Vibes 1(2)
Transcript of Vibes 1(2)
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MECH ENG 3028
DYNAMICS & CONTROL 2Vibrations
Mr. Gart! "ri#$s En$inrin$ So%t! S32'
email: [email protected]
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.
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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
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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
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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
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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.
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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
+=