power steering abe 435 october 21, 2005. ackerman geometry basic layout for passenger cars, trucks,...
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Power Steering
ABE 435October 21, 2005
AckermanAckerman GeometryGeometry
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
δo δi
L
t
R
Figure 1.1. Pivoting Spindle
Turn Center
Center of Gravity
δo
δi
(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 angle.
t
αV
Fy
Figure 1.2. Fy
acts at a distance (t) from the wheel center known as the pneumatic trail
(Milliken, et. al., 2002)
yFC (1)
Slip Angles The slip angle (α) is the angle at which a tire rolls
and is determined by the following equations:
RgC
VW
f
ff **
* 2
RgC
VW
r
rr **
* 2
W = weight on tires
C α= Cornering Stiffness
g = acceleration of gravity
V = vehicle velocity
(2)
(3)
(Gillespie, 1992)
t
αV
Fy
Figure 1.2. Repeated
Steering angle The steering angle (δ) is also known as the Ackerman angle
and is the average of the front wheel angles For low speeds it is:
For high speeds it is:
R
L
rfR
L
(4)
(5)
αf=front slip angle
αr=rear slip angle
(Gillespie, 1992)
t
δi
LR
δo
δi
δo
Figure 1.1. Repeated
Center
of
Gravity
Three Wheel
Easier to determine steer angle Turn center is the intersection of
just two lines
δ
R
Figure 1.3. Three wheel vehicle with turn radius and steering angle shown
Both axles pivot
Only two lines determine steering angle and turning radius
Can have a shorter turning radius
δR
Figure 1.5. Both axles pivot with turn radius and steering angle shown
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.
tFM yz * (6)
t
αV
Fy MzFigure 1.7. Top view of a tire showing the aligning torque.
(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
(Milliken, et. al., 2002)
Figure 1.8. Camber angle
Camber ThrustCamber Thrust Camber thrust (FCamber thrust (FYcYc) is ) is
due to the wheel rolling due to the wheel rolling at the camber angleat the camber angle
The thrust occurs at The thrust occurs at small distance (tsmall distance (tcc) from ) from the wheel centerthe wheel center
A camber torque is A camber torque is then produced (Mthen produced (MZcZc))
Fyc
tcMzc
(Milliken, et. al., 2002)
Figure 1.9. Camber thrust and torque
Camber on Ag TractorPivot 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
(Milliken, et. al., 2002)
Pivot Axis
Figure 1.11. Wheel caster creating stability
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) times the lateral acceleration.
yaKR
L* (7)
(Gillespie, 1992)
t
αV
ay
Figure 1.2. Repeated
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:
r
r
f
f
C
W
C
WK
(8)
Rg
Vay *
2
Since
(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.
K
gLVchar
**3.57 (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.
K
gLVcrit
**3.57
(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
LgKVLg
Vay
3.571
3.572
2
(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:
Loadlb/tire
Cornering Stiffnesslbs/deg
Cornering Coefficientlb/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.
rad.117.075
12/105
R
L
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)
A
B
DE
F
G
CH
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:
The flow from the supply to the valve can be calculated as:
(Merritt, 1967)
QL=flow to the load from the valve A1=larger valve orifice
QS=flow to the valve from the supply A2=smaller valve orifice
Cd=discharge coefficient ρ=fluid density
PS=pressure at the supply PL=pressure at the load
(1)
(2)
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:
QL=flow from the valve to the load
Xv=displacement from null position
(3)
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:
U=Underlap of valve
(10)
(11)
(Merritt, 1967)
If PL and xv are taken to be 0 then, the leakage flow is:
Open Center Flow Gain
In the null position, the flow gain can be determined by (Merritt, pg. 97):
The variables are the same as defined in the previous slide.
(12)
(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:
U = underlap
(Merritt, 1967)
(13)
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, Non-Reversing Non-Reversing-
Cylinder ports are blocked in neutral valve position, the operator must steer the wheel back to straight
Metering Section
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 equipment and utility vehicles
Auxiliary Port
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 Calculations-Hydraguide
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
(Parker, 2000)
Figure 3.10. Coefficient of Friction Chart and Kingpin Diagram (Parker)
Kingpin Torque Information about the tire is needed. If
we assume a uniform tire pressure then the following equation can be used.
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
2** EA
IWT o (1)
(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.
A
Ik o2
22
8E
BW*μTk
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)
(3)
(Parker, 2000)
Calculate Approximate Cylinder Force (Fc)
R
TF KC
Fc= Cylinder Force (lbs)
R = Minimum Radius ArmTK= Kingpin Torque
(4)
(Parker, 2000)
Figure 3.11 Geometry Diagram (Parker)
Calculate Cylinder Area (Ac)
P
FA cc
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
(5)
(Parker, 2000)
Determine Cylinder Stroke (S)
(Parker, 2000)
Figure 3.11 Geometry Diagram (Parker) Repeated
Swept Volume (Vs) of Cylinder
Swept Volume (in3) One Balanced Cylinder
SDDV RBS *)(*4
22
DB=Diameter of boreDR=Diameter of rodS = StrokeVs = Swept volume
(6)
(Parker, 2000)
Swept Volume of Cylinder
Two Unbalanced Cylinders
)*2(4
* 22RBs DD
SV
One Unbalanced Cylinder– Head Side
– Rod Side
-Same as one balanced
SD
V Bs *
4
* 2
(7)
(8)
(Parker, 2000)
Displacement
n
VD s
D= Displacementn= Number of steering wheel turns lock to lockVs = Volume swept
(9)
(Parker, 2000)
Minimum Pump Flow
231
* sNDQ
Ns = steering speed in revolutions per minuteQ = Pump Flow is in gpm per revolutionD = Displacement
(10)
(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