7.1 Introduction

The mechanical nonlinearities taken into account for the study of the dynamic behavior of a system have various well-known origins:

• nonlinearities of a geometrical origin introduced for example by a formulation in great displacements or the expression of the curvature;

• nonlinearities resulting from the constitutive laws (mathematically smooth or not), for example due to nonlinear elasticity;

• nonlinearities at the interface, introduced by joints and resulting for example in phenomena of friction or impact. In addition, nonlinearities resulting due to boundary conditions.

These nonlinearities are often introduced into the model via the experimental data. Fitting of these data in order to study a dynamic behavior leads a designer to apply mathematically regular interpolations (polynomial in practice) that enable convenient analytical studies or, at the same time, numerical investigations. It is the case in dynamics of the structures in civil engineering to represent the action of the ground on a foundation starting from an in situ experiment by a spring with a return strength expressed as a polynomial function of displacement. In addition, simple systems described by piecewise linear models were abundantly studied in the literature. The system of Chua [Madan (1993)], [Chua et. al. (1986a)], [Komuro et. al. (1991)] constitutes for example a paradigm for the study of chaos, and the studies of global dynamic behavior.

Here we are interested in a model of an arch subject to vibration [Lamarque and Malasoma (1992)], [Szempli?ska-Stupnicka (1969)], [Fung and Kaplan (1952)]: The exact model is described by the equation...