Heavy top

In gyro theory , the heavy gyroscope is a gyroscope in which an external torque acting on it comes from its weight . The classic gyro theory is almost exclusively dedicated to the heavy gyroscope with a base and a lot of effort has been and will be put into finding exact solutions to the equations of motion in the form of the Euler-Poisson equations .

Due to the omnipresent gravity on earth , the heavy tops are particularly relevant.

Moment of gravity and angular momentum

The severity moment is calculated from the cross product × of the lever arm from the base to the center of mass with the weight to ${\ displaystyle {\ vec {s}}}$ ${\ displaystyle {\ vec {G}}}$

{\ displaystyle {\ vec {M}} = {\ vec {s}} \ times {\ vec {G}} = {\ vec {s}} \ times (-mg {\ hat {e}} _ {e.g. })}

In it is

m the mass,
g is the acceleration due to gravity and
ê _{z is} the unit vector pointing upwards antiparallel to the weight force .

The moment is thus oriented horizontally perpendicular to the weight force and, according to the principle of twist, is equal to the speed of the end point of the angular momentum . The end point of the heavy gyro with a support point therefore moves in a horizontal plane, the distance to the support point being an integral of the movement.

Integrals of motion

With every heavy top, the norm of the weight force , the angular momentum in the vertical direction and the total energy E are constant. The constants are called integrals in gyro theory , the first two also Casimir invariants . The second integral L _z is also called the twist or surface integral after the theorem of swirl or area . The total energy is constant because of the law of conservation of energy , see Euler-Poisson equations .

Equations of motion

Euler's gyroscopic equations

If the moment of gravity is used in Euler's gyroscopic equations, the result is

{\ displaystyle {\ begin {aligned} \ Theta _ {1} {\ dot {\ omega}} _ {1} + (\ Theta _ {3} - \ Theta _ {2}) \ omega _ {2} \ omega _ {3} = {\ dot {L}} _ {1} + \ left ({\ frac {1} {\ Theta _ {2}}} - {\ frac {1} {\ Theta _ {3} }} \ right) L_ {2} L_ {3} = & mg (n_ {2} s_ {3} -n_ {3} s_ {2}) \\\ Theta _ {2} {\ dot {\ omega}} _ {2} + (\ Theta _ {1} - \ Theta _ {3}) \ omega _ {3} \ omega _ {1} = {\ dot {L}} _ {2} + \ left ({\ frac {1} {\ Theta _ {3}}} - {\ frac {1} {\ Theta _ {1}}} \ right) L_ {3} L_ {1} = & mg (n_ {3} s_ {1 } -n_ {1} s_ {3}) \\\ Theta _ {3} {\ dot {\ omega}} _ {3} + (\ Theta _ {2} - \ Theta _ {1}) \ omega _ {1} \ omega _ {2} = {\ dot {L}} _ {3} + \ left ({\ frac {1} {\ Theta _ {1}}} - {\ frac {1} {\ Theta _ {2}}} \ right) L_ {1} L_ {2} = & mg (n_ {1} s_ {2} -n_ {2} s_ {1}) \ end {aligned}}}

The superpoint forms the time derivative , mg is the weight force and for k = 1,2,3 is in each case

ω _{k is} the component of the angular velocity ,

L _{k is} the component of the angular momentum ,

s _{k is} the component of the center of mass,

n _{k is} the direction cosine of the unit vector ê _z and
Θ _{k is} a principal moment of inertia of the top

in the main axis system. Become common

the main moments of inertia Θ _1,2,3 with A, B or C,
the angular velocities ω _1,2,3 with p, q or r and
the components n _1,2,3 with γ _1,2,3 or γ, γ ', γ ", occasionally with the opposite sign

designated.

The angular velocities and accelerations as well as the direction cosines can be expressed with the Euler or Cardan angles , see e.g. B. Euler angles in gyroscopic theory , which leads to differential equations of the second order in the three angles.

Alternatively, the direction cosines can be introduced as independent unknowns, resulting in the Euler-Poisson equations , which are a system of six first-order differential equations.

Wilhelm Hess was able to use the #integrals of motion in 1890 to express the direction cosines with the angular momentum, which results in three differential equations for the angular momentum, see Euler-Poisson equations .

Symmetrical heavy top

If the figure axis ê _{3 is of} primary interest in the symmetrical top and the center of mass lies on it, the coordinate-independent vector equations apply with regard to the center of mass

{\ displaystyle {\ begin {aligned} {\ dot {\ vec {L}}} = & s {\ hat {e}} _ {3} \ times (-mg {\ hat {e}} _ {z}) \\ {\ dot {\ hat {e}}} _ {3} = & {\ frac {1} {A}} {\ vec {L}} \ times {\ hat {e}} _ {3} \ end {aligned}}}

It is a system of six coupled first-order differential equations. The relationship exploited here

{\ displaystyle {\ vec {\ omega}} = {\ frac {1} {A}} {\ vec {L}} + {\ frac {AC} {AC}} L_ {3} {\ hat {e} } _ {3}}

is justified by the symmetrical top . There are C the axial and A the equatorial principal moment of inertia of the gyro.

Lagrange and Hamilton functions of the top

The Lagrangian of the top is the difference between the rotational energy and the positional energy

{\ displaystyle {\ mathcal {L}} = E _ {\ mathrm {rot}} -E _ {\ mathrm {pot}} = {\ frac {L_ {1} ^ {2}} {2 \ Theta _ {1} }} + {\ frac {L_ {2} ^ {2}} {2 \ Theta _ {2}}} + {\ frac {L_ {3} ^ {2}} {2 \ Theta _ {3}}} -mg (s_ {1} n_ {1} + s_ {2} n_ {2} + s_ {3} n_ {3})}

Here mg is the weight force and each is for k = 1,2,3

Θ _{k are} the main moments of inertia,

L _{k is} the angular momentum,

n _k , the direction cosines of the perpendicular to the top facing the unit vector and

s _{k are} the constant coordinates of the center of mass

in the main axis system.

The Lagrangian can be expressed with the Euler or Cardan angles , the time course of which then results from the Lagrangian equations.

The Hamilton function H of the gyro is the sum of the rotational and positional energy:

{\ displaystyle H = E _ {\ mathrm {rot}} + E _ {\ mathrm {pot}} = {\ frac {L_ {1} ^ {2}} {2 \ Theta _ {1}}} + {\ frac {L_ {2} ^ {2}} {2 \ Theta _ {2}}} + {\ frac {L_ {3} ^ {2}} {2 \ Theta _ {3}}} + mg (s_ {1 } n_ {1} + s_ {2} n_ {2} + s_ {3} n_ {3})}

The Euler-Poisson equations can be expressed with it and the Poisson bracket {}:

{\ displaystyle {\ dot {L}} _ {i} = \ {L_ {i}, H \}, \ quad {\ dot {n}} _ {i} = \ {n_ {i}, H \} }

for i = 1, 2, 3

These equations of motion can also be expressed in vector form

{\ displaystyle {\ begin {aligned} {\ dot {M}} = & M \ times {\ frac {\ partial H} {\ partial M}} + \ gamma \ times {\ frac {\ partial H} {\ partial \ gamma}} \\ {\ dot {\ gamma}} = & \ gamma \ times {\ frac {\ partial H} {\ partial M}} \ end {aligned}}}

write. Here, M is the coordinate vector of the angular momentum and γ the coordinate vector of the unit vector in the direction of the acceleration due to gravity in each case with respect to the main axis system.

The Poisson algebra e (3) of this variable is given by

{\ displaystyle \ {L_ {i}, L_ {j} \} = - \ varepsilon _ {ijk} L_ {k}, \ quad \ {L_ {i}, n_ {j} \} = - \ varepsilon _ { ijk} n_ {k}, \ quad \ {n_ {i}, n_ {j} \} = 0}

Here ε _{ijk is} the Levi-Civita symbol . There are two Casimir functions F ₁ = M · γ and F ₂ = γ², which commute with every function of M and γ with respect to the Poisson bracket and which are also integrals of the Hamilton function.

Individual evidence

↑ Magnus (1971), p. 105.

↑ Magnus (1971), p. 109.

↑ Magnus (1971), p. 51.

↑ S. Rauch-Wojciechowski, M. Sköldstam, T. Glad: Mathematical Analysis of the Stehaufkreisels . In: Regular and Chaotic Dynamics . tape 10 , no. 4 . Springer Nature, 2005, ISSN 1468-4845 , p. 335 , doi : 10.1070 / RD2005v010n04ABEH000319 (English, turpion.org [accessed on December 15, 2018] Original title: Mathematical analysis of the tippe top .).

↑ The Fréchet derivative of a scalar function with respect to a vector is the vector for which - if it exists - applies: ${\ displaystyle f ({\ vec {v}})}$ ${\ displaystyle {\ vec {v}}}$ ${\ displaystyle {\ vec {A}}}$
${\ displaystyle {\ vec {A}} \ cdot {\ vec {h}} = \ left. {\ frac {\ mathrm {d}} {\ mathrm {d} s}} f ({\ vec {v} } + s {\ vec {h}}) \ right | _ {s = 0} = \ lim _ {s \ rightarrow 0} {\ frac {f ({\ vec {v}} + s {\ vec {h }}) - f ({\ vec {v}})} {s}} \ quad \ forall \; {\ vec {h}}}$
In it and “·” is the Frobenius scalar product . Then will too ${\ displaystyle s \ in \ mathbb {R}}$
${\ displaystyle {\ frac {\ partial f} {\ partial {\ vec {v}}}} = {\ vec {A}}}$
written.

↑ AV Borisov, IS Mamaev: Euler-Poisson equations and integrable cases . 2001, p. 254 , doi : 10.1070 / RD2001v006n03ABEH000176 , arxiv : nlin / 0502030 (English, original title: Euler-Poisson Equations and Integrable Cases . Contains a detailed description of solutions to the Euler-Poisson equations and further references.).

literature

K. Magnus : Kreisel: Theory and Applications . Springer, 1971, ISBN 978-3-642-52163-8 , pp. 105 ff . ( limited preview in Google Book Search [accessed February 7, 2019]).
R. Grammel : The top . Its theory and its applications. 2. revised Edition volume 1 . Springer, Berlin, Göttingen, Heidelberg 1950.
Eugene Leimanis: The general problem of the movement of coupled rigid bodies around a fixed point . Springer Verlag, Berlin, Heidelberg 1965, ISBN 978-3-642-88414-6 , p. 7 , doi : 10.1007 / 978-3-642-88412-2 (English, limited preview in Google Book Search [accessed on February 7, 2019] Original title: The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point . ).

[1] Magnus (1971), p. 105.

[2] Magnus (1971), p. 109.

[3] Magnus (1971), p. 51.

[4] S. Rauch-Wojciechowski, M. Sköldstam, T. Glad: Mathematical Analysis of the Stehaufkreisels . In: Regular and Chaotic Dynamics . tape 10 , no. 4 . Springer Nature, 2005, ISSN 1468-4845 , p. 335 , doi : 10.1070 / RD2005v010n04ABEH000319 (English, turpion.org [accessed on December 15, 2018] Original title: Mathematical analysis of the tippe top .).