Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades

TLDR

The equations of motion for twisted nonuniform rotor blades are derived using Hamilton’s principle and a Newtonian approach, employing an ordering scheme that neglects only squared bending slopes, torsion deformation, and chord/radius and thickness/radius ratios while retaining all other nonlinear terms, and are based on exact strain–displacement relations between deformed and undeformed coordinates. The derived equations, valid to second order for long, slender beams with various offsets, nonuniform properties, variable pretwist, and small precone angles, reveal that retained nonlinear structural and inertial terms can significantly affect the aeroelastic stability and response of hingeless helicopter rotor blades.

Abstract

The equations of motion are developed by two complementary methods, Hamilton's principle and the Newtonian method. The resulting equations are valid to second order for long, straight, slender, homogeneous, isotropic beams undergoing moderate displacements. The ordering scheme is based on the restriction that squares of the bending slopes, the torsion deformation, and the chord/radius and thickness/radius ratios are negligible with respect to unity. All remaining nonlinear terms are retained. The equations are valid for beams with mass centroid axis and area centroid (tension) axis offsets from the elastic axis, nonuniform mass and stiffness section properties, variable pretwist, and a small precone angle. The strain-displacement relations are developed from an exact transformation between the deformed and undeformed coordinate systems. These nonlinear relations form an important contribution to the final equations. Several nonlinear structural and inertial terms in the final equations are identified that can substantially influence the aeroelastic stability and response of hingeless helicopter rotor blades.