Page 213 - Mechatronics with Experiments

MECHANISMS FOR MOTION TRANSMISSION 199
conditions, the following fundamental relations hold,

T out1 = T out2 ; unlocked differential mode (3.329)
w out1 = w out2 ; locked differential mode (3.330)

where T out1 , T out2 are output torques, and w out1 , w out2 are output shaft speeds of the differ-
ential.
When in locked condition (or unlocked condition and the planetary gear speed around
its carrier axis is zero, hence both output shaft speeds are same), the differential’s gear ratio
is (Figure 3.35)
N 1
N diff = (3.331)
N 2
w out1 = w out2 = N diff ⋅ w in (3.332)

When it is in unlocked condition, output speeds can be different, which is determined by
the dynamic interactions of the load and traction conditions and the type motion path the
vehicle is following (i.e., a curved path),

w = N ⋅ w +Δw (3.333)
out1 diff in
w = N ⋅ w −Δw (3.334)
out2 diff in
where Δw speed is added and subtracted from two output shafts due to the rotation of the
planetary gear about its carrier axis, which results in “differential” speeds between the two
output shafts.
Normally, a differential operates in the unlocked case. If the vehicle travels on curved
paths, the differential allows the outside wheel to rotate faster than the inside wheel in
order to accomodate the speed differential needed. This is provided by the rotation of the
planetary gear about its own axis. This is the main function of an unlocked differential
operation, reducing tire slip and wear in curved paths.
The drawback of the unlocked differential condition is that the torques transmitted
to each output shaft are the same. The implication of this is that if the two wheels have
different traction conditions on the ground, then one of the wheels which has less traction
will spin much faster than the other wheel. Eventually, the maximum torque that can be
delivered to each wheel is limited by the smallest of the traction torques available between
the tires and ground. Consider a case where a vehicle is on straight line path, but the left
wheel has almost no traction due to ice, while the right wheel has good ground traction.
If we start from a stopped condition, the left wheel spins at a high speed and provides not
much traction to the vehicle, and the right wheel will move very slowly or will not move at
all (or even rotate in reverse direction) depending on how small the traction torque available
between the tire and icy surfacis. The differential will not be able to direct more torque to
the wheel that has more traction. The maximum torque that the differential can transmit is
limited by the smaller of the two traction torques that can be supported by each tire–ground
interaction. Under these conditions, the differential is better to be locked to provide better
traction at the expense of giving up the differential function (different output shaft speeds)
temporarily. When the differential is locked, both output shafts must then rotate at the same
speed. This eliminates the condition that the maximum torque transmitted is limited by the
minimum traction. As a result, traction is improved. However, the penalty paid is that if
the vehicle is moving on a curved path, there will be tire slip and more wear on tires, and
steering quality will be poorer, since the wheel speeds are forced to be same by locking the
differential.