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Research Papers

# Global Analysis of Gravity Gradient Satellite's Pitch Motion in an Elliptic Orbit

[+] Author and Article Information
Dayung Koh

Department of Astronautical Engineering,
University of Southern California,
Los Angeles, CA 90089
e-mail: dayungko@usc.edu

Henryk Flashner

Professor
Department of Aerospace and
Mechanical Engineering,
University of Southern California,
Los Angeles, CA 90089
e-mail: hflashne@usc.edu

Manuscript received November 15, 2014; final manuscript received January 14, 2015; published online April 28, 2015. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 10(6), 061020 (Nov 01, 2015) (11 pages) Paper No: CND-14-1290; doi: 10.1115/1.4029621 History: Received November 15, 2014; Revised January 14, 2015; Online April 28, 2015

## Abstract

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 a small parameter and, therefore, no limitations are imposed on the magnitudes of eccentricity or amplitude of motion. This is in contrast to the perturbation based approaches that require assumptions of small orbital eccentricity and small motion. As a result, stable periodic solutions of twice and three times the orbital period are found for some different eccentricities and inertia parameters. Global behavior and evolution of periodic solutions are also demonstrated using invariant surfaces and bifurcation diagrams.

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## References

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## Figures

Fig. 1

Geometry of satellite motion

Fig. 2

Equilibrium point x* and P − 3 solutions X*(i), i = 1, …, K for K = 3

Fig. 3

Cell state space for N = 2

Fig. 4

Equilibrium cell z* and P − K cells z *(i), i = 1, …, K for K = 3

Fig. 5

Small perturbation z of periodic solutions z * at cell j. (a) Perturbation of P − 1 solution and (b) perturbation of P − K solution.

Fig. 6

Result of the cell mapping algorithm showing global behavior in a cell state space for e = 0.01, k2 = 0.1

Fig. 7

P − 2 cell groups for e = 0.01, k2 = 0.1 and convergence of solution B1 and B2 using extended cell mapping

Fig. 8

Three different ways to verify P − 3 solutions in Table 1 (e = 0.01, k2 = 0.1). (a) Phase plane trajectories, (b) fast Fourier analysis, (c) response of a stable solution S1, and (d) response of an unstable solution U1.

Fig. 9

P − 1 to P − 1 bifurcation for different eccentricities (solid line: stable solution, dashed line: unstable solution)

Fig. 10

Invariant surfaces in phase plane when bifurcation from a stable P − 1 solution to two stable P − 1 solutions and one unstable P − 1 solution occurs (e = 0.01)

Fig. 11

Bifurcation sequence P − 1 → P − 2 for k2 = 0.1. (a) Three-dimensional representation, (b) projection on e − x1 plane, and (c) projection on e − x2 plane.

Fig. 12

Invariant surface in phase plane for P − 1 to P − 2 bifurcation (k2 = 0.1)

Fig. 13

Floquet multipliers confirming the P − 1 → P − 2 bifurcation condition

Fig. 14

Bifurcation sequence P − 1 → P − 3 → P − 9 for e = 0.1. (a) Three-dimensional representation, (b) projection on x1 − k2 plane, and (c) projection on x2 − k2 plane.

Fig. 16

Invariant surfaces in phase plane when bifurcation from a stable P − 1 solution to two stable P − 3 solutions and then one unstable P − 9 solution occurs (e = 0.1)

Fig. 15

(a) Trajectory of P − 3 solutions in phase plane (gi) and trajectory of a P − 1 solution in phase plane (+), (b) pitch angle response versus true anomaly in number of revolutions for a P − 3 solution, (c) trajectory of P − 9 solutions in phase plane: numbers 0–9 show the points at n·T, and (d) pitch angle response versus true anomaly in number of revolutions for a P − 9 solution

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