Bobylew-Steklow solution

The Bobylew-Steklow solution ( English case of Bobylev – Steklov ) is a solution of the Euler-Poisson equations in the gyro theory that Wladimir Andrejewitsch Steklow and D. Bobylew found independently of one another in 1896 , which determine the rotation of heavy gyroscopes around a fixed point .

Bobylew and Steklow showed that for an asymmetrical top with main moments of inertia A, B and C, where C = 2A and the center of mass lies on the main axis of inertia belonging to C (C-axis for short), the Euler-Poisson equations can be integrated when the gyro initially has neither an angular velocity nor an angular acceleration about the B-axis.

Such a top never turns around the B-axis, has a constant angular velocity around the C-axis and all of its other state variables are Jacobian elliptic functions of time. The heavy asymmetrical top performs analytically representable periodic precession movements. Although the main moment of inertia B is not included in the analytical description, it influences the stability of the resulting movement.

Steklow found another solution three years later (1899), which was supplemented by PA Kuz'min in 1952 or by AP Markeev in 2005.

Bobylew-Steklow solution

Solution of the equations of motion

In the Bobylew-Steklow solution, the top is exclusively subject to gravity , so that its motion is determined by the Euler-Poisson equations. These equations have two conservation quantities called integrals , the total energy and the angular momentum in the perpendicular direction . In the Bobylew-Steklow case, the angular velocities around the 2- and 3-axis are still constant. These axles have in the vertical z-direction components of n _2.3 , the n as functions of z component ₁ prove the 1-axis. As a result, there is an autonomous differential equation for n ₁ which can be fulfilled with elliptic functions and whose solution can be used to represent all other state variables of the gyro.

Euler-Poisson equations

→ Main article : Euler-Poisson equations

The Euler-Poisson equations in the Bobylew-Steklow case are:

${\displaystyle {\begin{aligned}A{\dot {p}}=&(B-2A)qr+c_{0}n_{2}\\B{\dot {q}}=&Arp-c_{0}n_{1}\\2A{\dot {r}}=&(A-B)pq\\{\dot {n}}_{1}=&rn_{2}-qn_{3}\\{\dot {n}}_{2}=&pn_{3}-rn_{1}\\{\dot {n}}_{3}=&qn_{1}-pn_{2}\end{aligned}}}$

In this, n _{1,2,3 are} the coordinates of the unit vector pointing antiparallel to the weight force and p, q, r = ω _{1,2,3 are} the angular velocities in the main axis system . The support point moment c ₀  =  mgs is the product of the weight mg with the distance s between the center of mass and the support point along the 3-axis. The over point forms the time derivative .

Are as for any heavy gyro in the Bobylev-Steklov solution, the total energy and the angular momentum L _z in the perpendicular direction constant. The precession angle about the vertical cannot be determined from these equations, which can be looked up in the main article.

Initial conditions

With the initial conditions , the Euler-Poisson equations specialize in: ${\displaystyle q={\dot {q}}=0}$

${\displaystyle {\begin{aligned}A{\dot {p}}=&c_{0}n_{2}&(1)\\B{\dot {q}}=&Arp-c_{0}n_{1}=0&(2)\\2A{\dot {r}}=&0&(3)\\{\dot {n}}_{1}=&rn_{2}&(4)\\{\dot {n}}_{2}=&pn_{3}-rn_{1}&(5)\\{\dot {n}}_{3}=&-pn_{2}&(6)\end{aligned}}}$

The time derivative of equation (2) vanishes because of the conditions (1), (3) and (4), which is why (2), (3) and q  = 0 permanently hold. The six equations are therefore not only initially, but permanently in force. From (3) it can be seen that r is constant. At r  = 0 there is a pendulum- like rotation around the horizontal first main axis, which is not of interest here. Here will

${\displaystyle r=r_{0}={\sqrt {\frac {c_{0}}{A}}}\rho \neq 0}$

accepted. The dimensionless parameter introduced at this point is proportional to the angular momentum around the 3-axis. ${\displaystyle \rho :=r_{0}{\sqrt {\tfrac {A}{c_{0}}}}}$

Derivation of the elliptic integral

Using equations (6), (2) and (4) , n _{3 is} calculated:

${\displaystyle {\dot {n}}_{3}=-pn_{2}=-{\frac {c_{0}n_{1}}{Ar_{0}}}{\frac {{\dot {n}}_{1}}{r_{0}}}=-{\frac {(n_{1}^{2}{\dot {)\;}}}{2\rho ^{2}}}\quad \rightarrow \quad n_{3}={\frac {1}{2\rho ^{2}}}(j\rho -n_{1}^{2})}$

The constant of integration j is dimensionless and proportional to the perpendicular angular momentum L _z . The fifth equation delivers with what is found

${\displaystyle {\dot {n}}_{2}=r_{0}\left({\frac {j}{2\rho ^{3}}}-1\right)n_{1}-{\frac {r_{0}}{2\rho ^{4}}}n_{1}^{3}}$

and the time-derived equation (4) an autonomous differential equation

${\displaystyle {\ddot {n}}_{1}=r_{0}{\dot {n}}_{2}=r_{0}^{2}\left({\frac {j}{2\rho ^{3}}}-1\right)n_{1}-{\frac {r_{0}^{2}}{2\rho ^{4}}}n_{1}^{3}}$

in n ₁ . Multiplication by allows time integration : ${\displaystyle 2{\dot {n}}_{1}}$

${\displaystyle {\dot {n}}_{1}^{2}=D+r_{0}^{2}\left({\frac {j}{2\rho ^{3}}}-1\right)n_{1}^{2}-{\frac {r_{0}^{2}}{4\rho ^{4}}}n_{1}^{4}}$

Because of this , the constant of integration D is not arbitrary, but results from algebraic transformations ${\displaystyle {\dot {n}}_{1}=r_{0}n_{2},\,n_{3}={\tfrac {1}{2\rho ^{2}}}(j\rho -n_{1}^{2}),\,n_{1}^{2}+n_{2}^{2}+n_{3}^{2}=1}$

${\displaystyle {\dot {n}}_{1}=\pm {\frac {r_{0}}{2\rho ^{2}}}{\sqrt {4\rho ^{4}-j^{2}\rho ^{2}+(2j\rho -4\rho ^{4})n_{1}^{2}-n_{1}^{4}}}}$

Separation of the variables leads to an elliptical integral

${\displaystyle \pm {\frac {r_{0}}{2\rho ^{2}}}\int \mathrm {d} t=\int {\frac {\mathrm {d} n_{1}}{\sqrt {4\rho ^{4}-j^{2}\rho ^{2}+(2j\rho -4\rho ^{4})n_{1}^{2}-n_{1}^{4}}}}}$

which solves the problem in principle. The values ​​in it

${\displaystyle \rho ={\sqrt {\frac {A}{c_{0}}}}r_{0},\quad j={\frac {L_{z}}{\sqrt {Ac_{0}}}}=2\rho \left(n_{3}+{\frac {n_{1}^{2}}{2\rho ^{2}}}\right)}$

are constants and exist with the initial conditions in which

${\displaystyle q=0,\quad {\dot {q}}={\frac {1}{B}}(Arp-c_{0}n_{1})=0}$

must be set.

The total energy of the top results from the combined relationships

${\displaystyle E:={\frac {1}{2}}(Ap^{2}+Cr^{2})+c_{0}n_{3}={\frac {2\rho ^{3}+j}{2\rho }}c_{0}}$.

Representation with Jacobi's elliptical functions

Fig. 1: Elliptical module k in the analytical solutions

The polynomial under the root in the elliptic integral for n ₁ has the zeros

${\displaystyle {n_{1}}_{1,2,3,4}=\pm {\sqrt {\rho j-2\rho ^{4}\pm 2\rho ^{2}{\sqrt {\rho \left(\rho ^{3}+{\frac {1}{\rho }}-j\right)}}}}}$

Depending on whether two or four of these zeros are real, different solution functions arise:

• Region I with -2 ρ <j <2ρ only allows two real zeros.
• Region II with allows four real zeros. ${\displaystyle \rho <1,\;2\rho <j<\rho ^{3}+{\tfrac {1}{\rho }}}$

The elliptic moduli of the elliptic solution functions are shown in the picture. When k is close to 0, the elliptical functions deviate only slightly from the sine and cosine . The areas left white are not accessible to the gyroscope if an overlay resulting from the special initial conditions. Comparable solutions arise when ρ and j both change their sign at the same time. For the sake of simplicity, positive values ​​for ρ are assumed below . ${\displaystyle n_{3}={\tfrac {1}{2\rho ^{2}}}(j\rho -n_{1}^{2})}$

Region I.

In region I, | j | <2 ρ and the analytical solution is:

${\displaystyle {\begin{aligned}p=&{\hat {p}}\operatorname {cn} (u,k),&n_{1}=&{\hat {n}}_{1}\operatorname {cn} (u,k)\\q=&0,&n_{2}=&-2dk\operatorname {sn} (u,k)\operatorname {dn} (u,k)\\r=&r_{0},&n_{3}=&{\frac {j}{2\rho }}-2dk^{2}\operatorname {cn} ^{2}(u,k)\end{aligned}}}$

With

${\displaystyle {\begin{aligned}{\hat {p}}=&2k\omega ,&{\hat {n}}_{1}=&{\sqrt {\frac {A}{c_{0}}}}\rho {\hat {p}}\\u=&\pm \omega (t-t_{0})&\omega =&{\sqrt {{\frac {c_{0}}{A}}d}},\\k=&{\sqrt {{\frac {1}{2}}\left(1+{\frac {j-2\rho ^{3}}{2\rho d}}\right)}},&d=&{\sqrt {\rho \left(\rho ^{3}+{\frac {1}{\rho }}-j\right)}}\end{aligned}}}$

The constants j and ρ are given in the # derivation of the elliptic integrals and sn, cn and dn are three basic Jacobian elliptic functions . The angular velocity p has the period where K (k) is the complete elliptic integral of the I type . With this solution p and n _1,2 oscillate periodically around 0. ${\displaystyle {\tfrac {4}{\omega }}K(k)}$

In the limit case j = -2ρ or j  = 2 ρ  > 2, k  = 0 and Staude rotations occur with angular velocity around the vertical 3-axis with n ₃  = -1 or n ₃  = +1. ${\displaystyle r={\sqrt {\tfrac {c_{0}}{A}}}\rho }$

If j = 2ρ  <2, k  = 1 and this results in a #Aperiodic borderline case common to Region II .

Region II

In region II, ρ  <1, 2ρ <j <ρ ³  + 1 / ρ and the analytical solution is:

${\displaystyle {\begin{aligned}p=&{\hat {p}}\operatorname {dn} (u,k),&n_{1}=&{\hat {n}}_{1}\operatorname {dn} (u,k)\\q=&0,&n_{2}=&-2d\operatorname {sn} (u,k)\operatorname {cn} (u,k)\\r=&r_{0},&n_{3}=&{\frac {j}{2\rho }}-{\frac {2d}{k^{2}}}\operatorname {dn} ^{2}(u,k)\end{aligned}}}$

With

${\displaystyle {\begin{aligned}{\hat {p}}=&\pm 2\omega ,&{\hat {n}}_{1}=&\pm \rho ^{2}{\frac {\hat {p}}{r_{0}}}=\pm {\sqrt {\frac {2\rho (j-2\rho ^{3})}{2-k^{2}}}}\\u=&\omega (t-t_{0}),&\omega =&{\sqrt {\frac {j-2\rho ^{3}}{2\rho ^{3}(2-k^{2})}}}r_{0}\\k=&{\sqrt {\frac {4\rho d}{j-2\rho ^{3}+2\rho d}}},&d=&{\sqrt {\rho \left(\rho ^{3}+{\frac {1}{\rho }}-j\right)}}\end{aligned}}}$

The constants j and ρ are given in the # derivation of the elliptic integrals , sn, cn and dn are three basic Jacobian elliptic functions . The angular velocity p has the period where K (k) is the complete elliptic integral of the I type . With this solution, p and n ₁ never change signs and periodically oscillate around a value other than 0. ${\displaystyle {\tfrac {2}{\omega }}K(k)}$

At j = ρ ³  + 1 / ρ , k  = 0 and the top performs a Staude rotation around the vertical with n ₁  = ± √ (1 -  ρ ⁴ ), n ₂  = 0 and n ₃  =  ρ ² . If j  = 2 ρ , the aperiodic limit case occurs.

Aperiodic borderline case

The borderline case j  = 2 ρ  <2 is common to both regions I and II . There k  = 1 and the Jacobian elliptic functions become the aperiodic hyperbolic functions :

${\displaystyle {\begin{aligned}\\p=&{\frac {2\omega }{\cosh(u)}},&n_{1}=&{\frac {2\rho {\sqrt {1-\rho ^{2}}}}{\cosh(u)}}\\q=&0,&n_{2}=&-2(1-\rho ^{2}){\frac {\sinh(u)}{\cosh ^{2}(u)}}\\r=&r_{0},&n_{3}=&1-{\frac {2(1-\rho ^{2})}{\cosh ^{2}(u)}}\end{aligned}}}$

With

${\displaystyle u=\omega (t-t_{0})\quad {\text{und}}\quad \omega ={\sqrt {\frac {c_{0}}{A}}}{\sqrt {1-\rho ^{2}}}}$

The movement changes asymptotically into a Staude rotation with the angular velocity around the vertical 3-axis. ${\displaystyle r={\sqrt {\tfrac {c_{0}}{A}}}\rho }$

stability

The stability of the movement depends not only on the parameters ρ and j , but also on the mean main moment of inertia B. For a real top, this is in the interval A <B <3A. The zones of stable and unstable behavior are separated in the parameter space ρ, j, β = B / A by three-dimensionally curved surfaces. In Region I, tops with B> A that is too large are often unstable. In region II, tops with B close to 2A or ρ> 0.8 are often unstable.

Second solution from Steklow and Markeev

Elliptical module in the second solution from Steklow

In 1899 Steklow found a further solution in which he made the approach n ₂  ~  p · q and n ₃  ~  p · r . This time the center of mass lies on the first main axis belonging to A at a distance s from the support point, which means that the Euler-Poisson equations with the names from the section # Euler-Poisson equations

${\displaystyle {\begin{aligned}A{\dot {p}}=&(B-C)qr\\B{\dot {q}}=&c_{0}n_{3}+(C-A)rp\\C{\dot {r}}=&-c_{0}n_{2}+(A-B)pq\\{\dot {n}}_{1}=&rn_{2}-qn_{3}\\{\dot {n}}_{2}=&pn_{3}-rn_{1}\\{\dot {n}}_{3}=&qn_{1}-pn_{2}\end{aligned}}}$

write. In area I, see picture, the solution is

${\displaystyle {\begin{aligned}p=&-\mu {\sqrt {\frac {A(2B-A)(A-2C)}{(A-C)^{2}(B-A)}}}\operatorname {cn} (u,k)\\q=&\mu {\sqrt {\frac {A^{2}(A-2C)}{(B-A)(A-C)(B-C)}}}\operatorname {sn} (u,k)\\r=&\mu {\sqrt {\frac {A^{2}(2B-A)}{(A-C)^{2}(B-A)}}}\operatorname {dn} (u,k)\\n_{1}=&1-{\frac {A}{A-C}}\operatorname {cn} ^{2}(u,k)\\n_{2}=&{\sqrt {\frac {A(2B-A)}{(A-C)(B-C)}}}\operatorname {sn} (u,k)\operatorname {cn} (u,k)\\n_{3}=&-{\sqrt {\frac {A(A-2C)}{(A-C)^{2}}}}\operatorname {cn} (u,k)\operatorname {dn} (u,k)\end{aligned}}}$

With

${\displaystyle {\begin{aligned}u=&\mu {\sqrt {\frac {A(B-C)}{(A-C)(B-A)}}}(t-t_{0})\\\mu =&{\sqrt {\frac {c_{0}}{A}}}\\k=&{\sqrt {\frac {B-A}{B-C}}}\end{aligned}}}$

So that all coefficients are real, a real rigid body must have 2C <A <B <A + C. It turns out that the vertical angular momentum is zero and the total energy of the top

${\displaystyle E=c_{0}\left({\frac {A^{2}}{2(A-C)(B-A)}}+1\right)}$

becomes. This movement is always unstable.

PA Kuz'min (1952) or AP Markeev (2005) supplemented the solution in Area II:

${\displaystyle {\begin{aligned}p=&\mu {\sqrt {\frac {A(2B-A)(A-2C)}{(A-C)^{2}(A-B)}}}\operatorname {sn} (u,k)\\q=&\mu {\sqrt {\frac {A^{2}(A-2C)}{(A-B)(A-C)(B-C)}}}\operatorname {cn} (u,k)\\r=&\mu {\sqrt {\frac {A^{2}(2B-A)}{(A-B)(A-C)(B-C)}}}\operatorname {dn} (u,k)\\n_{1}=&{\frac {A}{A-C}}\operatorname {sn} ^{2}(u,k)-1\\n_{2}=&{\sqrt {\frac {A(2B-A)}{(A-C)(B-C)}}}\operatorname {sn} (u,k)\operatorname {cn} (u,k)\\n_{3}=&-{\sqrt {\frac {A(A-2C)}{(A-C)(B-C)}}}\operatorname {sn} (u,k)\operatorname {dn} (u,k)\end{aligned}}}$

With

${\displaystyle {\begin{aligned}u=&\mu {\sqrt {\frac {A}{A-B}}}(t-t_{0})\\\mu =&{\sqrt {\frac {c_{0}}{A}}}\\k=&{\sqrt {\frac {A-B}{A-C}}}\end{aligned}}}$

This time the main moments of inertia must meet the conditions 2C <A, B <A <B + C. The vertical angular momentum is zero again and the total energy of the top is:

${\displaystyle E=c_{0}\left({\frac {A^{2}}{2(A-B)(A-C)}}-1\right)}$

The movement in Area II is stable almost everywhere.

Since the trajectories are determined exclusively by the gyro properties , the initial conditions must be precisely matched to the gyro, as in Grioli's precession , because only the time parameter t _{0 is} free.

Individual evidence

1. ↑ K. Magnus : Kreisel: Theory and Applications . Springer, 1971, ISBN 3-642-52163-0 , pp. 131 ( limited preview in Google Book search).
2. ^ Yehia, Hassan, Shaheen (2015), pp. 1173-1185.
3. ↑ Leimanis (1965), p. 92.
4. ↑ Yehia, Hassan, Shaheen (2015), pp. 1180 ff.
5. ↑ ^{a b} Leimanis (1965), p. 96.
6. ↑ ^{a b c d} A. P. Markeev: On the Steklov case in rigid body dynamics . In: Regular And Chaotic Dynamics . tape 10 , no. 1 . Turpion-Moscow Ltd, 2005, ISSN  1560-3547 , p. 81–93 , doi : 10.1070 / RD2005v010n01ABEH000302 .
7. ↑ R. Grammel : The top . Its theory and its applications. Vieweg Verlag, Braunschweig 1920, p. 89 ( Textarchiv - Internet Archive - "Schwung" means angular momentum, "torsional shock" means torque and "torsional balance" means rotational energy).
8. ^ Yehia, Hassan, Shaheen (2015), p. 1177.
9. ↑ ^{a b} Yehia, Hassan, Shaheen (2015), p. 1176.
10. ^ Yehia, Hassan, Shaheen (2015), p. 1182.

literature

• HM Yehia, SZ Hassan, ME Shaheen: On the orbital stability of the motion of a rigid body in the case of Bobylev – Steklov . In: Nonlinear Dynamics . tape 80 . Springer Link, 2015, ISSN  1573-269X , p. 1173-1185 , doi : 10.1007 / s11071-015-1934-3 .
• Eugene Leimanis: The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point . Springer Verlag, Berlin, Heidelberg 1965, ISBN 3-642-88414-8 , p. 92 and 96 , doi : 10.1007 / 978-3-642-88412-2 (English, limited preview in Google Book Search [accessed on March 21, 2018]).