Damped Harmonic Oscillator Css3

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Mu Ku JuTamping

mu t Ju't Ku 0Photo credit Wikipedia

chareq Mr't Tr t ku o shock absorber

roots re 8174M2M

Ism I FF m

Overdampedy m o dire Foots

rqgcritically dampedW Im o repeated

real roots

2Im O ex roots

Oscillation Underdamped

mhm

Underdamped

cxroots r Im FEETEmt in

µGen Sohn

UH C e Etcos ut t a e Etsin ut

e Ent C cosbut t G Sin Mt

eEnt A cos but 8

Quasifreq.lk EmfLWo fEm

Quasi Period T2 To

2

Amplitudeenvelope I Ae Imt

First time touching the upper envelope A eEnt

µ t d o t Iµ

assuming we chose 2E e g e 2e which we did so

the Snf eastt o sit Alt 8 2 In happen when

A 2 It touches the envelopeA T I N I M r

like this hiriAco's I e Emt

not

Iu

Ae Ent

r

A r

Exercise At any time tp Sit cos Mtp 8 11

check that U'Itp to

Criticallydampede M o

Gen soln Uct c eEnt at e Emt

e Imt c CatUH i

U'lo Ofexp decay comment added after

u.ca em7Im Iiealso be 0 like

9 G IU O

Exercise check u lo Cr Imc

Can use u Iti o to figure out the peak or

bottom of the bumplocal max min

Overdamped

real roots r Em I f E

neut aemtfEFEIt c.etEm EEm t

both terms are exp decay

Im EEEThe graph looks exactly like the critically damped case

UH e Etf.eeEt

cze EFEntJ

e Etfftcosh EEnt Bsinh EET

Note about cosh Sinhnail nail

ooo

cosh x e teX

2

Sinh x ecosh x

2

Reedcos

eix e2

Sine't e

ix

2T

Comments added after lecture since someone asked

Notice cosh x Sinh x ex

cosh Sinh e

So c ex e

C coshxtsinhx 1 Cz coshx sinhx

it G cosh x Cc G Sinh X

A cosh x t B SinhX