Nonlinear Equations of Motion for Twisted Nonuniform Rotating Beams

Nonlinear partial differential equations of motion suitable for describing bending in two directions and torsion of a rotating cantilevered beam are derived by use of Hamilton's principle. A nonlinear strain displacement relation is developed based on a newly constructed coordinate transformation and forms an important contribution to the final equations. In determining the order of various terms in the derivation, it is assumed that the squares of the bending (flap and lead-lag) slopes, of the torsion deformation, and of the chord/radius and thickness/radius ratios are small compared to unity, but otherwise all nonlinear terms are retained. Thus the more important nonlinear terms are identified. In addition to the product of bending curvatures (discussed by earlier authors) which generates a twist about the blade elastic axis, it is found that a product of bending curvature in one direction and twist generates a bending moment for the other direction of bending. In general all such terms are of equal physical importance and could substantially influence the stability and response of hingless or cantilevered helicopter rotor blades, particularly when such blades are significantly deformed under normal operating loads.