ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Basic Vehicle Dynamics
Prof. R.G. Longoria
Updated for Spring 2010
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Lecture goals
• Review concepts from dynamics
• 2D (planar) dynamics problems
• Discuss cases studies and illustrate need for
introducing 3D dynamic effects
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Part 1: Review concepts in dynamics
• Basic coordinate systems
• Free body (or force) diagrams
• How to express position vectors in defined coordinate systems, and how to differentiate them to get velocity and acceleration if needed.
• Relative velocity/acceleration, transferring between coordinate systems.
• Mass properties (e.g., moments of inertia, inertia matrix, etc.)
• Particle and rigid body kinematics
• Newton’s laws, Euler’s equations
• Coordinate transformations are essential for some problems (e.g., turning) – reviewed later
• Bond graphs – optional (*-ed slides)
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Vehicle-fixed coordinate system
SAE vehicle axis system
x = forward, on the longitudinal
plane of symmetry
y = lateral out the right side of the
vehicle
z = downward with respect to the
vehicle
p = roll velocity about the x axis
q = pitch velocity about the y axis
r = yaw velocity about the z axis
Consider the standard SAE
coordinate system and
terminology.
Ground vehicle coordinate systems commonly employ a
coordinate system standardized by SAE.
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Earth fixed coordinate system
X = forward travel
Y = travel to the right
Z = vertical travel (+down)
ψ = heading angle (between x
and X in ground plane)
ν = course angle (between
vehicle velocity vector and X
axis)
β = sideslip angle (between x
and vehicle velocity vector)
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Example: 2 axle vehicle
FBD:
x
z
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Relative velocity of particles
It is recommended that you review these basic
kinematic concepts in a reference of your choice.
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Relative acceleration of particlesIt is recommended that you review these basic
kinematic concepts in a reference of your choice.
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Rigid body velocities
( )0
p o
p
V V V
A A R R
= + Ω×
= + Ω× + Ω× Ω×
ɺ
(cf. Karnopp&Margolis, eqs. 1.18)
(a) Transfer the c.g. velocity to body-fixed directions at the four wheels
(b) If each wheel is constrained to have no velocity perpendicular to the
plane of the wheel, state the kinematic constraints for each wheel.
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Translating and rotating ref. framesIt is helpful to have an understanding of the coordinate systems used
for rigid body analysis, and the terminology employed for these
applications. One of the key results is given below.
can be any vector quantity.
XYZ xyz
dV dVV
dt dt
V
= + Ω×
This relationship between
vector quantities in xyz and
XYZ will prove very useful.
Translating reference axes, with body
rotating with angular velocity, ωωωω.
Rotating reference axes rotate with
velocity ΩΩΩΩ, and body rotates with
angular velocity, ωωωω.
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Rigid body in 3-D motion
By using body fixed coordinates, the
rotational inertial properties remain fixed.
The products of inertia* are all zero, and this
makes it convenient for our purposes.
*See dynamics text to review inertial properties.
†
†
x y z
x y z
v v v =
= Ω Ω Ω
v
Ω
Newton:
xyz
m
d
dt
= + ×
p v
pF Ω p
≜
With v relative to rotating frame.
(trans. momentum)
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Basic equations for a rigid body
x x y z z y
y y z x x z
z z x y y x
F p p p
F p p p
F p p p
= + Ω − Ω
= + Ω − Ω
= + Ω − Ω
ɺ
ɺ
ɺ
x x y z z y
y y z x x z
z z x y y x
T h h h
T h h h
T h h h
= + Ω − Ω
= + Ω − Ω
= + Ω − Ω
ɺ
ɺ
ɺ
xyz
d
dt= + ×
pF Ω p
xyz
d
dt= + ×
hT Ω h
The complete equations for a rigid body are nonlinear,
coupled differential equations.
These are sometimes referred to as the Euler equations, often only when you let the
ref. axes coincide with the principal axes of inertia at the mass center or at a point
fixed to the body so the products of inertia go to zero – this leads to a simpler form.
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
System dynamics formulation
• State space formulation for the vehicle dynamic
states
• For modeling, use momentum states: p, h
• Can readily switch to velocity states as needed
x x y z z y
y y z x x z
z z x y y x
p F p p
p F p p
p F p p
= − Ω + Ω
= − Ω + Ω
= − Ω + Ω
ɺ
ɺ
ɺ
x x y z z y
y y z x x z
z z x y y x
h T h h
h T h h
h T h h
= − Ω + Ω
= − Ω + Ω
= − Ω + Ω
ɺ
ɺ
ɺ
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Value of understanding 6 DOF eqs.
• By having the full dynamic equations at your
disposal, you can:
– Examine effects that might be hard to ‘see’
intuitively or reliably
– You can ‘throw out’ terms that do not apply and
keep those that will impact the problem at hand.
x x y z z y
y y z x x z
z z x y y x
p F p p
p F p p
p F p p
= − Ω + Ω
= − Ω + Ω
= − Ω + Ω
ɺ
ɺ
ɺ
x x y z z y
y y z x x z
z z x y y x
h T h h
h T h h
h T h h
= − Ω + Ω
= − Ω + Ω
= − Ω + Ω
ɺ
ɺ
ɺ
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Rigid body - using bond graphs*
Rigid body motion in a body-centered coordinate system:
–vector angular velocity
–coordinate system moves
with body
–3 components of
translational momentum
–3 components of angular
momentum
–6 nonlinearly-coupled
DOF
Karnopp and Rosenberg (1968)
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Planar dynamics of a vehicle
Consider a car with total mass,
m, centered at CG reaches
maximum acceleration, and
assume the mass of the wheels
are small compared with the total
mass of the car. The coefficient
of static friction between the
road and the rear driving wheels
is assumed known as µ.
Find relations for the forces at
the front and rear pairs of wheels
the under condition of maximum
acceleration.
Must assume relation for friction force.
FBD:
x
z
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
planar dynamics (cont.)
0
0
x x y z z y
y y z x x z
z z x y y x
p F p p
p F p p
p F p p
= − Ω + Ω
= = − Ω + Ω
= = − Ω + Ω
ɺ
ɺ
ɺ
0
0
0
x x y z z y
y y z x x z
z z x y y x
h T h h
h T h h
h T h h
= = − Ω + Ω
= = − Ω + Ω
= = − Ω + Ω
ɺ
ɺ
ɺ
( )1 2 10y r r
T W l l W h mglµ+ = = + − −∑0z f r
F W W mg+ = = − −∑
Solve for the forces, apply to x-direction equation.
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Example: Deriving Bicycle ModelSymmetric vehicle, No Roll
•Represent the two wheels on the front and rear
axles of a two-axle vehicle by a single equivalent
wheel.
•The bicycle model has at least three states:
–forward CG translational momentum or velocity
–lateral CG translational momentum or velocity
–yaw angular momentum or velocity about CG
( )
( )rear drivefront drive lateral force effect
1 2 1
cos( ) sin( )
cos( ) sin( )
cos( ) sin( )
x y z xf f xr yf f
y x z yr yf f xf f
z z yf f yr xf f
m V V F F F
m V V F F F
I l F l F l F
δ δ
δ δ
δ δ
− Ω = + −
+ Ω = + +
Ω = − +
ɺ
ɺ
ɺ
Wong, Eqs. 5.25 – 5.27:
You should be able to see how the equations
shown here (from Wong) can be derived from the
basic Euler equations.
Note that the right-hand sides are basically just
the ‘external’ forces and torques (here applied by
the tire-surface interaction).
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Contrast D’Alembert Formulation
It is common in conducting some
basic rigid body analysis, to
employ kinetic diagrams to
visualize the effect of
translational and rotational
forces.
md
dt
− =
=
∑F a 0
ppɺ
The analysis breaks down, effectively, to a d’Alembert
formulation, where an ‘inertial force’ becomes part of a ‘dynamic
equilibrium’ analysis.
Rate of change of
momentum – ‘inertial
force’
1 I:mpɺ
V
1F
2F
nF
*
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Example of D’Alembert Approach
A bicyclist applies the brakes as he descends a
10° incline.
What deceleration a would cause the dangerous
condition of tipping about the front wheel A?
The combined center of mass of the rider and
bicycle is at G. Ans. a = 0.510g
Meriam & Kraige (6/3)
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Case study: 2D Vehicle Rollover
• Rollover can occur on flat and level surfaces (on-road). On-
road rollovers typically arise from loss of directional control,
which may result from driver steering actions.
• Off-road rollover may result from the cross-slope effect adding
to lateral forcing from curb impacts, soft ground/soil, or other
obstructions that “trip” the vehicle.
Marine, et al (1999)
Off-road path
Steering input to re-
enter roadway
Loss of directional control
due to excessive corrective
steering
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Rollover Classification
• Friction rollover - occurs due to high lateral
friction forces in tire-surface interaction without
any tripping
• Spin out rollover - rear outside tire saturates
before front leading to yaw instability
• Plow out rollover - front outside tire saturates
before rear leading to understeer and possibly
inability to steer out of accident
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Quasi-static Rollover
of a Rigid Vehicle
ϕ
cos sin sin cos 02 2
o y zi
t tM ma h mgh mg F tϕ ϕ ϕ ϕ
= − − − + =
∑
Rigid vehicle moving in a steady turn,
and assume there is no roll acceleration.
Take moments about outside wheel,
V
x
Assume that ϕ is small, then you can solve for the
ratio of lateral to gravitational acceleration,
1
2 2
y zia F tt
hg h t mg
ϕϕ
= + − −
Note, you assure tire contact forces are equal, or,1
2
ziF
mg=
by making,1
2
y
y
a g
at
h g
ϕ =
+
(cross-slope design)
‘vehicle-fixed’ axes
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Quasi-static Rollover
of a Rigid Vehicle
ϕ
The lateral acceleration at which rollover begins is
the “rollover threshold”.
The point where the inside contact force goes to
zero specifies,
Cross-slope angle can
counter lateral
acceleration
The Static Stability Factor (SSF) is defined for ϕ = 0, or
0, 02
zi
y
F
a tSSF
g hϕ= =
= ≜
This can also be referred to as “rollover threshold”.
0
1
2 2zi
y
F
a th
g h tϕ
ϕ=
= + −
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Rollover Threshold
Rollover Threshold (Gillespie, 1992)
Note that these values can exceed the
cornering capabilities that arise from friction
limits (about 0.8).
So vehicle could spin out in such a case,
implying rollover would not occur. We know
this is not true.
y
z
F
Fµ =
We have examined ‘rigid body’ rollover.
The effect of roll angle shows that, at least
for a simple steady-state case, there is more
to rollover prediction than this simple
analysis.
1 2tane
h
tφ −=
Roll Angle, φ
unstableLateral
Accel ay
Rollover threshold
0, 0
tan2
zi
y
F
a t
g hϕ
φ= =
= −
Roll lowers SSH
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Summary of dynamics review
• We need models for insight, basic analysis/simulation, and control design.
• Not possible to make comprehensive review. Instead, adopt dynamics concepts for vehicle system modeling on an ‘as needed’ basis, focusing on answering the questions asked.
• Rely on fundamental concepts such as relative velocity/acceleration.
• It can be helpful to understand the basic 3D rigid body equations as a basis for studying simple (e.g., 2D) problems.
• Example given of how vehicle static stability (rollover) can be evaluated with basic planar dynamics.
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
References
1. J.L. Meriam and L.G. Kraige, Engineering Mechanics: Dynamics (4th
ed.), Wiley and Sons, Inc., NY, 1997.
2. D.T. Greenwood, Principles of Dynamics, Prentice-Hall, 1965.
3. T.D. Gillespie, Fundamentals of Vehicle Dynamics, SAE, Warrendale,
PA, 1992.
4. J.Y. Wong, Theory of Ground Vehicles, John Wiley and Sons, Inc., New
York, 1993 (2nd) or 2001 (3rd) edition.
5. Hibbeler, Engineering Mechanics: Dynamics, 9th ed., Prentice-Hall.
6. J.P. Den Hartog, Mechanics, Dover edition.
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Appendix A: Example Problems
1. Anti-rollover control (gyro stabilizer)
2. Bus flywheel
3. Truck with trailer
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
1. Anti-rollover controlwith gyro stabilizer
An experimental car is equipped with a gyro stabilizer to counteract
completely the tendency of the car to tip when rounding a curve (no change
in normal force between tires and road).
The rotor of the gyro has a mass mo and a radius of gyration k, and is
mounted in fixed bearings on a shaft that is parallel to the rear axle of the
car. The center of mass of the car is a distance h above the road, and the car
is rounding an unbanked level turn at a speed v. At what speed p should the
rotor turn and in what direction to counteract completely the tendency of the
car to overturn for either a right or a left turn? The combined mass of car and
rotor is m.
We introduced this example to motivate the need to review 3D rigid body dynamics: useful for
‘back of the envelope’ analysis but also for building an understanding helpful for more complex
problems.
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Dynamics of Spinning Flywheel
With the symmetric flywheel spinning about
the z-axis, if the forces are applied about the
‘torque’ axis, the right-hand rule helps indicate
how the flywheel would precess.
However, we also know that if we spin the
flywheel and precess about the y-axis, a torque
will be applied about the x-axis. The applied
moment would be,
precess velocity
spin velocity
M I p
p
= Ω
Ω =
=Right-hand rule
This concept can be used to solve the gyro-
stabilizer problem.
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Solution from Meriam and KraigeThe sense of the spin actually can be inferred
by always making sure you form the right
hand system with spin-precession-torque.
Right-hand
turn
Left-hand
turnRotor should spin in a direction
opposite to rotation of wheels.
Here is the solution from the
Instructor’s manual.
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Finding Rollover Torque
Identify the relevant body velocity as you make a left turn, as shown
here, or a right turn. For this case the body has an angular rotational
velocity, ˆzkΩ = −Ω
When you apply Newton’s law for the y direction, you account for
the effect of this rotation (through the Euler equations), showing
how this so-called ‘centrifugal force’ arises,
2
00
y y x y z x
vR
mvF p m v m v
R== =
= − Ω + Ω =ɺ
So in a left-hand turn, this induced force generates a moment about the roll axis (x) that
tends to induce rollover. This rollover torque we are trying to control is,2
x y
mvT F h h
R= =
Note the sign change for a right-hand turn.
Note py here is
momentum, not spin
velocity (as in SAE).
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Applying Euler’s Equations
( )rollover
This term must canceltorque
the rollover torque
x x y z z y
x y z z z y y
x y z y z
h T h h
T I I
T I I
= − Ω + Ω
= − Ω Ω + Ω Ω
= + − Ω Ω
ɺ
Assume that Iy−Iz>0.
In a left-hand turn, Ωz<0, the torque to control
is positive, and we require Ωy>0.
In a right-hand turn, Ωz>0, the torque to
control is negative, and we require Ωy>0.
For both, the rotor should spin opposite to the
direction of rolling wheels.
( )2
0x
y z y z x
y y
h
I I T
v hI mv
R R
≈
− − Ω Ω =
Ω =
ɺ
y
y
mvh
IΩ =
Neglecting Iz:
Required spin
velocity:
cf. M&K solution
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Visualizing with a Bond Graph*
1
1 1
G
G
G
Tx
TzTy
ωy ωz
hy
hx
hz
I:Ix
I:IyI:Iz
ωxhx
hy
ωy
hzωy
ωz
hyωz
roll
spin
precession
The sum of torques at this 1-
junction reflects the relevant
dynamics. It is just the relation
we wrote before but now we
might write:
rollover spin precessiontorque torque torque
x x y z z yh T h h= − Ω + Ωɺ
Now visualize with a bond
graph.
Causally, you can see that a
torque from the vehicle
body induces precession of
the rotor, but it is the
angular velocity ωz that
leads to a torque about x.
The same can be said for the
torque induced by the spin
velocity ωy.
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
2. Bus FlywheelAn experimental antipollution bus is
powered by the kinetic energy stored in
a large flywheel that spins at a high
speed p in the direction indicated. As the
bus encounters a short upward ramp, the
front wheels rise, thus causing the
flywheel to precess. What changes occur
to the forces between the tires and the
road during this sudden change?
(Meriam and Kraige, 7/100)
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
A Bond Graph Perspectiveon the Bus Flywheel
• The bus flywheel is a good example of how rigid body bond
graphs can be used to represent or apply the rigid body
equations.
• One advantage is that graphical modeling can be used for
`intuitive' gain, and for some people this is helpful.
• As previously discussed, a spinning flywheel is mounted in a
bus or cart. The body fixed axes are mounted in the vehicle,
with the convention that z is positive into the ground.
• The bus or cart approaches a ramp, and the questions which
arise include whether any significant loads will be applied, what
their sense will be, and on which parameters or variables they
are dependent.
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Flywheel on BusBond Graph Representation
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
3. Truck with Trailer
A loaded pickup truck which weighs 3600 lb with mass center at G1, is hauling an 1800-lb
trailer with mass center at G2. While going down a 10-percent grade, the driver applies his
brakes and slows down from 60 mi/hr to 30 mi/hr in a distance of 360 ft. For this interval,
compute the x- and y-components of the force exerted on the trailer hitch at D by the
truck. Also find the corresponding normal force under each pair of wheels at B and C.
Neglect the rotational effect of the wheels.
Meriam & Kraige (6/26)
To find the unknown forces, need to determine
acceleration and then the inertial forces.
ME 379M/397 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
3. Truck with Trailer (cont)
Meriam & Kraige (6/26)
Compare to
Gillespie
example.