road roughness and suspensions

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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

3

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−=&&

4

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

&&

&&


15

� 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


� 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


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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


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


� 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)

18


� Human Tolerance Limit


� Measurement System

19


� 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


� 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

road roughness and suspensions

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