steering arc
DESCRIPTION
steering mechTRANSCRIPT
Geometry and Linkage
Lecture 1Day 1-Class 1
References Gillespie, T., The Fundamentals of Vehicle
Dynamics, Society of Automotive Engineers, Warrendale, PA, 1992.
Milliken, W.F. and Milliken, D.L., Chassis Design Principles and Analysis, Society of Automotive Engineers, Warrendale, PA, 2002.
Hunt, D., Farm Power and Machinery Management, Iowa State University Press, Ames, IA, 2001.
Ackerman Geometry
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
δoδi
(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:
RgCVW
f
ff **
* 2
RgCVW
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:
RL
rfRL
(4)
(5)
αf=front slip angle
αr=rear slip angle
(Gillespie, 1992)
t
δi
LR
Center of Gravity
δoδi
δo
Figure 1.1. Repeated
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
Pivoting Single Axle
Entire axle steers Simple to determine steering angle
δ
R
Figure 1.4. Pivoting single axle 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 do 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 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)
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.
yaKRL * (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
CW
CW
K
(8)
RgVay *
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.
KgLVchar
**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.
KgLVcrit
**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.
117.075
12/105
RL
Or 6.68 deg
Find the Understeer gradient The load on each front tire is 925 lbs and the
load on each rear tire is 775 lbs The front cornering stiffness is 218 lb/deg and
the rear cornering stiffness 187 lb/deg (by interpolation)
Using equation 7:
)deg(/099.0deg/187
775deg/218
925
glblb
lblb
CW
CW
Kr
r
f
f
Find the characteristic speed Use equation 8 plugging in the given wheelbase and the understeer
gradient
mphsft
ftinsftinrad
KgLVchar
275/404
deg099.0*/12/2.32*105*deg/3.57
**3.57
2
Determine the lateral acceleration gain if velocity is 55 mph Use equation 10
deg/391.0)/2.32)(/12/105(deg/3.57
)/81(deg/099.01
)/2.32)(/12/105(deg/3.57)/81(
3.571
3.57
2
2
2
2
gsftftininrad
sftgsftftininrad
sft
LgKVLg
Vay