10 Viscoelasticity Viscoelasticity time-dependence creep

Transcription

1 Vclacy h Lnar lac Sl ha bn h man maral ml analy n h bk hu far. I ha a lng hry an ll h m wly u ml n applcan ay. Vclacy h uy f maral whch hav a m-pnnc. Vca, a Frnch ngnr frm h Dparmn f Ra Cnrucn, nc n h 83 ha brgcabl cnnu lnga vr m vn hugh unr cnan la, a vclac phnmnn knwn a crp. Many hr nvgar, uch a Wbr an Blzmann, u vclacy hrughu h nnnh cnury, bu h ral rvng frc fr uy cam lar h ncra man fr pwr an h aca man fr maral whch wul an up mpraur an prur ha wn byn prvu xprnc. By hn ha bn rcgn ha gnfcan crp ccurr n mal a hgh mpraur. h hry vlp furhr wh h mrgnc f ynhc plymr plac, whch xhb rng vclac prpr. h uy f vclacy al mpran n Bmchanc, nc many bmaral rpn vclacally, fr xampl, har u, mucl u an carlag. Vclac maral ar fn n cn. an m vryay vclac maral an phnmna ar cu n cn.. h bac mchancal ml f vclacy, h Maxwll an Klvn ml, ar nruc n cn.3, a h gnral ffrnal quan frm f h lnar vclac law. h hrary ngral frm f h cnuv quan cu n cn.4 an hwn hw h Laplac ranfrm can b u lv lnar vclac prblm n cn.5. In cn.6, ynamc lang, mpac an vbran f vclac maral ar cnr. Fnally, n h la cn, mpraur ffc ar brfly cu, nclung h mpran cncp f hrmrhlgcally mpl maral. 83

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3 Scn.. h Rpn f Vclac Maral.. Vclac Maral h bac rpn f h vclac maral wa cu n cn Rpang wha wa a hr, h ypcal rpn f a vclac maral a kch n Fg.... h fllwng wll b n: h lang an unlang curv n cnc, Fg...a, bu frm a hyr lp hr a pnnc n h ra f ranng /, Fg...b; h far h rchng, h largr h r rqur hr may r may n b m prmann frman upn cmpl unlang, Fg...a la unla pbl prmann frman a b Fgur..: Rpn f a Vclac maral n h nn ; a lang an unlang wh pbl prmann frman nn-zr ran a zr r, b ffrn ra f rchng h ffc f ra f rchng hw ha h vclac maral pn n m. h cnra wh h lac maral, wh cnuv quan npnn f m, fr xampl mak n ffrnc whhr an lac maral la m gvn r lvl fr n cn r n ay, r la lwly r quckly; h rulng ran wll b h am. I wa hwn n Chapr 5 ha h ara bnah h r-ran curv h nrgy pr un vlum; urng lang, h nrgy r n a maral, urng unlang h nrgy rcvr. hr a ffrnc bwn h w fr h vclac maral, nca by h ha rgn n Fg...a. h ha rgn a maur f h nrgy l hrugh ha ranfr mchanm urng h frman. M ngnrng maral unrg placy, manng prmann frman ccur nc h r g abv h lac lm. h r-ran curv fr h maral can lk vry mlar ha f Fg...a, bu, n cnra vclacy, placy ra npnn. Placy wll b cu n chapr. Sl Mchanc Par I 85

4 Scn. Lnar Vclacy Lnar vclac maral ar h fr whch hr a lnar rlanhp bwn r an ran a any gvn m,. A mnn bfr, h rqur al ha h ran ar mall, ha h ngnrng ran maur can b u nc h xac ran nhrnly nn-lnar. Sran-m curv fr a lnar vclac maral ubjc varu cnan r ar hwn n Fg.... A any gvn m, ay, h ran prprnal r, ha h ran hr u 3 hr m h ran u. 3 Fgur..: Sran a a funcn f m a ffrn la Lnar vclacy a ranabl apprxman h m-pnn bhavur f mal an cramc a rlavly lw mpraur an unr rlavly lw r. Hwvr, m wpra applcan n h mllng f plymr... ng f Vclac Maral h nn crb n cn 5. h anar maral. A numbr f hr whch ar pcally uful fr h characran f vclac maral hav bn vlp, an h ar cu nx. h Crp an Rcvry h crp-rcvry nvlv lang a maral a cnan r, hlng ha r fr m lngh f m an hn rmvng h la. h rpn f a ypcal vclac maral h hw n Fg...3. Fr hr an nananu lac ranng, fllw by an vr-ncrang ran vr m knwn a crp ran. h crp ran uually ncra wh an vr crang ran ra ha vnually a mr-r-l cnan-ran ay a rach, bu many maral fn n rach uch a ncabl ay-a, vn afr a vry lng m. Sl Mchanc Par I 86

5 Scn. Whn unla, h lac ran rcvr mmaly. hr hn anlac rcvry ran rcvr vr m; h anlac ran uually vry mall fr mal, bu may b gnfcan n plymrc maral. A prmann ran may hn b lf n h maral. A whch fcu n h lang pha nly mply call h crp. r ran crp ran lac rcvry anlac rcvry nananu ran m prmann ran Fgur..3: Sran rpn h crp-rcvry Sr Rlaxan h r rlaxan nvlv ranng a maral a cnan ran an hn hlng ha ran, Fg...4. h r rqur hl h vclac maral a h cnan ran wll b fun cra vr m. h phnmnn call r rlaxan; u a r-arrangmn f h maral n h mlcular r mcr-cal. f h la abv h yl r, hn m f h prmann frman wll b nananu plac ra-npnn ran; h ubjc f h chapr cnfn maral whch ar la up a r blw any fnabl yl r; ra pnn maral wh a yl r abv whch prmann frman ak plac, h vcplac maral, ar cu n Chapr Sl Mchanc Par I 87

6 Scn. ran r r rlaxan m Fgur..4: Sr rpn h r-rlaxan h Cyclc h cyclc nvlv a rpang parn f lang-unlang, Fg...5 cn I can b ran-cnrll wh h rulng r brv, a n Fg...5, r r-cnrll wh h rulng ran brv. h rul f a cyclc can b qu cmplx, u h crp, r-rlaxan an prmann frman. ran m r ran Fgur..5: ypcal r rpn h cyclc Sl Mchanc Par I 88

7 Scn.. xampl an Applcan f Vclac Maral Sm f h prpr f vclac maral ar hr ably crp, rcvr, unrg r rlaxan an abrb nrgy. Sm xampl f h phnmna ar cu n h cn... Crp an Rcvry h k n h human pn ar vclac. Unr nrmal by wgh, h k crp, ha hy g hrr wh m. Lyng wn allw h pnal k rcvr an h man ha m ppl ar allr n h mrnng han n h vnng. Arnau hav gan up 5cm n hgh unr nar-zr gravy cnn. Skn u vclac. h can b n by pnchng h kn a h back f h han; ak m rcvr back rgnal fla pn. h lngr h kn hl n h pnch pn, h lngr ak rcvr. h mr raply pnch, h l m ak rcvr bhav mr lacally. Skn an agng maral, ha, phycal prpr chang vr m. Yungr kn rcvr mr raply han lr kn. W vclac. h bam f l wn hu can fn b n ag, bu h crpng unr h wgh f h rf an gravy can ak many ca r cnur b ncabl. Cncr an l ar hr maral whch crp, a c, whch ha cnqunc fr glacal mvmn. Maral whch bhav lacally a rm mpraur fn aan gnfcan vclac prpr whn ha. Such h ca wh mal urbn bla n j ngn, whch rach vry hgh mpraur an n whan vry hgh nl r. Cnvnnal mal can crp gnfcanly a hgh mpraur an h ha l h vlpmn f crp-ran ally; urbn bla ar nw fn ma f call uprally whch cnan m r all f nckl, cbal, chrmum, alumnum, anum, ungn an mlybnum. Crp al n f h prncpal cau f falur n h lcrc lgh bulb. h flamn n lgh bulb ar ma f ungn, a mal wh a vry hgh mlng pn >33 C; h nal bcau h flamn n b lcrcally ha a mpraur hgh nugh fr lgh mn C. If h flamn crp much ag an cl uch ach hr, lang a lcal hr crcu. Lgh bulb la lngr f h mpraur ruc, a n mm lgh. Crp can al b ruc by ang paum bubbl h ungn. Plymr fam u n a cuhn crp, allwng prgrv cnfrman f h cuhn h by hap. h cuhn hlp ruc h prur n h by an ar vry hlpful fr ppl cnfn whlchar r hpal b fr lnghy pr. qu a fw f h applcan an xampl hr ar akn frm Vclac Sl, by R. S. Lak, CRC Pr, 999 Sl Mchanc Par I 89

8 Scn. hy fn hav b rplac afr abu 6 mnh bcau crp cau hm bcm mr n an ff. A nwly brn baby ha vclac an ably crp an rcvr hlp n h brhng prc. Al, f a baby l n n pcfc pn fr lng, fr xampl h am way f lpng all h m, ha can bcm mhapn u crp frman. A baby' kull bcm mr l afr abu a yar. Vclacy al nvlv n h mvmn an bhavur f h cnc pla, h pla whch fla n an ravl npnnly vr h manl f h arh, an whch ar rpnbl fr arhquak, vlcan, c... Sr Rlaxan Guar rng ar vclac. Whn ghn hy ak up a nl r. Hwvr, whn fx a cnan lngh ran, r rlaxan ccur. h p f un n a rng c /, whr h r an h ny. h frquncy f c /, whr h wavlngh. h lngh f h rng L qual half h wavlngh: f / /L. h rucn n r hu mpl a rucn n frquncy an a lwrng f pch h guar g u f un. h rng f a Clacal Guar ar ma f Nyln, a ynhc plymr. h gra clacal guar f h 9 h Cnury n hav Nyln, nvn n 938, bu u Cagu rng, uually ma frm h nn f hp; Cagu a naural plymr. Mal guar rng n g u f un aly nc mal ar l vclac han plymr...3 nrgy Abrpn all bulng vbra whn ynamcally la by wn r arhquak. Vclac maral hav h prpry f abrbng uch vbranal nrgy ampng h vbran. Vclac ampr ar u n m all bulng, fr xampl n h Clumba Cnr n Sal, n whch h ampr cn f l pla ca wh a vclac plymr cmpun - h ampr ar fx m f h agnal bracng mmbr. Smm ncary cnrl vbran bu h u f a plymr napprpra - n h ca ncary u m hr maral wh g vbran-cnrl prpr. A g xampl h u f cppr-mangan ally ruc vbran an n frm naval hp prpllr. h ally ha al bn u n pnumac rck cruhr. Znc al rlavly vclac fr a mal an znc-alumnum ally ar u n pnumac rll - h ally amp h vbran an mak a ll l uncmfrabl fr anyn hlng a pnumac rll. Vclac maral ar al u ln h glv wrn by ppl wrkng wh pnumac rll an jackhammr. Hlcpr mak a l f n, whch cm manly frm h urbn rary ngn an gar, bu uually xacrba by rnanc f h fulag kn. Acuc blank cnng f a layr f fbrgla anwch bwn layr f vnyl clh, plac n h fulag, can ruc h n. Skrky, n hr HH-53C rcu hlcpr, ca a Sl Mchanc Par I 9

9 Scn. mall prn f h fulag kn wh ampng ramn, whch hlp ruc h hgh-frquncy n n h cabn by B. In quarz wach, vbran ar up n quarz cryal a ulranc frquncy khz. h vbran ar hn u gnra prc gnal, whch may b v n nrval f m, lk h cn. Quarz SO a vry lw l maral, manng ha vry un-vclac. h nur ha h vbran ar n ampn an h wach kp g m. unng frk ar fn ma f alumnum a al a lw-l maral. An alumnum unng frk wll cnnu vbrang fr qu a lng m afr bng ruck h vbran vnually wn bcau f un-nrgy l, bu al bcau f h mall nrgy l u vclacy whn h alumnum frk. Vclac maral ar xclln mpac abrbr. A pak mpac frc can b ruc by a facr f w f an mpac buffr ma f vclac, rahr han lac, maral. lamr ar hghly vclac an mak g mpac abrbr; h ar any f varu ubanc rmblng rubbr - hy hav ra nam lk Srbhan, Implu an Nn. Vclac maral ar u n aumbl bumpr, n cmpur rv prc frm mchancal hck, n hlm h fam pang n, n wrlng ma, c. Vclac maral ar al u n h nl ruc mpac ranm a prn' kln. h carlag a h n f h fmur an ba, n h kn jn, a naural hck abrbr. In an arhrc kn, h carlag ha gra mm h bn grn agan ach hr caung gra pan. Synhc vclac maral can b njc rcly n an arhrc kn, nvlpng carlag-fcn jn an acng a a lubrcan an hck abrbr. Sl Mchanc Par I 9

10 Scn.3.3 Rhlgcal Ml In h cn, a numbr f n-mnnal lnar vclac ml ar cu..3. Mchancal rhlgcal ml h wr vclac rv frm h wr "vcu" + "lac"; a vclac maral xhb bh vcu an lac bhavur a b lk a flu an a b lk a l. On can bul up a ml f lnar vclacy by cnrng cmbnan f h lnar lac prng an h lnar vcu ah-p. h ar knwn a rhlgcal ml r mchancal ml. h Lnar lac Sprng h cnuv quan fr a maral whch rpn a a lnar lac prng f ffn Fg h rpn f h maral a crp-rcvry unrg an nananu lac ran upn lang, manan ha ran lng a h la appl, an hn unrg an nananu -ranng upn rmval f h la. Fgur.3.: h lnar lac prng h Lnar Vcu Dah-p Imagn nx a maral whch rpn lk a vcu ah-p; h ah-p a pncylnr arrangmn, fll wh a vcu flu, Fg..3. a ran achv by raggng h pn hrugh h flu. By fnn, h ah-p rpn wh a ranra prprnal r:.3. whr h vcy f h maral. h h ypcal rpn f many flu; h largr h r, h far h ranng a can b n by puhng yur han hrugh war a ffrn p. Sl Mchanc Par I 9

11 Scn.3 Fgur.3.: h lnar ah-p h ran u a unly appl la may b ban by ngrang h cnuv quan.3.. Aumng zr nal ran, n ha.3.3 h ran n ncra lnarly an whu bun lng a h r appl, Fg N ha hr n mvmn f h ah-p a h n f la; ak m fr h ran bul up. Whn h la rmv, hr n r mv h pn back hrugh h flu, ha any ran bul up prmann. h lp f h crp-ln /. r appl r rmv Fgur.3.3: Crp-Rcvry Rpn f h Dah-p h rlanhp bwn h r an ran urng h crp- may b xpr n h frm,.3.4 hr call h crp cmplanc funcn / fr h lac prng..3. h Maxwll Ml Cnr nx a prng an ah-p n r, Fg h h Maxwll ml. On can v h al ran n n fr h prng an n fr h ah-p. Sl Mchanc Par I 93

12 Scn.3 qulbrum rqur ha h r b h am n bh lmn. On hu ha h fllwng hr quan n fur unknwn :,,.3.5 lmna an, ffrna h fr an hr quan, an pu h fr an cn n h hr: Maxwll Ml.3.6 h cnuv quan ha bn pu n wha knwn a anar frm r n lf, ran n rgh, ncrang rr f rvav frm lf rgh, an cffcn f. Fgur.3.4: h Maxwll Ml Crp-Rcvry Rpn Cnr nw a crp. Phycally, whn h Maxwll ml ubjc a r, h prng wll rch mmaly an h ah-p wll ak m rac. hu h nal ran /. Ung h a h nal cnn, an ngran f.3.6 wh a zr r-ra la If n cnr an acual prng f lngh L an a ahp f lngh L a n Fg..3.4, an crrpnng lngan an u ran ε an ε, h al lngan wul b L L L L, whch n qu h am a qn..3.5c. I b hnk f h Maxwll ml a: h al ran a a maral parcl can b cmp avly accrng qn..3.5c, wh h para ran bng lnar lac an vcu; Fg..3.4 mply an amp vual h cncp. hr a jump n r frm zr whn h la appl, mplyng an nfn r-ra. On n rally nr n h jump hr bcau h crrpnng jump n ran can b prc frm h phycal rpn f h prng. On mr nr n wha happn ju "afr" h la appl. In ha n, whn n pak f nal ran an r-ra, n man hr valu a, ju afr ; h r-ra zr frm n. b mr prc, n can al wh h un jump n r by ngrang h cnuv quan acr h pn a fllw: / / In h lm a, h ngral n zr fn, h valu f r an ran a h lm a frm h lf, ar zr. All ha rman ar h valu h rgh, gvng,.. n Sl Mchanc Par I 94

13 Scn.3 C.3.7 h crp-rpn can agan b xpr n rm f a crp cmplanc funcn: whr.3.8 Whn h la rmv, h prng agan rac mmaly, bu h ah-p ha n nncy rcvr. Hnc hr an mma lac rcvry /, wh h crp ran u h ah-p rmanng. h full crp an rcvry rpn hwn n Fg h Maxwll ml prc crp, bu n f h vr-crang ran-ra yp. hr n anlac rcvry, bu hr h lac rpn an a prmann ran. r appl r rmv Fgur.3.5: Crp-Rcvry Rpn f h Maxwll Ml Sr Rlaxan In h r rlaxan, h maral ubjc a cnan ran a. h Maxwll ml hn la { Prblm } whr / R, R.3.9 Analgu h crp funcn fr h crp, call h rlaxan mulu funcn., a xpc. On can al wh h un bhavur mr aly ung ngral frmulan r wh h Laplac ranfrm.4,.5 Sl Mchanc Par I 95

14 Scn.3 h paramr R call h rlaxan m f h maral an a maur f h m akn fr h r rlax; h hrr h rlaxan m, h mr rap h r rlaxan..3.3 h Klvn Vg Ml Cnr nx h hr w-lmn ml, h Klvn r Vg ml, whch cn f a prng an ah-p n paralll, Fg I aum hr n bnng n h yp f paralll arrangmn, ha h ran xprnc by h prng h am a ha xprnc by h ah-p. h m,,,.3. whr h r n h prng an h ah-p r. lmnang, lav h cnuv law Klvn Vg Ml.3. Fgur.3.6: h Klvn Vg Ml Crp-Rcvry Rpn If a la appl unly h Klvn ml, h prng wll wan rch, bu hl back by h ah-p, whch cann rac mmaly. Snc h prng n chang lngh, h r nally akn up by h ah-p. h crp curv hu ar wh an nal lp /. Sm ran hn ccur an m f h r ranfrr frm h ah-p h prng. h lp f h crp curv nw /, whr h r n h ahp, wh vr-crang. In h lm whn, h prng ak all h r an hu h maxmum ran /. Slvng h fr rr nn-hmgnu ffrnal quan.3. wh h nal cnn gv Sl Mchanc Par I 96

15 Scn.3 /.3. whch agr wh h abv phycal ranng; h crp cmplanc funcn nw, / R R.3.3 h paramr R, n cnra h rlaxan m f h Maxwll ml, hr call h raran m f h maral an a maur f h m akn fr h crp ran accumula; h hrr h raran m, h mr rap h crp ranng. Whn h Klvn ml unla, h prng wll wan cnrac bu agan h ah p wll hl back. h prng wll hwvr vnually pull h ah-p back rgnal zr pn gvn m an full rcvry ccur. Supp h maral unla a m. h cnuv law, wh zr r, ruc. Slvng la / C.3.4 whr C a cnan f ngran. h hr maur frm h pn whr "zr la" bgn. If n wan maur m frm h n f la, mu b rplac / wh. Frm qn..3., h ran a /. Ung h a h nal cnn, n fn ha / /,.3.5 h crp an rcvry rpn hwn n Fg hr a rann-yp crp an anlac rcvry, bu n nananu r prmann ran. r appl r rmv Fgur.3.7: Crp-Rcvry Rpn f h Klvn Vg Ml Sl Mchanc Par I 97

16 Scn.3 Sr Rlaxan Cnr nx a r-rlaxan. Sng h ran b a cnan, h cnuv law.3. ruc. hu h r akn up by h prng an cnan, hr n fac n r rlaxan vr m. Acually, n rr ha h Klvn ml unrg an nananu ran f, an nfn r n b appl, nc h ah-p wll n rpn nananuly a fn r hr lmn Ml h Maxwll an Klvn ml ar h mpl vclac ml. Mr ralc maral rpn can b mll ung mr lmn. h fur pbl hrlmn ml ar hwn n Fg..3.8 blw. h ml f Fg..3.8a-b ar rfrr a l nc hy rac nananuly a lac maral an rcvr cmplly upn unlang. h ml f Fg..3.8c- ar rfrr a flu nc hy nvlv ahp a h nal lang pha an n rcvr upn unlang. h ffrnal cnuv rlan fr h Maxwll an Klvn ml wr n ffcul rv. Hwvr, vn wh hr lmn, rvng hm can b a ffcul ak. h bcau n n lmna varabl frm a f quan, n r mr f whch a ffrnal quan fr xampl,.3.5. h ak mr aly accmplh ung ngral frmulan an h Laplac ranfrm, whch ar cu n Only rul ar gvn hr: h cnuv rlan fr h fur ml hwn n Fg..3.8 ar a b c.3.6 h rpn f h ml can b rmn by pcfyng r ran an lvng h ffrnal quan.3.6 fr ran r. 3 h r rqur, whr h Drac la funcn h can b rmn ung h ngral rprnan f.4 Sl Mchanc Par I 98

17 Scn.3 a b c Fgur.3.8: hr-lmn Ml: a Sanar Sl I, b Sanar Sl II, c Sanar Flu I, Sanar Flu II.3.5 h Crp Cmplanc an h Rlaxan Mulu h crp cmplanc funcn an h rlaxan mulu hav bn mnn n h cnx f h w-lmn ml cu abv. Mr gnrally, hy ar fn a fllw: h crp cmplanc h ran u un r:, whn Crp Cmplanc.3.7 h rlaxan mulu h r u un ran:, whn Rlaxan Mulu.3.8 Whra h crp funcn crb h rpn f a maral a crp, h rlaxan mulu crb h rpn a r-rlaxan. Sl Mchanc Par I 99

18 Scn Gnralz Ml Mr cmplx ml can b cnruc by ung mr an mr lmn. A cmplx vclac rhlgcal ml wll uually b f h frm f h gnralz Maxwll ml r h gnralz Klvn chan, hwn n Fg h gnralz Maxwll ml cn f N ffrn Maxwll un n paralll, ach un wh ffrn paramr valu. h abnc f h la prng wul nur flu-yp bhavur, whra h abnc f h la ah-p wul nur an nananu rpn. h gnral Klvn chan cn f a chan f Klvn un an agan h la prng may b m f a flu-yp rpn rqur. In gnral, h mr lmn n ha, h mr accura a ml wll b n crbng h rpn f ral maral. ha a, h mr cmplx h ml, h mr maral paramr hr ar whch n b valua by xprmn h rmnan f a larg numbr f maral paramr mgh b a ffcul, f n an mpbl, ak. Gnralz Maxwll Ml N N Gnralz Klvn Chan N N Fgur.3.9: Gnral Vclac Ml In gnral, a lnar vclac cnuv quan wll b f h gnral frm p p p p 3 p 4 IV q q q q q 3 4 IV.3.9 Sl Mchanc Par I 3

19 Scn.3 h mr lmn prng/ahp n u, h hghr h rr f h ffrnal quan. qn..3.9 mm wrn n h hr-han nan whr P an Q ar h lnar ffrnal prar P Q.3. n n P p, Q q.3. A vclac ml can b cra by mply nrng valu fr h cffcn p, q, n.3.9, whu rfrrng any parcular rhlgcal prng ahp arrangmn. In ha n, prng an ahp ar n ncary fr a ml, all n n a ffrnal quan f h frm.3.9. Hwvr, h u f prng an ahp hlpful a gv n a phycal fl fr h way a maral mgh rpn, rahr han mply ung an abrac mahmacal xprn uch a Nn-Lnar Ml Mr ralc maral rpn can b achv by ung nn-lnar ml. Fr xampl, h prng f h prvu cn can b rplac wh mr gnral nn-lnar r-ran rlan f h frm: n.3. Varu nn-lnar xprn fr ah-p can al b u, fr xampl,.3.3 n / A, A Maral aa wll cranly b br mach by uch nn-lnar xprn; hwvr, f cur, hy wll rul n nn-lnar ffrnal quan whch wll b mr ffcul lv han hr lnar cunrpar..3.8 Prblm. Drv h Rlaxan Mulu fr h Maxwll maral.. Wha ar h valu f h cffcn p, q n h gnral ffrnal quan.3.9 fr a h Maxwll ml an h Klvn ml? b h hr-lmn ml Sl Mchanc Par I 3

20 Scn.3.3b Raran an Rlaxan Spcra Gnral ml can cnan many paramr an wll xhb a whl array f rlaxan an raran m. Fr xampl, cnr w Klvn un n r, a n h gnral Klvn chan; h fr un ha prpr, an h cn un ha prpr,. Ung h mh cu n.4-.5, can b hwn ha h cnuv quan.3.4 Cnr h ca f pcf r, ha h a cn rr ffrnal quan n. h hmgnu lun { Prblm 3} / R / R h A B.3.5 whr R /, R / ar h gnvalu f.3.4. Fr a cnan la, h full lun { Prblm 3} / R / R.3.6 hu, whra h ngl Klvn un ha a ngl raran m, qn..3.3, h ml ha w raran m, whch ar h gnvalu f h ffrnal cnuv quan. h rm n h quar brack vnly h crp cmplanc f h ml. N ha, fr cnan ran, h ml prc a ac rpn wh n r rlaxan a n h ngl Klvn ml. In a mlar way, fr N un, can b hwn ha h rpn f h gnral Klvn chan a cnan la, nglcng h ffc f h fr prng/ahp, f h frm, N / R R.3.7 whr, ar h prng ffn an ahp vcy f Klvn lmn, N, Fg h rpn f ral maral can b mll by allwng fr a numbr f ffrn raran m f ffrn rr f magnu,.g.,,,,,. R If n cnr many lmn, qn..3.7 can b xpr a N / R R, R R.3.8 Sl Mchanc Par I 3

21 Scn.3 If n ban h am rr f magnu f ran fr appl r, h ' wll hav g mallr an mallr fr ncrang numbr f Klvn un. In h lm a N, lng / RR, n ha, changng h ummy varabl f ngran frm, an lng /, R R R /.3.9 h rprnan.3.9 allw fr a cnnuu raran m, n cnra h cr m f h ml.3.7. h funcn call h raran pcrum f h ml. Dffrn rpn can b mll by mply chng ffrn frm fr h raran pcrum. An alrnav frm f qn..3.9 fn u, ung h fac ha / ln whr. / : ln.3.3 A mlar analy can b carr u fr h Gnral Maxwll ml. Fr w Maxwll lmn n paralll, h cnuv quan can b hwn b.3.3 Cnr h ca f pcf ran, ha h a cn rr ffrnal quan n. h hmgnu lun, analgu.3.5, { Prblm 4} / R / R h A B.3.3 whr agan R /, R /, an ar h gnvalu f.3.3. Fr a cnan ran, h full lun { Prblm 4} / R / R.3.33 hu, whra h ngl Maxwll un ha a ngl rlaxan m, qn..3.9, h ml ha w rlaxan m, whch ar h gnvalu f h ffrnal cnuv quan. h rm n h quar brack vnly h rlaxan mulu f h ml. By cnrng a ml wh an nfn numbr f Maxwll un n paralll, ach wh vanhngly mall lac mul, n ha h xprn analgu.3.9, Sl Mchanc Par I 33

22 Scn.3 / R R R an R call h rlaxan pcrum f h ml cmpl h cn, n ha, fr h w Maxwll un n paralll, a cnan r la h crp ran { Prblm 5} / / R / R R / R,.3.35 Prblm 3. Cnr w Klvn un n r, a n h gnral Klvn chan; h fr un ha prpr, an h cn un ha prpr,. h cnuv quan gvn by qn a h hmgnu quan f h frm A B C. By cnrng h characrc quan A B C, hw ha h gnvalu ar /, / an hnc ha h hmgnu lun.3.5. b Cnr nw a cnan la. Shw ha h parcular lun /. c On nal cnn f h prblm ha. h cn cnn rul frm h fac ha nly h ahp rac a m quvalnly, n can ngra h cnuv quan acr a n h fn n.3.. Shw ha h cnn la /. U h nal cnn hw ha h cnan n.3.4 ar gvn by A /, B / an hnc ha h cmpl gvn by.3.6. Cnr agan h cnuv quan.3.4. Wha valu h cnan, ak ha ruc h ngl Klvn ml, qn Cnr w Maxwll un n paralll, a n h gnral Maxwll ml; h fr un ha prpr, an h cn un ha prpr,. h cnuv quan gvn by qn a Supp w hav a prcrb ran hry an w wan rmn h r. h hmgnu quan f h frm A B C. By cnrng h characrc quan A B C, hw ha h gnvalu ar R /, R / an hnc ha h hmgnu lun.3.3. b Cnr nw a cnan la. Shw ha h parcular lun zr. c On nal cnn rul frm h fac ha nly h prng rac a m,. A cn cnn can b whch la h cnn Sl Mchanc Par I 34

23 Scn.3 ban by ngrang h cnuv quan acr a n h fn n.3.. Shw ha h la h cnn / /. U h nal cnn hw ha h cnan n.3.3 ar gvn by A, B an hnc ha h cmpl lun gvn by Cnr agan h cnuv quan.3.3. Wha valu h cnan, ak ha ruc h ngl Maxwll ml, qn Cnr agan h w Maxwll un n paralll, a n Prblm 4. h m cnr a r-rvn prblm. a Frm h cnuv quan.3.3, h ffrnal quan b lv f h frm A B.... By cnrng h characrc quan A B, hw ha h gnvalu ar, / R an hnc ha h hmgnu lun C C whr R /. b Cnr nw a cnan r. By ung h cnn ha nly h prng rac a m, hw ha h parcular lun /. c On nal cnn rul frm h fac ha nly h prng rac a m, whch la h cnn /. A cn cnn can b ban by ngrang h cnuv quan acr a n h fn n.3.. B carful cnr all rm n.3.3. Shw ha h la h cnn / R / / / /. U h nal cnn hw ha h cmpl lun gvn by Sl Mchanc Par I 35

24 Scn.4.4 h Hrary Ingral In h prvu cn, wa hwn ha h cnuv rlan fr a lnar vclac maral can b xpr n h frm f a lnar ffrnal quan, qn Hr hwn ha h r-ran rlan can al b xpr n h frm f an ngral, call h hrary ngral. Qu a fw ffrn frm f h ngral ar cmmnly u; bgn h cn, h ffrn frm ar fr rv fr h Maxwll ml, bfr lkng a h mr gnral ca..4. An xampl: h Maxwll Ml Cnr h ffrnal quan fr h Maxwll ml, qn..3.6,.4. h fr rr ffrnal quan can b lv ung h anar ngrang facr mh. h cnvr.4. n an ngral quan. hr mlar ngral quan wll b rv n wha fllw. Hrary Ingral vr, I mm cnvnn rgar.4. a a ffrnal quan vr h m nrval,, vn hugh h m nrval f nr rally,. h can mak ar al wh un jump n r r ran a m. h nal cnn n.4. hn Ung h ngrang facr /..4., r-wr.4. n h frm Ingrang bh vr,ˆ gv /.4.3 / / / / ˆ ˆ.4.4 r n ha qn..4. prc ha un chang n h ran-ra,, wll la un chang n h r-ra,, bu h r wll rman cnnuu. h ran n appar xplcly n.4.; un chang n ran can b al wh by ngrang acr h pn whr h jump ccur, r ung p funcn an h ngral frmulan lar Sl Mchanc Par I 36

25 Scn.4 ˆ ˆ ˆ /.4.5 Changng h nan,.4.6 whr, h rlaxan mulu fr h Maxwll ml, /.4.7 h knwn a a hrary ngral; gvn h ran hry vr,, n can valua h r a h currn m. I h am cnuv quan a qn..4., nly n a ffrn frm. Hrary Ingral vr, h hrary ngral can al b xpr n rm f an ngral vr,. L hr b a un nn-zr ran a, wh h ran pbly varyng, bu cnnuuly, hrafr. h ran, whch n qn..4.6 b rgar a a ngl funcn vr, wh a jump a, kch n Fg..4.. Fgur.4.: Sran wh a un jump a nn-zr ran a hr ar w way prc. Fr, wr h ngral vr hr para nrval: lm.4.8 Wh,, h fr ngral zr. Wh a jump n ran nly a, h ngran n h hr ngral rman fn. h cn ngral can b valua by cnrng h funcn f llura n Fg..4., a ragh ln wh lp /. A, apprach h acual ran funcn, whch jump a. hn vr Sl Mchanc Par I 37

26 Scn.4 / / / lm lm.4.9 f Fgur.4.: A funcn u apprxma h ran fr a un jump Ung h apprxman x / x fr mall x, h valu f h ngral. hu qn..4.6 can b xpr a.4. By hr n h lwr lm f h ngral, n man, ju afr any pbl nnzr nal ran. In ha n, h ran n qn..4. b rgar a a cnnuu funcn,.. wh n jump vr,. ump n ran afr can b al wh n a mlar mannr. A cn an mr lgan way arrv a qn..4. r-xpr h abv analy n rm f h Hav p funcn H an h Drac la funcn h Appnx h cn fr a cun f h funcn. h funcn kch n Fg..4. can b xpr a H whr nw b rgar a a cnnuu funcn vr, h jump nw cnan whn h p funcn H. qn..4.6 nw bcm H H.4. h fr ngral bcm h ngral n.4.. Frm h brf cun n h Appnx h cn, h cn ngral bcm H.4. A hr Hrary Ingral Fnally, h ngral can al b xpr a a funcn f, rahr han rvav. achv h, n can ngra.4. by par: Sl Mchanc Par I 38

27 Scn h can b xpr a R.4.4 whr R /. N ha ngran by par nly pbl whn hr ar n jump n h funcn unr h ngral gn an h aum fr h ngran n.4.. If hr ar jump, h ngral can hr b pl n para ngral a n.4.8, r h funcn can b rprn n rm f p funcn, whch aumacally accun fr jump. h frmula.4.6,.4. an.4.4 gv h r a funcn f h ran. Smlar frmula can b rv fr h ran n rm f h r h Appnx h Scn. Rlaxan llura h u f h hrary ngral frmula, cnr a rlaxan, whr h ran hry gvn by,.4.5, hrw xprng h ran hry a H, qn..4.6 gv.4.6 Frm.4., wh h rvav n h ngran zr, n ha. R / Fnally, frm.4.4, wh /, n agan ha / / Hrary Ingral: Gnral Frmulan Alhugh rv fr h Maxwll m, h frmula qn..4.6,.4.,.4.4, ar n fac qu gnral, fr xampl hy can b rv frm h ffrnal quan fr h Klvn ml Appnx h cn. Sl Mchanc Par I 39

28 Scn.4 h hrary ngral wr rv rcly frm h Maxwll ml ffrnal quan a mpha ha hy ar n an h am cnuv quan. Hr hy ar rv mr gnrally frm fr prncpl. h ran u a cnan p la appl a m by fnn, whr h crp cmplanc funcn. h ran u a cn la, ay, appl a m lar m,. h al ran u bh la, Fg..4.3,.4.8 Fgur.4.3: Suprpn f la Gnralng an nfn numbr f appl la f nfnmal magnu, n ha,.4.9 In h lm, h umman bcm h ngral, r 3 Fg..4.4 Hrary Ingral fr Sran.4. h agan an applcan f h lnar uprpn prncpl, mnn n 6..; bcau h maral lnar an nly bcau lnar, h "ffc" f a um f "cau" qual h um f h nvual "ffc" f ach "cau" 3 h ngral quan allw fr a un nn-zr r a. Ohr jump n r a lar m can b allw fr n a mlar mannr n wul pl h ngral n para ngral a h pn whr h jump ccur Sl Mchanc Par I 3

29 Scn.4 Fgur.4.4: Frman f h hrary ngral On can al rv a crrpnng hrary ngral n rm f h rlaxan mulu { Prblm }: Hrary Ingral fr Sr.4. h qn..4., whch wa rv pcfcally frm h Maxwll ml. h hrary ngral nly rqur a knwlg f h crp funcn r rlaxan funcn. On n n cnruc a rhlgcal ml wh prng/ahp rmn a crp funcn. Fr xampl, h crp funcn fr a maral may b rmn frm -aa frm a crp. h hrary ngral frmulan hu n rrc parcular cmbnan f prng an ah-p. xampl Cnr h Maxwll ml an h w la hr hwn n Fg h maxmum r h am n bh, ˆ, bu la appl mr graually. ˆ ˆ xamn la fr. h r hry Fgur.4.5: w r hr Sl Mchanc Par I 3

30 Scn.4 Sl Mchanc Par I 3 ˆ,, ˆ In h hrary ngral.4., h crp cmplanc funcn gvn by.3.8, / /, an h r zr a m zr,. h ran hn : / / ˆ ˆ ˆ : / / ˆ ˆ Fr la hry, ˆ,, h ran hn fr. h hrary ngral.4. allw fr a jump a. Fr a jump frm zr r a nn-zr r a can b mf ˆ whch l han h ran u la. Alrnavly, n cul u h Hav p funcn an l ˆ H n.4., lang h am rul, ˆ ˆ. h xampl llura w pn: h maral ha a "mmry". I rmmbr h prvu lang hry, rpnng ffrnly ffrn lang hr h ra f lang mpran n vclac maral. h rul agr wh an brv phnmnn: h ran n vclac maral largr fr r whch grw graually hr fnal valu, rahr han whn appl mr quckly 4. 4 fr h Maxwll ml, f n appl h cn la a m /, ha h al r appl n an wa h am, n wul hav ban h am rpn afr m, bu h n h ca n gnral

31 Scn Nn-lnar Hrary Ingral h lnar vclac ml can b xn n h nn-lnar rang n a numbr f way. Fr xampl, gnralng xprn f h frm.4.4, R f f.4. whr f, f ar nn-lnar funcn f h ran hry. h rlaxan funcn can al b aum b a funcn f ran a wll a m: K, f f Prblm. Drv h hrary ngral.4.,. U h hrary ngral frm f h cnuv quan fr a lnar vclac maral, qn..4., valua h rpn f a maral wh crp cmplanc funcn ln a la B. Skch, whch f cur gv h ran rpn u a un la. Skch al h la an h calcula ran. [n: ln b x x b ln b b ln b 3. A crp wa carr u n a cran lnar vclac maral an h aa wa f apprxmaly by h funcn ˆ, whr ˆ wa h cnan appl la. Skch h ran rpn vr 3 vry rughly, wh ˆ. Whch f h fllwng hr rhlgcal ml cul b u ml h maral: a h full gnralz Klvn chan f Fg..3.9 b h Klvn chan mnu h fr prng c h gnralz Maxwll ml mnu h fr prng an fr ah-p Gv ran fr yur chc an ran fr cunng h hr w. Fr h rhlgcal ml yu ch n par, rughly kch h rpn a anar crp-rcvry h rpn urng h lang pha ha alray bn n n par. v Fn h maral rpn a la. ] Sl Mchanc Par I 33

32 Scn.4 4. Drmn h ran rpn f h Klvn ml a r hry whch rangular n m:, c /, c c /,, c.4.5 Appnx.4. h Hav Sp Funcn an h Drac Dla Funcn h Hav p funcn H fn hrugh, a H a /, a.4.4, a an llura n Fg..4.6a. h rvav f h Hav p funcn, H /, can b valua by cnrng H a b h lm f h funcn f hwn n Fg..4.6b a. h rvav f / hwn n Fg..4.6c an n h lm H a f lm a.4.5 whr h Drac la funcn fn hrugh h ngral hr a ha h ara uny, a llura n Fg..4.6c, a a, a hrw.4.6 Ingral nvlvng la funcn ar valua a fllw: cnr h ngral Sl Mchanc Par I 34

33 Scn.4 g b.4.7 h la funcn hr zr, an hnc h ngran zr, vrywhr xcp a b. hu h ngral b g b b g b b g g b.4.8 H a a a f b a a c f ara a a Fgur.4.6: h Hav Sp Funcn an valuan f rvav. h Maxwll Ml: Funcn f h Sr In.4., h hrary ngral fr h Maxwll ml wr rv fr h r n rm f ngral f h ran. Hr, hy ar rv fr h ran n rm f ngral f h r. Cnr agan h ffrnal quan fr h Maxwll ml, qn ,.4.9 Drc ngran gv.4.3 Sl Mchanc Par I 35

34 Scn.4 Sl Mchanc Par I 36 Ingrang by par la.4.3 Brngng h fr rm n h ngral,.4.3 r.4.33 whr h crp funcn / /. If hr a jump n r a,.4.33 can b xpr a an ngral vr, by valuang h cnrbun f h jump h ngral n.4.33: lm / lm lm.4.34 lang.4.35 Alrnavly, n cul al hav mply l H n.4.33, agan lang h rm. Fnally, ngrang by par, n al ha S.4.36 whr S /.

35 Scn.4 3. h Klvn Ml: Funcn f h Sr Cnr h ffrnal quan fr h Klvn ml, qn..3.,.4.37 Ung h ngrang facr /, n ha Ingrang bh vr,ˆ gv / /.4.38 / / / ˆ ˆ.4.39 r ˆ ˆ / ˆ.4.4 Changng h nan, An ngran by par la /.4.4 /.4.4 Fnally, akng h fr rm n h ngral: / whr /.4.43 h crp cmplanc funcn fr h Klvn ml. h hr vrn f h hrary ngral,. qn..4.,.4.4 can b rv frm h a bfr. Sl Mchanc Par I 37

36 Scn.5.5 Lnar Vclacy an h Laplac ranfrm h Laplac ranfrm vry uful n cnrucng an analyng lnar vclac ml..5. h Laplac ranfrm h frmula fr h Laplac ranfrm f h rvav f a funcn : L f f L f f f f f, c..5. whr h ranfrm varabl, h vrbar n h Laplac ranfrm f h funcn, an f h valu f h funcn a m. h Laplac ranfrm fn n uch a way ha f rfr, ha, ju bfr m zr. Sm hr mpran Laplac ranfrm ar ummar n abl.5., n whch a cnan. f f / H / / / / / / / n n!/ n, n,, abl.5.: Laplac ranfrm Anhr uful frmula h m-hfng frmula: L f H f Mchancal ml rv h Maxwll Ml h Maxwll ml gvrn by h f hr quan.3.5: h rul acually nly wrk fr funcn wh rvav ar cnnuu, alhugh h rvav f h funcn bng ranfrm may b pcw cnnuu. Dcnnu n h funcn r rvav nruc anal rm Sl Mchanc Par I 38

37 Scn.5,,.5.3 akng Laplac ranfrm gv,,.5.4 an ha bn aum ha h ran zr a. h hr ffrnal quan hav bn ruc a f hr algbrac quan, whch may nw b lv g ranfrmng back hn gv qn..3.6: Nw xamn h rpn a un la. Whn ung h Laplac ranfrm, h la wrn a H, whr H h Hav p funcn h Appnx h prvu cn. hn.5.6 ra H.5.7 Ung h Laplac ranfrm gv H.5.8 whch h am rul a bfr, qn Subqun unlang, a m ay, can b al wh m cnvnnly by uprmpng anhr la H n h fr. Pung h n h cnuv quan an ung h Laplac ranfrm gv.5.9 ranfrmng back, agan ung h m-hfng rul, gv H H.5. Sl Mchanc Par I 39

38 Scn.5 Sl Mchanc Par I 3 Ang h h ran u h fr la hn gv h xpc rul,,.5. h Klvn Ml akng Laplac ranfrm f h hr quan fr h Klvn ml, qn..3., gv, whch yl.3.,. h rpn a la H H / /.5. h rpn anhr la f magnu H / / H H.5.3 h rpn bh la nw gv h cmpl crp an rcvry rpn:,, / / /.5.4 analy h rpn a unly appl ran, ubu H n h cnuv quan g H, whch hw ha h rlaxan mulu f h Klvn ml.5.5 h Sanar Lnar Ml Cnr nx h anar lnar ml, whch cn f a prng n r wh a Klvn un, Fg..5. Fg..3.8a. Upn lang n xpc h lf-han prng rch mmaly. h ah p hn ak up h r, ranfrrng h la h cn prng a lwly pn vr m. Upn unlang n xpc h lf-han prng cnrac mmaly an fr h rgh-han prng lwly cnrac, bng hl back by h ah-p. h quan fr h ml ar, frm h fgur,

39 Scn.5 Sl Mchanc Par I Fgur.5.: h anar lnar ml On can lmna h fur unknwn frm h fv quan ung h Laplac ranfrm, gvng.5.7 whch ranfrm back n anar frm.5.8 whch qn..3.6a. h rpn a la H / / H.5.9 an h crp cmplanc

40 Scn.5 Sl Mchanc Par I 3 / /.5. N ha / a xpc. Fr rcvry n can uprmp an pp la n h fr, a m ay: / / / / H H.5. h rpn afr m hn / /.5. h, a xpc, mply h rcvry rpn f h Klvn un. h full rpn a hwn n Fg..5.. h m b farly cl nw h rpn f a ral maral a cu n., alhugh n allw fr a prmann ran. Fgur.5.: Crp-rcvry rpn f h anar lnar ml Nn-cnan Lang h rpn a cmplx lang hry can b valua by lvng h ffrnal cnuv quan r h crrpnng hrary ngral. h ffrnal quan can b m aly lv ung Laplac ranfrm. xampl Cnr h xampl ra arlr ung hrary ngral, a h n f.4.. La f Fg..4.5 can b hugh f a cnng f h w la a / ˆ an b / ˆ H appl a m. La cn f a cnan la appl a m. Fr la a,

41 Scn.5 Sl Mchanc Par I 33 3 ˆ ˆ ˆ ˆ ˆ whch gv h rpn fr. Fr la b n ha [n: L ] ˆ ˆ ˆ ˆ ˆ ˆ ˆ 3 H H H h rpn afr m hn gvn by ang h w rul: ˆ.5.3 Rlanhp bwn Crp an Rlaxan akng h Laplac ranfrm f h gnral cnuv quan.3.9, Q P, la q q q q q p p p p p.5.3 whch can al b wrn n h cnrac frm Q P.5.4 whr P an Q ar h plynmal n n q Q p P,.5.5 h Laplac ranfrm f h crp cmplanc an rlaxan mulu can b wrn n rm f h plynmal a fllw. Fr, h ran u a un la H. Snc /, ubun n h abv quan gv

42 Scn.5 P.5.6 Q Smlarly, h r u a un ran H an I fllw ha Q.5.7 P.5.8 hu, fr a lnar vclac maral, hr a unqu an mpl rlanhp bwn h crp an rlaxan bhavur..5.4 Prblm. Chck ha h rlan.5.8, /, hl fr h Klvn ml. a Drv h cnuv rlan n anar frm fr h hr-lmn ml hwn blw ung h Laplac ranfrm h h Sanar Flu II f Fg..3.8 an h cnuv rlan gvn by qn..3.6 b Drv h crp cmplanc by cnrng a unly appl la. Sl Mchanc Par I 34

43 Scn.6.6 Ocllary Sr, Dynamc Lang an Vbran Crp an rlaxan xprmn n prv cmpl nfrman cncrnng h mchancal bhavur f vclac maral. h xprmn uually prv aa n h m-rang frm cn yar. I fn f nr knw h rpn f maral la f vry hr uran. Fr xampl, uran f h mpac f a l ball n a vclac blck may b f h rr f 5 c. In rr b abl rmn h rpn fr uch cnn, ncary knw h bhavur f a maral a hgh ra f lang r hr uran lang. h chnqu an apparau fr nvgang h rpn f a maral vry hr rm lang ar ffrn h nvlv n lngr-rm ng. Fr vry hr m lang mr cnvnn u cllary han ac lang, an n rr prc h bhavur f a vclac maral ubjc an cllary la, n n frmula h hry ba n cllary r an ran..6. Ocllary Sr Cnr a ynamc la f h frm c.6. whr h r amplu an h angular frquncy. Aum ha h rulng ran f h frm 3 c.6. ha h ran an cllan a h am frquncy a h r bu lag bhn by a pha angl, Fg..6.. h angl rfrr a h l angl f h maral, fr ran whch wll bcm clar lar. xpanng h ran rgnmrc rm, c c n n.6.3 h fr rm hr cmplly n pha wh h npu; h cn rm cmplly u f pha wh h npu. If h pha angl zr, hn h r an ran ar n ynamc xprmn uually prv aa frm abu 8 c. abu 3 c. hr a mwha vrlappng rgn whr aa can b ban frm bh yp f xprmn whn an cllary frc fr appl, rann vbran rul a h naural frquncy f h maral h n u lavng h vbran a h urc frquncy 3 f n ubu.6. n h gnral cnuv quan.3.7, n ha h ran an rvav cnan n an cn rm, ha h ran mu b f h gnral frm A c B n, whr A an B ar cnan. Fr cnvnnc, h can b wrn a C c D whr C an D ar nw cnan Sl Mchanc Par I 35

44 Scn.6 pha a happn wh an al lac maral, whra f /, h r an ran ar cmplly u f pha. c Fgur.6.: Ocllary r an ran c h Cmplx Cmplanc Dfn c, n.6.4 ha c n.6.5 h quan an ar a maur f hw n, r u f, pha h r wh h ran. h frmr,, call h rag cmplanc an h lar,, call h l cmplanc. hy ar uually wrn a h cmpnn f a cmplx cmplanc, * : *.6.6 If n ha a r npu n h frm f a n funcn, hn n n c n n c n c.6.7 an agan h rag cmplanc a maur f h amun "n pha" an h l cmplanc a maur f h amun "u f pha". Sl Mchanc Par I 36

45 Scn.6 h Cmplx Mulu On can al rgar f h ran a h npu an h r a h upu. In ha ca n can wr agan h pha angl by whch h ran lag bhn h r c c c c n n.6.8 h n ffc h am r-ran rlanhp a ha u abv, nly h r/ran ar hf alng h -ax. Dfn nx h w nw quan c, n.6.9 ha c n.6. Agan, h quan ar a maur f hw much h rpn n pha wh h npu. h frmr,, call h rag mulu an h lar,, call h l mulu. A wh h cmplanc, hy ar uually wrn a h cmpnn f a cmplx mulu 4 *, : *.6. Agan, f n ha a nual ran a npu, n can wr n n c n n c n c.6. I apparn frm h abv ha * *.6.3 whch a much mplr rlanhp han ha bwn h crp cmplanc funcn an h rlaxan mulu whch nvlv Laplac ranfrm, qn ypcal valu fr h rag an l mul fr a plymr wul b arun MPa, *. MPa. h ra f h amplu call h ynamc mulu, /. Sl Mchanc Par I 37

46 Scn.6 Sl Mchanc Par I 38 Cmplx Frmulan h abv quan can b uccncly wrn ung a cmplx frmulan, ung ulr' frmula c n.6.4 Fr a r npu, * n c.6.5 h crp cmplanc funcn h ran rpn a un la. In h am way, frm.6.5, h cmplx cmplanc * can b nrpr a h ran amplu rpn a nual r npu f un magnu. Smlarly, fr a ran npu, n ha * n c.6.6 an h rm n brack, by fnn, h cmplx mulu *. h rlanhp bwn h cmplx cmplanc/mulu an h ffrnal cnuv quan Pung an h rulng ran n h gnral ffrnal prar frm f h cnuv quan.3.9, n ha q q q q p p p p * h quan hu gv h rlanhp bwn h cmplx cmplanc an h cnan q p,. A mlar rlanhp can b fun fr h cmplx mulu: * p p p p q q q q.6.8

47 Scn.6 * * Agan n ha. Frm.6.7-8, h cmplx cmplanc an cmplx mulu ar funcn f h frquncy, an hu, frm h fnn.6.4,.6.6,.6.9,.6., h pha angl. hu h prmary varabl nfluncng h vclac prpr whra m wa u fr h purp n h analy f ac lang. h rlanhp bwn h cmplx cmplanc/mulu an h crp cmplanc/ rlaxan mulu I can b hwn 5 ha h cmplx cmplanc * an h cmplx mulu * ar rla h crp cmplanc an rlaxan mulu hrugh * L * L Hr, h Laplac ranfrm fr akn an hn valua a A N n Frquncy, Frqunc blw. Hz ar aca wh mc wav. Vbran f rucur an l bjc ccur frm abu. Hz khz pnng n h z f h rucur. Sr wav frm Hz khz ar prcv a un - abv khz h ulranc rang. Frqunc abv Hz crrpn mlcular vbran an rprn an uppr lm fr r wav n ral l..6. xampl: h Maxwll Ml h cnuv quan fr h Maxwll ml gvn by qn..3.6,.6. Cnr an cllary r c. W hu hav 7 c n n c c n.6. 5 ung Furr ranfrm hry fr xampl 6 an ar al rla ach hr a ar an by an vn mr cmplca rul knwn a h Kramr-Krng rlan 7 h cnan f ngran zr aumng ha h nal ran ha n h prng, /. Sl Mchanc Par I 39

48 Scn.6 hu h cmplx cmplanc *.6. h rul can b ban mr aly ung h rlanhp bwn h cmplx cmplanc an h cnuv quan: h cnuv quan can b rwrn a p Frm qn..6.7, p q q, whr p, p, q, q.6.3 * p q p p / q q.6.4 Al, h cmplx mulu rla h cmplx cmplanc hrugh.6.3, * * /, ha *.6.5 Fr vry lw frqunc,, n / / /. ha fr a ac la,, an h rpn, a xpc, ruc Fr vry hgh frqunc, /, an h rpn / c. hu h ran cmplly n-pha wh h la, bu h ah-p n mvng ha n m rpn a uch hgh frqunc - h prng/ah-p ml racng lk an la prng, ha, lk a l, wh n flu bhavur..6.3 nrgy Dpan Bcau h quan.6. n, n.6.6 ar h paramrc quan fr an llp, ha, hy rac u an llp fr valu f, h r-ran curv fr an cllary r an llpc hyr lp, Fg..6.. h wrk n n rng a maral pr un vlum gvn by h nrgy l hu W.6.7 W hrugh nrnal frcn an ha gvn by h ara f h llp. Sl Mchanc Par I 33

49 Scn.6 W.6.8 whr m arng m an h pr f cllan, n qn..6.6 fr ran an r hn gv /. Subung W n c n n c n.6.9, /, /, /,3 /, akng hn gv8 Fgur.6.: llpc Sr-Sran Hyr Lp W n nrgy L.6.3 Whn, h nrgy pa zr, a n an lac maral. I can al b n ha W.6.3 an hnc h nam l mulu an l cmplanc. 8 h am rul ban fr n, n r whn h r an ran ar cn funcn Sl Mchanc Par I 33

50 Scn.6 Dampng nrgy h nrgy r afr n cmpl cycl zr nc h maral ha rurn rgnal cnfguran. h maxmum nrgy r urng any n cycl can b cmpu by ngrang h ncrmn f wrk frm zr up a maxmum r, ha vr n quarr h pr f n cycl. hu, ngrang frm / whr /, Fg c W n h cn rm n / 4, whch n quarr f h nrgy pa pr cycl, an can b cnr rprn h pa nrgy. h rmanng, fr, rm rprn h ara f h ha rangl n Fg..6.3 an can b cnr b h nrgy r, W c / ruc h lac lun W / whn. h ampng nrgy f a vclac maral fn a W /WS, whr W S h maxmum nrgy h ym can r n a gvn r/ran amplu. hu vng W by 4 cnn wh h ngran vr a quarr-cycl ban h r nrgy W W an Dampng nrgy.6.33 hu h ampng ably f a lnarly vclac maral nly pnn n h pha/l angl. / / c / / n n c / Fgur.6.3: llpc Sr-Sran Hyr Lp 9 r n cul ngra frm zr maxmum ran, vr, /, gvng h am rul Sl Mchanc Par I 33

51 Scn.6 h quany an knwn a h mchancal l, r h l angn. I can b cnr b h funamnal maur f ampng n a lnar maral hr maur, fr xampl, an, c., ar fn u. ypcal valu fr a rang f maral a varu mpraur an frqunc ar hwn n abl.6.. Maral mpraur Frquncy v L angn an Sapphr 4. K 3 khz.5 Sapphr r 3 khz 9 5 Slcn r khz 8 3 Quarz r MHz 7 Alumnum r khz 5 Cu-3%Zn r 6 khz 5 9 Sl r Hz. 5 Alumnum r Hz. F-.6%V 33 C.95 Hz. 6 Baal r.-.5 Hz. 7 Gran r.-.5 Hz. 3 Gla r Hz. 43 W r Hz. Bn 37 C - Hz. La r -5 Hz. 9 PMMA r Hz. abl.6.: L angn f Cmmn Maral.6.4 Impac Cnr h mpac f a vclac ball rpp frm a hgh h n a rg flr. Durng h mpac, a prprn f h nal pnal nrgy mgh, whch nw knc nrgy mv, whr v h vlcy a mpac, l an nly m r. h r nrgy cnvr back knc nrgy whch rv h ball up n h rbun, rachng a hgh h h, wh fnal pnal nrgy mgh r. h ra f h w hgh r f h h r r.6.34 mgh mgh W W W whr W h nrgy r an W h nrgy pa urng h mpac. m nvgar rcmmn ha n u h maxmum rabl nrgy whn, n whch ca h r nrgy / an h ampng maur wul b W / W n / frm abl 7. f Vclac Sl, by R. S. Lak, CRC Pr, 999 h cffcn f run fn a h ra f h vlc bfr an afr mpac, v r / v, f. S Sl Mchanc Par I 333

52 Scn.6 h mpac vn can b apprxma by a half-cycl f h cllary r-ran curv, Fg Ingrang vr, / r /, /, n ha 3 W c n.6.35 an h hgh l gvn by W W f an.6.36 W W W cllary half-cycl mpac rcvry afr mpac Fgur.6.4: Impac apprxma a a half-cycl f cllary r an ran N m hr apprxman ma: nrgy l u ar ranc, frcn an raan f un nrgy urng mpac hav bn nglc n a ral mpac, h r an ran ar bh nally zr. In h currn analy, whn n f h quan zr, h hr fn, an h wll nvably nruc m rrr Dampng f Vbran h nral frc n many applcan can b nglc. Hwvr, whn alng wh vbran, h pruc f acclran m ma can b apprcabl whn cmpar h hr frc prn. Vbranal ampng can b xamn by lkng a a mpl cllar wh n gr f frm, Fg A ma m cnnc a wall by a vclac bar f lngh L an cr cnal ara A. h mn f h ym crb by h quan 3 alhugh mgh b mr accura ngra vr, / 4 a mnn, hr a rann rm nvlv n h cllan whch ha bn gnr, an whch u vr m, lavng h ran lag bhn h r a a cnan pha angl Sl Mchanc Par I 334

53 Scn.6 Dynamc quan: m x F Knmac rlan: x / L Cnuv rlan: pn n ml Aumng an cllary mn, x x, an ung h fr w f h, mx A x L m L A L m A.6.37 h quany n brack h cmplx mulu A an xampl, fr h Maxwll ml qn..6.4 * qn an *.6.38 A Lm,.6.39 whch can b lv g A Lm L x vclac bar If m mall r larg an ha an n ha h amp vbran Fgur.6.5: Vbran / n larg h r ha a ral par, v ay, / v.6.4 Sl Mchanc Par I 335

54 Scn.6 x x x x c / / c c v c v Ac v Bn v.6.4 If, n h hr han, h ma larg r h prng cmplan, n g a pur magnary r, / v, ha ral an l han zr an n ha h aprc ampng x x c / v c / v Prblm. U h ffrnal frm f h cnuv quan fr a lnarly vclac maral rv h cmplx cmplanc, h cmplx mulu, an h l angn fr a Klvn maral. pu h fr w n h frm. U yur xprn fr h cmplx cmplanc rv h ran rpn a r c, n rm f,,,,, n h frm Ac Bn Wha happn a vry lw frqunc? Sl Mchanc Par I 336

55 Scn.7.7 mpraur-pnn Vclac Maral Many maral, fr xampl plymrc maral, hav a rpn whch rngly mpraur-pnn. mpraur ffc can b ncrpra n h hry cu hu far n a mpl way by allwng fr h cffcn f h ffrnal cnuv quan b funcn f mpraur. hu, qn..3.9 can b xpr mr gnrally a p p p q q q.7. whr n mpraur. quvalnly, n can allw fr h crp an rlaxan funcn b funcn f mpraur n h hrary ngral frmulan. hu qn..4.- ra,,,,,, xampl: h Maxwll Ml Cnr a Maxwll maral wh ah-p vcy a funcn f mpraur. h ffrnal cnuv quan hn.7.3 whr h mpraur-npnn prng ffn. h quan a funcn f bh mpraur an m. Wh mpraur a funcn f m,, a lnar ffrnal quan wh nn-cnan cffcn. Fr cnan mpraur, ha cnan cffcn. Cnr fr h ca f cnan mpraur. h rlaxan mulu an crp cmplanc funcn can b valua by applyng un ran an un r. Frm h prvu wrk, n ha,, / R, R.7.4 hu any gvn maral ha mpraur-pnn rlaxan an crp funcn. Cnr nw h chang f varabl Sl Mchanc Par I 337

56 Scn.7 A.7.5 whr A any cnan whch can b chn arbrarly fr cnvnnc lar. h ranfrm qn..7.3 n A A.7.6 h nw an quan wh pnnc n nly n varabl,. Frm h quan, n ban rlaxan an crp funcn / R A, R A.7.7 h quan gnra mar curv frm whch h ffrn mpraur-pnn curv.7.4 can b ban. xampl Daa Fr xampl, cnr a vcy whch var lnarly vr h rang C C accrng h rlan A.7.8 whr a cnan vcy, A. an C a rfrnc mpraur a. h funcn pl n Fg..7. blw. whch / Fgur.7.: lnar pnnc f vcy n mpraur Sl Mchanc Par I 338

57 Scn.7 Al, l / m. h rulng rlaxan an crp funcn f qn..7.4 ar pl n Fg..7. blw fr m 5. a, b 5, Fgur.7.: mpraur-pnn funcn; a rlaxan mulu, b crp cmplanc N h fllwng, rfrrng Fg..7.: fr mpraur grar han h rfrnc mpraur qn..7.8, h vcy. h mpl ha, fr, h rlaxan m ar hrr han fr qn..7.4a, Fg..7.a, an h lp f h crp curv grar han fr qn..7.4b., Fg..7.b. fr mpraur mallr han h rfrnc mpraur,. hu, fr, h rlaxan m ar lngr han fr an h lp f h crp curv mallr han fr. Sl Mchanc Par I 339

58 Scn.7 Nw ch h cnan A n qn..7.5 b qual. h nur ha a h rfrnc mpraur.7.8. In hr wr, h mar curv f qn..7.7 an h funcn.7.4 crrpnng cnc wh h ax an ax cncn. h mar rlaxan an crp curv f qn..7.7 ar nw / / m. h ar pl n Fg..7.3 blw fr m 5. / m an a b Fgur.7.3: mar curv; a rlaxan mulu, b crp cmplanc All h curv f Fg..7. cllap n h mar curv f Fg..7.3 a fllw: h curv crrpnng h rfrnc mpraur,, n Fg..7. l n h mar curv wh h ax an ax cncn fr a curv wh, f h m ax f Fg..7.a,b rch accrng.7.5, h curv wll cm l alng h curv an hnc n h mar Sl Mchanc Par I 34

59 Scn.7 curv; fr a curv wh, f h m ax f Fg..7.a hrunk accrng.7.5, h curv wll cm l alng h curv an hnc n h mar curv.7. hrmrhlgcally Smpl Maral h fac ha h rlaxan an crp curv f Fg..7. cllap n h mar curv f Fg..7.3 rl n h chang f varabl, qn..7.5, rucng h m an mpraur pnn cnuv rlan.7.3 an quan n n varabl,, nly, qn h n urn pn crcally n h frm f h ffrnal quan.7.3. Fr xampl, f h prng ffn n h Maxwll ml mpraurpnn, h cllapng f curv n pbl. mpraur-pnn vclac maral fr whch h cllapng f curv pbl ar call hrmrhlgcally mpl maral. In h cnx, h paramr call h ruc m. Mr gnrally, h ranfrman.7.5 xpr n h frm.7.9 a an h funcn a call h hf facr funcn. h hf facr chn ha h rlaxan an crp curv crrpnng h chn rfrnc mpraur cnc a n h Maxwll ml xampl abv,.. ha a. h rlaxan an crp funcn nw ranfrm a,,,,,.7. Fr mpraur blw h rfrnc mpraur,, a wll b grar han, an h crrpnng rlaxan/crp curv cllap n h mar curv by hrnkng h m ax, whch lk lk a hfng f h curv h lf n h curv. On h hr han, fr, a, an h crrpnng curv cllap by a rchng f h m ax, whch lk lk a hfng f h curv h rgh n h mar curv. h ummar n Fg..7.4 blw. h rul f h ha maral a hgh mpraur an hgh ran ra bhav mlarly maral a lw mpraur an lw ran ra. h mh cu can al b u whn h mpraur m-pnn, fr hn h ranfrman can b xpr a a.7. Sl Mchanc Par I 34

60 Scn.7 ha.7. a lang h am ruc ffrnal quan., a a a hrnk rch, Fgur.7.4: Rlaxan mulu, a a funcn f a m, b ruc m h abv cun ha rla h ffrnal cnuv quan.7.. h analy can al b xpr n rm f hrary ngral f h frm.7.. Fr xampl, h quvaln hrary ngral n rm f ruc m, crrpnng h ruc ffrnal quan qn..7.6 fr h Maxwll ml quan whr.7.3 a bfr qn..7.7 fr h Maxwll ml xprn. a Sl Mchanc Par I 34