Steering Class

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)

 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 s lip α angle. V

C  



F y  

Fy (1)

t

Figure 1.2. Fy acts at a distance (t) from the wheel center  known as the pneumatic trail

Slip Angles 

The slip angle (α) is the angle at which a tire rolls and is determined by the following equations:   f 

 r 





W  f  *V 

2

C   f  * g * R

W r  *V 2 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) Center 

For high speeds it is:  



 L  R

   f    r 

L

of  Gravity

R δi

(5)

δo

αf =front slip angle αr =rear slip angle

t

Figure 1.1. Repeated

(Gillespie, 1992)

Three Wheel

R

Figure 1.3. Three wheel vehicle with turn radius and steering angle shown 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



F  y * t  Figure 1.7. Top view of a tire showing the aligning torque.

α

(6)

V Fy

t Mz

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

Camber Thrust Camber thrust (F Yc) 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

Camber on Ag Tractor Pivot Axis Φ

Figure 1.10. Camber angle on an actual tractor 

Wheel Caster Pivot Axis 

 

The axle is placed some distance behind the pivot axis Promotes stability Steering becomes more difficult

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) α times the lateral acceleration.

  

 L  R



K * a y

(7)

ay

t

Figure 1.2. Repeated (Gillespie, 1992)

V

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 



W   f   C    f  



 

W r  C 

(8)

r 

 

Since

a y

V 

2



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. V char  

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.

V crit 



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

V  a y  



2

57.3 Lg 1

KV 

(12)

2

57.3 Lg (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  R



105 / 12

Or 6.68 deg

75



0.117 rad.

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 X =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, NonReversing Metering 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. T   W  *  *

 I o  A



E 2

(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 can be computed. The following relationship can be applied. k 



2



 I o (2)

 A

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

T k 



W* μ

 B

2

8



2

 E 

(3)

(Parker, 2000)

Calculate Approximate Cylinder Force (Fc)

F C  

T K 

(4)

 R

Fc= Cylinder Force (lbs) R = Minimum Radius Arm TK= Kingpin Torque

Figure 3.11 Geometry Diagram (Parker)

(Parker, 2000)

Calculate Cylinder Area (Ac)  Ac   



F c

(5)

P

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 (V s) of Cylinder 

Swept Volume (in3) One Balanced Cylinder

V S 

  

4

2  B 

* ( D

2  R

 D ) * S 

(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

V s



  * DB2 4

* S 

(7)

 – Rod Side -Same as one balanced 

Two Unbalanced Cylinders

V s



  * S  4

2  B 

(2 * D

2  R

D )

(8)

(Parker, 2000)

Displacement

 D



V s

(9)

n

D= Displacement n= Number of steering wheel turns lock to lock Vs = Volume swept (Parker, 2000)

Minimum Pump Flow

Q

 D * N s

(10)

231

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