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Intelligent Mechanics Lab. 1
Fundamentals of Vibration
(2006, 1)
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Intelligent Mechanics Lab. 2
() : () () (), , , ,
( ): ) (pendulum) , (electron) , , rolling, (wave motion) Oscillation
( ): ( ) , , , : Hz kHz ( ) : 1 Hz 100 Hz Vibration
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Intelligent Mechanics Lab. 3 1
(, system) : , (device) (ISO ) : , , ,
(analysis) : . .
System Input Output
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Intelligent Mechanics Lab. 4 1
(linear system)
() (Linear system) () (Non-linear system) () (Time-invariant system) () (Time-variant system)
(time-variant system) : , k(t)
( ) ( ) ( ) ( )t t t t+ + =xM C Kx x F
( ) ( ) ( ) )( ) (t t t tt+ + =Mx C xKx F
x
F F = k x, k = constant F = k(x) x, k = ax + bx2
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Intelligent Mechanics Lab. 5 1
(vibrating system) : , (mass), (stiffness), (damping)
1(1 degree of freedom system) : (multi-degree of freedom system) : 2 () (discrete system) : () (continuous system) :
() ()
1 ()
x q(x, t)
q1(t) q1(t) q2(t) q3(t)
3 DOF 1 DOF
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Intelligent Mechanics Lab. 6 1
Discrete model (1 DOF)
x F(t)
Inertia force distribution Over-estimated Over-estimated, better
Continuous model Discrete model (3 DOF)
Items Continuous model Discrete model
Equation of motion Partial differential equation Ordinary differential equation
Variables Time, geometry (x, y, z) Time only
Solution accuracy Exact solution Approximate solution
Solution type Analytical solution Numerical solution
Solution procedure Very complex, limited Easy, Computer used
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Intelligent Mechanics Lab. 7 1
(natural frequency) (natural mode)
(free vibrations) (forced vibrations) (self-excited vibrations)
(harmonic forces) (periodic forces) (impulsive forces) (random forces)
(mass) (stiffness) (damping)
(mass) : (inertia force) () (stiffness) : () x () F (k = F/x) (elastic restoring force) (damping) : (dissipation), (damping force): :
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Intelligent Mechanics Lab. 8 1
)( tFq Kq CqM =++
)(tF )(tq
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Intelligent Mechanics Lab. 9 1
HP Rotor LP Rotor
Turbine/Generator Rotor System
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Intelligent Mechanics Lab. 10
(sine wave) :
-180 0 180 360 540A
(f = 1 / T)
T
Angle (deg)
y
t
3
)2sin( += ftAy
(frequency) f :
(amplitude) A : ,
(phase) f :
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Intelligent Mechanics Lab. 11
(Unit) : Hz (Hertz) : 1 (cps ; cycle per second), f cpm (cycle per minute) : 1 , N n N = 60 f (cpm) (angular frequency) : , (omega) = 2f (rad/s) (period) : 1 () (Hz) T = 1 / f (s)
() : f
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Intelligent Mechanics Lab. 12
peak
(displacement), (velocity), (acceleration) , , ,
peak-to-peak rms
peak 0-to-peak(, p) : , peak-to-peak (, , p-p) : rms (root mean square, ) : , ,
, (= 0.707 peak)
(magnitude)
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Intelligent Mechanics Lab. 13
y1(t), y2(t) : deg, rad
(Phase)
y1(t)
y2(t)
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Intelligent Mechanics Lab. 14
: : mm/s, IPS(in/s)
:
: m/s2, g( G)
22 // dtyddtdva ==
dtdyv /=)sin( += tAy
)cos( += tAv
)sin(2 += tAa
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Intelligent Mechanics Lab. 15
Newton 2: Newton : m d 2x/dt2 = F Newton ()
m d 2x/dt2 = F = Fspring + Fdamping + Fexternal = kx c dx/dt + F(t)
maF =Fma =
)(tFkxxcxm =++
Equation of Motion
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Intelligent Mechanics Lab. 16
3
M ( )
kg ( )
kgm2
C (/)
Ns/m (/)
Nms/rad
K (/)
N/m (/)
Nm/rad
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Intelligent Mechanics Lab. 17
1
(F = 0, c =0),
0=+ kxxm
tXtx sin)( =
tXtx sin)( 2=
0sin)( 2 =+ tXkm
X 0 ( )
02 =+ km mk /=
, 2
,
(Natural frequency)
= n
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Intelligent Mechanics Lab. 18 1
m
k
x(t)
m
c k
x(t)
taetx =)(
+= + tjtjt eaeaetx
22 12
11)(
+= + ttt eaeaetx 12
11
22
)(
( ) tetaatx += 21)(
1
( ) ( ) 0mx t kx t+ =
1 2( )j t j tx t a e a e + = +
( ) tx t Ae=
( )( ) ( ) 0mx t kxcx t t+ + =
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Intelligent Mechanics Lab. 19 1
tjtj eaeatx + += 21)(
+= + tjtjt eaeaetx
22 12
11)(
( ) += tAtx sin)(
tAtAtx sincos)( 21 += ( )tAtAetx ddt sincos)( 21 +=
( ) += tAetx dt sin)(
211 aaA += ( ) jaaA 212 =
22
21 AAA += ( )211 /tan AA=
221
1jAAa +=
221
2jAAa =
.
1
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Intelligent Mechanics Lab. 20
Potential energy (PE)
Kinetic energy (KE)
Total energy (TE)
m
k
n = 2 fn = (elasticity / inertia)1/2
x(t) = X cos (n t + )
PE = k x2 = k X 2 cos2 (n t + )
KE = mv2 = m (dx/dt)2
= m n2 X 2 sin2 (n t + )
TE = P.E. + K.E. = k x2 + mv2
= k X 2 cos2 (n t + )
+ m n2 X 2 sin2 (n t + )
= k X 2 (cos2 n t + sin2 n t )
= k X 2 = mV 2
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Intelligent Mechanics Lab. 21
(natural frequency) ? () (m) (k)
() :
:
?
(resonance), (critical speed)
mkn /=
mkfn 2
1=
(rad/s)
(Hz)
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Intelligent Mechanics Lab. 22 1
m
k
x(t)
F(t)
m
c k
x(t)
F(t) tFtF drcos)( 0=
1
0( ) ( ) cos drmx t kx t F t+ = 0( ) ( ) ( ) cos drmx t cx t kx t F t+ + =
Harmonic excitation
Undamped Vibration Model Damped Vibration Model
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Intelligent Mechanics Lab. 23 1
tAx drp cos0=
tf
tx drdrn
p cos)( 22
0
=
mF
f 00 = mk
n =tAtAtx nnh cossin)( 21 +=
tf
tAtAtx drdrn
nn coscossin)( 22
021
++=
m
k
x(t)
F(t)
1
(Undamped Vibration Model)
0( ) ( ) cos drmx t kx t F t+ =
Forced response Free response
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Intelligent Mechanics Lab. 24 1
220
20 )0(drn
fAxx
+== 10 )0( Axv n==
tf
tf
xtv
tx drdrn
ndrn
nn
coscossin)( 220
220
00
+
+=
1
: 01.00 =x01.00 =v
ndr
0f
= 1 rad/s
= 0.1 N/kg
m m/s
= 2 rad/s
, 2 (n, dr) (2 )
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Intelligent Mechanics Lab. 25 1
(Beat)
000 == vx
( ))cos()cos()( 22 0 ttftx ndr
drn
=
+
= tt
ftx drndrn
drn 2sin
2sin
2)( 22
0
2 (n dr)
, ,
: 2 ( fn fdr) 1
2n dr n drf f f
= =
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Intelligent Mechanics Lab. 26 1
(resonance)
ttAtx drp sin)( 0=
ttf
tx drn
p sin
2)( 0=
ttf
tAtAtx drn
nn sin
2cossin)( 021 ++=
ttf
txtv
tx nrn
nnn
sin2
cossin)( 000 ++=
2 (n = dr)
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Intelligent Mechanics Lab. 27 1
mF
f 00 = mk
n =
m
c k
x(t)
F(t)
nmc
2
=
)cos()( 0 = tAtx drp
+= 22
1
2222
0 2tancos)2()(
)(drn
drndr
drndrn
p tf
tx
)cos()sin()( 0 ++= drd
t AtAetx
1
(Damped Vibration Model)
0( ) ( ) cos( ) drmx t kx t Fc t tx + + =
20( ) 2 ( ) ( ) cosn n drx t x t x t f t + + =
Forced response Free response
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Intelligent Mechanics Lab. 28 1
)cos()sin()( 0 ++= drd
t AtAetx
2222
00
)2()( drndrn
fA
+=
221 2tan
drn
drn
=
2220
20
0
0
)2()1(1
rrfA
FkA n
+== 2
1
12tan
rr
=
2220
0)()( drdr cmk
FA +
=2
1tandr
dr
mkc
=
1
Forced response/ Steady state response
Free response/ Transient response
Amplification factor (dynamic/static) Phase angle
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Intelligent Mechanics Lab. 29
:
:
tFkxxcxm sin0=++
)sin(0 = txx
22220
2220
0)/2(])/(1[
/)()( n
kFcmk
Fx +
=+
=
kmc
cc
r 2==
211
)/(1/2tan)(tan
n
n
mkc
=
=
(damping ratio)
,
n n
frf
= =
(frequency ratio)
Static response
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Intelligent Mechanics Lab. 30
() :
:
2220
0
)]/(2[])/(1[1
/nn ffffkF
xM+
==
21
)/(1)/(2tan
n
n
ffff
=
nff /
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Intelligent Mechanics Lab. 31
()
0 1 2 3 4 5
Frequency Ratio
1
3
5
7
9
0
2
4
6
8
10
Mag
nifie
r
0.00.05
0.10
0.15
0.250.3750.501.0
Dynamic Magnification Factor
Damping Ratio
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Intelligent Mechanics Lab. 32
0.0 1.0 2.0 3.0 4.0 5.0Frequency Ratio
0
30
60
90
120
150
180
Phas
e (d
eg)
0.0 0.050.10
0.150.250.375
0.501.0
Response Phase Delay
Damping Ratio
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Intelligent Mechanics Lab. 33
(resonance) ? f fn ( f = fn)
() (quality factor: Q factor)
2/1=Q
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Intelligent Mechanics Lab. 34
, (resonance) F , 1 X
Resonance (f = fn)
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Intelligent Mechanics Lab. 35
: 2 . : (system) M M , fn fn fe fn , Xe . Xe
( )
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Intelligent Mechanics Lab. 36
X B B+B , Xe Xe fn (viscous damper)
( )
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Intelligent Mechanics Lab. 37
, Xe -() , fn 2 ( f1n f2n ) M C B
( )
Secondary system
Main vibration system
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Intelligent Mechanics Lab. 38
Vibration Isolation
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Intelligent Mechanics Lab. 39
F (t) (foundation) ,
( / )
)(tFkxxcxm =++
222
2
)]/(2[])/(1[
)]/(2[1
nn
nT
ffff
ffFFTR
+
+==
)(tFkxxc T=+
XkicmF )( 2 ++=
XkicFT )( +=
:
:
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Intelligent Mechanics Lab. 40
(foundation)
,
( / )
0)()( =++ bb xxkxxcxm
bb kxxckxxcxm +=++
bXkicXkicm )()(2 +=++
222
2
)]/(2[])/(1[
)]/(2[1
nn
n
b ffff
ffXXTR
+
+==
tieXx = tibb eXx=
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Intelligent Mechanics Lab. 41
(Transmissibility)
,
1 .
, 1 .
.
222
2
)]/(2[])/(1[
)]/(2[1
nn
n
ffff
ffTR
+
+=
)/( nff 2
)/( nff 2
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Intelligent Mechanics Lab. 42
0 1 2 3 4 5Frequency Ratio
1
3
5
0
2
4
Tran
smiss
ibilit
y
0.00.05
0.10
0.15
0.25
0.3750.501.0
Vibration Transmissibility
Damping Ratio
2
(Transmissibility)
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Intelligent Mechanics Lab. 43
, 1
(resilient mounting design)
mounting , ()
3, 0.1
(Vibration Isolation)
2
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Intelligent Mechanics Lab. 44
(Dynamic Absorber)
Main mass
Absorber mass
F0 sin t
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Intelligent Mechanics Lab. 45
(Harmonic Analysis)
(periodic signal) :
: Fourier
: , , , ,
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Intelligent Mechanics Lab. 46
Diesel engine combustion chamber Pressure change
Gas turbine Shaft whirling
Ship propulsion shaft Torsional vibration
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Intelligent Mechanics Lab. 47
(Harmonic Analysis)
Joseph Fourier : " ." ,
....3sin 2sin sin ....3cos2coscos)(
321
3210
++++++++=
tbtbtbtatataatx
=T
dttxT
a00
)(1
=T
n dttntxTa
0 cos)(2
=T
n dttntxTb
0 sin)(2
:
(n = 1, 2, 3, )
(n = 1, 2, 3, ) Joseph Fourier (17681830)
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Intelligent Mechanics Lab. 48
n :
(harmonic coefficient)
Fourier series, harmonics
)cos(sincos nnnn tnctnbtna =+
, 22 nnn bac += )/(tan1
nnn ab=
nc n (nth order component)
nn ba ,
nnc ,( )
( )
?
!
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Intelligent Mechanics Lab. 49
(fundamental frequency) :
(high order) : ( 2, 3, 4 )
(order) : (1, 2, 3 )
: " (sine wave)
/ ? " sine (sine cosine ) , "
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Intelligent Mechanics Lab. 50
-4 -3 -2 -1 0 1 2 3 4-1.5
-1
-0.5
0
0.5
1
1.5
Time
Am
plitu
de
Fourier series Diagram
(1)
(rectangular wave)
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Intelligent Mechanics Lab. 51
(rectangular wave)
1st order
3rd order
5th order
7th order
9th order
11th order
13th order
Original
Sum up to each order
15th order
Original
(2)
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Intelligent Mechanics Lab. 52
Output torque of single cylinder engine
1st order
2nd order
3rd order
4th order
5th order
6th order
7th order
Original
Sum up to each order
8th order
Original
(3)
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Intelligent Mechanics Lab. 53
(continuous system)
(discrete system)
()
(analytical) (numerical)
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Intelligent Mechanics Lab. 54
(mode)
(vibration mode) ?
Mode, Normal Mode, Mode Shape, Mode Vector, Eigenvector, Elastic Curve, Relative Amplitude
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Intelligent Mechanics Lab. 55
(1 ) (1 degree of freedom system) 1 . (multi-DOF system) .
.
.
.
.
1st mode
2nd mode
3rd mode
(mode)
1 DOF 2 DOF 3 DOF
1 DOF 2 DOF
3 DOF
q1(t) q2(t) q3(t)
3 DOF
q1(t) q2(t)
2 DOF
q1(t)
1 DOF
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Intelligent Mechanics Lab. 56
(string)
:
:
/Tl
nn =
lxnxYn sin)( =
( l : , T : , : ) 1st Mode
2nd Mode3rd Mode
Node
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Intelligent Mechanics Lab. 57
3
:
x 1 x 2 x 3
m m m k k k
mk /4450.01 =
mk /2470.12 =
mk /8019.13 =
1
2
3
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Intelligent Mechanics Lab. 58
(2, 1) (1, 2) (2, 1)+(1, 2) (2, 1)-(1, 2)
(3, 1) (1, 3) (3, 1)+(1, 3) (3, 1)-(1, 3)
(1, 1) (2, 2) Plate
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Intelligent Mechanics Lab. 59
n (nth order) n (nth mode)
/
: 2 0
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Intelligent Mechanics Lab. 60
(modal analysis)
Excitation
Transducer
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Intelligent Mechanics Lab. 61
K C
M
F x
x F
t
F
x
x
x
t
t
t
?
?
Input (Force)
Output (Response) System
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Intelligent Mechanics Lab. 62
,
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Intelligent Mechanics Lab. 63
(exciting force)
(harmonics)
(natural frequency) (mode)
=K x F
( ) ( ) ( ) ( )t t t t+ + =M x Cx K x F
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Intelligent Mechanics Lab. 64
(Vibration assessment) :
:
(condition monitoring & diagnosis) (prediction & maintenance)
() :
(vibration quality)
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Intelligent Mechanics Lab. 65
Assignment
Self-study & Presentation Presentation material using Power Point File
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