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Mechatronics in
2020 - 2021
Part 3
Driveline Dynamics
3. Driveline Dynamics
The maximum achievable acceleration of a vehicle
is limited by two factors: maximum torque at
driving wheels, and maximum traction force at
tireprints. The first one depends on engine and
transmission performance, and the second one
depends on tire-road friction. In this part, we
examine engine and transmission performance.
3.1 Engine Dynamics
The maximum attainable power Pe of an internal
combustion engine is a function of the engine angular
velocity ωe. This function must be determined
experimentally, however, the function Pe = Pe (ωe), which
is called the power performance function, can be estimated
by a third-order polynomial
If we use ωM to indicate the angular velocity, measured in
[ rad/s], at which the engine power reaches the maximum
value PM , measured in [Watt = Nm/s], then for spark
ignition engines we use:
The following figure illustrates a sample for power
performance of a spark ignition engine that provides PM =
50kW at ωM = 586 rad/ s ≈ 5600 rpm. The curve begins at an
angular velocity at which the engine starts running
A sample of power and torque performances
for a spark ignition engine.
For indirect injection Diesel engines we use
and for direct injection Diesel engines we use
The driving torque of the engine Te is the torque that
provides Pe
120 Porsche 911T M and Corvette Z06T M engines.
A model of Porsche 911 turbo has a flat-6 cylinder, twinturbo engine with 3596 cm3 total displacement. The
engine provides a maximum power PM = 353kW ≈ 480
hp at ωM = 6000 rpm ≈ 628 rad/s, and a maximum torque
TM = 620Nm at ωe = 5000rpm ≈ 523 rad/s. The car
weighs around 1585 kg and can move from 0 to 96 km/ h
in 3.7 s. Porsche 911 has a top speed of 310 km/ h.
The power performance equation for the Porsche 911
engine has the coefficients
and, its power performance function is
A model of Corvette Z06 uses a V 8 engine with 6997
cm3 total displacement. The engine provides a maximum
power PM = 377 kW ≈ 512 hp at ωM = 6300 rpm ≈ 660
rad/ s, and a maximum torque TM = 637Nm at ωe = 4800
rpm ≈ 502 rad/ s. The Corvette weighs around 1418 kg
and can move from 0 to 100km/ h in 3.9 s in first gear. Its
top speed is 320 km/ h.
The power performance equation for the engine of
Corvette Z06 has the coefficients
and, its power performance function is
The power performance curves for the Porsche 911 and
Corvette Z06 are plotted in the following figure.
Power performance curves for
the Porsche 911 and Corvette Z06.
Although there is almost no limit for developing a
powerful engine, any engine with power around 100 hp
would be enough for street cars with normal applications.
It seems that engines with 600 hp reach the limit of
application for street cars. However, race cars may have
higher power depending on the race regulations. As an
example, formula 1 regulations dictates the type of engine
permitted. It must be a four-stroke engine, less than 3000
cm3 swept volume, no more than ten cylinders, and no
more than five valves per cylinder, but there is no limit
for power.
4.2 Driveline and Efficiency
We use the word driveline, equivalent to transmission, to
call the systems and devices that transfer torque and power
from the engine to the drive wheels of a vehicle. Most
vehicles use one of two common transmission types:
manual gear transmission, and automatic transmission with
torque convertor. A driveline includes the engine, clutch,
gearbox, propeller shaft, differential, drive shafts, and
drive wheels.
The following figure illustrates how the driveline for a
rear-wheel-drive vehicle is assembled.
Driveline components of a rear wheel drive vehicle.
1. The engine is the power source in the driveline. The output from the
engine is an engine torque Te, at an associated engine speed ωe.
2. The clutch connects and disconnects the engine to the rest of the
driveline when the vehicle is equipped with a manual gearbox.
3. The gearbox can be used to change the transmission ratio between
the engine and the drive wheels.
4. The propeller shaft connects the gearbox to the differential. The
propeller shaft does not exist in front-engined front-wheel-drive and
rear-engined rear-wheel-drive vehicles. In those vehicles, the
differential is integrated with the gearbox in a unit that is called the
5. The differential is a constant transmission ratio gearbox that allows
the drive wheels to have different speeds. So, they can handle the car
in a curve.
6. The drive shafts connect the differential to the drive wheels.
7. The drive wheels transform the engine torque to a traction force on
the road.
The input and output torque and angular velocity for each
device in a driveline are indicated in the following figure.
The input and output torque and angular velocity
of each driveline component.
The available power at the drive wheels is
where η < 1 indicates the overall efficiency between the
engine and the drive wheels
ηc < 1 is the convertor efficiency and ηt < 1 is the
transmission efficiency.
The relationship between the angular velocity of the
engine and the velocity of the vehicle is
where ng is the transmission ratio of the gearbox, nd is the
transmission ratio of the differential, ωe is the engine
angular velocity, and Rw is the effective tire radius.
Transmission ratio or gear reduction ratio of a gearing
device, n, is the ratio of the input velocity to the output
while the speed ratio ωr is the ratio of the output velocity
to the input velocity.
4.3 Gearbox and Clutch Dynamics
The internal combustion engine cannot operate below a
minimum engine speed ωmin. Consequently, the vehicle
cannot move slower than a minimum speed vmin while the
engine is connected to the drive wheels.
At starting and stopping stages of motion, the vehicle
needs to have speeds less than vmin. A clutch or a torque
converter must be used for starting, stopping, and gear
Consider a vehicle with only one drive wheel. Then, the
forward velocity vx of the vehicle is proportional to the
angular velocity of the engine ωe, and the tire traction force
Fx is proportional to the engine torque Te
where Rw is the effective tire radius, nd is the differential
transmission ratio, ni is the gearbox transmission ratio in
gear number i, and η is the overall driveline efficiency.
These two equations are called the speed equation, and the
traction equation.
The froward velocity vx of a driving wheel with radius Rw
and the traction force Fx on the driving wheel is:
Tw is the applied spin torque on the wheel, and ωw is the
wheel angular velocity.
The wheel inputs Tw and ωw are the output torque and
angular velocity of differential. The differential input
torque Td and angular velocity ωd are:
where nd is the differential transmission ratio and ηd is the
differential efficiency.
The differential inputs Td and ωd are the output torque and
angular velocity of the vehicle’s gearbox.
The engine’s torque Te and angular velocity ωe are the
inputs of the gearbox. The input-output relationships for a
gearbox depend on the engaged gear ratio ni.
ηg is the gearbox efficiency, and ni is the gear reduction
ratio in the gear number i.
Therefore, the forward velocity of a driving wheel vx, is
proportional to the engine angular velocity ωe, and the tire
traction force Fx is proportional to the engine torque Te,
when the driveline is engaged to the engine.
Having the torque performance function Te = Te (ωe)
enables us to determine the wheel torque Tw as a function
of vehicle speed vx at each gear ratio ni.
Using the approximate equation for Te:
A six-gear gearbox.
Consider an inefficient passenger car with the following
m = 1550 kg
Rw = 0.326 m
η = 0.24
torque = 392Nm at 4400 rpm ≈ 460.7 rad/ s
power = 206000W at 6800 rpm ≈ 712.1 rad/ s
1st gear ratio = n1 = 3.827
2nd gear ratio = n2 = 2.36
3rd gear ratio = n3 = 1.685
4th gear ratio = n4 = 1.312
5th gear ratio = n5 = 1
6th gear ratio = n6 = 0.793
reverse gear ratio = nr = 3.28
final drive ratio = nd = 3.5451
Based on the speed equation,
The angular velocities associated to maximum power and
maximum torque are indicated by dashed lines.
The power and torque performance equations for the
engine can be approximated by
Using the above torque and traction equations, we can
plot the wheel torque as a function of vehicle speed at
different gears.
The following figure shows the wheel torque-speed at
each gear ni.
The envelope curve for the series of torque-speed
equations is similar to the torque curve of a constant
power ideal engine.
Wheel torque-speed at each gear ni of a gearbox,
and the envelope curve simulating an ideal engine behavior.
4.4 Gearbox Design
The speed and traction equations:
are used to calculate the gear ratios of a gearbox as well as
vehicle performance. Theoretically the engine should work
at its maximum power to have the best performance.
However, to control the speed of the vehicle, we need to
vary the engine’s angular velocity. Hence, we pick an
angular velocity range (ω1, ω2) around ωM , which is
associated to the maximum power PM , and sweep the
range repeatedly at different gears. The range (ω1, ω2) is
called the engine’s working range.
As a general guideline, we may use the following
recommendations to design the transmission ratios of a
vehicle gearbox:
1. We may design the differential transmission ratio nd and
the final gear nn such that the final gear nn is a direct gear,
nn = 1, when the vehicle is moving at the moderate
highway speed. Using nn = 1 implies that the input and
output of the gearbox are directly connected with each
other. Direct engagement maximizes the mechanical
efficiency of the gearbox.
2. We may design the differential transmission ratio nd and
the final gear nn such that the final gear nn is a direct gear,
nn = 1, when the vehicle is moving at the maximum
attainable speed.
3. The first gear n1 may be designed by the maximum
desired torque at driving wheels. Maximum torque is
determined by the slope of a desired climbing road.
4. We can find the intermediate gears using the gear
stability condition.
Stability condition provides that the engine speed must not
exceed the maximum permissible speed if we gear down
from ni to ni−1, when the engine is working at the
maximum torque in ni.
5. The value of cg for relative gear ratios
can be chosen in the range.
4.4.1 Geometric Ratio Gearbox Design
When the jump of engine speed in any two successive
gears is constant at a vehicle speed, we call the gearbox
geometric. The design condition for a geometric gearbox
where cg is the constant relative gear ratio and is called
step jump.
A geometric gearbox has constant engine speed jump in any
gear shift. So, a geometric gearbox must have a gear-speed
plot such as that shown in the following figure.
A gear-speed plot for a geometric gearbox design.
The engine working range is defined by two speeds (ω 1, ω2)
When the engine reaches the maximum speed ω2 in the
gear number i with ratio ni, we gear up to ni+1 to jump the
engine speed down to ω1. The engine’s speed jump is kept
constant for any gear change from ni to ni+1.
Employing the speed equation,
, we have
and therefore,
Let’s indicate the maximum vehicle speed in gear ni by vi
and in gear ni−1 by vi−1, then,
and therefore, the maximum speed in gear i to the
maximum speed in gear i − 1 is inverse of the gear ratios
The change in vehicle speed between gear ni−1 and ni is
indicated by:
and is called speed span.
Having the step jump cg , and knowing the maximum
speed vi of the vehicle in gear ni, are enough to find the
maximum velocity of the car in the other gears
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