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416. Rotations of a dumbbell equipped with ‘the leier constraint’ A. V. Rodnikov Bauman Moscow State Technical University, Russia e-mail: avrodnikov@yandex.ru (Received: 23 September; accepted: 02 December) Abstract. We consider a special space tethered system consisting of a dumbbell-shaped rigid body and a particle. The particle coast along on the cable. The cable ends are placed in the dumbbell endpoints. We call such system ‘the system with leier constraint (the Dutch term ‘leier’ means the rope with both fixed ends). We assume that the system mass center moves along the circular orbit in the Newtonian Central Force Field. We study the dumbbell's relative motion caused by the particle of small mass in the orbital frame of reference. We deduce a sufficient condition for librations of the dumbbell about its stable equilibrium. We find a family of the dumbbell's asymptotic motions tending to librations about unstable equilibrium. The surface of such asymptotic motions is an interstream separating the areas of the dumbbell's right-hand and left-hand rotations. We deduce an equation of this surface. Key words: space tethered system, leier constraint, asymptotic solution, circular orbit Introduction a dumbbell, i.e. it is composed of particles with masses m1 and m 2 connecting by weightless rod of length 2c. Space tethered systems are one of the most interesting topics in dynamics. For the first time the motion of a particle tethered to a spacecraft has been suggested in [1, 2]. Presently there are hundreds papers devoted to various aspects of the motion of tethered satellites. In this paper we study some generalization of the classic couple. We consider the system that moves in the Newtonian Central Force Field and consists of a dumbbell-shaped rigid body and a particle. The particle coasts along on the cable with ends placed in the dumbbell endpoints. We call such cable ‘a leier’. (the Dutch maritime term ‘leier’ means the rope with both fixed ends). We assume the system mass center describes circular orbit, the cable length is small in comparison with orbit radius, the particle mass is small in comparison with the dumbbell mass, the cable do not leave the orbit plane. We study the dumbbell rotation caused by the small particle in the orbital frame of reference. It is well known that the dumbbell-shaped satellite has two types of relative equilibria. There are the stable ‘vertical’ equilibria and unstable ‘horizontal’ equilibria. We claim that the small particle sufficiently influence the dumbbell relative motion Fig. 1. only if the dumbbell is initially quasi-horizontal. We prove that if the system Jacobi’s integral less than some constant Without loss of generality, m2 ≥ m1 . Suppose the then only librations about the ‘vertical’ equilibrium are particle m3 coast along on the cable with ends fixed to the possible. We note that there exist a set of the dumbbell dumbbell endpoints (Fig.1). This cable can be called ‘a relative motions tending to librations about the ‘horizontal’ leier’. Denote by 2a the cable length. Let C be the mass equilibria. Factually, these asymptotic motions form the center for considered system and O1 be the attracting surface being an interstream between areas of left-hand center. Suppose C moves along the circular orbit, i.e. and right-hand rotations of the dumbbell. We deduce the O1C=r=const and the particles m1 , m 2 , m3 do not leave equation of this interstream. the plane of this orbit. Moreover assume a<<r. Denote by Designations and parameters ϕ the angle between O1C and the rod. Consider a mechanical system consisting of a rigid Evidently, the particle m3 cannot leave the ellipse with body and a particle with mass m3 . Assume that the body is foci in the dumbbell endpoints. The ellipse has eccentricity 557 VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 2008 DECEMBER, VOLUME 10, ISSUE 4, ISSN 1392-8716 416. ROTATIONS OF A DUMBBELL EQUIPPED WITH ‘THE LEIER CONSTRAINT’. A. V. RODNIKOV e = c / a and semi-axises a and b = a 2 − c 2 . Let Oxy be 1 − e 2 ( 1 − e µ cos γ )( ϕ ′ + 1 )2 + a coordinate system with origin in the dumbbell midpoint (see fig.1). Clearly, if x and y is the coordinates of the + 2( 1 − e 2 cos 2 γ )( ϕ ′ + 1 )γ ′ + particle m3 the inequality 3 1 − e2 x + dy − a ≤ 0 ; d = a / b 2 2 2 2 2 (1) + 1 − e 2 γ ′2 − ( 1 − e 2 ) sin 2γ sin 2ϕ + ⋅ 2 2 is valid. The motion of m3 is called the constrained one if ⋅ ( 3 cos 2ϕ cos 2γ + 1 − eµ cos γ ( 1 + 3 cos 2ϕ )) ≥ 0. (1) is equality. In this case the coordinates of m3 can be determine by formulae The dumbbell rotations caused by the small particle x = a cos γ , y = b sin γ (2) Let the mass m3 be small in comparison with the where γ is an eccentric anomaly of the mentioned ellipse. If m3 moves inside the ellipse then we say that the motion dumbbell mass, i.e. k << 1. It is well-known that there exist two types of is the unconstrained one (or the free one). stationary motions of the dumbbell-shaped satellite. There Let µ = ( m2 − m1 ) /( m2 + m1 ) and are ‘the vertical’ equilibria ( ϕ = 0 or ϕ = π ) and ‘the ν = m3 /( m2 + m1 ) . Trivially, 0 < µ < 1 , 0 < e < 1 , ν > 0 , horizontal’ equilibria ( ϕ = ±π / 2 ). It is clear that the dimensionless parameters µ, ν, e and Obviously ‘the vertical’ equilibria are stable. The the variables ϕ, γ determine the considered system particle motion does not destroy these equilibria. Only dynamics completely in the case of constrained motion. some librations of the dumbbell about ‘vertical’ position are possible in this case. Lagrangian and Jacobi’s integral Lagrangian for relative motion of the considered couple has a form [3,4] L = L 2 + L1 + L 0 (3) where L2 = 1 2 { ϕ ′ 2 + k [( 1 − 2 eµ cos γ + e 2 cos 2 γ )ϕ ′ 2 + } { + 1 − e 2 ( 1 − 2 e µ cos γ )ϕ ′γ ′ + ( 1 − e 2 cos 2 γ )γ ′ 2 } L1 = ke cos γ ( e cos γ − 2 µ )ϕ ′ 3 L 0 = −W = cos 2 ϕ + 2 Fig. 2. 9 3 3 + k e 2 cos ϕ − e µ cos γ + e 2 cos 2γ − It can easily be checked that if the dumbbell is 8 2 8 ‘quasi-horizontal’ initially then the particle motion along − 3 4 [ e µ ( 1 − 1 − e 2 ) cos( 2ϕ − γ ) + the leier force the upturning of the dumbbell. The further motion of the dumbbell belongs to one of three types. [+ ( 1 + 1 − e ) cos( 2ϕ + γ )]+ 2 There are ‘the libratory motion’ about the ‘vertical’ equilibria, ‘the rotary motion’ about mass center, the + 3 16 [( 1 − 1 − e ) cos( 2ϕ − γ ) + 2 2 complicated ‘tumbling motion’ consisting of libratory and rotary segments. [+ ( 1 + 1 − e ) cos( 2ϕ + γ )]}. 2 2 Let us remark that the dumbbell tends to librations about its ‘horizontal’ equilibria for some singular initial ν values of ( γ ,γ ′ ) . k= . e (1 − µ 2 ) 2 A sufficient condition for the dumbbell libration Hence we have Jacobi’s integral L2 + W = h. The prime _' ’ ‘ denotes the derivative w.r.t. It is not hard to prove that if Jacobi’s integral constant dimensionless time τ = G M r 1/ 2 1 / 2 −3 / 2 t , where G is the h is smaller than h * = 3 / 8 ⋅ k ( 5 e 2 − 2 ) then only `the gravity constant, M is the mass of the attracting center. libratory motion' is possible. Consider a plot of W (Fig. 2). We see a mountain country consisting of parallel ridges The constrained motion condition ϕ = π / 2 + π k and valleys ϕ = π k , where k is integer. The ridge is the sequence of ‘peaks’ γ = π k and saddle- Note also that the constrained motion is possible only if points γ = π / 2 + π k . In the saddle-point W = h* . [3,4] 558 VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 2008 DECEMBER, VOLUME 10, ISSUE 4, ISSN 1392-8716 416. ROTATIONS OF A DUMBBELL EQUIPPED WITH ‘THE LEIER CONSTRAINT’. A. V. RODNIKOV Therefore if h < h* then the dumbbell ‘cannot pass through where σ ( τ ,γ 1 , h2 ) is T-periodic function of τ . Thus if the the ridge’ and rotations on complete angle are impossible. motion is constrained then For instance, the libratory motion is observed for any D = D ( γ ′′( τ , ,γ 1 , h2 ), γ ′( τ , ,γ 1 , h2 ), γ ( τ , ,γ 1 , h2 ) = D1 ( τ ) is T- initial value of ϕ and zero initial velocities if initial value periodic function of τ . (Here h2 depends on ( γ 1 , γ 1 ) ). ′ of γ is about π / 2 . It can be shown numerically that ‘the rotary motion’ is guaranteed only if the initial value of γ ′ The reduced equations’ solutions is sufficiently big. Solutions of (4) can be represented in a form The motion equations reduction for the symmetric ψ ( τ ) = p( τ ) + q( τ ) , where dumbbell p( τ ) = 2 3 k ( +∞ exp( τ 3 )∫τ exp( −ξ 3 )D1( ξ )dξ + ) Note that ‘the tumbling motion’ is a set of right-hand and left-hand rotations with close to 1800 angles. Factually, ( τ + exp( −τ 3 )∫−∞ exp( ξ 3 )D1( ξ )dξ , ) chaotic rotations of the dumbbell are obtained. Consider a single rotation from this set. Let q( τ ) = 1 2 3 ( C1 exp( τ 3 ) + C2 exp( −τ 3 ) ) (7) ′ ′ ( γ 1 ,γ 1 ,ϕ1 ,ϕ 1 ) be values of ( γ , γ ′ ,ϕ ,ϕ ′ ) in the beginning From equalities of this rotation. It is clear that ϕ ′ ≈ 0 and ϕ 1 ≈ ± π / 2 . p( τ + T ) = (exp((τ + T ) (Without loss of generality it can be assumed that k +∞ ϕ 1 ≈ − π 2 ). It is obvious that the motion in the vicinity of = 3 )∫τ +T exp( −ξ 3 )D1( ξ )dξ + 2 3 ) `horizontal' equilibrium determine the direction of the τ +T considered rotation. Substituting ϕ ≈ −π / 2 + kψ in the + exp( −( τ + T ) 3 )∫−∞ exp( ξ 3 )D1( ξ )dξ = ( ) dumbbell's motion equation we obtain k ψ ′ − 3ψ + D k = 0 , (4) = exp( τ + T ) 3 ⋅ 2 3 2( 1 − e 2 cos 2 γ )γ ′ + e 2 γ ′ sin 2 γ − +∞ (5) ⋅ ∫τ exp( −( ζ + T ) 3 )D1( ζ + T )dζ + − 3( 1 − e ) sin γ = 0 2 + exp( −( τ + T ) 3 ) ⋅ where D = 1 − e γ ′′ − e 2 γ ′ sin 2γ − 3 / 2 1 − e 2 sin 2γ . (⋅ ∫ ) 2 τ +T −∞ exp((ζ + T ) 3 ) D1(ζ + T )dζ = Here we are restricted to a case of symmetric dumbbell µ = 0 ⇔ m1 = m 2 and neglect the terms of order higher than k . = k 2 3 ( +∞ expτ 3 )∫τ exp( −ξ 3 )D1( ξ )dξ + ) Note that (5) is equation of motion for the particle if the dumbbell is fixed in the ‘horizontal’ position [3]. τ + exp( −τ 3 )∫−∞ exp( ξ 3 )D1( ξ )dξ = p( τ ) ) Equality it follows that p( τ ) is T-periodic function of τ . Constants ( 1 − e 2 cos 2 γ )γ ′ 2 + 3( 1 − e 2 ) cos 2 γ = h2 (6) C1 and C2 are defined by formulae is the Jacobi’s integral for (5). Analyzing phase portrait of 3π (6) we see that there exist three types of equation (5) C1 = k −1 / 2 3ϕ1 + + ϕ1 − k 1 / 2 A , ′ 2 solutions. Solutions of the first type correspond to librations about ± π / 2 . They are periodic functions of 3π C2 = k −1 / 2 3ϕ1 + − ϕ1 − k 1 / 2 B , ′ τ with period 2 dγ T =∫ . where γ ′( γ , h2 ) +∞ ξ A = ∫0 e −ξ 3 D1( ξ )dξ , B = ∫− ∞ eξ 3 D1( ξ )dξ . Solutions of the second type correspond to the asymptotic motions tending to γ = 0 or γ = π . It can easily be checked that in this case the cable weakens. Let us remark A surface of asymptotic motions as an interstream that such effect is also observed for the motions in some Clearly, if C1>0 then the dumbbell will turn vicinity of the separatrix. Solutions of the third type counterclockwise and if C1<0 then the dumbbell will turn correspond to rotations about the dumbbell. In this case clockwise. Certainly, this criterion is valid only for the derivative of γ w.r.t. τ is the periodic function with period constrained motion. π dγ If C1=0 then the dumbbell remain in the vicinity of T =∫ . 0γ ′( γ , h2 ) horizontal equilibrium, i.e. we have the dumbbell Moreover, solutions of (5) can be represented in a form asymptotic motion tending to librations about ϕ = −π / 2 π (or ϕ = π / 2 ). Clearly, this asymptotic motion is unstable. ′ γ = γ ( τ , γ 1 , h2 ( γ 1 ,γ 1 )) = τ + σ ( τ ,γ 1 , h2 ) T Thus the equation 559 VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 2008 DECEMBER, VOLUME 10, ISSUE 4, ISSN 1392-8716 416. ROTATIONS OF A DUMBBELL EQUIPPED WITH ‘THE LEIER CONSTRAINT’. A. V. RODNIKOV 3π +∞ The similar interstream for ϕ 1 = − 90 ° 3′ ; ϕ1 = 0 is 3ϕ1 + +ϕ ′1= k ∫0 e −ξ 3 D1( ξ )dξ (8) depicted in Figure 4. Here also e=1/2. 2 define a surface of asymptotic motions in the four- In figures 2 and 3 the shadowed area corresponds to the motion with the weakened cable. ′ ′ dimensional space of ( γ 1 ,γ 1 ,ϕ1 ,ϕ 1 ) . In other words, (8) is the equation of an original interstream dividing the On integral A computation space of initial values into the areas of rotations clockwise and rotations counterclockwise. Note also that if C1= C2=0 Finally note that the infinite integral A is reduced up to then we have the dumbbell periodic motion about definite. It follows from equalities +∞ horizontal equilibrium. A = ∫0 exp( −ξ 3 )D1( ξ )dξ = ∞ ( n +1 )T =∑ ∫ exp( −ξ 3 )D1( ξ )dξ = n =0 nT ∞ T = ∑ ∫ exp( −( ζ + nT ) 3 )D1( ζ + nT )dξ = n =0 0 T ∞ = ∫ exp( −ζ 3 )D1( ζ )dζ ∑ exp( − nT 3 ) = 0 n =0 T 1 = ∫ exp( −τ 3 )D1( τ )dτ 1 − exp( −T 3 ) 0 Further, using dτ = γ ′( τ , ,γ 1 , h2 )dγ we can change the variable in the last integral. For instance, consider the area of the particle ‘positive rotations’. In this area h2 > 3( 1 − e 2 ) and γ ′ > 0 . Here using (6) we get Fig. 3. γ ′ = r( h2 ,γ ) , (9) where h2 − 3( 1 − e 2 ) r( h2 ,γ ) = , 1 − e 2 cos 2 γ h2 = ( 1 − e 2 cos 2 γ 1 )γ 1 2 + 3( 1 − e 2 ) cos 2 γ 1 . ′ From (9) it follows that γ dγ T = ∫γ 1 r ( h 2 ,γ ) and π dγ T = ∫0 . r ( h2 , γ ) Hence 1 Fig. 4. A= . 1 − exp( −T 3 ) Examples of interstreams π +γ1 γ dξ D ( h2 ,γ ) ⋅ ∫γ exp − 3 ∫ dγ , The right side of (8) depends only on γ 1 ,γ 1 and left ′ 1 γ 1 r ( h2 ,γ ) r ( h2 ,γ ) ′ side depends only on ϕ1 ,ϕ1 . Therefore the interstreams can be depicted in the plane ( γ 1 ,γ 1 ) for fixed values of ′ where ϕ1 ,ϕ1′ . In particular, if the dumbbell is precisely horizontal D 2 ( h2 ,γ ) = − e 2 sin 2γ ⋅ ( ϕ1 = − π / 2 ; ϕ1 = 0 at the beginning of considered ′ rotation then (8) is reduced up to the equality A=0. The ⋅ ( 1 − e 2 3 sin 2 γ + r ( h ,γ ) 2 + ) r ( h2 , γ ) . corresponding interstream is depicted in Fig. 3 for e=1/2. 2( 1 − e cos γ ) 2 2 In this figure the areas of right-hand and left-hand rotations are marked by the circular arrows. 560 VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 2008 DECEMBER, VOLUME 10, ISSUE 4, ISSN 1392-8716 416. ROTATIONS OF A DUMBBELL EQUIPPED WITH ‘THE LEIER CONSTRAINT’. A. V. RODNIKOV Conclusions References In this paper the space tethered system consisting of the dumbbell-shaped rigid body and the particle of small mass [1] Beletsky, V. V. and Novikova, E. T. On the Relative Motion is considered. The particle moves along the cable with of Two Tethered Bodies on an Orbit. //Cosmic Research, v.7, ends fixed in the body. The dumbbell rotations caused by No 3, pp.377-384. (1969). the particle are studied. The sufficient condition of the [2] Beletsky, V. V. On the Relative Motion of Two Tethered Bodies on an Orbit II. //Cosmic Reseach ,v.7,No 6. (1969). dumbbell librations about its stable equilibrium is [3] Rodnikov, A. V. Equilibrium Positions of a Weight on a obtained. The family of asymptotic motions tending to Cable Fixed to a Dumbbell-Shaped Space Station moving librations about unstable equilibria is found. This family along a circular geocentric orbit.// Cosmic Reseach,v. 44,No forms the interstream separating the area of the dumbbell 1, pp. 58-68. (2006). rotations clockwise from the area of rotations [4] Rodnikov A. V. The algorithms for capture of the space counterclockwise. The equation of the interstream is garbage using ‘leier constraint’.: Regular and Chaotic deduced. Dynamics, v.11, 4, pp. 483-489. (2006). Acknowledgment The author thanks V.V.Beletsky, Yu.F.Golubev, I.I.Kosenko and V.V.Sazonov for useful discussions. 561 VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 2008 DECEMBER, VOLUME 10, ISSUE 4, ISSN 1392-8716