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차량동역학차량동역학차량동역학차량동역학차량동역학차량동역학차량동역학차량동역학 Lecture 5Lecture 5
2008. 4. 112008. 4. 11
Spring ‘2008
Midterm Midterm Midterm Midterm - 추후 수업 시간에 실시- 따라서, 4월 25일은 정상 수업중간중간중간중간 고사고사고사고사 개요개요개요개요- Closed book- 간단한 식의 유도 및 활용, 물리적 의미- 주요한 개념들 (암기보다는 이해)- 간단한 계산 (수학적 처리 능력 불요)
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Chap 5 Chap 5 Chap 5 Chap 5 – RideRideRideRideChap 5 Chap 5 Chap 5 Chap 5 – RideRideRideRide22222222
RideRideRideRide
� NVH (Noise Vibration and Harshness)
� Ride: 0 ~ 25 Hz
� Noise: 25 ~ 20,000 Hz
� Subjective Rating
� Ride: 0 ~ 25 Hz
� Noise: 25 ~ 20,000 Hz
� Ride Dynamic System
� Ride excitation sources
� Vehicle vibration response
� Human perception
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Excitation SourcesExcitation SourcesExcitation SourcesExcitation Sources
� Road Roughness
� Elevation profile
� Broad band random signals
� Power Spectral Density (PSD)
� Road Properties
( )2
2
2
1
)(πν
ν
ν
ν
+
=
o
oz GG
where, Gz(ν) = PSD amplitude (feet2/cycle/foot)
ν = Wavenumber (cycles/ft)
Go = Roughness magnitude parameter
(roughness level)
= 1.25 × 105 for rough roads
= 1.25 × 106 for smooth roads
νo = Cutoff wavenumber
= 0.05 cycle/foot for bituminous roads
= 0.02 cycle/foot for Portland Cement Concrete roads
Road RoughnessRoad RoughnessRoad RoughnessRoad Roughness
� Road Input
� Road profile � (differentiate) � Velocity � (differentiate) � Acceleration
� Large acceleration input @ high frequency
� ‘Ride isolation’
)2sin()2sin( VtAXAZr πνπν == )2sin()2( 2 VtAVZr πνπν−=&&
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Road RoughnessRoad RoughnessRoad RoughnessRoad Roughness
� Vertical Input
� Excite bounce and pitch motions
� Roll Input
� For most vehicles, bounce is more dominant response.
� At low speed, roll input is comparable to vertical one.
Tire/Wheel Assembly Tire/Wheel Assembly Tire/Wheel Assembly Tire/Wheel Assembly ---- ImbalanceImbalanceImbalanceImbalance
� Non-uniformity of Tire/Wheel Assembly
� Mass imbalance
� Dimensional variations
� Stiffness variations
� Imbalance
� Static imbalance
� Dynamic imbalance
� Overturning moment
2ωrmFi =
where, Fi = Imbalance force
ω = Rotational speed (rad/sec)
m
e
M-m
z(t)z(t)
mω
d
ω x(t)y(t)
o o
ω 2me sin tω
ω 2me
θ x θ z
Lower plane
m U
d
ω
M U
ω
M L
m COUPLE
ω
M COUPLE
+ =
m COUPLE
θ U θ L
mL
θCOUPLE
Upper plane
5
Tire/Wheel Assembly Tire/Wheel Assembly Tire/Wheel Assembly Tire/Wheel Assembly – Force VariationsForce VariationsForce VariationsForce Variations
� Radial Force Variation, RFV
� Harmonics
� Tractive Force Variation, TFV
� Lateral Force Variation, LFV
Tire/Wheel Assembly Tire/Wheel Assembly Tire/Wheel Assembly Tire/Wheel Assembly ---- HarmonicsHarmonicsHarmonicsHarmonics
� Eccentricity
� Eccentricity of tire, wheel and hubs
� 10 ~ 15 Hz @ normal highway speedds
� ‘Matching mounting’
� Ovality
� Twice frequency of 1st harmonic
� Higher order variations
� May arise from construction
method
<NOTE> Excitation force is not equivalent to the force variation
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Driveline ExcitationDriveline ExcitationDriveline ExcitationDriveline Excitation
� Driveshaft
� Mass Imbalance
� Asymmetry of the rotating parts
� Off-centered
� Straightness
� Running clearance
� Deflection of the shaft
� Secondary Couples
� Universal joint
θβ
θ
ω
ω22
sinsin1
cos
⋅−=
i
o
where, θ = Angle of the U-joint
β = Angle of rotation of the driving yoke
Driveline ExcitationDriveline ExcitationDriveline ExcitationDriveline Excitation
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Engine/TransmissionEngine/TransmissionEngine/TransmissionEngine/Transmission
� Torque Variation
� Cyclic process
� Flywheel acts as an inertial damper
� Engine Mounting
� 3 translational and 3 rotational directions
� Roll direction is the most important
� Isolating
Vehicle Response PropertiesVehicle Response PropertiesVehicle Response PropertiesVehicle Response Properties
� Rigid Body Motion
� Low frequency
� Sprung and unsprung masses
� Structural Modes of Vibration
� Resonance
� Input-Output Relationship
� Gain: Ratio of output and input amplitudes
� Transmissibility: Nondimensional ratio of response amplitude
to excitation amplitude for a system in steady-state forced
vibration
� Transfer function
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Quarter Car ModelQuarter Car ModelQuarter Car ModelQuarter Car ModelQuarter Car ModelQuarter Car ModelQuarter Car ModelQuarter Car Model11111111
Suspension IsolationSuspension IsolationSuspension IsolationSuspension Isolation
� Ride Isolation
� Suspension: Stiffness, Damping
� Tire: Stiffness, (Damping)
� Quarter Car Model
� Ride rate
� Bounce natural frequency
� Damped natural frequency
� Natural Frequency vs. Static Deflection
Quarter car model
ts
ts
KK
KKPR
+=
M
PRn =ω
21 snd ζωω −=
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SDOF ModelSDOF ModelSDOF ModelSDOF Model
� EOM
� Magnification Factor
222)2()1(
1
rr ζ+−
)()()()( tFtkxtxctxm =++ &&&
)(1
)()(2)( 2 tFm
txtxtx nn =++ ωζω &&&
n
rω
ω=where,
2
1
1
2tan
r
r
−= − ζ
φ
� Vibrations of ¼ Car Model
� Simple example,
M = 240 kg, m = 36 kg
ks= 16 kN/m, C
s = 980 N-s/m
kT
= 160 kN/m
� Road roughness input
� Tire/wheel excitation input
� Direct force excitation input
5 10 15 20 250
0.5
1
1.5
2
2.5
Z M
Fb
Z M
Fw
Z
Zr
zr
M
m
zFb
Fw
Frequency (Hz)
Gain
Suspension IsolationSuspension IsolationSuspension IsolationSuspension Isolation
� Steady-State Vibration
( ) wrTssuTsusu
bususss
FzkzkzCzkkzCzm
FzkzCzkzCzM
+++=+++
++=++
&&&&
&&&& where, z = Sprung mass displacement
zu = Unsprung mass displacement
zr = Road displacement
Fb = Force on the sprung mass
Fw = Force on the unsprung mass
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Suspension StiffnessSuspension StiffnessSuspension StiffnessSuspension Stiffness
� Basics of Suspension Design
� Keeping low stiffness
� To minimize natural frequency because road input increases @ high freq.
� 1 ~ 1.5 Hz range for ride
� 2 ~ 2.5 Ha for handling
Sprung Mass Natural Freq.
2 Hz
1.75 Hz
1.5 Hz
Frequency (Hz)
Gain
5 10 15 20 250
1
2
3
4
5
1.25 Hz
1 Hz
1 10. 100.
0.01
0.1
10
1
Frequency (Hz)
Gain
Sprung MassNatural Freq.
2 Hz
1.75 Hz
1.5 Hz
1.25 Hz
1 Hz
Suspension DampingSuspension DampingSuspension DampingSuspension Damping
Frequency (Hz)
Gain
Suspension Damping
10 %
40 %
100 %
200 %
5 10 15 20 250
0.5
1
1.5
2
2.5
3
Frequency (Hz)
Gain
1 10. 100.
0.001
0.01
0.1
1
10.
Suspension Damping
10 %
40 %
100 %
200 %
� Shock Absorber
� Dissipate the energy put into the system by the bump
� Jounce (compression) and Rebound (extension)
� Nonlinearity
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Active ControlActive ControlActive ControlActive Control
� Passive and Active Systems
� Performance Variables
� Vibration isolation: Unsprung mass acceleration ( )
� Suspension travel: Deflection of suspension (Z1)
� Tire load constancy: Deflection of the tire (Z3)
� Characteristic Parameters
� Mass ratio: χ = m / M
� Stiffness ratio: rk = Kt / Ks
� Damping ratio:
� Natural freq. of unsprung mass:
2Z&&
MK2
Cζ
s
ss =
m
Kω t
u =
Active ControlActive ControlActive ControlActive Control
� Active Suspension
� Force generation depending on accel. and displ.
� Effects of Stiffness and Damping
� Stiffness: Sports car vs. luxury car
� Damping: Suspension travel vs. damping force
� Limitation: Suspension stroke
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5 10 15 20 250
0.5
1
1.5
2
2.5
Z M
Fb
Z M
Fw
Z
Zr
zr
M
m
zFb
Fw
Frequency (Hz)
Gain
5 10 15 20 250
0.5
1
1.5
2
2.5
Unsprung Mass
Heavy
Typical
Light
Frequency (Hz)
Gain
Wheel Hop ResonanceWheel Hop ResonanceWheel Hop ResonanceWheel Hop Resonance
1 10. 100.
0.001
0.01
0.1
1
Unsprung Mass
Heavy
Typical
Light
Frequency (Hz)
Gain 10
� Wheel Hop Frequency
� Unsprung Mass
� Wheels/tire, axle/spindle, brakes and suspension components
asta WgKKf /)(159.0 += where, Wa = Axle weight = 0.1~0.5 GAWR
Suspension NonlinearitySuspension NonlinearitySuspension NonlinearitySuspension Nonlinearity
� Hysteresis
� Higher effective stiffness
� Road types: Damping and stiffness change
� What else?
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Pitch / Bounce MotionsPitch / Bounce MotionsPitch / Bounce MotionsPitch / Bounce MotionsPitch / Bounce MotionsPitch / Bounce MotionsPitch / Bounce MotionsPitch / Bounce Motions22222222
Rigid Body Bounce/Pitch MotionsRigid Body Bounce/Pitch MotionsRigid Body Bounce/Pitch MotionsRigid Body Bounce/Pitch Motions
� Bounce and Pitch Motion
� Combination of vertical and longitudinal vibration
� Wheelbase filtering
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� Ride Rate
� EOM
� Bounce
� Pitch
� Spring Center
( ) ( )
RF
FR
RF
kk
kakbc
cbkcakM
+
−=⇒
−=+=∑ :00
TR
TRR
TF
TFF
kk
kkk
kk
kkk
+
⋅=
+
⋅= ,
( ) ( ) 0=−+++ θRFRF
kbkazkkzM &&
022
=
−+
++ z
J
kbka
J
kbkaRFRF θθ&&
Road
M
kRkF
θz
a b
c
(Spring center)C.G 0
Bounce/Pitch FrequenciesBounce/Pitch FrequenciesBounce/Pitch FrequenciesBounce/Pitch Frequencies
Angular rate about C.Gγ
Total ride rate evaluated at C.Gβ
Total spring rate evaluated at spring centerα
MeaningFormulaParameters
� System Parameters
M
kk RF +
M
kbka RF −
J
kbka RF
22 +
� EOM
gyration) of (Radius where, 2
M
Jr =0
0
2=++
=++
r
z
zz
βθγθ
θβα
&&
&&
Bounce/Pitch FrequenciesBounce/Pitch FrequenciesBounce/Pitch FrequenciesBounce/Pitch Frequencies
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� Natural Frequencies
( ) ( ) 02
222 =−−⋅−⇒
r
βωγωα
( )2
22
2
2,142 r
βγαγαω +
−±
+=∴
22
2
2
2
22
1
2
1 21
, rlrl BPβ
γω
αω
β
β
γω
αω
β
ωωωω
−=
−==
−=
−==
== Θ
Ζ
Θ
Ζ
� Oscillation Center
0=++ θβα zz&&
02
=++r
zβθγθ&&
tz ωcosZ=
tωθ cosΘ=
( ) 02 =+− ΘZ βωα
( ) 02
=−+ 2222ωγβ
ΘZr
αω
β
−=
2Θ
Z
2rβ
γω −=
2222
Θ
Z
Bounce center
lB
Pitch center
lP
Bounce/Pitch FrequenciesBounce/Pitch FrequenciesBounce/Pitch FrequenciesBounce/Pitch Frequencies
� Rule of Thumbs
1) kF < 0.3 ⋅ kR: Flat Ride Tuning
2) fP ≈ fB: fB < 1.2 ⋅ fP
3) fP, fB < 1.3 Hz
4) froll ≈ fP, fB
� Flat Ride Tuning
time
front suspension
rear suspension
Bounce/Pitch FrequenciesBounce/Pitch FrequenciesBounce/Pitch FrequenciesBounce/Pitch Frequencies
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� Special Cases
� Uncoupled case, when β=0
� Dynamic Index = 1, when r 2 / ab = 1
blalB
B
P
P =−
=−=−
=αω
β
αω
β22
,
Car of the past
C.G
Modern car design
C.G
a b baOverhanging
masses
00 =+
−=∴=
−=
RF
FRRF
kk
kakbc
M
kbkaβ � C.G = Spring center
Bounce/Pitch FrequenciesBounce/Pitch FrequenciesBounce/Pitch FrequenciesBounce/Pitch Frequencies
Perception of RidePerception of RidePerception of RidePerception of RidePerception of RidePerception of RidePerception of RidePerception of Ride33333333
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Perception of RidePerception of RidePerception of RidePerception of Ride
� ISO Standard
Coordinate system for mechanical vibration
influencing humans as defined in ISO 2631
Reduced comfort boundaries for translational
vibration as defined in International Standard 2631
Perception of RidePerception of RidePerception of RidePerception of Ride
� British Standard
Median experimental equivalent comfort contours for 12 axes of
vibration of the seated body ( - - - ) compared with asymptotic
contours ( ------- ) as defined in British Standard 6841
A 12-axis basicentric coordinate system
(M. J. Griffin, Handbook of Human Vibration)
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Perception of RidePerception of RidePerception of RidePerception of Ride
� Human Tolerance Limit
Perception of RidePerception of RidePerception of RidePerception of Ride
� Measurement System
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Perception of RidePerception of RidePerception of RidePerception of Ride
� Evaluation of Ride
� RMS
� Crest factor
� Vibration dose value
� Estimated vibration dose value
� SEAT (seat effective amplitude transmissibility)
( )RMSN
x ii
= ∑1 2
( )( )Crest Factor
x i
RMS=
max
( )VT
Nx iVDV
s
i
= ∑ 44
V RMS TeVDV s= ⋅ ⋅14 4.
∫
∫⋅
⋅⋅=
1
1
)()(
)()()(
2
22
f
fff
f
fff
o
o
dffSfG
dffSfHfGSEAT
where,
Gff(f) = power spectrum of floor vibration
H(f) = seat transfer function
S(f) = frequency weighting of human
response to vibration
Perception of RidePerception of RidePerception of RidePerception of Ride
� Examples of Measured Data and Analysis
AccelerationTime
History
FrequencyWeighting
AxisMultiplying
Factor
ComponentRideValue
PointRideValue
axf
ayf
azf
r.m.s
r.m.s
r.m.s
FEET
Wb
Wb
Wb
0.25
0.25
0.40
r.s.s
axs
ays
azs
arx
ary
arz
axb
ayb
azb
r.s.s
r.m.s
r.m.s
r.m.s
r.m.s
r.m.s
r.m.s
r.m.s
r.m.s
r.m.s
SEAT
BACK
Wd
Wd
Wb
We
We
We
Wc
Wd
Wd
1.0
1.0
1.0
0.63
0.40
0.20
0.80
0.40
0.50
r.s.s
r.s.s
r.s.s
OverallRideValue
Original Data Weighted Data
FEET
SEAT
BACK
WeightingMultiplying
FactorRide
Value
Time (sec.) Frequency (Hz) Time (sec.)
0 10.5 10.50.1 1 10 100
10
10
10
10
10
10
10
10
10
10
10
10
0.25
0.25
0.40
1.0
1.0
1.0
0.63
0.40
0.20
0.80
0.50
0.40
0.220
0.197
0.502
0.822
0.584
1.151
0.583
0.442
0.219
1.554
0.646
0.562
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