Section 2 3 pr off circuits and

Resonanceplaini Springs and circuits

Forced OscillationsFy

Non Resonant

Resonant

o

Springs and Circuits

Spreng Oscillations LC Series Circuit

K M fCt L C Vctwent

f a b y ICE electoricJct current7 0

Li Inductor coolNr Friction

C i capacitor conductorSeparated byinsulator

Ct Voltagesource powert t

Mathematically EquivalentNew ton's Ef Kirchhoff's Ef

my b Y t fCt L I t f I V'Ct

Spring externalforce force 2 1 I 25 f E 44

Mel the 25

Y t 25 Y f Ct Y t 25 Y fCt

commen ti Friction Terms

K m fCt R L C Vct

friction That 1 05 111yet y ICH one

7 0

With Friction d 0 Resistance 12.20

H ILI.it 2I'td I V'C

O

tree oscillations fCt o

r2 t 25 0

It si

y c cos 5t tc2SinC5 III cf ref

Forged Oscillations Non Resonant

fCt cos VE Of 5 w

dr m

I lfett.gg ycosCvttcosC5fFo

ced Oscillations Resonant

fCt Cos4T yo8 52

s t253 o4 I'g jyctl tsinC5t

Forged Oscillations Non Resonant

Example Solve the IVP

y t 25 y advt V 5

y O e O Y o

soliy 25 y cos Vt

Ley

1 Sods H m.EE Y t 253 0 Y Ert

Lt 25 o r V

i

5 it sitFund Sols Yt e Y e

Real Fund Sols Ya Sin 5

2 Sec Ion Heon EffCt cos t YpCt k cos Dt Ks Sin t

Recall i f 5 L YpO_ Yp GoodGuess

3 Compute k kz from L Tpi f

Yip kid sin 6T t ka V cos t

Yp k 02 cos t t ka tu D Seis t

Soi

k C02 cos t t ka Cv2 Sin t t 25 K and't t by Sin

cos t

cos t V k t 52k l t Seis ut V2 ka t 5 ka so

cosCvt K C52 v2 l t Seis Cvt Rz 52 V2 o

for all t

t I

so

1 95 3 I 5 Ik

µpH L ym

Gen Soc Jct e arrest Ca Seis Est t c.EE j

I C O Y o C t O t 152 D2

a L_

Y'Ct 5C sirs C5E t 5 Cz cos 69 r jI C O Y'co O t 5 Cz O

lc

Yet costs t tg g

an t

Jlt Fay coset costs1 1

Forged Oscillations Resonant

Example Solve the IVP

y t 25 y costsE 0 5

y O 0 Y'co O

Sol Y t25Y cos 5

Ley

L Hom Ef Y t 25 Y O T Y Ert

r2 t 25 o r I5i_

12 5454 1q

2 NonHom EfFirst Guess S D

sin 64f cos 58 Jp k Cortot t k

But now Yp o Yp i W rong Guess

3 Second Guess

Jp k t cosest t ka t Seists E L Ypa 401Ip k cost54 t k Sin 5t t kit C5 Sirs E t Kat 5 Cos64

Y 2 k C5 SeisCSE t ka coset

I kibtcoz SEJ kz52tsin.GL

10 k sirs t t 10 Kz cos6T k 52 55 k 554t 25 k 58 t Katt cos t

lo k Sin 5T t lo k cos4 t cos 6 t

10K Seis C5t t fo ka 1 cos E 0 acc

Y µF

Yp to Sin 5 t1 1Gen Soc YCt 9as 5t tCzSin 5EJ Seis

5 II C O Yco G t O t O 1 12

Y Ct 5C sirs E t 5 S Cosest

fo sirs 64 t 5,01 coset

O Y'co O t 5 Cz t t O fc2TO

yct o SeisC5