Kinetics of Rigid Bodies in Three Dimensions
The concepts of plane rigid body motion can be extended to the
more complicated problems in three dimensions, such as of gyroscopes
and jet engines. This section brie?y covers some fundamental topics.
There are many additional topics and useful methods that are included
in the technical literature.
Angular Momentum in Three Dimensions
For analyzing three-dimensional angular momentum, three special
de?nitions are used. These can be visualized by considering a spinning top
(Figure 1.3.22).
Precession rotation of the angular velocity vector about the y axis.
Space Cone locus of the absolute positions of the instantaneous axis
of rotation.
Body Cone locus of the positions of the instantaneous axis relative
to the body. The body cone appears to roll on the space cone
(not shown here).
FIGURE 1.3.22 Motion of an inclined, spinning top.
Equations 1.3.73 provide the scalar components of the total
angular momentum.
Impulse and Momentum of a Rigid Body
in Three-Dimensional Motion The extension of the planar motion
equations of impulse and momentum to three dimensions is
straight-forward.
where G and H have different units. The principle of
impulse and momentum is applied for the period
of time t1 to t2,
Kinetic Energy of a Rigid Body in Three-Dimensional Motion
The total kinetic energy of a rigid body in three dimensions is
For a rigid body that has a ?xed point O,
Equations of Motion in Three Dimensions
The equations of motion for a rigid body in three dimensions
are extensions of the equations previously stated.
where aC = acceleration of mass center
HC = angular momentum of the body about its mass center
xyz = frame ?xed in the body with origin at the mass center
? = angular velocity of the xyz frame with respect to
a ?xed XYZ frame
Note that an arbitrary ?xed point O may be used for reference
if done consistently.
Eulers Equations of Motion
Eulers equations of motion result from the simpli?cation of allowing
the xyz axes to coincide with the principal axes of inertia of the body.
where all quantities must be evaluated with respect to the
appropriate principal axes.
Solution of Problems in Three-Dimensional Motion
In order to solve a three-dimensional problem it is necessary to
apply the six independent scalar equations.
These equations are valid in general. Some common cases are
brie?y stated. Unconstrained motion. The six governing equations
should be used with xyz axes attached at the center of mass of
the body. Motion of a body about a ?xed point. The governing equations
are valid for a body rotating about a noncentroidal ?xed point O.
The reference axes xyz must pass through the ?xed point to allow
using a set of moment equations that do not involve the unknown
reactions at O. Motion of a body about a ?xed axis. This is the
generalized form of plane motion of an arbitrary rigid body. The analysis
of unbalanced wheels and shafts and corresponding bearing reactions
falls in this category.
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