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ME542 Vehicle Dynamics-Lecture 1- 1
ME542 Vehicle Dynamics
Winter 2014
Tu & Th 1:30‐3:00pm
Huei Peng 3012 PMLDepartment of Mechanical Engineering
[email protected], 1-734-769-6553
Office hours: Mon. 4-5pm (Peng), Wed. 1-2pm (Peng)
GSI office hours to be announced
ME542 Vehicle Dynamics-Lecture 1- 2
Vehicle Dynamics and Safety
• Accident occurred in May 2006 on I-96, caught by police dash-cam.
• Induced by an initial light side impact, SUV rolled over multiple times.
ME542 Vehicle Dynamics-Lecture 1- 3
Lecture 1‐‐Introduction and Motivation(and administrative stuff)
• About this course
– Introduction and administrative information
– Major course content
– Grading policy
– MATLAB/SIMULINK
• Review of rigid body dynamics
ME542 Vehicle Dynamics-Lecture 1- 4
Administrative Information• Textbook (not required)
– J.Y. Wong, Theory of Ground Vehicles, John Wiley &Sons, Inc, 4th
edition, 2008. (http://www.wiley.com/WileyCDA/WileyTitle/productCd‐
0470170387.html)
• Course material will be made available on the Ctools site (PPT files should be downloaded and printed before the lectures, HW, solutions, example MATLAB programs will be distributed using this site)
ME542 Vehicle Dynamics-Lecture 1- 5
Course Requirements
• Prerequisites
– Knowledge in Newtonian Dynamics (ME240 level) is essential
– That of Automotive Engineering (ME458) and Intermediate Dynamics (ME440) are helpful but not required.
– Familiarity with Matlab/Simulink, since Matlab/Simulink is used extensively in the lecture examples and homework assignments.
ME542 Vehicle Dynamics-Lecture 1- 6
Major course content
Background
Tire
Handling
Ride
Lec. Date Lecture contents 1 1/9 Introduction, motivation 2 1/14 Review of Rigid Body Dynamics 3 1/16 MATLAB‐SIMULINK review 4 1/21 Tire Models: Overview, Terminology, Definitions 5 1/23 Brush Tire Model (Lateral) 6 1/28 Brush Tire Model (Lateral) 7 1/30 Brush Tire Model (Longitudinal) 8 2/4 Combined‐Slip Tire Model 9 2/6 (Lateral) Taut String Tire Model, Magic Formula Tire Model 10 2/11 Off‐road tire model 11 2/13 Steady‐State Handling 12 2/18 Steady‐State Handling: Understeering and Oversteering13 2/20 Transient Handling 14 2/25 Lateral‐Yaw‐Roll model 15 2/27 Lateral‐Yaw‐Roll model16 3/11 Four‐Wheel Steering 3/13 Midterm (in class) 17 3/18 Guest lecture: chassis control systems 18 3/20 On‐Center Handling, Steady‐State And Transient Handling of Articulated
Vehicles 19 3/25 Driver‐vehicle interaction 20 3/30 Cross‐wind stability 21 4/1 Ride dynamics—Principle and Vibration Isolation and human perception 22 4/3 Ride dynamics—road excitations 23 4/8 Ride dynamics—Quarter‐car suspension Model 24 4/10 Ride dynamics—Quarter‐car suspension Model 25 4/15 Ride dynamics—Bounce and Pitch Model 26 4/17 Ride dynamics—Active Suspensions 27 4/22 Summary 4/25 Final exam (4‐6pm)
ME542 Vehicle Dynamics-Lecture 1- 7
Related Courses
• ME458 Automotive EngineeringEmphasizes the vehicle as an engineering system and
review design consideration associated with all major systems including the vehicle structure, powertrain, suspension, steering and braking.
• ME568 Vehicle Control Systems
Covers control issues for all major vehicle control systems including engine control, cruise control, ABS/traction control, four‐wheel steering, active suspension and advanced control systems for Intelligent Transportation Systems.
ME542 Vehicle Dynamics-Lecture 1- 8
Grading Policy
• Grading: 5‐6 Homework 55%
Midterm exam 20%
Final exam 25%
• Homework: Must be handed in on the due date in class (on‐campus
students) or uploaded to Ctools (distance learning students). Latehomework will be accepted up to 48 hours late with a 20% penalty foreach 24 hours (rounded up, i.e., 0 to 24 hours late = 24 hours late). Allproblem sets (home work assignments) are to be completed on yourown. You may discuss homework assignments with your fellowstudents at the conceptual level, but must complete all calculations andwrite‐up, from scrap to final form, on your own. Verbatim copying ofanother student's work is forbidden. If you have any questions aboutthis policy, please do not hesitate to contact the instructor.
• Exams: Midterm: in class, Dynamics, Tire, HandlingFinal exam: 2 hours, Accumulative but more weight on Handling and Ride
ME542 Vehicle Dynamics-Lecture 1- 9
MATLAB/SIMULINK Tutorial
http://www.engin.umich.edu/class/ctms/
ME542 Vehicle Dynamics-Lecture 1- 10
Getting a Copy of MATLAB/SIMULINK
http://www.mathworks.com/products/studentversion/
CAEN labs
Student version:
Virtual Site
http://virtualsites.umich.edu/
ME542 Vehicle Dynamics-Lecture 1- 11
MATLAB/SIMULINK
• Example MATLAB/SIMULINK programs will be distributed, which you can freely use/modify.
• MATLAB is used for simple (usually linear) vehicle dynamic simulations and analysis for this course.
• SIMULINK: A GUI based simulation program. Strength:
– Realistic dynamic phenomenon such as nonlinearity, quantization, noise, switches, look‐up tables, delays, etc. can be simulated easily.
– GUI makes it easy to recycle vehicle simulation modules
ME542 Vehicle Dynamics-Lecture 1- 12
Review of Rigid Body Dynamics
• Vehicle Coordinate Systems
• Newton/Euler Formulation
• Lagrange Formulation
ME542 Vehicle Dynamics-Lecture 1- 13
Vehicle Coordinate System
• First step in deriving vehicle dynamic equations.
• The Society of Automotive Engineers (SAE) has introduced standard coordinates and notations for describing vehicle dynamics
x
y
z
longitudinal roll
lateral pitch
vertical yaw
ISO coordinate: x is the same but y and z are reversed.
ME542 Vehicle Dynamics-Lecture 1- 14
SAE Vehicle‐Fixed Coordinate System‐‐Symbols and Definitions
Pitch angle: the angle between x‐axis and the horizontal plane. Roll angle: the angle between y‐axis and the horizontal plane. Yaw angle: the angle between x‐axis and the X‐axis of a inertia frame
Axis TranslationalVelocity
AngularDisplacement
AngularVelocity
ForceComponent
MomentComponent
x u (forward) p or (roll) Fx Mx
y v (lateral) q or (pitch) Fy My
z w (vertical) r or (yaw) Fz Mz
/ˈθiːtə/, US /ˈθeɪtə/
/ˈfaɪ/
/ˈsaɪ/, /ˈpsaɪ/
ME542 Vehicle Dynamics-Lecture 1- 15
Earth Fixed Coordinate and Vehicle Slip
Course angle = +
OXYZ fixed on Earth (does not turn with the vehicle)
Heading angle
Sideslip angle
O
In the left figure, sideslip angle is negative)
tanv
u
ME542 Vehicle Dynamics-Lecture 1- 16
Tire Slip
o: Center of tire contact patchZ: Perpendicular to the ground plane.X and Y axes: On the ground plane. Tire camber angle: Angle between the XZ plane and the
wheel plane.
Source: Wong page 7
ME542 Vehicle Dynamics-Lecture 1- 17
Ride/Handling Dynamics
SPRUNG MASS RIDE HANDLING longitudinal
lateral vertical pitch roll yaw
UNSPRUNG MASS vertical
wheel rotation
Involves vehicle motions in several directions
( )
ME542 Vehicle Dynamics-Lecture 1- 18
Newton/Euler Formulation
Consider a rigid body of mass m, with c.g. at point o, subject to N external forces.
For unconstrained motion, a rigid body possesses 6 DOF, 3 translational (x, y, z) and 3 rotational ().
oi j
k1F
2F
NF
ME542 Vehicle Dynamics-Lecture 1- 19
Let V and to denote the absolute velocity of mass center and angular velocity of the rigid body, ( ) be unit vectors of the body-fixed coordinate oxyz. The Newton/Euler equations of motion are then
Newton/Euler Equations
ˆ i , ˆ j , ˆ k
1
N
ii
dV d LF F m
dt dt
ri Fii1
N
Mo d H
dt
Where L is the linear momentum and H is the angular momentum of the rigid body about the mass center o.
Newton’s2nd law:
oi
j
k1F
2F
NF
1r
2r Nr
Euler
ME542 Vehicle Dynamics-Lecture 1- 20
Angular Momentum
Let H Hxˆ i Hy
ˆ j Hzˆ k
x
y
z
H
H I
H
wherexx xy xz
xy yy yz
xz yz zz
I I I
I I I I
I I I
2 2( )xx
V
I y z dV xy
V
I xy dV etc.
x xx xy xz xx xy xz
y xy yy yz xy yy yz
z xz yz zz xz yz zz
H I I I p I p I q I r
H H I I I q I p I q I r
H I I I r I p I q I r
ME542 Vehicle Dynamics-Lecture 1- 21
Rigid Body Motion of a Vehicle(a key concept: rate of change of a vector with respect to a rotating frame)
Translational motion:
ˆˆ ˆDF ma m ui vj wk
Dt
( )
( )
( )
x
y
z
qw rv
ru pw
p
u
v
w v q
F
F m
F m u
m
F ma
d
dt t
D
D
Cross Product
Change of the unit vectors
Rate of change wrt a fixed frame
Rate of change wrt a rotating frame
Rate of change due to the rotation of the frame
ˆˆˆ ˆˆˆ ui vj wkui vj km w
ud
vdt
w
p u
q v
r
m
w
ME542 Vehicle Dynamics-Lecture 1- 22
Rigid Body Motion of a Vehicle (cont.)
Rotational motion:
xx xy xz
xy yy yz
xx xy xz
xy yy yz
xzxz y yz zzz zz
p I p I q I r
q I p I q I r
r I p I q I
I p I q I rd
I p I q I rdt
I rp I q I r
M
2 2
2 2
2 2
xz yz zz xy yy yz
xx x
xx
y xz xz yz zz
xy yy y
x
y
z xx xy xz
xy xz
xy yy yz
xz yzz zz
I pq I q I rq I pr I qr I r
I p
I p I q I r
I r I qr I r I p I qp I rpp I q I r
I p I qp I r p I pq II p I q Ir r
M
M qq
M
I
xx xy xz
xy yy yz
xz yz zz
I p I q I r
H I p I q I r
I p I q I r
D d
Dt dt
M H
ME542 Vehicle Dynamics-Lecture 1- 23
Equations of Vehicle Rigid Body Motion
2 2
2 2
2
( )
( )
( )
x
y
z
x xx xy xz xz yz zz xy yy yz
y xy yy yz xx xy xz xz yz zz
z xz yz zz xy yy yz
F m u qw rv
F m v ru pw
F m w pv qu
M I p I q I r I pq I q I rq I pr I qr I r
M I p I q I r I pr I qr I r I p I qp I rp
M I p I q I r I p I qp I r p
2xx xy xzI pq I q I rq
ME542 Vehicle Dynamics-Lecture 1- 24
Simplified Equations of Motion• Assume
– vehicle is symmetric wrt the xz plane (Ixy = Iyz =0)
– p,q,r,v, and w are small, i.e., their higher order terms are negligible.
– u=uo + u’, u’ is small compared with uo.
Linear!
Naturally groupsinto 3 sets
'
( )
( )
x
y o
z o
x xx xz
y yy
z xz zz
F mu
F m v ru
F m w qu
M I p I r
M I q
M I p I r
)( rvqwumFx
)( pwruvmFy
)( qupvwmFz
.
.
.
ME542 Vehicle Dynamics-Lecture 1- 25
Review of Rigid Body Kinematics
• Kinematics: study of the geometry of motions.
• Quantities: position, velocity and acceleration.
oi j
k
OI J
Kp
or
pr
relr
Given: Kinematics quantities of onepoint (mass center), and the angular motion of the rigid body.
Find: Those at another point on thesame rigid body.
Application: e.g., use motion of c.g.to infer motion of other points on the vehicle
ME542 Vehicle Dynamics-Lecture 1- 26
Rigid Body Kinematics
Position:
p o relr r r oi j
k
OI J
K
p
or
pr
relr
p o relr r r Velocity:
p o relV V r
p o rel relV V r r
Acceleration:
p rel rel( )oa a r r
ME542 Vehicle Dynamics-Lecture 1- 27
Example 1
A car is traveling in the xz plane, pitching, bouncing
o pGiven:
Find:
o
rel
ˆˆ
ˆ
ˆ
V ui wk
qj
r li
pV pa
l
ME542 Vehicle Dynamics-Lecture 1- 28
Example 1 Solution
o p o
rel
ˆˆ
ˆ
ˆ
V ui wk
qj
r li
l
p o rel
0
0 0 0
0 0
ˆˆ ( )
u l u
V V r q
w w ql
ui w ql k
ME542 Vehicle Dynamics-Lecture 1- 29
Example 1 Solution
p r rel
2
el ( )
0 0
0
0
0
0
0
0 0
0
0
0 0
oa r
q
ql
u qw q l
w q
a r
l
q
u
u
q
w
q
u
l
w
o
rel
ˆˆ
ˆ
ˆ
V ui wk
qj
r li
ME542 Vehicle Dynamics-Lecture 1- 30
Another Kinematic Example (hw1)
• Vehicle Bicycle Model
• Vehicle is driving in the xy plane at a constant forward speed, and turning. All other motions are ignored
• Find the speed and acceleration at the center of the front axle and the rear left tire
ME542 Vehicle Dynamics-Lecture 1- 31
Example 2
• Vehicle Bicycle Model
• Vehicle is driving in the xy plane, and yawing. All other motions are ignored
• Find the dynamic equations
vr
a
b
f
u
1xF
2xF
1yF
2yF
ME542 Vehicle Dynamics-Lecture 1- 32
Example 2 Solution( )
( )
( )
x
y
z
F m u qw rv
F m v ru pw
F m w pv qu
2 2
2 2
2 2
x xx xy xz xz yz zz xy yy yz
y xy yy yz xx xy xz xz yz zz
z xz yz zz xy yy yz xx xy xz
M I p I q I r I pq I q I rq I pr I qr I r
M I p I q I r I pr I qr I r I p I qp I rp
M I p I q I r I p I qp I r p I pq I q I rq
?
ME542 Vehicle Dynamics-Lecture 1- 33
Lagrange’s Formulation • An alternative way to formulate the equations of motion.
• Starts from “energy” (scalar!) rather than vectors.
• We first define three terms
Degrees Of Freedom (DOF):
Constraints
Generalized coordinates
Number of independent coordinates required to uniquely define the motion of a particle, a body or a system of bodies.
Kinematic relationship that limit possible motions.
A set of independent coordinates whose number is equal to the DOF and which uniquely define the position and orientation of the rigid body.
ME542 Vehicle Dynamics-Lecture 1- 34
Examples for DOF, Constraints, and GC
Position of a particle in space: 3DOF
Position of a particle on a cylindrical surface
constraint
2DOF, position uniquely described by 2 coordinates, which are referred to as generalized coordinates (q1, q2).
Rigid wheel rolling on flat surface without slip, DOF=? Rigid wheel rolling on flat surface with slip, DOF=?
ME542 Vehicle Dynamics-Lecture 1- 35
Lagrange EquationsConsider a system of particles or rigid bodies with n DOFdescribed by n generalized coordinates {q1(t), …, qn(t)}.The Lagrange's equation is then
where
ii i i
d T T VQ
dt q q q
i 1n
( , )i iT T q q
V V(qi )
: kinetic energy
: potential energy
Qi : the ith generalized non‐conservative force.
Qi F jj1
k
r j
qi
ME542 Vehicle Dynamics-Lecture 1- 36
Example 3: Compound Pendulum
Derive the equation of motion for the compound pendulum shown below for large angle q.
mb
Ib
L
Is
msr
g
ME542 Vehicle Dynamics-Lecture 1- 37
mb
Ib
L
Is
msr
g
DOF: only 1 (q)
Kinetic energy
Potential energy
2 2 2 2 21 1 1 1( ) ( )
2 2 2 2 2s s b b
LT m L r I m I
disk bar
( )(1 cos ) (1 cos )2s b
LV m g L r m g
disk bar
i
d T T VQ
dt
0
2 2( ) ( ) sin ( ) 02 2s b s b s b
L Lm L r m I I g m L r m
ME542 Vehicle Dynamics-Lecture 1- 38
Example 4: Inverted Pendulum
• Later in the term, we will apply the Lagrange’s method to derive the lateral‐yaw‐roll model.
• Here we will start from a simpler system, which only involves lateral and roll (yaw is ignored for now)
mc
x
Lp,mp
F
ME542 Vehicle Dynamics-Lecture 1- 39
Example 4 Solution
mc
x
Lp,mp
pm g
xF
yF
mc
xF
yF
F
Newtonian Method: for the pendulum
sin cos2 2
p pp p
L Lx x y
2
2( ) ( cos )
2p
x p p p
Ld dF m x m x
dt dt
2( cos sin )2 2
p pp
L Lm x
2
2( ) ( sin )
2p
y p p p p
Ld dF m g m y m
dt dt
2( sin cos )2
pp
Lm
x-direction
y-direction
roll-direction
sin cos2 2
p py x p
L LF F I 21
12p p pI m L
F
ME542 Vehicle Dynamics-Lecture 1- 40
Example 4 Solution (cont.)Newtonian Method: for the cart
x cF F m x
2( ) cos sin2 2
p pc p p p
L LF m m x m m
2sin cos
3p p p pm g m x m L
sin cos2 2
p py x p
L LF F I Plug-in equations from x- and y- direction
We wrote down four equations, two of them used to cancel the internal forces Fx and Fy.
mc
xF
yF
F
ME542 Vehicle Dynamics-Lecture 1- 41
Example 4 Solution (cont.)
sin cos2 2
cos sin2 2
p pp p
p pp p
L Lx x x x
L Ly y
Lagrange’s method
The kinetic energy and potential energy are then
2 2 2 2 2 2 2 21 1 1 1 1 1[ ] [ 2 cos ]
2 2 2 2 2 2 3p
c p p p p c p p
LT m x m x y I m x m x x L
cos2
pp
LV m g
For x-direction: x
d T T VQ F
dt x x x
2( ) cos sin2 2
p pc p p p
L LF m m x m m
For -direction: 0d T T V
Qdt
2sin cos
3p p p pm g m x m L We worked with two equations
ME542 Vehicle Dynamics-Lecture 1- 42
Example 5: Double Pendulum
gl1
m1
l2
k
2
x
1
The double pendulum system consists of a mass‐less rigid bar of length l1, a point‐mass m1, a spring with free length l2 and spring constant k, and another point‐mass m2. Use Lagrange's equation to derive the equations of motion.
2m
ME542 Vehicle Dynamics-Lecture 1- 43
DOF: 3 (1, 2, x)
Kinetic energy
Potential energy
2
1 1 1
2 2
2 1 1 2 1 2 2 1 1 2 1
1( )
21
sin( ) ( ) cos( )2
T m l
m x l l x l
gl1
m1
l2
k
2
x
1
21 2 1 1 2 2 2 2 2
1( ) (1 cos ) (1 cos ) cos
2V m m gl m gl m gx kx
A
B
1 1l
2 2( )l x
x
2 1
ME542 Vehicle Dynamics-Lecture 1- 44
1 1 1
0d T T V
dt
2 21 1 1 2 1 2 2 1 2 1 2 1 2 1 1
22 1 2 2 2 1 2 1 2 2 2 1 1 2 1 1
2 cos( ) sin( )
( ) sin( ) ( ) cos( ) ( ) sin( ) 0
m l m l x m l x m l
m l l x m l l x m m gl
2 2 2
0d T T V
dt
2 22 2 2 2 1 2 1 2 1 2 1 2 1 2 1
2 2 2 2 2 2
( ) ( ) cos( ) ( ) sin( )
2 ( ) ( )sin( ) 0
m l x m l l x m l l x
m x l x m g l x
0x
d T T V
d xt x
22 2 1 1 2 1 2 1 1 2 1
22 2 2 2 2
sin( ) cos( )
cos ( ) 0
m x m l m l
m g kx m l x
ME542 Vehicle Dynamics-Lecture 1- 45
Appendix
ME542 Vehicle Dynamics-Lecture 1- 46
Rotation about a fixed axis
Beer, Johnson And Clausen, Vector Mechanics for Engineers, Dynamics, 8th edition, page 920
di dj dk
i j kdt dt dt
ˆˆ ˆu p u
v q v
w
ui v
w
j k
r
w
ME542 Vehicle Dynamics-Lecture 1- 47
Cross Product
• An operation on two vectors in the three‐dimensional space
http://en.wikipedia.org/wiki/Cross_product
ME542 Vehicle Dynamics-Lecture 1- 48
x
y
ij
k
dij
dt
dji
dt
ˆˆ ˆ
ˆ ˆ0 0
1 0 0
i j kd
i jdt
ˆˆ ˆ
ˆ ˆ0 0
0 1 0
i j kd
j idt