Skidding Vehicle Simulation
The figure below is a schematic representation of a vehicle
that consists of a rigid chassis A
and two identical wheels, B and C.
It is assumed that B and C roll without slip on a horizontal plane
N, and are completely free to rotate relative to A on an axle
whose midpoint is D.
The other two wheels of this vehicle are ``locked up'' and can be considered
to be rigidly attached to A and sliding on N without friction.
To assist in the analysis,
unit vectors a1, a2, and
a3
are fixed in A with
a3 perpendicular to N,
a2 parallel to the axle of A, and
a1 =
a2 x a3.
The following identifiers are useful in describing this system.
| Description |
Symbol |
Value |
| Radius of wheels B and C |
R |
0.35 m |
| Distance from center of each wheel to point D |
b |
0.75 m |
| Distance from D to the center of mass of A |
a |
1.64 m |
| Mass of A |
mA |
640 kg |
| Mass of B and C |
m |
30 kg |
| Central moment of inertia of A parallel to a3 |
IA |
166.6 kg*m2 |
| Axial moment of inertia of B and C |
J |
2.0 kg*m2 |
| Transverse moment of inertia of B and C |
K |
1.0 kg*m2 |
| a3 measure number
of the angular velocity of A in N |
w |
0.01 rad/sec (initial value) |
| a1 measure number
of the velocity of D in N |
v |
25 m/sec (initial value) |
| Time |
t |
seconds |
The Autolev file
vehicleSkid.al
is a complete listing of the Autolev commands to:
- generate the vehicle's exact nonlinear symbolic equations of motion
- produce C, Matlab, or Fortran code
for exceptionally fast numerical motion simulation
- produce a nicely-formatted C/Fortran input file
vehicleSkid.in
for multiple simulations
- generate symbolic linearized equations of motion for stability analysis
- record Autolev's input and output responses in the file
vehicleSkid.all
The sequence of photos shown below match one second of the vehicles motion
on the onset of skidding.
These photos result from numerically integrating the
nonlinear equations of motion with the given values of m, R, etc.
These results are useful in predicting the behavior of a car
with its back wheels rolling and front wheels sliding
(when v(0)>0 Autolev's linearization and stability analysis predicts that perturbations decay exponentially)
or with its front wheels rolling and back wheels sliding
(when v(0)<0 Autolev's stability analysis predicts that perturbations grow exponentially).
This and similar analyses give insights into a variety of other vehicle phenomena.
For example,
- There are two brake-lines on cars because if one punctures,
the second one can still stop the vehicle.
Normally, one brake-line brakes the front right and rear left wheels and
the other brakes the front left and rear right wheels.
It is easier to build a car that uses one brake-line for the two front wheels
and the other for the two rear wheels.
However, a punctured front brake-line results in unstable braking
(see the previous figure which shows the car moving into another lane of traffic
and spinning out of control).
This information is useful for creating dramatic car-chase scenes in movies.
- This phenomenon is one reason that modern aircraft
have a single nose wheel and two tail wheels rather
than two nose wheels and a single tail wheel.
The DC-3 used the unstable configuration and frequently did ground-loops.
- The analysis gives insight into why it is easier to push (rather than pull)
a tricycle, shopping cart, or golf cart.