Forced Vibrations 3.8

car mframe

y 3kAK

sudden bump at t o gives nonzero initial condition

mu ka 8 u U displacement from equit

mu t Ju't Ku 0

carcar M frameframe

spring displacement f up isu oy o j k Ult from equilibrium positive

httmm nh O

uneven ground starting at b o

m fear frameacceleration Ku Ju

m ath Ku ru

mu t ru't Ku _mnonhomg termeffective external

driving force

There are many other examples of forced vibrationse.g anytime sound wave hits an object

mu t Ju't Ku 7thconsider periodic driving force

f t Fo cos wt Fo w 0

or to sin wtconstants

Undamped Forced Vibration

mu t Ku Fo cos wt

Gensoln

Ult Ci cost wot 1 Cz sin wot t Uplt wo fkm

Icase 1 W Wo Beats

Ansatz Upa A costwt t B sin wt

Fo cos E Mup t Kup

mf Awhos t Buisincwt tk Acostwt t Bsinlwt

f MWZ 1k Acos wt t f muftis Bsin wt

A to FoMWZ 1k m WI WT B O

Cco's wot 8

Gen Sohn UH Ccostwot tasin wot 1 mfwotz wyco.rs wt

sum of waves ofdifferent frequencies

Amplitude of Uplt Fom wordWI

n

ifW

Wo

Useful Trig identitiessin 0 I sing 2 sin 02 cos OFIco50 cosy 2 sin Off sin 021CosO t C054 2 cos 021 cos 021

Exampled Let Ulo O U lo O

c FMWf uf C z O

ult Fomo2 w2J Costcot cos wot

m wFsinlwoyst sinlwot.LI

M 737at mzEoEwgsiiw s

actas an amplitude envelope

sinkoogst

T

mE sinkoogst

Amplitude modulation

Example 2 cost t cos lot

Example U 125 u cos3T UCO O U lo O

chareq R2 125 0

r2 25

r t5i

UH C cos5T Cz sin 5T t Uplt

Ansate Uplt Acos3tt Bsin3t

Up It 9Acos3t 913 sin 3T

Up t 25Up cos 3T

9Acos3t 9Bsin3t 25 Aco53ttBsin3t cos3t

16A cos3T 116135in3T cos3T

16A 1 1613 0

A to 13 0

UH C cos5T Cz sin 5T tf cos3T

O Uto C If C 76

U'It 5C sin 5T 15oz cos 5T 36T Sin 3 t

O U o 5 Cz Cz 0

Ult IT cos 5T It cos 3T

t cos5T cos 3T

Is sin 5322 sin 15 31 2

att f sent sin 4T

Amplitude period of sin At2

NUH

f Sint

N

taint

Case 2 W Wo Resonancei

mu t Ku f cos wot

Ansatz Uplt At cos wot t Bt sincwot

Focostwot Mup t KUp

AfterM Up 1 WEup

simplification Mf 2AWosin ot7t2BWoCo5Cwotf

7A o.B E2M Wo

Gen Solis UH CCos wot 195in wot 1 2mW tsin wot

Ceos Wot 8

UpIt wot sin wotFoTmwot

If

It2MWo

Purple 2 cost t Sint It sint

Orange 12 tsint

Green 2 cost Sint

Periodic driving force damping

mu t Ju't Ku Fo cos t

Gen Sol n i ult Guilt t Cz Udt UpIt

Gen s da to mu Yu't Ku 0

always decays called transient solution

Ansatz UpLt Cao wt t D sin wt

Upits is called the steady state soln

left WC sin wt t w D cos cut

Up t w2C cos wt t w Dsn wt

Fo cos wt M Up t tap KupK Mw Ct 8WD cos wt t K mw 3D 8Wc sinwt

K mw2 Ct ow D Fo

rwc ick.me oJcfEIi54IzD8wFoCkmw74 rwJ2

Write Up Coos wt D sin wt Acost wt 5

A2 c2 p2 K Mur to ow 2FIK mw

2t yw 2 2

FoZ

K mat 2t 8wF

Fo

my wi w'T

Amplitude AFo

m4woZwWTrdenominator

always to if 81 0

Remark This calculation can also be done byAnsatz Up A cos wt 8 Re Ae

wt 8

Ttp

Re Foe'wt Re mup trip't Kuip

Plot A vs w

FoA Iwf

m2 Wf w t

w o A ITw a A o

A has a max when mywi w t WYhas a min

ddw m2 wi wz t w

f 2m21 we wz 182 2W

W O W 2 M2

real when WE IMI z owe In wi Enzomu t Ju't k U o

underdamped

Acw

racticalt t resonance

jfw m

i

i1I 7 ww wtf L Wo

By adjusting damping 8 can change resonance frequency

EI U t 4 u t Bu 20 Cos wt

charege r't 4 r t 13 O

r 21 3 i

U C e Hoos t t Ge Hsin 3T t Uptt

my

Resonance happens when

W fT3 IF I 2.24

mmmm

A U t 4 u't 134 20 cos t Uco lo Uto O

ult 8.5 e 2tcosbtj 5 se tsinlzt t3z costtzsintI 6coslt ov3

Blue transient soln

Orange steady state solnBlack Ut

b U 144 t 134 20 cos t t 20 cos Bt

Uplt cost Isin t t 3zcosfbt.int

L 6 cos t 0.3 0.1 cos 13T 3.4