Geometrically Exact Beam Analysis Based on the Exponential Map Finite Rotations

Abstract

This paper proposes a nonlinear finite-element analysis for slender composite structures with anisotropic material properties using the mixed geometrically exact beam formulation and exponential map finite rotation. Rodrigues finite rotation expression used in the existing mixed-variational formulation of the geometrically exact beam is further improved by replacing it with the exponential map finite rotation, which is free of singularities. As a result, the existing mixed-variational formulation is further simplified by the matrix arithmetic, so that numerical implementation of the resulting equations may become more compact. Using the present beam formulation, nonlinear static displacements of a straight beam, twisted straight beam, and initially curved beam are precisely predicted within smaller number of iterations compared to the existing formulations. In addition, the present eigenmode analysis is capable of providing the rotating natural frequencies for a full-scale realistic rotor blade accurately. As a result, the present analysis is applicable for large deflection behavior of a composite structure without the singularities in a better convergence.