Perennial rotation

The Staude rotations according to Otto Staude are in the top theory uniform rotations of a heavy asymmetrical top around a vertical , fixed body but freely rotatable axis. No conditions are placed on the main moments of inertia and the location of the center of gravity . Under these circumstances, uniform rotations are only possible with a certain rotational speed around axes of rotation that lie on an elliptical cone fixed to the body , the Staude cone , which also depends largely on the position of the center of gravity and called the Staude itself the center of gravity cone .

The rotation of the asymmetrical top around a vertical main axis has a certain technical significance. Staude discovered this analytical solution to Euler's gyroscopic equations in 1894.

General

The gyro equations cannot be solved analytically in the general case of the asymmetrical heavy gyro. The three cases that can always be solved analytically with any initial conditions, the Euler top , the Lagrange top and the Kowalewskaja top each place three conditions on the mass distribution in the top. In the case of the perennial twists, four demands are made exclusively on the initial conditions .

Edward Routh proved in 1892 that gravity can only maintain a uniform precession of the asymmetrical top, in which a main axis OC describes a straight cone around the vertical, if the top rotates around a vertical, fixed axis. Because the torque of the centrifugal forces , in short the centrifugal moment in the rotating top, fluctuates at twice the frequency of the top's own rotation around OC, whereas the gravitational moment of the weight, regardless of the position of the center of gravity, pulsates at most with the single frequency of the natural rotation. Therefore, no uniform self-rotation about OC can take place. Routh further concluded that the axis of rotation must be vertical and solid.

The centrifugal moment is perpendicular to this axis with a uniform rotation around a fixed axis of rotation. The gravitational moment is always perpendicular to the vertical axis and in dynamic equilibrium with the centrifugal moment, which is why a spatially fixed axis of rotation must be perpendicular for a uniform rotation.

When rotating around a vertical body-fixed axis, the angular velocity is parallel to this and because the inertia tensor is constant in the body-fixed system , the angular momentum is also fixed there. The gyroscopic effect resulting from its rotation around the vertical axis is in dynamic equilibrium with the gravitational moment, which is why the angular velocity, the angular momentum and the center of gravity axis are coplanar from the reference point to the center of gravity. All angular velocities for which this applies form the center of gravity cone. It turns out that the center of gravity, which is not on any main axis, can only be at the highest or lowest point - i.e. on the axis of rotation - when the machine is stationary. In addition, the main axes are the only surface lines of the center of gravity cone that are not permanent axes of rotation, unless they carry the center of gravity. Then, however, the gyro can revolve around this axis at any speed.

The Griolische precession after MP Guljaew the only possible dynamic precession of the unbalanced gyroscope with autorotation. In that precession, neither the precession axis nor the angular velocity is fixed in space. In 1959, EI Harlamowa constructed a solution to the gyroscopic equations that represents the only dynamically possible precession of an asymmetrical body around a vertical axis. The angular velocity is also not constant here and the main moments of inertia must meet the condition C> 2A> 2B, which is why the solution does not describe a rigid body , because A + B> CMP Guljaew noticed that a rigid body with cavities, which with a incompressible ideal fluid filled that condition can adhere to.

Condition equation for the permanent vertical axes

Fig. 1: Rotational cone of angular momentum around the vertical axis of rotation.

The Staude top rotates with a uniform angular velocity Ω around a vertical , i.e. fixed, and at the same time body-fixed axis

${\displaystyle {\hat {n}}=n_{1}{\hat {e}}_{1}+n_{2}{\hat {e}}_{2}+n_{3}{\hat {e}}_{3}=\varepsilon {\hat {e}}_{z}}$

where n _{1,2,3 are} the constant coordinates of the body-fixed unit vector in the main axis system, points vertically upwards and ε = ± 1 indicates the orientation of the body-fixed axis. The angular velocity and the angular momentum are also fixed to the body. Here Θ is the inertia tensor of the top with respect to the support point. Because of the spatially fixed axis of rotation, the angular momentum describes a spatially fixed cone around the perpendicular direction, see Fig. 1. The change in angular momentum causes a gyroscopic effect , which can be specified in this case: ${\displaystyle {\hat {n}}}$ ${\displaystyle {\hat {e}}_{1,2,3}}$ ${\displaystyle {\hat {e}}_{z}}$ ${\displaystyle {\vec {\Omega }}=\Omega {\hat {n}}}$ ${\displaystyle {\vec {L}}:=\mathbf {\Theta } \cdot {\vec {\Omega }}=\Omega \mathbf {\Theta } \cdot {\hat {n}}}$ ${\displaystyle {\vec {K}}}$

${\displaystyle {\dot {\vec {L}}}={\vec {\Omega }}\times {\vec {L}}=\Omega ^{2}{\hat {n}}\times \mathbf {\Theta } \cdot {\hat {n}}=-{\vec {K}}}$

This is analogous to , where the speed for a pure rotation with angular speed results from the cross product × with the distance to a fixed point on the axis of rotation. According to the law of swirl , this change in angular momentum corresponds to the applied torque, which here is from the moment of gravity ${\displaystyle {\vec {v}}={\vec {\omega }}\times {\vec {r}}}$ ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {\omega }}}$ ${\displaystyle {\vec {r}}}$

${\displaystyle {\vec {M}}={\vec {s}}\times m{\vec {g}}=\varepsilon mg\,{\hat {n}}\times {\vec {s}}}$

is provided. This is the center of gravity, m is the mass, the acceleration due to gravity and g is its amount. The gyroscopic action and moment are in dynamic equilibrium ( ) with the consequence that the center of gravity, the rotational speed and the angular momentum are in one plane: ${\displaystyle {\vec {s}}}$ ${\displaystyle {\vec {g}}=-g{\hat {e}}_{z}=-\varepsilon g{\hat {n}}}$ ${\displaystyle {\vec {M}}+{\vec {K}}={\vec {0}}}$

${\displaystyle \Omega {\hat {n}}\times (\varepsilon mg{\vec {s}}-\Omega ^{2}\mathbf {\Theta } \cdot {\hat {n}})={\vec {\Omega }}\times \left(\varepsilon mg{\vec {s}}-\Omega {\vec {L}}\right)={\vec {0}}}$

If, in compliance with this condition, the gyro rotates around the vertical axis fixed to the body at angular velocity Ω , then it remains in the state of motion thus established. The components of the above vector equation in the main axis system can be resolved for Ω in the asymmetrical top, if none of the three n _1,2,3 vanishes: ${\displaystyle {\hat {n}}}$

${\displaystyle \Omega ^{2}=\varepsilon {\frac {mg(n_{3}s_{2}-n_{2}s_{3})}{(B-C)n_{3}n_{2}}}=\varepsilon {\frac {mg(n_{1}s_{3}-n_{3}s_{1})}{(C-A)n_{1}n_{3}}}=\varepsilon {\frac {mg(n_{2}s_{1}-n_{1}s_{2})}{(A-B)n_{2}n_{1}}}}$

Are in it

• s _1,2,3 the coordinates of the center of gravity,
• n _1,2,3 the coordinates of the perpendicular direction and
• A, B, C are the three main moments of inertia

in the main axis system fixed to the body. Of the two possible values ​​for the sign ε , only the one that leads to non-negative Ω² is valid, see also the section #Properties of the center of gravity cone . The sign of Ω, i.e. the direction of rotation, remains arbitrary. The angular speeds in the direction of the main axes result from the scalar product of the angular speed ε  Ω ê _z with the main axes ê _1,2,3 zu

ω _1,2,3 = Ω n _1,2,3 .

Staude's center of gravity plane and cone

Fig. 2: Staude's center of gravity (at ε = +1)

The perpendicular direction, the angular velocity, the center of gravity and the angular momentum are in one plane, the Staude's center of gravity plane, see Fig. 2. The following special cases can occur here:

1. The vertical is the main axis. If the center of gravity is also on the main axis, then rotations with any angular velocity are possible. Otherwise, Ω approaches infinity, because the sides of the triangle in the directions of the rotational speed and the angular momentum are then parallel. Because the gyroscopic moment of the centrifugal forces around the main axes disappears, a compensation of the gravitational moment cannot take place at finite rotational speeds.
2. The focus is on the vertical, which is not a major axis. Then Ω = 0.

Scalar multiplication of the above equation with or gives the late product ${\displaystyle {\vec {\Omega }}\times ({\color {blue}\varepsilon mg{\vec {s}}}-{\color {green}\Omega {\vec {L}}})={\vec {0}}}$ ${\displaystyle {\vec {s}}}$ ${\displaystyle {\vec {L}}}$

${\displaystyle ({\vec {\Omega }}\times {\vec {L}})\cdot {\vec {s}}=\Omega ^{2}({\hat {n}}\times \mathbf {\Theta } \cdot {\hat {n}})\cdot {\vec {s}}=0}$

or in components in the main axis system

${\displaystyle {\begin{vmatrix}s_{1}&s_{2}&s_{3}\\n_{1}&n_{2}&n_{3}\\An_{1}&Bn_{2}&Cn_{3}\end{vmatrix}}=(C-B)s_{1}n_{2}n_{3}+(A-C)s_{2}n_{1}n_{3}+(B-A)s_{3}n_{1}n_{2}=0}$

With fixed n _1,2,3 and variable s _1,2,3 this equation defines a plane, the plane of the center of gravity. Conversely, if the center of gravity is given with components s _1,2,3 and the n _1,2,3 free, then the equation represents an elliptical cone, the Staude's center of gravity cone. The center of gravity cone consists of the angular velocities which, with the associated angular momentum, span a plane that contains the center of gravity . In the symmetrical top, where two of the three main moments of inertia coincide, the cone degenerates into two mutually perpendicular planes, see the section #Symmetrical top below. ${\displaystyle {\vec {\Omega }}}$ ${\displaystyle {\vec {L}}}$ ${\displaystyle {\vec {s}}}$

Properties of the center of gravity cone

Fig. 3: Staudescher center of gravity cone

The equation, which is quadratic in the coordinates of the perpendicular direction, defines an elliptical cone for an asymmetrical top whose center of gravity is not on any main axis, which consists of two half-cones that are connected at their apex, see Fig. 3. An elliptical cone is clearly defined by five surface lines, which can be specified here. All three main axes of the top are surface lines (black), because then two of the coordinates n _{1,2,3 are} equal to zero. The focus (blue) and (purple) are also on surface lines. Staude's center of gravity cone can be constructed as soon as the mass distribution of the top is known. ${\displaystyle {\vec {s}}}$ ${\displaystyle {\vec {s}}\,'=\mathbf {\Theta } ^{-1}\cdot {\vec {s}}}$

According to Fig. 2 , the moment of gravity in the z-direction is equal to m g s sinϑ, if s is the distance between the center of gravity and the support point and ϑ is the angle of inclination between and the vertical. The gyroscopic effect of the angular momentum, on the other hand, is equal to Ω L sinα where α is the inclination of the angular momentum in relation to the perpendicular direction and L is the amount of the angular momentum that increases linearly with the speed of rotation: L = J Ω with J> 0. The gravity moment and gyroscopic effect are equal: ${\displaystyle {\vec {s}}}$

mgs sinϑ = J Ω² sinα

Of the two half-rays with angles α and π + α in radians , only one is permitted, depending on the sign of sinϑ. If the rotation axis now moves along the Staude cone, then sinϑ changes the sign when the rotation axis passes the center of gravity axis , and sinα when a main axis is reached. The two hemispheres are divided by the center of gravity axis and the three main axes into four areas each, which alternately contain the permissible and the impermissible half-ray of the axis of rotation. In Fig. 3, the sectors with the permitted half-rays are colored green and those with the impermissible rays are colored red. Curves drawn in red lie in the permissible sectors of the hemispheres and the distance from one of their points P to the tip of the cone Q is proportional to the possible angular velocity Ω when rotating around PQ. On the principal axes with α = 0 Ω grows over all limits if sinϑ ≠ 0. On the surface line containing the center of gravity, sinϑ = 0 and Ω = 0 if sinα ≠ 0. Find out more about the center of gravity cone, especially its degeneracies in the original work. ${\displaystyle {\vec {s}}}$

Stability of the perennial twists

The perennial twists can be stable or unstable. The investigation of stability requires many distinctions between cases and has at first turned out to be a task with almost hopeless difficulty. The solution can be given in full.

It turns out that a body that rotates stably as a force-free top can become unstable in certain speed ranges when it hangs, i.e. is statically stable. Conversely, a gyroscope that rotates unstably around its main axis with the medium-sized moment of inertia without force can also become stable at certain speeds under the effect of gravity if it is upright, i.e. statically unstable. This can also be confirmed experimentally. In the case of an asymmetrical top that revolves around a main axis, its main moment of inertia must not be greater than twice as large as the other two main moments of inertia, otherwise the rotation is unstable

Special cases

Special focal points

• If the center of gravity lies in a main plane , that is, a plane generated by two main axes, then the center of gravity cone breaks up into two planes, namely into this plane and one through the main axis perpendicular to it.
• If the center of gravity is on a main axis, then the cone breaks up into the two main planes that intersect in this main axis.

The following sections deal with the conditions in symmetrical or force-free gyroscopes.

Symmetrical tops

Fig. 4: Stable carousel movement of the Kovalevskaya roundabout

With symmetrical tops the center of gravity cone degenerates. If the center of gravity is on the figure axis of the top , it is a Lagrange top without any rotation and the center of gravity cone is no longer defined. The top, which is now a spherical pendulum , can precess regularly around any axis .

In the more general case, where the center of gravity is not on the figure axis of the top, the center of gravity cone degenerates into two mutually perpendicular planes: the equatorial plane perpendicular to the figure axis and the center of gravity plane, which contains the center of gravity and the figure axis. However, the equatorial plane is ruled out again, as it consists of nothing but main axes that do not carry the center of gravity. The center of gravity cone thus withers to the plane that contains the center of gravity and the figure axis.

Because the main axes are oriented in any way in the equatorial plane, the 1-axis can be placed so that the center of gravity is on it at a distance s ₁  > 0 from the figure axis.

The conditional equation for the permanent vertical axes leads to three equations with the equatorial main moments of inertia A = B in the main axis system of the gyro ${\displaystyle {\hat {n}}\times (\varepsilon mg{\vec {s}}-\Omega ^{2}\mathbf {\Theta } \cdot {\hat {n}})={\vec {0}}}$

ê ₁ : A [ ε κ c ₃ + Ω² (1 - κ) n ₃ ] n ₂  = 0

ê ₂ : A [ ε κ ( c ₁ n ₃ - c ₃ n ₁ ) + Ω² (κ - 1) n ₁ n ₃ ] = 0

ê ₃ : -ε mgs ₁ n ₂  = 0

In it mean

• κ = ^C ⁄ _A the ratio of the axial to the equatorial main moment of inertia and
• c _1.3 = mgs _1.3 / C are constants with the dimension T ^–2 .

The third equation immediately follows n ₂ = 0, which then also satisfies the first identity. The angular velocity around the z-axis is calculated from the second equation:

${\displaystyle \Omega ={\sqrt {\varepsilon {\frac {\kappa (c_{1}n_{3}-c_{3}n_{1})}{(1-\kappa )n_{1}n_{3}}}}}}$

The sign ε has to be chosen so that the radical under the root does not become negative. With the carousel movements of the Kovalevskaya gyroscope as in Fig. 4, κ = ½ and c ₃  = 0.

Fig. 5: Permissible (green) and impermissible (red) angles of inclination ϑ for the flattened top

When using the Euler angles ψ, ϑ and φ , see Euler angles in gyro theory , n ₂  = 0 can be satisfied with cos ( φ ) = 0 and η  : = sin ( φ ) = ± 1. The gravity axis from the reference point to the center of gravity at the distance s includes the angle λ with the 3-axis: s ₃  = η s  cos (λ) and s ₁  = η s  sin (λ). Then we get:

${\displaystyle \Omega ={\sqrt {\varepsilon {\frac {2mgs}{C-A}}{\frac {\sin(\vartheta -\lambda )}{\sin(2\vartheta )}}}}}$

With the flattened top ( C> A ) the radicand is in the areas

λ <  ϑ  <  ^π ⁄ ₂ , π <  ϑ  <λ + π or   ^3π ⁄ ₂  <  ϑ  <2π

positive. In Fig. 5 these permissible ranges are marked in green on the abscissa . With the straight top, C - A changes the sign and only the complementary, red marked areas are allowed.

Force-free top

With the force-free Euler gyro , g s _1,2,3  = 0 and the center of gravity cone is meaningless. The #conditional equation for the permanent vertical axes is reduced to ${\displaystyle {\hat {n}}\times (\varepsilon mg{\vec {s}}-\Omega ^{2}\mathbf {\Theta } \cdot {\hat {n}})={\vec {0}}}$

${\displaystyle {\hat {n}}\times \Omega ^{2}\mathbf {\Theta } \cdot {\hat {n}}={\vec {0}}}$

what is fulfilled when the top stands still or rotates around one of its main axes, see ellipsoid of inertia .

example

The center of gravity cone from Fig. 3 can be compared with the data from the table

A.	B.	C.	s 1	s 2	s 3
8th	11	16	-6	-1	-8th

construct in the main axis system as follows.

The cone equation is expressed using a symmetric matrix M :

${\displaystyle an_{2}n_{3}+bn_{1}n_{3}+cn_{1}n_{2}={\frac {1}{2}}{\begin{pmatrix}n_{1}&n_{2}&n_{3}\end{pmatrix}}\underbrace {\begin{pmatrix}0&c&b\\c&0&a\\b&a&0\end{pmatrix}} _{M}{\begin{pmatrix}n_{1}\\n_{2}\\n_{3}\end{pmatrix}}=0}$

With

${\displaystyle a=(C-B)s_{1},\quad b=(A-C)s_{2},\quad c=(B-A)s_{3}}$

The main axis transformation of this matrix succeeds with its characteristic polynomial

${\displaystyle \operatorname {det} (M-\lambda E)=-\lambda ^{3}+(a^{2}+b^{2}+c^{2})\lambda +2abc}$

where E is the identity matrix and det gives the determinant . The polynomial has the real zeros

${\displaystyle {\begin{aligned}\lambda _{i}=&r\cos \left({\frac {\varphi +2\pi (i-1)}{3}}\right),\quad i=1,2,3\\r=&{\sqrt {{\frac {4}{3}}(a^{2}+b^{2}+c^{2})}},\quad \varphi =\arccos \left({\frac {8}{r^{3}}}abc\right)\end{aligned}}}$

Here cos is the cosine and arccos is its inverse function . The eigenvalues λ _1,2,3 are sorted so that the first two have the same sign. The eigenvectors normalized to length one are calculated for the eigenvalues

${\displaystyle {\hat {v}}_{i}={\frac {1}{\sqrt {(\lambda _{i}^{2}-a^{2})^{2}+(ab+c\lambda _{i})^{2}+(ac+b\lambda _{i})^{2}}}}{\begin{pmatrix}\lambda _{i}^{2}-a^{2}\\ab+c\lambda _{i}\\ac+b\lambda _{i}\end{pmatrix}},\quad i=1,2,3}$

If these do not form a legal system , the first two eigenvalues ​​and vectors are exchanged. The values ​​from the table are rounded to four significant digits :

${\displaystyle {\begin{aligned}\lambda _{1}=&-34{,}77,&{\hat {v}}_{1}=&{\begin{pmatrix}0{,}3847\\0{,}7408\\0{,}5507\end{pmatrix}}\\\lambda _{2}=&-7{,}787,&{\hat {v}}_{2}=&{\begin{pmatrix}-0{,}7862\\-0{,}04974\\0{,}6160\end{pmatrix}}\\\lambda _{3}=&42{,}55,&{\hat {v}}_{3}=&{\begin{pmatrix}0{,}4837\\-0{,}6699\\0{,}5632\end{pmatrix}}\end{aligned}}}$

Each point on the perennial cone is represented by a vector

${\displaystyle {\vec {v}}=x{\hat {v}}_{1}+y{\hat {v}}_{2}+z{\hat {v}}_{3}}$

where the coordinates x, y, z of the quadric

${\displaystyle -(\lambda _{1}x^{2}+\lambda _{2}y^{2}+\lambda _{3}z^{2})={\frac {x^{2}}{0{,}1696^{2}}}+{\frac {y^{2}}{0{,}3584^{2}}}-{\frac {z^{2}}{0{,}1533^{2}}}=0}$

to obey. An ellipse as a section of this elliptical cone is created at a constant z .

Footnotes

1. ↑ Grammel (1920), p. 132, Grammel (1950), p. 172, Magnus (1971), p. 132, Klein and Sommerfeld (2010), p. 386.
2. ↑ Grammel (1920), p. 132, Magnus (1971), p. 134
3. ↑ Staude (1894), p. 321
4. ↑ Magnus (1971), p. 136
5. ↑ ^{a b} Otto Staude: About permanent axes of rotation when moving a heavy body around a fixed point , see literature.
6. ↑ Klein and Sommerfeld (2010), p. 378 and p. 581.
7. ↑ Edward Routh : The Dynamics of Rigid Body Systems . The higher dynamic. tape 2 . BG Teubner, Leipzig 1898, p. 163 ( archive.org - The sentence appeared as early as 1892 in the 5th edition of The advanced part of A treatise on the dynamics of a system of rigid bodies by Routh, pp. 142f.).
8. ↑ Grammel (1920), p. 130, Grammel (1950), p. 171f.
9. ↑ Grammel (1950), p. 171 ff.
10. ↑ Magnus (1971), p. 143.
11. ↑ Leimanis (1965), p. 116.
12. ↑ Staude (1894), p. 321
13. ↑ Magnus (1971), p. 136.
14. ↑ Grammel (1920), p. 132.
15. ↑ Staude (1894), p. 322
16. ↑ Magnus (1971), p. 134.
17. ↑ A: B: C = 8: 11: 16, s ₁  : s ₂  : s ₃ = -6: -1: -8
18. ↑ Klein and Sommerfeld (2010), p. 388.
19. ↑ Klein and Sommerfeld (2010), p. 389. The starting position was chosen as in Fig. 2 so that initially ε  = +1.
20. ↑ Magnus (1971), p. 136, Grammel (1950), p. 178ff. and Leimanis (1965)
21. ↑ Grammel (1950), pp. 178f.
22. ↑ Magnus (1971), p. 139.
23. ↑ Magnus (1971), p. 141. For more details, see Grammel (1950), p. 191 ff.
24. ↑ ^{a b} Grammel (1950), p. 177 f.
25. ↑ Magnus (1971), p. 134.
26. ↑ Magnus (1970), pp. 135f. There is ψ = λ, φ = ϑ and on p. 136 Magnus confuses the stretched top with A> C with the flattened top with A <C .

literature

• Otto Staude : About permanent axes of rotation when moving a heavy body around a fixed point . In: Journal for pure and applied mathematics . tape 114 , 1894, pp. 318-334 ( digizeitschriften.de [accessed on 14 April 2018]).
• K. Magnus : Kreisel: Theory and Applications . Springer, 1971, ISBN 978-3-642-52163-8 , pp. 132 ff . ( Limited preview in Google Book Search [accessed February 20, 2018]).
• R. Grammel : The top . Its theory and its applications. Vieweg Verlag, Braunschweig 1920, DNB  451641280 , p. 130 ff . ( archive.org - "swing" means angular momentum, "torsional shock" torque and "torsional balance" rotational energy). or R. Grammel : The top . Its theory and its applications. 2. revised Edition volume
  
   1 . Springer, Berlin, Göttingen, Heidelberg 1950, pp. 171 ff .
• F. Klein , A. Sommerfeld : The Theory of the Top . Development of the Theory in the Case of the Heavy Symmetric Top. Volume II. Birkhäuser, Boston 2010, ISBN 978-0-8176-4824-4 , pp. 386 ff ., doi : 10.1007 / 978-0-8176-4827-5 (English, symbols are explained on p. 197ff., in particular p. 200).
• 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. 67 , doi : 10.1007 / 978-3-642-88412-2 (English, limited preview in Google Book Search [accessed on March 21, 2018]).