr fn lmn mol, ach rm n h Prony srs as a subsanal numbr of global varabls. Thus, a shor Prony srs, whch can accuraly rrsn h maral, s srabl. Nonlnar rgrsson mhos can hl wh rmnng a shor an accura Prony srs. Th uros of hs ar s o rsn a mho for nclung h loang an unloang aa, along wh h rlaxaon aa, n a nonlnar rgrsson analyss o oban h Prony srs. Th rsulng vscolasc maral mol s hn caabl of smulang h loang sgmns as wll as h rlaxaon sgmns. Ths s an mrovmn whn molng hysrc ffcs s moran. Th analycal soluon for loang an/or unloang s rmn hrn an mloy n a nonlnar rgrsson analyss o rmn h Prony srs. In aon, aa wghng funcons ar nvsga an ar shown o mrov h

7 f n h bgnnng of rlaxaon ro. Agan, hs mho allows all h masur aa o b nclu an rsuls n an mrov consuv mol. Hrary Ingrals for lnar vscolascy Dal scrons of lnar vscolascy can b foun n h lraur., An ovrvw of lnar vscolascy s rov hr n orr o nrouc h hrary ngral mho whch s us blow o rmn h analycal soluon for h loang an unloang sgmns. Lnar vscolasc consuv mols ar rrsn by sml hyscal mols comos of srngs an ashos. Th srng s h lnar-lasc comonn, an s consuv quaon s σ Th asho s h vscous comonn, an s consuv quaon s η σ whr η s h vscosy consan. Lnar vscolasc consuv mols ar consruc by surmosng comonns wh consuv quaons gvn by quaons an. Snc h mchancal rsons of h asho s m nn, h bhavor of a vscolasc maral ha s mol by aralll an/or srs combnaons of srngs an ashos s also m nn. Th cr s consss of a consan srss, σ, al o a scmn for a ro of m whl s sran s rcor Fgur a. In a rlaxaon s, h scmn s sran,, s hl consan for a ro of m whl h srss s rcor Fgur b. In Fgur, an σ ar h nal sran an srss, rscvly. For h rlaxaon s, a consuv rlaon for h ro of consan sran can b wrn as follows: σ Y whr Y s a rlaxaon funcon. Whn h maral s assum o b a gnral Maxwll sol, h rlaxaon funcon s ycally mol wh a Prony srs as follows,

8 Y n / whr: s h h Prony consan,,. s h h Prony raraon m consan,, 4 s h nsananous moulus of h maral For m, Y an for, Y -Σ. In h cas of a cr s, a cr comlanc funcon, J, s fn as follows. J σ 5 Th comlanc funcon s hn rmn by rocurs analogous o hos scrb abov. To rmn h srss sa n a vscolasc maral a a gvn m, h formaon hsory mus b consr. For lnar vscolasc marals, a suroson of hrary ngrals scrbs h m nn rsons. If a scmn s loa fr ror o h m, a whch a srss, σ σ, s al h sran for m > can b rrsn as follows. σ σ J J 6 whr J s h comlanc funcon of h maral an σ / s h srss ra. A smlar quaon can b us for h rlaxaon mol o oban h srss funcon nrouc by an arbrary sran funcon σ Y Y 7 whr Y s h rlaxaon funcon quaon 4 an / s h sran ra. An xaml of alyng hrary ngrals for a mull loang sgmn rocss s shown n nx scon. 4

9 5 Hrary ngrals for a mull loang rocss Hrary ngrals wh Prony srs krnls can b al o mol a loang rocss such as h on shown n Fgur. Th rocss n Fgur s v no four sgmns for whch sran an sran ra funcons ar fn. Th funcons ar: 4 / /, 4 / / 8 whr an. For a maral wh a rlaxaon funcon n h form of a Prony srs quaon 4, h srss funcons of h loang rocss can b rv as follows: S. Subsu quaons 4 an frs sran an sran ra funcons of quaon 8 no quaon 7 an oban: n Y Y σ 9 whr n s h numbr of rms n h Prony srs. To smlfy h xrsson, n wll no b shown n followng quaons. an ar h consans n h -h rm of h Prony srs. S. Usng h scon sran ra funcon oban:

10 6 Y Y Y σ S. Th hr sran ra funcon yls: Y Y Y Y σ No, h frs oron of quaon s qual o quaon. Thus, σ σ whr σ s quaon an s funcon of m.

11 S 4. 4 Smlarly, h quaon for h fourh s can b wrn as follows: σ σ A numrcal xaml of a mull loang sgmn rocss usng MATHCAD sofwar s shown n Anx A. In h xaml, h srss funcon was calcula bas on h sran an sran ra funcons shown abov, an mloy a wo-rm Prony srs. Th rsuls of h vscolasc analyss ar shown n h srss-m an srss-sran los. Ths worksh can b us o gnra aa n a aramrc suy nvolvng vscolasc marals. Th worksh s also us as ar of h wgh nonlnar rgrsson algorhm as s shown n h followng scons. Wgh Nonlnar Rgrsson Th Prony srs coffcns an raraon ms aarng n quaon 4 n o b rmn n a rgrsson analyss. Hr, a sanar nonlnar rgrsson mho h Marquar-Lvnbrg Mho,4 s us o rform h aa fng. In h nonlnar rgrsson, an rror funcon χ wh rsc o h unknown consans s fn as, χ ; N y y x a a 4 σ whr x an y ar h xrmnal aa, funcon yx ;a s h mol o b f, an σ s h sanar vaon of masurmn rror of -h aa on. A s of unknown consans a wll b rmn ha mnmz h rror funcon χ. Th rror funcon χ s aroxma by s Taylor srs wh h quarac form: χ a c a a D a 5 whr c s a consan an s h gran of χ wh rsc o h aramrs a, whch wll b zro whr χ s mnmum. Marx D s h scon aral rvav of χ wh rsc o h aramrs a. Inal ral valus of a ar scf an mrov valus ar rmn 7

12 by h nonlnar rgrsson algorhm. Iraon s connu unl h rror funcon, χ, ffcvly sos crasng. Snc ach Prony rm nclus wo varabls an an snc h nsananous moulus mus b rmn, h oal numbr of varabls n h rgrsson s n. Bas on hrmoynamc rncls, svral consran conons mus b al: P, P,, > 6 In aon, h srbuon of h sanar vaon of masurmn rror σ s no asly rmn bas on h rror of aa acquson qumn an h rror of s machn, h rror s usually assum o b unform for all aa ons σ. As s wll known, h vscolasc ffcs ar mos sgnfcan a h bgnnng of h rlaxaon ro, h fng rror n hs rgon s sgnfcan. Snc h rcnag of h numbr of aa ons a h bgnnng of h rlaxaon ro s lss, h rror funcon χ s omna by a long unform al rgon of h rlaxaon ro. To ruc h rror an mrov h f a bgnnng of h rlaxaon ro, a wgh funcon w /σ, σ s us. Th largr h wgh facor a a aa on, h br h curv f h aa on. Thr s no analycal mho o rmn h wgh funcon, hus a ral-an-rror mho s us. Th accanc of h wgh funcon s bas on a grah of h aa an rgrsson mol rsuls. Wgh nonlnar rgrsson rqurs mor raons han unwgh nonlnar rgrsson, bu can rov a br f o h xrmnal aa n h rgon of mos nrs. Wgh Nonlnar Rgrsson for Rlaxaon Ts A hr-on bnng rlaxaon s of a comos maral was rform 5. Th scmn n. x n. x.768 n. was loa n scons o a maxmum flcon of. n. a h ml of h san. Thn h flcon was hl for,7 scons Fgur. Th Marquar-Lvnbrg nonlnar rgrsson mho was al by h commrcal sofwar SgmaPlo o a hrary ngral mol usng wo sgmns loang an holng o oban h Prony srs coffcns for h aa. Snc no analycal mho xss o form h wgh funcon, a ral-an-rror mho bas on maral rors was us o oban a f curv. Th vscolasc srss cays xonnally n h rlaxaon s, 8

13 hrfor h vscolasc ffc s mor sgnfcan urng h loang ro an a h bgnnng scons of h holng ro. Th numbr of aa ons n hs ros s much lss han h numbr of aa ons n al rgon of h rlaxaon ro > 5. Th rror funcon χ wll b omna by a long unform al rgon of h rlaxaon ro f a unform wgh funcon s al. Thrfor, a cws wgh funcon was us o oban br fs for hs ros an mrov h accuracy of h rgrsson. Fgur 4 shows h loa rlaxaon a h bgnnng of h rocss. Th os rrsn h s aa. Thr rgrsson rsuls ar shown. Th ash-o curv s h rsul of a rgrsson analyss whou h wgh funcon w/o WF for a wo-rm Prony srs. Th long-ash curv s h rsul for a wo-rm Prony srs wh wgh funcon numbr WF shown blow: w 7 Th shor-ash curv s h rsul for a hr-rm Prony srs wh wgh funcon WF as follows: 6 w / 8 7 Snc an nal valu s rqur for ach of h varabls, a ral aa s bas on h s aa was assum. Th sum of h P consans shoul b abou.9 snc h loa a h n of rlaxaon ro s 9% lowr han h loa a h bgnnng. Th raraon m consans can b s o owrs of n. As long as h nal ral valus ar rasonabl, convrgnc wll b achv. As s wll known, h rgrsson aa fs br n a arcular rgon f h rlav wgh facor wgh facor / sum of wgh facors n ha rgon s grar han h avrag valu. In Fgur 4, h curvs wh wgh funcons wr closr o h aa ons nar h bgnnng of h rlaxaon ro. Snc h wgh facor a scons of funcon 66 s grar han h valu of h wgh funcon, h curv of funcon s a br mach o h aa han curv of funcon a h bgnnng of rlaxaon. 9

14 Howvr, as shown n h Fgur, wgh funcon os no f h aa as wll as h ohr curvs afr scons. I aars ha funcon was ovr wgh a h bgnnng of h rlaxaon an h rlav wgh facor a h ohr rgon was oo small. Th Prony consans of h rgrsson ar shown n Tabl. Tabl. Th Prony consans for h rgrsson Moulus P sc. P sc. P sc. w/o WF WF WF Th rsuls show ha a wgh funcon shoul b slc bas on h maral rors an s aa srbuon. Prorly slcng h wgh funcon n h mos sgnfcan rgon can mrov accuracy of h rgrsson. Wgh Nonlnar Rgrsson for a Mull Loang Procss In a rcn suy 5, a hck comos anl rson vscolascly whn was s. In orr o characrz h rors of h anl, a hr-on bnng s wh mull loang sgmn rocsss was rform on. Snc h sffnss of h anl was qu hgh, h s was conuc wh a larg hyraulc sng machn. Hgh formaon ras wr no avalabl wh hs loang machn. Th loang ha slacmn schul of h machn s shown n Fgur 5. Th schul s unlk a sanar rlaxaon s. Though loa rlaxaon ros xs, h m rqur o aly a full loa was clarly long whn comar o h rlaxaon ros. As mnon n h rvous scons, n orr o nclu h aa of loang an unloang sgmns, a nonlnar rgrsson combn wh h hrary ngral was us o smula h aa. Snc h formaon was vry small comar o h scmn mnsons a lnar mol was us for h calculaons. Th rsulng slacmns an h loas ar lnarly rla o h srans an srsss. Thus, h slacmn an loa aa wr us as sran an srss aa n h quaons. Th slacmn sran schul n Fgur 5 was v no ss. Th cws connuous funcon for h frs ss was assum o b aroxmaly lnar, smlar o h rvous xaml quaon 8. Th aa for s was f o a cubc

15 olynomal for a mor rcs rgrsson mol. Th sran an sran ra funcons wr fn as follows, / b a L L L 9 / b a L L L Th aramrs of h olynomal us n s wr , a.976, b an c Th cws connuous srss funcon for frs ss, bas on h hrary ngral, was rv by h sam chnqu as shown from quaon 9 o quaon. Th loa funcon for s was rv as follows: [ ] [ / / / n c b a c b a c b a c b a c b a c b a c b a Y σ σ σ σ σ whr σ s h srss funcon from s. A cws wgh funcon was gnra n orr o ncras h accuracy of rgrsson. Agan, h wgh funcon was rmn by a ral-an-rror mho. Afr

16 svral raons, h consan wgh facors us for ach s wr rmn as: W [., 5.,.,.5,., 7.,.,.5,., 5.,.] an h analyss rsuls wr shown n Fgur 6. Th wgh facors wr qual o. for h loang an unloang ss, an wr grar han. for h holng ss. Th oal numbr of varabls n h Prony srs maral mol s n, whr n s h numbr of Prony rms. In hs suy, a wo-rm Prony srs was suffcn o f h aa. Th rgrsson analyss rsuls an s aa ar shown n Fgurs 6 an 7. Th only ffrnc bwn h rsuls of h wo analyss, shown n h fgurs, was h slacmn funcon of s. On of h analyss assum ha h slacmn funcon s lnar an ohr assum was cubc. Boh mol rsuls agr closly wh h xrmnal aa. Snc h cubc olynomal scrb slacmn n s mor rcsly, h on wh cubc slacmn funcon f br n s han h on wh lnar funcon. Th rror funcon χ of h on wh cubc funcon s % lowr han h on wh lnar funcon. CONCLUDING RMARKS: A mho of rmnng h coffcns n a Prony srs rrsnaon of a vscolasc moulus from ra nn aa has bn rsn. Th hrary ngral mho was mloy o oban an analycal rrsnaon of maral rsons whn s subjc o ra nn loang. Th analycal rrsnaon was us n a nonlnar rgrsson analyss, wh masur aa, o valua h Prony srs consans. Svral rgrsson analyss wr rform usng ffrn wgh funcons. For h aa analyz n hs suy, mrov smulaons of h hysrss ffcs wr oban whn h aa a h bgnnng of ach rlaxaon ro was aroraly wgh. No, h aa analyz hr ha loang an rlaxaon rgons of smlar lngh n m. Ohr wghng funcons may b n for ffrn loang schuls. Th mho rsn hr rov a hghly accura rrsnaon of h maral bhavor n h ra nn loang rgon. I can also rrsn h rsons of a vscolasc maral for ohr unqu loang schuls. For xaml, can b us for schuls n whch h maral s no allow o rlax bwn subsqun loang changs.

17 Rfrncs. Fl gg, W. Vscolascy, Blasll Publshng Co., Massachuss, Crsnsn, R. M. Thory of Vscolascy n on, Acamc Prss, Nw York, 98.. Schary, R. A., Mchancs of Comos Marals, Vol.,. Snckyj, G. P., Acamc Prss, Nw York, Johnson, A. R., an Qugly, C. J., A Vscohyrlasc Maxwll Mol for Rubbr Vscolascy, Rubbr Chmsry an Tchnology, Vol. 65, No., Johnson, A. R., Qugly, C.J., Young D.G., an Dank,J.A. Vscohyrlasc Molng of Rubbr Vulcanzas, Tr Scnc an Tchnology, TSTCA,Vol., No., July-S. 99, Hll, S.A., Th Analycal Rrsnaon of Vscolasc Maral Prors Usng Omzaon Tchnqus, NASA TM-894, Fbruary, Bowr, M.V. an Gan, D.F., Srss Rlaxaon Funcons: Mho of Aroxmaon, NASA-CR-958, Arl, ABAQUS/Sanar Usr s Manual, Hbb, Karlsson an Sornsn, Inc Johnson, A.R. Molng Vscolasc Marals Usng Inrnal Varabls, Th Shock an Vbraon Dgs, Vol.,No. 999,. 9-. Lsur, G.A. an Govnswamy, K., fn lmns molng of frquncynn an mraur nn ynamc bhavor of vscolasc marals n sml shar, In. J. Sols Srucurs, 996, Johnson, A. R., Tsslr, A., an Dambach, M. Dynamcs of hck vscolasc bams, ASM J. of ng. Ma. An Tch., 9, 997, Mahca Usr s Gu, Mahca Plus 6., Mahsof Inc., Transforms & Rgrssons, Rfrnc Manual of Sgmalo 4. for Wnows, SPSS Inc., Prss, W., Plannry, B., Tukolsky, S. an Vrlng, W., Numrcal Rcs, Cambrg Unvrsy Prss, Chn, T.K., Dávla, C. an Bakr, D. Analyss of Tl-Rnforc Comos Armor. Par : Vscolasc Rsons Molng, Proc. s Army Scnc Confrnc, Norfolk, VA. 998

18 Anx A Numrcal Soluon for Mull Loang Sgmn Procss Th uros of hs anx s o rsn an nu fl for MATHCAD whch can b us o numrcally comu rsuls from quaons 9 o for h mull loang sgmn rocss. Two Prony rms ar us n hs smulaon. Th loang rocss s fn by Fgur. In h aa fl blow, x wh a bol fon rrsns a commn an wh a normal fon rrsns a comman. Th rsul of h smulaon s sav o a fl OUTPUT.PRN whch s shown a n of h anx. Inu fl Hrary.MCD Th rlaxaon funcon of h maral s as follows: Y - P - -/ - P - -/ Vscolasc Maral Consans: : 9 P :. : P :. : whr s h moulus, P an ar Prony consans Loang m: Scons l : 5 hol : 5 : : l : hol : l 4 : hol Pcws Sran Funcon: Mull Loang Procss :. :. :.. 4 sran funcon. f. f. f 4 ohrws f.5 55 Hrary Ingral for h srss funcon S Loang o :.. σ : P P P x P P P x S Holng h loa for 5 scons :.. σ : P P x P x P P x P x 4

19 S :.. Unloa o σ : P P P P P P P x P x σ : σ σ _ S 4 4 Holng for 5s :, σ 4 _ : P P x P P x P P P x P x σ 4 : σ σ 4 _ Pcws srss funcon srss funcon σ σ f σ f σ f σ4 f 4 ohrws σ 5 6 S u a marx o rn ou of aa x, x, x, σ Wr h rsul fl OUTPUT.PRN WRITPRNouu : x n of Hrary.MCD Ouu fl OUTPUT.PRN Thr columns of aa m, sran an srss ar nclu n h ouu fl as shown blow: m sran srss σ Srss vs. Sran.5. 5

20 a σ σ Cr Ts Tm Tm b Rlaxaon Ts σ σ σ Tm Tm Fgur. Vscolasc maral characrzaon ss: a cr s, b rlaxaon s. 6

21 ..8 Sran Tm scons 4 Fgur. Mull-sgmn loang rocss s loa w/o WF rgrsson WF rgrsson WF rgrsson Loa, lb Tm, sc. Fgur. Th rgrsson rsuls of rlaxaon s 7

22 s loa w/o WF rgrsson WF rgrsson WF rgrsson Loa, lb Tm, sc. Fgur 4. Th rgrsson rsuls a bgnnng of rlaxaon s. 8

23 Dslacmn, n Tm Sc. Ts Daa Cubc funcon for s Lnar funcon for s Fgur 5. Ts an smulaon of loang schul 9

24 ..9.8 Normalz Loa Ts Daa Nonlnar Rgrsson wh cubc olynomal sran for s Nonlnar Rgrsson wh lnar sran funcon for s Tm Sc. Fgur 6. Normalz loa vs. m for h s aa an nonlnar rgrsson rsuls.

25 ..9.8 Normalz Loa Ts Daa Cubc funcon for s Lnar funcon for s Dslacmn,n. Fgur 7. Ts an rgrsson smulaon of Normalz loa-slacmn for CAV anl

26 RPORT DOCUMNTATION PAG Form Arov OMB No Publc rorng burn for hs collcon of nformaon s sma o avrag hour r rsons, nclung h m for rvwng nsrucons, sarchng xsng aa sourcs, gahrng an mananng h aa n, an comlng an rvwng h collcon of nformaon. Sn commns rgarng hs burn sma or any ohr asc of hs collcon of nformaon, nclung suggsons for rucng hs burn, o Washngon Haquarrs Srvcs, Drcora for Informaon Oraons an Rors, 5 Jffrson Davs Hghway, Su 4, Arlngon, VA -4, an o h Offc of Managmn an Bug, Parwork Rucon Projc 74-88, Washngon, DC 5.. AGNCY US ONLY Lav blank. RPORT DAT May 4. TITL AND SUBTITL Drmnng a Prony Srs for a Vscolasc Maral From Tm Varyng Sran Daa. RPORT TYP AND DATS COVRD Tchncal Mmoranum 5. FUNDING NUMBRS WU AUTHORS Tzkang Chn 7. PRFORMING ORGANIZATION NAMS AND ADDRSSS NASA Langly Rsarch Cnr Hamon, VA U.S. Army Rsarch Laboraory Vhcl Tchnology Drcora NASA Langly Rsarch Cnr Hamon, VA PRFORMING ORGANIZATION RPORT NUMBR L SPONSORING/MONITORING AGNCY NAMS AND ADDRSSS Naonal Aronaucs an Sac Amnsraon Washngon, DC 546- an U.S. Army Rsarch Laboraory Alh, MD SUPPLMNTARY NOTS. SPONSORING/MONITORING AGNCY RPORT NUMBR NASA/TM-- ARL-TR-6 a. DISTRIBUTION/AVAILABILITY STATMNT Unclassf-Unlm Subjc Cagory 64 Dsrbuon: Nonsanar Avalably: NASA CASI 6-9 b. DISTRIBUTION COD. ABSTRACT Maxmum wors In hs suy a mho of rmnng h coffcns n a Prony srs rrsnaon of a vscolasc moulus from ra nn aa s rsn. Loa vrsus m s aa for a squnc of ffrn ra loang sgmns s las-squars f o a Prony srs hrary ngral mol of h maral s. A nonlnar las squars rgrsson algorhm s mloy. Th masur aa nclus ram loang, rlaxaon, an unloang srss-sran aa. Th rsulng Prony srs whch caurs sran ra loang an unloang ffcs, roucs an xclln f o h comlx loang squnc. 4. SUBJCT TRMS hrary ngral, vscolascy, wgh nonlnar rgrsson, Prony srs, mull loang sgmns 7. SC U RITY CL ASSIF IC AT ION O F RPO R T Unclassf 8. SC U RITY CL ASSIF IC AT ION O F TH IS PA G Unclassf 9. SCURITY CLASSIFICATION OF ABSTRACT Unclassf 5. NUMBR OF PAGS 6 6. PRIC COD A. LIMITATION OF ABSTRACT UL NSN Sanar Form 98 Rv. -89 Prscrb by ANSI S. Z