Page 149 - Mechatronics with Experiments

MECHANISMS FOR MOTION TRANSMISSION 135
be the kinetic energy of the load with inertia J and output speed ̇ out . In order to provide
l
such an energy, the actuator must provide a kinetic energy plus the losses. Hence,
KE = ⋅ KE in (3.7)
l
1
KE = ⋅ J ⋅ ̇ 2 out (3.8)
l
l
2
1
= ⋅ ⋅ J in,eff ⋅ ̇ 2 (3.9)
2 in
1 2
̇
= ⋅ J ⋅ ( ∕N) (3.10)
l
in
2
Hence, the effective reflected inertia is
1
J = ⋅ J (3.11)
in,eff 2 l
⋅ N
In summary, an ideal motion transmission mechanism (efficiency is 100%, = 1.0)
has the following reflection properties between its input and output shafts
̇
= N ⋅ ̇ out (3.12)
in
1
J in,eff = ⋅ J l (3.13)
N 2
1
T in,eff = ⋅ T l (3.14)
N
where N is the effective gear ratio. The efficiency factor of the motion transmission mech-
anism is often taken into account by a relatively large safety factor. If the efficiency factor
is to be explicitly included in the actuator sizing calculations, then the following relations
hold,
̇
= N ⋅ ̇ out (3.15)
in
1
J in,eff = ⋅ J l (3.16)
2
N ⋅
1
T in,eff = ⋅ T l (3.17)
N ⋅
It is important to note that in either direction of power transmission, the efficiency fac-
tor is in the denominator of the equations which indicates loss of power due to transmission
efficiency in either direction.
A motion transmission mechanism is characterized by the following parameters:

1. The main characteristic of a motion transmission mechanism is its gear ratio. This
is sometimes called the effective gear ratio since the motion conversion may not
necessarily be performed by gears.
2. Efficiency: efficiency of a real gear ratio is always less than 100%. For most gear
mechanisms, forward and back drive efficiencies are same except for the lead-screw
and ball-screw type mechanisms. In such mechanisms, it is appropriate to talk about
rotarytolinearmotionconversionefficiency, ,andlineartorotarymotionconversion
f
efficiency, . For ball-screw, the typical values of the efficiency coefficients are
b
= 0.9, = 0.8, and for lead-screws they vary as a function of the lead angle. The
b
f
lead angle is defined as the angle the lead helix makes with a line perpendicular to
the axis of rotation. As the lead (linear distance traveled per rotation) increases, so
does the lead angle, hence the efficiency.