A new flexible body dynamic formulation for beam structures undergoing large overall motion.

A new flexible body dynamic formulation, called the Augmented Imbedded Geometric Constraint (AIGC) approach, for beam structures undergoing large overall motion is developed. It is restricted to small elastic deformations of the beam about the large overall motion. The formulation outlined herein pertains to Go-dimensional motion and deformation of a single beam when the overall motion is prescribed as a function of time. The formulation can be easily extended to beam assemblies undergoing arbitrary motion in three-dimensional space. Elastic deformation is characterized by the superposition of a number of assumed global shape functions developed from a substructuring method. The motion of the system is governed by a set of differential and algebraic equations. The algebraic constraints arise from enforcement of the boundary conditions. The AIGC approach improves upon two existing approaches by allowing the solution of two disparate classes of elasto-dynamics problems with a single formulation, demonstrated by simulations for several verification problems. The problems are ones in which the lateral deformation of the beam is dominated by either bending or membrane behavior. Because the new formulation is problem independent, it is applicable to beam problems where the dominant stiffness effects are not known beforehand. The study of the coupling between overall dynamic motion and local deformation of structures has become important with the advent of the space-age, since the interaction is more pronounced with the relatively flexible structures common in spacecraft design. The effects of such coupling are important in the aeronautics industry and can be seen, for example, in helicopter blade response. High speed motion of robotic arms and rapid ground transportation systems are other areas in which the coupling effects are imnortant. One approach to studying flexible body dynamics is through the use of finite element methods ( see for example, Simo and Vu Quoc1s2 and Christensen and Lee3). Another strategy is to use rigid body dynamic approaches which have been modified to include the flexibility effects. Kane, Ryan and Banne rjee4 used this strategy to study beams undergoing large overall motion of a prescribed nature. The technique introduced in reference [4] was restricted to systems with known overall motion. Ryan5 extended that formulation to allow solutions when forcesltorques are applied. Subsequently, Yoo6 has shown that the approach in references [4-51, which he refers to as the Imbedded Geometric Constraint (IGC) approach, fails to produce the correct result for problems where the lateral deformation of the beam is dominated by membrane stiffness. Yoo demonstrated that his formalism, which he refers to as the Nonlinear Strain Displacement (NSD) approach, handles such problems quite successfully. However the NSD method does not reliably solve problems in which lateral deformations are dominated by bending stiffness, which are handled very well by the IGC approach. A new approach called the Augmented Imbedded Geometric Constraint (AIGC) approach, is presented herein. It allows the solution of problems where the lateral deflection of the beam is dominated by either bending or membrane stiffness. This formulation is a -cation of the IGC approach. It is problem independent and, therefore, is applicable to structural dynamics problems where the dominant effects are not known before hand. Only small local deformations of the beam are considered. An Euler-Bernoulli model of the beam transverse flexibility, assuming linear elastic, isotropic behavior, is used. A set of ordinary differential equations (ODES) describing the flexible body dynamic behavior of the beam is developed using Kane's7 method. That pomon of the development of the AIGC approach is identical to that for the IGC approach. Differential algebraic eauations (DAEs) of motion for the AIGC a ~ ~ r o a c h a& generat& by de;eloping a set of algebraic constra& enforcing the physical boundary conditions for the beam. The spatial rLpresen&tion of the deformation is achieved through the use of global shape functions which are based on the substructuring techniques of Craig and Bampton8. Copyright c 1992 by the American Institute of Aeronautics 14 15 and Astronautics, Inc. All rights reserved. The plan of the paper is as follows. In the next section the system differential equations are derived. Then a general method for enforcing boundary conditions is described and implemented, resulting in a set of differential algebraic equations of motion. Numerical results for two benchmark problems, generated using several formulations, are then presented and compared. The model for a two-dimensional beam undergoing large overall motion and small local deformation is shown in Figure 1. This model consists of a rigid body A and a flexible beam B of length L. A dextral set of mutually perpendicular unit . +-. vectors, al,az,as, are fmed in A and directed as shown in Figure 1. The centroidal axis of the beam is assumed to be coincident with the elastic axis, and is parallel to the $ direction when the beam is undeformed. Point Po located a distance x along the undeformed centroidal axis represents a generic point on the beam. After deformation, that point lies at a new position which is labeled point P. The position vector from point 0 to point P is given by: where u 1 and u2 are the $ and x2 measures of the beam deformation. An additional variable of interest is s, the stretch of the centroidal axis of the beam. After deformation, point P is located at a distance x + s measured along the defonned centroidal axis.