2dof Forced vibration of two degrees of freedom system

Twodegreeoffreedomsystems
•Equationsofmotionforforcedvibration
•Freevibrationanalysisofanundampedsystem
Introduction
Introduction
h dd d dbh
• Systemst
h
atrequiretwoin
d
epen
d
entcoor
d
inatesto
d
escri
b
et
h
eir
motionarecalledtwodegreeoffreedomsystems.
f
Nb
masseach ofmotion of s
y
stemin the s
y
stem theof
typespossible ofnumber masses ofNumber freedom of degrees
o

f
N
um
b
er

y
y
Introduction
Introduction
h f d f fd f h
• T
h
erearetwoequations
f
oratwo
d
egreeo
f

f
ree
d
omsystem,one
f
oreac
h

mass(preciselyoneforeachdegreeoffreedom).
• Theyaregenerallyintheformofcoupleddifferentialequations‐thatis,

eachequationinvolvesallthecoordinates.
• Ifaharmonicsolutionisassumedforeachcoordinate,theequationsof
motionleadtoafre
q
uenc
y
e
q
uationthat
g
ivestwonaturalfre
q
uenciesof
qyq g q
thesystem.
Introduction
Introduction
If i it bl iitil it ti th t ib t t f th
•
If
weg
i
vesu
it
a
bl
e
i
n
iti

a
l
exc
it
a
ti
on,
th
esys
t
emv
ib
ra
t
esa
t
oneo
f

th
ese
naturalfrequencies.Duringfreevibrationatoneofthenatural
frequencies,theamplitudesofthetwodegreesoffreedom(coordinates)
are related in a specified manner and the configuration is called a
normal
are

related

in


a

specified

manner

and

the

configuration

is

called

a

normal

mode,principlemode,ornaturalmodeofvibration.
• Thusatwodegreeoffreedomsystemhastwonormalmodesofvibration
correspondingtotwonaturalfrequencies.
• Ifwegiveanarbitraryinitialex citationtothesystem,theresultingfree
vibrationwillbeasuperpositionofthetwonormalmodesofvibration.
However
,
ifthes
y

stemvibratesundertheactionofanexternalharmonic
, y
force,theresultingforcedharmonicvibrationtak esplaceatthefrequency
oftheappliedforce.
Introduction
Introduction
d f h h h f h f f
• Asisevi
d
ent
f
romt
h
esystemss
h
ownint
h
e
f
igures,t
h
econ
f
igurationo
f
a
systemcanbespecifiedbyasetofindependentcoordinatessuchas
length,angleorsomeotherphysicalparameters.Anysuchsetof
coordinatesiscalledgeneralizedcoordinates.
•

Although the equations of motion of a two degree of freedom system are
•
Although

the

equations

of

motion

of

a

two

degree

of

freedom

system

are

generallycoupledsothateachequationinvolvesallcoordinates,itis
alwayspossibletofindaparticularsetofcoordinatessuchthateach

i f i i l di Th i f i
equat
i
ono
f
mot
i
onconta
i
nson
l
yonecoor
di
nate.
Th
eequat
i
onso
f
mot
i
on
arethenuncoupled andcanbesolvedindependentlyofeachother.Such
asetofcoordinates,whichleadstoanuncoupledsystemofequations,is
calledprinciplecopordinates.
Equationsofmotionforforced
vibration
d l ddd f fd
• Consi
d

eraviscous
l
y
d
ampe
d
two
d
egreeo
f

f
ree
d
omspring‐masssystem
showninthefigure.
• Themotionofthesystemiscompletelydescribedbythecoordinatesx1(t)
andx
2(t),whichdefinethepositionsofthemassesm1 andm2 atanytimet
from the respective equilibrium positions
from

the

respective

equilibrium

positions
.

Equationsofmotionforforced
vibration
h l f
d
h
d
l
• T
h
eexterna
l

f
orcesF1 an
d
F2 actont
h
emassesm1 an
d
m2,respective
l
y.
Thefreebodydiagramsofthemassesareshowninthefigure.
• Thea
pp
licationofNewton’ssecondlawofmotiontoeachofthemasses
pp
givestheequationofmotion:
Equationsofmotionforforced
vibration

b h h f l
h
• Itcan
b
eseent
h
att
h
e
f
irstequationcontainstermsinvo
l
vingx2,w
h
ereas
thesecondequationcontainstermsinvolvingx
1.Hence,theyrepresenta
systemoftwocoupledsecond‐orderdifferentialequations.Wecan
thereforeexpectthatthemotionofthem1 willinfluencethemotionof
m
2,andvicaver sa.
Equationsofmotionforforced
vibration
h b f
• T
h
eequationscan
b
ewritteninmatrix
f

ormas:
where [m] [c] and [k] are mass damping and stiffness matrices
where

[m]
,
[c]

and

[k]

are

mass
,
damping

and

stiffness

matrices
,
respectivelyandx(t)andF(t)arecalledthedisplacementandforce
vectors,respectively. whicharegivenby:
Equationsofmotionforforced
vibration
b h h [][] d [k] ll h
• Itcan

b
eseent
h
att
h
ematrices
[
m
]
,
[
c
]
an
d

[k]
area
ll
2x2matricesw
h
ose
elementsaretheknownmasses,dampingcoefficienst,andstiffnessofthe
system,respectively.
• Further,thesematricescanbeseentobesymmetric,sothat:
Freevibrationanalysisofanundampedsystem
•
For the free vibration analysis of the system shown in the figure we set
For


the

free

vibration

analysis

of

the

system

shown

in

the

figure
,
we

set

F
1(t)=F2(t)=0.Further,ifthedampingisdisregarded,c1=c2=c3=0,andthe
equationsofmotionreduceto:
Freevibrationanalysisofan

undampedsystem
d k hh
d
ll
• Weareintereste
d
in
k
nowingw
h
et
h
erm1 an
d
m2 canosci
ll
ate
harmonicallywiththesamefrequencyandphaseanglebutwithdifferent
amplitudes.Assumingthatitispossibletohaveharmonicmotionofm
1
andm2 atthesamefrequency

andthesamephaseangle

,wetakethe
solutionstotheequations
as:
h X
d X
h d h i li d f

w
h
ere
X
1 an
d

X
2 areconstantst
h
at
d
enotet
h
emax
i
mumamp
li
tu
d
eso
f

x
1(t)andx2(t)and isthephaseangle.Substitutingtheabovetwo
solutionsintothefirsttwoequations,wehave:
Freevibrationanalysisofan
undampedsystem
• Sincetheaboveequationsmustbesatisfiedforallvaluesoftimet,the
termsbetweenbracketsmustbezero.Thisyields,

which represents two simultaneous homogeneous algebraic equations in
which

represents

two

simultaneous

homogeneous

algebraic

equations

in

theunknownsX
1 andX2.Itcanbeseenthattheaboveequationcanbe
satisfiedbythetrivialsoutionX
1=X2=0,whichimpliesthatthereisno
ib i F iil li f X
d X
h di f
v
ib
rat
i
on.
F

oranontr
i
v
i
a
l
so
l
ut
i
ono
f

X
1 an
d

X
2,t
h
e
d
eterm
i
nanto
f

coefficientsofX
1 andX2 mustbezero.
Freevibrationanalysisofan

undampedsystem
• Theaboveequationiscalledthefrequency orcharacteristicequation


0}))({(})({)(
2
23221
2
132221
4
21
 kkkkkmkkmkkmm

becausesolutionofthisequationyieldsthefrequenciesofthe
characteristicvaluesofthesystem.Therootsoftheaboveequationare
g
ivenb
y
:
g y
Freevibrationanalysisofan
undampedsystem
Thi h th t it i ibl f th t t h tii lhi
•
Thi
ss
h
ows
th
a

t

it

i
sposs
ibl
e
f
or
th
esys
t
em
t
o
h
aveanon
t
r
i
v
i
a
l

h
armon
i
c

solutionoftheform
when=1 and=2 givenby:
WeshalldenotethevaluesofX1 andX2 correspondingto1 as
andthosecorrespondingto
2 as.
Freevibrationanalysisofan
Fth i
undampedsystem
•
F
ur
th
er,s
i
nce
theaboveequationishomogeneous,onlytheratiosand
r2=canbefound.For,theequations
give:
• Noticethatthetworatiosareidentical.
Freevibrationanalysisofan
undampedsystem
Th l d f ib ti di t b
•
Th
enorma
l
mo
d
eso
f

v
ib
ra
ti
oncorrespon
di
ng
t
ocan
b
e
expressed,respectively,as:
• Thevector s,whichdenotethenormalmodesofvibrationare
knownasthemodalvect orsofthes
y
stem.Thefreevibrationsolutionor
y
themotionintimecanbe expressedusing
as:
as:
wheretheconstantsaredeterminedbytheinitial
conditions.
Freevibrationanalysisofan
undampedsystem
Iitil diti
I
n
iti
a
l

con
diti
ons:
Eachofthetwoequationsofmotion,
involvessecondordertimederivatives;henceweneedtospecifytwo
initial conditions for each mass
initial

conditions

for

each

mass
.
Thesystemcanbemadetovibrateinitsithnormalmode(i=1,2)by
subjectingittothespecificinitialconditions.
However,foranyothergeneralinitialconditions,bothmodeswillbe
excited.Theresultingmotion,whichisgivenbythegeneralsolutionofthe
equations
canbeobtainedbyalinearsuperpositionoftwonormalmodes.
Freevibrationanalysisofan
undampedsystem
Iitil diti
I
n
iti
a
l

con
diti
ons:
Sinceandalreadyinvolvetheunknownconstantsand
constants. arec and c where
)()()(
21
2211
txctxctx





wecanchoosec1=c2=1withnolossofgenerality.Thus,thecomponentsofthe
vectorcanbeexpressedas:
)(tx

wheretheunknowncanbedeterminedfromtheinitial
conditions
Freevibrationanalysisofan
undampedsystem
Freevibrationanalysisofan
undampedsystem
Freevibrationanalysisofan
undampedsystem
Frequencies of a mass
spring system
Frequencies


of

a

mass
‐
spring

system
l
d h l f d
Examp
l
e:Fin
d
t
h
enatura
l

f
requenciesan
d

modeshapesofaspringmasssystem,which
isconstrainedtomoveinthevertical
direction.
Solution:Theequationsofmotionaregiven
by:
by:

Byassumingharmonicsolutionas:
thefrequencyequationcanbeobtainedby:
Frequencies of a mass
spring system
Frequencies

of

a

mass
‐
spring

system
• Thesolutiontotheaboveequationgivesthenaturalfrequencies:
Frequencies of a mass
spring system
Frequencies

of

a

mass
‐
spring

system
• From

theamplituderatiosaregivenby:
Frequencies of a mass
spring system
Frequencies

of

a

mass
‐
spring

system
• From
• Thenaturalmodesaregivenby