Power Steering
ABE 435
October 21, 2005
Ackerman Geometry
δo




Basic layout for
passenger cars, trucks,
and ag tractors
δo = outer steering
angle and δi = inner
steering angle
R= turn radius
L= wheelbase and
t=distance between
tires
L
δi
Center of
Gravity
Turn
Center
R
δi
t
δo
Figure 1.1.
Pivoting
Spindle
(Gillespie, 1992)
Cornering Stiffness and
Lateral Force of a Single
Tire


Lateral force (Fy) is the force produced by
the tire due to the slip angle.
The cornering stiffness (Cα) is the rate of
change of the lateral force with the slip
α
angle.
C 
V
Fy

Fy
(1)
t
Figure 1.2. Fy
acts at a
distance (t) from
the wheel center
known as the
pneumatic trail
(Milliken, et. al., 2002)
Slip Angles

The slip angle (α) is the angle at which a tire rolls
and is determined by the following equations:
f 
W f *V 2
Cf * g * R
Wr *V 2
r 
Cr * g * R
α
(2)
(3)
V
Fy
t
W = weight on tires
C α= Cornering Stiffness
g = acceleration of gravity
Figure 1.2.
Repeated
V = vehicle velocity
(Gillespie, 1992)
Steering angle


The steering angle (δ) is also known as the
Ackerman angle and is the average of the front
wheel angles
δi
δo
For low speeds it is:
L
 
R

(4)
For high speeds it is:
L
    f r
R
Center
L
of
Gravity
R
δi
(5)
δo
αf=front slip angle
αr=rear slip angle
t
Figure 1.1.
Repeated
(Gillespie, 1992)
Three Wheel
Figure 1.3. Three wheel
vehicle with turn radius
and steering angle
shown
R
δ


Easier to determine steer angle
Turn center is the intersection of just
two lines
Both axles pivot
R
δ


Figure 1.5. Both axles
pivot with turn radius
and steering angle
shown
Only two lines determine steering
angle and turning radius
Can have a shorter turning radius
Articulated


Can have
shorter
turning
radius
Allows front
and back
axle to be
solid
Figure 1.6. Articulated
vehicle with turn radius
and steering angle
shown
Aligning Torque of a
Single Tire

Aligning Torque (Mz) is the resultant
moment about the center of the
wheel due to the lateral force.
M z  Fy * t
Figure 1.7. Top
view of a tire
showing the
aligning torque.
α
(6)
V
Fy
t
Mz
(Milliken, et. al., 2002)
Camber Angle


Camber angle (Φ)
is the angle
between the wheel
center and the
vertical.
It can also be
referred to as
inclination angle
(γ).
Φ
Figure 1.8.
Camber angle
(Milliken, et. al., 2002)
Camber Thrust



Camber thrust (FYc) is
due to the wheel rolling
at the camber angle
The thrust occurs at
small distance (tc) from
the wheel center
A camber torque is then
produced (MZc)
Mzc
tc
Fyc
Figure 1.9. Camber thrust and torque
(Milliken, et. al., 2002)
Camber on Ag Tractor
Pivot Axis
Φ
Figure 1.10.
Camber angle on
an actual tractor
Wheel Caster



The axle is placed
some distance
behind the pivot
axis
Promotes stability
Steering becomes
more difficult
Pivot Axis
Figure 1.11. Wheel
caster creating
stability
(Milliken, et. al., 2002)
Neutral Steer



No change in the steer angle is
necessary as speed changes
The steer angle will then be equal to
the
Ackerman angle.
Front and rear slip angles are equal
(Gillespie, 1992)
Understeer


The steered wheels must be steered to a
greater angle than the rear wheels
The steer angle on a constant radius turn is
increased by the understeer gradient (K)
α
times the lateral acceleration.
V
L
   K * ay
R
(7)
ay
t
Figure 1.2.
Repeated
(Gillespie, 1992)
Understeer Gradient


If we set equation 6 equal to equation 2 we can see that
K*ay is equal to the difference in front and rear slip
angles.
Substituting equations 3 and 4 in for the slip angles yields:
K 
Wf
Cf
Wr

Cr
(8)
Since
2
V
ay 
g*R
(9)
(Gillespie, 1992)
Characteristic Speed


The characteristic speed is a way
to quantify understeer.
Speed at which the steer angle is
twice the Ackerman angle.
Vchar 
57.3 * L * g
K
(10)
(Gillespie, 1992)
Oversteer



The vehicle is such that the steering
wheel must be turned so that the
steering angle decreases as speed is
increased
The steering angle is decreased by the
understeer gradient times the lateral
acceleration, meaning the understeer
gradient is negative
Front steer angle is less than rear steer
angle
(Gillespie, 1992)
Critical Speed

The critical speed is the speed
where an oversteer vehicle is
no longer directionally stable.
Vcrit 
57.3 * L * g
K
(11)
Note: K is negative in oversteer case
(Gillespie, 1992)
Lateral Acceleration Gain


Lateral acceleration gain is the ratio of
lateral acceleration to the steering angle.
Helps to quantify the performance of the
system by telling us how much lateral
acceleration is achieved per degree of
steer angle
V2
ay
57.3Lg

2
KV

1
57.3Lg
(12)
(Gillespie, 1992)
Example Problem

A car has a weight of 1850 lb front axle and 1550 lb
on the rear with a wheelbase of 105 inches. The tires
have the cornering stiffness values given below:
Load
lb/tire
Cornering
Stiffness
lbs/deg
Cornering
Coefficient
lb/lb/deg
225
74
0.284
425
115
0.272
625
156
0.260
925
218
0.242
1125
260
0.230
Determine the steer angle if the
minimum turn radius is 75 ft:

We just use equation 1.
L 105 / 12
 
 0.117 rad.
R
75
Or 6.68 deg
Basic System Components






Steering Valve
Cylinder/Actuator
Filter
Reservoir
Steering Pump
Relief Valve
– Can be built into
pump
Pump


Driven by direct or indirect coupling
with the engine or electric motor
The type depends on pressure and
displacement requirements,
permissible noise levels, and circuit
type
Actuators

There are three types of actuators
– Rack and pinion
– Cylinder
– Vane

The possible travel of the actuator is
limited by the steering geometry
Cylinders




Between the steered wheels
Always double acting
Can be one or two cylinders
Recommended that the stroke to bore
ratio be between 5 and 8 (Whittren)
Hydrostatic Steering Valve


Consists of two sections
– Fluid control
– Fluid metering
Contains the following
– Linear spool (A)
– Drive link (B)
– Rotor and stator set
(C)
– Manifold (D)
– Commutator ring (E)
– Commutator (F)
– Input shaft (G)
– Torsion bar (H)
E
D
A
G
F
C
B
H
Steering Valve
Characteristics








Usually six way
Commonly spool valves
Closed Center, Open Center, or Critical Center
Must provide an appropriate flow gain
Must be sized to achieve suitable pressure losses
at maximum flow
No float or lash
No internal leakage to or from the cylinder
Must not be sticky
Valve Flows

The flow to the load from the valve can be
calculated as:
(1)

The flow from the supply to the valve can
be calculated as:
(2)
QL=flow to the load from the valve
QS=flow to the valve from the supply
Cd=discharge coefficient
PS=pressure at the supply
A1=larger valve orifice
A2=smaller valve orifice
ρ=fluid density
PL=pressure at the load
(Merritt, 1967)
Flow Gain


Flow gain is the ratio of flow
increment to valve travel at a given
pressure drop (Wittren, 1975)
It is determined by the following
equation:
(3)
QL=flow from the valve to the load
Xv=displacement from null position
Flow Gain
Lands ground to
change area
gradient
Open Center Valve Flow

The following equation represents the flow to the
load for an open center valve:
(10)

If PL and xv are taken to be 0 then, the leakage
flow is:
(11)
U=Underlap of valve
(Merritt, 1967)
Open Center Flow Gain

In the null position, the flow gain
can be determined by (Merritt,
pg. 97):
(12)
The variables are the same as defined in the
previous slide.
(Merritt, 1967)
Pressure Sensitivity


Pressure sensitivity is an indication
of the effect of spool movement on
pressure
It is given by the following equation
from Merritt:
(4)
Open Center Pressure
Sensitivity

In the null position, the open center
pressure sensitivity is:
(13)
U = underlap
(Merritt, 1967)
Open Center System

Fixed Displacement Pump
– Continuously supplies flow
to the steering valve
– Gear or Vane


Simple and economical
Works the best on smaller
vehicles
Open Center Circuit, NonMetering
Reversing
Section

Non-ReversingCylinder ports are
blocked in neutral
valve position, the
operator must steer
the wheel back to
straight
Open Center Circuit,
Reversing

Reversing –
Wheels
automatically
return to
straight
Open Center Circuit,
Power Beyond


Any flow not used
by steering goes
to secondary
function
Good for lawn
and garden
Auxiliary
equipment and Port
utility vehicles
Open Center Demand
Circuit



Contains closed center load
sensing valve and open
center auxiliary circuit valve
When vehicle is steered,
steering valve lets pressure
to priority demand valve,
increasing pressure at
priority valve causes flow to
shift
Uses fixed displacement
pump
Closed Center System


Pump-variable delivery, constant pressure
– Commonly an axial piston pump with
variable swash plate
– A compensator controls output flow
maintaining constant pressure at the
steering unit
– Usually high pressure systems
Possible to share the pump with other
hydraulic functions
– Must have a priority valve for the
steering system
Closed Center Circuit,
Non-Reversing



Variable displacement
pump
All valve ports blocked
when vehicle is not
being steered
Amount of flow
dependent on steering
speed and
displacement of
steering valve
Closed Center Circuit with
priority valve

With steering
priority valve
– Variable volume,
pressure
compensating
pump
– Priority valve
ensures adequate
flow to steering
valve
Closed Center Load
Sensing Circuit




A special load sensing
valve is used to operate
the actuator
Load variations in the
steering circuit do not
affect axle response or
steering rate
Only the flow required by
the steering circuit is
sent to it
Priority valve ensures the
steering circuit has
adequate flow and
pressure
Arrangements


Steering valve
and metering
unit as one
linked to
steering wheel
Metering unit at
steering wheel,
steering valve
remote linked
Design CalculationsHydraguide









Calculate Kingpin Torque
Determine Cylinder Force
Calculate Cylinder Area
Determine Cylinder Stroke
Calculate Swept Volume
Calculate Displacement
Calculate Minimum Pump Flow
Decide if pressure is suitable
Select Relief Valve Setting
(Parker, 2000)
Kingpin Torque (Tk)

First determine
the coefficient
of friction (μ)
using the chart.
E (in) is the
Kingpin offset
and B (in) is the
nominal tire
width
Figure 3.10.
Coefficient of
Friction Chart
and Kingpin
Diagram
(Parker)
(Parker, 2000)
Kingpin Torque

Information about the tire is needed. If we
assume a uniform tire pressure then the
following equation can be used.
Io
T W * *
 E2
A
(1)
W=Weight on steered axle (lbs)
Io=Polar moment of inertia of tire print
A=area of tire print
μ.= Friction Coefficient
E= Kingpin Offset
(Parker, 2000)
Kingpin Torque

If the pressure distribution is known then the
radius of gyration (k) can be computed. The
following relationship can be applied.
k

2
Io

A
(2)
If there is no information available about the tire
print, then a circular tire print can be assumed using
the nominal tire width as the diameter
2
B
Tk  W*μ
 E2
8
(3)
(Parker, 2000)
Calculate Approximate
Cylinder Force (Fc)
TK
FC 
R
(4)
Fc= Cylinder Force (lbs)
R = Minimum Radius Arm
TK= Kingpin Torque
Figure 3.11 Geometry
Diagram (Parker)
(Parker, 2000)
Calculate Cylinder Area (Ac)
Ac



Fc

P
(5)
Fc=Cylinder Force (lbs)
P=Pressure rating of steering valve
Select the next larger cylinder size
-For a single cylinder use only the rod area
-For a double cylinder use the rod end area plus
the bore area
(Parker, 2000)
Determine Cylinder Stroke (S)
Figure 3.11 Geometry
Diagram (Parker)
Repeated
(Parker, 2000)
Swept Volume (Vs) of Cylinder

Swept Volume (in3) One Balanced Cylinder
VS 

4
* (D  D ) * S
2
B
2
R
(6)
DB=Diameter of bore
DR=Diameter of rod
S = Stroke
Vs = Swept volume
(Parker, 2000)
Swept Volume of Cylinder

One Unbalanced Cylinder
– Head Side
Vs 
 * DB2
4
*S
(7)
– Rod Side
-Same as one balanced
Two Unbalanced Cylinders
 *S
2
2
Vs 
(2 * DB  DR )
(8)
4

(Parker, 2000)
Displacement
Vs
D
n
(9)
D= Displacement
n= Number of steering wheel turns lock to lock
Vs = Volume swept
(Parker, 2000)
Minimum Pump Flow
D * Ns
Q
231
(10)
Ns = steering speed in revolutions per minute
Q = Pump Flow is in gpm per revolution
D = Displacement
(Parker, 2000)
Steering Speed



The ideal steering speed is 120 rpm,
which is considered the maximum
input achievable by an average person
The minimum normally considered is
usually 60 rpm
90 rpm is common
(Parker, 2000)
Hydraulic Power Assist


Hydraulic power assist means that a
hydraulic system is incorporated with
mechanical steering
This is the type of power steering used
on most on-highway vehicles
Full Time Part Time
Power Steering

Part Time
– The force of the center springs of the valve gives
the driver the “feel” of the road at the steering
wheel.

Full Time
– The valve is installed without centering springs.
Any movement of the steering wheel results in
hydraulic boost being applied.
(Vickers, 1967)
Electrohydraulic Steering

Electrohydraulic steering can refer
to
– A hydraulic power steering system
driven with and electric motor
– A power steering system that uses
wires to sense the steering wheel
input and actuate the steering valve
Electric Motor

An electric motor can be used to
power the steering pump instead of
the engine
– Lowers fuel consumption
– Allows for more flexibility of design
SKF Electro-hydraulic Steering
Considerations for E-H
System Design





Simulation of end stops
Operational environment
Safety
Steering functions
Force feedback