ME 4590 Dynamics of Machinery
Equations of Motion for Fixed Axis Rotation
The figure at the right shows a rigid body rotating
about a fixed vertical axis. As the body rotates,
bearing loads may be generated in the X  and Y 
directions resulting from 1) the mass center G being
off the axis of rotation, and 2) the asymmetric shape
of the body. The forces associated with these effects
may be calculated using the equations of motion
presented below.
In general, if the mass center G is not on the axis
of rotation, then the Newton/Euler equations of
motion are
 F  ma
i
G
 m(r i   r 2 j)
i
and
 M 
G i
 IG   B   B  HG
i
or
 M 
P i
 I P B  B  H P
i
where
H G  IG   B
H P  IP B
and where P is any point on the axis of rotation, and I G and I P are inertia matrices for a
set of axes passing through G and P, respectively.
If the mass center G is on the axis of rotation then, it can be shown, that the
equations of motion can be written
F
i
i
0
and
 M 
P i
 I P   B   B  H P  IG   B   B  HG
i
Here again, P is any point along the axis of rotation.
Kamman – ME 4590: page 1/1