Chapter 11 Dynamics of Rigid Bodies

11.2 Inertia Tensor Rigid body in rotation For a rigid body, each particle does not move in the frame attached to the body, thus

fixed rot

thus switch notation note

use

Define with which is a generalization of

 is a 2nd rank tensor can be treated as a 3X3 matrix. noted  units [mass][length] 2 called Inertia tensor Clearly

 11,  22,  33 moments of inertia about x 1,x 2, and x 2 axes. -  12,  13, … products of inertia Note  ji =  ij Clearly, generalization leads to

Example Cube of uniform density b b b x1x1 x2x2 x3x3

11.7 Eulerian Angles Three angles are needed to define a full rotation matrix. Eulerian angles defined as follows: 1) First rotate ccw through  about x 3

2) 2nd rotate ccw through  about x  1

3) 3rd rotate ccw through  about x  3 thus

In the body system (x i )

Euler Eqs for force free motion

Notes: The motion of a rigid body depends on its structure/shape only through three numbers: I 1, I 2, I 3. Two different bodies with different shapes but identical principal moments of inertia will move in the same manner. Friction effects may however vary with shape details. The simplest geometrical shape with 3 different moments is a homogeneous ellipsoid. The motion of any rigid body can thus be represented by the motion of an equivalent ellipsoid.

Example: Dumbbell of Section Find the torque required to keep the motion. 0

0

11.9 Force-free motion of a symmetric top Assume, without loss of generality, the body’s center of mass is at rest, and at the origin of the fixed system.

A solution of the form:

Case of the Earth The Earth’s rotation axis is NOT perfectly aligned with one of its principal axes. - Only a slight deviation. The Earth is slightly flattened near the poles because of centrifugal effects - its shape is that of an oblate spheroid.

The observed precession has an irregular period of  50% greater. The deviation arises because (1)the Earth is not rigid; (2) its shape is not exactly that of an oblate spheroid but rather a higher order deformation (it actually resembles a flattened pear). The bulge (of the Earth) and the inclined orbit (23.5º) produces a gravitational torque which produces a slow precession of the Earth’s axis with a period of 26,000 years.