UNIVERSITY OF CINCINNATI. I, Joon-Hyun Lee, hereby submit this as part of the requirement for the degree of: Ph.D.

Transcription

1 UNIVESITY OF CINCINNATI March 8, I, Joo-Hyu, hrby submt ths as part of th rqurmt for th dgr of: Ph.D. : Mchacal Egrg It s ttld: DEVEOPMENT OF NEW TECHNIQUE FO DAMPING IDENTIFICATION AND SOUND TANSMISSION ANAYSIS OF VAIOUS STUCTUES Approvd by: Dr. Jay H. Km Dr. Davd. Brow Dr. Mld Jog Dr. Yju u

2 DEVEOPMENT OF NEW TECHNIQUE FO DAMPING IDENTIFICATION AND SOUND TANSMISSION ANAYSIS OF VAIOUS STUCTUES A dssrtato submttd to th Dvso of sarch ad Advacd Studs of th Uvrsty of Ccat I partal fulfllmt of th rqurmts for th dgr of DOCTOATE OF PHIOSOPHY (Ph.D.) th Dpartmt of Mchacal, Idustral ad Nuclar Egrg of th Collg of Egrg by Joo-Hyu B.S.M.E., Yos Uvrsty, 988 M.S.M.E., Yos Uvrsty, 99 Commtt Char: Dr. Jay H. Km

3 Abstract A w xprmtal mthod to dtfy dampg charactrstcs of dyamc systms ad uqu aalytcal tchqus to study soud trasmsso charactrstcs of varous structurs wr dvlopd ths dssrtato wor. Th dampg dtfcato mthod dvlopd ths wor dtfs dampg charactrstcs of a dyamc systm matrx forms from masurd frqucy rspos fuctos. A thortcal xampl was usd to valdat th mthod ad study th os ffct o th dtfcato rsults. A uqu st of xprmtal masurmts was dvsd to vrfy th practcalty of th mthod ral grg applcatos. Som pottal applcatos ad possbl mprovmts of th mthod wr dscussd. Whl soud trasmsso charactrstcs of structurs ar mportat basc formato os cotrol, th rlatd aalyss usually bcoms a vry dffcult tas bcaus of th complcatd tractos btw th structurs ad acoustc mda. Soluto tchqus wr dvlopd to study soud trasmsso charactrstcs of varous cyldrcal structurs wth a sgl wall, doubl walls, ad doubl walls ld wth porous matral for th frst tm ths study. Grally th systm was dald as a ftly log crcular cyldr subjctd to a pla cdt wav. Th soluto tchqu was xtdd to solv th soud trasmsso problms of prodcally stffd pals ad cyldrs. For all cass, xact solutos wr obtad by usg th full shll vbrato quatos coupld wth th acoustc wav quatos usg th mod suprposto mthod.

4 A approxmat aalyss tchqu was proposd to calculat soud trasmsso through cyldrcal walls ld wth porous matral. Bcaus th porous matral has both sold ad flud phass, whch mas th rlatd aalyss vry complcatd. Th uqu approxmat mthod was dvlopd as a two-stp aalyss allowg rlatvly asy calculato of th soud trasmsso such structurs. A aalyss mthod for stffd plats ad stffd crcular cyldrcal shlls was also dvlopd for th frst tm ths wor. Th spac harmoc xpaso mthod was usd to solv for th prodcally stffd structurs. I all cass of th soud trasmsso studs, aalytcal solutos wr compard to corrspodg masurd rsults xcpt for th prodcally stffd structurs. Th ffcts of mportat dsg paramtrs ar studd to obta usful dsg gudls.

5 Acowldgmts Brg Glory ad Spcal Thas to God! I would l to rmmbr ad acowldg th may dvduals who gav m vry frutful dscussos ad suggstos. I partcular, I would l to xprss my dpst grattud to my advsor Dr. Jay H. Km, whos xcllt gudac, brllat das ad ddcato playd mor tha a sgfcat rol th succss of ths davor. Wthout hs grat gudac, ths dssrtato would ot b possbl. I also wsh to acowldg Dr. Davd. Brow, Dr. Mld Jog, ad Dr. Yju u for bg my dssrtato commtt mmbrs provdg valuabl hlp ad gudac. I would l to tha Dr. I-Ch Wag ad Dr. adall J. Allmag for thr grat tachg ad thoughtful cocrs. I am truly gratful for th mmbrs of UC SD, o spcfc ordr, Dr. Ally Phllps, Bll Fladug, Dr. Doug Adams, Matt Wttr, Da aor, Erc Fr, Jff Hylo, Amt Shula, ad Tom Trrll who hav provdd m wth whatvr I dd, whch abld m to joy my graduat studt lf whl pursug my dgr. Espcally, I wsh to tha Bruc Fouts for Eglsh proofradg ad hs frdshp. I tha hoda Chrstma for hr xcllt scrtaral hlp durg my graduat studt lf at UC SD. My spcal thas go to Yogho, Hsu ho, Bagyog Kum, ad Jsoo Km who hav always b thr wh I dd thm ad provdd stady, costat ad grat frdshp ovr th last 3 yars. I also wsh to acowldg Dr. Xyu Dou ad Dr. Shaoha Ch at Motorola for hlpg m to fd th job opportuty at Motorola. v

6 ast, but by o mas last, I am most gratful to my parts ad sstrs for thr vrdg lov, couragmt, support, ad udrstadg for th last 36 yars, wthout whch ths dssrtato could ot hav b accomplshd. Ths wor was fudd by ArvMrtor Idustrs. I wsh to tha obrt T. Uslma, Dr. Ha-Ju Km, Scott Grr, Vasudva Kothamasu, Dmg Wa, Stv Fag, Dc, ad c amog may othrs at ArvMrtor Advacd Egrg ot oly for th facal support but also for th prsoal frdshp ad couragmt. v

7 To My Parts v

8 Tabl of Cotts TABE OF CONTENTS...I IST OF FIGUES...5 IST OF TABES... CHAPTE INTODUCTION Gral Structur of th Dssrtato... 4 CHAPTE IDENTIFICATION OF DAMPING MATICES OF DYNAMIC SYSTEMS Itroducto Idtfcato Thory Thortcal Valdato of th Procdur Study of Error du to th Nos FFs latv Magtud of Dffrt Dampg Mchasms Study of Nos Effct Exprmtal Valdato of Idtfcato Thory Stratgy for Exprmtal Valdato Ncssary Masurmt ad Sgal Procssg Issus Exprmtal sults Coclusos CHAPTE 3 SOUND TANSMISSION THOUGH SINGE-WAED CYINDICA SHES Itroducto Formulato of th Problm Th Systm Modl Vbro-Acoustc Equatos Soluto Procss Soluto of th Equatos Trasmsso oss (T)... 76

9 3-4. Covrgc Chcg Comparso to a Equvalt -D Modl Formulato of Equatos Soluto Procdur Comparso of th -D ad -D Solutos Comparso to Exprmtal Masurmts Paramtr Studs Effct of th Icdc Agl Effct of Dffrt Matrals adus Effcts Thcss Effcts Coclusos CHAPTE 4 - SOUND TANSMISSION THOUGH DOUBE-WAED CYINDICA SHES Itroducto Aalytcal Soluto Procdur Soud Trasmsso by Bdg Wavs Shlls Soud Trasmsso Aalyss by Pla Wav Modl Combd Solutos Comparso to Exprmtal sults Masurmt Stup Sgl Shll Masurmt Doubl Shll Masurmt Paramtr Studs Choc of Icdt Agl Aalyss Effct of th Doubl-Wall Costructo Effct of Thcss Effct of th Argap Coclusos... 3 CHAPTE 5 SOUND TANSMISSION THOUGH DOUBE-PANES INED WITH EASTIC POOUS MATEIA Itroducto Dvlopmt of Smplfd Aalyss Mthod vw of th Full Thory Dvlopmt of Approxmat Aalyss Procdur Comparso of th Solutos from th Smpl ad Full Aalyss Formulato of th Problm Comparso of th Approxmat Soluto to th Soluto from th Full Thory... 58

10 5-4. Coclusos... 6 CHAPTE 6 SOUND TANSMISSION THOUGH DOUBE-WAED CYINDES INED WITH EASTIC POOUS COE Itroducto Dscrpto of th Problm Drvato of th Systm Equato Doubl Shll wth Bodd-Bodd Porous Matral ayr Doubl Shll wth Bodd-Ubodd Porous Matral ayr Calculato of Trasmsso oss (T) B-B Shll B-U Shll U-U Shll Dscussos Coclusos CHAPTE 7 SOUND TANSMISSION THOUGH STIFFENED PANES Itroducto Formulato of th Systm Equato Soluto Procdur Soluto of th Govrg Equato Th Trasmsso oss (T) Obtad from th Soluto Covrgc of th Soluto Paramtr Studs Paramtrs latd to Modlg Paramtr latd to Dsg Coclusos... 3 CHAPTE 8 SOUND TANSMISSION THOUGH STIFFENED CYINDICA SHES Itroducto Formulato of th Systm Equato Assumd Solutos Boudary Codtos at th Structur-Acoustc Itrfacs Equatos of Moto of th Systm Calculato of Trasmsso oss Covrgc of th Soluto

11 8-5. Paramtr Studs Paramtrs latd to Modlg Study of Dsg Paramtrs Coclusos... 6 CHAPTE 9 CONCUSIONS Summary Cotrbutos commdatos for Furthr sarch BIBIOGAPHY

12 st of Fgurs Fgur.. Thr DOF lumpd paramtr modl... Fgur.. Vscous dampg dx... 5 Fgur.3. Structural dampg dx... 6 Fgur.4. FF of th systm wth.% vscous ad structural dampg ad.% os... 6 Fgur.5-a. Error rato dagram of a systm wth.% dampg ad.% os (a) th dtfd vscous dampg... 9 Fgur.5-b. Error rato dagram of a systm wth.% dampg ad.% os (b) th dtfd structural dampg... 3 Fgur.6. FF of th systm wth.% vscous ad structural dampg ad.5% os... 3 Fgur.7-a. Error rato dagram of a systm wth.% dampg ad.5% os (a) th dtfd vscous dampg... 3 Fgur.7-b. Error rato dagram of a systm wth.% dampg ad.5% os (b) th dtfd structural dampg... 3 Fgur.8. FF of th systm wth.% vscous ad structural dampg ad. % os... 3 Fgur.9-a. Error rato dagram of a systm wth.% dampg ad.% os (a) th dtfd vscous dampg... 3 Fgur.9-b. Error rato dagram of a systm wth.% dampg ad.% os (b) th dtfd structural dampg Fgur.. FF of th systm wth.5% vscous ad structural dampg ad.% os Fgur.-a. Error rato dagram of a systm wth.5% dampg ad.% os (a) th dtfd vscous dampg Fgur.-b. Error rato dagram of a systm wth.5% dampg ad.% os (b) th dtfd structural dampg Fgur.. FF of th systm wth.5% vscous ad structural dampg ad.5% os

13 Fgur.3-a. Error rato dagram of a systm wth.5% dampg ad.5% os (a) th dtfd vscous dampg Fgur.3-b. Error rato dagram of a systm wth.5% dampg ad.5% os (b) th dtfd structural dampg Fgur.4. FF of th systm wth.5% vscous ad structural dampg ad.% os Fgur.5-a. Error rato dagram of a systm wth.5% dampg ad.% os (a) th dtfd vscous dampg Fgur.5-b. Error rato dagram of a systm wth.5% dampg ad.% os (b) th dtfd structural dampg Fgur.6-a. Exprmtal stup (a) clampd bam wthout a dampr Fgur.6-b. Exprmtal stup (b) clampd bam wth a dampr Fgur.7-a. Tst stup (a) schmatc dagram... 4 Fgur.7-b. Tst stup (b) gomtry... 4 Fgur.8. Implcato of dtfyg dampg th vscous ad structural dampg matrcs Fgur.9. A typcal FF Bod plot Fgur.. Illustrato to xpla th msta to comb C ad D matrcs dtfd usg dffrt bads Fgur.. Phas corrcto of FF: Fgur.. Sgl DOF systm Fgur.3. Illustrato of dampg dtfcato usg Argad plot Fgur.4-a. Errors dtfd dampg as a fucto of th phas rror Fgur.4-b. Errors dtfd dampg as a fucto of th phas rror... 5 Fgur.5-a. Errors othr dtfd structural paramtrs as a fucto of th phas rror... 5 Fgur.5-b. Errors othr dtfd structural paramtrs as a fucto of th phas rror... 5 Fgur.5-c. Errors othr dtfd structural paramtrs as a fucto of th phas rror

14 Fgur.6. Phas msmatch foud from th calbrato... 5 Fgur 3.. Schmatc dagram of th sgl cyldrcal shll: -D modl... 7 Fgur 3.. Algorthm for dtfyg th optmum mod umbr Fgur 3.3. Mod Covrgc Dagram for th sgl shll ( =. m, h =. mm) at, H Fgur 3.4. Mod Covrgc Dagram for th sgl shll ( =. m, h =. mm) at, H Fgur 3.5. Schmatc dagram of a ft bam modl... 8 Fgur 3.6. T curvs of th ft bam ad sgl shll Fgur 3.7. Exprmtal stup of T masurmt Fgur 3.8. Calculat T avragd for radom cdt agls of th sgl shll compard wth masurd T Fgur 3.9. T curvs for th sgl shll wth rspct to cdc agl... 9 Fgur 3.. T curvs for th sgl shll wth rspct to matral... 9 Fgur 3.. T curvs for th sgl shll wth rspct to radus... 9 Fgur 3.. T curvs for th sgl shll wth rspct to thcss Fgur 4.. Schmatc dscrpto of th problm: -D modl Fgur 4.. Ts calculatd from -D modl at mufflr codto... 8 Fgur 4.3. Schmatc dscrpto of th problm: -D modl... Fgur 4.4. Ts calculatd from -D modl at mufflr codto... Fgur 4.5. Combd Ts from Fgurs 4. ad Fgur 4.6. Exprmtal stup of T masurmt... 3 Fgur 4.7-a. Calculatd T compard wth masurd T (a) sgl shll, sourc sd... 6 Fgur 4.7-b. Calculatd T compard wth masurd T (b) sgl shll, sourc outsd... 6 Fgur 4.7-c. Calculatd T compard wth masurd T (c) sgl shll, T avragd for radom cdt agls

15 Fgur 4.8-a. Calculatd T compard wth masurd T (a) doubl shll, sourc sd... 7 Fgur 4.8-b. Calculatd T compard wth masurd T (b) doubl shll, sourc outsd... 8 Fgur 4.8-c. Calculatd T compard wth masurd T (c) doubl shll, T avragd for radom cdt agls... 8 Fgur 4.9. Effct of th cdc agl Ts of th doubl shll ( =.m, h =.6mm, h =.4mm)... Fgur 4.. Ts of th sgl shll (=.m, h=.mm) ad doubl shll ( =.m, h =h =.mm)... Fgur 4.. Ts of th doubl shll ( =.m) wth rspct to thcss combato... Fgur 4.. Effct of argap Ts of th doubl shll ( =.m, h =.6mm, h =.4mm)... 3 Fgur 5.. Illustrato of wav propagato th porous layr... 3 Fgur 5.-a. Illustrato of wav propagato th B-B doubl-pal Fgur 5.-b. Illustrato of wav propagato th B-U doubl-pal Fgur 5.-c. Illustrato of wav propagato th U-U doubl-pal Fgur 5.3. Dtald cross-sctoal vw of th op surfac of a porous layr [48] Fgur 5.4. Dtald cross-sctoal vw of porous layr drctly attachd to a pal [48] Fgur 5.5. Fram ad shar wav cotrbutos to th flud ad sold dsplacmts th y-drcto for th B-B doubl-pal (ormald by th strgth of th arbor wav):... 4 Fgur 5.6. Fram ad shar wav cotrbutos to th flud ad sold stra rgs th y-drcto for th B-B doubl-pal (ormald by th rgy of th arbor wav): Fgur 5.7. Fram ad shar wav cotrbutos to th flud ad sold stra rgs th y-drcto for th B-U doubl-pal (ormald by th rgy of th arbor wav): Fgur 5.8. Fram ad shar wav cotrbutos to th flud ad sold stra rgs th y-drcto for th U-U doubl-pal (ormald by th rgy of th arbor wav):

16 Fgur 5.9. Smplfd modl of th B-B doubl-pal Fgur 5.. Smplfd modl of th B-U doubl-pal... 5 Fgur 5.. Smplfd modl of th U-U doubl-pal Fgur 5.. Comparso of Ts of th B-B doubl-pal... 6 Fgur 5.3. Comparso of calculatd Ts of th B-B doubl-pal... 6 Fgur 5.4. Comparso of Ts of th B-U doubl-pal... 6 Fgur 5.5. Comparso of Ts of th U-U doubl-pal... 6 Fgur 6.. Schmatc dagram of th doubl shll wth porous layr Fgur 6.. Cross-sctoal vw of th shll wth a B-B porous layr Fgur 6.3. Cross-sctoal vw of th shll wth a B-U porous layr Fgur 6.4. Th cdt, rflctd, ad trasmttd wavs of th B-U cofgurato th r- pla Fgur 6.5. Cross-sctoal vw of th shll wth a U-U porous layr... 8 Fgur 6.6. Th cdt, rflctd, ad trasmttd wavs of th U-U cofgurato th r- pla... 8 Fgur 6.7. Fram ad shar wav cotrbutos to th flud ad sold stra rgs th y-drcto for th B-B doubl-pal cofgurato (ormald by th rgy of th arbor wav): Fgur 6.8. Comparso of th calculatd Ts of cyldrcal doubl-walld shlls B-B cofgurato Fgur 6.9. Fram ad shar wav cotrbutos to th flud ad sold stra rgs th y-drcto for th B-U doubl-pal (ormald by th rgy of th arbor wav):... 9 Fgur 6.. Comparso of th calculatd Ts of cyldrcal doubl-walld shlls B-U cofgurato... 9 Fgur 6.. Fram ad shar wav cotrbutos to th flud ad sold stra rgs th y-drcto for th U-U doubl-pal (ormald by th rgy of th arbor wav): Fgur 6.. Comparso of th calculatd Ts of cyldrcal doubl-walld shlls U-U cofgurato

17 Fgur 6.3. Comparso of th calculatd Ts of th thr doubl-walld shlls wth thr porous layrs Fgur 7.. Schmatc rprstato of a stffd pal... Fgur 7.. Comparso of th prdctd avragd Ts btw th stffd ad th ustffd pals... Fgur 7.3. Comparso of th prdctd Ts btw th stffd ad th ustffd pals o whch a pla wav s cdt wth a agl Fgur 7.4. Coffct covrgc dagram for th stffd pal (t=.7mm) at 3, H... 3 Fgur 7.5. T curvs for th stffd pal wth rspct to cdc agl... 5 Fgur 7.6. T curvs for th stffd pal wth rspct to phas attuato agl.. 5 Fgur 7.7. T curvs for th stffd pal wth rspct to loss factor... 6 Fgur 7.8. T curvs for th stffd pal wth rspct to stffr mass (K t =3.6 9 N/m)... 8 Fgur 7.9. T curvs for th stffd pal wth rspct to stffr mass (K t =. 5 N/m)... 8 Fgur 7.. T curvs for th stffd pal wth rspct to plat matral... Fgur 7.. T curvs for th stffd pal wth rspct to thcss of th pal... Fgur 7.. T curvs for th stffd pal wth rspct to stffr spacg... Fgur 7.3. T curvs for th stffd pal wth rspct to rotatoal stffss of th stffr... Fgur 7.4. T curvs for th stffd pal wth rspct to traslatoal stffss of th stffr... 3 Fgur 8.. Schmatc rprstato of a stffd shll... 8 Fgur 8.. Comparso of th prdctd avragd Ts btw th stffd ad th ustffd shlls... 5 Fgur 8.3. Comparso of th prdctd Ts btw th stffd ad th ustffd shlls o whch a pla wav s cdt wth a agl Fgur 8.4. T covrgc dagram for th stffd cyldrcal shll (=.m, t=. mm) at 3, H... 53

18 Fgur 8.5. T curvs for th stffd shll wth rspct to cdc agl Fgur 8.6. T curvs for th stffd shll wth rspct to phas attuato agl Fgur 8.7. T curvs for th stffd shll wth rspct to loss factor Fgur 8.8. T curvs for th stffd shll wth rspct to shll matral Fgur 8.9. T curvs for th stffd shll wth rspct to shll thcss Fgur 8.. T curvs for th stffd shll wth rspct to stffr spacg Fgur 8.. T curvs for th stffd shll wth rspct to traslatoal stffss of th stffr... 6

19 st of Tabls Tabl.. Comparso of th dtfcato mthods: ffct of os o th dtfd matrcs... Tabl.. Matrcs of th 3 DOF lumpd paramtr systm... 3 Tabl.3-a. Summary of xprmtal comparsos (a) summary of Tabls 3 to Tabl.3-b. Summary of xprmtal comparsos (b) purposs of comparsos Tabl.4. Dampg matrcs dtfd usg sd bad (35-44 H), thr phas matchd or FM codtod Tabl.5. Dampg matrcs dtfd usg sd bad (35-44 H), phas matchd but ot FM codtod Tabl.6. Dampg matrcs dtfd usg sd bad (35-44 H), phas matchd ad FM codtod Tabl.7. Dampg matrcs dtfd usg low bad (5- H), phas matchd ad FM codtod Tabl.8. Dampg matrcs dtfd usg a dffrt low bad (5- H), phas matchd ad FM codtod... 6 Tabl.9. Dampg matrcs dtfd usg a dffrt sd bad (3-48 H), phas matchd ad FM codtod... 6 Tabl.. Dampg matrcs dtfd usg wd bad (5-8 H), phas matchd ad FM codtod... 6 Tabl 3.. Physcal dmsos ad smulato codtos Tabl 3.. Matral proprts... 9 Tabl 4.. Paramtrs to calculat Ts of th doubl shll at th mufflr codto... 8 Tabl 4.. Paramtrs to calculat Ts of th doubl shll at th tst codto... 4 Tabl 7.. Dmsos of th pal ad smulato codtos... Tabl 7.. Matral proprts of th stffd pal... 9 Tabl 8.. Dmsos of th cyldrcal shll ad smulato codtos... 5 Tabl 8.. Matral proprts of th stffd shll... 57

20 Chaptr Itroducto -. Gral O major accomplshmt of ths rsarch s dvlopg a w mthod to dtfy dampg charactrstcs of a dyamc systm dampg matrcs, whch rprst spatal dstrbutos of th dampg as wll as dffrt dampg mchasms. Dampg paramtrs hav b of rlatvly mor cocr to grs compard to othr modal paramtrs. Oft dampg charactrstcs ar dtfd as modal dampg ratos, whch rprst quvalt ffcts of may dffrt dampg mchasms wthout ay spatal formato. Ths approach s oly vald for small dampg rato, for low os ad low modal dsty. For ths rasos, a w dampg dtfcato thory ad a xprmtal mthod ar proposd ths rsarch to dtfy dampg charactrstcs of a dyamc systm dampg matrcs. Th othr major accomplshmt of ths dssrtato s dvlopg gral aalyss mthods for acoustc-structur tracto problms, spcally for varous cyldrcal structurs. Vbratos acoustc mda ad lastc structurs sstally volv propagato of wav motos. I dalg wth a vbro-acoustc problm basd o th wav dscrpto, th followg thr catgors of practcal problms ar addrssd ths dssrtato. A sgl-walld or doubl-walld cyldrcal shll s commoly usd to cof strog acoustc prssur flld wth a cavty, thrfor th shll must provd good os sulato ovr broad audo-frqucs. It s tdd to dvlop smpl aalytcal 3

21 procdurs for th rlatv comparso of dsg altratvs to complmt dtal aalyss tools basd o a umrcal aalyss tchqu. Vbro-acoustc aalyss of soud trasmsso through structurs ld wth porous matral stll rmas to b a vry dffcult tas may practcal applcatos du to hrt complxty of th mult-wav propagato thory lastc porous matral. Th dsr to fd a smplfd modl srvd as th practcal motvato of studyg ths topc. I ths dssrtato, th smplfd mthod s dvlopd ad th appld to prdct th soud trasmsso through a doubl-walld cyldrcal shll sadwchg porous matral as a practcal applcato. For varous rasos stffd structurs ar oft usd as foud a arcraft ad buldg structurs. Aalyss for vbrato charactrstcs of such structurs s a formdabl tas bcaus of th d to modl th tractos btw stffr ad structur. Vbro-acoustc aalyss to study th soud trasmsso charactrstcs through such structurs obvously bcoms mor dffcult, thrfor ar rarly foud, mostly umrcal wor, ay rportd studs. Ths rsarch s substatatd by th scarcty of th rsarch assocatd wth th vbro-acoustc aalyss of stffd structurs. -. Structur of th Dssrtato Ths dssrtato s composd of chaptrs ad orgad th followg way. I Chaptr, a w xprmtal mthod to dtfy dampg matrcs s dscrbd. I Chaptr 3, a vbro-acoustc aalyss to fd soud trasmsso through a cyldrcal sgl-walld shll s prstd alog wth th xprmtal rsults. I Chaptr 4, soud trasmsso through a doubl-walld shll s aalyd aalytcally ad xprmtally. 4

22 Chaptrs 5 ad 6 prst a smplfd mthod to rprst mult-wav thory dscrbg a lastc porous matral. I Chaptr 5, dvlopmt of th smplfd mthod s xplad ad vrfd by comparg th Ts from th smplfd mthod wth th rportd rsults from th full modl ad masurmts. I Chaptr 6, th smplfd mthod s appld to calculat th Ts of a foam-ld cyldrcal doubl-walld shll for thr dffrt typs of th porous cor. Chaptrs 7 ad 8 ar about dvlopmt of th vbro-acoustc aalyss mthod to calculat th soud trasmsso through a prodcally stffd pal ad a prodcally stffd cyldrcal shll. I Chaptr 7, a vbroacoustc aalytcal modl for stffd pals, whch s supportd by ft stffss lmts at rgular trvals, s dvlopd by applyg th prcpl of vrtual wor ad spac harmoc xpaso mthod. I Chaptr 8, trasmsso of soud s aalyd for a stffd cyldrcal shll, whch s stffd by qually spacd stffg rgs, by th sam mthods as th stffd pal. I Chaptr 9, ovrall cotrbutos of ths vstgato to th stat of art vbro-acoustc aalyss as wll as dampg dtfcato ar summard, ad futur wors that ca b xtdd from ths vstgato ar addrssd. Bcaus two major dffrt studs (dampg ad vbro-acoustcs) compassg sv dffrt subjcts ar rportd ths dssrtato, th ltratur survy s cludd th troducto scto of ach chaptr. 5

23 Chaptr Idtfcato of Dampg Matrcs of Dyamc Systms -. Itroducto A vscous or structural dampg modl dscrbs th rgy loss mchasm a vbratg systm a smpl mathmatcal form []. Th modal dampg or proportoal dampg cocpt furthr uss a assumpto that th spatal dstrbuto of dampg follows th mod shap (modal dampg) or th systm gomtry (proportoal dampg). Such assumptos ar obvously ot always vald. For xampl, wh a catlvr s assmbld to ts bas structur, a rlatvly larg rgy loss mchasm wll xst alog th trfac. If th dampg dstrbuto of such a structur s ow mor dtal, mor accurat strss aalyss of th structur wll b possbl, whch wll bft a hgh cycl fatgu (HCF) aalyss of th structur (.g., a turb blad). I a hgh-spd rotor systm, dffrt dampg mchasms hav dffrt ffcts o th systm stablty [-4]. Thrfor, fdg dffrt dampg mchasms rspctv matrcs wll mprov th qualty of th smulato modl of such a systm. I most past wors, th dampg matrx of a structur has b dtfd usg FFs drctly. Typcally, modal paramtrs such as atural frqucs ad mods ar xtractd frst, th th mass, stffss ad dampg matrcs usg thos dtfd paramtrs [5-]. Sc dampg matrcs hav much smallr ffct o th systm rsposs compard to th mass ad stffss matrcs, th dampg matrcs dtfd ths mar bcom accurat. I a typcal xprmtal modal aalyss [, ], dtal formato of th dampg ffct s usually ot a ma cocr. 6

24 Ovr th past dcad, xtsv rsarch actvts hav b mad th modl updatg, whch th dampg matrx s dtfd as a part of th rsult. For xampl, a crmtal last-squars mthod was usd for th modl updatg by M. Dalbrg [] ad H. G. t al. [3]. Th basc da of modl updatg tchqus s fdg a thortcal modl whos rspos s bst matchd wth th masurd systm rspos. Th dampg matrcs ar dtfd to match th systm rspos of th xprmtal ad thortcal modls, but thr uquss s ot guaratd. Tsu t al. [4-6] dvlopd a mthod that wors drctly o FFs to fd th dampg matrcs as th prmary objctvs of dtfcato. ad Km [7-8] coductd a thortcal valdato of th mthod ad rlatd os study, ad also attmptd a xprmtal valdato of th tchqu. Whl worg to coduct a xprmtal valdato of th mthod proposd by Tsu, t was rald that a much smplr algorthm could b usd. Th mthod uss a dyamc stffss matrx (DSM), or th vrs of FM. Th mthod s vry smpl, rqurg far fwr stps of umrcal opratos compard to th prvously usd mthod. Owg to ths smplcty, th dtfcato rsult s much lss flucd by th masurmt rrors ad oss. A thortcal xampl s usd to valdat th algorthm, study th os ffct o th dtfcato rsults ad dmostrat advatags of th w mthod ovr th prvously usd mthod. A st of xprmtal masurmts s dvsd ad coductd to valdat th practcalty of th mthod. 7

25 -. Idtfcato Thory Th quato of moto of a dgrs of frdom (DOF) dyamc systm subjctd to a harmoc put forc s: Mx jωt ( t) + Cx ( t) + ( jd+ K) x( t) = f ( t) = F(ω) (.) whr, M, C, D ad K ar th mass, vscous dampg, structural dampg ad stffss matrcs rspctvly, j =, ad x(t) ad f(t) ar th dsplacmt ad forc vctors. ttg x jωt ( t) = X (ω), Equato (.) bcoms: ( K Mω ) + j( ωc+ D) X( ω) = F( ω) (.) Th dyamc stffss matrx (DSM) s dfd as: [ H( ω) C ] ( K Mω ) j( ωc D) = + + (.3) C whr, H ( ω ) s th frqucy rspos matrx (FM) dfd as: C C H ( ω ) = Hj = X / Fj, j=,,3,... (.4) I Equato (.4), th suprscrpt C s to dcat th varabl s a complx quatty, ad C H j s th frqucy rspos fucto (FF) masurd btw th ods ad j. Bcaus FM s much asr to masur tha th DSM, th DSM s obtad by vrtg th masurd FM. If th DSM s avalabl, Equato (.3) ca b rwrtt: C mag( H ( ω) ) ωc D = + (.5) C ral( H ( ω) ) K ω M = (.6) 8

26 whr, mag ad ral stad for th magary ad ral parts rspctvly. For xampl, C mag( H ( ω) ) s th matrx composd of th magary part of th DSM matrx C H ( ω). Equatos (.5) ad (.6) ca b put to: D = C C [ I ω] mag H ω ( ( ) ) (.7) whr, I s a dtty matrx, ad: K I ω = M C ral H ω ( ( ) ) (.8) Thrfor, th systm dampg matrcs C ad D ca b foud by a psudo-vrs procdur of Equato (.7) as follows: D C + C ( ( ω) ) C ( ( ω) ) I ω I mag H I ω I mag H = I ωi mag H C ( ( ω ) ) (.9) whr, + mas th psudo-vrs of th matrx. If cssary, th stffss ad mass matrcs ca also b foud: K M + C ω C ω I ω I ral( H ( ) ) I ω I ral( H ( ) ) = I ω I ral( H ( ω ) ) C (.) Equatos (.9) ad (.) hav to b st up at last at two frqucs (=) to ma th quatos solvabl. Usually th quatos ar ovr-dtrmd by usg mor frqucs tha dd. 9

27 As t was show, th procdur tslf s surprsgly smpl, loog almost l a obvous dtty. Howvr th author could ot fd ay prvous wors that usd ths rlatoshp to fd dampg matrcs. Th procdur proposd by Tsu t al. [4-6], whch also fds th dampg matrcs from masurd FFs, may b compard to th proposd mthod. I th mthod, C ad D matrcs ar foud by solvg th followg quato. ωh ( ω) H ( ω) = G( ω) D (.) C N N N whr, H ( ω) th ormal FF, whch s dfd as: H N [ K M ] ( ω ) = ω (.) Th ormal FF s obtad as: ( ω) ( ω) ( ω) ( ω) ( ω) H = H + H H H (.3) N C C C C I I whr, subscrpts I ad stad for th magary ad ral parts, rspctvly, ad C ( ) H ω s th vrs of th ral part of th FM,.. ( ( C )) G( ω ) s dfd as: C C ( ω) ( ω) ( ω) I ral H. G = H H (.4) Th abov mthod s obvously mor volvd. Th objctvs of th dtfcato, lmts of th dampg matrcs, hav physcally small ffct o th FM, thrfor ach of ths xtra stps amplfs th ffct of masurmt rrors or oss. If oly th vscous dampg s usd th modlg, a quvalt vscous dampg matrx C q, that rprsts th tr rgy loss mchasm systm, ca b obtad by solvg:

28 + C I mag( H ( ω) ) C I mag( H ( ω) ) ω ω C q =.... ω I C mag( H ( ω ) ) (.5) If oly th structural dampg s usd th modl, a quvalt structural dampg matrx D q ca b obtad by solvg: + C I mag( H ( ω) ) C I mag( H ( ω) ) D q =.... I C mag( H ( ω ) ) (.6) -3. Thortcal Valdato of th Procdur A 3 DOF systm show Fgur. was usd to compar th proposd mthod wth th Tsu s mthod. Th 3 DOF systm show Fgur. s dfd by th lumpd masss m, m ad m 3 of g, 4 g ad g, ad th sprg costats,, ad 3 of, N/m, 3, N/m ad,5 N/m, th vscous dampg coffcts c, c, ad c 3 of N s/m, 3 N s/m ad 5 N s/m, ad th structural dampg coffcts d, d ad d 3 of N/m, 5 N/m ad N/m rspctvly. Tabl. compars th dtfd rsults from th two mthods wh.5% radom oss ar mxd FFs for two cass, wh th FM s codtod ad ot codtod. Codtog FM volvs mag th matrx symmtrc, to utl th fact that th FM s thortcally symmtrc. Scto ca b rfrrd to for th ffct of ths codtog. Th comparso shows that th rsult from th w mthod s much lss sstv to th masurmt os, gvg much bttr dtfcato rsults. If th FM s codtod, th w mthod dtfs

29 th matrcs symmtrc forms, howvr th othr mthod dos ot, whch dcats that xtra stps th lattr dtrorat th accuracy. K K K 3 C C C M 3 M M 3 D x (t) D D 3 x (t) x 3 (t) Fgur.. Thr DOF lumpd paramtr modl Tabl.. Comparso of th dtfcato mthods: ffct of os o th dtfd matrcs Estmato of Dampg Matrcs From th Thortcal Data wth.5% Nos Estmato Vscous Dampg [C] Structural Dampg [D] Mthod Thortcal Matrx Tsu s Mthod (Ucodtod) Nw Mthod (Ucodtod) Tsu s Mthod (Codtod) Nw Mthod (Codtod)

30 -4. Study of Error du to th Nos FFs Th 3 DOF systm show Fgur. s dfd by th vscous dampg coffcts c, c, ad c 3 of N s/m, 3 N s/m ad.5 N s/m wth th sam lumpd masss, th stffss costats ad th structural dampg coffcts as Scto -3. Th lmts of th mass, vscous dampg, stffss ad structural dampg matrcs of th systm ar calculatd as Tabl.. Tabl.. Matrcs of th 3 DOF lumpd paramtr systm Mass Matrx Vscous Dampg Matrx (N s/m) Stffss Matrx Structural Dampg Matrx (N/m) (g) (N/m) [M] [C] [K] [D] latv Magtud of Dffrt Dampg Mchasms It s xpctd that th ffct of th os o th accuracy of th dampg dtfcato wll b dpdt o ot oly th os lvl but also th magtud of th dampg. For xampl, f th structur s havly dampd, th objcts of th dtfcato (lmts of th dampg matrcs) ar larg, thrfor th accuracy of th dtfcato wll b lss sstv to th os lvl. Thrfor, t s cssary to cross-compar th magtuds of th dampg ad th os lvl. Comparg th os lvl wth th dampg rato mas ss bcaus both ar o-dmsoal paramtrs. For xampl, w may say that a % os s larg compard to a % dampg rato. Sc th dampg rato s th cocpt basd o th proportoal vscous dampg that dos ot hav ay spatal 3

31 formato, t s cssary to df a w cocpt to assss th rlatv magtud of th lmts of gral dampg matrcs. Th dampg forcs ducd by th vscous ad structural dampg mchasms assocatd wth spcfc dgrs of frdom ad j ca b cosdrd as Thrfor, f o dfs a frqucy matrx such as: ωcj ad D j. D j ω j = Cj (.7) Each lmt of ths matrx rprsts th frqucy blow whch th structural dampg ffct s bggr tha that of th vscous dampg. For xampl, ωof th systm our xampl s foud to b 5 rad/s, whch mas that blow 5 rad/s th ffct of th structural dampg s a mor domat rgy dsspato mchasm for th moto ducd at th od by th xctato forc appld at th od. Fgurs. ad.3 show trstg cocpts: th vscous dampg dx ad th structural dampg dx, whch ar calculatd for th systm bg studd. Fgur. (Fgur.3) plots th prctag of th lmts of th matrx [ ω j ], whos valus ar lowr (hghr) tha th frqucy usd as th abscssa. Thrfor, for stac, th structural (vscous) dampg dx s 45 % (55 %) at H Fgur.3, whch ca b trprtd as approxmatly 45 % (55 %) of th systm DOFs (4 out of 9 lmts ths cas) s dampd mor structurally tha vscously. Thrfor, from Fgurs. ad.3, t ca b sad that at or abov H th systm s dampd prmarly by th vscous dampg mchasm, ad at or blow 8 H by th structural dampg. Th frqucy rag btw may b cosdrd as th trasto rag. 4

32 Now, th cocpt of th dampg rato [] s grald. Th lmts of th vscous ad structural dampg matrcs ar dfd trms of th dampg rato as follows. c = ζ m,, j =,, 3. (.8) j j j j d = c ω,, j =,, 3. (.9) j j j Th abov quatos ar usd to rlat th gral dampg matrcs to th dampg rato. For xampl, % vscous dampg matrx ca b dtrmd by Equato (.8) usg ζ j =.. Th, usg th vscous dampg lmts drvd as such, % structural dampg lmts ar dtrmd by Equato (.9). Fgur.4 compars H obtad for th systm wth.% structural ad vscous dampg dfd ths way wth. % os ad % os. Pctag of Vscous dampd Elmts (%) Frqucy(H) Fgur.. Vscous dampg dx 5

33 9 Pctag of Structural dampd Elmts (%) Frqucy(H) Fgur.3. Structural dampg dx - - Ampltud Frqucy (rad/sc) Fgur.4. FF of th systm wth.% vscous ad structural dampg ad.% os , wthout os ;,.% os 6

34 -4.. Study of Nos Effct Th 3 DOF systms wth th sam M ad K matrcs but wth two dffrt lvls of dampg ratos,.% ad.5%, ar cosdrd. For ach cas, 9 FFs ar calculatd, to whch thr lvls of th radom oss,.%,.5% ad %, ar addd. Ths rsults 6 combatos of dffrt os lvls rlatv to th dampg lvls. For xampl, th combato of. % dampg ad % os rprsts a systm, whch has vry small dampg, ad whos masurd FFs ar sgfcatly tatd wth oss. I ths cas a accurat dtfcato s ot xpctd. Furthr, a rror vctor s dfd, whos ach lmt s th dffrc btw th xact valu ad th dtfd valu dvdd by th largst lmt of th dampg matrx. For ths xampl, whch has 9 matrx lmts, th rror vctor s dfd % ut: E ( c E ( c E ( c 3 E4 ( c E5 = ( c E 6 ( c E7 ( c E 8 ( c E 9 ( c N N 3N N N 3N 3N 3N 33N c c c c c c c c c ) ) ) ) ) ) ) ) ) Extractd max([ c j ]) (.) whr, subscrpt N dcats th xact valu. Fgur.5 shows th dtfcato rror dfd as abov for th cas wth. % dampg ad. % os. Wh Equato (.9) s appld to dtfy dampg matrcs, o ca us FFs at frqucs ovr th tr rag or at frqucs aroud th rsoac pas. Th rsult obtad usg th tr frqucy rag s mard as tr frqucy rag ad that obtad usg th frqucs ar rsoac 7

35 frqucs s mard as pa bad Fgur.5 ad othr fgurs to follow. Cosdrg th fact that th dampg ffct s mor prooucd at frqucs ar th rsoac pas, t was xpctd that pa bad cass would provd bttr rsults. Ths turs out to b ot tru for th structural dampg, bcaus ts ffct s prdomatly th low frqucy rag. Fgur.5 shows that th maxmum rror as w dfd s about % wh both th dampg rato ad os ar small but th sam ordr (. % ad. %). Fgurs.6 ad.7 show th cas wth. % dampg rato ad.5 % os lvl. Fgurs.8 ad.9 show th cas wth. % dampg ad % os. As o ca s from vry larg rrors ths fgurs, f th os s sgfcatly largr tha th dampg rato, 5 to ad to ths cass, th dtfcato rsult bcoms uslss. Fgurs. ad. ar wh th dampg s larg compard to th os,.5% dampg ad.% os. I ths cas, accurat rsults ar obtad wth maxmum rror lss tha %. Fgurs. ad.3 ar for th cas wh both th dampg ad os ar larg but at th sam lvl:.5% ad.5%. Th stmato rror bcoms almost th sam ordr (maxmum s about %) as th cas Fgur.5. Fgurs.4 ad.5 ar for th cas wh th dampg s rlatvly larg (.5%), but th os s v largr (%). Aga, Fgur.5 shows that th dtfcato rsult cotas too larg rrors to b usful. Gral coclusos ca b mad from th rror study. Th accuracy of th drct dampg dtfcato algorthm dvlopd ths wor dpds o th magtud of th os rlatv to th dampg magtud. Th dtfcato mthod wors accuratly f th os lvl s th sam as or lowr tha th dampg rato. 8

36 Usg th data from FFs oly aroud th rsoac pa mprovs th accuracy of th dtfd vscous dampg matrx slghtly but ot th structural dampg matrx. 5 Dffrc ato(%) Elmt Numbr Fgur.5-a. Error rato dagram of a systm wth.% dampg ad.% os (a) th dtfd vscous dampg, pa bad;, tr frqucy rag 9

37 5 Dffrc ato(%) Elmt Numbr Fgur.5-b. Error rato dagram of a systm wth.% dampg ad.% os (b) th dtfd structural dampg, pa bad;, tr frqucy rag - - Ampltud Frqucy (rad/sc) Fgur.6. FF of th systm wth.% vscous ad structural dampg ad.5% os , W/O os;,.5% os 3

38 7 x Dffrc ato(%) Elmt Numbr Fgur.7-a. Error rato dagram of a systm wth.% dampg ad.5% os (a) th dtfd vscous dampg, pa bad;, tr frqucy rag 6 x 5 4 Dffrc ato(%) Elmt Numbr Fgur.7-b. Error rato dagram of a systm wth.% dampg ad.5% os (b) th dtfd structural dampg, pa bad;, tr frqucy rag 3

39 - - Ampltud Frqucy (rad/sc) Fgur.8. FF of th systm wth.% vscous ad structural dampg ad. % os , W/O os;,.% os x 4 8 Dffrc ato(%) Elmt Numbr Fgur.9-a. Error rato dagram of a systm wth.% dampg ad.% os (a) th dtfd vscous dampg, pa bad;, tr frqucy rag 3

40 5 x 4 Dffrc ato(%) Elmt Numbr Fgur.9-b. Error rato dagram of a systm wth.% dampg ad.% os (b) th dtfd structural dampg, pa bad;, tr frqucy rag - - Ampltud Frqucy (rad/sc) Fgur.. FF of th systm wth.5% vscous ad structural dampg ad.% os , W/O os;,.% os 33

41 5 5 Dffrc ato(%) Elmt Numbr Fgur.-a. Error rato dagram of a systm wth.5% dampg ad.% os (a) th dtfd vscous dampg, pa bad;, tr frqucy rag 5 Dffrc ato(%) Elmt Numbr Fgur.-b. Error rato dagram of a systm wth.5% dampg ad.% os (b) th dtfd structural dampg, pa bad;, tr frqucy rag 34

42 - - Ampltud Frqucy (rad/sc) Fgur.. FF of th systm wth.5% vscous ad structural dampg ad.5% os , W/O os;,.5% os 8 6 Dffrc ato(%) Elmt Numbr Fgur.3-a. Error rato dagram of a systm wth.5% dampg ad.5% os (a) th dtfd vscous dampg, pa bad;, tr frqucy rag 35

43 4 Dffrc ato(%) Elmt Numbr Fgur.3-b. Error rato dagram of a systm wth.5% dampg ad.5% os (b) th dtfd structural dampg, pa bad;, tr frqucy rag - - Ampltud Frqucy (rad/sc) Fgur.4. FF of th systm wth.5% vscous ad structural dampg ad.% os -----, W/O os;,.% os 36

44 5 Dffrc ato(%) Elmt Numbr Fgur.5-a. Error rato dagram of a systm wth.5% dampg ad.% os (a) th dtfd vscous dampg, pa bad;, tr frqucy rag Dffrc ato(%) Elmt Numbr Fgur.5-b. Error rato dagram of a systm wth.5% dampg ad.% os (b) th dtfd structural dampg, pa bad;, tr frqucy rag 37

45 -5. Exprmtal Valdato of Idtfcato Thory Th fact that a xprmtal dtfcato mthod s worg a thortcal problm s maglss ulss t also wors ral xprmtal cass. A xprmtal valdato wll b cssary to prov th practcalty of th mthod. Howvr, th dffculty ths cas was fdg a dyamc systm whos xact (or thortcal) dampg matrcs ar ow. If such a systm xstd, th valdato ca b do a much smlar way as th thortcal valdato dscussd th prvous scto, comparg xprmtally dtfd dampg matrcs wth th thortcal matrcs. Not owg such a systm, a drct, partal valdato of th dtfcato mthod was dvsd. Fgur.6 shows two systms usd ths xprmt, a bam cofgurd two dffrt ways. Th systm show Fgur.6-a s a uform wdth bam whos ds ar clampd. Th systm Fgur.6-b s obtad by attachg a vscous dampr to th bam show Fgur.6-a. Four odal pots ar usd to df th systm as show Fgur.7, whch mas that th dampg matrcs wll b dtfd as 4 by 4 matrcs. Th vscous dampr th lattr systm was attachd btw th ods 3 ad 4 as show th fgur. Acclratos ar masurd at four odal pots, whch ar tgratd twc to ma th FFs trms of complacs. Th mult-rfrc- mpacttstg (MIT) schm [9] was usd to obta FFs. ovg th xctato to ach odal pot, 6 FFs ar obtad, whch comprs a 4 4 FM. Th FM s vrtd to obta th DSM. 38

46 Fgur.6-a. Exprmtal stup (a) clampd bam wthout a dampr Vscous Dampr Fgur.6-b. Exprmtal stup (b) clampd bam wth a dampr 39

47 Frqucy (rad/sc) Impact Hammr Computr Ampltud Clampd Bam HP-VXI Data-Acqusto Ma Fram Fgur.7-a. Tst stup (a) schmatc dagram =7 mm t=6 mm d=54 mm Acclromtr W=9 mm 3 4 Vscous Dampr Fgur.7-b. Tst stup (b) gomtry 4

48 -5.. Stratgy for Exprmtal Valdato Th valdato stratgy s sstally to obsrv f th dtfd dampg matrcs proprly rflct th udrlyg physcs ad th cofguratos of th two modls, spcally f th followg facts ar obsrvd.. Th dagoal lmts of th dampg matrcs ar postv.. Th systm wth th dampr (Fgur.6-b) shows largr dampg matrcs, spcally vscous dampg matrx, tha th systm wthout a dampr. 3. Th lmts of th dampg matrcs of th systm wth a dampr corrspodg to th ods 3 ad 4 ar rlatvly larg. Satsfyg abov codtos s oly a partal valdato of th dtfcato thory by tslf. Howvr, bcaus th dtfcato algorthm tslf was valdatd thortcally, ths partal valdato s cosdrd ough from th practcal stadpot. Bsds th abov thr codtos, o may b tmptd to us th symmtry of th dampg matrcs as aothr obsrvato pot. Dampg matrcs wll b dtfd symmtrc forms f th FM s symmtrc. Th FM, whch s thortcally a symmtrc matrx, s masurd slghtly o-symmtrc. Ths dvato from th symmtry ca b cosdrd th rflcto of th qualty of th masurmt. Thrfor, th symmtry of th dampg matrcs may b usful to valuat th qualty of th masurmt but ot th qualty of th dtfcato. Ev for that purpos, usg th FM wll b a bttr opto. Th FM may b codtod to a symmtrc form, whch sms to mprov th dtfcato rsult sgfcatly as wll b xplad Scto

49 -5.. Ncssary Masurmt ad Sgal Procssg Issus Svral masurmt ad sgal procssg ssus, som of whch ar ot mportat a covtoal modal aalyss, wr rald to b crtcal dampg matrcs dtfcato aftr may tral ad rrors durg th xprmt. Ths tchcal ssus wll b xplad o by o DOFs of th Exprmtal Modl Th dmso of dampg matrcs to b dtfd s dtrmd by th DOFs of th xprmtal modl. For xampl, f FFs ar masurd at four ods as show Fgur.7, th matrcs ar dtfd as 4 4 matrcs. Usg mor DOFs would provd bttr spatal rsoluto of th dampg formato, howvr at th cost of crasd xprmtal ffort. Also, mor paramtrs (lmts of dampg matrcs) to b foud wll rqur hghr accuracy th masurmt. DOFs of th xprmtal modl wll hav to b dtrmd cosdrg th cssary spatal rsoluto ad practcal lmtatos Slcto of Frqucy ag Modlg th systm quato usg C ad D matrcs mpls that th dampg forc s modld as a lar fucto of frqucy. As Fgur.8 llustrats, th dtfcato procss ca b cosdrd tryg to fd a bst fttg straght l from scattrd xprmtal data pots rprstg th dampg forc. From th fgur, t s asy to s that th matrx D wll b foud mor accuratly f th FM data ar ta from th low frqucy rag to form th dtfcato quato (Equato (.9)). Howvr, acclromtrs grally hav poor accuracy th low frqucy rag, whch s furthr 4

50 dtroratd wh tgratg acclrato to dsplacmt. Fgur.9 s o of th masurd FFs, whch shows that th data blow 5 H ar ot accurat. Th dampg ffct o th systm rspos s mor prooucd aroud th rsoac frqucy (about 383 H ths cas as s Fgur.9). Thrfor, th masurd data hav ffctvly hghr sgal to os ratos aroud th rsoac frqucy. Ths s why sd bads (frqucy rags btw half powr pots) hav b usd dampg dtfcatos. Cosdrg ths facts, th frqucy rag was chos as follows ths wor. Data blow 5 H ar dscardd. Th low frqucy rag s dfd as 5 H to H. Usg data from ths rag s xpctd to provd a mor accurat D matrx. Th sd bad was obsrvd as 378 H to 389 H for th udampd bam ad 374 H to 4 H for th dampd bam. As a comproms (ad also dg ovr-dtrmato of th dtfcato quato), th sd bad ths xprmt s dfd as th rag 35 H to 44 H. Usg ths bad s xpctd to provd a mor accurat C matrx. Usg th frqucy rag of trst of th partcular problm may also b a opto, spcally practcal stuatos. O may b tmptd to comb th C matrx dtfd by usg th sd bad ad th D matrx dtfd usg th low frqucy bad. Fgur. llustrats th problm ths approach, whch wll ovrstmat or udrstmat th dampg forc. 43

51 F d D q Bst fttg l: D q C D Bst fttg l: Cω+D C q Bst fttg l: C q ω Fgur.8. Implcato of dtfyg dampg th vscous ad structural dampg matrcs ω - Ampltud -4-6 Bad spos Sgal Frqucy (H) Phas Agl(dgr) Bad Phas Sgal Frqucy (H) Fgur.9. A typcal FF Bod plot 44

52 Imag(H C (ω) - ) =Cw +D C D C C D Fgur.. Illustrato to xpla th msta to comb C ad D matrcs dtfd usg dffrt bads, dtfd usg low bad; -----, dtfd usg sd bad;, combd modl ω Sg Covto of FFs Th sg covto of FFs s ot mportat a typcal modal aalyss as log as t s usd cossttly. I othr words, cosstt us of thr -X/F or X/F would ot caus ay problm fdg atural frqucs ad mod shaps. Howvr, th dtfcato procdur xplad Scto -, usg X/F wll rvrs th sg of th matrcs to b dtfd. Espcally, mxd us of sg covtos wll ma th dtfcato rsult vald. Th problm ca b avodd by mag th acclrato ad xctato drctos th sam at all masurmt pots. For a pot whr ths s ot possbl, th phas agl of th corrspodg FF has to b corrctd umrcally. I practc, t wll b prudtal to chc all FFs ad ma sur that thy all start wth ro phas agl at th low frqucy rag by addg or subtractg 8 o f cssary. Fgur. shows such a corrcto that w dd for o of th masurd FFs. 45

53 - Ampltud Frqucy (H) Phas Agl(dgr) Frqucy (H) Fgur.. Phas corrcto of FF:, bfor corrcto; -----, aftr corrcto Phas Matchg btw th Forc ad Moto Trasducrs Bcaus th dtfcato uss th magary part of th DSM, th FFs hav to b obtad wth accurat phas agl, whch rqurs a accurat phas matchg btw th forc ad moto trasducrs. Itally th mportac of th phas matchg was ot rald bcaus t sldom bcoms a mportat ssu covtoal modal tstg. Ths problm ca b bst xplad by a sgl DOF xampl show Fgur.. Th FF of th systm s: X K M j C D H( ω ) = = F ( K ω M) + ( ωc+ D) ( ω ) ( ω + ) (.) Fgur.3 shows th Argad plot [] of ths FF. As th frqucy crass, th plot starts from /K th ral axs ad crosss th magary axs at pot P, whos coordat, ( ωc+ D), s usd to fd th quvalt vscous dampg C q = ω C+ D. 46

54 Now, suppos that thr s a phas agl rror of φ rada btw th forc ad dsplacmt sgals. Th FF wll b masurd as: ( K ω M) j( ωc + D) H( ω ) = ( K ω M) + ( ωc+ D) ( K ω M) + φω ( C+ D) + j[( K ω M) φ ( ωc+ D)] jφ ( K ω M) + ( ωc+ D) (.) Th Argad curv crosss th magary axs wh th ral part bcoms ro, thrfor: K ω M φωc D = ( + ) (.3) By substtutg ths to Equato (.), t s rald that th coordat of ths pot rmas th sam bcaus: ( + φ )( ωc+ D) ( + φ )( ωc + D) ( ωc + D) H( ω ) = = (.4) K =5 N/m C = N s/m M = g D =5 N/m x (t) Fgur.. Sgl DOF systm 47

55 /K al H(ω) ( ωc+ D) P Imagary Fgur.3. Illustrato of dampg dtfcato usg Argad plot Thrfor, t s s that th phas msmatch would ot affct th dampg paramtr ths mthod. To s th ffct of th phas msmatch o th dampg matrcs dtfd by th proposd mthod, lt th paramtrs M, K, C ad D of th systm Fgur. b Kg, 5, N/m, N s/m ad 5 N/m rspctvly. Th th proposd mthod (Equato (.9)) s appld to fd C, D, M ad K for varous phas msmatchs (φ rada). Fgurs.4-a ad b show th rrors th dtfd C ad D matrcs as fuctos of th phas agl rror prctag,.., rror /xact valu. Fgurs.5-a,b,c rprst th rrors th dtfd M, K, ad th atural frqucy. As t s show, th phas msmatch causs much largr rrors C ad D compard to othr modal paramtrs. 48

56 Fgur.6 shows th phas agl btw th sgals from th forc trasducr ad o of th acclromtrs usg th rato calbrato stup [9]. Th phas msmatch Fgur.6 s compsatd umrcally at ach frqucy to corrct 4 FFs obtad from ths st of th acclromtrs ad forc trasducr. All 6 FFs ar rcostructd ths way bfor thy ar usd to dtfy th dampg matrcs Error ato(%) Phas Shft (Dgr) Fgur.4-a. Errors dtfd dampg as a fucto of th phas rror -----, xact valu;, dtfd valu, (a) vscous dampg C 49

57 5 5 Error ato(%) Phas Shft (Dgr) Fgur.4-b. Errors dtfd dampg as a fucto of th phas rror -----, xact valu;, dtfd valu, (b) structural dampg D - - Error ato(%) Phas Shft (Dgr) Fgur.5-a. Errors othr dtfd structural paramtrs as a fucto of th phas rror -----, xact valu;, dtfd valu, (a) mass M 5

58 - - Error ato(%) Phas Shft (Dgr) Fgur.5-b. Errors othr dtfd structural paramtrs as a fucto of th phas rror -----, xact valu;, dtfd valu, (b) stffss K..5 Error ato(%) Phas Shft (Dgr) Fgur.5-c. Errors othr dtfd structural paramtrs as a fucto of th phas rror -----, xact valu;, dtfd valu, (c) atural frqucy ω 5

59 5 4 3 Phas (dgr) Frqucy (H) Fgur.6. Phas msmatch foud from th calbrato Codtog of th FM A FM (or DSM) s always masurd slghtly o-symmtrc, whl t s thortcally symmtrc. Th FM ca b mad symmtrc by avragg two FFs, usg (H j + H j )/ for both H j ad H j. It was foud that ths codtog ot oly mas th dtfd matrcs symmtrc but also mprovs th qualty of th dtfcato rsults, prhaps bcaus of th avragg ffct. Itrstgly ths codtog dd ot wor wll wth Tsu s mthod. Th mthod dtfs o-symmtrc dampg matrcs dspt usg a codtod FM, whch may hav b causd by accumulato of umrcal rrors du to xtra stps of th mthod. -6. Exprmtal sults Tabl.3 summars all th subsqut tabls, how thy wr obtad ad compard to o aothr. For xampl, th tabl shows that th dampg matrcs Tabl.4 wr dtfd usg th FM thr phas matchd or codtod ad usg th sd bad 5

60 (35 H ad 44 H). Tabl.3-b summars th purposs of th comparsos mad. All tabls show th dampg matrcs th sam format, lstg C, D, C q ad D q matrcs for two systms. Grally, th dffrt cofguratos of th two systms ar rflctd rasoably wll all cass. For xampl, dampg matrcs of th systm wth a dampr hav largr dampg matrcs all tabls. Th ffct of th phas matchg ca b s by comparg Tabls.4 ad.5, ad th ffct of th FM codtog ca b s by comparg Tabls.5 ad.6. Phas matchg mprovs th rsult gral, spcally judgg from th quvalt matrcs dtfd, whos dagoal lmts bcom arly all postv Tabl.5 as th phas s matchd. Comparg Tabls.5 ad.6 shows that th ffct of th FM codtog mprovs th dtfcato rsults a ovrall ss from th thr obsrvato pot of vws dscrbd Scto -5.. It s blvd that th avragg ffct of th FM codtog mprovs th rsult addto to th obvous ffct of mag th matrcs symmtrc. By comparg Tabls.6 ad.7, t ca b s that th structural dampg matrx obtad ar of hghr qualty f th lowr frqucy bad data s usd. Substatally dffrt matrcs ar obtad dpdg o whthr th sd bad or th low bad s usd. Ths dcats that th actual dampg mchasm of th systm s ot a lar fucto but a hghr ordr fucto of frqucy. For xampl, th small dampr usd th xprmt wll b thr vscous or costat. Cosdrg ths, o may us th frqucy bad of trst to dtfy th dampg matrcs. If th rag s too wd, a pcws lar modl or a o-lar dampg modl may hav to b usd. For th lattr, th dtfcato mthod wll hav to b modfd to clud hghr ordr trms. 53

61 Comparsos of Tabls.7 ad.8, Tabls.6 ad.9 show that a small chag th frqucy rag rsults also small chags th dtfd matrcs. Ths dcats that th larg dffrcs btw th rsults Tabls.6 (obtad usg th sd bad) ad.7 (obtad usg th low bad) wr ot causd by a umrcal problm but by th atur of th systm. Tabl. s th dtfcato rsults obtad usg a wd frqucy rag (5-8 H), whch cluds both th sd bad ad low bad. Such a rsult may b usd to rprst th systm a avrag ss for a wd frqucy rag of trst as a altratv to a pcws or o-lar modl. Aothr trstg obsrvato s that whl th bam wthout a dampr has a symmtrc gomtry, th dtfd rsult dos ot rflct th symmtry (.g., D s qut dffrt from D 44 Tabl.6). Howvr, th gomtrc symmtry s bttr rprstd th C matrx wh th sd bad s usd (s Tabl.6), ad D matrx wh th low bad s usd (s Tabl.7), whch s cosstt wth th prvous dscussos. Durg our xprmt, t was obsrvd that v a small dstorto of th systm rsults substatally dffrt dampg matrcs, whch also do ot show ay gomtrc symmtry. Ths may b xplad by th fact that a varato of th gomtry or clampg codtos, v f thy ar small, ca caus sgfcat chags th rgy loss mchasm. Ths fatur may b xplotd for a postv purpos. For xampl dtfd dampg matrcs may b usd to spct th qualty of th assmbly of hgh prcso qupmt. 54

62 Tabl.3-a. Summary of xprmtal comparsos (a) summary of Tabls.3 to.9 Tabl No. Frqucy ag(h) Phas Match FM codtog Tabl (sd bad) No No Tabl (sd bad) Ys No Tabl (sd bad) Ys Ys Tabl.7 5- (low bad) Ys Ys Tabl.8 5- (low bad) Ys Ys Tabl Ys Ys Tabl. 5-8 Ys Ys Tabl.3-b. Summary of xprmtal comparsos (b) purposs of comparsos Comparso Tabl.4 vs Tabl.5 Tabl.5 vs Tabl.6 Tabl.6 vs Tabl.7 Tabl.7 vs Tabl.8, Tabl.6 vs Tabl.9 Tabl. vs Tabl.6, Tabl. vs Tabl.7 Effct to dscuss Phas matchg FM codtog ow frqucy vs sd bad Gral frqucy dpdc Wd rag vs low rag vs hgh rag 55