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MECH ENG 3028

DYNAMICS & CONTROL 2Vibrations

Mr. Gart! "ri#$s En$inrin$ So%t! S32'

email: gareth.bridges@adelaide.edu.au

Lectures (for Dynamics & Control II)

Wednesday 11am 1 !m" #arr $mith $outh

%' (Control utorial)

hursday 11 am 1!m" *a!ier 1'(+ibrations Lectures)

,ny changes -ill be announced.

+ibrations utorials,erage 1 hour !er three -ee/s as arranged

*otes: ,ailable from I&CC

,dditional *otes -ill be made aailable

on 0yni.

1

1

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To,i+s

Introduction to ibrations

2undamentals of ibration

2ree ibration of single degreeoffreedomsystems

2orced ibrations

Dam!ed ibrations

+ibration isolation

-o degreeoffreedom systems

0ulti degreeoffreedom systems

+ibration of continuous systems

Determination of natural fre7uencies and mode

sha!es

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Intro#%+tion to Vibration

,ny system haing mass and stiffness is

ca!able of ibratory motion

,nalysis of ibration in design is im!ortant for4structural integrity

e3cessie ibration can lead to

!remature failure through fatigue

structures such as bridges and

buildings

4!erformance

ibration can limit the 7uality of ride in a

car" motorcycle or aircraft

ibrations of a s!ea/er cone determine

7uality of re!roduction of music

+ibration can also sere useful !ur!oses4ibratory coneyors

4ho!!ers

4siees

4sonic !ile driers

E

E

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;3am!le of im!ortance of ibration

48n *oember 4" 1>" at a!!ro3imately11: ,0" the first acoma *arro-s

sus!ension bridge colla!sed due to -ind

induced ibrations. $ituated on the acoma

*arro-s in =uget $ound" near the city of

acoma" Washington" the bridge had only

been o!en to traffic for a fe- months. #ridge failed because of a ibration induced

structural failure

4 the fre7uency of the e3citation forces

resulting from ortices shed from the

structure matched a bridge resonance

fre7uency for a common -ind elocity Wind testing no- standard

4

4

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=icture from F%G.

=icture from F%G.

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=icture from F%G.

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-%n#a(ntas o* Vibration

Consider a sim!le masss!ring system

4mass" m" is assumed to be a rigid body

4s!ring" k" is elastic and assumed to be ofnegligible mass

4 to! of s!ring is attached to a fi3ed obHect

4static e7uilibrium

mgFk =

k

m 0

mg

kF

1

1

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;3amine static deflection

$!ring assumed to be linear

8nly linear dis!lacements and forces -ill be

considered in this course

Fk

linear

nonlinear

k

m

mm

1

2

0kk

= kFk

11

11

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*o- consider dynamic deflections about the

static e7uilibrium !osition"

x(t) is the dynamic dis!lacement

$umming forces

4static com!onents

4dynamic com!onents

his is an e3am!le of an undam!ed single

degree of freedom masss!ring system

))(()( txkmgtxm +=&&

k

m

0

mg

kF

)(tx+

= kmg0

)()( tkxtxm =&&

0)()( =+ tkxtxm&&

1'

1'

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0)()( =+ tkxtxm&&

$ole differential e7uation

-ith initial conditions

$olution 14assume solution of form

-here

00)( xtxt =

=

00)( vtxt =

=&

)sin()( += ttx

= ma3imum am!litude of the function FmG= natural fre7uency FradAsG= !hase FradG

m

k=

1%

1%

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7uadrantIofbecarefulbuttan

sin

cos

&

&1

&

&

'&

''&

=

=

=

+=

v

x

Ax

Av

vxA

Amplitude and Phase

cos cos J

sin J /2 < <

x K

v

< < /2

x K

v K

sin < < 3/2

x v

3/2 < < 2

x v K

1>

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6es!onse of a $D82 undam!ed masss!ring

system

t

)(tx

2=T

0x

)sin()( += ttx

=T !eriod of the function FsG

2

1==

Tf

=f fre7uency of the function in 9ertM F9MG

1

1

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Differentiate dis!lacement

-ith res!ect to time to determine4elocity

4acceleration

)cos()( += ttx&

)sin()( 2 += tAtx&&

)sin()( += tt

t

)(tx

t

)(tx&

t

)(tx&&2

2

Dis!lacement

,cceleration

+elocity

1E

1E

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;3am!le

4, small s!ring % mm long is -elded to theunderside of a table so that it is fi3ed at the

!oint of contact" -ith a 1' mm bolt -elded

to the free end. he bolt has a mass of

grams and the s!ring stiffness is measured

to be *Am. he initial dis!lacement

from the static e7uilibrium !osition is 1mm.

4Calculate the natural fre7uency" the !eriod"

and the ma3 am!litude of the res!onse.

14

14

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1

1

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Ot!r so%tions to #s+rib

si(, !ar(oni+ (otion

6ecall $olution 1

$olution '

$olution %

ransformations bet-een each solution

)sin()( += tt

0)()( =+ tkxtxm &&

)sin()cos()( 21 tBtBtx +=

=+=

2

1122

21 tan

BBBBA

m

k=

0

201

vBxB ==

tjtj

eCeCtx

+= 21)(

1=j

jCCBCCB )( 212211 =+=

22

212

211

jC

jC

+=

=

1

1

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Rotationa S/st(s

Consider the rotational system

4rigid body has !olar mass moment ofinertia"J F/gm'G

4 rod has torsional stiffness" kT" is elastic and

assumed to be of negligible mass

4 to! of rod is attached to a fi3ed obHect

G N shear modulus of elasticity F*Am'G

Jp N !olar moment of inertia of the rod Fm>G

lN length of rod FmG

)(t

Tk

'

l

GJk

pT=

'

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2or measured from the static e7uilibrium

!osition" summing moments about the centreof graity gies

6ecall for rectilinear undam!ed $D82 mass

s!ring system

6otational system

$olution 1

,s on $lide 1>" consideration must be ta/en of

7uadrant.

kT=

)sin()( += tt

=

+=

0

0120

220 tan

&

&A

)(t

Tk

'1

)(t

0)()( =+= tktJM T&&

0)()( =+= tkxtxmF &&

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''

0201

&

== BB

)sin()cos()( 21 tBtBt +=

$olution '

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n#%%(

Consider the !endulum sho-n belo-

4 ta/ing moments about the !iot !oint

4assuming that is small"

40oment of Inertia of mass rotating about a

!oint"

4 rearranging gies

2mlI=

)(sin)(0 tmgltIM == &&)()(sin tt )(t

)()(2

tmgltml =&&

0)()( =+ tl

gt &&

)(t

m

l0

mg

'%

'%

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D+ib 1#"

Consider a harmonic function for

dis!lacement" say

for undam!ed free res!onse

he !ea/ alue is defined as the ma3imum

dis!lacement

4magnitudeA of dis!lacement e7uation

+elocity and acceleration gien by

=ea/ elocity and !ea/ acceleration gien by

=ea/ acceleration a factor of larger than

the !ea/ dis!lacement

4conenient to com!ress scale of data for

com!arison of different alues

( ) += ttx sin)(

( ) += tAtx cos)(&

( ) += tAtx sin)( 2&&

xpeak =&

Axpeak2

=&&

2

'>

'>

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,nother 7uantity -e might consider in

describing ibration is the aerage alue"

4defined as

#ut the aerage alue of

oer one !eriod of oscillation is Mero $ince the s7uare of the dis!lacement is

associated -ith the systems !otential energy"

the aerage of the dis!lacement s7uared is a

useful !ro!erty to consider

he mean s7uare alue of the dis!lacement is

defined by

he s7uare root of this alue" called the root

mean s7uare (rms) alue is commonly used ins!ecifying ibration

4denoted as

2or harmonic motion only

'

x

=T

dttxT

x0

)(1

( ) += ttx sin)(

=T

dttxT

x0

22 )(1

rmsx

22

Axx

peakrms ==

'

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*o-" the decibel is defined as

and alues are sho-n follo-ed by d#" e.g.

d# rex2

a reduction of % d#

'E

=

=

2

110

2

2

110 log20log10

xx

xxdB

'E

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Mo#in$ %sin$ Enr$/

Mt!o#s

2or masss!ring system -e analysed the

motion follo-ing *e-tons second la- a!!lied

in thex direction

2or bodies that are free to rotate about a fi3ed

a3is

2or more com!le3 systems it may be more

difficult to determine forces andAor momentsacting on the com!onents

4 total energy of a system is constant

xmF && =

&&00 JM =

.constT =+=T ?inetic energy

= =otential energy

'4

'4

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;3am!le 1 using ;nergy 0ethods

t! is the time -hen the mass is !assing through

its static e7uilibrium !osition

4! " #

t2 is the time -hen the mass is at its ma3imum

dis!lacement

4T2 " #

4=otential energy change at t2 gien by

strain energy in s!ring

4?inetic energy change at t! gien by

t

)(tx

2x

1x1t 2t

k

m

max2max1 T =

21max1

2

1xmT &=

2

2max2 2

1kx =

'

'

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;3am!le ' using ;nergy 0ethods

Determine natural fre7uency of system sho-nbelo-

4static e7uilibrium

4ma3 dis!lacement

ma3imum /inetic energy

ma3imum !otential energy

!rinci!le of energy conseration

( ) 11 =t

( )21121max1

2

1

2

1 && rmJT +=

( )22222max2

2

1

2

1rkkx ==

( ) 22 =t

k

m

1r 2r 1

2

)(tx

)(t

%1

max2max1 T =

%1

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6ecall ma3imum alues for harmonic function

$ubstituting into energy e7uation

;7uation of motion for undam!ed system

( ) ( )2222

1121

2

1

2

1

2

1 rkrmJ =+ &&

k

m

1r 2r 1

2

)(tx

)(t

max2max1 T =

2

1

22

mrJ

kr

+=

)sin()( += tt A= 2

)cos()( += tAt& A = 1&

22

22

21

21 )( krmrJ =+ &

222

221 ))(( AkrAmrJ =+

( )21

2

22

22

mrJA

krA

+=

e$

e$

m

k=

02

2

2

1 =++ krmrJ &&

0=+ e$e$ km &&

%'

%'

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,lternatiely can use relationshi!

Oields same relations for e7uation of motion

max2max1 T = 22

212

&

=21

22

mrJ

kr

+=

e$

e$

m

k=

02221 =++ krmrJ &&

0=+ e$e$ km &&

%%

%%

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What ha!!ened to !otential energy of mass m

in energy e7uationsP4Lets e3amine energy terms more closely

,t static e7uilibrium

,t time t! (static e7uilibrium)

k

m

1r 2r

)(t

0

)(t +

0

)(tx+

0 )(t%+

1r=

2r=

1

2

r

r=

0210 == rkmgrM

21 rkmgr =

02

1 2 =mgk

01 =+= gravspr&ng

%>

%>

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,t time t2

loss of !otential energy of mass m is cancelled

by the -or/ done by the e7uilibrium force of

the s!ring

k

m

1r 2r

)(t

0

)(t +

0

)(tx+

0 )(t%+

)()(2

1

2

2

2 xmg%k ++=

gravspr&ng +=2

2222

2 22

1

2

1

2

1mgxmg%kk%k ++=

21

2222

2

2

1

2

1mgx

r

rkk%mgk ++=

21

21

22

210 mgx

rmgrk% ++=

22

2

1k%=

%

%

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;3am!le % using ;nergy 0ethods

Cylinder of radius rrolls in channel of radius'-ithout sli!!ing

What is the natural fre7uency of this systemP

se ;nergy 0ethod

4ma3 dis!lacement at t2

4 from aylor $eries

[ ]2max2 cos)()( r'r'mg =

[ ])cos1)(( 2= r'mg

2

22

2)(

smallforr'mg =

.....!6!4!2

1cos642

++=

m

1

2

cos)()( rr( =)(t

r

(

%E

%E

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;3am!le % (continued)

elocity at centre of cylinder is

angular elocity of cylinder is

4static e7uilibrium at t!

22max1

2

1

2

1c%lJmvT +=

max2max1 T =

m

12

)(t

r

v

&)( r'v =

rr'

rvc%l

&

)( ==

2

2

1

221

2 )(

2

1)(2

1

r

r'Jr'm

&

&

+=

2)(

)(

2

1)(

2

1 222

21

221

2 r'mgr

r'Jr'm =

+

&&

( )222

212

)( rJmr'

mg

+==

&

%4

%4

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;3am!le > using ;nergy 0ethods

consider the !endulum sho-n belo-

4ma3 !otential energy is at !oint

4ma3 /inetic energy at static e7uilibrium

2mlI=

)(t

m

l0

12

cosll

2

[ ] [ ]22max2 cos1cos == mglllmg

2

22

2

smallformgl=

21max1

2

1&IT =

1

max2max1 T =

222

212 2

2 ml

mgl==

&

l

g=

%

%

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Lets calculate the effectie mass of the s!ring

in the sim!le masss!ring system" assuming it

is not small enough to neglect

0a3imum !otential energy stored in the s!ring

during a cycle

0a3imum /inetic energy stored in the s!ring

-hen the s!ring is undistorted and !asses

through the static e7uilibrium !osition

;stimate the /inetic energy in the s!ring by

considering4 lN undistorted s!ring length

4s!ring elongates linearly so that

dis!lacement at any !ointx along its

length is

%

2maxmax

2

1k% =

%

l

x=

k

m%

0

lx

dx

%

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he /inetic energy of a small section x long at

x is

-here the mass !er unit length of the s!ring is

he total /inetic energy" T" of the s!ring is the

integral along the length of the s!ring

0ass m is attached to the end of the ibrating

s!ring" and the mass of the s!ring" ms N l

he total /inetic energy of the mass and s!ring

is

2ollo-ing 6ayleigh -e assume sinusoidal

motion so that

>

22

2

2

1

2

1%

l

xxxT &&

==

k

m%

0

lx

dx

621

22

02

2

%ldx%lxT

l

s && ==

max%% =&

22

62

1%

m%mTTT ssm && +=+=

dt

d%%=&

>

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Conseration of energy allo-s us to set the

ma3imum /inetic energy e7ual to the ma3imum

!otential energy

$oling for the angular fre7uency gies

4*ote that the effectie mass of the s!ring is

1A% of its mass

>1

2max

22max

22max

62

1

2

1%

m%mk% s +=

maxmax T =

smm

k

31

2

+=