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Space and Body Cones: Object Undergoes The Same Motion As The Body Cone

This document summarizes three-dimensional kinematics and dynamics of rigid bodies using space and body cones and Euler angles. It describes: 1) How a precessing object undergoes the same motion as a body cone rolling on the surface of a fixed space cone, with the nutation angle related to the angle between the cones. 2) How Euler angles define the orientation of a rigid body using successive rotations about a body-fixed coordinate system and a space-fixed coordinate system, including precession, nutation, and spin angles. 3) Diagrams illustrating space and body cones modeling precession and the Euler angle rotations of a rigid body from an initial to final orientation.

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Kacou Fatim
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226 views1 page

Space and Body Cones: Object Undergoes The Same Motion As The Body Cone

This document summarizes three-dimensional kinematics and dynamics of rigid bodies using space and body cones and Euler angles. It describes: 1) How a precessing object undergoes the same motion as a body cone rolling on the surface of a fixed space cone, with the nutation angle related to the angle between the cones. 2) How Euler angles define the orientation of a rigid body using successive rotations about a body-fixed coordinate system and a space-fixed coordinate system, including precession, nutation, and spin angles. 3) Diagrams illustrating space and body cones modeling precession and the Euler angle rotations of a rigid body from an initial to final orientation.

Uploaded by

Kacou Fatim
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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522 Chapter 20 Three-Dimensional Kinematics and Dynamics of Rigid Bodies

Space and Body Cones


Z
A way to visualize moment-free Z Space
steady precession. The space Space u cone
u cone
cone is fixed in space. The body
cone rolls on the surface of the z
y z
b
space cone. The precessing
object undergoes the same f
b
motion as the body cone. The
Body
angle b is related to the nutation
Y cone
angle by
Izz
tan b # ! "
Ixx
tan u. Body
cone
c x

X
Z
z
u Space
b cone
y
If b $ u, the interior surface of
the body cone rolls on the
exterior surface of the space
cone.
Y
Body
cone
c x

Euler Angles for an Arbitrary Object


Z, z Z , z!

y!

Y, y Y

The xyz coordinate system is body


fixed and aligned with the primary X, x X c x!
XYZ coordinate system in the initial
(a) (b)
orientation (a). The precession
angle c is a rotation of the object
about the Z axis (b). The nutation Z , z! y
y" y"
angle u is a rotation of the object
about the x¿ axis (c). The spin angle
z", z x
f is a rotation of the object about u
the z– axis (d). z"
y!
f
Y

x"
X c x!, x"

(c) (d)

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