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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