8170341
Stability properties of equilibrium sets of non-linear mechanical systems with dry friction and impact
In this paper, we will give conditions under which the equilibrium set of multi-degree-of-freedom non-linear mechanical systems with an arbitrary number of frictional unilateral constraints is attractive. The theorems for attractivity are proved by using the framework of measure differential inclusions together with a Lyapunov-type stability analysis and a generalisation of LaSalle’s invariance principle for non-smooth systems. The special structure of mechanical multi-body systems allows for a natural Lyapunov function and an elegant derivation of the proof. Moreover, an instability theorem for assessing the instability of equilibrium sets of non-linear mechanical systems with frictional bilateral constraints is formulated. These results are illustrated by means of examples with both unilateral and bilateral frictional constraints.
5058649
Performance of an Automatic Ball Balancer with dry Friction
In many industrial applications imbalance is a major cause for unwanted vibrations. One way to compensate for an unknown imbalance is the implementation of an automatic ball balancer. Oil-lubricated automatic ball balancers are applied in hand-held tools and washers. However, in applications such as optical drives fluid lubrication is highly undesirable since it may destroy the optical system upon leakage. Therefore, in this paper, the balancing performance of an automatic ball balancer without fluid lubrication is investigated. The absence of fluid lubrication gives rise to dry friction phenomena which cause the existence of equilibrium sets of the balls in the automatic ball balancer. A model of the system with dry friction, modeled by a set-valued force law, is built based on dedicated experiments. The resulting equilibrium sets and their dependency on system parameters are studied and the consequences for the balancing performance is assessed. Based on these results, it can be concluded that in parts of such equilibrium sets the balancing performance deteriorates when compared to the system without automatic ball balancer; in other words, the balancing performance is endangered by the presence of dry friction. This conclusion is supported by both numerical and experimental results.
15659486
Computation of piecewise quadratic Lyapunov functions for hybrid systems
This paper presents a computational approach to stability analysis of nonlinear and hybrid systems. The search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities. Several examples are included to demonstrate the flexibility and power of the approach.
30772424
Stability of Stationary Sets in Control Systems With Discontinuous Nonlinearities
Foundations of Theory of Differential Equations with Discontinuous Right-Hand Sides Auxiliary Algebraic Statements on Solutions of Matrix Inequalities of a Special Type Dichotomy and Stability of Nonlinear Systems with Multiple Equilibria Stability of Equilibria Sets of Pendulum-Like Systems.
122751535
Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings
Considers the extension of the absolute stability problem and of the Lagrange-Dirichlet theorem, to the case when nonlinearities are multivalued monotone mappings. Some of the considered systems may also be recast in the framework of complementarity systems, which in turn may be seen as a special class of hybrid dynamical systems. Another, fruitful point of view, is that of differential inclusions, with or without measures. This class of systems encompasses mechanical systems with unilateral constraints and impacts.
121557301
A stability theory for second-order nonsmooth dynamical systems with application to friction problems
Abstract A LaSalle's Invariance Theory for a class of first-order evolution variational inequalities is developed. Using this approach, stability and asymptotic properties of important classes of second-order dynamic systems are studied. The theoretical results of the paper are supported by examples in nonsmooth Mechanics and some numerical simulations.
56114756
Necessary Conditions of asymptotic stability for unilateral dynamical systems
In this paper, we develop a mathematical tool that can be used to state necessary conditions of asymptotic stability of isolated stationary solutions of a class of unilateral dynamical systems. More precisely, nonlinear evolution variational inequalities are considered. Instability criteria are also given. Applications can be found in mechanics or electrical circuits.
48362648
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 LaSalle's 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.
118838812
Unilateral Contact and Dry Friction in Finite Freedom Dynamics
An approach to the dynamics of mechanical systems with a finite number of degrees of freedom, involving unilateral constraints, is developed. In the n-dimensional linear spaces of forces and velocities, some classical concepts of Convex Analysis are used, but no convexity assumption is made concerning the constraint inequalities. The velocity is not supposed to be a differentiable function of time, but only to have locally bounded variation, so the role of the acceleration is held by a n-dimensional measure on the considered time interval. Dynamics is then governed by measure differential inclusions, which treat possible velocity jumps on the same footing as smooth motions. Possible collisions are described as soft, thus dissipative. Friction is taken into account under a recently proposed expression of Coulomb’s law. These formulations have the advantage of generating numerical algorithms of time-discretization, able to handle, in particular, the nonsmooth effects arising from unilaterality and from dry friction.
2325426
Switched networks and complementarity
A modeling framework is proposed for circuits that are subject both to externally induced switches (time events) and to state events. The framework applies to switched networks with linear and piecewise-linear elements, including diodes. We show that the linear complementarity formulation, which already has proved effective for piecewise-linear networks, can be extended in a natural way to also cover switching circuits. To achieve this, we use a generalization of the linear complementarity problem known as the cone-complementarity problem. We show that the proposed framework is sound in the sense that existence and uniqueness of solutions is guaranteed under a passivity assumption. We prove that only first-order impulses occur and characterize all situations that give rise to a state jump; moreover, we provide rules that determine the jump. Finally, we show that within our framework, energy cannot increase as a result of a jump, and we derive a stability result from this.
26769529
Stability Theory for Hybrid Dynamical Systems
This entry provides a short introduction to modeling of hybrid dynamical systems and then focuses on stability theory for these systems. It provides definitions of asymptotic stability, basin of attraction, and uniform asymptotic stability for a compact set. It points out mild assumptions under which different characterizations of asymptotic stability are equivalent, as well as when an asymptotically stable compact set exists. It also summarizes necessary and sufficient conditions for asymptotic stability in terms of Lyapunov functions.
53998525
On the control of finite-dimensional mechanical systems with unilateral constraints
This paper focuses on the problem of the control of a class of mechanical systems with a finite number of degrees-of-freedom, subject to unilateral constraints on the position. Roughly speaking, those systems are described by a set of ordinary differential equations that represent smooth dynamics, together with an algebraic inequality condition F(q)/spl ges/0 (where q is the vector of generalized coordinates) and an impact rule relating the interaction impulse and the velocity. Nonsmooth dynamics is at the core of the study of such systems. This implies that one can suitably define solutions and stability concepts that fit with the considered model. We then discuss the closed-loop control problem, and analyze various switching control strategies.
219614
Modeling and control of impact in mechanical systems: theory and experimental results
Considers the equations of motion of mechanical systems subject to inequality constraints, which can be obtained by looking for the stationary value of the action integral. Two different methods are used to take into account the inequality constraints in the computation of the stationary value of the action integral: the Valentine variables method and the penalty functions method. The equations of motion resulting from the application of the Valentine variables method, which introduces the concept of "nonsmooth" impacts, constitute the exact model of the constrained mechanical system; such a model is suitable to be employed when the impacting parts of the actual mechanical system are very stiff. The equations of motion resulting from the application of the penalty functions method, which introduces the concept of "smooth impacts," constitute an approximate model of the constrained mechanical system; such a model is suitable to be employed when the impacting parts of the actual mechanical system show some flexibility. Various feedback control laws from the natural outputs and from their time derivatives are studied with reference to both models of impact; the closed-loop systems resulting from the application of the same control law to both models show pretty much the same global asymptotic stability properties. The proposed control laws are only concerned with regulation problems in the presence of possible contacts and impacts among parts of the mechanical system or with the external environment. The effectiveness of the proposed control structure has been tested experimentally with reference to a single-link robot arm, showing a valuable behavior.
15278288
Lasalle's Invariance Theorem for Nonsmooth Lagrangian Dynamical Systems
A key condition for the statement of Lasalle's invariance theorem is the continuity of the trajectories of the dynamical systems with respect to initial conditions. Systems with discontinuous ﬂows generally don't present such a continuity, but in the particular case of nonsmooth Lagrangian dynamical systems we will see that in fact this continuity property holds in most cases. We will then be able to propose a a LaSalle's invariance theorem for nonsmooth dynamical systems satisfying this property through the use of general time-invariant ﬂows.orem for nonsmooth dynamical systems still needs to be stated, so we propose here to extend this theorem through the framework of nonsmooth dynamical systems.
36891633
The Krakovskii-LaSalle Invariance Principle for a Class of Unilateral Dynamical Systems
Abstract.This paper is devoted to the study of the extension of the invariance lemma to a class of hybrid dynamical systems, namely evolution variational inequalities. Applications can be found in models of electrical circuits with ideal diodes or oligopolistic market equilibrium.
6921154
Stability and instability matrices for linear evolution variational inequalities
This paper deals with the characterization of the stability and instability matrices for a class of unilaterally constrained dynamical systems, represented as linear evolution variational inequalities (LEVI). Such systems can also be seen as a sort of differential inclusion, or (in special cases) as linear complementarity systems, which in turn are a class of hybrid dynamical systems. Examples show that the stability of the unconstrained system and that of the constrained system, may drastically differ. Various criteria are proposed to characterize the stability or the instability of LEVI.
123305256
Multibody Dynamics with Unilateral Contacts
Unilateral Contact - Problem Formulation (A. Curnier).- Scalar Force Potentials in Rigid Multibody Systems (Ch. Glocker).- Contact, Friction and Discrete Mechanical Structures: Analogies and Dynamic Problems (A. Klarbring).- Non-smooth Dynamical Systems (T. Kupper).- Unilateral Problems of Dynamics (F. Pfeiffer).
16642323
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.
56276434
On frictionless impact models in rigid-body systems
The paper reviews the frictionless collision problem in rigid–body dynamics. After having established the contact kinematical equations and the impact equations, Newton's and Poisson's impact laws are stated in inequality form for one collision point. This situation is extended by superposition to multi–contact configurations. The concept of superposition together with the equations for a single contact defines the mechanical impact theory. It is valid only within a certain restricted framework specified by several physical assumptions, which are discussed to some extent.
123068247
Oblique Frictional Impact of a Bar: Analysis and Comparison of Different Impact Laws
In this paper a basic, easily to multi-contact problems extendable, non-smooth approach is applied to analyze a bar striking an inelastic half-space. Coulomb contact is assumed and modeled by using set-valued Newtonian impact laws in normal as well as in tangential direction. The resulting linear complementarity problem contains all possible impact states and provides an instantaneous collision operator that respects all inequality constraints. This operator depends on the orientation of the bar and determines uniquely the post-impact velocities as functions of the pre-impact state. Different types of solutions may occur, including “stick’’ and “slip’’. In this context, stick and slip have to be understood as the two cases characterized by the tangential impulsive force as an element of either the set-valued or of the single-valued domain of the friction law. Depending on the choice of parameters, sign reversal of the tangential contact velocity is possible. For certain inertia properties and initial conditions, the collision operator yields an impact, even for initially vanishing normal contact velocity. This phenomenon is well known as the Painlevé paradox. The results obtained by this fully non-smooth rigid body approach are compared with those of other impact models, such as a lumped mass model with compliance elements, and a collision operator used for particle interactions in flows.
118748321
Non-Smooth Dynamical Systems
Some general theory of differential inclusions.- Bounded, unbounded, periodic, and almost periodic solutions.- Lyapunov exponents for non-smooth dynamical systems.- On the application of Conley index theory to non-smooth dynamical systems.- On the application of KAM theory to non-smooth dynamical systems.- Planar non-smooth dynamical systems.- Melnikov's method for non-smooth dynamical systems.- Further topics and notes.
18318616
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.
119360732
Hybrid system stability and robustness verification using linear matrix inequalities
This paper addresses issues concerning exponential stability and robustness of hybrid systems. Stability conditions using Lyapunov techniques are given. The search for the Lyapunov functions is formulated as a linear matrix inequality (LMI) problem for hybrid systems with affine as well as non-linear vector fields. It is shown how the Lyapunov approach most advantageously also can be used to guarantee stability despite the presence of model uncertainties. Several examples are given to illustrate the theory.
117760159
Nonsmooth Mechanics of Solids
Collisions. Thermal effects. Collisions of deformable solids and collisions of solids and fluids.- An Introduction to Impacts.- Approximation of variational and hemivariational inequalities of elliptic type. Applications to contact problems with friction.- Semicoercive Hemivariational Inequalities, Regularization Methods, Applications on Mechanics.- Mathematical Programs with Equilibrium Constraints: Theory and Numerical Methods.- Applied Nonsmooth Mechanics of Deformable Bodies.
55708331
An Introduction to Impacts
Different methods to model and solve multi-contact collisions are presented in this report. The standard impact constitutive laws from non-smooth dynamics are reviewed for planar frictional collisions and formulated in terms of set-valued maps and linear complementarity. For the frictionless case, a geometric concept based on kinematic, kinetic and energetic compatibility is developed, which provides access to non-standard impact events as in Newton’s cradle. Within this context, Moreau’s impact law is reviewed and stated in various ways, providing even access to the collision problem at re-entrant corner points as an extension. Based on Moreau’s law, a geometric classification of impacts is proposed. Several examples are presented, such as the frictional reversible impact at a super ball, Newton’s cradle and the rocking rod.
198120391
Variational Analysis
Errata and Additions: December 2013 p. 10, l. 14 xν → x should be xν → x̄ p. 31, l. 10 argmin f should be argmin f̄ p. 45, Fig.2-7 the x-axis should be just IR not IRn p. 108, l. -6 insert ‘to’ after ‘said’ p. 136, l. -11 Proposition 4.37, replace xinIB by x ∈ IB p. 278, l. 13 replace ‘neq’ by 6= p. 514, l. 19 replace φ(x) by φ(y) p. 644, l. -7 S(B) should be S(B) p. 677, l. -1 replace Sε : T 7→ IR n by Sε : T → IR n p. 678, l. 4 in the expression for Aν replace κ by ν p. 734, l. -2 replace 38 by 37