Spectral Factorization and Homogenization Methods for Modeling and Control of Flexible Structures

摘要

This report describes continuum modeling and vibration control of flexible structures with application to active control of vibrations in large space structures. A comprehensive methodology is discussed for the construction of effective (linear) models for large composite structures consisting of various flexible members(e.g. beams, trusses, etc.) and rigid body elements. It is convenient to concentrate on frequency domain modeling. A systematic procedure is shown for computing the irrational transfer functions. Then by standard transform methods a complete hybrid model is developed. The methods were coded in a computer algebra system (SMP was used) which automated the model building process and produced Fortran code for numerical evaluation of the frequency responses. Effective continuum models of lattice structures with regular infrastructure can be obtained by a systematic procedure based on an asymptotic analysis of multiple scales called homogenization. This method is applied to several examples and accurate computation made of the required parameters of such continuum models somewhat more subtle than merely averaging over lattice cells. For the computation of distributed parameter control an optimal frequency domain method is based on solving an associated Wiener Hopf problem. The method employs effective numerical algorithms (e.g. FFT, etc.) to compute a certain spectral factorization of a possibly matrix valued (in the multiple control case) Hermittian, positive definite transform by sampling the frequency response. The control laws considered in this report take the form of distributed state feedback with respect to a naturally defined, distributed state space of functions over the spatial domain of the structure.