Curve gyro

Realization of a curve gyro (yellow) with center of gravity (blue) and guide curve (red).

The curve gyro or perimetric gyro is a gyro mounted in its center of gravity , which has a materially formed axis that rolls on a spatially fixed, also materially formed curve, see picture.

Under certain conditions the gyro axis follows the curve itself through turns and corners. Because of the frictional force , the axis moves along the curve and is pivoted in the process. As a result of this, gyroscopic effects arise which, with a suitable mass distribution of the gyro, press the gyro axis against the curve and sometimes unexpectedly violently.

In Koller mills the contact pressure is technically exploited. The curve gyro was invented by G. Sire.

Analytical foundation

Coordinate systems used with the curved top

The theory is less complicated if the curve top is a symmetrical top like in the picture, the figure axis of which rolls on the curve without slipping , and that's what is assumed here.

Then the track cone and pole cone are given by the guide curve or the cross section of the figure axis at the level of the guide curve in connection with the support point. When the cones roll over one another without sliding, the axis of rotation and with it the axial angular momentum are pivoted at a known speed. The conservation of angular momentum , according to a case arises gyro moment from which the perpendicular to the cornering force component can be determined. If it points from the curve to the figure axis, then the curve presses against the axis and vice versa, so that the top lies against the curve.

The spatially fixed curve is applied as a spherical curve with a function θ (ψ) in spherical coordinates { r, θ, ψ }, see figure. The intermediate system used for the curve gyro (blue in the picture) reads in these spherical coordinates:

${\displaystyle {\hat {e}}_{r}={\begin{pmatrix}\sin(\theta )\cos(\psi )\\\sin(\theta )\sin(\psi )\\\cos(\theta )\end{pmatrix}},\;{\hat {e}}_{\theta }={\begin{pmatrix}\cos(\theta )\cos(\psi )\\\cos(\theta )\sin(\psi )\\-\sin(\theta )\end{pmatrix}},\;{\hat {e}}_{\psi }={\begin{pmatrix}-\sin(\psi )\\\cos(\psi )\\0\end{pmatrix}}}$

It is placed in such a way that ê _{r is} the figure axis and ê _{ψ is} parallel to the tangent to the given curve, at the point where the figure axis touches it. In the body-fixed principal axes system ê _1,2,3 (red in the image) rotate the equatorial main axes ê _1.2 around the body axis at the angle φ :

ê ₁  = cos (φ) ê _ψ  - sin (φ) ê _θ

ê ₂  = -sin (φ) ê _ψ  - cos (φ) ê _θ

ê ₃  =  ê _r

The angular velocity runs smoothly on the guide curve from the support point to it and forms the angle θ _k  = θ - α with the z-axis , where θ _{k is} the opening angle of the toe cone (between the axis of rotation and the plumb line) and α that of the pole cone ( between rotation axis and figure axis). The angular velocity is written in the main axis system

${\displaystyle {\begin{aligned}{\vec {\omega }}=&\omega _{1}{\hat {e}}_{1}+\omega _{2}{\hat {e}}_{2}+\omega _{3}{\hat {e}}_{3}\\\omega _{1}=&{\dot {\theta }}\cos(\varphi )+{\dot {\psi }}\sin(\theta )\sin(\varphi )\\\omega _{2}=&-{\dot {\theta }}\sin(\varphi )+{\dot {\psi }}\sin(\theta )\cos(\varphi )\\\omega _{3}=&{\dot {\varphi }}+{\dot {\psi }}\cos(\theta )\end{aligned}}}$

and from it the angular momentum results by means of the main moments of inertia A, A or C around the main axes:

${\displaystyle {\vec {L}}=A\omega _{1}{\hat {e}}_{1}+A\omega _{2}{\hat {e}}_{2}+C\omega _{3}{\hat {e}}_{3}}$

Its time derivative can now be calculated using. With φ  = 0 the following results in the intermediate system: ${\displaystyle {\dot {\hat {e}}}_{1,2,3}={\vec {\omega }}\times {\hat {e}}_{1,2,3}}$

${\displaystyle {\begin{aligned}{\vec {M}}={\dot {\vec {L}}}=&C[{\ddot {\varphi }}+{\ddot {\psi }}\cos(\theta )-{\dot {\psi }}{\dot {\theta }}\sin(\theta )]{\hat {e}}_{r}\\&+\{C{\dot {\theta }}[{\dot {\varphi }}+{\dot {\psi }}\cos(\theta )]-A[{\ddot {\psi }}\sin(\theta )+2{\dot {\psi }}{\dot {\theta }}\cos(\theta )]\}{\hat {e}}_{\theta }\\&+\{A{\ddot {\theta }}+C{\dot {\psi }}\sin(\theta )[{\dot {\varphi }}+{\dot {\psi }}\cos(\theta )]-A{\dot {\psi }}^{2}\sin(\theta )\cos(\theta )\}{\hat {e}}_{\psi }\end{aligned}}}$

If, as designed, the ψ direction is tangential to the guide curve, then the components of the moment have the following effects:

1. The moment in the r direction accelerates or decelerates the rotary motion, which is why it is generally not constant. ${\displaystyle {\dot {\varphi }}}$
2. The moment in the θ direction has to be applied by the frictional force, which is not given if the contact pressure is too low, which is why the figure axis then slides along the curve.
3. The component in the ψ direction presses the top against the curve or releases it from it.

Two special cases are considered in more detail.

Orbit on a meridian

Let it be on a meridian and the top rotates with constant speed in a plane perpendicular to ê _ψ . Then the above moment becomes . If R is the radius of the meridian and r is the radius of the figure axis at the level of the meridian, then the contact force of the figure axis has the lever arm ρ ê _r with ρ² = R² - r² and for the meridian the direction ê _ψ : ${\displaystyle {\dot {\psi }}={\ddot {\psi }}=0}$ ${\displaystyle ({\ddot {\theta }}={\ddot {\varphi }}=0)}$ ${\displaystyle {\vec {M}}=C{\dot {\theta }}{\dot {\varphi }}{\hat {e}}_{\theta }}$

${\displaystyle {\vec {M}}=C{\dot {\theta }}{\dot {\varphi }}{\hat {e}}_{\theta }=(\rho {\hat {e}}_{r})\times (F{\hat {e}}_{\psi })=(\rho {\hat {e}}_{r})\times {\frac {-C{\dot {\theta }}{\dot {\varphi }}}{\rho }}{\hat {e}}_{\psi }}$

If the distance vector from the center of the figure axis to the curve is r ê _ψ , then the rolling condition applies . The resulting force is directed from the curve to the axis, which in turn presses on the curve according to the Actio and Reactio principle . The compressive force is proportional to the square of the rotational speed . ${\displaystyle r{\dot {\varphi }}=\rho {\dot {\theta }}}$ ${\displaystyle F=-{\tfrac {C}{r}}{\dot {\theta }}^{2}}$ ${\displaystyle {\dot {\theta }}}$

Conversely, if - r ê _{ψ points} from the center of the figure axis to the curve, then the rolling condition is written , which is why the top also presses against the curve here and with the same force as in the first case. ${\displaystyle r{\dot {\varphi }}=-\rho {\dot {\theta }}}$

Orbiting on a parallel

In this special case, the θ is constant , where 0 ≤  θ  ≤ 90 ° can be assumed, and is with uniform rolling . This results in the above moment in the intermediate system to ${\displaystyle ({\dot {\theta }}={\ddot {\theta }}=0)}$ ${\displaystyle {\ddot {\psi }}={\ddot {\varphi }}=0}$

${\displaystyle {\vec {M}}=[C{\dot {\varphi }}+(C-A){\dot {\psi }}\cos(\theta )]{\dot {\psi }}\sin(\theta ){\hat {e}}_{\psi }}$.

Outer circulation

If the top rolls on the outside of the curve, the rolling condition says , where R is the radius of the sphere on which the circle of latitude lies, r is the radius of the figure axis at the height of the circle of latitude. The track cone and pole cone have the opening angles θ _K and α , respectively , so that the figure axis assumes the angle θ = θ _K  + α with the vertical. Here, too, the moment results from a force ${\displaystyle r{\dot {\varphi }}=R\sin(\theta _{K}){\dot {\psi }}}$ ${\displaystyle {\vec {M}}={\sqrt {R^{2}-r^{2}}}{\hat {e}}_{r}\times (F{\hat {e}}_{\theta })}$

${\displaystyle F={\frac {R\sin(\theta _{k})-\rho _{0}}{r{\sqrt {R^{2}-r^{2}}}}}C{\dot {\psi }}^{2}\sin(\theta )\quad {\text{mit}}\quad \rho _{0}:={\frac {A-C}{C}}r\cos(\theta )}$

If the force F is positive, then the curve presses against the top and vice versa. Again, the compressive force is proportional to the square of the angular velocity.

With a flattened top, C> A , ρ ₀  <0 and the numerator in the fraction is always positive, so that the flattened top follows the curve in any case, even in sharp corners.

For the straight top, A> C , ρ ₀  > 0 and the top only leans towards the latitude if R sin (θ _K )> ρ ₀ . This condition can be interpreted in such a way that the top only adheres to the circle of latitude if its opening angle θ _{K is} greater than that of the nutation cone, which the power -free top would follow freely at a given speed of rotation.

Because the power-free top is

${\displaystyle {\dot {\psi }}={\frac {C\omega _{3}}{A\cos(\theta )}}\,,\quad {\dot {\varphi }}={\frac {A-C}{A}}\omega _{3}\quad \rightarrow \quad {\frac {\dot {\varphi }}{\dot {\psi }}}={\frac {A-C}{C}}\cos(\theta )}$

If the force-free top rolls on a circle of latitude, then the roll condition is fulfilled and the radius of the circle of latitude results ${\displaystyle r{\dot {\varphi }}=R\sin(\theta _{K}){\dot {\psi }}}$

${\displaystyle R\sin(\theta _{K})=r{\frac {\dot {\varphi }}{\dot {\psi }}}=r{\frac {A-C}{C}}\cos(\theta )}$

thus precisely the critical radius ρ _{0 of} the parallel for the elongated top of the curve.

Inner circulation

If the top of the curve runs along the inside of the circle of latitude, then due to θ = θ _K  - α the rolling condition changes from in and the above moment becomes ${\displaystyle r{\dot {\varphi }}=-R\sin(\theta _{K}){\dot {\psi }}}$

${\displaystyle {\vec {M}}=[C{\dot {\varphi }}+(C-A){\dot {\psi }}\cos(\theta )]{\dot {\psi }}\sin(\theta ){\hat {e}}_{\psi }=-{\frac {R\sin(\theta _{K})+\rho _{0}}{r}}C{\dot {\psi }}^{2}\sin(\theta ){\hat {e}}_{\psi }}$

with ρ ₀ as in the previous section. The strength results from here ${\displaystyle {\vec {M}}={\sqrt {R^{2}-r^{2}}}{\hat {e}}_{r}\times (F{\hat {e}}_{\theta })}$

${\displaystyle F=-{\frac {R\sin(\theta _{K})+\rho _{0}}{r{\sqrt {R^{2}-r^{2}}}}}C{\dot {\psi }}^{2}\sin(\theta )}$

If this is negative, then the curve presses against the top and vice versa. Here it is the elongated top that  always follows the curve inside because of ρ ₀ > 0. But the flattened top does that too, because because of C> A and | cos ( θ ) | ≤ 1 is | ρ ₀ | <  r . But this means that R  sin ( θ _K )> | ρ ₀ |, because for kinematic reasons, R  sin ( θ _K )>  r must be guaranteed if rolling is to take place on the concave side of the guide curve.

The top of the curve rolling on the inside of a concave guide curve never leaves this curve.

Pan grinder

→ Main article : pan grinder

Pan mill near Sax-Farben in Urdorf, Canton of Zurich

The pan grinder, as you can see in the picture, is a technical application of the curve gyro. This makes use of the fact that the force acting on the grist is increased by the gyroscopic effect.

In the section # Outer rotation around a circle of latitude, the compressive force resulted

${\displaystyle {\vec {F}}={\frac {R\sin(\theta _{k})-\rho _{0}}{r{\sqrt {R^{2}-r^{2}}}}}C{\dot {\psi }}^{2}\sin(\theta ){\hat {e}}_{\theta }}$ With  ${\displaystyle \rho _{0}:={\frac {A-C}{C}}r\cos(\theta )}$

The vertical component of this force acts from the latitude downwards on the runner, that with the opposite force of the same magnitude

${\displaystyle N={\frac {R\sin(\theta _{k})-\rho _{0}}{r{\sqrt {R^{2}-r^{2}}}}}C{\dot {\psi }}^{2}}$

presses on the latitude or the base plate. With the given dimensions and certain rotational speed, an optimum angle of inclination θ can be calculated from this, possibly taking into account the weight of the rotor .

At θ  =  ^π ⁄ ₂  = 90 ° and is like the #circulation on a meridian ${\displaystyle R\sin(\theta _{k})={\sqrt {R^{2}-r^{2}}}}$

${\displaystyle N={\frac {C{\dot {\psi }}^{2}}{r}}}$

For a full cylinder with weight G = mg , which is composed of the mass m and the gravitational acceleration g , C = mr² / 2 = Gr² / (2g) . Thus results

${\displaystyle {\frac {N}{G}}={\frac {r}{2g}}{\dot {\psi }}^{2}}$

The ratio of contact pressure to weight is independent of the radius R.

At r  = 0.2 m and a speed of 100 rpm, the normal force N already exceeds the weight force. Since the speed is added to the square, the grinding pressure can be increased considerably by increasing the rotational speed.

Footnotes

1. ↑ Magnus (1974), Magnus (1971), Wilhelm H. Westphal (1952), Grammel (1920), see literature.
2. ↑ Magnus (1971), p. 94. The different signs are explained because Magnus uses the gyroscopic effect K = -M and ê _θ  = -ê ₂
3. ↑ see Magnus (1974)
4. ↑ ^{a b} Magnus (1971), p. 95.
5. ↑ Magnus (1971), p. 96.
6. ↑ Magnus (1971), p. 97.
7. ↑ Magnus (1971), p. 98, Grammel (1920), p. 168.
8. ↑ Magnus (1971), p. 98.

literature

• K. Magnus : Kreisel: Theory and Applications . Springer, 1971, ISBN 978-3-642-52163-8 , pp. 92 ff . ( Limited preview in Google Book Search [accessed October 21, 2018]).
• K. Magnus : Contributions to the kinetics of the curve top . In: Ingenieur-Archiv (today Archive of Applied Mechanics) . tape 43 , no. 2-3 . Springer-Verlag, 1974, ISSN  1432-0681 , p. 145–157 , doi : 10.1007 / BF00537147 (Detailed analysis that also considers the sliding of the top along the curve.).
• R. Grammel : The top . Its theory and its applications. Vieweg Verlag, Braunschweig 1920, DNB  451641280 , p. 72 ff. and 166 ff . ( Archive.org - "swing" means angular momentum, " rotational shock as" torque and "rotational force" rotational energy).
• Wilhelm H. Westphal (Hrsg.): Physical handbook . Springer-Verlag, Berlin, Heidelberg 1952, ISBN 978-3-662-12706-3 , pp. 759 , doi : 10.1007 / 978-3-662-12706-3 ( limited preview in Google book search).