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Pitch motion of a gravity gradient satellite in an elliptical orbit is studied. The cell mapping method is employed to find periodic solutions and analyze the global behavior of the system. Stability characteristics of the solutions are established using a point mapping approximation algorithm. The proposed approach does not depend on existence of...
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... a gravity gradient stabilized satellite in an elliptical orbit. The position of the satellite in the orbit is defined by a radius vector R c from the Earth's center to the spacecraft's center of mass and true anomaly as shown in Fig. 1. Using Euler's equation of motion with kinematic relations, pitch motion of the satellite is decoupled from roll and yaw motions when they are initially quiescent, ...
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... Analysis for e ¼ 0.01. To understand the dynamics around the bifurcation point more carefully, the invariant surface is studied. The 2p-period solutions and invariant surfaces sur- rounding the solutions for the case of e ¼ 0.01 are shown in Fig. 10. There exists an uniform invariant surface surrounds a sta- ble PÀ1 solution for k 2 ¼ 0.3 as shown in Fig. 10(a). However, in Table 1 ...
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... for e ¼ 0.01. To understand the dynamics around the bifurcation point more carefully, the invariant surface is studied. The 2p-period solutions and invariant surfaces sur- rounding the solutions for the case of e ¼ 0.01 are shown in Fig. 10. There exists an uniform invariant surface surrounds a sta- ble PÀ1 solution for k 2 ¼ 0.3 as shown in Fig. 10(a). However, in Table 1 ...
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... 0.1681185 Transactions of the ASME the next case, k 2 ¼ 0.35, the process of movement and deformation of inside the surface is found as shown in Fig. 10(b). The charac- teristics of the surface are changing across the inner curve. While the inner surface is deforming, another stable P À 1 solution is appeared under the deformed region in Fig. 10(c). As shown in Figs. 10(d)-10(f), the invariant region around the new born P À 1 solutions is getting bigger and pushing the inner surface ...
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... 0.1681185 Transactions of the ASME the next case, k 2 ¼ 0.35, the process of movement and deformation of inside the surface is found as shown in Fig. 10(b). The charac- teristics of the surface are changing across the inner curve. While the inner surface is deforming, another stable P À 1 solution is appeared under the deformed region in Fig. 10(c). As shown in Figs. 10(d)-10(f), the invariant region around the new born P À 1 solutions is getting bigger and pushing the inner surface surround- ing the another stable solution away as k 2 is increasing. Another feature to note is that an unstable P À 1 solution is emerging at k 2 ¼ 0.38, Fig. 10(d), marked as an open circle. The ...
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... is appeared under the deformed region in Fig. 10(c). As shown in Figs. 10(d)-10(f), the invariant region around the new born P À 1 solutions is getting bigger and pushing the inner surface surround- ing the another stable solution away as k 2 is increasing. Another feature to note is that an unstable P À 1 solution is emerging at k 2 ¼ 0.38, Fig. 10(d), marked as an open circle. The invariant sur- face gives ideas of the attitude motion around the periodic ...
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... Analysis for k 2 ¼ 0.1. We show a case of k 2 ¼ 0.1 and e varies from 0.01 to 0.7 in this analysis. In the case of e ¼ 0.01, a stable P À 1 solution and stable and unstable P À 2 solutions were found as we discussed in Fig. 6. We are interested in how the P À 2 solutions are emerged. There are two bifurcation points, bfc1 and bfc2, shown in Fig. 11. This is explicitly examined with projections on x 1 and x 2 axis, respectively, for details. Consider, Table 2 Stability analysis using eigenvalues of matrix H for the periodic solutions in Table 1 x* H detH eigH Journal of Computational and Nonlinear Dynamics NOVEMBER 2015, Vol. 10 / 061020-7 first, the plane x 2 versus e for the ...
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... explicitly examined with projections on x 1 and x 2 axis, respectively, for details. Consider, Table 2 Stability analysis using eigenvalues of matrix H for the periodic solutions in Table 1 x* H detH eigH Journal of Computational and Nonlinear Dynamics NOVEMBER 2015, Vol. 10 / 061020-7 first, the plane x 2 versus e for the unstable P À 2 branch, Fig. 11(c). As e decreases from 0.37 to 0.01, x 2 of P À 1 solution decreases and the decreasing unstable P À 1 solution becomes sta- ble and unstable P À 2 solutions emerge at e ¼ 0.1, at bfc1. The stable P À 2 branch is interpreted in the plane of x 1 versus e, Fig. 11(b). As e decreases from 0.7 to 0.01, the stable P À 1 solu- tion loses the ...
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... Vol. 10 / 061020-7 first, the plane x 2 versus e for the unstable P À 2 branch, Fig. 11(c). As e decreases from 0.37 to 0.01, x 2 of P À 1 solution decreases and the decreasing unstable P À 1 solution becomes sta- ble and unstable P À 2 solutions emerge at e ¼ 0.1, at bfc1. The stable P À 2 branch is interpreted in the plane of x 1 versus e, Fig. 11(b). As e decreases from 0.7 to 0.01, the stable P À 1 solu- tion loses the stability and stable P À 2 solutions are emerged at e ¼ 0.37 which is bfc2. The left end tips in Figs. 11(b) and 11(c) which are (B 1 , B 2 ) and (C 1 , C 2 ) match with the P À 2 solutions of Fig. 6. It is also important to see the invariant surface surrounding ...
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... P À 1 solu- tion loses the stability and stable P À 2 solutions are emerged at e ¼ 0.37 which is bfc2. The left end tips in Figs. 11(b) and 11(c) which are (B 1 , B 2 ) and (C 1 , C 2 ) match with the P À 2 solutions of Fig. 6. It is also important to see the invariant surface surrounding the periodic solutions. Around the periodic solutions in Fig. 12, closed curves show invariant surface which contain quasi-periodic motions. The lager e we choose, the smaller bounded region there exists. Where e > 0.1, there are no more bounded regions, but the periodic ...
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... movement of Floquet multipliers for the point bfc1, as an example, is shown in Fig. 13 clarifying the P À 1 to P À 2 bifurca- tion. The multipliers follow the unit circle in the complex plane until they leave the unit circle through À1 at bfc1, satisfying the bifurcation condition in Eq. ...
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... Analysis for e ¼ 0.1. For a case of e ¼ 0.1, P À 1 solu- tions are already found and there exists one stable P À 1 solution when k 2 < 0.537 in Fig. 9. The stable P À 1 solution bifurcates into two PÀ3 solutions at k 2 % 0.16: one is stable, g i , and the other one is unstable, h i , where i ¼ 1, 2, 3. Bifurcation diagrams are given in Fig. 14 showing a three-dimensional representation and its projections on x 1 and x 2 , ...
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... continuous trajectories associated with the P À 3 solutions show the periodicity explicitly. As an example, a stable trajectory associated with g i , i ¼ 1, 2, 3, is examined for the case of k 2 ¼ 0.19 (see Fig. 15(a)). Integration of one period of time with the initial point g 1 results g 2 , integration of one period of time with g 2 results g 3 , and the same step for g 3 -g 1 , and the mapping repeats after all. For comparison of the trajectories, the P À 1 solu- tion is also shown in Fig. 15(a) with a "þ" sign. The continuous response as true ...
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... i , i ¼ 1, 2, 3, is examined for the case of k 2 ¼ 0.19 (see Fig. 15(a)). Integration of one period of time with the initial point g 1 results g 2 , integration of one period of time with g 2 results g 3 , and the same step for g 3 -g 1 , and the mapping repeats after all. For comparison of the trajectories, the P À 1 solu- tion is also shown in Fig. 15(a) with a "þ" sign. The continuous response as true anomaly progresses is given in Fig. 15(b). The initial point g 1 is integrated for t ¼ 54p and the response shows a period of three ...
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... period of time with the initial point g 1 results g 2 , integration of one period of time with g 2 results g 3 , and the same step for g 3 -g 1 , and the mapping repeats after all. For comparison of the trajectories, the P À 1 solu- tion is also shown in Fig. 15(a) with a "þ" sign. The continuous response as true anomaly progresses is given in Fig. 15(b). The initial point g 1 is integrated for t ¼ 54p and the response shows a period of three ...
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... new stable P À 3 solution bifurcates into unstable P À 9 solutions (k i , i ¼ 1, …, 9) at k 2 % 0.20 as shown in Fig. 14. The continuous trajectories associated with the P À 9 solutions are pre- sented in Fig. 15(c). The initial point denoted as "0" is integrated for 9 T and it is mapped back to the same point. The true anomaly Fig. 10 Invariant surfaces in phase plane when bifurcation from a stable P 2 1 solution to two stable P 2 1 solutions and one ...
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... new stable P À 3 solution bifurcates into unstable P À 9 solutions (k i , i ¼ 1, …, 9) at k 2 % 0.20 as shown in Fig. 14. The continuous trajectories associated with the P À 9 solutions are pre- sented in Fig. 15(c). The initial point denoted as "0" is integrated for 9 T and it is mapped back to the same point. The true anomaly Fig. 10 Invariant surfaces in phase plane when bifurcation from a stable P 2 1 solution to two stable P 2 1 solutions and one unstable P 2 1 solution occurs (e 5 0.01) response shows that the angular and angular rate motion ...
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... new stable P À 3 solution bifurcates into unstable P À 9 solutions (k i , i ¼ 1, …, 9) at k 2 % 0.20 as shown in Fig. 14. The continuous trajectories associated with the P À 9 solutions are pre- sented in Fig. 15(c). The initial point denoted as "0" is integrated for 9 T and it is mapped back to the same point. The true anomaly Fig. 10 Invariant surfaces in phase plane when bifurcation from a stable P 2 1 solution to two stable P 2 1 solutions and one unstable P 2 1 solution occurs (e 5 0.01) response shows that the angular and angular rate motion repeat after 9 T but start to be perturbed and diversed during its second orbit as shown in Fig. ...
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... the same point. The true anomaly Fig. 10 Invariant surfaces in phase plane when bifurcation from a stable P 2 1 solution to two stable P 2 1 solutions and one unstable P 2 1 solution occurs (e 5 0.01) response shows that the angular and angular rate motion repeat after 9 T but start to be perturbed and diversed during its second orbit as shown in Fig. ...
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... variations of invariant surfaces according to the bifurca- tions are shown in Fig. 16. When new P À 3 solutions emerged at k 2 ¼ 0.175, the associated deformation of invariant surfaces can be seen. Following the track of stable P À 3 solutions as k 2 is grad- ually increasing, they move apart and the size of surrounding invariant surfaces is decreasing. Finally, in between k 2 ¼ 0.2 and k 2 ¼ 0.21, the invariant ...
Citations
... Moreover, there were several investigations [12,25,26] about the existence and stability of periodic motions of a dumbbell satellite which consist of two point masses connected by a massless rod. For the past few years, an approach with the combination of point mapping and cell mapping method was employed by Dayung Koh and Henryk Flshner to find periodic solutions and analyzed the global behavior of gravity gradient satellites [27] in an elliptical orbit. ...
... Based on the particular solutions (27) and (29), if γ * = 3, q * 1 = ν/2, the Hamiltonian (36) corresponds to the particular solution (27); if γ * = 1, q * 1 = −ν/2, it corresponds to another particular solution (29). According to Eq. (34), the relations (26) and (28) can be expressed, respectively, as ...
... The stability analysis of the particular solutions (27) and (29) starts with the investigation of the linearized system ...
This paper deals with periodic motions and their stability of a flexible connected two-body system with respect to its center of mass in a central Newtonian gravitational field on an elliptical orbit. Equations of motion are derived in a Hamiltonian form, and two periodic solutions as well as the necessary conditions for their existence are acquired. By analyzing linearized equations of perturbed motions, Lyapunov instability domains and domains of stability in the first approximation are obtained. In addition, the third- and fourth-order resonances are investigated in linear stability domains. A constructive algorithm based on a symplectic map is used to calculate the coefficients of the normalized Hamiltonian. Then, a nonlinear stability analysis for two periodic solutions is performed in the third- and fourth-order resonance cases as well as in the nonresonance case.
... For the past few years, an approach with the combination of point mapping and cell mapping method was employed by Dayung Koh and Henryk Flshner to find periodic solutions and analyzed the global behavior of gravity gradient satellites [26] in an elliptical orbit. ...
This paper deals with periodic motions and their stability of a flexible connected two-body system with respect to its center of mass in a central Newtonian gravitational field on an elliptical orbit. Equations of motion are derived in a Hamiltonian form and two periodic solutions as well as the necessary conditions for their existence are acquired. By analyzing linearized equations of perturbed motions, Lyapunov instability domains and domains of stability in the first approximation are obtained. In addition, the third and fourth order resonances are investigated in linear stability domains. A constructive algorithm based on a symplectic map is used to calculate the coeffcients of the normalized Hamiltonian. Then a nonlinear stability analysis for two periodic solutions is performed in the third and fourth order resonance cases as well as in the nonresonance case.
... This approach has been successfully applied to a wide range of problems. The method was used for planning optimum trajectories of multiple robot arms by Wang and Lever, 31 analyzing the pitch motion of a gravity gradient satellite in elliptic orbits by Koh and Flashner,32 and studying coupled orbit-attitude dynamics in the elliptic three-body problem by Koh and Anderson. 33 Ding et al. 34 investigated the global behavior including limit cycle oscillation, periodic motions, and domains of attraction of a bilinear stiffness aeroelastic system by an improved cell-mapping method. ...
In this study, a cell-mapping approach is applied to various systems in the circular restricted three-body problem to obtain a rapid understanding of the global dynamics. The method is generic for various classes of problems including non-autonomous systems and different types of periodic solutions. The cell-mapping method also does not require previously known solutions as inputs, which is typical of continuation approaches, and no symmetric constraints are imposed. This method is especially applicable to a systematic periodic orbit search over a region of interest at one-period of integration. As additional strengths of the method, multiple-period solutions and bifurcation studies can be easily performed. In this study, the initial orbit search is applied to obtain an understanding of the orbit trade space at Europa and Enceladus.
A cell-mapping approach is implemented and parallelized to analyze three-body problem orbits in the vicinity of icy moons (Europa and Enceladus). The cell-mapping method is developed for studying nonlinear dynamics with periodic motions. The method does not require previously known solutions as inputs, which is an essential requirement of continuation approaches, and does not impose symmetric constraints. As major strengths of the method, multiple-period periodic solutions and bifurcation studies can be easily performed. This method is especially applicable to a systematic periodic orbit search over a region of interest using an integration time of one period. The parallelized cell-mapping method facilitates a rapid understanding of the global dynamics.
The pitch motion of spacecraft in the planar elliptic restricted three-body system is studied. Previous studies laid the foundation for spacecraft stability analysis with a small perturbation to the zero pitch motion. In this study, a cell-mapping approach that combines analytical and numerical techniques is used to study the global behavior of the full nonlinear spacecraft attitude in which coupling between orbital dynamics and attitude occurs. Spacecraft placed at the Lagrangian points and some of the reference trajectories are considered to study the effect of varying gravity gradient torque to pitch motion in the three-body system. The spacecraft configuration and orbital eccentricity are also taken into account as parameters for the study. Multiple-period periodic solutions and invariant surfaces are presented for different cases.
