07 Nutation (2)

Aim¶

To give a geometric description of nutation

Subjects¶

• 1Q50 (Gyros)

Diagram¶

Equipment¶

• Model (see Diagram and Figures).

Presentation¶

Watching a nutating object we observe that the body-axis makes a conical movement (see Nutation. This movement of the body-axis is visualized in our model by rotating the $L$-axis by hand (see Figures).

Explanation¶

$L$, wand body-axis are in one plane. While moving, $L$ remains fixed in space, so movement of that plane has to take place around $L$.

The fixed cone (so-called “space cone”) contains wand this $\omega$ turns around $L$. To show the position of $\omega$ with rerspect to the body axis, a cone around the body-axis (so-called body-cone) is visualized. This body-cone also contains $\omega$. So in our model the position of $\omega$ is seen where the two cones touch each other. While rotating, the movement of the body-axis around $L$ (nutation) and the movement of the momentary rotation-axis $(\omega)$ around the symmetry-axis can be observed.

Remarks¶

• The model is made in such a way that the two cones grip each other (teeth on the inside of the rim of the body-cone grip the wooden bars of the space-cone), so that the cones are not slipping. This is needed since there is only one $\omega$ while in our model $\omega$ is in two cones.

• Our model represents the movement of a disk-shaped nutating object $\left(I_{3}>I_{1}\right)$. Visualizing of a nutating bar-shaped object $\left(I_{1}>I_{3}\right)$ needs a model having the body-cone revolving with its outside around a fixed space-cone.

Sources¶

• Borghouts, A.N., Inleiding in de Mechanica, pag. 224-227

• Roest, R., Inleiding Mechanica, pag. 222-226 Nutation (2)