Chapter2FreeVibratio
Noexter
alexcitatio
ObjectivesEquatio
ofmotio
Naturalfreque
cyDampi
gPri
cipleofco
servatio
ofe
ergy21VIBRATIONMODELMasskglbs2i
Masslessspri
gli
earspri
gwithFkxkNmlbi
Viscousdampi
gfcxcNmslbi
s22EQSOFMOTIONNATURALFREQUENCYSimpleU
dampedSpri
gMassSystemAssumedtomoveo
lyalo
gtheverticaldirectio

fisthedeformatio
ofthespri
gi
thestaticequilibriumpositio
kmg221xmFmgkxxmkx222Withthestaticequilibriumpositio
asrefere
ceforxtheresulta
tforceo
missimplythespri
gforceduetothedisplaceme
txk2
223m2xEq222ca
bewritte
as
x0x0si
tx0cos
tx226

The
aturalperiodoftheoscillatio
mka
dthe
aturalfreque
cyis11kf
2mI
termsof1gf
2
2
227
228
229
fExample221A1kgmassissuspe
dedbyaspri
g4havi
gastiff
essof01533NmmDetermi
eits
aturalfreque
cyi
cyclesperseco
dDetermi
eitsstaticdeflectio

fExample222Determi
ethe
aturalfreque
cyofthemassMo
thee
dofca
tileverbeamof
egligiblemassshow
i
Fig222
fExample223A
automobilewheela
dtirearesuspe
dedbyasteelrod050cmi
diametera
d2mlo
gasshow
i
Fig223Whe
thewheelisgive
a
a
gulardisplaceme
ta
dreleaseditmakes10oscillatio
si
302sDetermi
ethepolarmome
tofi
ertiaofthewheela
dtire
fExample224Figure224showsau
iformbarpivotedaboutpoi
tOwithspri
gsofequalstiff
esskateache
dThebarishorizo
tali
theequilibriumpositio
withspri
gforcesP1a
dP2Determi
etheequatio
ofmotio
a
dits
aturalfreque
cy
f23ENERGYMETHODThediffere
tialequatio
motio
bythepri
cipleofco
servatio
ofe
ergyTheki
etice
ergyTStoredi
themassbyvirtueofitsvelocityThepote
tiale
ergyUStoredi
theformofstrai
e
ergyi
elasticdeformatio
orbyaspri
gorWorkdo
ei
aforcefieldsuchasgravityThetotale
ergybei
gco
sta
titsrateofcha
geiszeroTU0231dTU0232dtFor
aturalfreque
cyofthesystemT1U1T2U2Letsubscript‘1’bethetimewhe
themassispassi
gthroughitsstaticequilibriumU10Letsubscript‘2’bethetimecorrespo
di
gtothemaximumdisplaceme
tofthemassT20T100U2Ifthesystemisu
dergoi
gharmo
icmotio
T1a
dU2aremaximumvaluesTmaxUmax
fExample231Determi
ethe
aturalfreque
cyofthesystemshow
i
Fig231
fExample232Acyli
derofweightwr