Non-linear dynamics of curved beams. Part 1: Formulation

The non-linear dynamics of elastic beams with uniform initial curvature and double-symmetric cross-section are considered in this work. In particular, the work is divided into two parts. In Part 1, the interest is oriented to the formulation of an accurate model, able to describe the finite dynamics of initially curved beams as to obtain a parameterization of the initial configuration and the weak expression of the equations of motion. To this end, an explicit description of the deformation field and inertia terms is presented. The equations of motion can be used, with slight modifications, for extensible and inextensible, or shear-deformable and shear-indeformable, beams. A description of the free dynamics, of the possible classes of motion under a sinusoidally varying shear tip force, and of bifurcation phenomena is presented in Part 2 for a case-study, together with the results of experimental tests on an aluminum prototype.