Remco I. Leine

Dynamics and bifurcations of non-smooth mechanical systems

A weatherstrip tape includes an elongated body of resilient plastic foam material rectangular in cross section and having a pair of opposed faces. A pair of longitudinally extending parallel coplaner anchor strips of plastic material along their inner edge portions overly one face of said body and are secured thereto. Outer edge portions of said anchor strips extend laterally from opposite sides of said body. Said anchor strips are adapted for manual folding into parallel engagement, with said outer edge portions adapted for projection into and anchoring within a clearance space with the elongated body adapted for sealing over such clearance space.

Bifurcations in Nonlinear Discontinuous Systems

This paper treats bifurcations of periodic solutions indiscontinuous systems of the Filippov type. Furthermore, bifurcations offixed points in non-smooth continuous systems are addressed. Filippovstheory for the definition of solutions of discontinuous systems issurveyed and jumps in fundamental solution matrices are discussed. It isshown how jumps in the fundamental solution matrix lead to jumps of theFloquet multipliers of periodic solutions. The Floquet multipliers canjump through the unit circle causing discontinuous bifurcations.Numerical examples are treated which show various discontinuousbifurcations. Also infinitely unstable periodic solutions are addressed.

Stick-Slip Vibrations Induced by Alternate Friction Models

In the present paper a simple and efficient alternate friction model is presented to simulate stick-slip vibrations. The alternate friction model consists of a set of ordinary non-stiff differential equations and has the advantage that the system can be integrated with any standard ODE-solver. Comparison with a smoothing method reveals that the alternate friction model is more efficient from a computational point of view. A shooting method for calculating limit cycles, based on the alternate friction model, is presented. Time-dependent static friction is studied as well as application in a system with 2-DOF.

Stick-slip Whirl Interaction in Drillstring Dynamics

A Stick-slip Whirl Model is presented which is a simplification of an oilwell drillstring confined in a borehole with drilling fluid. The disappearance of stick-slip vibration when whirl vibration appears is explained by bifurcation theory. The numerical results are compared with the experimental data from a full-scale drilling rig.

Stability and Convergence of Mechanical Systems with Unilateral Constraints

Stability of motion is a central theme in the dynamics of mechanical systems. While the stability theory for systems with bilateral constraints is a well-established field, this monograph represents a systematic study of mechanical systems with unilateral constraints, such as unilateral contact, impact and friction. Such unilateral constraints give rise to non-smooth dynamical models for which stability theory is developed in this work. The book starts with the treatise of the mathematical background on non-smooth analysis, measure and integration theory and an introduction to the field of non-smooth dynamical systems. The unilateral constraints are modelled in the framework of set-valued force laws developed in the field of non-smooth mechanics. The embedding of these constitutive models in the dynamics of mechanical systems gives rises to dynamical models with impulsive phenomena. This book uses the mathematical framework of measure differential inclusions to formalise such models. The book proceeds with the presentation of stability results for measure differential inclusions. These stability results are then applied to nonlinear mechanical systems with unilateral constraints. The book closes with the study of the convergence property for a class of measure differential inclusions; a stability property for systems with time-varying inputs which is shown to be highly instrumental in the context of the control of mechanical systems with unilateral constraints. While the book presents a profound stability theory for mechanical systems with unilateral constraints, it also has a tutorial value on the modelling of such systems in the framework of measure differential inclusions. The work will be of interest to engineers, scientists and students working in the field of non-smooth mechanics and dynamics.

Snake Robot Obstacle-Aided Locomotion: Modeling, Simulations, and Experiments

Snakes utilize irregularities in the terrain, such as rocks and vegetation, for faster and more efficient locomotion. This motivates the development of snake robots that actively use the terrain for locomotion, i.e., obstacle-aided locomotion. In order to accurately model and understand this phenomenon, this paper presents a novel nonsmooth (hybrid) mathematical model for wheel-less snake robots, which allows the snake robot to push against external obstacles apart from a flat ground. The framework of nonsmooth dynamics and convex analysis allows us to systematically and accurately incorporate both unilateral contact forces (from the obstacles) and isotropic friction forces based on Coulombs law using set-valued force laws. The mathematical model is verified through experiments. In particular, a back-to-back comparison between numerical simulations and experimental results is presented. It is, furthermore, shown that the snake robot is able to move forward faster and more robustly by exploiting obstacles.

Modeling and Analysis of Rigid Multibody Systems with Translational Clearance Joints Based on the Nonsmooth Dynamics Approach

The main purpose of this paper is to present and discuss a methodology for a dynamic modeling and analysis of rigid multibody systems with translational clearance joints. The methodology is based on the non-smooth dynamics approach, in which the interaction of the elements that constitute a translational clearance joint is modeled with multiple frictional unilateral constraints. In the following, the most fundamental issues of the non-smooth dynamics theory are revised. The dynamics of rigid multibody systems are stated as an equality of measures, which are formulated at the velocity-impulse level. The equations of motion are complemented with constitutive laws for the normal and tangential directions. In this work, the unilateral constraints are described by a set-valued force law of the type of Signorini’s condition, while the frictional contacts are characterized by a set-valued force law of the type of Coulomb’s law for dry friction. The resulting contact-impact problem is formulated and solved as a linear complementarity problem, which is embedded in the Moreau time-stepping method. Finally, the classical slider-crank mechanism is considered as a demonstrative application example and numerical results are presented. The results obtained show that the existence of clearance joints in the modeling of multibody systems influences their dynamics response.

A set-valued force law for spatial Coulomb-Contensou friction

The aim of this paper is to develop a set-valued contact law for combined spatial Coulomb–Contensou friction, taking into account a normal friction torque (drilling friction) and spin. The set-valued Coulomb–Contensou friction law is derived from a non-smooth velocity pseudo potential. A higher-order Runge–Kutta time-stepping method is presented for the numerical simulation of rigid bodies with Coulomb–Contensou friction. The algebraic inclusion describing the contact problem is solved with an Augmented Lagrangian approach. The theory and numerical methods are applied to the Tippe-Top. The analysis and numerical results on the Tippe-Top illustrate the importance of Coulomb–Contensou friction for the dynamics of systems with friction.

Attractivity of Equilibrium Sets of Systems with Dry Friction

The dynamics of mechanical systems with dry friction elements, modelledby set-valued force laws, can be described by differential inclusions.An equilibrium set of such a differential inclusion corresponds to astationary mode for which the friction elements are sticking. Theattractivity properties of the equilibrium set are of major importancefor the overall dynamic behaviour of this type of systems. Conditionsfor the attractivity of the equilibrium set of MDOF mechanical systemswith multiple friction elements are presented. These results areobtained by application of a generalisation of LaSalles principle fordifferential inclusions of Filippov-type. Besides passive systems, alsosystems with negative viscous damping are considered. For such systems,only local attractivity of the equilibrium set can be assured undercertain conditions. Moreover, an estimate for the region of attractionis given for these cases. The effectiveness of the results isillustrated by means of both 1DOF and MDOF examples.

Periodic motion and bifurcations induced by the Painlevé paradox

In this paper we study the periodic motion and bifurcations of the Frictional Impact Oscillator, which consists of an object with normal and tangential degrees of freedom that comes in contact with a rigid surface. The Frictional Impact Oscillator contains the basic mechanism for a hopping phenomenon observed in many practical applications. We will show that the hopping or bouncing motion in this type of systems is closely related to the Painleve paradox. A dynamical system exhibiting the Painleve paradox has nonuniqueness and nonexistence of solutions in certain sliding modes. Furthermore, we will show that this type of systems can exhibit the Painleve paradox for physically realistic values of the friction coefficient.