Behavior of a Free Dual-Spin Gyrostat with Different Ratios of Inertia Moments

The attitude motion is studied of asymmetric dual-spin gyrostats which may be modeled as free systems of two rigid bodies, one asymmetric and one axisymmetric. Exact analytical solutions of the attitude motion are presented for all possible ratios of inertia moments of these bodies. The dynamics of free gyrostats with zero internal torque is considered. The dimensionless nonlinear equations of the gyrostat arewritten in Serret-Andoyer canonical variables.Thepreviously known exact solutions are complemented by new several solutions in terms of Jacobi elliptic functions. The results of the study can be useful for the analysis of dual-spin spacecraft dynamics.


Introduction
The dynamics of rotating bodies is a classic topic of study in mechanics.In the eighteenth and nineteenth centuries, several aspects of the motion of a rotating rigid body were studied by such famous mathematicians as Euler, Cauchy, Jacobi, Poinsot, Lagrange, and Kovalevskaya.However, the study of the dynamics of rotating bodies is still very important for numerous applications such as the dynamics of satellite gyrostat, spacecraft, robotics, and the like.In this paper we deal with a particular case of gyrostat; it consists of a platform with a triaxial ellipsoid of inertia and a symmetrical rotor whose axis of symmetry is aligned along one of the principal axes of inertia of the platform.
The paper deals with the analytical solutions of the attitude motion of the axial gyrostats.The analytical solutions of determining the attitude motion of the axial gyrostat with a free rotor (see, e.g., [1]) are relatively straightforward.Furthermore, Kane [2] has given the solution for the attitude motion of an axial gyrostat with a driven rotor.In many papers (see, e.g., [3][4][5][6][7][8][9][10][11][12][13][14][15][16]), this modern problem is usually performed by following two distinct steps.The dynamics of the rotation are represented by using differential equations written for the components of the angular momentum, and then the kinematic equations are utilized to transform the angular momentum.While this classical formulation is widespread, scientists also exhibit a preference for the Serret-Andoyer variables (SA) variables, which allow performing convenient analysis of the attitude motion of the gyrostats.Serret [17] discovered these variables by solving the Hamilton-Jacobi equation written in terms of Eulerian coordinates.Andoyer [18] used spherical trigonometry to show that the Serret transformation was simply a change of Eulerian coordinates that depended upon the angular momentum components.Deprit [19] and Deprit and Elipe [20] established the canonicity of the Serret transformation by using differential forms and without resorting to finding a generating function.SA variables are used also in Hamiltonian structure of an asymmetric gyrostat in the gravitational field [21].The SA variables are used to describe the Hamiltonian equations for attitude motions of the axial free gyrostat [22][23][24][25][26]. Euler angles and angular velocities of the gyrostat as functions of the SA variables are given in Aslanov [27].
Hall [6] proposed a procedure based upon the global analysis of the rotational dynamics.Hall and Rand [7] considered spinup dynamics of classical axial gyrostat composed of an asymmetric platform and an axisymmetric rotor.They obtained the averaged equations for slowly varying relative rotation of the bodies and analytical solutions for the projections of angular momentum in the case of constant relative rotation.Cochran et al. [3] have obtained analytical 2 Advances in Mathematical Physics solutions of the attitude motion of the free gyrostats.Elipe [8] investigated a free gyrostat with three flywheels rotating about the three principal axes of inertia and without any external forces or torques.Elipe and Lanchares [28] have introduced coordinates to represent the orbits of constant angular momentum as a flow on a sphere.This representation has shown that the problem is equivalent to a quadratic Hamiltonian depending on two parameters.
This paper focuses on the study of the attitude motion of the gyrostat + consisting of an axisymmetric rotor  (  =   ,   ) and of an unbalanced platform  (  >   ,   ).This paper is a development of Aslanov's works [23,24,27]  +  ), and we do not impose restrictions on the angular velocity of rotation of the rotor relative to the platform as in the papers [23,24].
The paper is divided into five sections.In Section 2 the dimensionless differential equations of the motion variables and the stationary solutions are presented in terms of the SA variables.In Section 3 we find singular points on a phase plane for various types of the gyrostats.Section 4 focuses on an integration of the motion equations in terms of Jacobi elliptic functions.Section 5 presents a complete set of the phase trajectories and the exact solutions.

Canonical Equations and Stationary Solutions
Let us consider a motion of a gyrostat in the absence of external torque and internal torque applied by  on  about axis .We set the platform's frame Oxyz relative to an inertial frame OXYZ, using the canonical SA variables [24]: , , h, , , H.The orientation of the rotor relative to the platform's frame Oxyz is defined by the angle  (Figure 1).The Hamiltonian for the torque-free motion with use of the SA variables and pair of variables where   = / δ is the generalized momentum and  is the kinetic energy of the gyrostat.Since , h, and  are cyclic and H does not appear in the Hamiltonian, thus , h, H, and   are constants, which means that the angular momentum vector is conserved and the first integrals of the motion are where components of the angular momentum vector in the coordinate system Oxyz are Since the number of degrees of freedom of the torque-free motion of the gyrostat is reduced to one, the motion The transformation of (4) to a dimensionless form is obtained by scaling the two momentums and time, as follows: Derivatives with respect to  are denoted by a derivative sign:   = /.The change of variables (5) leads to the to the equivalent set of dimensionless equations: Here  =   / and  =   / are the relative reciprocals of the principal moments of inertia of the gyrostat  =   +   and  =   +   , respectively. is Hamiltonian by [24] Solving expression (8) with respect to cos 2, we obtain the phase trajectories: By equating to zero ( 6) and ( 7), we get Solving (10), we have four stationary solutions [24]: cos cos The first stationary solution (11) is stable if  > 1 (  > ) and is unstable if  < 1 (  < ).The second stationary solution ( 12) is stable if  < 1 (  < ) and unstable if  > 1 (  > ).The third and fourth stationary solutions ( 13) and ( 14) are always unstable.
Figure 2 shows phase space on plane and unit sphere for solutions (13).An angle between the tangents to the separatrix and to the line  = 0 in the vicinity  * = ±1 is equal to the angle  =  * .Next we will study the trajectories in the phase space, except the separatrices which contain singular points ( 11)- (14).
It is obvious that the roots of ( 13) and ( 14) give ambiguous solutions when  = 0 (Euler's case, Figure 3), so this particular case is excluded from consideration.The wellknown solutions of Euler's Equations of motion of a rigid body are studied in terms of the SA variables in the papers [29,30].
Other authors [3,7,9] formulated the problem in terms of the angular momentum components (3): which avoid the singularity in the points ( 13) and ( 14) The Hamiltonian (8) has the form for the axisymmetric platform   =   ⇒ ( = ): In this case the attitude motion is a precession around the vector of the angular momentum ⃗  with a constant angle  = arccos().

Types of the Gyrostats and Singular Points in the Phase Space
The five basic types of the gyrostats are determined by the following ratios of the inertia moments: (1) oblate gyrostat:   >  >  ( >  > 1), (2) oblate-intermediate gyrostat: The singular points corresponding to stationary solutions (11)-( 14) for all types of gyrostats are shown in Table 1.The table notations are as follows:

Integration of the Canonical Equations by Quadrature.
In this formulation the exact solutions of ( 6) and ( 7) are presented only for the three subtypes [24]: Oa, Ib, and Pb.To find all possible kinds of the solutions we use ( 7) and ( 9): Function ( 20) is a polynomial of the third degree for the two subtypes Oi ( = 1) or Pi ( = 1): For all other subtypes function ( 20) is a polynomial of degree four: Real roots are numbered as follows: Complex conjugate roots are written as Let us denote the roots of the degree four (20): 4.2.Various Types of the Exact Solutions.Case (() =  4 (),  0 < 0).The integral (19) can be written as where Two Real Roots and Two Complex Conjugate Roots.The roots (25) include two real roots  1 >  2 and two complex conjugate roots  3,4 =   ±   .Change of variable [31] (tan  2 ) converts the integral (26) to the Legendre normal form where ( 1 ,  2 are acute angles).
Case (() =  4 (),  0 > 0).The integral ( 19) can be written as where Four Real Roots.If the roots of ( 25) are real and only two roots correspond to the real motion The variable substitution [31] The arbitrary constant  0 is equal to zero if  0 =  3 .The solutions obtained earlier by Elipe and Lanchares [28] and the solutions ( 31), (32), and (36) have similar structure.
Case (() =  3 ()).The polynomial ( 20) is the third-degree polynomial Three Real Roots.We are only interested of the three real roots, which according to ( 25) and (38) can be written as A positive value of the polynomial (38) and the condition  ∈ [1, −1] correspond to the real motion of the gyrostat.

Index of the region in phase space Solution Oi
Libration We denote the roots (39) so that the roots are located in the following sequence:  1 >  2 >  3 .The variable substitution [31] for the integral (19) gives the solutions: where The arbitrary constant  0 is equal to zero if  0 =  2 for (40) and if  0 =  3 for (41).

Pi
Libration

A Complete Set of the Phase Trajectories and the Exact Solutions
In this section the phase portraits are shown in the case  > 0 only.Keeping in mind that at  < 0 the stationary points are obtained as a mirror image relative to abscissa axis according to the solutions ( 11)-( 14), we use the following values of the gyrostat's parameters for the numerical calculations  = 0.85 kg × m 2 ,  = 0.65 kg × m 2 ,  = 0.05  −1 . (44) 5.1.Phase Space Structure for the Oblate Gyrostats.The solutions for the oblate gyrostats are given in Table 2. Figure 4 illustrates the case when the saddles are on verticals   = ±/2 (|  | < 1).The saddles can reach the horizontal   = 1 as shown in Figure 5.

Phase Space
Structure for the Oblate-Intermediate Gyrostats.Table 3 shows the solutions for the oblate-intermediate gyrostats ( = 1).The corresponding phase structure is shown on Figure 6.

Phase Space Structure for the Intermediate Gyrostats.
Other solutions for this type of the gyrostats are presented in Table 4 and the phase structures are shown in Figures 6-8.The phase structure for the Subtype Ia is similar to the Rotation phase portrait in Figure 6.We see a doubling of centers and saddles ("old" and "new") for the Subtype Ib (Figure 7).The "old" centers and "old" saddles disappear for the Subtype Ic (Figure 8).gyrostats are presented in Table 5.The motions of the gyrostats for the type Pi are qualitatively similar to the motions of the Subtype Ic (Figure 8).

Phase Space
Structure for the Prolate Gyrostats.Table 6 shows the solutions for the prolate gyrostats.We can see the similarity between the Subtype Ic the the Subtypes Pa (Figure 8).In case of the Subtype Pb we observe a similarity with the Subtype Oa (Figure 4), when the phase portrait is shifted along the horizontal axis at ±/2 (Figure 9).Thus we have observed the evolution of the phase space, depending on the parameters of the gyrostat.

Conclusion
This paper is a development of the results obtained in Aslanov [24,27] of the study can be useful for the analysis of dual-spin spacecraft dynamics.

Figure 2 :Figure 3 :
Figure 2: Separatrices in the plane and on the unit sphere.

Table 1 :
The singular points.

Table 2 :
The exact solutions for the oblate gyrostats.

Table 3 :
The exact solutions for the oblate-intermediate gyrostats.

Table 4 :
The exact solutions for the intermediate gyrostats.

Table 5 :
The exact solutions for the prolate-intermediate gyrostats.

Table 6 :
The exact solutions for the prolate gyrostats.