Main axis of inertia

A main axis of inertia , often also abbreviated main axis , of a body is an axis of rotation around which the body can continue to rotate without a dynamic imbalance occurring. The direction of the axis therefore remains constant without an external torque having to act.

At least three main axes of inertia always go through every point inside or outside the body. Most of the time, however, the term is only used for the main axes of inertia that go through the center of gravity of the body.

In the case of rotary axes that do not go through the center of gravity, the static unbalance always occurs regardless of the possible dynamic unbalance . This is not expressed by a torque on the axis, but by a force that does not act in the sense of a change in direction, but in the sense of a parallel displacement of the axis.

Relationship with the main moment of inertia

The moment of inertia associated with a principal axis of inertia is called a principal moment of inertia of the body.

Are the three main moments of inertia for a particular point the same for a body, such as B. for the center of a sphere or a cube, then every other axis through this point is also a main axis of inertia and has the same main moment of inertia.

For bodies with less rotational symmetry , at most two main moments of inertia are generally equal.

Main axes of inertia with different moments of inertia are perpendicular to each other. If all three main moments of inertia are different, then apart from the three relevant main axes of inertia, which are perpendicular to each other, there are no other.

For rotations about axes that run through the point in question, one of the three main moments of inertia is always the largest possible moment of inertia of the body, while another is the smallest possible.

More detailed description and examples

When a body rotates freely (i.e. without further application of a force or torque), the axis of rotation always passes through the center of gravity. If it is not a main axis of inertia, then the rotation is a tumbling movement , in which the direction of the axis is constantly changing both in space and in relation to the body. This is e.g. B. easy to demonstrate using cuboids with different side lengths.

On the other hand, free rotation around a main axis of inertia is ideally a stable state of motion. If all three main moments of inertia are not the same, the reaction to an external disturbance can be very different:

• If the disturbance is a small deflection of the axis of rotation from the direction of the main axis of inertia with the largest or smallest of the three main moments of inertia, then the axis of rotation moves around the relevant main axis of inertia, but always remains in its vicinity. You can see that B. the constant "egg" of a thrown football , which rotates quickly, but not exactly, around the longitudinal axis, which here is the axis with the smallest moment of inertia. The rotation around the axis with the mean moment of inertia is unstable.
• If the disturbance consists of a constant withdrawal of rotational energy and / or increasing angular momentum , then only the rotation around the axis with the greatest main moment of inertia is stable, because it enables the greatest angular momentum for a given rotational energy. This is e.g. This can be seen, for example, in stones that roll down a slope and seem to stand up "by themselves", even if they have a rather flat shape.

If the rotation is to take place around a fixed axis of rotation through the center of gravity, which is not a main axis of inertia, the direction of the axis must be kept constant by a bearing that exerts a torque on the axis. The required bearing forces increase with the square of the speed :

${\displaystyle F_{\text{Lager}}\sim n^{2}}$

In everyday life, this can be seen well with unbalanced car wheels or unevenly filled spin dryers . During balancing, the spatial mass distribution of the body is changed in such a way that the desired axis of rotation is made a main axis of inertia.

Explanation

The whole behavior is explained by the fact that only when a body rotates around one of its main axes of inertia is the angular momentum parallel to the axis of rotation and both maintain their direction without external forces.

When rotating around other axes, the angular momentum and axis of rotation form an angle. If the axis is then to remain fixed, the angular momentum must rotate with the body, i.e. change its direction, which, according to the principle of twist, can only be brought about by an external torque. But if there are no external torques, the angular momentum remains constant in terms of direction and magnitude, so that the axis of rotation is now moved around it.

One finds the main axes of inertia of a body as the main axes of its inertia tensor , see calculation there.

The wobbling of the axis of rotation when a free body rotates around an axis that is not the main axis of inertia can also be explained in the co-rotating reference system : in this, all rotating parts of the body generate centrifugal forces , which together can form a torque around the center of gravity. If this is nonzero, it will tilt the axis. In contrast, when rotating around a main axis of inertia, the moments of the centrifugal forces add up to zero. This is shown by the fact that the corresponding moments of deviation ( secondary diagonals in the inertia tensor) are zero.

literature

• Holzmann / Meyer / Schumpich - Technical Mechanics Volume 2 , BG Teubner Stuttgart
• Technical mechanics , Martin Mayr, Hanser-Verlag, ISBN 3-446-22608-7
• Classical mechanics , Herbert Goldstein, Charles P. Poole, John L. Safko, Wiley-VCH Weinheim 2006
• Technical mechanics 2. Elastostatics , Christian Spura, Springer Verlag, 2019, ISBN 978-3-658-19979-1

Individual evidence

1. ↑ Carsten Timm: Theoretical Mechanics , Chapter 5.3.1 Rotation about Free Axes , July 18, 2011, Technische Universität Dresden, Institute for Theoretical Physics, accessed on February 3, 2017
2. ↑ Brandt, Dahmen: Mechanics: An introduction to experiment and theory . 3. Edition. Springer, 1996, ISBN 978-3-540-59319-5 , pp. 174 ( limited preview in Google Book search).