Symmetrical top

In the gyro theory, the symmetrical top is a top with two equal main moments of inertia . Typical symmetrical tops are the Lagrange top and many play tops . An important special case are homogeneous body of revolution , with respect to the on the figure axis give symmetrical reference points lying roundabout.

Symmetrical gyroscopes are widely used in the spin stabilization of ships ( ship gyroscopes ), spacecraft , gyroscopic instruments and inertial navigation systems and are used in astronomy and ballistics .

General

Every rigid body has three main moments of inertia and three associated main axes of inertia or, for short, main axes, which are determined by solving the eigenvalue problem of the inertia tensor .

• A top with three equal main moments of inertia is a special symmetrical top and is called a spherical top .
• A top with three different main moments of inertia is an asymmetrical top .

The equations of motion decisive for the rotation around a fixed point or the center of mass are the Euler gyroscopic equations or, for the heavy gyro, the Euler-Poisson equations .

The figure axis ê ₃ in the symmetrical top is its axis of symmetry, with respect to which it has the third or axial main moment of inertia. The ellipsoid of inertia is rotationally symmetrical about this axis . The main axes  ê _1,2 with the two corresponding equatorial main moments of inertia are perpendicular to the figure axis in the equatorial plane and are freely oriented there. Two axes are selected so that the main axes  ê _{1,2,3 form} a right-handed orthonormal system. Depending on whether the main axial moment of inertia is greater or less than the equatorial, the top is called flattened or stretched .

The symmetry does not require that the gyroscope be somehow symmetrical in the geometric sense . In particular, according to Steiner's theorem , the symmetry depends on the reference point; a top can therefore be a symmetrical top with respect to one point and an asymmetrical top with respect to another point.

The symmetry of the top is independent of the position of the center of mass . The Kovalevskaya and Goryachev Tschaplygin gyroscopes , each with a center of gravity away from the figure axis, are nevertheless symmetrical gyroscopes.

Main moments of inertia

The symmetrical top has a double, equatorial main moment of inertia A and a third, axial C. In a rigid body , the main moments of inertia satisfy the triangle inequalities

A + C> A and A + A> C

see moment of inertia . While the first inequality always applies, the second means 2A> C or A> C / 2. Then there can be a symmetrical top with the main moments of inertia A and C.

Gyroscopic or spin stabilization

→ Main article : Gyroscopic stabilization

Fig. 1: Flywheel to explain the gyroscopic effect

One of the technically most valuable properties of symmetrical gyroscopes is the ability to use them to stabilize bodies in their spatial alignment. This is used in ships, spacecraft and projectiles. The twist stabilization is based on gyroscopic effects .

For example, if the flywheel rotating around the y-axis is affected by a moment M _z perpendicular to the freely movable axis of rotation, which is not too great , then the flywheel does not begin to rotate around z, but instead oscillates around z using the oscillation equation

${\displaystyle {\ddot {\psi }}+\left({\frac {L}{A}}\right)^{2}\psi ={\frac {M_{z}}{A}}}$

Therein is L , the axial angular momentum to the figure axis. The oscillation equation is an approximation that is only valid for a small deflection ψ .

The free rotation of the figure axis around the equatorial axes is decisive for the stabilizing gyroscopic effect. If the axis of rotation is tied to the xy plane by bearings , the moments of inertia cannot develop their potential and twist stabilization does not occur.

In a more complicated mechanism, however, twist stabilization is not always possible. William Thomson, 1st Baron Kelvin and Peter Guthrie Tait were able to show

1. that only systems with an even number of unstable degrees of freedom can be gyroscopically stabilized, with indifferent degrees of freedom generally counting among the unstable ones,
2. that if there is no damping, the stabilization of an even number of unstable degrees of freedom can always be enforced and
3. that with existing damping gyroscopic stabilization is only possible with the help of artificially stimulated degrees of freedom.

The angles of rotation around the figure axis (more precisely the cyclic coordinates ) of the gyroscope are excluded from the degrees of freedom addressed here .

Angular velocity and angular momentum

With a symmetrical top, the angular velocity can advantageously be expressed in terms of the angular momentum :

${\displaystyle {\begin{aligned}{\vec {\omega }}=&p{\hat {e}}_{1}+q{\hat {e}}_{2}+r{\hat {e}}_{3}\\=&{\frac {1}{A}}(Ap{\hat {e}}_{1}+Aq{\hat {e}}_{2}+Cr{\hat {e}}_{3})+\left(1-{\frac {C}{A}}\right){\frac {Cr}{C}}{\hat {e}}_{3}\\=&{\frac {1}{A}}{\vec {L}}+{\frac {A-C}{AC}}L_{3}{\hat {e}}_{3}\end{aligned}}}$

Designate here

• ê _1,2,3 the main axes ,
• ${\displaystyle {\vec {\omega }}}$ the angular velocity,
• p, q, r = ω _{1,2,3 are} the angular velocities in the main axis system,
• A, C are the main moments of inertia in 1- / 2- or 3-direction,
• ${\displaystyle {\vec {L}}}$ the angular momentum,
• ${\displaystyle L_{3}={\vec {L}}\cdot {\hat {e}}_{3}}$ the angular momentum in 3-direction and in the following
• "×" the cross product and "·" the scalar product .

This shows that with a symmetrical top, the angular velocity, the angular momentum and the figure axis are always coplanar. The law of swirl also gives:

${\displaystyle {\begin{aligned}{\vec {M}}=&{\dot {\vec {L}}}={\frac {\mathrm {d} _{r}{\vec {L}}}{\mathrm {d} t}}+{\vec {\omega }}\times {\vec {L}}={\frac {\mathrm {d} _{r}{\vec {L}}}{\mathrm {d} t}}+{\frac {A-C}{AC}}L_{3}{\hat {e}}_{3}\times {\vec {L}}\\\rightarrow \;{\vec {M}}\cdot {\hat {e}}_{3}=&{\frac {\mathrm {d} _{r}{\vec {L}}}{\mathrm {d} t}}\cdot {\hat {e}}_{3}={\dot {L}}_{3}\end{aligned}}}$

This forms the relative time derivative in the main axis system. If the external moment has no component in the direction of the figure axis, which is the case with the symmetrical Euler top , the Lagrange top and the regular precession around the perpendicular, then the angular momentum around the figure axis is constant over time. ${\displaystyle {\tfrac {\mathrm {d} _{r}}{\mathrm {d} t}}}$ ${\displaystyle {\vec {M}}}$

The Euler gyroscopic equations following from the first line of equations are reduced to a single differential equation

${\displaystyle 2f(t){\ddot {p}}(t)+{\dot {f}}(t){\dot {p}}(t)+2p(t)=g(t)}$

where f and g are known functions of time t , if the torques M _1,2,3 in the principal axis system are known functions of time, and p is the angular velocity about a principal axis. The differential equation is further reduced to the integrable form

${\displaystyle {\frac {\mathrm {d} ^{2}p(\tau )}{\mathrm {d} \tau ^{2}}}+p(\tau )=G(\tau )}$

when the time is substituted for the new independent variable . ${\displaystyle \textstyle \tau =\int [f(t)]^{-{\frac {1}{2}}}\;\mathrm {d} t}$

Regular precession around the perpendicular

→ Main article : Regular precession

Fig. 2: Regular precession of a symmetrical top

As in the animation in Fig. 2, the symmetrical top performs a regular precession around the plumb line when it rotates at a constant angular velocity around a fixed precession axis and an axis fixed to the body, which enclose a constant angle. The gyroscopic effect of the axial angular momentum is directed against its speed and horizontally. With heavy gyroscopes, this gyroscopic effect can only be dynamically balanced by the horizontal torque of the vertical weight if

1. the center of mass lies on the figure axis or
2. the gyro has no self-rotation about the body axis and the center of mass in the body axis and the perpendicular direction spanned precession is located.

The nutation of the symmetrical Euler gyro can be understood as a moment-free special case of the first possibility if the vertical is aligned parallel to the angular momentum. The second case is a rotation of a perennial.

Perennial twists

→ Main article : Perennial rotation

Fig. 3: Carousel movement of a Kovalevskaya roundabout

If the center of mass is not on the figure axis, the symmetrical top can then perform a regular movement comparable to the regular precession if the center of mass is still in the plane of precession. This is the case, for example, with the Kovalevskaya roundabout , when, as in the animation in Fig. 3, it performs carousel movements, see main article.

Web links

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

• K. Lüders, RO Pohl, G. Beuermann, K. Samwer: Precession of a rotating wheel. ( MP4 ) Institute for Scientific Film (IWF) , 2003, accessed on April 13, 2018 (Film about a wheel precessing with a horizontal axis of rotation.). 
• K. Lüders, RO Pohl, G. Beuermann, K. Samwer: Gyro compass. ( MP4 ) Institute for Scientific Film (IWF) , 2003, accessed on December 6, 2019 (The principle of the gyro compass is demonstrated with the help of a spoked wheel on a swivel chair.). 
• K. Lüders, RO Pohl, G. Beuermann, K. Samwer: Stabilization with the help of a gyro ("monorail"). ( MP4 ) Institute for Scientific Film (IWF) , 2003, accessed on December 6, 2019 (The precession phenomenon is exploited to prevent a very top-heavy system from tipping over.). 

Individual evidence

1. ↑ Magnus (1971), p. 20, Grammel (1920), p. 39, see literature.
2. ↑ Magnus (1971), p. 20.
3. ↑ Magnus (1971), pp. 20 and 126. 21 years before Magnus, Richard Grammel demanded that the center of mass of the symmetrical heavy top should be on the figure axis, see Grammel (1920), p. 88 or Grammel (1950), p. 78.
4. ↑ Klein and Sommerfeld (1910), pp. 767f.
5. ↑ Grammel (1950), p. 261 f.
6. ↑ Rauch-Wojciechowski, Sköldstam and Glad (2005), p. 335.
7. ↑ Leimanis (1965), p. 7.

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 .
• R. Grammel : The top . Its theory and its applications. Vieweg Verlag, Braunschweig 1920, DNB  451641280 , p. 39 ( archive.org - "swing" means angular momentum, "rotary shock" means torque and "torsional balance" means rotational energy).
• F. Klein , A. Sommerfeld : Theory of the gyro . The technical applications of the gyro theory. Book IV. Teubner, Leipzig 1910, p. 767 ( archive.org [accessed October 21, 2017]).
• S. Rauch-Wojciechowski, M. Sköldstam, T. Glad: Mathematical analysis of the tippe top . In: Regular and Chaotic Dynamics . tape 10 , no. 4 . Springer Nature, 2005, ISSN  1468-4845 , p. 333–362 , doi : 10.1070 / RD2005v010n04ABEH000319 ( turpion.org [accessed December 15, 2018]).