Bifurcation analysis of damped visco-elastic planar beams under simultaneous gravitational and follower forces

"The mechanical behavior of a non-conservative non-linear beam, internally and externally. damped, undergoing codimension-1 (static or dynamic) and codimension-2 (doublezero). bifurcations, is analyzed. The system consists of a purely flexible, planar, viscoelastic. beam, fixed at one end, loaded at the tip by a follower force and a dead load,. acting simultaneously. An integro-differential equation of motion in the transversal displacement,. with relevant boundary conditions, is derived. Then, the linear stability diagram. of the trivial rectilinear configuration is built-up in the space of the two loading. parameters. Attention is then focused on the double-zero bifurcation, for which a postcritical. analysis is carried out without any a-priori discretization. An adapted version of. the Multiple Scale Method, based on a fractional series expansion in the perturbation parameter,. is employed to derive the bifurcation equations. Finally, bifurcation charts are. evaluated, able to illustrate the system behavior around the codimension-2 bifurcation. point."