Lecture 17

(The driven harmonic oscillator)

ReviewEixample find the general Lohr ofLaps9yttbjtloy-etlacoscstltbsinbtlkjssociyititeys.laheld. eteif'lap :#4yitttgyaitypltibel2-q.ge#ilEfbp=RefGedtiky-

inZ

9ltftttzp.tt/otps(2.gfgecH3ilt9(llxsiPGeHHt)t6Cdt3i1GelHHt)t1oCGe4t3iH){ 9LH3ittbctt3.lt/o3GetH3iHsf2.6i)e4t3iHG..l2-36i19Ht3ift6LHHHO

Review,

Example Find the general Lohr ofLCyt-9ytbjtlog-etlacoscstltbsinbc.tl Greff!%%Solution than-

.

etbcoscstltbb.nl#)=RefC2.6dedt3iH/ gloss C, -0.065--0

let CHIHytd-Refcq-c.gl/gttdtZpH=GeWant

Lap]=f9(Its Schilt.to/GeH3dt(-0.0651-0.02344+31)=6

= (2,6,it = ft g - 0.13=0

So G=2-6 9=0.065{9(H3dY6lHttlo)II -

0.0651-0.023i

£0.156

0.069 edith.

ypkt-Rekptth-o.069e.us/3tt2.8lylHsAEtbcosH.y)

+ o.69etcoxstt2.gl .

=Re{G-igettitttfo.ossto.ona.de#dtfI Itransient quasi-periodic

Beads

•

A-wgiydentiticoscmtcoslBls-hcas.CA#usfAtrProof:lotas Atf , b s AGBThen A = atb

,B s a- b

cos (A) = cos lathis costal costs) - Sin la) fin (b){ cos (B) s cos ca-b) s coldcold thin Cassin lb)

So cos (A) twoCB) : 2 cos (alcohol

= 2 cos (Agf) cos CALI ) AM

Specially• cos CA) - cos CB) = cos (A) t cos (BHT) • Lin IA) - sin CB) = cos (A -I) t Cp (Bt't)--

= 2 cos (A;fBcs(AtBtt)= 2 us (BLAIK) cos(AIB)

= -2 sin (AI) sina.gg#-=-2sin(B-Ag)asCAIB)= a sin (BIA)sin (ATL) = 2tin (Agb) cos(Atg)

Application. codwt) - cool wot) = 2 sinfwgwtt) hilwotgwlt)4thwotw)as

--- ----" ''n,

-

--

i

w -wosma "

:÷÷¥..twos

Thedrivenharnnoniiator

Viagra .

d'

o

dfat Wo ⑤ = A coolwt)

→ imdb.tkas kyllt)m d} s - blog -g)At

AkkaAt

Resonance-

costal - costs)=2sin(bga)sinqtbg)mix"

tbxsfuscwtl

Yi!! .tl/7ttt.:E:sinCwoEdsinCwotSt)Xp# Reff wtf(Geiwtftw.YGeiutt.to/mewtwg-I low frequencyPenitent) - { wotan high frequency .÷÷÷i÷÷÷÷wtq÷÷÷÷:i:xth.Relkiislewotttughnguslwtlwlisq.t.gg#tIst wot ,

Alo)-0-5 C,# Fob so

So

wow x#→ Etmtsinlwotl- j2h00x'lol ⇒ Swot 0=0 which satisfies the IVP

Hence,

mnitkxsfocoskgth

µ# , Edm codwtl-Ywotlxloso.nl/d--owo2-w2

t¥Eo ?

Mx"t8x'thx > Eascwth.sk/FoeiwtfTo maimze 1Gt ,

minimizettzp,# feint xpH=Re{ tpktlftp.msn/wow4tt8w2wantmIptttptk2p=Foewtftp.2mfwo?w4t2w)fm.cws7oliwlthfGewtsEeiht+ zyawso

So - 2 2 I

Gs E- -- to ⇒ -Ymlwo - w4t2V=o

Ivmkwj.wwrswre.cwd-Vwi.Tmiwj.VEEmas to

nmlwj.ci)Therefore .

x # = Aeimtcoscwdt -yl .

ItransientFo

+# coslwt - a)vmki.wH.hr-

periodic .

mnitdnitkx-fcn.cat#mni=-kx-8x'tFoasEotlxpHsteady state Rlwlcolwt - y)

inqgain phase shift.

7ps Geiwt

Mtp"tftp.tbzp-feiwtf-mwtfwitb/Geiwtseeiwt-

G : F Rhys lfcwys FilmD--

-G-mw' tfwi

=Hm_Awful:(guy

Gm-w't 1mW 's

= Tim , Re"4

(Wf- watt Imwi