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The equilibrium positions of a satellite carrying a three-degree-of-freedom powered gyroscope in a central gravitational field
Journal of Applied Mathematics and Mechanics 77 (2013) 181–189
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The equilibrium positions of a satellite carrying a three-degree-of-freedom powered gyroscope in a central gravitational field夽 N.I. Amel’kin Moscow, Russia
a r t i c l e
i n f o
Article history: Received 6 December 2011
a b s t r a c t For a satellite, carrying an arbitrary number of three-degree-of-freedom powered gyroscopes, the whole set of equilibrium positions in a central gravitational field in a circular orbit is determined and a detailed analysis of their secular stability is presented. The asymptotic properties of the satellite motions when there is dissipation in the axes of the gyroscope frames are investigated. © 2013 Elsevier Ltd. All rights reserved.
The equilibrium positions of a satellite with a two-degree-of-freedom powered gyroscope in a circular orbit were investigated in Refs 1-3. A similar problem was solved for a satellite with a three-degree-of-freedom gyroscope in gimbals,4 and the case when the “satellitegyroscope” system does not possess the property of a gyrostat was considered, but it was assumed that the axis of the outer frame of the gyroscope is parallel to one of the principal central axes of inertia of the satellite body. In this paper we investigate the equilibrium positions of a satellite with an arbitrary number of three-degree-of-freedom powered gyroscopes, set up in an arbitrary way in the carrier, assuming that the “carrier–gyroscope” system possesses the property of a gyrostat. 1. The equations of motion of the satellite Consider a satellite consisting of a carrying rigid body and N three-degree-of-freedom powered gyroscopes in gimbals (Fig. 1). We will denote by ik , sk and hk unit vectors which indicate the directions of the axes of the outer frames, the inner frames and the rotors, respectively. The axes of the outer frames ik are fixed in the carrier, the positions of the axes of the inner frames sk are defined by the precession angles k , while the positions of the rotor axes hk are defined by the angles of precession k and nutation k . We will assume that, for each gyroscope, the axis of the inner frame is orthogonal to the axis of the outer frame and to the rotor axis, i.e., (1.1) while the angular velocities of rotation of the rotors ϕ˙ k are maintained constant. Then, we will have for the angular momenta of natural rotation of the rotors
We will also assume that the “carrier – gyroscope” system possesses the property of a gyrostat. This property exists when the following conditions are satisfied:5 1) the rotor of each gyroscope is statically balanced and dynamically symmetric about the hk axis; 2) the “inner frame – rotor” system is statically balanced and dynamically symmetric about the sk axis; we will denote the moment of inertia of this system about the sk axis by Jk ; 3) the weight of the gyroscope is statically balanced and dynamically symmetric about the axis of the outer frame ik ; we will denote the moment of inertia of the gyroscope about the ik axis by Ik .
夽 Prikl. Mat. Mekh., Vol. 77, No. 2, pp. 251–262, 2013. E-mail address: [email protected] 0021-8928/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jappmathmech.2013.07.008
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Fig. 1.
When the above conditions are satisfied, the inertia tensor of the “carrier – gyroscope” system in any basis connected with the body, will not change when the frames of the gyroscopes rotate, i.e., the system will be a gyrostat. This tensor, calculated in the basis of the principal central axes of inertia of the satellite e1 , e2 and e3 , will be denoted by J, while the responding resultant moments of inertia will be denoted by A, B and C. We will analyse the motions of the satellite within the framework of the limited circular problem, i.e., assuming that the centre of mass of the satellite moves in a Kepler circular orbit. The orbital basis is defined by the mutually orthogonal unit vectors r, n and = n × r, directed along the orbital radius, along the normal to the orbital plane and tangential to the orbit, respectively. The angular velocity of the orbital basis 0 = 0 n is constant in an inertial basis. The equations of motion of the satellite can be obtained from the theorem of the change in the angular momentum, and using it for the whole system and for individual components. The equations of motion in a uniform external field (when there are no external force moments) were obtained in Ref. 5. For the case of a central gravitational field, considered in this paper, these equations take the form
(1.2)
(1.3)
(1.4) Here is the vector of the absolute angular velocity of the body of the satellite and G is the total angular momentum of the gyroscope rotors. These equations differ from the equations of motion in a uniform field5 solely by the presence of the gravitational moment on the righthand side of Eq. (1.2), which acts on the satellite. For conditions 1-3 mentioned above, the central field forces do not produce moments about the ik and sk axes. Hence M1k and M2k are the moments of the internal forces in the precession axis ik and the nutation axis sk , respectively. For passive gyroscopes these moments may be due to the presence of springs and dampers in the frame axes. From Eqs (1.2)–(1.4) we can obtain equations describing the motions of the satellite about the orbital basis, if we express the absolute angular velocity and absolute angular acceleration of the satellite by the formulae (1.5) ˙ are the angular velocity and angular acceleration of the satellite about the orbital basis. In Eqs (1.2)–(1.4) the vectors Here and r and n are specified in a basis connected with the body of the satellite. Hence, to obtain a closed system we must add Poisson’s kinetic equations to these equations.
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2. Equilibrium positions We will now consider a satellite with passive powered gyroscopes, i.e., we will assume that only the moments of potential and dissipative forces can act in the axes of the frames of the gyroscopes. Equations, defining the relative equilibrium positions of the satellite can be obtained from Eqs (1.2)–(1.4), by putting
where is the energy of the potential forces acting on the axes of the frames. These equilibrium positions correspond to stationary points of modified potential energy1
(2.1) The “reduced” angular momenta of the rotors with a dimension of moment of inertia are denoted by Hk = Gk /0 . We will assume further that only moments of dissipative forces can act in the axes of the gyroscope frames, i.e., ≡ 0. Since the vectors n and r are unit vectors and mutually orthogonal, the stationary points of function (2.1) correspond to stationary points of the Lagrange function with factors (2.2) Taking the following equalities into account (2.3) we obtain the following system of equations for the equilibrium positions (2.4) (2.5) (2.6) (2.7) Equations (2.6) and (2.7) describe the equilibrium conditions of the gyroscopes with respect to the satellite body, while Eqs (2.4) and (2.5) are equivalent to the single equation (2.8) which describes the equilibrium condition relative to the orbital basis of the satellite with a rotor having reduced angular momentum H. It follows from Eqs (2.6) and (2.7) that, in equilibrium positions, each vector Hk should satisfy one of the conditions (2.9) (2.10) Actually, condition (2.9) does not contradict Eqs (2.6)–(2.8). If we assume that ik × Hk = / 0, then, taking conditions (1.1) into account, we will have from Eq. (2.6) nT sk = 0. Since, by virtue of Eq. (2.7), the vector n lies in the plane of the mutually orthogonal vectors sk and Hk , it follows from the equality nT sk = 0 that the vector n is parallel to the vector Hk , i.e., condition (2.10) is satisfied. Note that, when conditions (2.9) are satisfied the frames of the corresponding gyroscopes “add together”, i.e., the inner frame is parallel to the outer frame. We will determine the equilibria for which all the vectors Hk satisfy condition (2.10). In this case Eq. (2.8) takes the form
It describes relative equilibria of the rigid body in a circular orbit and has 24 different solutions {n, r}, for each of which the principal axes of inertia of the body are parallel to the axes of the orbital basis. All the equilibria of the “carrier – gyroscopes” system, corresponding to the case considered, are described by the formulae (2.11) Here and henceforth we will use the symbols ␦k to describe multivalued solutions, each of which can take two values: +1 and −1. We will investigate the solutions for which at least one of the vectors Hj satisfies condition (2.9). Suppose, to be specific, (2.12)
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From Eq. (2.8) we obtain the equation
(2.13) which describes the equilibria of the satellite with a rotor having a fixed angular momentum H*. it is well-known,6,7 that, depending on the value of the vector H* and its direction in the basis of the principal axes of inertia of the satellite, Eq. (2.13) has from 8 to 24 solutions {n, r}, which define the orientation of the housing of the satellite with respect to the orbital basis. After determining the solutions using Eqs (2.7), we determine the values of the angles of precession (the positions of the axes sj ) for the first m gyroscopes, and on the basis of conditions (2.10) and Eqs (2.6) and (2.7), we obtain the precession and nutation angles for the remaining gyroscopes. The number of different values of the vector H* is
while the number of different orientations of the satellite {n, r}, determined from Eqs (2.13), may range from 8S to 24S. 3. Analysis of the secular stability The equilibrium position of the satellite will be stable in the secular sense (it will satisfy the sufficient stability conditions), if, at the point considered, function (2.1) has a strict minimum, and unstable in the secular sense if there is no minimum. The nature of the secular stability is determined by analysing the values of the second differential of function (2.2) in the set (3.1) The second differential of function (2.2) is described by the expression
Using equalities (3.1), the variations of the vectors of the orbital basis can be expressed in terms of the independent variations u, v and w by the following formulae (3.2) We will then have (3.3) The Lagrange multipliers are expressed from Eqs (2.4) and (2.5) by the formulae (3.4) Taking into account relations (3.2)–(3.4) and the inequality form of the independent variables u, , w, dk , dk (k = 1,. . ., N)
T
J r = 0, which follows from Eq. (2.5), we obtain the following quadratic
(3.5) Equilibria for which quadratic form (3.5) is strictly positive definite will possess secular stability. If, among the eigenvalues of the matrix of quadratic form (3.5), there are negative values, the corresponding equilibria will be unstable in the secular sense. We will investigate the secular stability of solutions (2.12). Consider the set of variations
where j is the number of the gyroscope which satisfies condition (2.9). In this set, quadratic form (3.5) becomes (3.6)
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Fig. 2.
When nT sj = / 0 it is indeterminate, i.e., the equilibrium is unstable in the secular sense. When nT sj = 0, taking Eqs (2.7) into account, we obtain n × Hj = 0, i.e., condition (2.10) is satisfied. Hence, the necessary condition for secular stability is the satisfaction of conditions (2.10) for all the vectors Hk , i.e., equilibria that are stable in the secular sense exist among solutions (2.11). We will investigate the secular stability of solutions (2.11). To be specific, we will consider the solutions (3.7) We will denote by ␣k and k the angles which specify the position of the precession axis of the k-th gyroscope in the basis of the principal axes of inertia of the satellite (Fig. 2). For solutions (3.7), by virtue of conditions (1.1), we have sTk n = sTk e3 = 0, i.e., all the vectors sk lie in the principal plane e1 , e2 . Quadratic form (3.5) then becomes
(3.8) In the case when, for all the gyroscopes, the axes of the outer frames are not parallel to the principal axis of inertia of the satellite e3 , i.e., sink = / 0, by a non-degenerate conversion of the variables (3.9) this quadratic form is reduced to the form
(3.10) Hence we obtain that secular stability of the solutions considered occurs when the following conditions are satisfied (3.11) It follows from these that the satellite has only four stable equilibrium positions in the secular sense (3.12) which coincide with the stable equilibrium positions of a rigid body in a circular orbit. For these the e3 axis of the greatest moment of inertia of the satellite is directed along the normal to the orbital plane, while the e1 axis of the least moment of inertia is directed along the orbital radius. The angular momentum of the rotor of each gyroscope is then parallel to the normal to the orbital plane and has a positive projection onto this normal. If, for certain gyroscopes, the axis of the outer frame is parallel to the principal axis of inertia of the satellite e3 , i.e., sinj = 0, quadratic form (3.8) can be written as
(3.13)
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In this case conditions (3.11) ensure secular stability of solutions (3.7) with respect to all the phase variables, apart from the precession angles j . 4. The asymptotic stability of the equilibria When there is dissipation in the axes of the gyroscope frames, a more detailed analysis of the stability of the equilibria obtained is possible using the Barbashin–Krasovskii theorem.8 This theorem determines, for systems with partial dissipation, the conditions for which its asymptotic stability follows from the secular stability of the equilibrium, and for which Lyapunov instability follows from the secular instability. It was shown previously,2 that the conditions of the Barbashin–Krasovskii theorem are necessarily satisfied if the equilibrium considered is not a bifurcation point, while the truncated equations of the linear approximation have only a trivial solution. The truncated equations of the linear approximation are an overdetermined system. They are obtained by linearization of the equations of motion for fixed values of those variables for which dissipative forces act. We will write the truncated equations of the linear approximation for the problem in question, assuming that dissipative moments act in the axis of the frames of all the gyroscopes. We will denote by the vector of the small rotation of the housing of the satellite about the orbital basis. Then, in the linear approximation, we will have (4.1) while the variations of the vectors of the orbital basis about the housing of the satellite will be expressed by the formulae (4.2) For the variations in the angular velocity and angular acceleration of the satellite we obtain (4.3) In Eqs (4.2) and (4.3), n and r mean the values of the orbital balance vectors in the equilibrium position. Taking these equalities into ˙ = 0 for fixed values of the precession and nutation account, and linearizing Eqs (1.2)–(1.4) in the neighbourhood of equilibrium = 0, angles, we arrive at the following system of equations
(4.4) The primes denote derivatives with respect to the dimensionless time = 0 t, while the vectors f() and F(U) are found from the formulae2 (4.5) In these equations only the components of the vectors and U are the variables, while the vectors n, r, Hk and sk are fixed and equal to their values in the equilibrium position. Equations (4.4) form an overdefined system of 2N + 6 scalar equations for six variables. We will investigate the question of the asymptotic stability of equilibrium positions (3.12), which satisfy the sufficient conditions for stability (Fig. 2). For these, system of equations (4.4) takes the form (4.6) (4.7) (4.8) (4.9) (4.10)
(4.11) It is easy to see that for the conditions (4.12) system (4.6)–(4.11) allows of the following non-trivial solutions (4.13) They correspond to in-plane oscillations of the satellite about the normal to the orbital plane. It should be noted that, for conditions (4.12) and nonlinear system (1.2)–(1.4), similar oscillations are allowed for the fixed frames of the gyroscopes, i.e. equilibrium positions (3.12) in this case will not be asymptotically stable. We will show that if the inequality cosm = / 0 holds for at least one of the gyroscopes, equilibrium positions (3.12) may be asymptotically stable.
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For simplicity, consider the case
Then subsystem (4.10), (4.11) will contain the equations (4.14) (4.15) When the inequality
is satisfied, from Eqs (4.6), (4.14), (4.9) and (4.7) we will have (4.16) (4.17) Differentiating equality (4.17) and taking equalities (4.16) and (4.9) into account, we obtain (4.18) (4.19) Since, for the equilibra considered, the inequality
is satisfied, and it follows from Eqs (4.17) and (4.19) that (4.20) In turn, taking (4.16) and (4.20) into account, we obtain from Eqs (4.15), (4.8) and (4.9) (4.21) Hence, in the example considered, system (4.6)–(4.11) has only the trivial solution U = = 0, as a consequence of which when there is dissipation in the axes of the gyroscope frames, equilibrium position (3.12) is asymptotically stable. A detailed investigation of the asymptotic properties of the satellite motions in the neighbourhood of equilibrium positions (3.12) was carried out using numerical analysis. We investigated the roots S of the characteristic equation of the complete system of equations of the linear approximation and we determined the degree of stability (4.22) i.e., the value of the real part of the most right-hand root, taken with the opposite sign. We analysed how the degree of stability depends on the values of the satellite parameters – the resultant central moments of inertia of the satellite, the moments of inertia of the gyroscopes Ik and Jk , the angular momenta of the rotors Hk , the angles ␣k and k of the setting of the axes of precession of the gyroscopes and the damping factors in the frame axes. The linearized equations of motion of the satellite, written in the neighbourhood of the equilibria, were obtained from Eqs (1.2)–(1.4), taking equalities (2.6)–(2.8) and (4.2)–(4.4) into account, and have the form
(4.23)
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Here xk and yk are the deviations of the precession and nutation angles from their values in the equilibrium position, and 1k and 2k are the damping factors in the precession and nutation axes, having the dimension of the moment of inertia, while the primes denote derivatives with respect to the dimensionless time = 0 t. For equilibrium positions (3.12), Eqs (4.23) take the form
(4.24) The vectors ik , sk , jk , f() and F(U) are expressed by the formulae
Here 1 , 2 , 3 and U1 , U2 and U3 are the components of the vectors and U in the basis of the principal axes of inertia of the satellite. The results of a numerical analysis showed that the degree of stability of equilibrium positions (3.12) depends considerably on the value of the ratio
where Im are the greatest of the moments of inertia of the gyroscopes Ik (k = 1,. . ., N). For the overwhelming majority of values of the satellite parameters this relation has the quadratic form (4.25) while for individual (optimal) values of the parameters it has the linear form (4.26) Here the value of the degree of stability is very sensitive to deviations of the parameters from their optimum values, i.e., small deviations lead to a sharp reduction in the degree of stability. As a consequence of this, the values of the degree of stability obtained in practice are described by relation (4.25). It follows from the results obtained that, for practical satellites, in which the value of ␥ does not exceed 10−4 , the degree of stability of equilibrium positions (3.12) with respect to all the phase variables is comparatively small. Even for optimum values of the parameters, which are difficult to obtain in practice, it does not exceed a value of = 10−4 , i.e., the amplitude of small oscillations is reduced by a factor of e after 10−4 /(2) revolutions of the satellite. For comparison we note that, when using a two-degree-of-freedom gyroscope one can obtain stable equilibrium positions of the satellite with a degree of stability = 0.42.3 In the problem considered the comparatively low value of the degree of stability is due to damping of the satellite oscillations about the normal to the orbital plane that is too weak. Note that the conclusion regarding the weak damping of the satellite oscillations about the normal to the orbital plane can be drawn on the basis of an analysis of the precession equations of motion of the satellite, which are obtained from Eqs (1.2)–(1.4) when Ik = Jk = 0. The asymptotic stability of equilibrium positions (3.12) does not follow from these equations, since they allow of solutions corresponding to unattenuating oscillations of the satellite about the normal to the orbital plane for fixed frames of the gyroscopes. We will analyse the asymptotic properties of the satellite motions using the precession equations, linearized in the neighbourhood of equilibrium positions (3.12). Assuming Ik = Jk = 0 in Eqs (4.24), we obtain a system, which can be decomposed into two independent subsystems,
(4.27)
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(4.28) Subsystem (4.28) describes the law of variation of the angle of rotation of the satellite about the normal to the orbital plane, while subsystem (4.27) describes the law of motion of the satellite with respect to the remaining variables. It is obvious that the equilibria considered do not possess asymptotic stability with respect to the angle 3 . But the angle 3 and its derivative ’3 = U3 do not occur in subsystem (4.27), which enables the asymptotic stability to be analysed with respect to the remaining variables, i.e., one can investigate the speed at which the satellite tends towards rotations about the normal to the orbital plane. We investigated the roots of the subsystem characteristic equation (4.27) as a function of the values of the parameters for a satellite with a single gyroscope. We obtained a value of the local maximum of the degree of stability max = 0.35, which is obtained for the following values of the parameters
For such values of the parameters, the amplitude of small oscillations of the satellite with respect to the variables characterizing a departure from rotations about the normal to the orbital plane, is reduced by a factor of e due to semi-rotation of the satellite. Acknowledgements This research was supported financially by the Government of the Russian Federation within the framework of contracts with the Ministry of Science No. 13.G25.31.0028 and No. 14.740.11.0149 of the Federal Special Purpose Programme “Scientific and ScientificPedagogic Teams of Innovative Russia” in 2009-2013. References 1. Amel’kin NI. Steady motions of a satellite with a two-degree-of-freedom powered gyroscope in a central gravitational field and their stability. J Appl Math Mech 2009;73(2):169–78. 2. Amel’kin NI. Analysis of the stability of the equilibria of a satellite carrying a two-degree-of-freedom powered gyroscope with dissipation in the axes of the frame. J Appl Math Mech 2010;74(4):406–15. 3. Amel’kin NI. The equilibria and stability of a dynamically symmetrical satellite with a two-degree-of-freedom powered gyroscope. J Appl Math Mech 2010;74(5):513–23. 4. Sulikashvili RS. The stability of the steady motions of a satellite with a gyroscope in a central gravitational field. Zh Vychisl Mat Mat Fiz 1969;9(2):479–82. 5. Amel’kin NI. The steady motions of a rigid body carrying a three-degree-of-freedom powered gyroscope, and their stability. Izv Ross Akad Nauk MTT 2011;3:3–17. 6. Sarychev VA, Gutnik SA. The relative equilibra of a satellite-gyrostat. Kosmich Issled 1984;22(3):323–6. 7. Rubanovskii VN. Bifurcation and stability of the relative equilibria of a gyrostat. satellite. J Appl Math Mech 1991;55(4):450–5. 8. Rouche N, Habets P, Lalou M. Stability Theory by Liapunov’s Direct Method. N.Y: Springer; 1977.
Translated by R.C.G.
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The equilibrium positions of a satellite carrying a three-degree-of-freedom powered gyroscope in a central gravitational field夽 N.I. Amel’kin Moscow, Russia
a r t i c l e
i n f o
Article history: Received 6 December 2011
a b s t r a c t For a satellite, carrying an arbitrary number of three-degree-of-freedom powered gyroscopes, the whole set of equilibrium positions in a central gravitational field in a circular orbit is determined and a detailed analysis of their secular stability is presented. The asymptotic properties of the satellite motions when there is dissipation in the axes of the gyroscope frames are investigated. © 2013 Elsevier Ltd. All rights reserved.
The equilibrium positions of a satellite with a two-degree-of-freedom powered gyroscope in a circular orbit were investigated in Refs 1-3. A similar problem was solved for a satellite with a three-degree-of-freedom gyroscope in gimbals,4 and the case when the “satellitegyroscope” system does not possess the property of a gyrostat was considered, but it was assumed that the axis of the outer frame of the gyroscope is parallel to one of the principal central axes of inertia of the satellite body. In this paper we investigate the equilibrium positions of a satellite with an arbitrary number of three-degree-of-freedom powered gyroscopes, set up in an arbitrary way in the carrier, assuming that the “carrier–gyroscope” system possesses the property of a gyrostat. 1. The equations of motion of the satellite Consider a satellite consisting of a carrying rigid body and N three-degree-of-freedom powered gyroscopes in gimbals (Fig. 1). We will denote by ik , sk and hk unit vectors which indicate the directions of the axes of the outer frames, the inner frames and the rotors, respectively. The axes of the outer frames ik are fixed in the carrier, the positions of the axes of the inner frames sk are defined by the precession angles k , while the positions of the rotor axes hk are defined by the angles of precession k and nutation k . We will assume that, for each gyroscope, the axis of the inner frame is orthogonal to the axis of the outer frame and to the rotor axis, i.e., (1.1) while the angular velocities of rotation of the rotors ϕ˙ k are maintained constant. Then, we will have for the angular momenta of natural rotation of the rotors
We will also assume that the “carrier – gyroscope” system possesses the property of a gyrostat. This property exists when the following conditions are satisfied:5 1) the rotor of each gyroscope is statically balanced and dynamically symmetric about the hk axis; 2) the “inner frame – rotor” system is statically balanced and dynamically symmetric about the sk axis; we will denote the moment of inertia of this system about the sk axis by Jk ; 3) the weight of the gyroscope is statically balanced and dynamically symmetric about the axis of the outer frame ik ; we will denote the moment of inertia of the gyroscope about the ik axis by Ik .
夽 Prikl. Mat. Mekh., Vol. 77, No. 2, pp. 251–262, 2013. E-mail address: [email protected] 0021-8928/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jappmathmech.2013.07.008
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Fig. 1.
When the above conditions are satisfied, the inertia tensor of the “carrier – gyroscope” system in any basis connected with the body, will not change when the frames of the gyroscopes rotate, i.e., the system will be a gyrostat. This tensor, calculated in the basis of the principal central axes of inertia of the satellite e1 , e2 and e3 , will be denoted by J, while the responding resultant moments of inertia will be denoted by A, B and C. We will analyse the motions of the satellite within the framework of the limited circular problem, i.e., assuming that the centre of mass of the satellite moves in a Kepler circular orbit. The orbital basis is defined by the mutually orthogonal unit vectors r, n and = n × r, directed along the orbital radius, along the normal to the orbital plane and tangential to the orbit, respectively. The angular velocity of the orbital basis 0 = 0 n is constant in an inertial basis. The equations of motion of the satellite can be obtained from the theorem of the change in the angular momentum, and using it for the whole system and for individual components. The equations of motion in a uniform external field (when there are no external force moments) were obtained in Ref. 5. For the case of a central gravitational field, considered in this paper, these equations take the form
(1.2)
(1.3)
(1.4) Here is the vector of the absolute angular velocity of the body of the satellite and G is the total angular momentum of the gyroscope rotors. These equations differ from the equations of motion in a uniform field5 solely by the presence of the gravitational moment on the righthand side of Eq. (1.2), which acts on the satellite. For conditions 1-3 mentioned above, the central field forces do not produce moments about the ik and sk axes. Hence M1k and M2k are the moments of the internal forces in the precession axis ik and the nutation axis sk , respectively. For passive gyroscopes these moments may be due to the presence of springs and dampers in the frame axes. From Eqs (1.2)–(1.4) we can obtain equations describing the motions of the satellite about the orbital basis, if we express the absolute angular velocity and absolute angular acceleration of the satellite by the formulae (1.5) ˙ are the angular velocity and angular acceleration of the satellite about the orbital basis. In Eqs (1.2)–(1.4) the vectors Here and r and n are specified in a basis connected with the body of the satellite. Hence, to obtain a closed system we must add Poisson’s kinetic equations to these equations.
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2. Equilibrium positions We will now consider a satellite with passive powered gyroscopes, i.e., we will assume that only the moments of potential and dissipative forces can act in the axes of the frames of the gyroscopes. Equations, defining the relative equilibrium positions of the satellite can be obtained from Eqs (1.2)–(1.4), by putting
where is the energy of the potential forces acting on the axes of the frames. These equilibrium positions correspond to stationary points of modified potential energy1
(2.1) The “reduced” angular momenta of the rotors with a dimension of moment of inertia are denoted by Hk = Gk /0 . We will assume further that only moments of dissipative forces can act in the axes of the gyroscope frames, i.e., ≡ 0. Since the vectors n and r are unit vectors and mutually orthogonal, the stationary points of function (2.1) correspond to stationary points of the Lagrange function with factors (2.2) Taking the following equalities into account (2.3) we obtain the following system of equations for the equilibrium positions (2.4) (2.5) (2.6) (2.7) Equations (2.6) and (2.7) describe the equilibrium conditions of the gyroscopes with respect to the satellite body, while Eqs (2.4) and (2.5) are equivalent to the single equation (2.8) which describes the equilibrium condition relative to the orbital basis of the satellite with a rotor having reduced angular momentum H. It follows from Eqs (2.6) and (2.7) that, in equilibrium positions, each vector Hk should satisfy one of the conditions (2.9) (2.10) Actually, condition (2.9) does not contradict Eqs (2.6)–(2.8). If we assume that ik × Hk = / 0, then, taking conditions (1.1) into account, we will have from Eq. (2.6) nT sk = 0. Since, by virtue of Eq. (2.7), the vector n lies in the plane of the mutually orthogonal vectors sk and Hk , it follows from the equality nT sk = 0 that the vector n is parallel to the vector Hk , i.e., condition (2.10) is satisfied. Note that, when conditions (2.9) are satisfied the frames of the corresponding gyroscopes “add together”, i.e., the inner frame is parallel to the outer frame. We will determine the equilibria for which all the vectors Hk satisfy condition (2.10). In this case Eq. (2.8) takes the form
It describes relative equilibria of the rigid body in a circular orbit and has 24 different solutions {n, r}, for each of which the principal axes of inertia of the body are parallel to the axes of the orbital basis. All the equilibria of the “carrier – gyroscopes” system, corresponding to the case considered, are described by the formulae (2.11) Here and henceforth we will use the symbols ␦k to describe multivalued solutions, each of which can take two values: +1 and −1. We will investigate the solutions for which at least one of the vectors Hj satisfies condition (2.9). Suppose, to be specific, (2.12)
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From Eq. (2.8) we obtain the equation
(2.13) which describes the equilibria of the satellite with a rotor having a fixed angular momentum H*. it is well-known,6,7 that, depending on the value of the vector H* and its direction in the basis of the principal axes of inertia of the satellite, Eq. (2.13) has from 8 to 24 solutions {n, r}, which define the orientation of the housing of the satellite with respect to the orbital basis. After determining the solutions using Eqs (2.7), we determine the values of the angles of precession (the positions of the axes sj ) for the first m gyroscopes, and on the basis of conditions (2.10) and Eqs (2.6) and (2.7), we obtain the precession and nutation angles for the remaining gyroscopes. The number of different values of the vector H* is
while the number of different orientations of the satellite {n, r}, determined from Eqs (2.13), may range from 8S to 24S. 3. Analysis of the secular stability The equilibrium position of the satellite will be stable in the secular sense (it will satisfy the sufficient stability conditions), if, at the point considered, function (2.1) has a strict minimum, and unstable in the secular sense if there is no minimum. The nature of the secular stability is determined by analysing the values of the second differential of function (2.2) in the set (3.1) The second differential of function (2.2) is described by the expression
Using equalities (3.1), the variations of the vectors of the orbital basis can be expressed in terms of the independent variations u, v and w by the following formulae (3.2) We will then have (3.3) The Lagrange multipliers are expressed from Eqs (2.4) and (2.5) by the formulae (3.4) Taking into account relations (3.2)–(3.4) and the inequality form of the independent variables u, , w, dk , dk (k = 1,. . ., N)
T
J r = 0, which follows from Eq. (2.5), we obtain the following quadratic
(3.5) Equilibria for which quadratic form (3.5) is strictly positive definite will possess secular stability. If, among the eigenvalues of the matrix of quadratic form (3.5), there are negative values, the corresponding equilibria will be unstable in the secular sense. We will investigate the secular stability of solutions (2.12). Consider the set of variations
where j is the number of the gyroscope which satisfies condition (2.9). In this set, quadratic form (3.5) becomes (3.6)
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Fig. 2.
When nT sj = / 0 it is indeterminate, i.e., the equilibrium is unstable in the secular sense. When nT sj = 0, taking Eqs (2.7) into account, we obtain n × Hj = 0, i.e., condition (2.10) is satisfied. Hence, the necessary condition for secular stability is the satisfaction of conditions (2.10) for all the vectors Hk , i.e., equilibria that are stable in the secular sense exist among solutions (2.11). We will investigate the secular stability of solutions (2.11). To be specific, we will consider the solutions (3.7) We will denote by ␣k and k the angles which specify the position of the precession axis of the k-th gyroscope in the basis of the principal axes of inertia of the satellite (Fig. 2). For solutions (3.7), by virtue of conditions (1.1), we have sTk n = sTk e3 = 0, i.e., all the vectors sk lie in the principal plane e1 , e2 . Quadratic form (3.5) then becomes
(3.8) In the case when, for all the gyroscopes, the axes of the outer frames are not parallel to the principal axis of inertia of the satellite e3 , i.e., sink = / 0, by a non-degenerate conversion of the variables (3.9) this quadratic form is reduced to the form
(3.10) Hence we obtain that secular stability of the solutions considered occurs when the following conditions are satisfied (3.11) It follows from these that the satellite has only four stable equilibrium positions in the secular sense (3.12) which coincide with the stable equilibrium positions of a rigid body in a circular orbit. For these the e3 axis of the greatest moment of inertia of the satellite is directed along the normal to the orbital plane, while the e1 axis of the least moment of inertia is directed along the orbital radius. The angular momentum of the rotor of each gyroscope is then parallel to the normal to the orbital plane and has a positive projection onto this normal. If, for certain gyroscopes, the axis of the outer frame is parallel to the principal axis of inertia of the satellite e3 , i.e., sinj = 0, quadratic form (3.8) can be written as
(3.13)
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In this case conditions (3.11) ensure secular stability of solutions (3.7) with respect to all the phase variables, apart from the precession angles j . 4. The asymptotic stability of the equilibria When there is dissipation in the axes of the gyroscope frames, a more detailed analysis of the stability of the equilibria obtained is possible using the Barbashin–Krasovskii theorem.8 This theorem determines, for systems with partial dissipation, the conditions for which its asymptotic stability follows from the secular stability of the equilibrium, and for which Lyapunov instability follows from the secular instability. It was shown previously,2 that the conditions of the Barbashin–Krasovskii theorem are necessarily satisfied if the equilibrium considered is not a bifurcation point, while the truncated equations of the linear approximation have only a trivial solution. The truncated equations of the linear approximation are an overdetermined system. They are obtained by linearization of the equations of motion for fixed values of those variables for which dissipative forces act. We will write the truncated equations of the linear approximation for the problem in question, assuming that dissipative moments act in the axis of the frames of all the gyroscopes. We will denote by the vector of the small rotation of the housing of the satellite about the orbital basis. Then, in the linear approximation, we will have (4.1) while the variations of the vectors of the orbital basis about the housing of the satellite will be expressed by the formulae (4.2) For the variations in the angular velocity and angular acceleration of the satellite we obtain (4.3) In Eqs (4.2) and (4.3), n and r mean the values of the orbital balance vectors in the equilibrium position. Taking these equalities into ˙ = 0 for fixed values of the precession and nutation account, and linearizing Eqs (1.2)–(1.4) in the neighbourhood of equilibrium = 0, angles, we arrive at the following system of equations
(4.4) The primes denote derivatives with respect to the dimensionless time = 0 t, while the vectors f() and F(U) are found from the formulae2 (4.5) In these equations only the components of the vectors and U are the variables, while the vectors n, r, Hk and sk are fixed and equal to their values in the equilibrium position. Equations (4.4) form an overdefined system of 2N + 6 scalar equations for six variables. We will investigate the question of the asymptotic stability of equilibrium positions (3.12), which satisfy the sufficient conditions for stability (Fig. 2). For these, system of equations (4.4) takes the form (4.6) (4.7) (4.8) (4.9) (4.10)
(4.11) It is easy to see that for the conditions (4.12) system (4.6)–(4.11) allows of the following non-trivial solutions (4.13) They correspond to in-plane oscillations of the satellite about the normal to the orbital plane. It should be noted that, for conditions (4.12) and nonlinear system (1.2)–(1.4), similar oscillations are allowed for the fixed frames of the gyroscopes, i.e. equilibrium positions (3.12) in this case will not be asymptotically stable. We will show that if the inequality cosm = / 0 holds for at least one of the gyroscopes, equilibrium positions (3.12) may be asymptotically stable.
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For simplicity, consider the case
Then subsystem (4.10), (4.11) will contain the equations (4.14) (4.15) When the inequality
is satisfied, from Eqs (4.6), (4.14), (4.9) and (4.7) we will have (4.16) (4.17) Differentiating equality (4.17) and taking equalities (4.16) and (4.9) into account, we obtain (4.18) (4.19) Since, for the equilibra considered, the inequality
is satisfied, and it follows from Eqs (4.17) and (4.19) that (4.20) In turn, taking (4.16) and (4.20) into account, we obtain from Eqs (4.15), (4.8) and (4.9) (4.21) Hence, in the example considered, system (4.6)–(4.11) has only the trivial solution U = = 0, as a consequence of which when there is dissipation in the axes of the gyroscope frames, equilibrium position (3.12) is asymptotically stable. A detailed investigation of the asymptotic properties of the satellite motions in the neighbourhood of equilibrium positions (3.12) was carried out using numerical analysis. We investigated the roots S of the characteristic equation of the complete system of equations of the linear approximation and we determined the degree of stability (4.22) i.e., the value of the real part of the most right-hand root, taken with the opposite sign. We analysed how the degree of stability depends on the values of the satellite parameters – the resultant central moments of inertia of the satellite, the moments of inertia of the gyroscopes Ik and Jk , the angular momenta of the rotors Hk , the angles ␣k and k of the setting of the axes of precession of the gyroscopes and the damping factors in the frame axes. The linearized equations of motion of the satellite, written in the neighbourhood of the equilibria, were obtained from Eqs (1.2)–(1.4), taking equalities (2.6)–(2.8) and (4.2)–(4.4) into account, and have the form
(4.23)
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Here xk and yk are the deviations of the precession and nutation angles from their values in the equilibrium position, and 1k and 2k are the damping factors in the precession and nutation axes, having the dimension of the moment of inertia, while the primes denote derivatives with respect to the dimensionless time = 0 t. For equilibrium positions (3.12), Eqs (4.23) take the form
(4.24) The vectors ik , sk , jk , f() and F(U) are expressed by the formulae
Here 1 , 2 , 3 and U1 , U2 and U3 are the components of the vectors and U in the basis of the principal axes of inertia of the satellite. The results of a numerical analysis showed that the degree of stability of equilibrium positions (3.12) depends considerably on the value of the ratio
where Im are the greatest of the moments of inertia of the gyroscopes Ik (k = 1,. . ., N). For the overwhelming majority of values of the satellite parameters this relation has the quadratic form (4.25) while for individual (optimal) values of the parameters it has the linear form (4.26) Here the value of the degree of stability is very sensitive to deviations of the parameters from their optimum values, i.e., small deviations lead to a sharp reduction in the degree of stability. As a consequence of this, the values of the degree of stability obtained in practice are described by relation (4.25). It follows from the results obtained that, for practical satellites, in which the value of ␥ does not exceed 10−4 , the degree of stability of equilibrium positions (3.12) with respect to all the phase variables is comparatively small. Even for optimum values of the parameters, which are difficult to obtain in practice, it does not exceed a value of = 10−4 , i.e., the amplitude of small oscillations is reduced by a factor of e after 10−4 /(2) revolutions of the satellite. For comparison we note that, when using a two-degree-of-freedom gyroscope one can obtain stable equilibrium positions of the satellite with a degree of stability = 0.42.3 In the problem considered the comparatively low value of the degree of stability is due to damping of the satellite oscillations about the normal to the orbital plane that is too weak. Note that the conclusion regarding the weak damping of the satellite oscillations about the normal to the orbital plane can be drawn on the basis of an analysis of the precession equations of motion of the satellite, which are obtained from Eqs (1.2)–(1.4) when Ik = Jk = 0. The asymptotic stability of equilibrium positions (3.12) does not follow from these equations, since they allow of solutions corresponding to unattenuating oscillations of the satellite about the normal to the orbital plane for fixed frames of the gyroscopes. We will analyse the asymptotic properties of the satellite motions using the precession equations, linearized in the neighbourhood of equilibrium positions (3.12). Assuming Ik = Jk = 0 in Eqs (4.24), we obtain a system, which can be decomposed into two independent subsystems,
(4.27)
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(4.28) Subsystem (4.28) describes the law of variation of the angle of rotation of the satellite about the normal to the orbital plane, while subsystem (4.27) describes the law of motion of the satellite with respect to the remaining variables. It is obvious that the equilibria considered do not possess asymptotic stability with respect to the angle 3 . But the angle 3 and its derivative ’3 = U3 do not occur in subsystem (4.27), which enables the asymptotic stability to be analysed with respect to the remaining variables, i.e., one can investigate the speed at which the satellite tends towards rotations about the normal to the orbital plane. We investigated the roots of the subsystem characteristic equation (4.27) as a function of the values of the parameters for a satellite with a single gyroscope. We obtained a value of the local maximum of the degree of stability max = 0.35, which is obtained for the following values of the parameters
For such values of the parameters, the amplitude of small oscillations of the satellite with respect to the variables characterizing a departure from rotations about the normal to the orbital plane, is reduced by a factor of e due to semi-rotation of the satellite. Acknowledgements This research was supported financially by the Government of the Russian Federation within the framework of contracts with the Ministry of Science No. 13.G25.31.0028 and No. 14.740.11.0149 of the Federal Special Purpose Programme “Scientific and ScientificPedagogic Teams of Innovative Russia” in 2009-2013. References 1. Amel’kin NI. Steady motions of a satellite with a two-degree-of-freedom powered gyroscope in a central gravitational field and their stability. J Appl Math Mech 2009;73(2):169–78. 2. Amel’kin NI. Analysis of the stability of the equilibria of a satellite carrying a two-degree-of-freedom powered gyroscope with dissipation in the axes of the frame. J Appl Math Mech 2010;74(4):406–15. 3. Amel’kin NI. The equilibria and stability of a dynamically symmetrical satellite with a two-degree-of-freedom powered gyroscope. J Appl Math Mech 2010;74(5):513–23. 4. Sulikashvili RS. The stability of the steady motions of a satellite with a gyroscope in a central gravitational field. Zh Vychisl Mat Mat Fiz 1969;9(2):479–82. 5. Amel’kin NI. The steady motions of a rigid body carrying a three-degree-of-freedom powered gyroscope, and their stability. Izv Ross Akad Nauk MTT 2011;3:3–17. 6. Sarychev VA, Gutnik SA. The relative equilibra of a satellite-gyrostat. Kosmich Issled 1984;22(3):323–6. 7. Rubanovskii VN. Bifurcation and stability of the relative equilibria of a gyrostat. satellite. J Appl Math Mech 1991;55(4):450–5. 8. Rouche N, Habets P, Lalou M. Stability Theory by Liapunov’s Direct Method. N.Y: Springer; 1977.
Translated by R.C.G.