07-10-2014, 10:49 AM
Designing Stable Three Wheeled Vehicles,
With Application to Solar Powered Racing Cars
[attachment=68853]
INTRODUCTION
Many of the vehicles that participate in solar car racing have three wheels,
arranged with two in front and one at the rear. There were some incidents in Sunrayce
’95 involving such vehicles, but also many of the top finishing cars were three wheelers.
This suggests that there are probably some “do’s and don’ts” regarding the design of
these cars. This paper will discuss the dynamics of three wheeled vehicles, and show
how improved stability can be designed in.
A crucial vehicle property is the location of the vehicle center of gravity (CG). If
it is located properly, the vehicle will be “stable” in terms of:
a. Resistance to “losing the rear end” in turns and crosswinds.
b. Ability to travel at high speed without continual steering corrections to counteract
weaving.
c. Resistance to tipping over in turns and in encountering changes in road surfaces if
sliding.
d. Resistance to swapping ends in hard braking due to weight transfer from the rear
to front.
If the CG is in the “wrong place”, the vehicle may exhibit all these unstable behaviors.
During Sunrayce ’95, I asked many advisors and students if they knew where the
CG was on their cars. Very few knew. This is most disturbing. The location of the CG
should be a design specification. Components should be arranged to achieve a specified
location of the CG. It appears that few teams approach it this way. Rather, I suspect that
components are arranged through an ad hoc process of fitting things where it is
convenient or accessible or to solve some interference problem arising from previous ad
hoc choices. Now that Sunraycers are capable of traveling at posted speed limits on
sunny days, it is crucial that faculty and student designers understand how the location of
the CG can influence vehicle stability.
This paper illustrates how stability can be designed into three wheeled vehicles
through thoughtful choice of the CG location, both longitudinally and in height, the front
track and the wheelbase. The approach employs the simplest models of vehicle dynamics
and utilizes undergraduate level physics and mathematics. The simple models are not the
complete story of vehicle response, but do capture the main factors of vehicle behavior.
Specifically, there are no suspension systems in the models. Suspension will generally
soften and delay the responses, but the tendency toward stable or unstable behavior will
still be present.
The outcome will be the ability to state desirable vehicle responses in terms of
yaw, tipping and braking weight transfer. There will follow a few inequalities that
CENTER OF GRAVITY LOCATION AND YAW RESPONSE
The yaw response of the vehicle refers its tendency to rotate about a vertical axis
through the CG, or “spin”. A stable vehicle can undergo side loads as in cornering or
wind gusts, and not suddenly yaw in such a way as to amplify the tendency to spin. It is
possible to yaw slightly in a self-corrective manner. The type of response depends
largely upon the location of the CG. The following sections will describe factors that
connect CG location with yaw response. It first requires a brief description of how a
pneumatic tire responds to side loads.
WEIGHT TRANSFER IN BRAKING
For a three wheeled vehicle that has 2/3 or more of the weight on the front two
wheels and a high CG, one could envision that the forward weight transfer under hard
braking may severely reduce the vertical load at the rear. This could be studied as a
potential yaw situation in vehicle dynamics, but in keeping with the use of simple
models, a quasi-static approach will be used to determine the fraction of the static weight
at the rear that is transferred to the front in braking. Figure 8 shows a side view of a
vehicle under braking. It could have 3 or 4 wheels. The braking deceleration is denoted
by FB as a fraction of the acceleration of gravity. The static vertical load at the front is WF
and on the rear is WR. Weight transferred to the front during braking is denoted as ∆WR,
which is a positive value. The weight transfer from the rear to the front, expressed as a
fraction of the initial weight on the rear is
CONCLUSIONS
This paper presented a tutorial treatment of elementary vehicle dynamics models
in order to show how certain vehicle parameters affect vehicle stability. The concepts of
slip angle, cornering stiffness, yaw response, neutral steer point, Static Margin (SM),
Understeer Gradient (K) and tipping threshold were described. The “bicycle model” was
introduced and its yaw response was described when subjected to both a side load while
running in a straight line, and to a centrifugal force while cornering. Vehicles with
are considered to have a stable yaw response. The expressions for K
and SM involve the fore-aft location of the vehicle CG, expressed as distance LG or
equivalently and , and the tire cornering stiffness values. These stiffness values
may not be available data, but if the same type of tire is used at each location, then the
threshold conditions of K=0, SM=0 can be achieved by choosing the fore-aft location of
the CG as follows:
With Application to Solar Powered Racing Cars
[attachment=68853]
INTRODUCTION
Many of the vehicles that participate in solar car racing have three wheels,
arranged with two in front and one at the rear. There were some incidents in Sunrayce
’95 involving such vehicles, but also many of the top finishing cars were three wheelers.
This suggests that there are probably some “do’s and don’ts” regarding the design of
these cars. This paper will discuss the dynamics of three wheeled vehicles, and show
how improved stability can be designed in.
A crucial vehicle property is the location of the vehicle center of gravity (CG). If
it is located properly, the vehicle will be “stable” in terms of:
a. Resistance to “losing the rear end” in turns and crosswinds.
b. Ability to travel at high speed without continual steering corrections to counteract
weaving.
c. Resistance to tipping over in turns and in encountering changes in road surfaces if
sliding.
d. Resistance to swapping ends in hard braking due to weight transfer from the rear
to front.
If the CG is in the “wrong place”, the vehicle may exhibit all these unstable behaviors.
During Sunrayce ’95, I asked many advisors and students if they knew where the
CG was on their cars. Very few knew. This is most disturbing. The location of the CG
should be a design specification. Components should be arranged to achieve a specified
location of the CG. It appears that few teams approach it this way. Rather, I suspect that
components are arranged through an ad hoc process of fitting things where it is
convenient or accessible or to solve some interference problem arising from previous ad
hoc choices. Now that Sunraycers are capable of traveling at posted speed limits on
sunny days, it is crucial that faculty and student designers understand how the location of
the CG can influence vehicle stability.
This paper illustrates how stability can be designed into three wheeled vehicles
through thoughtful choice of the CG location, both longitudinally and in height, the front
track and the wheelbase. The approach employs the simplest models of vehicle dynamics
and utilizes undergraduate level physics and mathematics. The simple models are not the
complete story of vehicle response, but do capture the main factors of vehicle behavior.
Specifically, there are no suspension systems in the models. Suspension will generally
soften and delay the responses, but the tendency toward stable or unstable behavior will
still be present.
The outcome will be the ability to state desirable vehicle responses in terms of
yaw, tipping and braking weight transfer. There will follow a few inequalities that
CENTER OF GRAVITY LOCATION AND YAW RESPONSE
The yaw response of the vehicle refers its tendency to rotate about a vertical axis
through the CG, or “spin”. A stable vehicle can undergo side loads as in cornering or
wind gusts, and not suddenly yaw in such a way as to amplify the tendency to spin. It is
possible to yaw slightly in a self-corrective manner. The type of response depends
largely upon the location of the CG. The following sections will describe factors that
connect CG location with yaw response. It first requires a brief description of how a
pneumatic tire responds to side loads.
WEIGHT TRANSFER IN BRAKING
For a three wheeled vehicle that has 2/3 or more of the weight on the front two
wheels and a high CG, one could envision that the forward weight transfer under hard
braking may severely reduce the vertical load at the rear. This could be studied as a
potential yaw situation in vehicle dynamics, but in keeping with the use of simple
models, a quasi-static approach will be used to determine the fraction of the static weight
at the rear that is transferred to the front in braking. Figure 8 shows a side view of a
vehicle under braking. It could have 3 or 4 wheels. The braking deceleration is denoted
by FB as a fraction of the acceleration of gravity. The static vertical load at the front is WF
and on the rear is WR. Weight transferred to the front during braking is denoted as ∆WR,
which is a positive value. The weight transfer from the rear to the front, expressed as a
fraction of the initial weight on the rear is
CONCLUSIONS
This paper presented a tutorial treatment of elementary vehicle dynamics models
in order to show how certain vehicle parameters affect vehicle stability. The concepts of
slip angle, cornering stiffness, yaw response, neutral steer point, Static Margin (SM),
Understeer Gradient (K) and tipping threshold were described. The “bicycle model” was
introduced and its yaw response was described when subjected to both a side load while
running in a straight line, and to a centrifugal force while cornering. Vehicles with
are considered to have a stable yaw response. The expressions for K
and SM involve the fore-aft location of the vehicle CG, expressed as distance LG or
equivalently and , and the tire cornering stiffness values. These stiffness values
may not be available data, but if the same type of tire is used at each location, then the
threshold conditions of K=0, SM=0 can be achieved by choosing the fore-aft location of
the CG as follows: