rigid body motion

موضوع: rigid body motion السبت ماي 17, 2008 8:10 pm

[center]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

.

Here

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

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 the principal moments of inertia. Then

.

Definitions:

Asymmetrical top:
Symmetrical top:
Spherical top:

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

,

which in component form yields

.

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 [/center].
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موضوع: رد: rigid body motion الخميس ماي 22, 2008 9:21 pm

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