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UBC Theses and Dissertations

Mathematical modelling of flexible multibody dynamics with application to orbiting systems Ibrahim, Ahmed El-Hady M.


A relatively general formulation for the governing equations of motion, applicable to a large class of flexible multibody systems, is developed using a concise matrix format. The model considered consists of a number of arbitrarily connected flexible deployable members forming branched and closed loop configurations. Joints between bodies are permitted up to six degrees of freedom in translation and rotation. To be effective, the matrix-Lagrangian formulation necessitates development of the kinetic energy expression in a quadratic form in terms of the system velocities. The mass matrix associated with such a quadratic form is known for simple systems such as a collection of point masses, a group of connected rigid bodies, and a discretized flexible structure. However, for a multibody system, where the contributing forces arise from system's translation, rotation, elasticity, deployment, and their interactions, such an expression is not available. To fill this gap, multibody kinematics is developed in terms of the elements of the geometry matrix, which uniquely describes the configuration of branched systems. The characteristic dynamical quantities, i.e., elements of the mass matrix, are identified and the formulation is approached in an increasing order of complexity. The concept of specified and generalized coordinates together with established procedures of analytical dynamics lead to characteristic quantities ( Lagrangian, Hamiltonian, etc. ) and finally result in governing equations of motion which are new to the multibody dynamics. To account for flexibility in a consistent manner, a second-degree nonlinear displacement field is permitted. Alternatively, a linear displacement field can be used if the nonlinear terms up to the fourth-degree are preserved in the strain energy. An algorithm for calculating the stiffness matrix of a flexible element is developed, where terms up to the third-degree of nonlinearity in displacement are retained. Application of this versatile formulation is illustrated through a set of examples of contemporary interest. They pertain to a spacecraft comprising of a central rigid body with attached flexible appendages. The configuration corresponds to a large class of present and planned communication satellites. It can also represent the Space Shuttle based deployment of beam and plate type appendages aimed at scientific experiments or construction of the proposed Space Station. The system static equilibrium and stability are discussed. A computer code is developed and specialized to the specific cases in hand. Typical results of an extensive parametric study are presented for two particular situations : (i) the Space Shuttle based deployment of a beam or a plate type structural member; (ii) the configuration similar to the Waves In Space Plasma (WISP) experiment jointly proposed by Canada and the U.S.A. The problems are analyzed systematically, through progressive introduction of complexity, to help appreciate interactions between librational dynamics, flexibility, deployment, inertia parameters, orbit eccentricity, initial conditions, appendage orientation, etc. The information is fundamental to the missions concerned and essential to help develop appropriate control strategies.

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