Perturbation methods for nonlinear continuous systems: 

statics, buckling, resonances and dynamic bifurcations

Perturbation methods supply approximate analytical solutions to weakly nonlinear equations. They have been more often applied to discrete, rather than continuous systems. It is customary, indeed, to formulate a continuous model and then discretize it via the Galerkin method, before performing a perturbation analysis. Such an approach, however, suffers from having to choose in advance the pattern of the displacement field, very often taken as a combination of linear modes, thus renouncing to capture the modification of the deflection shape with the amplitude of the response (as predicted by the Center Manifold philosophy). A direct approach, therefore, in which the partial differential equations of motion are directly attacked, thus removing the limitation of the Galerkin method, is preferable, although not fully accepted, yet, in the international community. 

This talk is a contribution to spread the knowledge of perturbation methods for continuous systems. Algorithms are introduced for a metamodel and then applied to strings, cables, beams, membrane and plates. Several problems are addressed: nonlinear elasto-statics, elastic buckling and post-buckling, nonlinear external resonances, parametric excitation, dynamic and static bifurcations of nonconservative systems.