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Nonlinear Analysis of a Two-Parachute System Undergoing Pendulum Motion

Motion resembling that of a pendulum undergoing large-amplitude limit cycle oscillation was observed during a series of ﬂight tests of an unoccupied Orion Capsule Parachute Assembly System (CPAS) comprised of two parachutes and a capsule payload. Large excursions away from vertical by the capsule could cause it to strike the ground or ocean at a large angle with respect to vertical, or at a large horizontal speed. These conditions are undesirable because they would endanger the occupants of the capsule in an actual mission. A simpliﬁed planar dynamics model in conjunction with a nonlinear normal force coeﬃcient vs. angle of attack model serves as the basis of an analytical investigation of the fundamental dynamics of this pendulum motion. Output error methodology from system identiﬁcation theory was used to identify the parameters of the nonlinear aerodynamics model. The identiﬁed model yielded excellent comparison with portions of ﬂight test data where the pendulum motion occurred. Due to the inherent nonlinear nature of the pendulum motion limit cycle, traditional nonlinear analysis techniques were applied to gain further insight into the system. Lyapunov’s direct method provided mathematical proof in the absolute stability of the pendulum mode. Describing Function method was used to predict the amplitude and frequency of the limit cycle oscillation. Finally, phase plane analysis allowed easy visualization on the size and shape of the limit cycle with respect to variations in key aerodynamic parameters.