Summary

The most important concepts discussed and results achieved in the book are resumed in this chapter. Perturbation methods for nonlinear continuous systems have been illustrated. Reference has initially been made to a sample model, consisting of a linear beam resting on nonlinear Winkler soil. The response of the structure in five classes of problems has been analyzed, namely: (a) nonlinear elastostatics, (b) buckling of conservative static systems, (c) nonlinear primary, sub-harmonic and super-harmonic external resonances of dynamical systems, (d) dynamic bifurcation of nonconservative dynamical systems, (e) static bifurcation of nonconservative dynamical systems. General and symmetric systems have been distinguished in the algorithms, these latter requiring simpler analyses. The meaning of the bifurcation equations has been commented as a result of a reduction process. The analysis has successively been extended to a large class of viscoelastic systems, represented by a metamodel, for which the algorithms have been rediscussed. Finally, specific minimal models for one-dimensional (string, cable and beam) and two-dimensional (membrane and plate) structural elements have been formulated. Several examples, relevant to them, have been worked out by perturbation methods and proposed to the reader as solved problems. The Galerkin discretization method has also been commented in an Appendix, and all the perturbation algorithms reformulated for finite-dimensional systems.