Nonintrusive Computation of Rigid-Body/Flexible-Body Coupling Integrals

Abstract

The coupled equations of motion of an unrestrained flexible body when derived using multibody dynamics or mean axes formulation consist of three mass coupling integrals, namely, the first-order and second-order inertia effects as well as the second-order momentum-coupling vectors. The computation of these coupling integrals requires access to a finite element formulation [source code that uses shape functions to generate the finite element method (FEM) model], which is often unavailable. This work, using modal superposition, presents a mathematical formulation that allows computation of these integrals for a given finite element model (FEM model) of an unrestrained deformable vehicle nonintrusively: that is, without access to the actual finite element source code. It shows that the required integrals can be computed if one can access the consistent mass matrix of the vehicle from the FEM model. The formulation has been verified with an example of an unrestrained crossbeam model using Rayleigh–Ritz and the finite element method for computing the coupling integrals. In the Rayleigh–Ritz method, the coupling integrals are computed using the integral equations; in the FEM, they are computed using the consistent mass matrix. The use of a consistent mass matrix to compute the integrals agrees well with those obtained using volumetric integration in the Rayleigh–Ritz method. Finally, these integrals were calculated for two flexible unmanned-aerial–vehicle-type aircraft: mAEWing1 and mAEWing2. The results help in determination of the significance of these terms in flight dynamics.