ABB
Abb AC Servo motors
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71
5
Figure 2
where
J =
inertia
W =
weight
R =
radius
g =
gravitational constant (386 in/s
2
)(980 cm/s
2
)
L =
length
p =
density
For a known weight and radius:
J =
1
W R
2
2
g
For a known density, radius, and length:
J =
1 π LpR
4
2
g
where
J =
inertia
W =
weight
Ro =
outer radius
Ri =
inner radius
g =
gravitational constant (386 in/s
2
)(980 cm/s
2
)
L =
length
p =
density
For a known weight and radius
J =
1
W
(Ro
2
+ Ri
2
)
2
g
For a known density, radius, and length:
J =
1
πLp
(Ro
4
- Ri
4
)
2
g
Solid cylinder:
Inertia can be calculated if either the weight and radius are known; or the density, radius, and length are known. Figure 2 presents the equations.
As an example, if the cylinder were a lead screw with a radius of .312 inches (0.79 cm) and a length of 22 inches (55.8 cm), then the
inertia would be:
J =
1
π L p R
4
=
1
π (22) (.28) (.312)
4
=
0.000237 lb-in-s
2
2
g
2
386
Metric
=
1
π (55.8) (7.75) (0.79)
4
=
0.26
gm-cm-s
2
2
980
These equations are important since the inertia of mechanical components (i.e. shafts, gears, drive rollers, leadscrews, etc.) can be
calculated by using them. Once the inertia is determined, it becomes just a task of reflecting that load inertia and friction through the
mechanical linkages to what the motor will “see”.
Material
Density (lb/in
3
)
gm/cm
3
Aluminum
.098
2.72
Copper
.322
8.91
Plastic
.040
1.11
Steel
.280
7.78
Wood
.029
0.8
Inertia
Abb AC Servo motors