GENERALIZED FREQUENCY DOMAIN STATE-SPACE MODELS FOR ANALYZING FLEXIBLE ROTATING SPACECRAFT

摘要

The mathematical model for a flexible spacecraft that is rotating about a singlernaxis rotation is described by coupled rigid and flexible body degrees-of-freedom,rnwhere the equations of motion are modeled by integro-partial differentialrnequations. Beam-like structures are often useful for analyzing boom-like flexiblernappendages. The equations of motion are analyzed by introducing generalizedrnFourier series for the deformational coordinate that transforms the governingrnequations into a system of ordinary differential equations. Though technicallyrnstraightforward, two problems arise with this approach: (1) the model isrnfrequency-truncated because a finite number of series terms are retained in thernmodel, and (2) computationally intense matrix-valued transfer function calculationsrnare required for understanding the frequency domain behavior of the system.rnBoth of these problems are resolved by: (1) computing the Laplace transformrnof the governing integro-partial differential equation of motion; and (2)rnintroducing a generalized state space (consisting of the deformational coordinaternand three spatial partial derivatives, as well as single and double spatialrnintegrals of the deformational coordinate). The Laplace domain equation ofrnmotion is cast in the form of a linear state-space differential equation that isrnsolved in terms of a matrix exponential and convolution integral. The structuralrnboundary conditions defined by Hamilton’s principle are enforced on thernclosed-form solution for the generalized state space. The generalized staternspace model is then manipulated to provide analytic scalar transfer functionrnmodels for the original integro-partial differential system dynamics. Symbolicrnmethods are used to obtain closed-form eigen decomposition–based solutionsrnfor the matrix exponential/convolution integral algorithm. Numerical results arernpresented that compare the classical series based approach with the generalizedrnstate space approach for computing representative spacecraft transfer functionrnmodels.