Advanced Topics in Nonsmooth Dynamics

This chapter is dedicated to comparisons of three well-known models that apply to multiple (that is, simultaneous) collisions: Moreau’s law, the binary collision law, and the LZB model. First, a brief recall of these three models and the way in which their numerical implementation is done. Then, an analysis based on numerical simulations, in which the LZB outcome is considered as the reference outcome, is presented. It is shown that Moreau’s law and the binary collision model possess good prediction capabilities in some few “extreme” cases. The comparisons are made for free chains of aligned grains, and for chains impacting a wall. The elasticity coefficient, coefficients of restitution, mass ratios and contact equivalent stiffnesses are used as varying parameters.

This chapter deals with frictionless instantaneous impacts in rigid multibody dynamics. For autonomous multibody systems which are subjected to perfect unilateral constraints, a geometric description of the impacts on the respective tangent space to the configuration manifold is presented. The mass matrix of a mechanical system endows the configuration manifold with the structure of a Riemannian manifold and provides an isomorphism between the tangent space and the cotangent space at each point of the configuration manifold. Kinematic quantities (virtual displacements, velocities) are elements of the tangent space, while kinetic quantities (forces, impulsive forces) live in the cotangent space, the dual space of the tangent space. Impact laws, as constitutive laws relating primal and dual quantities, are introduced as set-valued mappings between these two spaces. Methods from Convex Analysis permit to study what the implications are if the impact law is maximal monotone. Finally, the generalized Newton’s and the generalized Poisson’s impact law are considered as illustrative examples.

We study the existence and stability of time-periodic oscillations in a chain of coupled impact oscillators, for rigid impacts without energy dissipation. We formulate the search for periodic solutions as a boundary value problem incorporating unilateral constraints. This problem is solved analytically in the vicinity of the uncoupled limit and numerically for larger coupling constants. Different solution branches corresponding to nonlinear localized modes (breathers) and normal modes are computed.

We consider in this chapter the dynamics of rigid multibody systems subjected to frictionless unilateral constraints. Starting from the mechanical description of the problem, we derive its formulation as a second-order Measure Differential Inclusion and we introduce the corresponding mathematical framework, namely functions of Bounded Variation and Stieltjes measures. Then, the main difficulties in the study of vibro-impact problems are described and an overview of the state of the art about existence results and relevant numerical methods (penalty approach, time-stepping schemes at the position or velocity level) is proposed. Throughout this chapter, the bouncing ball model problem is considered to highlight the key points of the mathematical analysis without too many technicalities.

First, we define the bipotential and discuss some fundamental aspects concerning the existence and construction of a bipotential generating a given constitutive law. After a quick review of applications to solid mechanics, we highlight, in particular, problems of unilateral contact with isotropic and anisotropic Coulomb dry friction. The second part is devoted to variational methods and numerical algorithms inspired by the bipotential, illustrated, in particular, to multi-body systems. Extended limit analysis techniques are used to determine the collapse load of structures with plasticity and friction contact.

In this chapter, we consider a master system consisting of a nonlinear differential inclusion and an algebraic equation of constraint (resulting in a Differential Algebraic Inclusion (DAI) system). This system is coupled to a nonlinear energy sink (NES) corresponding to a one degree-of-freedom essentially nonlinear differential equation. We examine how a resonance capture can lead to a reduced order dynamical system. To obtain this reduced order model, we describe a multiple time scale analysis governed by the introduction of multi-timescales via a small parameter \(\varepsilon \) that is finite and strictly positive. The mass of the NES is small versus the mass of the master system, and it governs a mass ratio defining the small parameter \(\varepsilon \). The first timescale is the fast scale. Introducing the Manevitch complexification leads to the definition of slow time envelope coordinates. These envelope coordinates either do not directly depend on the fast time scale or do not depend on this fast time scale via introduction of the so-called Slow Invariant Manifold (SIM). The slow time dynamics of the master system components is analyzed through introduction of equilibrium points, corresponding to periodic solutions, or singular points (governing bifurcations around the SIM), corresponding to quasi-periodic behaviors. We present a simple example of semi-implicit Differential Algebraic Equation (DAE), including a friction term coupled to a cubic NES. Analytical developments of a 1:1:1 resonance case permit us to predict passive control of a DAI by a NES.

Torsional stick-slip vibrations decrease the performance, reliability and fail-safety of drilling systems used for the exploration and harvesting of oil, gas, minerals and geo-thermal energy. Current industrial controllers regularly fail to eliminate stick-slip vibrations, especially when multiple torsional flexibility modes in the drill-string dynamics play a role in the onset of stick-slip vibrations. This chapter presents the experimental validation of novel robust output-feedback controllers designed to eliminate stick-slip vibrations in the presence of multiple dominant torsional flexibility modes. For this purpose, a representative experimental test setup is designed, using a model of a real-life drilling rig as a basis. The model of the dynamics of the experimental setup can be cast in Lur’e-type form with set-valued nonlinearities representing an (uncertain) model for the complex bit-rock interaction and the interaction between the drill-string and the borehole. The proposed controller design strategy is based on skewed-\(\mu \)-DK-iteration and aims at optimizing the robustness with respect to uncertainty in the non-smooth bit-rock interaction. Moreover, a closed-loop stability analysis for the non-smooth drill-string model is provided. Experimental results confirm that stick-slip vibrations are indeed eliminated using the designed controller in realistic drilling scenarios in which state-of-practice controllers have failed to achieve the same.

The simulation of flexible multibody systems with unilateral contact conditions and impacts requires advanced numerical methods. The nonsmooth generalized-\(\alpha \) method was developed in order to combine an accurate and second-order time discretization of the smoother part of the dynamics and a consistent but first-order time discretization of the impulsive contributions. Compared to the Moreau-Jean scheme, this approach improves the quality of the numerical solution, especially for the representation of the vibrating response of flexible bodies. It relies on the formal definition of a so-called smooth motion that captures a non-impulsive part of the total nonsmooth motion. This definition may account for some contributions of the bilateral constraints and/or of the active unilateral constraints at the velocity or at the acceleration level. This chapter shows that the formulation of the constraints strongly influences the numerical stability and the computational cost of the method. A strategy for enforcing the bilateral and unilateral constraints simultaneously at the position, velocity and acceleration levels is also established, with a careful formulation of the activation criteria based on augmented Lagrange multipliers. In the special case of smooth systems, a comparison is made with more standard solvers for differential-algebraic equations. The properties of this method are demonstrated using illustrative numerical examples of smooth and nonsmooth mechanical systems.

In this chapter, we review several formulations of the discrete frictional contact problem that arises in space and time discretized mechanical systems with unilateral contact and three-dimensional Coulomb’s friction. Most of these formulations are well–known concepts in the optimization community, or more generally, in the mathematical programming community. To cite a few, the discrete frictional contact problem can be formulated as variational inequalities, generalized or semi–smooth equations, second–order cone complementarity problems, or optimization problems, such as quadratic programming problems over second-order cones. Thanks to these multiple formulations, various numerical methods emerge naturally for solving the problem. We review the main numerical techniques that are well-known in the literature, and we also propose new applications of methods such as the fixed point and extra-gradient methods with self-adaptive step rules for variational inequalities or the proximal point algorithm for generalized equations. All these numerical techniques are compared over a large set of test examples using performance profiles. One of the main conclusions is that there is no universal solver. Nevertheless, we are able to give some hints for choosing a solver with respect to the main characteristics of the set of tests.