Polar Moment of Inertia Machinery's Handbook, 31st Edition
255
Cone:
A
3
M r
2
With reference to axis AA : J = With reference to axis BB (through the center of gravity): J M r h 20 3 4 M 2 2 = + a k 10 M
h
B
B
A A
r
B
B
r
Frustum of Cone:
R
r
3 3 5 5
R r M R r 10 3 − − ^ ^
h
A
A
r
With reference to axis AA : J
=
A
M
h
R
Moments of Inertia of Complex Areas and Masses may be evaluated by the addition and subtraction of elementary areas and masses. For example, the accompanying fig- ure shows a complex mass at (1); its mass polar moment of inertia can be determined by adding the moments of inertia of the bodies shown at (2) and (3), and subtracting that at (4). Thus, J M 1 = J M 2 + J M 3 − J M 4 . All of these moments of inertia are with respect to the axis of rotation z − z . Formulas for J M 2 and J M 3 can be obtained from the tables beginning on page 253 . The moment of inertia for the body at (4) can be evaluated by using the fol- lowing transfer-axis equation: J M 4 = J M 4 ′ + d 2 M . The term J M 4 ′ is the moment of inertia with respect to axis z ′ − z ′ ; it may be evaluated using the same equation that applies to J M 2 where d is the distance between the z − z and the z ′ − z ′ axes, and M is the mass of the body (= weight in lbs ÷ g ).
z
z
z
z
(2)
(1)
z¢
z¢
d
z
z
z
z
(4)
(3) Moments of Inertia of Complex Masses
Similar calculations can be made when calculating I and J for complex areas using the appropriate transfer-axis equations I = I ′ + d 2 A and J = J ′ + d 2 A . The primed term, I ′ or J ′ , is with respect to the center of gravity of the corresponding area A ; d is the distance between the axis through the center of gravity and the axis to which I or J is referred.
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