Poinsot's ellipsoid

An ellipsoid of inertia is an illustration of the inertial property of a rigid body. It is the normalized form of the Energieellipsoids.

Energieellipsoid

The energy of rotation of a rigid body is a quadratic form of the components of the angular velocity. A positive definite ( the principal moments of inertia are non- negative) quadratic form generally describes a triaxial ellipsoid, which here means Energieellipsoid:

(: Rotational energy: inertia tensor, : angular velocity) is announced The square shape componentwise

Since is constant, the tip of the vector moves on the surface of the ellipsoid, if the start point of the vector is at the origin ( = Ellipsoidzentrum ).

If a surface is given by, so the gradient is at any point of this surface perpendicular to the surface. The result for the considered here Energieellipsoid:

Thus, it follows that the angular momentum is always perpendicular to the Energieellipsoid. The angular momentum is therefore parallel to the normal of the ellipsoid at the point where the tip of the angular velocity vector touches the ellipsoid. Thus be seen that, and only along the principal axes of inertia are in parallel. The six independent components of the inertia tensor correspond to the three principal moments of inertia and the orientation of the principal axes of inertia, ie, the shape and orientation of the ellipsoid.

Using a principal axis transformation, one obtains the principal moments of inertia ( eigenvalues ​​of the inertia tensor ) and the principal axes of inertia ( eigenvectors ). The rotational energy is obtained in the new coordinates ( the principal axes of inertia are the basis vectors of the rotated coordinate system):

Or formed

From the last equation, the semi-axes of Energieellipsoids can be read directly:

Of inertia

The moment of inertia with respect to an arbitrary axis of rotation is:

With the vector one obtains a quadratic form with respect to the ellipsoid of inertia:

This normalization does not change the shape of the Energieellipsoids, but only its size. That reflects the inertia ellipsoid of inertia properties of a rigid body and is independent of its movement. An arbitrary rotation axis intersects the ellipsoid at a distance from the center of the ellipsoid.

Componentwise wrote:

In the plane spanned by the principal axes coordinate system ( principal axis transformation ), the ellipsoid of inertia writes with the principal moments of inertia:

The corresponding energy and Trägheitsellipsoide have the same orientation in space, but the lengths of the principal axes are different:

  • Of inertia:
  • Energieellipsoid:

Trägheitsellipsoide special body

  • Unbalanced centrifugal own a ' real ' ellipsoid of inertia than since. Example: Rectangular with three unequal sides or angled molecules such as NO2.
  • Symmetric top have an ellipsoid of revolution ellipsoid of inertia than as two principal moments of inertia are equal, eg. For rotationally symmetric bodies, the axis of symmetry is always a principal axis of inertia, the two principal moments of inertia about any axis perpendicular to it are the same. Examples: circular cylinder, linear molecules. It should be noted however, that not every symmetric top is rotationally symmetrical. Example: Square column. Prolate gyro has an elongated ellipsoid of revolution along the axis of symmetry of the ellipsoid. Example: cigar.
  • Oblate rotor is equipped with a shrink spheroid along the symmetry axis of the ellipsoid. Examples: Puck, approximately the oblate Earth.
  • Spherical top or spherical gyroscope have a ball as the ellipsoid of inertia since. Has a body with respect to three different axes equal moments of inertia, the ellipsoid of inertia must be a ball. As a result, the moment of inertia with respect to any axis is the same. However, the shape of the body does not have to be that of a sphere. For example, a cube three equal moments of inertia with respect to the surface normal at the midpoint of the side surfaces. Consequently, there is a spherical top and the moment of inertia around the space diagonal is also the same. Also, a regular tetrahedron a spherical top.
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