7-8
7.1.3.2 Acceleration and deceleration time calculation
When an object whose moment of inertia is J (kg·m
2
) rotates at the speed N (r/min), it has the following
kinetic energy:
)J(
)
60
Nʌ2
(
2
J
E
2
x
x
(7.5)
To accelerate the above rotational object, the kinetic energy will be increased; to decelerate the object, the
kinetic energy must be discharged. The torque required for acceleration and deceleration can be expressed
as follows:
)mN()
dt
dN
(
60
ʌ2
J
IJ xx
(7.6)
This way, the mechanical moment of inertia is an important element in the acceleration and deceleration.
First, calculation method of moment of inertia is described, then those for acceleration and deceleration
time are explained.
[ 1 ] Calculation of moment of inertia
For an object that rotates around the rotation axis, virtually divide the object into small segments and
square the distance from the rotation axis to each segment. Then, sum the squares of the distances and the
masses of the segments to calculate the moment of inertia.
㩷
)mkg()
2
r
i
W
i
(J
2
xx
¦
(7.7)
The following describes equations to calculate moment of inertia having different shaped loads or load
systems.
(1) Hollow cylinder and solid cylinder
The common shape of a rotating body is hollow cylinder. The moment of inertia around the hollow
cylinder center axis can be calculated as follows, where the outer and inner diameters are D
1 and D2 (m)
and total mass is W (kg) in Figure 7.8.
)mkg(
8
)
D
2
D
1
(W
J
2
22
x
x
(7.8)
For a similar shape, a solid cylinder, calculate the moment of inertia as D
2
is 0.
Figure 7.8 Hollow Cylinder
(2) For a general rotating body
Table 7.1 lists the calculation equations of moment of inertia of various rotating bodies including the
above cylindrical rotating body.
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