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
Keep the tires in contact with the road with minimal
load variations and resist roll of the chassis.
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.
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Primary function of a suspension
system
Comfort
Safety
Handling
Provide vertical compliance so the wheels can follow
the uneven road, isolating the chassis from the
roughness of the road.
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Suspension System Parts
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Suspension
System
Mechanism
Spring
Damper
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Suspension System Mechanism
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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.
Prescribes the kinematic of the wheel relatively to chassis.
Determines the suspension geometry and specifies the kinematics of the wheel
points in the vertical and lateral movements.
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Coil Spring
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Coil Spring
4
3
64
m
G d
k
i r
cv

=

Spring rate
G: shear modulus of spring material.
i: number of active coils of the spring.
d: wire diameter.
rm: average radius of the coils of the spring.
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Equivalent Spring Rate
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Parallel springs
Springs in series
Spring with inclination
1 2 eq
k k k = +
1 2
1 2
eq
k k
k
k k

=
+
2
sin
eq
k k u =
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Rubber Spring
1.79
2
1
1
(4 7,5 ) 2 (1 )
0.5( ), , ,
4 4
d d
a
d
d
d
E A F
k
h
E c G E G
A d d
rubber c A A dh
A h
o
v
t
v t
= =
= + > = +
= = = = =
Axial spring
d: deal elasticity modulus
d: Cross section area
h: Spring height
G: Shear modulus of spring material
d
G A F
k
h o

= =
Shear Spring
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Damped spring and mass system
Free Vibration
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m :
Mass
c :
Damping coefficient
k :
Spring rate
Equation of motion
0 m y c y k y + + =
Inertial force
Damping force Spring force
Free body diagramm
s
F k y =
d
F c y =
I
F m y =
0 0
0
( ) ( cos sin )
t
d d
d
y
y t e y t t
o
v o
e e
e

+
= +
Initial conditions
0 0
, v y
2
d
d
T
t
e
=
Period
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Damped spring and mass system
Free Vibration
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m :
Mass
c :
Damping coefficient
k :
Spring rate
System parameters
0
/ k m e =
1 2
crit crit
crit
c
c k m
c
, , = = =
2
c
k m
, =

oCritical damping coefficient

oDamping ratio
oUndamped natural frequency
-1.2
-1
-0.8
-0.6
-0.4
-0.2
1E-16
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
=0,1 =0,4 =1
t/T0
Y
D
i
s
p
l
a
c
e
m
e
n
t
2
0
1
d
e e , =
oDamped natural frequency
0
0
2
T
t
e
=
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Forced Vibration (base excitation)
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m :
Mass
c :
Damping coefficient
k :
Spring rate
s :
Road input
Equation of motion
m y c y k y cs ks + + = +
0
/ k m e =
2
c
k m
, =

oDamping ratio
0
s ( ) s s in t = O
2
2 2 2
0
1 (2 )
(1 ) (2 )
y
Y
s
,q
q ,q
+
= =
+
0
q
e
O
=
oResonant frequency
Important designe parameters
Frequency ratio
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Quarter-car modal
ms:
Sprung mass
mus:
Unsprung mass
ks:
Suspension stiffness
cs:
Suspension damping coefficient
kt:
Tire stiffness
ys:
Sprung mass displacement
yus:
Unsprung mass displacement
s:
Road input
Sprung Mass
Unsprung Mass
Is the portion of the vehicle's total mass that is supported above the suspension. (chassis,
engine, differential system, passengers, cargo-luggage) .
Is the mass of the components suspended below the suspension (tire, control arms,
braking system, etc).
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Quarter-car modal
2
2 2
1,2
4 1
[ ( ) ]
2
s t s s t s s
us s us s s us
k k k k k k k
m m m m m m
e
+ +
= + +
Undamped natural frequencies
Equations of motion
0
s s s s s us s s s us
m y c y c y k y k y + + =
( )
us us s s s us s s s t us t
m y c y c y k y k k y k s + + + =
0 0
0
s s s s s s s s
us us s s us s s t us t
m y c c y k k y
m y c c y k k k y k s

| | | | | | | | | | | | | |
+ + =
| | | | | | |
+
\ . \ . \ . \ . \ . \ . \ .
for undamped system
, ( )
T
s us
M y C y K y f y y y + + = =
2
det( ) 0
n
K Me =
Ride Rate
The effective stiffness of the suspension and tire springs in series
s t
eq
s t
k k
k
k k

=
+
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Suspension stiffness
Suspension stifness ks Ride rate Bounce natural frequency
Road amplitude increase at high frequencies Keep 0 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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Suspension Damping
Low damping ratio
(-) high vertical acceleration at low frequencies
(+) High attenuation at high frequencies
So stiff damper
the suspension no longer moves
Vehicle bounces on its tires.
For good ride
Damping ratio =0.2 - 0.4
Damping ratio >1
Higher damping ratio values
(-) higher vertical acceleration in high
frequency range
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Measurement of Damper Characteristics
n
D D
F c v =
FD: Damping force
c: Damping coefficient
vD:
velocity
:
stroke
n:
rev speed
max
( / sec)
60
D
n
v m
t
=
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Quarter-car modal
2
2 2 1 2 1 1 2 1 1
1,2
2 1 2 1 1 2
4 1
[ ( ) ]
2
k k k k k k k
m m m m mm
e
+ +
= + +
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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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Bounce and Pitch vibration modes
Undamped natural frequencies
Equations of motion
2 2
( ) ( ) 0
( ) ( ) 0
f r r f
y r f r f
M z k k z k a k b
I k a k b z k a k b
u
u u
+ + + =

`
+ + + =

)
2 2
0
0
0
0
f r r f
y r f r f
M k k k a k b
z z
I k a k b k a k b
u u
+
| | | |
| | | | | |
+ =
| |
| | |
+
\ . \ . \ .
\ . \ .
for undamped system
0, ( )
T
M y C y K y y z u + + = =
2
det( ) 0
n
K Me =
for the case that =b=wheel base/2 and kf=kr=k
2
2
( 2 ) 0
2
,
( / 2) 0 2
b p
y y
M z k z
k k
I k M I
e e
u u
+ =


= =
`
+ =

)
f
r
f r
f r
k
k
m m
e e = = =
The bounce and pitch modes are decoupled and pure bounce and pitch motions result.
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Bounce and Pitch vibration modes
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Oscillation center
( / )( ) 0 Z u e >
The center is ahead of CG by a distancex=z/
( / )( ) 0 Z u e <
The center is behind the CG by a distancex=z/
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 f and r
f r
e e =
The center is at CG (decoupled motions)
f r
e e >
Bounce center ahead of the front axle
Pitch center toward the rear axle(coupled motions)
f r
e e <
Bounce center behind the rear axle
Pitch center forward near the front axle(coupled motions)
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Design criteria
Time lag between the axles
the front suspension should have lower natural frequency
The lower front to rear ratio of frequencies will tend to induce bounce
/ t v =
v:
Car speed
c :
Wheelbase
,
f r
b a
W W W W = =