A fully nonlinear theory of curved and twisted composite rotor blades accounting for warpings and three-dimensional stress effects

Abstract

A new approach is used to develop a geometrically exact nonlinear beam model for naturally curved and twisted solid composite rotor blades undergoing large vibrations in three-dimensional space. A combination of the new concepts of local displacements and local engineering stresses and strains, a new interpretation and manipulation of the virtual local rotations, an exact coordinate transformation, and the extended Hamilton principle is used to derive six fully nonlinear equations of motion describing one extension, two bending, one torsion, and two shearing vibrations of composite beams. The formulation is based on an energy approach, but the derivation is fully correlated with the Newtonian approach and provides a straightforward explanation of all nonlinear structural terms without using complex tensor operations or asymptotic expansions. The theory accounts for in-plane and out-of-plane warpings due to bending, extensional, shearing and torsional loadings, elastic couplings among warpings, and three-dimensional stress effects by using the results of a two-dimensional, static, sectional, finite element analysis. Also, the theory fully accounts for extensionality, initial curvatures and geometric nonlinearities. The equations display linear elastic couplings due to structural anisotropy and initial curvatures and nonlinear geometric couplings. The theory contains most of the existing beam theories as special cases, and the final equations of motion are put in compact matrix form.