The loads induced on the spacecraft orbiting the Earth by the deploying elastic arm are investigated. The coupled equations of motion of the arm with the vehicle orbital mechanics are used to describe the 3D dynamic behavior of the flexible-appendage and the related disturbing loads induced on the spacecraft. To this end, an equivalent dynamical system is derived for the arm by applying an attached Non-Newtonian Reference Frame which is subjected to the orbital motion and geocentric pointing maneuver of the spacecraft. With the help of the Assumed Modes Method, the behavior of the arm attached to the spacecraft in Keplerian orbits is studied. The results show that deploying the arm in some specific directions relative to the orbital plane leads to serious coupling between two lateral displacements. In addition, the effects of specific orbital parameters on arm responses and resulting induced loads are studied for the cases of "True Anomaly of spacecraft at deployment time, and Eccentricity of elliptical orbits". The prediction of disturbing loads induced on spacecraft helps design the robust attitude control system. Further, the positioning accuracy of the payloads (installed on the arm-tip) can be estimated by employing the obtained arm responses in the orbital motion, which enables us to determine the undesirable motions and predict any required control system for the arm.

Approximate solutions for vibrations of flexible beam-type appendages subjected to tip mass are studied while uniform and exponential profiles for arm deployment are simulated. Applying an equivalent dynamical system and following Lagrangian approach, the equations of motion of the system are derived as nonlinear ordinary differential equations (ODEs) (with time-varying coefficients), in which the effect of the tip mass can be considered as some nonlinearity added to the 'no tip mass' case dynamics. The approximate closed-form solutions are obtained through a novel methodology using a computer algorithm, in which the solutions of the 'no tip mass' case are expanded by imposing quadratic perturbations on the independent variable. The mean square of errors (MSEs) for the obtained approximate analytical solution is computed. Using this method, the amplitude and frequency of the arm response are presented by the algebraic equations, which help the parametric design of such systems. In addition, effects of tip mass as an indicator of nonlinearities added to the system dynamics, on the amplitude and frequency of the beam response, are investigated during arm deployment.

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Problems related to the three-dimensional (3D) dynamics of the deploying flexible arms subjected to base angular motions are studied with simulated tip payloads and actual deployment trajectories. To facilitate the solution, an equivalent dynamical system is developed by introducing the inertial reaction forces on the arm, while the equations of motion are derived in the non-Newtonian reference frame attached to the arm. The dynamic behavior of the arm is investigated both by the finite element and assumed Modes methods for the purpose of verification. This study reveals that base angular motions lead to considerable couplings between the two lateral displacements and axial motions. Meanwhile, the induced loadings on the flexible arm due to the base angular motions are obtained, which are useful for the design of more efficient arms. Furthermore, one may use the resulting arm–tip position envelop to predict the antenna positioning accuracy, which paves the way for possible control systems to limit undesirable motions.

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