1-56 Section 1Equations of Motion in Three Dimensions The equations of motion for a rigid body in three dimensions are extensions of the equations previously stated.. 1.3.78 where aC = a
Trang 11-56 Section 1
Equations of Motion in Three Dimensions
The equations of motion for a rigid body in three dimensions are extensions of the equations previously stated
(1.3.78)
where aC = acceleration of mass center
HC = angular momentum of the body about its mass center
xyz = frame fixed in the body with origin at the mass center
Ω = angular velocity of the xyz frame with respect to a fixed XYZ frame
Note that an arbitrary fixed point O may be used for reference if done consistently.
Euler’s Equations of Motion
Euler’s equations of motion result from the simplification of allowing the xyz axes to coincide with the
principal axes of inertia of the body
(1.3.79)
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
(1.3.80)
These equations are valid in general Some common cases are briefly 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 fixed point The governing equations are valid for a body rotating about a noncentroidal fixed point O The reference axes xyz must pass through the fixed point to allow using a set of moment equations that do not involve the unknown reactions at O.
Motion of a body about a fixed 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
F a
M H H H
∑
∑
=
m C
C ˙C ˙C xyz Ω C
∑
∑
∑
˙
˙
˙
F x ma C F y ma C F z ma C
∑
∑
∑
˙
˙
˙