Erasmus LLP Intensive Programme
Suspension System
Dr. Paul J. Aisopoulos
Department of Vehicles
Alexander Technological Educational
Institute of Thessaloniki
Greece
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Erasmus LLP Intensive Programme
Primary function of a suspension
system
 Comfort
Provide vertical compliance so the wheels can follow
the uneven road, isolating the chassis from the
roughness of the road.
 Safety
React to the control forces produced by the tires
longitudinal, lateral forces, braking and driving
torques, in purpose to protect the passengers, the
luggage and the suspension system itself.
 Handling
Keep the tires in contact with the road with minimal
load variations and resist roll of the chassis.
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Erasmus LLP Intensive Programme
Suspension System Parts
Mechanism
Suspension
System
Spring
Damper
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Erasmus LLP Intensive Programme
Suspension System Mechanism
Determines the suspension geometry and specifies the kinematics of the wheel
points in the vertical and lateral movements.
Prescribes the kinematic of the wheel relatively to chassis.
The part on which the wheel and of the braking and steering system are
installed.
Linking mechanism of knuckle with the control arms.
Supplementary units for absorbing the
vibration.
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Erasmus LLP Intensive Programme
Coil Spring
 Spring
rate
Gd4
k
64  i  rm3
G:
shear modulus of spring material.
iεν:
number of active coils of the spring.
d:
rm:
wire diameter.
average radius of the coils of the spring.
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Erasmus LLP Intensive Programme
Equivalent Spring Rate
Parallel springs
keq  k1  k2
Springs in series
keq 
k1  k2
k1  k2
Spring with inclination
keq  k  sin 2 
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Erasmus LLP Intensive Programme
Rubber Spring
 Axial spring
Ed Ad

h
Ed  (4  7,5  c1.79 )  G  E  2G (1   )
ka 
F

Ad
d
d2
  0.5(rubber ), c 
 , Ad 
, A1   dh
A1 4h
4
 Shear
Spring
k
F


G  Ad
h
Εd: Ιdeal elasticity modulus
Αd: Cross section area
h: Spring height
G: Shear modulus of spring material
Powering the Future With Zero Emission and Human Powered Vehicles – Terrassa 2011
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Erasmus LLP Intensive Programme
Damped spring and mass system
Free Vibration
Free body diagramm
m:
Inertial force
FI  m  y
Mass
c:
Damping coefficient
k:
Spring rate
Spring force
Damping force
Fs  k  y
Fd  c  y
Equation of motion
m y  c y  k  y  0
 0   y0
y (t )  e ( y0 cos d t 
sin d t )
d
 t
v0 , y0 Initial conditions
2
Period
Td 
d
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Erasmus LLP Intensive Programme
Damped spring and mass system
Free Vibration
System parameters
m:
oUndamped natural frequency
0  k / m
d  0 1   2
oDamping ratio
 
c
2 k m
oCritical damping coefficient
Y Displacement
oDamped natural frequency
Mass
c:
Damping coefficient
k:
Spring rate
1
0.8
0.6
0.4
0.2
1E-16
-0.2
-0.4
-0.6
-0.8
-1
-1.2
 crit  1  ccrit  2 k  m   
c
0
0.5
1
1.5
ζ=0,1
ζ=0,4
ccrit
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2
ζ=1
2.5
3
t/T0
T0 
2
0
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Erasmus LLP Intensive Programme
Forced Vibration (base excitation)
Equation of motion
m  y  c  y  k  y  cs  ks
s  s0 s in(t ) 
1  (2 ) 2
y
Y 
s0
(1   2 ) 2  (2 ) 2


0
Frequency ratio
Important designe parameters
oResonant frequency
oDamping ratio
 
0  k / m
c
2 k m
m:
Mass
c:
Damping coefficient
k:
Spring rate
s:
Road input
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Erasmus LLP Intensive Programme
Quarter-car modal
ms:
mus:
Unsprung mass
ks:
Suspension stiffness
cs:
Suspension damping coefficient
k t:
Tire stiffness
ys:
Sprung mass displacement
yus:
Sprung Mass
Sprung mass
s:
Unsprung mass displacement
Road input
Is the portion of the vehicle's total mass that is supported above the suspension. (chassis,
engine, differential system, passengers, cargo-luggage) .
Unsprung Mass
Is the mass of the components suspended below the suspension (tire, control arms,
braking system, etc).
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Erasmus LLP Intensive Programme
Quarter-car modal
Equations of motion
ms  ys  cs  ys  cs  yus  ks  ys  ks  yus  0
mus  yus  cs  ys  cs  yus  ks  ys  ( ks  kt )  yus  kt  s
 ms
 0

0   ys   cs


mus   yus   cs
cs   ys   ks


cs   yus   ks
M  y C y  K  y  f ,
for undamped system
 k s   ys   0 


ks  kt   yus   kt  s 
y  ( ys yus )T
det( K  M n2 )  0 
Undamped natural frequencies
k s  kt k s 2
4ks2
1 ks  kt ks
  [

 (
 ) 
]
2 mus
ms
mus
ms
ms mus
2
1,2
Ride Rate
The effective stiffness of the suspension and tire springs in series
keq 
Powering the Future With Zero Emission and Human Powered Vehicles – Terrassa 2011
k s  kt
k s  kt
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Erasmus LLP Intensive Programme
Suspension stiffness
Suspension stifness ks
Ride rate
Road amplitude increase at high frequencies
Bounce natural frequency
Keep ω0 as low as possible
Lower suspension stiffness ks
Higher ks elevate the frequency of the wheel hop to higher frequencies
More acceleration transmission in high frequency rage
•Keep the suspension soft for ride isolation
•Frequencies (1 - 1.3 Hz)
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Erasmus LLP Intensive Programme
Suspension Damping
Low damping ratio ζ
• (-) high vertical acceleration at low frequencies
• (+) High attenuation at high frequencies
For good ride
Damping ratio ζ=0.2 - 0.4
Higher damping ratio values
(-) higher vertical acceleration in high
frequency range
Damping ratio >1
•So stiff damper
•the suspension no longer moves
•Vehicle bounces on its tires.
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Erasmus LLP Intensive Programme
Measurement of Damper Characteristics
vD max 
  n
60
(m / sec)
FD  c  vDn
FD: Damping force
c: Damping coefficient
vD:
:
n:
velocity
stroke
rev speed
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Erasmus LLP Intensive Programme
Quarter-car modal
k1  k2 k1 2 4k12
1 k1  k2 k1
  [

 (
 ) 
]
2 m2
m1
m2
m1
m1m2
2
1,2
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Erasmus LLP Intensive Programme
Rigid Body motions
Bounce/Pitch
Combination of bounce and pitch determine the vertical and longitudinal vehicle vibrations.
Depending on the rode and speed conditions, one or the other motions may be largly absent.
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Erasmus LLP Intensive Programme
Bounce and Pitch vibration modes
Equations of motion
M  z  ( k f  kr )  z  ( kr  a  k f  b)    0


I y    ( kr  a  k f  b)  z  ( kr  a 2  k f  b 2 )    0 
M
0

0   z   k f  kr
kr  a  k f  b   z   0 




I y      kr  a  k f  b kr  a 2  k f  b 2      0 
M  y  C  y  K  y  0, y  ( z  )T
•for undamped system
det( K  M n2 )  0  Undamped natural frequencies
•for the case that α=b=wheel base/2 and kf=kr=k
M  z  ( 2k )  z  0


  b 
I y    ( k  2 / 2)    0 

2k
, p 
M
k 2
2I y
f 
kf
mf
 r 
kr
mr
The bounce and pitch modes are decoupled and pure bounce and pitch motions result.
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Erasmus LLP Intensive Programme
Bounce and Pitch vibration modes
Oscillation center
(Z /  )( )  0
The center is ahead of CG by a distancex=z/θ
(Z /  )( )  0
The center is behind the CG by a distancex=z/θ
The center is outside the wheelbase
bounce
The center is within the wheelbase
pitch
The center location depends on the values of the natural frequencies ωf and ωr
 f  r
 f  r
 f  r
The center is at CG (decoupled motions)
Bounce center ahead of the front axle
Pitch center toward the rear axle(coupled motions)
Bounce center behind the rear axle
Pitch center forward near the front axle(coupled motions)
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Erasmus LLP Intensive Programme
Design criteria
b
a
W f  W , Wr  W
the front suspension should have lower natural frequency
The lower front to rear ratio of frequencies will tend to induce bounce
Time lag between the axles t  / v
v:
Car speed
c:
Wheelbase
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