Modes of interconnected lattice trusses using continuum models, part 1

This represents a continuing systematic attempt to explore the use of continuum models--in contrast to the Finite Element Models currently universally in use--to develop feedback control laws for stability enhancement of structures, particularly large structures, for deployment in space. We shall show that for the control objective, continuum models do offer unique advantages. It must be admitted of course that developing continuum models for arbitrary structures is no easy task. In this paper we take advantage of the special nature of current Large Space Structures--typified by the NASA-LaRC Evolutionary Model which will be our main concern--which consists of interconnected orthogonal lattice trusses each with identical bays. Using an equivalent one-dimensional Timoshenko beam model, we develop an almost complete continuum model for the evolutionary structure. We do this in stages, beginning only with the main bus as flexible and then going on to make all the appendages also flexible-except for the antenna structure. Based on these models we proceed to develop formulas for mode frequencies and shapes. These are shown to be the roots of the determinant of a matrix of small dimension compared with mode calculations using Finite Element Models, even though the matrix involves transcendental functions. The formulas allow us to study asymptotic properties of the modes and how they evolve as we increase the number of bodies which are treated as flexible. The asymptotics, in fact, become simpler.