Rigid body motion

موضوع: Rigid body motion السبت 27 أكتوبر 2007, 11:13 am

Rigid body
motion

A rigid body is a system of particles in which the distances between the
particles do not vary. To describe the motion of a rigid body we use two
systems of coordinates, a space-fixed system X, Y, Z, and a moving system x, y,
z, which is rigidly fixed in the body and participates in its motion.

Let the origin of the body-fixed system be the
body’s center of mass (CM). The orientation of the axes of that system
relative to the axes of the space-fixed system is given by three independent angles.
The vector R points from the origin of the spaced-fixed system to the CM
of the body. Thus a rigid body is a mechanical system with six degrees of freedom.

Let r denote the position of an arbitrary
point P in the body-fixed system. In the space fixed system its position
is given by r + R, and its velocity is

v = d(R + r)/dt =
dR/dt + dr/dt = V + W
´ r.

Here V is the velocity of the CM and W is the angular velocity of the body. The
direction of W is along the axis of rotation
and W = df/dt.

The kinetic energy of the body is

T = (1/2)Smivi2 = (1/2)Smi(V + W
´ r)2.

We rewrite

T = (1/2)MV2 +
(1/2)Smi(W2ri2 - (W×ri)2), M = Smi,
Smiri =
0.

We find T = TCM + Trot,
i.e. the kinetic energy is the sum of the kinetic energy of the motion of the
CM and the kinetic energy of the rotation about the CM. In
component form we write

Rigid body motion Clip_image001

.

Here

is the inertia tensor. The Wi are the components of W along the axis of the body fixed system.
For a continuous system Rigid body motion Clip_image003

.

By appropriate choice of the orientation of the
body-fixed coordinate system the inertia tensor can be reduced to diagonal
form. The directions of the axes xi are then called the principal axes of inertia and the diagonal components
of the tensor are then called theprincipal moments of inertia. Then

Rigid body motion Clip_image004

.

Definitions:

Asymmetrical top:

Rigid body motion Clip_image005

Symmetrical top:

Rigid body motion Clip_image006

Spherical top:

Rigid body motion Clip_image007

Let L denote the angular momentum about the CM of the body.

Rigid body motion Clip_image008

,

which in component form yields

Rigid body motion Clip_image009

.

If x1, x2, and x3
are the principal axes of inertia, then

L1 = I1W1, L2 = I2W2, L3 = I3W3.

The Lagrangian of a rigid body is

Rigid body motion Clip_image010

.

The equations of
motion

The equations of motion of a rigid body are dP/dt
= F, where P = total momentum and F = external forces, and
dL/dt = t , where L = angular
momentum about CM and t = total torque produced by
external forces.

Let A be an arbitrary vector and dA/dt
its rate of change with respect to the space fixed axes, d'A/dt its rate
of change with respect to the body fixed axes.

dA/dt = d'A/dt +
W ´ A.

Therefore

dP/dt = d'P/dt +
W ´ P = F and dL/dt = d'L/dt
+ W ´ L = t .

Let the body fixed axes be the principal axes of
inertia of the body. Then

Li = IiWi and d'Li/dt = IidWi/dt,

and we have Euler’s equations:

Rigid body motion Clip_image012

The Eulerian angles

The orientation of the body-fixed coordinate system
with respect to the space-fixed coordinate system is described by three
angles. These angles are often taken as the Eulerian angles, defined in
the figure.

Rigid body motion Clip_image015

We can express the components of the angular
velocity W along the body-fixed axes x,
y, z in terms of the Eulerian angles and their derivatives.

Formulas

Center of mass:

Rigid body motion Clip_image019

رابط هذه الصفحة

http://electron6.phys.utk.edu/phys594/Tools/mechanics/summary/rigid/rigid.htm