Phys 331: Ch 5. Damped & Driven Harmonic Oscillator 1

Transcription

1 Phys 33: Ch 5 Damp & Dvn Hamnc Osclla Thus / F / Mn /5 Tus /6 W /7 Thus /8 F / Damp & Dvn Oscllans 55-6 Rsnanc 57-8 Fu Ss 6- Calculus Vaans Eul-Lagang HW5a 5 5 HW5b HW5c Pjc Bblgaphy Mass n spng wh c pb Why n w npnn sls? Wha s? Gaph n 86 Damp Oscllans: S a w hav jus cns a sng c I w nclu lna ssv c h smpls cas h n c n a pacl mvng n -D s: Nwn s scn law gvs: F k b m k b nng h naual qcy k m qcy whu ampng an h ampng cnsan b m h w maks la sul na w g: N: As h bk pns u an anyn wh s akn u Elcncs cus wul cgnz Th m h quan h chag n a capac q n an RLC ccu s h sam plac wh q chag plac m wh L nucanc plac wh R ssanc an plac k wh /C capacanc Scn- nal quan Thnk abu whn yu slv h vy smpl scn nal quan y/ = Yu nga nc an g v = v hn yu nga agan an g y = y + v Nc ha alng h way yu nuc w cnsans ha can nly b s by knwng sm bay cnns; n hs cas h nal psn an nal vlcy h bjc Tha s gnally u a n - nal quan yu gnal sl mus hav w paams wh whch spcc bay cnns

2 W wll nc agan guss ha h sls a h m s an Subsg hs n u nal quan an canclng u m ach m yls h aulay quan: Applyng h quaac quan slvs hs wh: 4 Cancl u a h asn n s n h nn an n: As lng as an a npnn sls hy a n cnsan mulpls ach h S ms gnally a sl can b lna cmbnan hm In h cas ha hs gvs: C C C C Nc ha hs gvs us u w paams C an C Ths shul b sucn ng any spcc scna; any spcc nal psn an nal vlcy ampl Th ampng cnsan s al an psv s n c h ssv c s mak h sz cas v m Qualavly n Cass Th a 4 qualavly n pssbls = = < an > Ths wll gv qualavly n mahmacal suls / physcal bhavs L s lk a h u pssbl cass: Unamp Oscllans: I h s n ampng an: Whch can b wn as C C ACs as n h s scn hs chap Ths scbs an scllan wh an angula qcy an a cnsan amplu W alay scuss h many ways wng hs sl Wak Dampng / Unamp Oscllans: Th bvus hng cmpa h ampng cnsan wh s h naual qcy snc hy appa gh h squa

3 W wll say ha h ampng cnsan s small squa s ngav s: In ha cas h quany h wh Ths qcy s lss han h naual qcy bu hn Th gnal sl s: C C whch can als b wn as h pa n backs s h sam m as h amp sl: A cs S hs lks lk snusal scllan wh an pnnally-cayng amplu Smms s call h cay paam hs yp mn bcaus mns hw quckly h amplu cass 3 Sng Dampng / Ovamp Oscllans: W wll say ha h ampng cnsan s lag In ha cas h quany h squa s psv an h pnns a al s h gnal sl s: C Ths sl s n sclla bcaus bh ms a cayng pnnals Th ac n backs s small h s m s cays m slwly an mns h mn a lag ms Th h cay cnsan vamp mn s Th ga h ampng cnsan h lng aks h sysm qulbum nc s bn splac hnk a mass n a spng hangng n l C 4 Ccal Dampng: Th las cas cns s h hshl bwn hs w pvus cass whn In hs cas whch mans ha u w pvuslynpnn sls an s npnn anym an w ll n g hng a nw 3

4 sl h mus b w bcaus hs s a scn lna nal quan an w n hav u w cnsans wh whch all pssbl nal cnns W knw ha h s n sl h m Th bk ppss ha quan phas usng ha s als a sl O cus w can plug hs back n h n hs spcal cas L s n hs w vavs an hn plug h pssns n us h puc ul: Okay pluggng hs n h quan: whch wks? S h gnal sl n hs cas s: C C C C Bh ms cay wh h sam cay paam Gaphcally hs sls a smla h vamp sls Th a n scllans Ths s acually h cas ha cays h qucks s cay a pns n jus as h amp cas hwv w v cank up h valu b as lag as can b whu lppng v h vamp cas whch hn cays accng Summay: Th a w ccns n ach sl b mn by h nal cnns usually an a pacula pblm F cass an 4 wh h cay paam s whch s z cas an cas 4 F cas 3 wh h cay paam s whch cass as ncass! 4

5 cay paam T much ampng vampng s n caus h mn cay m quckly! I yu wan an bjc subjc a sng c cm s as quckly as pssbl yu shul mak su Applcans nclu h nl n a m an h mn a ca by lav s whls spngs an shcks n bwn Eampl #: F an amp cas shw ha h a h amplus a w succssv mama n h splacmn s cnsan Th mama n ccu wh cnacs h cuv A Fn h ms whn: A cs s mamum by sng h vav qual z: m m m A cs m A sn m sn m cs m an m m an Acually w v h ms mama an mnma s vy h sl s a mamum Th angn pas vy aans an an wh I s h m h s mamum h subsq mama a a: : n n n n Th a h amplus a w succssv mama s: n n A n cs n A n cs n 5

6 bcaus cs cs Eampl #: Supps h angula qcy a amp sclla s 68 Hz qcy Hz an h a h amplus a w succssv mama s n hal Wha s h naual qcy? Usng h sul Eampl n : n n ln ln 68 Hz ln 693 Hz Th amp qcy s la h naual qcy by: s: 68 Hz 693 Hz 63 Hz Lna Dnal Opas: Th bk ss up h scussn h vn sclla by s pnng u sm gnal pps lna nal pas I m n su ha ha scussn sn m wk han s wh Th l s h nal quan w slv ay: can b hugh as h sul an pa acng n h cn Dn h nal pa D by: Whn hs acs n gvs: D D s h nal quan abv can b wn as D D a b ad bd Th pa s lna : W us h ac ha h pa w v bn alng wh s lna whn w m gnal sls 6

7 Dvn Damp Oscllans: B w mbak n hs l s paus an hnk abu cmpl numbs I yu w u h Tayl ss all vn ms wll b al an all ms wll b magnay gahng all hs vn/al ms gh yu ll cgnz h Tayl pansn cs ; smlaly yu gah all h magnay/ ms gh yu ll cgnz h Tayl pansn sn S sang bu u cs sn Rah vally w mulply hs by sm cnsan cal hn w hav cs sn al magnay I hs bcaus h pssn n h gh lks kn amla; lks a l lk h w cmpnns a -D vc: cs ˆ sn yˆ W can push hs analgy uh by psnng n h cmpl plan: ha s Imagnay mag sn al cs Ral Tha mgh hlp yu mmb hw ansla bwn h w psnans I wh I * an an I / Dam-Dvn Osclla Okay hs wll cm n hany shly Nw n h ask a han Supps a amp sclla s vn by an anal m-pnn c F s h n c n h sclla s k bv F Nwn s scn law gvs: k bv F ma 7

8 : m b k F I w n h c p mass F m an call h nn h ampng cnsan b m hn w can w: Fu s hm can b phas as sayng ha any cmplca cn can b phas as a lna cmbnan cmpl pnnal ms G lv pnnals s s asy ak h vavs! O cus h vng c can hav any cnal pnnc n m; hwv w ll cus n a spcc vng c Fu s hm lls us ha any cmplca cn can b bul a cmbnan sns an csns S a naual plac sa s slvng hs pblm a csn vng c: c c c cs O mayb a sn vng c s s s sn O h s a cazy hugh snc hs s a lna nal quan a hs w quans an smulanusly n a sl a csn an sn vng cs hck h csn an sn hav n amplus say by a ac w cul mulply h scn quan by an hn a : c a s c a s c a s cs asn Okay s w cul ha bu why? Wll nly maks l as an h s n chc a ha s mak l as: Th acual pblm w ll ackl s S ah pvsly s acually gng b as us slv hs quan a -D vc han a scala Th sls w ll g wll als b cmpl s hy cul b wn n h w ms z y * wh y z z z an an y / Fmally hn say w n a cmpl sl ha s z z z 8

9 9 Thn ha s quvaln sayng sn cs y y y w can bak hs n vc quan n w cmpnn quans: n lang jus h al ms an n lang jus h magnay ms cs sn y y y S w ll acually slv h pblm a cmpl vng c bu m ha w ll b abl ac h sl a csn vng c h sl h sn vng c an cus w can bul any cmplca vng c an s sls m lna cmbnans hs Okay h w g; u quan has h m z z z I s ha magn bng abl gna a chaacsc quan lss u sl ha h sam m pnnc as s h vng c ha s C z wh C s an mn cmpl cnsan Subsu hs n h nal quan g: C C C C Ths s a g sl : C Lkng a h nmna ha has n w classc ms a cmpl numb an w can w n h h: y z wh * z z z y an y / an C wh

10 an S hn u sl s z Th amplu gs lag an h phas sh gs small whn na h naual qcy W ll alk m abu ha n m s h vng qcy s z C A h al pa cspnng whn a csn vng c s appl s: p Acs Smlaly a sn vng c s appl: y p Asn S hs slvs h nal quan Bu ha s n h cmplly gnal sl y Thn agan call ha h sls h vn quan plugg n h l han s wll gv Whch mans ha w can a hm hs sl an g anh sl! Tha s Wh w ha C C An nw w v ha v v v cs Wh w v nw ha v Acs Bu ang hs w quans gh w als g ha v v v cs S appanly sl s als a sl hs vn cas Tha s u ms gnal v

11 v Acs C C Th bk cuchs hs sul n m gnal nal-quans languag; many yu I hp ha hlps mak cnncns an san hs b I v av usng ha languag hmgnus nhmgnus pacula n cas was m usul jus cusng n h spcc cas a han Th hmgnus sl mn by h nal cnns mus b a hs g h gnal sl N ha h hmgnus sl cays s can als b call h ansn sl Only h aly mn h sclla pns n hw sas u Th pacula sl s an scllay sl wh h sam qcy as h vng qcy whch can als b call h say-sa sl Th mn lag ms nly pn n h paams h sysm nclg h vng c n h nal cnns Eampl: smla E 53 Supps a/s a/s m/s an 4 a/s nly nc m E 53 I h sclla sas a s a h gn n an pl h cn psn as a cn m Cmpa wh h suls E 53 Th qcy h vn sclla an h hmgnus sl s: Th amplu h pacula sl s: A an h phas angl s: a/s m/s 77 m 4 an an aans Th gnal sl an amp vn sclla can b wn as: Acs B cs B sn wh h ccns B an B mus b mn m h nal cnns v Fm h quan abv: Acs B B Acs 77 m cs 465 a 76 m Takng h vav gvs: v Asn B cs B sn B sn B cs B v Asn B 9987 v Asn B B 4 77 m sn 465 a 76 m

12 B 87 m Th gaph h sl s shwn blw sl ln alng wh h sl E 53 ash ln wh h vng qcy s a/s Th say sa sl hs ampl has a slghly lag amplu bcaus h vng qcy s cls h naual qcy I als lags a ll ah bhn h vng c In h mal languag Dnal Equans wha I jus agu s ha yu hav a pacula lna nal quan Dˆ p usng h Quanum nan ha an pa Thn h gnal sl s h lna cmbnan a pacula sl hs quan p an h sls hs smpl hmgnus quan ˆ D h Snc Dˆ p h Dˆ p Dˆ h