Elaine Pratt

1 – The Model

: This chapter contains the different mechanical models to which the analysis presented in this book is applied. Two types of nonlinearities are considered here. Some arise from the constitutive laws others result from the geometry of the models.

Exploring the dynamics of a simple system involving Coulomb friction

The work presented here consists in an exhaustive study of a simple mass-spring system involving Coulomb friction. The aim was to gain some insight into the behaviour of a chain of masses in frictionnal contact. If it is simple to explicit the analytical solution of a single mass system, the analytical solution for a two mass system is already far more complicated. We thus consider two masses linked by a spring in bilateral contact with Coulomb friction and submitted to an external force applied onto one of the masses. The existence, uniqueness and regularity of the dynamics of the system is established through a recent paper [1]. Once the uniqueness is ensured it is simple to exhibit the explicit solution for certain values of the external force (i.e. when the amplitude of the force is either small or large). When the amplitude of the external force belongs to a certain intermediate range the dynamics turns out to be more interesting. The solution can be calculated analytically, however as the computation becomes rapidly tiresome, we use a symbolic calculus tool to compute a solution corresponding to a given external force.

7 – Open Problems and Challenges

This chapter summarizes the results presented in this book, then suggests different types of future work that could be considered as complementary to the book. The first line of study consists of adding results from sections that have been dropped or removed from the book, either to prevent overloading, or, when no remaining difficulties persist, because those additional calculations are more or less identical to those already presented in the book. Such problems cannot therefore be considered as open. The second line of study really consists of open problems. Different parts of the book give rise to these open problems and some of them might be both long term and difficult ones.

The Case of the Nonlinear Restoring Force

This chapter investigates the dynamics of the simple mass-spring system when the restoring force is nonlinear but still involves non-regularized unilateral contact and Coulomb friction. As in the previous chapter, the response of the system when submitted to an oscillating excitation will be studied. The main qualitative differences with the case of a linear restoring force are due to the shape of the set of equilibrium states. As classically done in qualitative analysis of dynamical systems, this chapter aims at an investigation of the { period, amplitude } plane of the excitation.

The Equilibrium States

In this chapter, the equilibrium states are determined and then classified. Whether in the case where the restoring force is linear or in the general large strains case, the methods applied to find the equilibrium states are identical. The method consists of solving the equilibrium equations and then determining in the first place under which conditions the normal components of the solutions of the equilibrium equations are strictly negative, i.e. are not in contact, and then, in the second place, under which conditions the solutions of the equilibrium equations in contact, that is such that the normal component of the displacement is equal to zero, satisfy the Coulomb friction conditions.

2 – Mathematical Formulation

: The term nonsmooth nonlinearity has already been introduced in Chapter 2. However, it is important to give a more accurate characterization of nonsmoothness by highlighting two points: – the first point is discussed in Chapter 2. The graphs shown in were introduced as constitutive laws, which usually means that the reaction of the obstacle is given by a function of the displacement, so the displacement is the single unknown of the equilibrium or dynamical problems, as it is the case in elasticity, whether linear or not. But specifically the graphs in are not those of functions. This is not exceptional, it occurs also, for example, in plasticity, but the effects of this nonsmoothness are important whether in statics or in dynamics, in particular the equilibrium problem may have infinitely many solutions; – the second point is related to dynamics and is qualitatively illustrated. Assume that a single particle is moving in the plane when submitted to a force F(t), and that a part of this plane is forbidden to the particle. The curve representing the boundary of this part of the plane can consequently be seen as an obstacle: the particle must remain on the same side of this obstacle. Then, either the particle is not in contact with the obstacle and its velocity is a continuous function of time whatever the smooth function F(t), or the particle enters into contact with the obstacle, and there is a jump in the velocity at the time of impact.