Road Vehicle Performance: Tractive
Effort and Acceleration
TRANSPORTATION ENGINEERING
ASSIST. PROF. DR. M. ALI MOSABERPANAH
Available Tractive Effort
Tractive effort available to overcome resistance forces and/or to
accelerate the vehicle is determined by either:
◦ The force generated by the vehicle’s engine
◦ Some maximum value that is a function of the vehicle’s weight distribution
and the characteristics of the pavement/tire interface
Maximum Tractive Effort
There is a point at which additional engine-generated tractive effort is
not productive
Eventually your tires will vaporize
To determine the point of maximum tractive effort (i.e., the value of
impending tire spin), we can use a force and moment-generating
diagram
Maximum Tractive Effort
Fig. 2.2
L = wheelbase
h = height of the center of gravity
lf, lr = distance from the front, rear axle to the CG
Wf, Wr = weight of vehicle on front, rear axle
Maximum Tractive Effort
To determine the max tractive effort that the tire/pavement
interface can support, we must examine the normal load on
the drive axle.
Assuming a rear-wheel drive car, the following equation can
be used to determine the normal load on the rear axle:
Wr 
Ra h  Wl f cos g  mah  Wh sin  g
L
Grade moment: + for incline
Eq. 2.10
Maximum Tractive Effort
Rearranging terms, assuming
cos g = 1, and then substituting into Eq. 2.2 (
), yields:
F = ma + Ra + Rrl + Rg
lf
h
Wr  W  F  Rrl 
L
L
Eq. 2.11
Maximum Tractive Effort
And from basic physics we know
(for a rear-wheel-drive car):
Fmax  Wr
 = coefficient of road adhesion
Eq. 2.12
Maximum Tractive Effort
Substituting Eq. 2.11 into Eq. 2.12 yields:
W l f  f rl h  / L
Fmax 
1  h / L

Eq. 2.14
Similarly, by summing moments about the
rear axle, we have the following formula for a
front-wheel-drive vehicle:
W lr  f rl h  / L
Fmax 
1  h / L
Eq. 2.15
Engine-Generated
Tractive Effort
Two most common measures of engine output are torque and power
Torque is the work generated by the engine (the twisting moment), and is
expressed in newton-meters (n-m).
Engine-Generated
Tractive Effort
Power is the rate of engine work, and is related to
torque by the following equation:
2M e n e
Pe =
1000
Pe = engine-generated power in kW,
Me = engine torque in N-m, and
ne = engine speed in crankshaft revolutions per second.
Pe =
2M e n e
550
Pe = engine-generated power in hp,
Me = engine torque in lb-ft, and
ne = engine speed in crankshaft revolutions per second.
Eq. 2.16
Engine-Generated
Tractive Effort
Typical torque-power curve for a gasoline-powered engine
(Fig. 2.3)
Engine-Generated
Tractive Effort
Tractive effort needed for acceptable vehicle acceleration is greater at
lower vehicle speeds, but max torque is developed at high engine
speeds.
We need to use gear reductions to provide a mechanical advantage for
acceptable performance over a range of vehicle speeds.
Engine-Generated
Tractive Effort
Fig. 2.4
Engine-Generated
Tractive Effort
Of course, not all engine generated power will reach the wheels
There are mechanical losses along the way (e.g., transmission,
differential)
◦ Typically 5-25% of the engine-generated tractive effort is lost in the driveline
◦ We use a mechanical efficiency term, d, to denote this
Engine-Generated
Tractive Effort
Additionally, the overall gear reduction ratio, 0, is an important
consideration
0 refers to the relationship between the revolutions of the engine’s
crankshaft and the revolutions of the drive wheels.
◦ An 0 of 3 means that the engine’s crankshaft turns 3 revolutions for every 1
revolution of the drive wheels
Engine-Generated
Tractive Effort
The engine-generated tractive effort reaching the drive wheels is given by
the following equation:
M e 0d
Fe 
r
Eq. 2.17
Engine-Generated
Tractive Effort
The relationship between vehicle speed and engine speed is:
V
2rne (1  i)
0
V = vehicle speed in ft/s
ne = engine speed in rev/s
i = driveline slippage (generally taken as
2-5% for passenger cars)
r= Radius of wheel in ft
Eq. 2.18
Tractive Effort
The available tractive effort (F in Eq. 2.2) at any given speed is the lesser
of:
◦ maximum tractive effort (Fmax)
◦ engine-generated tractive effort (Fe)
Vehicle Acceleration
Eq. 2.2 can be used again, with an additional term added
F = mma + Ra + Rrl + Rg

m is called the mass factor, and
accounts for the inertia of the vehicle’s
rotating parts that must be overcome
during acceleration
Vehicle Acceleration
Rearranging Eq. 2.2 with the mass factor included
gives:
F   R  γ m ma

Eq. 2.19
The mass factor is approximated as:
γm = 1.04  0.0025ε
2
0
Eq. 2.20
Vehicle Acceleration
The force available to accelerate is given by:
Fnet  F   R
When Fnet = 0, vehicle cannot accelerate,
and is at its top speed
 Relationship between Fnet, F (lesser of Fmax
and Fe) and R is shown in Fig. 2.5

Vehicle Acceleration
Fig. 2.5