A Generalized Formulation of the Vectorial Equations of Motion for Nonprismatic Thin Space Beams

The formulation of a generalized vectorial equation of motion for small vibrations of any nonprismatic thin beam, which center line is an arbitrary space curve, is presented. The thin beam is such that any characteristic dimension of any cross section is assumed to be small compared to the local radii of curvature and geometric torsion. The equations of motion are given in terms of two independent vectors; a linear displacement vector of the centroid of the cross section and a rotation displacement vector about the centroid. A brief discussion of the boundary conditions in terms of these two vectors is given. The effects of rotary inertia and the shear deformation upon the general derived expressions are discussed.