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