Nutation (physics)

This article covers nutation in gyro theory. For the meaning in astronomy, see nutation (astronomy) .

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Nutation of a rotating CD player in weightlessness

The nutation is the movement of the body axis of a force-free top when the angular momentum not parallel to one of the main axes is aligned with the gyroscope.

In a symmetrical top, the nutation causes the figure axis to sweep over a cone with the angular momentum as the axis. Due to the conservation of angular momentum , the angular momentum remains constant in terms of amount and direction.

Nutation can be provoked by bumping into a top that rotates stably around its axis.

In addition to nutation, a top, on which a torque acts, can also perform a precession movement.

Symmetrical top

The symmetrical top is an important special case that simplifies the consideration of nutation.

A further simplification arises when the reference system for momentary observations is aligned with the gyro. A coordinate axis  (z) lies along the figure axis, so that the inertia tensor appears as a diagonal matrix . The next coordinate axis x is chosen so that a plane is spanned in which the angular momentum is located, i.e. the angular momentum vector takes on the value zero in one dimension: L _y  = 0. ${\displaystyle {\bar {I}}}$

Vector decomposition of the motion parameters on the flattened top

Vector decomposition of the motion parameters on the extended top

Now only two variable components remain in the angular momentum law :

${\displaystyle {\vec {L}}={\bar {I}}\,{\vec {\omega }}}$.

Here it can be seen that the angular velocity is not parallel to the angular momentum fixed in space , but deviates from it and thus changes over time. ${\displaystyle \omega }$ ${\displaystyle {\vec {L}}}$

However, the movement behavior of the gyroscope can be described better by clever graphic vector decomposition. The vector component  ω _Fig is chosen so that it lies parallel to the figure axis, and the second vector component  ω _Nut such that it lies parallel to the angular momentum. Because when a symmetrical top rotates around its figure axis, neither its orientation in space nor the inertia tensor changes, the movement  ω _{Fig is} considered “neutral”.

In contrast, the angular velocity ω _groove is more exciting  ; with it, the gyroscope and the coordinate system defined at the beginning are pivoted around the angular momentum vector. This shows that the figure axis, angular momentum and angular velocity of the symmetrical top are in constant spatial relationship to one another and always lie in one plane. The figure axis and the angular velocity each sweep over the surface of a cone, the cone axis of which forms the angular momentum.

Using the graphic for vector decomposition, the following equations can be found:

${\displaystyle \omega _{\text{x}}=\sin \alpha \,\omega _{\text{Nut}}}$

${\displaystyle L_{\text{x}}=\sin \alpha \,L}$

Skillful insertion of the angular momentum principle results in:

${\displaystyle L_{\text{x}}=\omega _{\text{x}}\,I_{\text{x}}\Leftrightarrow \omega _{\text{x}}={\frac {L_{\text{x}}}{I_{\text{x}}}}={\frac {\sin \alpha \,L}{I_{\text{x}}}}=\sin \alpha \,\omega _{\text{Nut}}}$

If applies , then the following approximate calculation can be made: ${\textstyle \omega _{\text{z}}\gg \omega _{\text{x}}}$

${\displaystyle \omega _{\text{Nut}}={\frac {L}{I_{\text{x}}}}\approx {\frac {I_{\text{z}}}{I_{\text{x}}}}\,\omega }$

A flattened top that has been bumped will then flutter at a frequency above its rotational frequency . The high frequency usually dampens the nutation quickly , and the figure axis soon aligns itself with the angular momentum.

A detailed mathematical description of the gyroscopic motion is made possible by Euler's equations .

meaning

• Astronomical observations
• Atomic physics (e.g. MRI )

Web links

• Top with rotation, precession and nutation