Rayleigh-Ritz based substructure synthesis for flexible multibody systems

Abstract

This paper is concerned with the modeling of flexible multibody systems by a Rayleigh-Ritz based substructure synthesis method, so that certain advantages can be accrued by using the variational approach to derive the eigenvalue problem. As with the classical Rayleigh-Ritz method, if the admissible functions used to represent the motion of the substructures are not chosen properly, convergence can suffer. This paper presents a new substructure synthesis method with superior convergence characteristics achieved by representing the motion by means of a recently developed class of functions, namely, the class of quasi-compari son functions. This improved convergence is shown to be related to improved approximation of both the differential equations and the natural boundary conditions. The theory is demonstrated by means of a numerical example. I. Introduction T HIS investigation is concerned with the modeling of flexible multibody systems. Many structures, such as fixedwing aircraft, helicopters, flexible spacecraft, flexible robots, etc., can be modeled as assemblages of interacting flexible bodies. A method proposed by Hurty1'2 in the early 1960s, and known as component-mode synthesis, consists of modeling the motion of the individual substructures, referred to as components, by means of He uses three types of modes to represent the motion of a substructure: rigid-body modes, constraint modes, and normal modes. To define constraint modes, Hurty divides the constraints into two classes, statically determinate and redundant. The constraint modes are equal in number to the number of redundant constraints and are defined by producing a unit displacement on each redundant constraint in turn, with all other constraints fixed. The normal modes represent the modes of vibration of the components with all constraints fixed. An approach by Craig and Hampton3 differs from that of Hurty 1'2 mainly in the selection of the component modes. Indeed, Craig and Hampton suggest two types of component modes: boundary modes providing for displacements and rotations at points along the substructure boundaries, and related to the constraint modes of Hurty, and substructure modes corresponding to completely restrained boundaries. The substructures modeled individually in Refs. 1-3 are made to act together as a single structure by eliminating the redundant generalized coordinates arising from the fact that displacements at points common to two adjacent substructures are included twice in the overall problem formulation, once for each substructure. The elimination process is based on the use of constraint equations resulting from the enforcement of compatibility conditions, which amounts to saying that the displacement of a boundary point shared by two substructures is the same. In another approach to the problem, proposed by