Extrusion of a beam from a rotating base

Abstract

This paper deals with a new method for simulating motions of a beam that is being extruded from, or retracted into, a rigid body rotating in a general manner. In essence, the method consists of modeling the beam as a series of elastically connected rigid links and then working with equations of motion linearized in the modal coordinates for the links outside the rigid body at a given time. The theory is applied to a problem of current interest, the extrusion/retraction of the WISP antenna from the Shuttle. Simulation results show that beam tip deflections are highly sensitive to the total time devoted to extrusion /retraction, with retraction the less stable process of the two; and the nature of the angular velocity of the rigid body from /into which the beam is being extruded/retracted is found to affect beam behavior significantly. I. Introduction C URRENT interest in dynamics of extrusion of a beam from a rotating base stems from project WISP (Waves In Space Plasma), which involves extending two booms from the Shuttle, each to a final length of 150 m and having a tubular cross section with a 0.0635-m diam and a 0.00254-m wall thickness, while the Shuttle is rotating at a rate of 1 deg/s. Clearly, it is helpful in a project of this kind to be able to perform simulations leading to results such as those shown in Figs. 1 and 2, the first of which contains time plots of the tip deflection, the root bending moment, and the length of each boom during an extrusion operation, whereas the second deals with the same quantities during retraction. It is the purpose of this paper to provide an algorithm for the performance of such simulations. In previous attacks15 on the problem under consideration, beams were modeled as continua, and deflections were described by linear combinations of modal functions weighted by time-dependen t generalized coordinates, an approach familiar from the classical vibrations literature. However, whereas the arguments of the classical modal functions are independent, purely spatial variables, those of the modal functions used in Refs. 1-5 are treated as time-dependen t spatial variables. The deployment of tethers, 6 plates,7 and other structures8 has been treated similarly. Regarding the soundness of this approach as questionable, we adopt a new technique, one that can be applied not only to the title problem but also to the simulation of motions of continua other than beams undergoing extrusion from, or retraction into, moving bases. In what follows, a simulation algorithm is described in sufficient detail to permit a reader to construct a simulation program; numerical results in addition to those plotted in Figs. 1 and 2 are reported and discussed, and the rationale underlying the algorithm is explained.