Fundamentals of Vehicle Dynamics

Fundamentals of
Vehicle Dynamics
Thomas D. Gillespie

Puhlilhed by:
Society of AuromOli,'e Engm""", Ilic.
400 Commonweal(h Drive
Warrendale. PA 151l96-01.m

LIST OF SYMBOLS
a
b
A

Tire cornering stiffness parameter
Tire cornering stiffness parameter
Frontal area of a vehicle
Lateral force compliance steer coefficient on the front axle
Af
Lateral force compliance steer coefficient on the rear axle
Ar

Acceleration in the x-direction
ax
Acceleration in the lateral direction
ay
b
Longitudinal distance from front axle to center of gravity
c
Longitudinal distance from center of gravity to rear axle
Cornering stiffness of the tires on an axle
Ca
Ca' Cornering stiffness of one tire
CC a Tire cornering coefficient
Tire camber stiffness
Cy
Aerodynamic drag coefficient
Co
Road surface rolling resistance coefficient
Ch
Aerodynamic lift coefficient
CL

CPM Aerodynamic pitching moment coefficient
CRM Aerodynamic rolling moment coefficient
CYM Aerodynamic yawing moment coefficient
Cs
Suspension damping coefficient
Aerodynamic side force coefficient
Cs
CP
Center of pressure location of aerodynamic side force
d
Lateral distance between steering axis and center of tire contact at the ground
Distance from axle to the hitch point
dh
dns
Distance from center of mass to the neutral steer point
D
Tire diameter
Dynamic index
DI
Linear

deceleration
Dx
Aerodynamic drag force
DA
e
Height of the pivot for an "equivalent torque arm"
Drum brake geometry factor
E[y2] Mean square vibration response
f
Longitudinal length for an "equivalent torque arm"
Wheel hop resonant frequency (vertical)
fa
vii

FUNDAMENTALS OF VEHICLE DYNAMICS

fn
fr
Fb

Undamped natural frequency of a suspension system (Hz)
Rolling resistance coefficient
Braking force
Vertical disturbance force on the sprung mass
Fi
Imbalance force in a tire
Fx
Force in the x-direction (tractive force)
Fxm Maximum brake force on an axle
Fxt
Total force in the x-direction
Fy
Force in the y-direction (lateral force)
Lateral force on an axle
F'
Lateral force on one tire
Y
Force in the z-direction ( vertical force)
Fz
Vertical force on inside tire in a turn

Fzi
Vertical force on outside tire in a turn
Fzo
Tire/wheel nonuniformity force on the unsprung mass
Fw
g
Acceleration of gravity (32.2 ftlsec 2 , 9.81 mlsec 2 )
G
Brake gain
Road roughness magnitude parameter
Go
Gz
Power spectral density amplitude of road roughness
G zs
Power spectral density amplitude of sprung mass acceleration
Center of gravity height
h
Height of the aerodynamic drag force
ha
Hitch height

hh
Height of the sprung mass center of gravity above the roll axis
hI
Height of suspension roll center
hr
Tire section height
ht
HP
Engine or brake horsepower
HPA Aerodynamic horsepower
HI'R Rolling resistance horsepower
HPRL Road load horsepower
Response gain function
Hv
Moment of inertia of the driveshaft
Id
Moment of inertia of the engine
Ie
Moment of inertia of the transmission
It

Moment of inertia of the wheels
Iw
Moment of inertia about the x-axis
Ixx
viii

LIST OF SYMBOLS
Moment of inertia about the y-axis
Moment of inertia about the z-axis
lzz
Radius of gyration
k
U ndersteer gradient
K
Understeer gradient due to aligning torque
Kat
Understeer gradient due to lateral load transfer on the axles
Kilt
Klfes Understeer gradient due to lateral force compliance steer
Vertical stiffness of a suspension

Ks
Steering system stiffness
Kss
Kstrg Understeer gradient due to the steering system
Vertical stiffness of a tire
Kt
Suspension roll stiffness
Kq,
L
Wheelbase
Aerodynamic lift force
LA
m
Drum brake geometry parameter
M
Mass of the vehicle
MAT Moment around the steer axis due to tire aligning torques
Moment around the steer axis due to tire lateral forces
ML
Equivalent mass of the rotating components

Mr
MSA Moment around the steer axis due to front-wheel-drive forces and torques
Moment around the steer axis due to tire tractive forces
MT
Moment around the steer axis due to tire vertical forces
Mv
M Rolling moment
n
Drum brake geometry parameter
N
Normal force
Numerical ratio of the transmission
Nt
Numerical ratio of the final drive
Nf
Numerical
ratio of the combined transmission and final drive
Ntf
NSP Neutral steer point
Pneumatic trail

p
Pa
Brake application pressure/effort
Patm Atmospheric pressure
Front brake application pressure
Pf
Pr
Rear brake application pressure
Ps
Static pressure
Total pressure
Pt

Iyy

ix

FUNDAMENTALS OF VEHICLE DYNAMICS

PM

P

Aerodynamic pitching moment
Roll velocity about the x-axis of the vehicle
q
Pitch velocity about the y-axis of the vehicle
q
Dynamic pressure
r
Yaw velocity about the z-axis of the vehicle
r
Rolling radius of the tires
Ratio of tire to suspension stiffness
rk
R
Radius of tum
Hitch force
Rh
Grade force
Rg
Rolling resistance force
Rx
RRL Road load
RM
Aerodynamic rolling moment
RR
Ride rate of a tire/suspension system
Rep Roll rate of the sprung mass
s
Lateral separation between suspension springs
Aerodynamic side force
SA
Spectral density of white-noise
So
SD
Stopping distance
t
Tread
ts
Length of time of a brake application
Torque in the axle
Ta
Brake torque
Tb
Torque at the clutch
Tc
Torque in the driveshaft
Td
Torque of the engine
Te
Roll torque in a front suspension
Tsf
Roll torque in a rear suspension
Tsr
Tamb Ambient temperature
Torque about the x-axis
Tx
V
Forward velocity
Ambient wind velocity
Vw
Vf
Final velocity resulting from a brake application
Initial velocity in a brake application
Vo
w
Tire section width
W
Weight of the vehicle
x

LIST OF SYMBOLS

Wa
Wd
Wf
Wr
W rr
W fs
W rs

Wy
x
y

YM
Z

X
Y
Z
Zr
Zu
a
acw
ad

ae

aw
ax
~

Y
Yg

Yb
Ii
lie
Iii

Axle weight
Dynamic load transfer
Dynamic weight on the front axle
Dynamic weight on the rear axle
Dynamic weight on the right rear wheel
Static weight on the front axle
Static weight on the rear axle
Lateral weight transfer on an axle
Forward direction on the longitudinal axis of the vehicle
Lateral direction out the right side of the vehicle
Aerodynamic yawing moment
Vertical direction with respect to the plane of the vehicle
Forward direction of travel
Lateral direction of travel
Vertical direction of travel
Vertical displacement of the sprung mass
Road profile elevation
Vertical displacement of the unsprung mass
Tire slip angle
Coefficient in the pitch plane equations
Aerodynamic wind angle
Rotational acceleration of the dri veshaft
Rotational acceleration of the engine
Rotational acceleration of the wheels
Rotational acceleration about the x-axis
Sideslip angle
Rotation angle of aU-joint
Coefficient in the pitch plane equations
Camber angle
Coefficient in the pitch plane equations
Wheel camber with respect to the ground
Wheel camber with respect to the vehicle body
Steer angle
Compliance steer
Steer angle of the inside wheel in a tum
xi

d

FUNDAMENTALS OF VEHICLE DYNAMICS

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