ME 563 MECHANICAL VIBRATIONS

Transcription

1 E 56 echcl Vb Fll 8 E 56 EHANIAL VIBRATIONS Fll 8 Pe WF :.m.-:.m. Iuc: Pf. D. E. Am Rm: E 6 Eml: em@uue.eu Phe:

2 E 56 echcl Vb Fll 8 Iuc echcl Vb. B vb, g vb, he le f ly Vb e cll mechcl ymc yem. Alhugh y yem c clle whe fce eelly, he em vb mechcl egeeg fe eeve f yem h c clle feely whu le fce. Smeme hee vb cue m eu efmce fey blem egeee yem. F ce, whe cf wg vbe ecevely, ege he cf becme ucmfble eeclly whe he fequece f vb ce ul fequece f he hum by g. I fc, well kw h he e fequecy f he hum el c. 4-8 Hz hul be ve ll c whe egg hgh efmce cf euble luch vehcle becue ue eue c cue eu el um Lehew Demey, 976 NASA TN D-888. If cf wg vbe lge mlue f eee e f me, he wg wll eveully eeece fgue flue f me k, whch wul elly cue he cf ch eulg jue / fle. Wg vb f h ye e uully ce wh he we vey f flue heme bugh by flu-ucue ec. The m fmu egeeg e f ll me w he Tcm Nw Bge e 94 ee Fgue. belw. I fle ue he me ye f elf-ece vb behv h ccu cf wg. Fgue.: lef Vew f Tcm Nw Bge lg eck; gh vew f l vb I eg bk echcl e vb clug he evu gh, egeeg ue e uully lef wh he me h ll vb e emel becue m ublcze wk cue vb euc e fm he. Bu -

3 E 56 echcl Vb Fll 8 vb c l be beefcl. F ce, my ffee ye f mg e ely fg vb hugh whch ffee ze cle e e ug vb. I ue, vb e l ue by ll k f ffee ece he ly lve. Ob web e, f emle, ue vb he web eec he eece f fle he ec hey uggle fe beg cue he web f f. The e h mechcl yem vbe feely becue eegy echge bewee he yem el me eleme elc g eleme. Fee vb uully cee fe ce legh f me becue mg eleme yem e eegy cvee bck--fh bewee kec eegy el eegy. The le f mechcl vb ly hul be ue mhemcl l f melg ecg el vb blem lu, whch e uully bvu elmy egeeg eg. If blem c be ece, he eg c be mfe mge vb blem befe yem e mufcue. Vb c l be elly uce eg ke vge f beef f elve mechcl m ee yem e.g., cg mccy. Ufuely, kwlege f vb elmy mechcl eg ely cee eel, my vb ue e ce u ly fe yem e mufcue. I hee ce, vb blem mu be ee ug ve cve eg mfc. Smeme eg mfc my be mle hcke chge vbg el; e hcke e uh he e fequece f el hghe leg le vb he eg fequecy ge. Deg mfc c l be cmlce eg elec-helgcl ER flu me mechcl yem ke eegy wy fm vbg yem ecfc me ug he m. The hee h eg chge mufcue e le eeve me effecve h eg mfc e le.. elg ue elg uully 95% f he eff el-wl mechcl vb blem; hweve, h cue wll fcu mly he ev f equ f m, fee ee fce ee ly, me lu meh f vbg yem. Fgue. llue e emle f why melg c be chllegg mechcl vbg yem. A lge ce hg ck hw lg/ulg ckge fm cg h. I e ble vb ce, he cble mgh be elze mle he ce elze g. I -

4 E 56 echcl Vb Fll 8 h mle ce, he ckge ce bh clle g be; he ckge clle bu he e f he ce he ce clle bu be f he w echge eegy. Thee vb wul m lkely ce elvely lw fequece wul ke lce he g ymcl m f he ce ckge. Tw cule y ffeel equ wul be eee h ce mel he cee, eee m f he ce ckge. Th mel mgh be uffce me ce, bu wh f he m f he cble cmble he m f he ckge? I h ce, he ce ckge ll behve lke g be, bu he cble wll bbly vbe ehe veely lgully cuu by lg legh. Thee hghe fequecy vb wul eque h bh y ffeel equ f he ce ckge l ffeel equ f he cble be ue mel he ee yem. Fuheme, f he um f gy he ce wee l ele, he wul ee be mele wh l ffeel equ. All f hee cmlc wul be ueme f he mle g by ymc f he ce ckge. Fgue.: e f lg/ulg ckge fm cg h. Dffee egme f e eque ffee level f hc he mechcl vb mel. We wll hve he uy cu melg ce hughu he cue whe ce ue f vb heme e ue efce heecl cce ly ceue. Befe g lyze yem, we mu be ble eve ffeel equ -4

5 E 56 echcl Vb Fll 8 f m h equely ecbe he yem. Thee e my ffee meh f g h; hee e cue he.. Le ue wkg cle We c cu eveyhg h cue. I cul, hee uffce me ee le le meh f vb ly. Theefe, he cue wll mly fcu le vbg yem le che ly. Oly ce ecl chcec f le yem wll be uce ug he emee. Becue he ec h bee me lk mly bu le yem, he cle f ue wll hl evey blem h cue. Ie f g h cle he begg f he cue, efeecg whe eee f ev, we wll vew me wkg cle. I he w, le ue wll gue u u ly f fee fce le vb. Whe we beg lyze vb, we wll lk he cle f ue hel u mve fw u lye. Recll h mhemcl e, L[ ], whch bey he cle f le ue by ef fe he fllwg w ee: [ ] L[ ] L [ by] L[ ] L[ by] L.. whee L ee he w ffee fuc, y, b e c. Eq.. he cle f hmgeey Eq.. he cle f vy. Thee w ee my eem vl bvu, bu hey wll fc be ely ueful le he cue. The m emembe h le yem, whch e gvee by le e, L[ ], e equl he um f he. Alhugh h eme fu my eve be fu f hlhcl cu, he mv f ug le vb he ce f le ue hee h mke vb ly fee fce yem much ee evel ue. e wll be bu ue he. -5

6 E 56 echcl Vb Fll 8.4 Revew f kemc geelze ce ec Th ec wll evew me f he fumel echque cle kemc. Ne h h ly evew em beg me evew eveyhg hee. Sue hul ke h uy efeh he meme f uegue cue mechc. Geelze ce e he b f u kemc ec f vbg be. Geelze ce e uully ehe vble e.g.,, y, z,, gul vble e.g., φ, θ, α, cmb hee f e.g., cφ. We mu che u e f geelze vble ech blem equely ecbe he e f ll be he mechcl yem f ee. Ne h he e e bh m becue bh f hee ce e ce wh kec el eegy ge. The mmum umbe f ce eque equl he egee-f-feem DOF. Smeme he umbe f DOF bvu. F emle, Fgue. llue eulum-c yem h h my lg g e. The que : Hw my geelze ce e eque lce e ll f he e? Fgue.: -g-me-eulum c yem I e lce e evey by, e f e ce eee f, e ce f, e ce f, w ce f 4, w ce f 5 : l f eve ce e eee. Bu ll f hee ce e cully eque becue hee e me c bewee hem. F, he l f equl he -6

7 E 56 echcl Vb Fll 8 m f he cee-f-m f 4. Sec, he l f 4 l, 4 R 4 θ 4. Th, he l f 5 equl he l f lu cme ue he f 5. Wh l f hee c, he umbe f DOF euce fllw: ce kemc / gemec # DOF # # che c equ 7 4 Smly u, h eme mle h hee hul be fu ffeel equ f m f h yem. The f e y lycl vb blem hul be cmue he umbe f DOF. A e, he umbe f DOF.e., ½ he e f he yem mu l lwy be eme lyg eemel vb echque. Thee e my ffee cmm e f geelze ce ue f mechcl vb ly. If vec,, f cle cee-f-m we em f ce geelze bly cuvle ce, q, fllw: q, q q, he he ffeel ge vec,, gve by,. q q q q hq hq h q h h q e h q e h q e q q q h q h q.4 whee he geelze ce u vec, e k, e gve ehe he ec f hee equ e che be hgl m vb blem, he h kk e cle fc ce wh he geelze ce ffeel, q k. F emle, he cylcl ce yem em f he e ce u vec gve by, -7

8 E 56 echcl Vb Fll 8 c θ θj zk.5 he ffeel vec, c θ θj θ cθj zk θ wh he fllwg cle fc hgl u vec:.6 h e cθ θj, e θ cθj, e k, h θθ, h zz θ z.7 The fumel heem f kemc c w be ue cmue he velcy vec ecly h equ ue le whe cmug velcy vec f he kec eegy em. The geel ecfc fm cylcl ce f he velcy, /, e gve belw: & hq& e hq& e & e & θe z& e h q& e.8 Thee e he ueful cle-fm ee f kemc vble lke ccele, f emle, whch ve gh vu meh f evg equ f m he e che. eve, we wll ee le h he ccele he e k ec c be cmue fllw: T T T & & k h q&, kk k qk e.g., cylcl θ & θ && θ & & θ.9-8

9 E 56 echcl Vb Fll 8 whch f ecly Lgge ch f evg ffeel equ f m. The m e hee h we mu be ble ecbe hw be mve e eve hycl equ f m. emc he f eel eleme f vb ly. I he e che, meh f cmbg he kemc hyc ug New vecl meh lycl meh e.g., eegy/we, Lgge e cue..5 Revew f eegy we ee Th ec wll evew me m e eegy we mechcl yem. Thee e e m becue we wll ue lycl eegy meh my blem eve equ f m f vbg yem. If ceme f mechcl wk cle f m ee ΔW, he h mu f wk c be clcule ug he fce le he cle lg he cemel chge h,, f he cle fllw: Δ W F. whee he ee uc bewee he fce ffeel lceme vec. Ne h he cemel mu f wk cl. Scl que lke h e wll be ue he evel eleg meh f evg cmlce equ f m. Becue he kec eegy, T, f cle equl he e f chge ΔW wh me, cvee ee f T c be eve fllw: T T T ΔW lm Δ Δ F & & F f c. -9

10 E 56 echcl Vb Fll 8 whee he h equ he equece w be ug New ec lw ee Sec. belw. Th ee f he kec eegy f cle wk f y che e f ce wh eec el efeece fme. Whe ug cmue mul lve ymc/vb blem, cmm cce mleme Eq..8 e clcule he velcy vec. F g by, whch fe umbe f uch cle, he kec eegy c be fu by ummg he kec eege f ll f he vul cle f m,. Th umm efme ug egl ve he ee by fllw T & & By. whee he vec,, he f he m cle, ee Fgue.. F e h wll becme cle le, be we wh eec he cee f m lc, R, fllw: R & R& R& / / / /. whee / he vec c legh w fm he feml cle he gul velcy f he g by. Remembe h he gul velcy f g by e ee ll m. Whe Eq.. ubue Eq.., he fllwg equece f e le geel ee f he kec eegy f g by: -

11 E 56 echcl Vb Fll 8 T R& R& / / By R& R& R& R& R H / By By / /.4 whee H he l gul mmeum f he g by bu he. F m he le wh gul velcy k velcy vec V V V y j, he l kec eegy, T v v H y y v v k I v v I k whee I he l m mme f e f he by bu he..5 The el eegy, V, f cle g by l m he wk fllw. e ce whee eel fce vec, F, c be we em f ecl el fuc, VV,, whch ly fuc f he cle ce el efeece fme, e, e, e, bly me, F V V e V e V e.6 I h ce, he l wk e he el eegy ecee clcule fllw, -

12 E 56 echcl Vb Fll 8 W V V V.7 F ce, le g h c e le fce ccg F-, whee he g c f he g fce/lceme, he clcul Eq..7 gve by, V W V V V.8 Thu, he el eegy e le g /, whee he fl lceme f he e f g h lly uefme. The el eegy e he gvl fel bewee m he eh gve by Vg, whee he vecl ce fm by um efeece hegh he m. Ne h bh he g he gvl el eege, cevve fce ce wh he e eegy. I he g he cevve fce F-, whee he gvl fel he fce F-gj. We wll ee he e he h by ecg ll cevve fce, F c, fm he l eel fce by, FF c F c, eeeg hem wh el eegy em, eegy/we meh f evg equ f m c be be que ely. We wll hve my ue cmue el eege ffee blem he e he. Equ f I e cu vb mechcl yem, we mu f eve mhemcl equ, whch c he be ue lyze he fee fce vb f ee. Thee equ h ecbe he hyc kemc f yem e clle he equ f m EO. Thee e my echque f evg hee equ mg hem New ec lw f m Eule equ, he cev f eegy f lw f hemymc, Hml cle, y f lycl vl meh ymc e.g., cle f -

13 E 56 echcl Vb Fll 8 Ju, Gu-Gbb clug Lgge equ cl I II, he. We wll ly lk bu few f hee. Sue e ecuge cul e ymc by Geew, ll, Thm, Te, Rch, f hugh evew f ech echque. The m hg ue hee hw hee echque e ffee. I he w, we w kw whch meh che f cul ye f blem why. F ce, New ec lw Eule equ ffe me gh my ce becue hey vlve vec, whch e fe ely vulze he cue f lvg he blem; ufuely, h gh cme ce: ll f he eel fce ech fee by gm mu be kw eeble em f he eee vble, he ccele f ech fee by mu be cmue. Fce el-wl blem uully vlve hyc h e well kw e.g., fc, eymc buy c, ec., whee ccele vlve eeve mu f kemc lgeb, whch c be cmlce whe g by ce yem e ue. The key lyg New meh elec he geelze ce blce he eff eque mhemclly ee he hyc ΣF kemc A V /. Dug h cue, we wll em llue g wy che meh f evg EO geel.. New ec lw Eule equ New ec lw f ll m f g by f c m ubjece eul eel fce vec, F, P & F R, whee P V. whee R he vec f he wh eec el efeece fme. Eule equ f he l m f g by, H, whee H R P. -

14 E 56 echcl Vb Fll 8 whee F he um f ll eel fce cg e, P he l le mmeum f e mvg wh velcy V, H he l gul mmeum f e bu w cee f m, R he f he cee f m el efeece fme, V R / he blue velcy f he cee f m wh eec he el efeece fme. The f hg emembe bu Eq.. h he ccele velcy mu be clcule wh eec el efeece fme. NEWTON S SEOND LAW IS NOT VALID FOR RELATIVE OORDINATES. The ec hg emembe bu Eq.. h F clue ALL eel fce cevve -cevve he by. Thu, he fee by gm FBD, whch hw ll eel fce mme cg he by, mu be ccue ele he eulg EO wll be wg. Alwy mke ue h he EO mch he FBD. We wll hve my ue efce hee w ule. Alhugh he fm f Eule equ Eq.. cec lwy wk, hee e leve fm, whch e ee ly my ce. If A che u whch evel Eule equ e f, he he mme equ c be we evel he fm fllw: A H H H A A R& A A/ A/ R& R&& R&& A whee he vec hee equ e efe Fgue...,b,c -4

15 E 56 echcl Vb Fll 8 Fgue.: e yem f evelg vu fm f Eule equ The e Eq..,b,c e ll ueful becue hey mlfy he kemc emeuly ce ce. e he yem Fgue. belw. A k ll cle ulle w he cle by gvy u he cle by he le g,. The ceg FBD hw he gh f he fgue. We wll w llue he meh f evg EO ug New vecl echque.. F, w chemc f he yem f e ve. Ag my ce yu ee efe he e f ll be he yem; mke ue lbel he ve ec f ll ce efe y u vec yu feel e e lve he blem. We e llwe ue elve ce he blem; hweve, we mu emembe h he ccele em New equ Eule equ mu be blue.e., wh eec el efeece fme. The mmum umbe f ce eee lce e ll be he yem by ef he umbe f DOF; yu hul lwy e h yu lu. The umbe f DOF he yem Fgue. e becue ly φ eee lce e he k. If yu e bluely ue bu he umbe f DOF, he fllwg fmul my hel yu f yu eu ce: ce kemc / gemec # DOF # # che c equ.4-5

16 E 56 echcl Vb Fll 8 F emle, f we che bh φ lce e he k, he we c ue Eq..4 b he umbe f DOF wh he gle c, -φ: #DOF#c.- #c-. We wll che he he gle, φ, u m ce. Th me we wll ee ue he c equ, -φ, elme ll ccuece f u equ. Se mly ccee wh he kemc f he yem. Fgue.: Lef Sug k cle gh fee by gm FBD. Se y um yu me wg he chemc y um yu wll mke evg he EO. F ce, he l c Fgue. me h he velcy f he k f cc wh he ufce ze.e., he velcy f he k elve he ufce ze, V cc/a, whee A e he cle he f cc. Th um m mu be e.. Dw he FBD f evey by he yem. Dw FBD eve f he be h yu beleve e eee. We mu u l f hugh FBD becue hey wll eeme whehe we ge he cec EO he e. I uully be e-w ce yem wh ech FBD ceg vul be he yem. mue fce g me by ug he elve m c he eleme; he ec f hee fce e eeme by emembeg h fce me e cee elve lceme velcy c hem. Whe we w vec he FBD, we lbel vec wh mgue he vec ke ce f ecly. -6

17 E 56 echcl Vb Fll 8 4. Se few mme hkg bu hw yu wll we w ly my equ yu ee f he EO. We cul we w evey equ we hk f, bu h uully effecve hkg bu whch equ we ee why. F emle, we cul we w bh he fce mme equ f he cle llg k Fgue.. The fce equ he ec wll vlve he fc fce ukw em f ce, gvy kw, he eg fce he g kw. The fce equ he y ec wll vlve he ml fce ukw gvl cme kw. The mme equ cul be we u evel ffee efeece : Au cee f m: H j f I & φk fk I && φk I && φ f.5 Au A f cc e he cle: A H A & A I & φk j gj j g αk R& k I && φk I & φ g α u H A u A A u.6 Au B eecul f cc: B H B & B R& I & φk j f j gj fk g αk H B k I && φk I && φ f g α u u B B u.7-7

18 E 56 echcl Vb Fll 8 I h ce, he mme equ bu w he ee ly; hweve, leve u wh ukw, f, whch c be fu ug New ec lw he ec: F A j u f g α & && g α f u &.8 5. Slve he eulg equ b he EO; hee hul be my EO hee e DOF fm Se. U ubug Eq..8 Eq..5, he fllwg EO be: I & φ φ gα u.9 Ne h he ly fce h wk he by e he g fce he gvl fce. Th me h he fc fce ml fce e bh eeu u ly. We ly ue hem lve f he EO; hey elly ly le u EO. We wll uy he meh f evg EO le h ue eeu fce lke h e clle el fce f c. 6. heck ee f he EO mke hycl ee. D fge h e; bbly he m m e f gue ue efm. Thee e few hg we c check: I he e ve? Ae he ffe mg ve bly? D he fce uh ull he yem he e ec? ec. We wll ee le h hee e evel me check f mulle DOF yem wh me h e EO. 7. Slve f he fee fce ee chcec ee he 4. heck mke ue hee chcec lu mke hycl ee. Se 5 bve mee h New echque eque h ll fce ech by e clue; hweve, ll f he fce geelly wk he yem. I fc, me f he fce mly hl he yem gehe.e., l c, ml fce bewee mh -8

19 E 56 echcl Vb Fll 8 ufce, ec.. F ce, he ml fce Fgue. e wk he yem becue he k e mve he j ec. If we cul ge ll f hee -clle elze c fce ly clue he fce h wk.e., cve fce, he wul be ee eve he EO my ccumce. Wh f we ju jec he EO he ec eecul he c fce? Th jec wul effecvely elme he c bu e he cve fce ee. F emle, f we jece New ec lw f m f he llg k he ec f m, δ, he we cul elme ll bu he m fce fllw: F A δ Fcve Fc A δ N g cα j f g α && I && φ u I && φ && u g α δ I && u u N g cα j g α && δ g α f by δ δ. whee δ/ δ/φ δφδ clle he l v f wh eec he ce φ. We ee h by jecg New ec lw δ we wee ble emve he ml fce f c. Ne h we ubue he he fce f c, f, fm Eq..8. Whe we ecgze he f me h he c wll fll u f he equ wh effec, we c we he -clle Alembe-Lgge cle f gle cle,, fllw, F A δ cve,. whee A he blue ccele f he cle, F cve, he l cve fce he cle, δ v f he vec f he cle. I l meme me cvee -9

20 E 56 echcl Vb Fll 8 elce he v, δ, wh he f ec evve f he v uce he geelze Alembe-Lgge cle f cle. Th echque c be eee mulle cle g be, bu we efe cu eegy meh f fllw.. ev f eegy f lw f hemymc Eegy cle e lwy lcble vb lg he ecey kec el eege c be clcule. Fuheme, eegy meh ly ke ccu cve fce; elze fce f c, whch wk, e ge. Th mlfc gfc vge yem wh my cme.e., DOF. F gle egee-ffeem SDOF yem, he we equ equl he f lw f hemymc, T V W c. whee W c he wk e by -cevve ve fce ug he m, T he kec eegy, V he el eegy f he cle by. F ce, he yem Fgue. h T/ /I φ/ /, V-g- u α e V ecee f ceg, -cevve fce. Wh hee eegy ee, Eq..8 yel, & I & g α u I && && g α I && g α u u & u W. Ne h mvg fm he h he fuh equ, hee wee w f fyg he equ, bu he ze velcy c vl ly ecl ce. Al e h befe ceeg ke evve, he l kec eegy w eee em f he ll ce ug he kemc c. We wll uully cy u h e whe -

21 E 56 echcl Vb Fll 8 lyg he we meh whe lyg geelze eegy meh he e ec; we wll ly he c befe kg evve mlfy he clculu. Eq.. w elvely ey b cm y f he New-be EO. Of cue, ecl Eq.. Eq..9 whe he kemc c efce; hweve, hee w ee clue ehe f he c fce becue ehe e f hem e wk he k. Lkewe, he cevve fce ue gvy he g e ccue f by he el eegy ee, V. Whe hee e -cevve fce cg, he he wk e by he fce mu be cmue. Th clcul efme ug vl clculu, whch mly jec he fce ffeel chge he vec. Aume h he -cevve fce gve by, F F F e e.4 whee e e e u ce vec F F e he cme f he fce. Al, ume h he vec f lc f he fce, A e e.5 whee bh e fuc f me geelze vble, q. The he l v f he vec gve by, δ A A q q e q q e q.6 Flly, he wk e by he -cevve fce fu by jecg he fce he ffeel Eq..6 : -

22 E 56 echcl Vb Fll 8 W F δ A F e F e F F q e q q q q e q.7 F emle, f fce ff c c he k he ve ec Fgue., he he ceme f wk e he k gve by, W F δ f f F c A.8 heefe he we equ Eq.. becme, & & I g α I && && g α u F & I g α && u c & F c u W.9 The m hg emembe bu he eegy meh ug he we equ, Eq.., h he fce f c wk e clue he ly. Al, e h he echque eee ly uce e EO, cequely, ly wk f SDOF yem. Ne, we wll lk bu geel eegy meh lycl ymc h c be ue f y umbe f DOF. Th echque ll eque u f he kec el -

23 E 56 echcl Vb Fll 8 eegy ee clcule he wk e by eel -cevve fce, bu geelly mle ly h New meh cmlce blem.. Lgge equ f cl II hlmc fm The eegy meh Sec. wke well w mle ly, bu w ly vl f SDOF yem;.e., we ly uce e EO. Lgge equ mly ee he eegy meh ccmme mulle egee-f-feem DOF yem. Recll fm Sec. h f we w ge he el fce f c, he we c jec New ec lw f m he vl lceme f he yem h e elme he c m he fce h wk he yem. If hee e N cle, he Eq.. eee N me: N k Fcve, & δ. whee A / he blue ccele f he h cle, F cve, he l cve fce h cle, δ k he k h e evve f he v f he vec,. The ef f he v, δ k q k k δq k. whee q he h geelze ce he l umbe f geelze ce. Ech f he δq k v clle by kemc v; hee v mu lwy fy he c he yem. Lgge equ f cl II fllw ecly fm Eq.. fe me vl clculu. F we mke he fllwg ubu, -

24 E 56 echcl Vb Fll 8-4 q q q & & & & & &&. Eq... Recll he mly f h eul h Eq..9 f he ccele cmu he geelze ce ec, q. Al, e h Eq.. equl he e f chge T wh ffeel, whch ff ffee mehlgy f evg Lgge equ. The we ecme he eul cve fce vec ech cle cevve lmell -cevve cme, F cve, F,c F,c, jec hem he ceg v b he -clle geelze fce ecbe he f em f Eq.. gve Eq.. : *,,,, k k N c k k N c c k k N cve Q q V q V q q Q F F F F. whee Q he h geelze fce, Q * he h geelze -cevve fce, V he el fuc ceg he cevve fce, V he ge f he el fuc. Lly, Eq.. ubue Eq.. lg wh Eq.. b he fllwg:

25 E 56 echcl Vb Fll 8-5 * *,,,, k k k k k N c c k k k N cve k N cve q Q q L q L q Q q V q T q T q q q q δ δ δ δ δ & & && && && F F F F.4 whee LT-V clle he Lgg. If ll f he geelze ce e eee f e he, whch hey c meme be f we e ceful bu chg geelze ce, he he hlmc fm f he Lgge equ f cl w e gve by, * Q q L q L &.5 If Eq..5 le he yem Fgue. whe fce, ff c, le he m he ec vcu me,, lce llel wh he g, he he kec el eegy ee e he me befe Eq.. Q * clcule Eq.. : F F q F q Q k k N k k N c & & & c c c, * F.6

26 E 56 echcl Vb Fll 8 The kec el eege c he be ubue Eq..5 : & & & & & & I I && & I I g & α g g α α F c & g α F c u u u u u u Q *.7 Th equ h he me fm he EO be he evu ec. I mgh be helful h eflec wh we ju. We le Lgge equ f cl II yem whee he hlmc c h ley bee le emve eeece bewee he geelze ce. Th ceue gve u EO h w ecl he EO we eve ug New-Eule vecl echque. The que we hul we w : Why hee w eemgly ffee echque yel he me equ? F f ll, we ue New ec lw he bece f el c eve Lgge equ ecll Eq.., hey hul be he me. Sec, we c ce em fllw he w echque cvce uelve h he w echque mu gve he me EO: F && T T q& q V Q * cve, q.8.4 ulle egee-f-feem DOF yem U h, we hve ly cue EO f SDOF yem. We fllw he me ceue h ec Sec.-. ly New, eegy/we, Lgg echque mulle egee-f-feem DOF yem. Ech DOF he yem -6

27 E 56 echcl Vb Fll 8-7 wll eque he EO ll f hee EO wll geel be cule me wy. F emle, he EO f he mle yem Fgue. wh vcu mg c be eve ug New Lgg echque. The FBD f h yem e hw he bm f he fgue. Fgue.: T hee egee-f-feem vbg yem bm FBD Whe lyg y EO ev echque, we f f he umbe f DOF. I h ce, hee ce e eque lce e ll me he yem: #DOF. The ce ef e gve he fgue. The f New ec lw, he fllwg equece f equ be: f f & & & & & && & & & & && & & & & & &&.9 Th equece c he be ege lce m fm fllw:

28 E 56 echcl Vb Fll f f & & & && && &&. Thee e hee equ becue hee e hee DOF. Th e f equ h me m ee h e cmm yem f h ye whee blue ce e ue eve he EO. F, e h he m, mg, ffe mce e ll ymmec; h ymmey chge whe elve ce e ue. Sec, e h he ff-gl f he m m e ze. Becue he m m gl, we y hee ymc culg h yem hugh hee ce. Dymc culg mle h he e ce wh e ce ecly uce e he ce. Becue he ffe m gl, we y h he yem clly cule hugh hee ce. Sc culg mle h eflec f e ce cue eflec he ce. Th, e h he fce vec ly h w ze ee f ec becue he h DOF h egeu fce le. Thee e fe umbe f he EO f h yem eeg he chce f ce. Sue e ecuge efe he w e f ce eve he EO eme h f hemelve. F emle, f elce wh *, whee * -, elve ce, he he EO m becme: * * * f f & & & && && &&.

29 E 56 echcl Vb Fll 8-9 Ne h he ymmey he m, mg, ffe mce h ee he ce hve becme ymclly cule well clly cule. We c l ue Lgge equ fm Eq..5 eve he EO f he yem Fgue.. T h, we f cmue he kec el eegy fuc he clle Rylegh fuc, R: R V T & & & & & & & & & & &. The Rylegh fuc cmm me f mlfyg he ev f EO me yem. I ee, mg ee ju lke kec el eegy lyclly. Whe R clue Lgge equ, Eq..5 mfe mewh, * Q q R q L q L & &. whee Q * w ue be he eme f he h geelze fce whe vcu mg h bee ccue f wh he fuc, R. I ummy, eve Lgge EO f yem, fllw Se hugh Sec. he cue wh he e belw: Afe efg he ce yem, fg he umbe f DOF, wg he FBD, clcule he el kec eegy fuc well he Rylegh fuc V, T, R. ENFORE THE ONSTRAINTS h elme eeece he elece geelze ce.

30 E 56 echcl Vb Fll 8 Fm he Lgg, L. Th e elly ecey f we che leve Eq.. he ee fm, ec he l le Eq..4, vlvg V T e. Tke he ecey l evve f V, T, R wh eec he geelze ce velce he ke he me evve f he kec eegy l. 4 F he geelze fce, Q *, Eq.. by jecg he -cevve fce ley ccue f Rylegh fuc he vl ce. 5 Subue he eul fm Se 4 Lgge equ, Eq....5 Lezg he ffeel equ f m The ffeel EO f ycl el-wl vbg yem e le. echcl yem c c my ffee ye f lee clug gemec, kemc, mel, he le cmm ye. Alhugh hee e weful echque f lyzg le vbg yem, h cue wll lgely ge he echque fcu e le echque. Thu, we mu wh le ffeel EO by lezg le EO. Thee e w che f g h: we c eve he full EO he leze hem; we c leze he ecey eegy fuc befe lyg Lgge ch eve he fl leze EO. We wll cu bh f hee echque, bu we che fcu he f meh becue lghly me uve f ucy mel. e he ufce ume mle eulum, whch h he fllwg EO: L & θ gl θ.4 whee he lume m f he eulum, g he gvl ccele c, L he ce fm he cee f he lume m, θ he gul ce h he eulum mke wh he vecl. I e leze Eq..4, we lk he le fuc, θ. Fgue.4 hw h whe he eulum clle clely u he ww equlbum.e., e θ, θ me well by he e gh le h -

31 E 56 echcl Vb Fll 8 e hugh he g. If he eulum clle f wy fm he g.e., f θ lge, he he gh le e ccuely ecbe he eul m. We c ue Tyl ee clu ee mhemclly leze he EO ug hee gume. The Tyl ee f θ bu θ clu ee, θ θ! θ θ θ θ θ θ L θ θ 6!.5 Fgue.4: Nle fuc wh w equlbum ce le chcec The fm f Eq..5 cfm u cu bve; lg he gul mll, he h hghe e em e eglgble cme he f e le em. Thu, f we e ju eee hw he eulum mve he eghbh f he g f θ <<, he we c efe ew ce, θ θ-θ θ-, he leze Eq..4 fllw, L && θ glθ L && θ gl θ && g θ θ L.6 -

32 E 56 echcl Vb Fll 8 We ee he h he gl f lez me le fuc EO wh le em e ce eg.e., equlbum e f ee. We cul ju ely hve leze Eq..4 bu θπ well: L && θ glθ L && θ gl θ π L && θ gl θ && g θ θ L.7 whch f cue ecbe he uble ue f he equlbum θπ. I ummy, ll f he ly echque we wll ue h cue e be he um h he EO le. We uully hve leze he EO befe lyg hee echque by efg u eg f ee he mg le fuc he EO wh le fuc he eghbh f u che eg. Alhugh he echque eee bve f lezg fe he fc he fl EO ueful elvely mle blem, wll becme uwely f me gfc blem. I he ce, we c ue he Lgge lu ceue fllw: f, we clcule he eegy ee f T, V, R he geelze fce, Q* befe; ec, we leze he eegy ee; lly, we we w he leze EO ecly. We beg by emg he hlmc Lgge equ f cl II equlbum whee q / f >. A he equlbum, he vlue f he geelze ce e lce he vec, q. Fm he fm f Lgge equ, we hve L L q& q V q R q& Eq. P..8 -

33 E 56 echcl Vb Fll 8 If we e g ly eee mll m u he equlbum, he we c e he el eegy fuc ug mul-vble Tyl ee h e Δq: V q V q q q q q q q V V q q q q mqqm, whee m q q q, qm qm qm m V q q m m q q Δ V q q m q q m m O Δq.9 Ne h we ue Eq..8 he c he evve f he geelze ce. Becue Eq..9 quc fm.e., q q m, we c mmeely we w he l evve h we ee f Lgge equ: V q V q q q m m q q m.4 whch wll ve he le ffe gu ll he culg ffee f he leze EO. A f he kec eegy, we c b ml ee by cmug he w e f l evve we ee f Lgge equ: wh T q& q& m q& m q& q& q j q q j m q j O q.4 -

34 E 56 echcl Vb Fll 8 T q q e q m m q& q& m.4 Rylegh fuc hle he me wy he evu ec. e he mle eulum g wh he el kec eege fllw: V T gl cθ L & θ.4 The fllwg ceue ue belw evel he leze EO f he mle eulum: F he el eegy fuc, Vq. Ne h h ceue ume h ll eg fce e ccue f V. F he equlbum, q, uch h Eq..8 fe. lcule he ymmec ffe ceffce, m, ug he fmul Eq..9 f he ffe m. F he kec eegy fuc, T, e quc fm vlvg uc f he geelze ce velce, q / q m /. Evlue he ceffce f hee quc em q clcule he m m f he leze EO. F he Rylegh fuc, R, e quc fm, efy he vcu mg ceffce cee he vcu mg m. 4 We w he leze EO em f he ymc ce.e., v f ce wy fm he equlbum ug he clcule m, mg, ffe mce cmue Se -. Smle eulum emle: -4

35 E 56 echcl Vb Fll 8 V θ V gl θ θ Sce T T T V & θ θ θ && θ θ, gl cθ, L whee θ 4 θ θ V gl O! 4! & θ, he whee θ θ θ θ 6 θ L.44 Ne h he em f e w lwe quc le e ke he el eegy becue hee e he em h wll uce le em he EO. We wll ly h ceue me cmlce emle hughu he cue..6 uu yem f ec fuh-e S f hee e we hve ly cue cee yem: me, me, g, whch wee ll lume eleme. Ne, we e cuu yem by llwg he yem m, mg, ffe vy cuu fuc f he geelze ce. yem e ely cuu yem, bu we e fe eee kw hw cee mel c be ue ge ccue me lu? We wll cu h le, bu f w we wh elvely mle emle f vee vb lg g he mve me cmlce emle. Reul hee wll llel he cuc. The ce f evg EO f cuu yem ecl h f cee yem Sec.-.4; hweve, he FBD wll lk me cmlce whe ely hey e. e he cuu g hw Fgue.5 belw wh m ey, ρ m/legh e, T fce. Ne h he ey e e bh llwe vy wh, he lgul f he mll ece f m lg he g c he fe m g ceffce he evu ec. -5

36 E 56 echcl Vb Fll 8-6 Fgue.5: Lef Sg fe bh e gh FBD f feml g eleme The FBD f feml g eleme hw he gh f he fgue. Ne h chge bh he e le lg he g e eee ug w em Tyl ee; em f hghe e h e e eglece umg mll eflec. New ec lw le he FBD Fgue.5 yel he fllwg: O f y T y f y T y T y T y f y T y y T T y,,,,,,,,,, ρ ρ ρ.45 The ec e em h l equ c be ge ju hey wee ge clculg he le f he g bh e f he feml egme Fgue.5. Of cue, h equ ly vl lg he legh f he g <<L mu be ccme by w buy c e hve hycl meg. Becue he equ f ec e y,, eque w buy c. I he ce hw Fgue.5, bh e e fe,,, L y y.46

37 E 56 echcl Vb Fll 8 Ohe ble buy c wul be cmue mlly em f he lceme, y, he e f he g. F emle, f he lef e fe he gh e me clle ccg he fuc g, he he buy c wul be y, yl,g. Befe ceeg wh he ev f he EO f cuu yem, we hul eflec he meg f Eq..45. Recll h e f he e evg EO vefy h he eul equ mke hycl ee. If we ewe h equ he fllwg fm: y, y, ρ T f,.47 he elvely cle h he EO f g ecbe blce bewee el, e, eel egeu fce. F emle, f he g mle he eel fce ze, he ρ f,. I h ecl ce, y, y, T T c.48 whch me h f c e g, Tc, he le f he g he me ll lg legh ech mme me. De h mke ee? Pve yuelf h e by cucg eeme f me k. Al, he ey e whe f,c hee mll mu f mg he g he el fce ge ze lg wh he ccele becue he m cee he e fce blce he eel fce: y, T f, ey e.49-7

38 E 56 echcl Vb Fll 8-8 A Eq..48, h equ hw h f c fce.e., gvy c e g, he e f chge le wh ce lg he g l f,. We c l eve EO f b eeecg beg. Fgue.5 hw he yem FBD f h ye f yem. I h ye f blem, he le m ey, m, fleul beg ffe, EI, e gve fuc f lg he b,, he lg bu, f,, gve well. The eflec lg he b he eflec f he eul f he bem, whch ume eeece eglgble he efm lg legh. Alc f New ec lw he vecl ec Eule equ bu he lef e f he feml eleme yel he fllwg w equ:,,,,,,,, f Q Q f Q Q Q y m.5,b By gg ec e em he g ce ubug he eul fm he ec f Eq..5,b, /Q, he f, he fllwg EO be:,, f y m.5 Becue u gl f EO f he eflec, y,, we w hve elme, em f y by ug he mme equ fm elemey B-E beg hey:,, y EI.5 whch gve he fl EO em f y,:

39 E 56 echcl Vb Fll 8 Fgue.4: Lef Beg b fe bh e gh FBD f feml b eleme y, y, m EI f,.5 Ne h h fuh e l ffeel equ, whch eque fu buy c w ech e f hycl lc. I Fgue.5, wh clme e he lef gh f he b, he buy c e gve by, y, y,, y, y L,, L.54 Ohe ye f buy c clue hge ze eflec mme fee e ze mme heg fce. Whe he buy c e eeme by he gemey f he blem, hey e clle gemec buy c. Whe he buy c e eeme by he hyc f he blem he bue, hey e clle ul. Ag, we hul ee Eq..5 eue h mke hycl ee. F, e h Eq..5 eelly ecbe blce bewee el, ffe, eel egeu fce. Sec, e h he EO ecbe yem uegg c beg eflec whe he el em ge. We c efm vu he mle hugh eeme vefy he cece f Eq

40 E 56 echcl Vb Fll 8 Alhugh we ue New ec lw eve bh Eq..45 Eq..5, we c l ue Hml meh eegy meh b he me eul. T h, we mu fmule he kec el eegy ee f cuu yem. All f hee ee e be he ceg cee ve, bu vlve egl. The kec eegy f cuu yem c be fu fllw: T L y, m.55 The el eegy f lgully vbg wh lceme, u,, gve by, V L u, EA.56 he el eegy f b beg wh vee eflec, y,, gve by, V L EI y,.57 Thee ee c be ue cjuc wh Hml cle eve he EO; hweve, h echque wll be eee hee becue eque cle wkg kwlege f vl clculu, whch bey he ce f h cue. Fee Vb Fee vb ccu whu he f eel fce. Nw h we hve eve EO f me cmm vbg yem, we e ey lyze he fee ee behv f hee yem. Recll h ee ly w he e e he EO ceue gve Sec -4

41 E 56 echcl Vb Fll 8.. The le f uyg he fee ee behv f he he fce ee behv ve by he cle f ue,.e., he l ee lu equl he um f he hmgeeu lu he -hmgeeu lu. We wll beg by uyg he fee ee f ume me SDOF yem. Dffee ye f mg he effec ee chcec wll he be cue. The we wll uy hw DOF yem c be ee ml wy ug he f le ue wkg cle. Afe cug he mce f egevlue egevec cee leze DOF yem, we wll uy egevlue egefuc he fee ee f cuu yem.. Fee ee f gle egee-f-feem yem wh vcu mg e he SDOF ffeel equ mel Eq..9 f he yem Fgue.4 he llg k cle eee belw f efeece. If we e he u ze, f, mu be fee vb blem, he he EO becme: I & & α g u. Th equ clle he hmgeeu, ufce, EO. We c ee mmeely h he ey e ee f h yem,, fe he l e h ecye ue mg, fu by eg / /.e., he e f chge f velcy e ze: u g α u g α. De h eul mke ee? Becue gvy cg w he cle, he ey e f he k equl he f he k whe he g uefme.e., V lu he efm he g whe blcg he fce ue gvy he k. Th ey e vlue m my lc; f ce, he c eflec -4

42 E 56 echcl Vb Fll 8 umble ue yem mgh be m becue ce hw much wkg ce lef f ymc eflec he u f hle he u. Hweve, vey cmm vb emve he c ey e f he ee by eefg he ce yem. I u ce, Eq.. ugge h he ew ce, ee he ymc ce hul be efe fllw: Δ. Whe h ubu me he EO Eq.., ly he vbl ec f he ee e ee: I & &.4 We wll w lyze he fee ee ecbe by h EO. Wh he ey e lu f h EO? Thee e my ffee wy lve Eq..4. We c gue lu f he fm, Ae ubue h gue Eq..4,.5 I Ae I Ae Ae Ae.6 he elec he ly -vl lu h fe he eulg chcec, uby, equ, Eq..6 : -4

43 E 56 echcl Vb Fll 8 A e A e whee, I ± I 4 I.7 The c e clle he, le, ml fequece f he yem ecbe by Eq..4. They ee ecly he m, mg, ffe mee; he w, he f yem e eeme lely by he yem. They wll ee wh k f u we meue eve wh ye f uu we meue. Thee e w ukw c h lu, A A, whch e eeme by he l c evve velcy: A & A A A.8 Oly he l c velcy e eee becue h ec e yem, whch h by ef w e f mce. If we kw hee e me ze hve vl EO, he we c lwy f he fee vb ee lu. Th ch fly ey ly lwy wk lg we emembe hw el wh yem vlvg eee.e.,. Recll fm he v f mee h f we e e fc f me,, we c e w eee lu Eq..7. I e f he lu Eq..4, whch le, we cul l ke he Llce fm f h equ lve f he Llce fm f he ee, X, fllw: -4

44 E 56 echcl Vb Fll 8-44 [ ] I I I I X I I X I X X X I I I & & &.9 Eq..9 he lu f he fequecy m ee meme clle he cmle fequecy. Th ceue h he vge h he l c e umclly he lu, bu h he vge h we mu fc he em f bh em.e., he chcec lyml he cy u l fc e b he me m ee. Regle f whch meh we ue b he ee, wll ee f he ml fequece he ly he l c. Theefe, mke ee uy f. We c h by lg he w he cmle le hw Fgue.. Reeeve me m fee ee e l hw he gh f he fgue. We wll w lyze he Eq..7 bee ue el Fgue..

45 E 56 echcl Vb Fll 8 Fgue.: Lef Pl f evel f he cmle le gh ceg me m fee ee llug he ul ee behv f ec e yem f ume, ueme, cclly me, veme ce. If we eme he Eq..7 clely, we ce he fllwg: R ec e yem cme. Fuheme, wheeve he e cmle el mgy mgy, hey mu be cmle cjuge f e he: bj,-bj bj,-bj. yu ee h Eq..7? Lk he - g. The el f he, whch we wll cll σ, he mg fc, eeme he e f ecy gwh f he fee ee. The mgy f he, whch we wll cll he me ul fequecy,, eeme he fequecy f cll. Th emlgy e becue whe we ubue he he eel lu, he el fce u he mgy v Eule fmul uce cll bewee u cu hw belw: -45

46 E 56 echcl Vb Fll 8-46 j e e e e e j e e e e e j j j j j σ σ σ σ σ σ σ c, c he, If *. We l e h he u f σ ce he f ccul fequecy, /. We c f he cul vlue f he mg fc me ul fequecy ug Eq..7 : ± ±, 4 whee 4 I I e mg I j I I I σ σ. R wh ve el e uble whee wh egve el e ble. Th eul mke ee becue e σ eeme he ecy/gwh f he fee ee. Whe he mg ve >, he e he lef hlf le, whe mg egve <, he e he gh hlf le. Recll h mg eeme he e whch eegy e l eeme he e whch eegy bbe by he yem. 4 Puely mgy uce ue hmc lu; hee lu hve c mlue ecy gw. Th chcec mke ee becue he eel f he lu, e σ, equl e whe he mg fc ze, σ. Al, whe he mg fc ze hee mg, he ume ul fequecy fm Eq.. e fu fllw:

47 E 56 echcl Vb Fll 8, ± ± j whee I 4 j I ffe ± j I e Δ ffe e he ume ul fequecy. 5 Whe he e cmle mgy, he c A A clle he eue mu be cmle cjuge f e he ju he e cmle cjuge. Why h ue? If hey e cmle cjuge, he whe we ubue Eq.. he lu,, we uce cmle lu. Th eul e mke hycl ee. The ly wy b el lu f he fee ee fy he c: A A *. Rel lu eque cmle cjuge eue. We l e h geel he c be ewe em f he me hemelgclly megful mee, he mg ume ul fequecy fm Eq.., gve belw:, I whee σ ς ς ς I ± j I 4 I c Dmg, whee σ ± c j ς ± j ς cl mg ceffce. -47

48 E 56 echcl Vb Fll 8 Eq.. hw h he ye f fee ee Fgue. c be clfe em f he ceg mg, ζ. Whe ζ, he ee ume, whe ζ<, he ee ueme, whe ζ>, he ee veme. Whe ζ<, he e he gh hlf le, whch ce h he ee uble. 6 Gve he elh evele cmme hugh 5, fe be l em f he -clle ml mee e.g., mg fc, me ul fequece, ume ul fequece hw Fgue. belw. The ue ecuge ve h he equ fllwg he fgue e ue. Fgue.: Pl f lc cmle le ug ml mee Gve Re ulu Agume,,,, σ ± ς, Im j ς ± σ ± c ς, ± j j ς ς.4 Theefe, h e f wy fm he g ce lge ume ul fequece, cle he mgy ce lghly me ueme ee, cle he el ce hevly me ee. -48

49 E 56 echcl Vb Fll 8 I ummy, he cmle le mle wy vul fee ee le vbg yem. By lg he f he chcec equ, uyg he el mgy, we c mmeely eeme whehe he ee ehb cll wh egee. We c l eeme whehe he ee ble uble. Sue e ecuge evew Fgue. hughly; g kwlege f hw le lc eeme fee ee chcec eble mechcl vb ly. The lu f SDOF yem ely we he fm gve Eq..7, he uully we he fllwg mlfe me: whee X e X X σ e & σx c ς cφ φ c cφ X ς φ φ.5 Th fm be ug Eule fmul wh he cmle cjuge c A A Eq..7. Afe lyg he l c, he fllwg vlue f X φ e be: φ X & σ & σ.6 Ne h ju he c A A Eq..7 wee eeme by he l c, e he c X φ Eq

50 E 56 echcl Vb Fll 8. Fee ee f gle egee-f-feem yem f by mg The SDOF yem Eq.. h vcu mg, whch e. Syem uully hve me by le eve le ye f mg mechm. F ce, whe we mel yem wh wheel, we uully ee mel he ulmb mg/fc he beg ccu f ufce--ufce ug cll. Whe we mel ucul vb wh flu.e., ff-he l g, we uully clue le quc mg ccu f ue mmeum fe he flu. We c l mel ucul hyeec mel mg blem wh hmc u by ug cmle ffe mee. Sucul mg ccu mel lye le ve e he ug vb. I m emembe h mg e f he m ffcul heme mel vbg yem. I fc, he wey ye fm , e wee ublhe he e f mg echlgy. ffe c bh be me he ccuely ug fe eleme mel, f ce; hweve, mg eme e uly ccue becue ffcul qufy lyclly. Dmg uully be eme eemelly. Alhugh mg mechm el yem e ely vcu, he ce lycl ee f vbg yem wh vcu mg e wh elg f ble. I fc, he cce f equvle vcu mg we ue wh he e & vb egeeg cmmuy. The equvle vcu mg mel efe uch h he l eegy e e cycle he mel he me he ue hycl vbg yem ue hmc ec fce ee be ee he e che. e ecfclly, f W c he mu f eegy e e cycle, he he equvle vcu mg, eq, efe fllw: W c Wc cycle F vcu cycle π / π, eq eq eq & X X gve X π / c eq & c.7-5

51 E 56 echcl Vb Fll 8 The equvle vcu mg che uch h f cll mlue X, he eegy e he vcu me he me h e he by me wh he yem. Ne h we e emuely ucg fce SDOF ee hmc ec e lk bu he fllwg m cmm mg mechm. F emle, yem wh ulmb fc gvee by he ye f mg chcec hw Fgue.4. Recll h he elze c ulmb fc fce equl he uc f he c fc ceffce bewee w ufce, μ, he ml fce bewee he ufce, N. A vcu mg chcec l hw he fgue f efeece. Ne h ulmb fc le k f mg;.e., he qulve ue f he mg chcec fuc f he elve velcy bewee he w ufce. F ve velce, he fc/mg fce ve e he m whee f egve velce h fce egve g e he m. I hee w vb egme, SDOF yem mel gve by he fllwg w equ, whee he ml fce h bee ume be equl he gvl fce he m: && μg && μg f f & > & <.8,b By lvg ech f hee equ ucce ug he SDOF fce ee lu echque cue he e che, we c hw h he e f ecy f he ek mlue le wh me he h eel wh vcu mg. I my l be cle fm Eq..8,b h he fequecy f cll/vb e ee he fc fce ceffce f fc; lwy equl /. A lg he eg fce he g lge eugh vecme he fc fce, he m wll cue clle hw Fgue.4. -5

52 E 56 echcl Vb Fll 8 Fgue.4: T lef ulmb fc chcec gh fee vb ee A ec emle f -vcu mg, ce he cmm ce f hyeec ucul mg. Recll h vcu mg fce e l he velcy c he me; he w, f SDOF yem me clle ce fequecy,, ccg X cφ, he he vcu mg fce : F vcu & φ X.9 whch me he fce he wh he velcy l he fequecy. Recll fm Eq..7 h he eegy e he vcu me e cycle l l he fequecy. I wely kw h eegy my k f mel ucul j l fequecy bu e he mlue f cll. Thee mg mechm e he me me he wh he velcy.e., 9 eg u f he wh he lceme. I e fy hee w beve heme, he vcu mg mel c be ue ecbe ucul mg lg he mg ceffce che uch h h/, whee h he hyeec mg ceffce. Th chce f le he fllwg hyee mg fce ue hmc ec: -5

53 E 56 echcl Vb Fll 8 F hyee & X hx φ φ. I m lc, hyeec ucul mg uully ce mel by efg -clle cmle mulu, jη, whee ηh/ clle he l ceffce. I hul be e h m mel me ve le eegy f hghe fequece, whch ce wh bh he vcu hyeec mel. Fgue.5 hw ycl l f l ceffce vcelc mel, whch by f he m cmm mg mel ue y. Ne h hee mum fequecy emeue whch he vcelc hul be ue e eegy. Fgue.5: hcec f cmle mulu vcelc mg mel. Fee ee f mulle egee-f-feem yem Alhugh my lc c be mele wh gle egee-f-feem, m yem c mulle egee-f-feem DOF. Smeme we ee clue ll f hee DOF. F ce, ffhe l lfm Fgue.6 vbe f wee SDOF yem -5

54 E 56 echcl Vb Fll 8 my ccumce becue he u c lgely ffe eleme he lfm c lke e he fequecy ge ce wh ycl wve. F hghe fequece ce wh eue uge ul he l elvey ele, he we behve me lke cllec f cllg be wh ffee cme he lfm cg he me f vb. Th ye f fm lw-e SDOF-lke hgh-e DOF-lke vb cmm m lc. We hul lwy hk bu he fequecy ge f ee befe we beg mel lyze vbg yem. Fgue.6: Offhe l lfm wh SDOF mel The b f u ly f fee ee DOF yem he cle f le ue. Recll fm Sec. h he fee ee f SDOF yem ccu he me ul fequecy f cll. Whe we lk he fee ee f DOF yem Fgue.7, we f h ech DOF ee c evel fequece. Ech f hee fequece c be ce wh SDOF yem. By g he fee ee fm ll f he SDOF yem, he DOF yem ee be. Thee SDOF yem h mke u he DOF yem e clle ml me f ecl ye f mg ju me f vb. Ech me h eml cme, he ml fequecy σ j, l cme, he ml vec ψ. Fgue.7 hw h geel he ml fequece e ffee e he ml vec. I h cul ce, he w DOF yem ecme w SDOF -54

55 E 56 echcl Vb Fll 8 yem wh -he lwe fequecy me f vb lwe-lef u-f-he hghe fequecy me f vb lwe-gh. Ou gl h ec jufy h e SDOF meh f ly wh DOF lycl echque. We wll be ug ege-lu meh f bg he ml ee, ue e ecuge evew he evu eeece wh meh le lgeb. Fgue.7: DOF yem ecbe ue f SDOF me f vb We wll by wg w he EO f he mle w DOF hw Fgue.7: && && & &. Ne h he eel egeu fce e ze h ce hey mu be fee vb blem. Thee e w wy cee fm hee: we c ue hycl gume f he ml ee we c ue mhemcl ege-meh clcule he ml ee. Bh che ffe gh geel DOF vbg yem, we wll cu bh f -55

56 E 56 echcl Vb Fll 8 hem. We wll beg wh he f ch becue me hycl uve ch fee vb ly DOF yem. The mel Eq.. m we vey f lu. Becue u gume bve Fgue.7 be he eme h DOF yem e cully cmb f SDOF ml me, we wll cee ecly he me wy we f he fee ee f SDOF yem. F, we mke gue f he fm, X X e. ubue u gue he EO, X X e. ck he ly -vl lu. I hee -vl lu h wll mke u he ml me we lke bu Fgue.7. F -vl lu f Eq.., X X, he m he lef h e mu be gul.e., mu hve -ze ull ce. Th c gve u he chcec equ f he w DOF yem:.4 whch h fu lu, σ j, σ -j, σ j, 4 σ -j. The el f hee lu eeme he ecy/gwh e f he fee ee whee he mgy eeme he fequece f cll ecll he cu uug Eq.. Sec.. F mlcy, we hve uee he ubc h ce h he mgy f he e me ul fequece. Ne h ech cmle cjuge f ml fequece ce wh ffee lu vec ml vec -56

57 E 56 echcl Vb Fll 8 Eq... Fm he fm f he gl gue, he lu c w be we ehe f he fllwg wy ug Eule fmul e ±j c ± j : * X X X X e * e e X X X X ψ σ ψ X e c ψ ψ * * e σ φ X e c φ 4.5,b Thee equ e he fm we cue evuly; w me f vb e ue ecbe he l lu. The c vec, {X X } T {X * X * } T, e cmle cjuge f e he e ce wh me. They ce he c, A A, he SDOF fee ee lu. The ml vec, {ψ ψ } T, he ec f Eq..5,b e eeme wh cle fc; hee vec ecbe hw he DOF mve elve e he he w me f vb,. I m e h he ml vec e uque. The e σ c φ fc Eq..5,b e eeme by he ml fequece/ fm Eq..4 ; hey ecbe he fequecy mg chcec f ech f he ml ee. Ju we gue Fgue.7, he ml fequece fm he chcec equ eeme he eml chcec f he lu he ml vec fm he ullce clcul Eq.. eeme he l chcec he lu. Th ecm f he DOF ee eml l ece ly ble wh le yem hk he cle f le ue. The ec equ Eq..5,b he fm we wll ue h cue becue m ely le fy he l c. Hw my l c we ee? Becue hee e fu c eeme, X, X, φ, φ, we ee fu l c he lceme velce f he w ce. Thee fu c e fe fllw: -57

58 E 56 echcl Vb Fll 8 ψ ψ A c φ A c φ ψ ψ & ψ ψ σ A c & ψ ψ OR ψ ψ A c φ ψ ψ A c φ & σ ψ σ ψ A c φ ψ & σ ψ σ ψ A c φ ψ ψ ψ φ A φ σ A c φ A φ ψ A ψ A φ φ ψ ψ.6,b,c, I e mke h cu le bc, ce he ecl ce whe N/m,. N/m, kg. The he lu Eq..5,b gve by,.68 X e..9 c..68 φ X e c.6 φ..68.7,b Thee w me f vb e llue Fgue.8 belw. Ne h he f me lwe fequecy h he ec me me lghly me well. Al e h he f me, he ce l le h lwe elve mlue h he ce e le. Flly, m e h he mlue f he ml cme hw Fgue.8 e by. They wee elece h he lge mlue wul hve vlue f uy. I e b he ec mlue, l c mu be le Eq..6,b,c, mu he be lve b he c he lu. The elve mlue hw Fgue.8 wul ll ly h ce. -58

59 E 56 echcl Vb Fll 8 e #:, Tme [ec] e #:, Tme [ec] Fgue.8: Tw me f vb w DOF yem We c l ue ege-ly echque f he ml fequece vec fllw. If he yem h mg, he Eq.. euce he fllwg: X X e.8 whch c be ely efmule he fllwg ege-blem, X X A λ X X [ ].9 by mvg he e em he gh he ffe em he lef. -59

60 E 56 echcl Vb Fll 8 X X X X X X λ X X / / X X λ / / X X. Nw we ju lve f he egevlue egevec f he m he lef h e f h equ. The vlue we b e gve belw: X X whch yel λ,,.8, λ, X.,. X.68 ± j.68 ± j ± j.68 ± j,4. Ne hw he egevlue f he yem Eq.. e ele he ml fequece f he yem. F evey f ml fequece, we hve e egevlue. Sce we ge he mg fmule he egevlue blem, we ee u wh uely mgy ml fequece, whch ce he ume ul fequece f he yem. The ume ul fequece Eq.. e vey cle he me ul fequece h wee fu Eq..7,b becue he yem lghly me.e., ζ<< wh he elece vlue f. I lge blem, we fe cy u he ceue bve f he ml fequece. Th, we ge mg emly fmule he egevlue blem, cmue he egevlue egevec, cmue he ume ml fequece, he eme he mg fc f ech le. I fc, h echque he m cmm e le fe eleme fwe. If we efee e mg hughu he egevlue blem, he we mu efmule he EO Eq.. he e ce. I he w, we efe w f he e -6

61 E 56 echcl Vb Fll 8-6 vble he lceme f he w DOF he he w he evve f hee vble. Th uce he Duc-ll fmul gve belw: 4 4 4,,, whee y y y y y y y y y y y y y y & & & & & & & &. The egevlue egevec f he yem c he be cmue befe ece w hee ee efm he uby clcul wh becue he yem fmule f e he h ec e. Sue e ecuge em h clcul cme he eul wh he e gve Eq..7,b..4 Pcl ce ml fm The echque ecbe bve f eg cee DOF yem lke cllec f SDOF yem, ech wh he w ce ml fequece vec, c be fmule me yemc wy. e g he w DOF yem fm Eq.., eee belw f cveece: & & && && If we ge mg g, f cveece ly, he u gl h ec wll be f ew e f ce efe by he fm, Ψ, whee he vec f cl ce. Ne h we e me ue le fm hee becue he yem bey he cle f ue. We w che he fm, Ψ, uch h he eulg EO em f cl ce e mle ble. Recll fm Fgue.7 h u el ec f DOF yem cllec f SDOF yem. Fuheme, we w hee SDOF yem be ucule fm e he h we c lve ech f hem eeely he he eul uce he l lu f he

62 E 56 echcl Vb Fll 8-6 DOF yem. Th fm wll be clle ml fm f he e ele belw. Fuely, we hve ley cheve h k f fm Eq..5,b, whch eee belw e-we ug he bve: c c c c e X e X e X e X Ψ φ φ ψ ψ ψ ψ φ ψ ψ φ ψ ψ σ σ σ σ. Thu, he cl ce we eek e ecely he me he eml cme he fee ee lu, whch we ley be Eq..5,b. The fm m, Ψ, w l be evuly mly he m f ml vec. Ag we ee h he cle f le ue wk: he fee ee f DOF yem he ue f he fee ee f he vul me f vb cl ce. If we ubue he ml vec fm Eq..7,b he ume ve f he w DOF yem, he we c cy u he fllwg mlfc ug le lgeb: Ψ Ψ Ψ Ψ T T T T && && && && && && && && ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ.4 If we w he cl ce ecule he DOF yem w SDOF yem, he he w ceffce mce, he -clle ml m ml ffe mce, Eq.

63 E 56 echcl Vb Fll 8.4 mu be gl. We c ve h hey e by eceg u ege-yem ly f h yem. F, ecll fm Eq.. h ech e f egevlue egevec, λ- {X X } T, fy he fllwg equ: X X λ X X.5 Ne, we h elh f he w ffee e f egevlue egevec, : ψ ψ λ ψ λ ψ.6 Nw we efm e equece f le lgeb e: we e-mully he f equ by he e f he egevec he ec by he e f he evevec, he ubc hee w eul b he fllwg eul: ψ ψ λ ψ ψ T T T λ λ ψ f T ψ ψ λ ψ ψ T ψ.7 The l f hee equ ve h f he w egevlue e c, he he ceg egevec e hgl wh eec he m m. Lkewe, hey e l hgl wh eec he ffe m: T λ λ ψ ψ f.8-6

64 E 56 echcl Vb Fll 8 Of cue, whe, Eq..7.8 e fe, he, hey uce he ml m ml ffe, eecvely. Ame wh h fm, we c ev Eq..4 cve h we wee ucceful fmg he DOF yem ucule SDOF yem f he fllwg fm: && whee ψ ψ T && ψ ψ T f,.9 The cl ce lu hee equ e he gve Eq... I ummy, we ue ce fm fm hycl cl ml ce ecule he DOF yem cllec f SDOF yem. The lu ech f hee SDOF EOS e he cmbe ug le ue ccg Eq... The ml ly ecbe bve eulg Eq..9 w ly vl f ume yem; hweve, ml lye c l be ce u f me yem wh ecl cle f mg. F emle, lly me yem hve vcu mg mce f he fllwg fm: α β.4 whee α β e el c. Syem wh mg f h ye ff he me f ml ecm Eq..9 wh he f e mg em: & && & f,.4 Whe yem e -lly me vcu, he he Duc-ll fmul Eq.. ue efm he ml fm uby f e vec ce. Whe he by fm f mg e.g., ucul e ecuee, he he equvle -64

65 E 56 echcl Vb Fll 8 vcu mg c be cmue f hmc u ecbe evuly Sec., he he Duc-ll fmul c be ue efm he ml fm. I he ee f cly, we hul ue f mme ce me f he v emlgy h we ee he leue. Smeme he cme wll be clle he ul ce e f he cl ce,, ml wy, we wll cll he vec Ψ he ul me e f he ml vec. Fuheme, f he ul me/ml vec e mlze y ce wy e.g., uy ml m wh eec he m ffe mce Eq..9 he we elce he w ul wh ml.e., ml ce ml me. We wll ee h me emlgy he ly f fee ee cuu yem..5 Buy c g by me em-efe DOF yem Thee e my c cu h cue, bu w f he m m e buy c he effec mechcl vb chcec. Wh we me by buy c? e he w DOF yem Fgue.7 ecll h hee e w e f g me. Oe e cule he w e.e., he w DOF gehe whee he he e hl he w DOF he me geel e hey vbe. I h ce, he g me gu he lef f DOF fm he buy c f h yem. Of cue, he ufce belw he w DOF l buy c, bu h ce vey eeg becue he wheel e fcle mle eeece l. We wll w ce hw he buy c he lef f m ffec he fee ee f he w DOF yem. Phyclly, we ee h f he ffe,, gu becme vey, vey lge, he he w DOF yem becme SDOF yem becue m lge ble mve; fe gu. De u ly bve evel h evy he buy c? Fm Eq.., he mfe ege-blem f vble ffe he buy c, buy / X X λ / X X.4-65

66 E 56 echcl Vb Fll 8 The ume ml fequece ceg h egevlue blem e hw belw Fgue.8. Ne fm he equ bve h he buy ffe cee, he ml fequecy f he ec me gw he fequecy f he f me level u. Th eul mke hycl ee becue f he ffe becme lge eugh, he f DOF wll em y he ec m vbe / ccg he chemc Fgue.7. 5 Ume. feq., [/] 5 5 Ache / buy [N/m] Fgue.8: Effec f B ffe ume ml fequece w DOF yem Wh he he ml fequece f we euce he buy ffe ze? Fgue.9 hw he me ml fequecy evy l f h ce Fgue.8. Ne h he f ml fequecy, he e whch ce he me f vb whee he w DOF mve he, che ze. Whe hee me f vb ze lke h, clle g by me he yem be em-efe. Phyclly, me h he yem e clle ll f h ml vec. Th eul hul ue u becue he yem Fgue.7 fee mve y lg he ec he c clle u h. Fm Eq.., whe he buy ffe ge ze, he ffe m cully becme gul. Th eul l mke hycl ee becue we hul -66

67 E 56 echcl Vb Fll 8 be ble lve f uque ey e equlbum h ce becue he yem uee fee-fee B. Rg by me e eel lycl eemel vb becue whe yem vbe, hey l uully eeece g by m. F ce, ege c c mve g be lg he ck cy ege fm e lce he, bu hey c l vbe eegy echge bewee he c. Bh ye f m e eee ecbe he geel m f he. I em f el eegy, he e c hve he me el eegy e he g/lk bewee he c y he f he lg he ck. The m cmm el-wl lc whch g by me e m g yem e.g., ge g ube ege, ve- vehcle. I hee lc, lge g by m f he e-hf emble e evee by beg..8 Ume. feq., [/] Ache buy [N/m] Fgue.9: Effec f B ffe ume ml fequece w DOF yem -67

68 E 56 echcl Vb Fll 8.6 Sec e cuu yem e f vble I he cee DOF ege-yem/ml ly ecbe bve, we wee ble ecme he l fee ee w cme: eml ece he cl ce l ece he ml vec. We fu h h ecm gely mlfe he EO becue effecvely ecule he equ, levg u ly wh mle ec e SDOF equ lve f he ml ee ml me he ume ce. We wul lke el he me f e f he eml l ece he fee ee f ec e cuu yem lke he g EO Eq..6 hmgeeu equ eee belw f cveece. Buy c wll be clue belw. y, y, ρ T.4 Becue he ee, y,, fuc f bh ce me, h blem cul be elly vey ffcul lve whe me ce e cule. Fuely, u ch f ug le ue h bee ecule he eml l cme by hkg em f ychu me f vb, whch ll eleme f yem clle gehe he ce ecl fequece ml fequece. Wh h ch m, we c ee he w vble by umg h he ee f he fllwg fm: y, Y G.44 I ucve cme Eq..44 wh Eq.. : he l fuc, Y, ecly lgu he m f ml vec/ul me, Ψ, he eml fuc, G, ecly lgu he cl ce/ul ce,. I fc, f we ju llw he umbe f DOF ch fy, he he ml vec hul u Y he cl ce hul u G. Ohe lge bewee cee cuu fee vb wll emege we cee. Whe we ubue Eq..44 Eq..4, we cve he l evve y evve ve he ee belw: -68

69 E 56 echcl Vb Fll 8-69 [ ] [ ] [ ] [ ] [ ] [ ] Y T Y G G Y T G G Y G Y T G Y ρ ρ ρ.45 Nw we mke e bev: becue he lef h e f h equ ly fuc f he gh h e ly fuc f, he ly wy hve equly f bh e equl he me c. Ou e ubu f h c wll be - : Y Y T G G ρ &&.46,b We hve cheve ecly wh we h he cheve Eq..46,b h he eml l cme he fee ee e ecule. F ce, we c mmeely we w he fm f G, he cl ce: G G φ c.47 whch he c, G φ, e ce wh he fm f Y. A f Y, we mu ly he buy c f he ble ul me by lvg he cuu egeblem. Oce we b he fe umbe f ul me, Y, he he lu, y,, c be we w em f he cl ce ul me: c, G Y G Y y φ.48

70 E 56 echcl Vb Fll 8 We meme ue ml me ml ce e he fee ee. Thee ml ml ee e be by efmg cuu ve f he cee hgly clcul he evu Sec.4. e ecfclly, u hgly f he ul me wh eec he ey fuc eee fllw: ρ Y L Y δ f f.49 whee δ he by ecke el fuc, he ceg hgly f he ul me wh eec he e eee fllw: L T Y Y δ.5 Whe hee ml me e ue e he fee vb ee, we c ewe Eq..48 fllw: y, Y g Y whee g&& g g c φ Y mlze bve.5 The cu bve c be me le bc by lvg f he fee ee f he g hw Fgue.5; h g fe bh e. We c l ume h he g h c ey e f u. Thee mlfc he ec f Eq..46,b yel he fllwg ec e ffeel equ: Y ρ Y wh Y Y L T.5-7

71 E 56 echcl Vb Fll 8 whch m he fllwg e f hmc lu: Y ρ Y T ρ ρ Y A c A T T Y A ρ Y L A L T Nul fequece, wh Y Y L π f < < L - vl lu, T ρl f ege,,l ρ L T.5 A befe he cee ce, we hve ecl ml/ul fequece wh he ceg ul me he. Whee he cee ml fequece wee equl he que f ffe ve by e, he fequece Eq..5 e equl he que f e e u legh ve by he m f he g. Thee fequece he e he ubue Eq..48 Eq..5 b he fee ee lu. Fgue. hw hw he fee ee f cuu g, whch fe bh e, lyze ug ul fequece me he. Sme cmm emlgy uce he fgue clug he f fumel fequece hmc vee. ucl ge ume e.g., vl, vl, cell, be e ecelle el-wl llu f he cu bve he fee vb ee f fe-fe g ee evch, 986. F emle, vl h fu g wh fu fumel fequece, whch he muc c chge wh e fgeg lg he g. I elvely ey ee h chge L v fgeg he bge T v gheg f cew bg bu chge he fumel fequece ee Fgue.. I eeg e, hweve, h mucl ume, he u fm fumel fequece e fe m he hghe hmc vee. The ub f eeve vl e.g., Svu, f ce, e ege cuce uce me leg, full u h he fm le eeve vl. Lkewe, he ub f Sewy g ege ee vee me effecvely h le eeve ume. -7

72 E 56 echcl Vb Fll 8 Fgue.: Illu f cuu fee vb lu ug ml chcec.7 Fuh e cuu yem e f vble We c cee ecly he me wy Sec.6 eve he fee ee lu f cuu yem f fuh e. Recll he EO fm Eq..5 f he b uegg vee vb. I he fee vb blem, we e f,, b he hmgeeu EO: m y, y, EI.54 We c beg by eg he ee y, l eml cme befe: y,yg. Afe ubug h fm Eq..54 mkg he me f gume uug Eq..45 bu equly ce me fuc, we b he fllwg w equ: -7

73 E 56 echcl Vb Fll 8-7 L Y Y L Y Y Y EI m Y G G ' ' ' ', wh ,b Ne we hve ume h he buy c he b e mly ue bh e ucg ze lceme ecve mme. The ul ce e fu befe Eq..47, he ul me e fu by fcg he lvg he ec f Eq..55,b. Ne h Eq..55,b ye f ege-blem f cuu yem. eve, cme h fm he fm f egevlue blem cee yem, λi-a Eq..5. I fc, h ee fe ewe ug e fllw emhze h lgy: 4 4 Y EI m.56 The ble lu egevec f h ee e gve belw: EI m A EI m A EI m A EI m A Y ch h c.57 We c ly he buy c Eq..55,b elec ly he ul me egevec h fy he hycl c he vbg b fllw:

74 E 56 echcl Vb Fll 8-74,,L f, h h h ch h Recll L EI m L EI m A L EI m A L EI m A EI m Y L EI m A L EI m A L Y EI m A EI m A Y A A A A EI m Y A A Y e e e e L π.58 Wh hee ul fequece me, we c e he fee ee f he b fllw: L L whee c, Y Y EI Y my g L ml g Y y δ δ φ π.59 The e Eq..59 ume h he ul me hve bee mlze efe he ec le f h ee. I e jufy h mlz ceue, we mu f ve h ul me cuu yem e hgl e he wh eec m ffe ju ml vec wee cee yem. We wll cee he me wy befe. F, we we w he ege-blem f he ul me f he b f w ffee me:

75 E 56 echcl Vb Fll 8 Y Y EI m Y EI m Y.6,b The we mully he ege-equ by Y ege by ve he legh f he b mulle he ege-equ by Y ege by. Whe we h, ubc he eul, ume h hee e buy c lke clme, hge, fee, he he fllwg hgly c e fe me clculu: L L EI If EI m Y Y f Y L m Y Y δ, L Y Y f Y δ he hgly Sffe hgly Nmlze.6 I ummy, we hve lve cuu egevlue blem f he ul fequece he he me wy h we lve he cee egevlue blem he evu ec f he ul fequece vec. The w ffeece bewee he cee cuu fee vb blem e h ml vec e fe mel whee cuu me he e fe mel, ml fequece e f fe umbe whee cuu ml fequece e f fe umbe..8 e ue fee vb We e w gg uy few emle f fee vb el-wl yem e efce he hey h w cue h che. Thee emle wll be e lgely cl; hweve, he blem eme e gve belw f efeece. -75

76 E 56 echcl Vb Fll 8 Vehcle ue vb e: Vehcle ue yem c vbe vey f wy. The by fme c buce eelly g be he e ue, he by fme c vbe cuu yem well. We wll mel lyze me f hee m cu el ue ce whe egg vehcle ue v e vb blem m hlg efmce. Becue buy c e eemely m cmle, mul-by yem lke h e, we wll fcu much f u cu wy ce eme he effec f ffee ye f buy c. Fgue.: Illu f vehcle ue mel Equvle me -el g: Whe we cue DOF yem, we hve mele mechcl yem ug el m, mg, ffe eleme. Of cue, he el wl, m cml me ee, g hve m, me hve bh m ffe. We e gg mel lyze he vlve f umve el cmbu ege ug eegy-be meh Rylegh que, whch we hve ye cue. -76