Machinery's Handbook, 31st Edition
238
Radius of Gyration
Sphere:
A
A
k r
5 2
k a r k a r 2 2 2 2 = + = +
2
. r 06325 5
h h
k
r
2 2 2 = =
r
5 2
k
A
A
Axis its diameter
Axis at a distance
Hollow Sphere and Thin Spherical Shell:
A
A
R r R r − −
3 3 5 5
. 06325 5 2 − − ^ ^
k
R
=
. r 08165 3
k
r
2 2 2 = =
r
k R
3 3 5 5
R r R r
k
h h
k
2
k
=
A
A
Hollow Sphere Axis its diameter
Thin Spherical Shell
Ellipsoid and Paraboloid:
k b
k b c k b c 5 2 2 2 = = + ^ h 2 2 +
k r
. r 05773 3
A
A
k
r
1 2 2 = =
A
A
k
5 1
a
c
Ellipsoid Axis through center
Paraboloid Axis through center
Center and Radius of Oscillation.— If a body oscillates about a horizontal axis which does not pass through its center of gravity, there will be a point on the line drawn from the center of gravity, perpendicular to the axis, the motion of which will be the same as if the whole mass were concentrated at that point. This point is called the center of oscillation . The radius of oscillation is the distance between the center of oscillation and the point of suspension. In a straight line, or in a bar of small diameter suspended at one end and oscillating about it, the center of oscillation is at 2 ⁄ 3 the length of the rod from the end by which it is suspended. When the vibrations are perpendicular to the plane of the figure, and the figure is sus pended by the vertex of an angle or its uppermost point, the radius of oscillation of an isosceles triangle is equal to 3 ∕ 4 of the height of the triangle; of a circle, 5 ∕ 8 of the diameter; of a parabola, 5 ∕ 7 of the height. If the vibrations are in the plane of the figure, then the radius of oscillation of a circle equals 3 ∕ 4 of the diameter; of a rectangle suspended at the vertex of one angle, 2 ∕ 3 of the diagonal. Center of Percussion.— For a body that moves without rotation, the resultant of all the forces acting on the body passes through the center of gravity. On the other hand, for a body that rotates about some fixed axis , the resultant of all the forces acting on it does not pass through the center of gravity of the body but through a point called the center of
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