Gyroscopic stabilization

The gyroscopic stabilization or twist stabilization is an effect in gyro theory with which inherently unstable systems are stabilized in their spatial alignment by built-in cyclical mechanisms. The cyclical mechanisms are mostly symmetrical gyroscopes and we will only talk about this. The effect can be experienced in everyday life with rapidly rotating game tops , which are remarkably insensitive to interference, or with a swinging wheel held in the hand that opposes the change in direction of its axis.

The theory of gyroscopic stabilization deals with the question under which circumstances stabilization succeeds, which is by no means 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 .

Gyroscopic stabilization is used in ships ( ship gyro ), spacecraft , gyroscopic instruments and inertial navigation systems as well as in ballistics , see #Applications .

Stabilization of a flywheel

Flywheel to explain the spin stabilization

The gyroscopic stabilization occurs with the flywheel , in which the center of mass is rotatably fixed in the coordinate origin and the figure axis (initially in the y-direction) is free so that it can change its direction at will, see picture. A constant moment M _z acts on this otherwise force-free flywheel for a short time in the z-direction, which causes the flywheel to rotate about z. This rotation is noticeable differently on the stationary and rotating flywheels:

1. If the flywheel is at rest, the moment causes it to rotate around z. Once the moment has ceased to function, the flywheel remains in the rotation for the rotation angle ψ of the figure axis about z increases monotonically to and un limited . The angular velocity and the angular momentum have only one component and they point in the z-direction. The angle of inclination ϑ between the figure axis and the moment axis z remains unchanged.
2. If the flywheel rotates sufficiently quickly around the figure axis at the beginning, a different picture emerges. The moment leads to a linear increase in the angular momentum in the z-direction, but because this component is vectorially added to the initial angular momentum in the y-direction (assumed to be much larger), i.e. the angular momentum continues to be oriented primarily in the y-direction , and angular momentum and angular velocity enclose an acute angle (see ellipsoid of inertia ), the flywheel continues to rotate mainly around the y-axis. As a result, the angle of rotation ψ of the figure axis remains limited by z. According to the rule of parallelism in the same direction , the top tries to adjust its rotation to the attacking moment, whereby the angle ϑ decreases.

The reason for the slight influence of the moment on the rotation of the rotating flywheel around z is the inertial forces which, as described below, build up counter torques. It can be seen that the figure axis oscillates around z under the moment:

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

Here L is the initial axial angular momentum around the figure axis and A the equatorial main moment of inertia . The oscillation equation is an approximation that is only valid for a small deflection ψ . From can with ψ also be calculated θ. ${\displaystyle {\dot {\vartheta }}=-{\tfrac {L}{A}}\psi }$

The ability to freely rotate the figure axis around the equatorial axes is decisive for stabilization. If the axis of rotation is bound to the xy plane by bearings, the moments of inertia cannot develop their potential and twist stabilization does not occur.

For the derivation of the oscillation equation , the usual case is assumed that the flywheel is an oblate top , i.e. its moment of inertia C around the figure axis is greater than the equatorial moments of inertia A. Otherwise the gyroscopic effects would be oriented the other way around in the x-direction. Unlike in the picture, the angle ψ should count from the y-axis and the relationship M = L ω known from the twist law is used, according to which a moment M leads to the rotational speed ω of an angular momentum L perpendicular to it and, conversely, such a rotation leads to a moment evokes.

1. The small moment M _z initially rotates the flywheel with angular momentum L in the y-direction (slowly) around the z-axis and the angle ψ to the figure axis increases according to the acceleration equation . The acceleration term is a gyroscopic effect in the -z direction, which is fed by Euler forces . ${\displaystyle A{\ddot {\psi }}=M_{z}}$ ${\displaystyle A{\ddot {\psi }}}$
2. The angular velocity has a small component in the z-direction and the inclination ϑ of the axis of rotation relative to the vertical is reduced accordingly . This angular acceleration by x results in Euler forces, which in total produce a gyroscopic effect in the -x direction. ${\displaystyle A{\ddot {\vartheta }}=-L{\dot {\psi }}\rightarrow {\dot {\vartheta }}=-{\tfrac {L}{A}}\psi }$
3. In the same way as the moment M _z causes the gyroscopic effect in the -x direction, the latter creates a further gyroscopic effect in the -z direction, which must be added to the acceleration equation in the first step: which leads to the above oscillation equation. ${\displaystyle L{\dot {\vartheta }}=-{\tfrac {L^{2}}{A}}\psi }$ ${\displaystyle A{\ddot {\psi }}=M_{z}-{\tfrac {L^{2}}{A}}\psi }$
4. Just like the moment M _z triggers an opposite gyroscopic effect, the gyroscopic effect in the -x direction also has an adversary in the + x direction, which is fed by the centrifugal forces in the flywheel and which also contributes to the gyroscopic effect in the -z direction.

While the gyroscopic effects in the -z direction ( and ) add up exactly to M _z , the gyroscopic effects in the x and y directions cancel each other out exactly. The moment of the Euler forces is there antiparallel to the moment of the centrifugal forces . In this way, the angular momentum in the x and y directions remains unchanged compared to the initial state. ${\displaystyle A{\ddot {\psi }}}$ ${\displaystyle {\tfrac {L^{2}}{A}}\psi }$

Kelvin-Tait equations

By exploiting the special properties of gyroscopic mechanisms, the Lagrange equations result in the Kelvin-Tait equations, with which the effect of built-in and invisible rotating gyroscopes is treated analytically.

Gyros, there are often coordinate φ _k , in the kinetic energy is not itself, but only with their time derivative occur. In total energy

${\displaystyle E={\frac {1}{2}}\left(A{\dot {\psi }}^{2}\sin(\vartheta )^{2}+A{\dot {\vartheta }}^{2}+C{\dot {\varphi }}^{2}\right)}$

above flywheel, see derivation of the motion function of the Lagrange gyro with c ₀  = 0, φ and ψ are such variables, where φ is also not noticeably influenced by the external moment M _z as long as the inclination angle ϑ is almost a right one .

In the Lagrange formalism , the φ _{k are called} cyclic coordinates and the associated generalized forces Q _k , as with the rotation angle φ of the flywheel above, are equal to zero, it is a cyclic system . There the generalized impulses Φ _{k are} constant at the cyclic coordinates φ _k (in the case of the flywheel, the axial angular momentum L is this constant.) By eliminating the cyclic coordinates in the Lagrange equations in favor of their constant generalized impulses - an idea based on Edward Routh goes back - the Kelvin-Tait equations arise

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial F_{1}({\dot {q}}_{1,2,3})}{\partial {\dot {q}}_{k}}}\right)-{\frac {\partial F_{1}({\dot {q}}_{1,2,3})}{\partial q_{k}}}+G_{k1}{\dot {q}}_{1}+G_{k2}{\dot {q}}_{2}+G_{k3}{\dot {q}}_{3}=Q_{k}+{\frac {\partial F_{2}(\Phi _{1,2})}{\partial q_{k}}}\;,\quad k=1,2,3}$

They are written for the special case of two cyclic φ _k and three further, non-cyclic, generalized coordinates q _k , which can be calculated from the three equations. The functions F _1,2 are additionally and the terms G _kl exclusively dependent on the q _1,2,3 , which was omitted from the equation for reasons of space. The gyroscopic terms G _kl are due to the antisymmetry

G _kl  = - G _lk and G _ll  = 0, k, l = 1,2,3

characterized. In the Kelvin-Tait equations, the cyclic coordinates φ _k no longer occur; they are therefore called hidden or kinosthenic coordinates in contrast to the visible coordinates q _k . Once the visible coordinates have been calculated from the Kelvin-Tait equations, the hidden coordinates can then be determined. If these also turn out to be fixed values, the system is called isocyclic .

The triple effect of built-in and invisible rotating gyroscopes can be read from the equations:

1. The inertia of the system has apparently changed, because the kinetic energy is replaced by the value F ₁ , to which increased inertia usually contribute.
2. An apparent force is added to the "visible" generalized force Q _k . ${\displaystyle {\tfrac {\partial F_{2}}{\partial q_{k}}}}$
3. The gyroscopic members G _kl mean a gyroscopic coupling created by the hidden movements between the visible coordinates. They appear as the gyroscopic forces of the hidden gyroscope, brought about by gyroscopic effects which, as a whole, do not produce any performance.

Requirements for gyroscopic stabilization

In this and the following section, the three theorems of Kelvin and Tait mentioned above are justified analytically.

In a system with n stable or unstable degrees of freedom q _k , in the absence of gyroscopic couplings, the equations of motion can be used for small disturbances of a state of equilibrium in the form

${\displaystyle B_{k}{\ddot {q}}_{k}+K_{k}{\dot {q}}_{k}+H_{k}q_{k}=0\,,\qquad k=1,2,\ldots ,n}$

to be written. Are in it

• B _k the always positive "inertia _coefficients ",
• K _k the mostly positive damping numbers and
• H _{k is} the "reset _coefficient ".

The solutions to these oscillation equations are

${\displaystyle q_{k}=b_{k}e^{\sigma t}\quad {\text{mit}}\quad \sigma _{1,2}={\frac {1}{2B_{k}}}\left(-K_{k}\pm {\sqrt {K_{k}^{2}-4B_{k}H_{k}}}\right)}$

Here e ^{x is} the exponential function and t is the time. In any case , if H _k is negative, one of the σ _{1,2 is} positive, which is an incessant Anwaxes of q _k , thus resulting in instability. At H _k  > 0, on the other hand, there is stability with q _k decaying aperiodically or oscillatingly .

In a gyroscopically coupled system, the Kelvin-Tait equations result in coupled oscillation equations

${\displaystyle B_{k}{\ddot {q}}_{k}+K_{k}{\dot {q}}_{k}+H_{k}q_{k}+\sum _{l}G_{kl}{\dot {q}}_{l}=0\,,\qquad k=1,2,\ldots n}$

The above approach results in n linear equations for the n coefficients b _k . So that these are not all zero, the determinant of the system of equations must be zero, which leads to an equation of order 2 n in σ: ${\displaystyle q_{k}=b_{k}e^{\sigma t}}$

${\displaystyle {\begin{aligned}{\begin{vmatrix}B_{1}\sigma ^{2}+K_{1}\sigma +H_{1}&G_{12}\sigma &\ldots &G_{1n}\sigma \\-G_{12}\sigma &B_{2}\sigma ^{2}+K_{2}\sigma +H_{2}&\ldots &G_{2n}\sigma \\\vdots &\vdots &\ddots &\vdots \\-G_{1n}\sigma &-G_{2n}\sigma &\ldots &B_{n}\sigma ^{2}+K_{n}\sigma +H_{n}\end{vmatrix}}&=\ldots \\\ldots =:a_{0}\sigma ^{2n}+a_{1}\sigma ^{2n-1}+\cdots +a_{2n-1}\sigma +a_{2n}&=0\end{aligned}}}$

With

${\displaystyle a_{0}=B_{1}B_{2}\ldots B_{n}\quad {\text{und}}\quad a_{2n}=H_{1}H_{2}\ldots H_{n}}$

The original state of equilibrium is stable if and only if no root σ has a positive real part, because a positive real part would mean a constant increase in at least one coordinate q _k and therefore instability.

Accordingly, there is stability precisely when the polynomial is a Hurwitz polynomial . In any case, the first coefficient a ₀ is positive. With the Hurwitz criterion it can be decided whether all zeros σ have negative real parts, and it turns out that a _{2n must} also be positive. This is only possible if at most an even number of the H _{k is} negative, i.e. at most an even number of the q _{k is} unstable, which justifies the first of the listed sentences . Whether the stabilization  can actually succeed at a _2n > 0 depends on the other Hurwitz conditions.

In a system with two degrees of freedom, these can be written down and fulfilled relatively easily and the plausibility of the other two sentences listed above can be checked .

Systems with two visible degrees of freedom

Systems with two visible degrees of freedom are common in technology and are therefore presented in detail here. Such a system produces a fourth degree polynomial

${\displaystyle a_{0}\sigma ^{4}+a_{1}\sigma ^{3}+a_{2}\sigma ^{2}+a_{3}\sigma +a_{4}=0}$

with the Hurwitz matrix

${\displaystyle H={\begin{pmatrix}a_{1}&a_{3}&0&0\\a_{0}&a_{2}&a_{4}&0\\0&a_{1}&a_{3}&0\\0&a_{0}&a_{2}&a_{4}\end{pmatrix}}}$

from which with a ₀  > 0 the Hurwitz criteria

${\displaystyle {\begin{aligned}D_{1}&:={\begin{vmatrix}a_{1}\end{vmatrix}}=a_{1}>0\\D_{2}&:={\begin{vmatrix}a_{1}&a_{3}\\a_{0}&a_{2}\end{vmatrix}}=a_{1}a_{2}-a_{0}a_{3}>0\\D_{3}&:={\begin{vmatrix}a_{1}&a_{3}&0\\a_{0}&a_{2}&a_{4}\\0&a_{1}&a_{3}\end{vmatrix}}=a_{3}D_{2}-a_{1}^{2}a_{4}>0\\D_{4}&:=|H|=a_{4}D_{3}>0\end{aligned}}}$

consequences. The vertical bars |… | denote here the determinant of the enclosed matrix. So that all these conditions are met is

${\displaystyle {\begin{aligned}a_{0}&=B_{1}B_{2}>0\\a_{1}&=B_{1}K_{2}+B_{2}K_{1}>0\\a_{2}&=B_{1}H_{2}+K_{1}K_{2}+B_{2}H_{1}+G^{2}>0\\a_{3}&=K_{1}H_{2}+K_{2}H_{1}>0\\a_{4}&=H_{1}H_{2}>0\\D_{3}&=a_{1}a_{2}a_{3}-a_{0}a_{3}^{2}-a_{1}^{2}a_{4}>0\end{aligned}}}$

necessary and sufficient . Here G  =  G _{12 is} the coupling link.

In the absence of damping ( K _1,2  = 0), a system with an even number of unstable degrees of freedom ( 1st sentence , a ₄  > 0) can be stabilized in accordance with the second sentence by a sufficiently strong top - i.e. a large G - .

In the case of unstable degrees of freedom ( H _1.2  <0) there are options for fulfilling the Hurwitz criteria

${\displaystyle {\begin{aligned}{\text{a)}}&{\frac {B_{1}}{B_{2}}}<\left|{\frac {H_{1}}{H_{2}}}\right|,K_{1}>0,K_{2}<0,\left|{\frac {H_{1}}{H_{2}}}\right|>{\frac {K_{1}}{|K_{2}|}}>{\frac {B_{1}}{B_{2}}}\\{\text{b)}}&{\frac {B_{1}}{B_{2}}}>\left|{\frac {H_{1}}{H_{2}}}\right|,K_{1}<0,K_{2}>0,\left|{\frac {H_{1}}{H_{2}}}\right|<{\frac {|K_{1}|}{K_{2}}}<{\frac {B_{1}}{B_{2}}}\end{aligned}}}$

In the case of a damped degree of freedom, for example in a) with K ₁  > 0, the other degree of freedom must be artificially increased so that in a) K ₂  <0, and the corresponding fourth inequality (here in case a) must also be satisfiable. This fact could be generalized by Kelvin and Tait to the third condition .

Applications

The twist stabilization is discussed by Richard Grammel (1920), see #Literature and what the page references refer to, in the following systems:

• Natural vibrations of airplanes (p. 208)
• Spinning discs (p. 231)
• Elastic bonds on gyroscopes (e.g. cardanic suspension that deforms elastically) (p. 243)
• Flywheel suspended from threads on the surface of the earth (p. 252)
• Single gyro compass (p. 258)
• Multi-gyro compass (p. 269)
• Cross stabilization on the aircraft gyro (p. 290)
• Support gyro in the Howell Torpedo (p. 312)
• Monorail especially the monorail to Brennan (p. 318)

Gyroscopic stabilization is also used

• in space travel, see stabilization (space travel) and, for example, low-density supersonic decelerator ,
• in gyroscopic instruments , see for example Artificial Horizon .
• in ballistics especially external ballistics , see bullet stabilization, which can also be used when throwing a javelin , for example .

Web links

• 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 November 23, 2019 . 
• K. Lüders, RO Pohl, G. Beuermann, K. Samwer: Gyro compass. ( MP4 ) Institute for Scientific Film (IWF) , 2003, accessed on November 24, 2019 . 

Individual evidence

1. ↑ ^{a b c} Grammel (1950), p. 258.
2. ↑ Grammel (1950), p. 261 f.
3. ↑ F. Klein, A. Sommerfeld: Theory of the gyro . The technical applications of the gyro theory. Book IV. Teubner, Leipzig 1910, p. 767 f . ( archive.org [accessed October 21, 2017]).
4. ↑ Grammel (1950), pp. 253ff. especially p. 257.
5. ↑ Grammel (1950), p. 257 f.
6. ↑ Grammel (1950), p. 259.
7. ^ Grammel (1950), p. 262.
8. ^ Howell torpedo. Wikipedia, accessed October 27, 2019 . 

literature

• R. Grammel : The top . Theory of the gyro. 2. revised Edition volume 1 .. Springer, Berlin, Göttingen, Heidelberg 1950, DNB  451641299 , p. 258 ff .
• R. Grammel : The top . Its theory and its applications. Vieweg Verlag, Braunschweig 1920, DNB  573533210 ( archive.org - "Schwung" means angular momentum, "torsional shock" torque and "torsional balance" rotational energy.).