GEOMETRICAL EFFECTS
ON
STEERING FORCES
HPV 2000 – Land Speed Record
VEEHHIICCLLEE DYYNNAAM
MIIC
CSS
TEECCHHNNIICCAALL COONNTTRRIIBBUUTTIIOONN
SPRING 2000
JOSHUA L. BENO
The forces and moments acting on the steering system of any vehicle come from
the tire-road interface. So, how does the geometry of the HPV steering system effect
these forces that the rider must respond to? We first need to consider what geometry is
being discussed. Secondly, we need to look at these forces and moments and the factors
that effect them. Third the 1999 CSU HPV “Two Timer” and the new chassis for 2000
will be analyzed and compared.
There are several angles and distances that can effect steering and handling.

Caster Angle

Kingpin Inclination Angle

Camber Angle

Toe-in

Kingpin Offset

Scrub Offset
Other factors include:

Steer Angle

Speed

Turning Radius

Tire Radius

Tire Pressure

Center of Gravity Location
The caster angle is the angle in side elevation between the steering axis, and the
vertical. This can be seen in Figure 1.
Figure 1
The kingpin inclination angle is the angle in front elevation between the steering
axis and the vertical as seen in Figure 2.
Figure 2
The camber angle is the inclination of the wheel plane to the vertical (Figure 3).
Figure 3
The toe angle of a wheel is the angle between a longitudinal axis of the vehicle
and the line of intersection of the wheel plane and the road surface (Figure 4).
Figure 4
Kingpin offset at the ground is the horizontal distance in front elevation between
the point where the steering axis intersects the ground and the center of tire contact.
Please refer to Figure 5.
Figure 5
The scrub or caster offset is the distance in side elevation between the point of
intersection of the steering axis and the ground, and the center of tire contact (Figure 6).
Figure 6
The steer angle is the angle between the projection of a longitudinal axis of the
vehicle and the line of intersection of the wheel plane and the road surface (Figure 7).
Figure 7
The speed is the forward velocity of the vehicle. The turning radius is the radius
of the turn encountered in steering. The tire radius is the distance from the wheel center
to the center of tire contact. This is usually approximated by half the wheel diameter.
Tire pressure is the inflation level of the tires usually in PSI.
The center of gravity location refers to the positioning of the combined center of
gravity of the rider and the vehicle. The vertical height of the center of gravity and its
position along the longitudinal axis (measured from the rear wheel center) of the vehicle
are both important.
The forces and moments to be considered can be seen in Figure 8 below. The
rolling resistance moment and the overturning moment seen below have little effect on
the steering system and will not be considered.
Figure 8
These forces and moments and their effect on the steering system are examined in
tables 1-5. After these tables the steering forces on the 1999 CSU HPV “Two Timer”
and the 2000 CSU HPV chassis will be examined.
Moment from Fz
Normal Force
Fz
Percentage of Weight
MV
Moment from 
-(Fzl+Fzr)d*sin()sin()
Moment from 
(Fzl-Fzr)d*sin()cos()
on Wheel
+
Moments on left
and right in opposite
direction.
Caster Angle
( )
+
Moments on both
wheels act together
to produce a centering
Kingpin Inclination
Angle ()
moment
Both sides of the
vehicle lift
Kingpin Offset at
Ground (d)
+
+
Imbalances result in
steering pull.
Speed
Moment depends on
lateral acceleration
level.
(V)
+
-
Turning Radius
(R)
+
-
Steering Angle
( )
+
Increases Torque.
CGz
+
CGx
Increases portion of
weight on front steering
wheels.
Table 1.
Lateral Force
Fy
Moment from Fy
ML
(Fyl+Fyr)r*tan()
2
Mass*(V) /R
+
Scrub Offset
With positive caster
produces a moment
attempting to steer
the vehicle out of the
turn.
(s)
+
Caster Angle
( )
+
Speed
(V)
Turning Radius
(R)
*
Steering Angle
( )
+
Tire Radius
r
Table 2.
Tractive Force
Fx
Moment from Fx
MT
(Fxl-Fxr)d
For steered wheels
Rolling resistance force
*
Kingpin Inclination
Angle ()
Moments are
opposite in direction.
Imbalances produce
steering moment.
Kingpin Offset at
Ground (d)
+
+
Camber
Increases Rolling
Resistance
Toe-In
Increases Rolling
Resistance
Tire Pressure
Decreases Rolling
Reesistance
+
-
+
CGx
Increases portion of
weight on front steering
wheels.
Table 3.
Aligning Torque
Mz
Moment from Mz
MAT
Caster Angle
( )
(Fy*Lc)/6
(Mzl+Mzr)cos(( + ) )
This is assuming the
force on the tire acts
over a triangular area
-
2
2 1/2
Resists any turning
motion. This has a
centering effect.
Both caster and
kingpin angles
decrease the moment.
Kingpin Inclination
Angle ()
+
Speed
(V)
Increases lateral
force.
Turning Radius
(R)
Decreases lateral
force.
Tire Pressure
Decreases contact
patch length (Lc).
-
-
Table 4.
The Two Timer’s geometry and dimensions are listed as follows:

Caster Angle = 19o

Kingpin Inclination Angle = 22 o

Camber Angle = -9 o

Toe-in ~ 5 o

Kingpin Offset = 2 in.

Scrub Offset = 3.5 in.

Tire Radius = 10 in.

Tire Pressure ~ 85 psi.

Length of Contact Patch = 2 in.

CGz = 17 in.

CGx = 25 in.

Wheel Base = 40 in.

Mass = 200 lb.

Coefficient of Rolling Resistance = 0.013
We will consider a turn at these conditions:

Steer Angle = 10 o

Speed = 20 mph.

Turning Radius = 30 ft.
The calculations of the steering forces and moments can be seen on the following two
pages.
Calculations of Steering Moments for the Two Timer.
Caster Angle
  19 deg
Kingpin Angle
  22 deg
Wheelbase
Camber Angle
c  9 deg
Toe-In
t  5 deg
Mass
Kingpin Offset
d  2 in
Scrub Offset
s  3.5 in
CG Height
Tire Radius
r  10 in
Tire Pressure
p  85 psi
CG Longitudinal
Steer Angle
  10 deg
Speed
V  20 mph
Turn Radius
Rolling Resistance
Cr  0.013
Length of Contact Patch
Lc  2 in
Assume (1/2) of total wieght on left wheel and (1/6) on right wheel
Normal Force
M l 
M
2
W l  M l g
M
M l  45.359 kg
M r 
W l  444.822 N
W r  M r g
M r  15.12 kg
6
W r  148.274 N
Moment from 
M    W l  W r  d  sin    sin  
M   1.96 N m
Moment from 
M    W l  W r  d  sin     sin  
M V  M   M 
M   0.852 N m
M V  1.108 N m
M V  9.809 in  lbf
Lateral Force
2
Fyl  M l
V
R
2
Fyl  396.536 N
M L   Fyl  Fyr  r tan   
Fyr  M r
V
R
M L  46.241 N m
Fyr  132.179 N
M L  409.267 in  lbf
W B  40 in
M  200 lb
CGz  17 in
CGx  25 in
R  30 ft
Tractive Force
Fxl  Cr W l
Fxl  5.783 N
M T   Fxl  Fxr  d
Fxr  Cr W r
M T  0.196 N m
Fxr  1.928 N
M T  1.733 in  lbf
Aligning Moment
M zl  Fyl
Lc
6
M zl  3.357 N m
M AT   M zl  M zr  cos

2


2
M zr  Fyr
Lc
6
M AT  3.913 N m
M zr  1.119 N m
M AT  34.629 in  lbf
Sum of Moments
M   M V  M L  M T  M AT
M   49.241 N m
Two Timer Steering System
(See Figure 9)
Length of steering arm
La  8 in
Force required by each arm of the rider to resist steering moment.
Fr 
M
2 La
Fr  121.164 N
Fr  27.239 lbf
M   435.821 in  lbf
The 2000 chassis’ geometry and dimensions are listed as follows:

Caster Angle = 6o

Kingpin Inclination Angle = 17.5 o

Camber Angle = 0 o

Toe-in ~ 0 o

Kingpin Offset = 0 in.

Scrub Offset = 1.5 in.

Tire Radius = 10 in.

Tire Pressure ~ 100 psi.

Length of Contact Patch = 1.5 in.

CGz = 23 in.

CGx = 36 in.

Wheel Base = 50 in.

Mass = 200 lb.

Coefficient of Rolling Resistance = 0.013
We will consider a turn at these conditions:

Steer Angle = 10 o

Speed = 20 mph.

Turning Radius = 30 ft.
The calculations of the steering forces and moments can be seen on the following two
pages.
Calculations of Steering Moments for the 2000 HPV prototype Chassis.
Caster Angle
  6 deg
Kingpin Angle
  17.5 deg
Wheelbase
W B  50 in
Camber Angle
c  0 deg
Toe-In
t  0 deg
Mass
M  216 lb
Kingpin Offset
d  0 in
Scrub Offset
s  1.5 in
CG Height
Tire Radius
r  10 in
Tire Pressure
p  100 psi
CG Longitudinal CGx  36.1 in
V  20 mph
Turn Radius
  10 deg
Steer Angle
Rolling Resistance Cr  0.0077
Speed
Length of Contact Patch
Lc  1.5 in
Assume (1/2) of total wieght on left wheel and (1/6) on right wheel
Normal Force
M l 
M
2
W l  M l g
M
M l  48.988 kg
M r 
W l  480.408 N
W r  M r g
M r  16.329 kg
6
W r  160.136 N
Moment from 
M    W l  W r  d  sin    sin  
M   0 N m
Moment from 
M    W l  W r  d  sin     sin  
M V  M   M 
M V  0 N m
M   0 N m
M V  0 in  lbf
Lateral Force
2
Fyl  M l
V
R
2
Fyl  428.259 N
M L   Fyl  Fyr  r tan   
Fyr  M r
V
R
M L  15.244 N m
Fyr  142.753 N
M L  134.921 in  lbf
CGz  23.32 in
R  30 ft
Tractive Force
Fxl  Cr W l
Fxl  3.699 N
M T   Fxl  Fxr  d
Fxr  Cr W r
M T  0 N m
Fxr  1.233 N
M T  0 in  lbf
Aligning Moment
M zl  Fyl
Lc
6
M zl  2.719 N m
M AT   M zl  M zr  cos

2

M zr  Fyr

2
Lc
M zr  0.906 N m
6
M AT  3.439 N m
M AT  30.434 in  lbf
Sum of Moments
M   M V  M L  M T  M AT
M   18.683 N m
M   165.354 in  lbf
HPV 2000 Steering System
(See Figure 10)
Length of Handle from top to steer rod
L1  .1905 m
L1  7.5 in
Length from pivot to steer rod
L2  .0254 m
L2  1 in
Length of Handle
Larm  .2159 m
Larm  8.5 in
Length of Steering Arm
L3  .0762 m
L3  3 in
Force required by each arm of the rider to resist steering moment.
 M  L2


 2   L3  Larm
FR  
1
FR  14.422 N
FR  3.242 lbf
Below we see the steering mechanisms for the two vehicles:
Steering Diagram for 1999 Two Timer.
L arm
Figure 9
Steering Diagram for 2000 Chassis
L1
L arm
L3
L2
Pivot
Figure 10
The 2000 Chassis employs a lever to add mechanical advantage to the steering system.
This lowers the arm force required by the rider as seen in the calculations.
So we see that the moment generated by the action of steering is greatly reduced
on the 2000 Chassis. This is due to the zeroing of the Camber angle, Toe-in, and kingpin
offset. It is apparent that these are important values to consider when designing a
vehicle. By accounting for the forces encountered in riding a more responsive and better
performing vehicle can be designed.
But, does the 2000 Chassis actually ride better? Yes it does. The steering is smooth
and responsive. It is easy to control and steer. Let’s summarize the conclusions
concerning steering geometry:

Caster Angle is important for returning the vehicle to a centered position. However, a
large amount should not be necessary. A value between 5 and 10 degrees is good.

Kingpin Inclination Angle produces a centering moment with the normal force. Its
value should be determined in order to make the Kingpin Offset zero. This will vary
from vehicle to vehicle.

Excessive Camber Angle can increase rolling resistance and should be avoided. A
good value to design for is zero degrees without a rider.

Toe-in also increases rolling resistance. A small amount can make the vehicle tend to
stay steered straight, so depending on the application a small amount may be desired,
but probably not more than a degree or two.

Kingpin Offset increases the steering moments and is ideally zero.

Scrub Offset is beneficial and should be determined from the designed caster angle.
It is important to consider these forces and the geometry behind them when
designing a HPV. There is much intuitive or experimental data available concerning
steering geometry and what is desirable for racing performance. However, it is important
to also consider the “engineering” behind the conclusions. Engineering involves
analytical models and calculations; numbers, to support design decisions. Determining
the proper geometry for steering is not something best left to chance or trial and error.
By analyzing the forces encountered in steering and the geometry that effects them a
satisfactory solution can be attained.
Bibliography
Gillespie, Thomas D., Fundamentals of Vehicle Dynamics, SAE, 1992.