Pseudoregular precession

In gyroscopic theory, pseudoregular precession is a gyroscopic movement that apparently resembles the regular precession of symmetrical gyroscopes, but where, on closer inspection, small, fast, overlapping oscillations that are barely perceptible to the naked eye take place around the precession cone. The angular momentum is oriented close to the main axis with the largest or the smallest main moment of inertia , and the top rotates around this main axis so fast that all of its other rotations, which it also performs, are slow compared to it.

The pseudoregular precession is an important point of the gyroscopic theory and has attracted the greatest interest from natural philosophy because of the frequency of its occurrence and its paradoxical properties . The pseudoregular precession occurs in more general circumstances than the regular precession. The vibrations of the axis of rotation are called nutations after a word borrowed from astronomy .

The term was coined by Felix Klein and Arnold Sommerfeld in 1898.

Paradox of pseudoregular precession

With a whip top or top that is set in rapid rotation by pulling a cord and then left to its own devices on a horizontal surface in the gravitational field of the earth, the figure axis revolves around a circular cone around the vertical at the same speed as in a regular precession . This fact is doubly astonishing because

1. In spite of all expectations, the points on the figure axis follow a circular path in a perpendicular, horizontal plane under the action of the perpendicular force of gravity and
2. the regular precession theoretically only occurs under special circumstances, but it appears in experiments with any choice of starting conditions.

While the former is a natural consequence of the interaction between gravity and angular momentum , the objection to the latter view is that the observation result is not precise. This is because the gyroscopic motion has only an external resemblance to the regular precession, because the small oscillations mentioned at the beginning are ignored. In contrast to the regular precession, the observed process is called pseudoregular. The illusion of resemblance to regular precession is reinforced by the fact that pseudoregular precession actually quickly changes into regular precession through secondary effects such as friction and elastic deformation. In addition, the initial conditions set for the whip or throwing top are by no means arbitrary, on the contrary, quite special: the powerful pulling of the cord always provides the top with a considerable angular momentum, which is aligned close to the figure axis. The frequency of its occurrence thus owes the pseudoregular precession to a selection or design effect .

Symmetrical tops

Lagrange top

In the pseudoregular precession of the symmetrical Lagrange top , the angular momentum is large and aligned near the figure axis . Proximity here means more precisely that the oscillations of the locus of the figure axis on the unit sphere cannot be perceived with the naked eye. The angular momentum L is of sufficient magnitude if L ²  > 100CGs, where C is the main axial moment of inertia , G is the weight and s is the distance between the center of mass and the support point. Because the angular momentum is dominated by the inherent angular momentum L ₃ , instead of the amount L of the angular momentum, the inherent angular momentum L ₃ can also be used for qualification.

The cone of rotation of the angular momentum, which in pseudoregular precession is close to the figure axis

The pseudoregular precession can be represented in a clearly simplified way, see picture. Because the moment of weight is always perpendicular to the figure axis, which is assumed in the vicinity of the angular momentum, the rate of change of the angular momentum is approximately perpendicular to it according to the principle of twist . Therefore, the angular momentum moves in good approximation on a cone, the precession cone , around the vertical precession axis with a precession angular frequency according to ${\displaystyle {\vec {M}}={\dot {\vec {L}}}}$ ${\displaystyle {\vec {\Omega }}}$

${\displaystyle {\vec {M}}={\dot {\vec {L}}}={\vec {\Omega }}\times {\vec {L}}}$

This is analogous to where the speed for a pure rotation with angular speed results from the distance to a fixed point on the axis of rotation. In components with and is written that ${\displaystyle {\vec {v}}={\vec {\omega }}\times {\vec {r}}}$ ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {\omega }}}$ ${\displaystyle {\vec {r}}}$ ${\displaystyle {\vec {M}}=Gs\sin(\alpha ){\hat {e}}_{N},\;{\vec {\Omega }}=\Omega {\hat {e}}_{z},\;{\vec {L}}\approx C\omega _{3}{\hat {e}}_{3}}$ ${\displaystyle {\hat {e}}_{z}\times {\hat {e}}_{3}=\sin(\alpha ){\hat {e}}_{N}}$

${\displaystyle M=Gs\sin(\alpha )=\Omega C\omega _{3}\sin(\alpha )\quad \rightarrow \quad \Omega ={\frac {Gs}{C\omega _{3}}}}$

Here ê _{N, z, 3 are} the direction vectors of the nodal line ê _z  × ê ₃ , the plumb line or the figure axis. The precession frequency Ω thus increases with the moment and is the smaller the greater the intrinsic angular momentum L ₃  = Cω ₃ around the figure or 3-axis. The precession frequency corresponds to that of the regular precession of the high-speed gyroscope , into which the gyro rapidly changes due to secondary effects such as friction and elastic deformation. Because the figure axis is only near the angular momentum vector, it quickly revolves around it on a narrow cone.

The rotation of the locus of the figure axis around the angular momentum vector can be approximated to the unit sphere by means of a cycloid in a tangential plane :

${\displaystyle \xi =r\omega t+\rho \sin(\omega t),\;\eta =r+\rho \cos(\omega t)}$

With

${\displaystyle r={\frac {AGs\sin(\vartheta _{0})}{L_{3}^{2}}},\;\rho ={\frac {L_{z}-L_{3}\cos(\vartheta _{0})}{L_{3}\sin(\vartheta _{0})}}-r,\;\omega ={\frac {L_{3}}{A}}}$

Here, ξ is the coordinate parallel to the circle of latitude on which the angular momentum approximately revolves, r is the radius of the circle, ω is the speed of the figure axis around the angular momentum on the circle and η is the deflection perpendicular to the circle of latitude. The index 0 marks the mean value of the angle of inclination.

The cosine u = cos (ϑ) of the inclination angle ϑ and the precession angle ψ are approximate

${\displaystyle u=u_{0}+{\hat {u}}\sin(\omega t),\;\psi ={\frac {Gs}{L_{3}}}t+{\frac {\hat {u}}{1-u_{0}^{2}}}\cos(\omega t)}$

In it is

${\displaystyle {\hat {u}}={\frac {L_{z}-L_{3}u_{0}}{L_{3}}}-{\frac {AGs}{L_{3}^{2}}}(1-u_{0}^{2}),\;\omega ={\frac {L_{3}}{A}}}$

The Spielkreisel can in the same size and the same initial conditions , an equally fast pseudo regular precession run, but make him both the center of mass and the contact point of fine fluctuations, so no more talk of a nutation cone be

Goryachev Chaplygin roundabout

If the Gorjatschew-Tchaplygin centrifugal the angular velocity of the center of gravity axis from the base is very large for emphasis, then the gyro moves similar to the pseudo regular precession of the Lagrange gyroscope.

Asymmetrical tops

The general asymmetrical top can only perform a uniform rotation in the form of the Staude rotation , because the Griolic precession is only possible with a special mass distribution. The asymmetrical top can execute a pseudoregular precession if it rotates so rapidly around a main axis of inertia e that all other rotations it performs are slow compared to this. The resulting precession is similar to that of a symmetrical top and also takes place with the same precession speed. However, this movement is only stable if - as with the Euler gyro - said main axis of inertia e has the largest or the smallest main moment of inertia.

Around this precession the axis of rotation of the top executes approximately epicycles , as they were used to explain the planetary orbits well into the 17th century. Four spherical circles roll on each other at a uniform angular speed, the first circle remaining on a horizontal circle of latitude and the axis of rotation on the fourth.

If the rapid self-rotation p around the 1-axis takes place with the main moment of inertia A, then the first circle rolls at the same speed

${\displaystyle \Omega ={\frac {Gs_{1}}{Ap}}}$

with which the # Lagrange top precesses pseudoregularly, where G is the weight force and s _{1 is} the distance between the center of mass and the support point along the 1-axis.

“Two of the four nutations of the fast asymmetrical top, namely the two with the orbital speeds p + σ and p - σ are dependent on the initial conditions, the ratio b / c of their spherical widths, however, only on the main rotating masses . The other two nutations cannot be influenced from outside; one of them takes place at the speed of rotation of the top, originating from the remote position (s ₂ , s ₃ ) of the center of gravity, and it disappears when the center of gravity is either on the axis of rotation or when it precesses horizontally; the other takes place at twice the speed of rotation of the top, stems from the difference in the rotating masses B and C of the other two main axes and disappears when either these two rotating masses are equal or the center of gravity is in the plane of the support point normal to the axis of rotation, and it is at greatest when the axis of rotation precesses horizontally. "

- Richard Grammel : Der Kreisel, his theory and his applications (1950) p. 211.

The sizes mentioned are

${\displaystyle \sigma =p{\sqrt {\frac {(A-B)(A-C)}{BC}}},\;{\frac {b}{c}}={\sqrt {{\frac {C}{B}}{\frac {C-A}{B-A}}}}}$

The parameters b and c are assumed to be small and remain indefinite in this context as long as their ratio has the required value. The radicands are positive if A is the largest or smallest of the main moments of inertia or rotating masses.

Web links

Commons : Gyroscope  - collection of images, videos and audio files

Individual evidence

1. ↑ Grammel (1950), pp. 75ff.
2. ^ Grammel (1950), p. 205.
3. ↑ Klein and Sommerfeld (2010), p. 291.
4. ^ Grammel (1950), p. 76.
5. ↑ Klein and Sommerfeld (2010), pp. 209 and 291.
6. ↑ Klein and Sommerfeld (2010), p. 291 f.
7. ↑ Klein and Sommerfeld (2010), p. 297.
8. ↑ Grammel (1950), p. 77.
9. ↑ Leimanis (1965), p. 92.
10. ↑ Grammel (1950), p. 204 ff.

literature

• K. Magnus : Kreisel: Theory and Applications . Springer, 1971, ISBN 978-3-642-52163-8 , pp. 20 ( limited preview in Google Book Search [accessed February 20, 2018]).
• R. Grammel : The top . Theory of the gyro. 2. revised Edition volume 1 . Springer, Berlin, Göttingen, Heidelberg 1950, DNB  451641299 , p. 75 ff., 204 ff . or
  R. Grammel : The top . Its theory and its applications. Vieweg Verlag, Braunschweig 1920, DNB  451641280 , p. 61 ff., 134 ff . ( archive.org - "swing" means angular momentum, "torsional shock" torque and "torsional balance" rotational energy).
• F. Klein , A. Sommerfeld : The Theory of the Top . Development of the Theory in the Case of the Heavy Symmetric Top. Vol. II. Springer, New York 2010, ISBN 978-0-8176-4824-4 , pp. 291 , doi : 10.1007 / 978-0-8176-4827-5 (translation of the original from 1897, see p. IX.).
• Eugene Leimanis: The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point . Springer Verlag, Berlin, Heidelberg 1965, ISBN 978-3-642-88414-6 , p. 92 , doi : 10.1007 / 978-3-642-88412-2 (English, limited preview in Google Book Search [accessed on March 21, 2018]).