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172 MECHATRONICS
T
T
The torque needed to balance a weight load, F = [F F ] = [0 − W] , at the tip is
x y
determined by
[ ] [ ][ ]
Torque J J 0.0
1 11 21
= (3.254)
Torque J J −W
2 12 22
which shows the necessary static torque at each joint to balance a weight at the tip for
different positions of the manipulator. Notice that since the Jacobian matrix is 2x2, it is
relatively simple to obtain the inverse Jacobian in analytical form.
[ l cos( + ) l ⋅ sin( + ) ]
J −1 = 1 2 1 2 2 1 2
l ⋅ l ⋅ sin( ) −l cos( ) − l cos( + ) l sin( ) − l ⋅ sin( + )
1 2 2 1 1 2 1 2 1 1 2 1 2
(3.255)
Notice that when = 0.0, the mechanism is at a singular point, which is indicated by the
2
sin( ) term in the denominator of the inverse Jacobian equation above.
2
3.8 A CASE STUDY: AUTOMOTIVE TRANSMISSION
AS A “GEAR REDUCER”
An automotive transmission is a gear reducer. It changes the speed ratio (gear ratio) between
the input and output shaft. The input shaft is connected to the crank shaft of the engine, the
output shaft is connected to the lower powertrain which is connected to the differential (or
differentials in four wheel drive vehicles) to drive the wheels. The gear ratio is not constant,
but can be one of a finite number of gear ratios (i.e., in five speed transmission, five different
the gear ratios can be engaged). In continuously variable transmissions, the gear ratio can
be changed to any value between a minimum and a maximum value, instead of one of a
finite number of selections. The mechanism by which the gear ratio is changed from one
ratio to another determines the type of transmission. In manual shift transmissions, the
gear is changed by the operator by moving a lever. In automatic transmissions, the gear is
changed by an electronic control module (ECM) which controls a set of electrically actuated
clutch/brakes or valves. In continuously variable transmissions (CVT), i.e. toroidal CVT,
the gear ratio is changed by continuously varying a drive diameter, hence the gear ratio
can be any value between a minimum and maximum value (hence the name “continuously
varying”), instead of having a finite number of discrete gear ratio numbers, that is, 5-speed
(or 6-speed) transmission meaning there are 5 (or 6) gear ratios to select from.
3.8.1 The Need for a Gearbox “Transmission”
in Automotive Applications
A gearbox is used to change the speed and torque ratio between the input and output shafts.
If we assume that the gearbox operates with 100% efficiency, the input power and output
power are equal to each other. As the gearbox changes the speed ratio between input–output
shafts, it also changes the torque ratio in the inverse ratio.
In automotive and mobile equipment motion, the power output from engine is defined
as a function of speed using the so called “lug-curve” (see Section 1.2). At a given speed,
the maximum power capacity of an engine is known. Hence, the maximum available torque
output from the engine is known as a function of engine speed. The gearbox (better known
as the transmission in automotive applications) essentially scales that lug-curve at different
gear ratios as shown in Figure 3.24, excluding the effect of the torque converter. A typical