Parallel axis theorem

The Steiner set ( also set by Steiner, Steiner rule or parallel-axis theorem) goes back to studies by Jakob Steiner and is used to calculate the moment of inertia of a rigid body for shifted parallel axes of rotation.

The moment of inertia is not the sole property of a body, but depending on the considered axis of rotation. If the moment of inertia of an axis of rotation through the center of mass is known, then the set of Steiner the moment of inertia for all axes of rotation are parallel to this, are calculated.

The phrase is also used to determine area moments of inertia of beam cross-sections.

Application to inertia moments

Moments of inertia are usually tabulated for rotary axes through the center of mass. If the moment of inertia for a rotation axis parallel thereto is required, the Steiner theorem can be applied and the moment of inertia is given by:

In this case, the moment of inertia of the body is connected to ground of the axis of rotation ( substantially equal to the center of gravity) passes through its center of mass at a distance from and parallel to the rotational axis.

For the application of Steiner 's theorem two things to consider:

• The moment of inertia of a body is at its lowest when the rotation axis passes through the center of gravity. This follows from the fact that the Steiner's share is always positive, if one performs a shift of emphasis away.
• With repeated application of Steiner 's theorem, the moment of inertia can be calculated to any parallel axis, even if the initially given moment of inertia does not pass through the center of mass.

Application to area moments of inertia

If the centroid of a body cross-section is not the origin of the coordinate system, its area moment of inertia can be calculated using to Steiner's theorem:

For the distance of the surface center of gravity is the origin squared, multiplied by the area of the cross section and is added to ( recorded in a table ), geometrical moment of inertia. It is apparent that invisible with the Steiner term.

It is handy that one complex (eg, T - carrier ) in simple body can (eg rectangles ) to share with these formulas, the area moment of inertia is already known.

For then applies, for example:

Wherein the area of ​​the figure, and to the damage caused by the cutting member surfaces.

Generalization by inertia tensors

If the inertia tensor of gravity known, the inertia tensor results from the parallel -shifted by the vector coordinate system by the sum and the inertia tensor of a particle of mass with the position vector:

That is:

In which

Or in summation convention with the totally antisymmetric ε - tensor

Therefore, the property applies

By moving, it may happen that the axes of the new coordinate system no longer coincide with the principal axes of inertia by the new point.

Derivation of Steiner 's theorem

Considering a rigid body in a coordinate system whose origin coincides with its center of mass and defines the axis of rotation in the z- direction, the moment of inertia of this axis as

Defined. Wherein the sum of all the mass points of the body passes, the location of each mass point is denoted by and the rotational axis on the straight line is parallel to the z-axis through the point. Multiplying out the brackets leads to

, The first term corresponds to the moment of inertia of the rotational axis through the center of gravity ( parallel to the z -axis). The second and third terms are zero because they correspond to the definition of the center of mass and this is just the origin. The fourth term is by Pythagoras just the inverse square of the rotational axis to the origin multiplied by the total mass of the body under consideration to. If we write the distance as, then the Steiner theorem follows as