Energetic consistency conditions for standard impacts: Part I: Newton-type inequality impact laws and Kane's example

The paper studies the conditions under which Newtonian impacts lead to an energetically consistent post-impact state, and identifies the mechanisms responsible for a potential violation. Unilateral and bilateral geometric and kinematic constraints, as well as various Coulomb type friction elements are equipped with a kinematic restitution coefficient and are taken into account in the impact problem as normal cone inclusions. Based on the condition number of the Delassus operator, bounds on the impact coefficients are derived that ensure energetic consistency. As counter-examples, a slide-push mechanism is presented, and Kane's example of a frictional impact at a double pendulum is analyze

Displacement Potentials in Non-Smooth Dynamics

International audienceThe paper treats the evaluation of the accelerations in rigid multibody systems which are subjected to displacement dependent set-valued force interactions. The interaction laws are represented by non-smooth displacement potentials and derived through generalized differentiation. The resulting multifunctions contain the cases of smooth force characteristics , bilateral constraints, as well as combinations of them like unilateral constraints or prestressed springs with play. Impacts are excluded. A generalization of the classical principles of d ' Alembert, Jourdain, and Gauss in terms of hemi-variational inequalities is given. A strictly convex minimization problem depending on the unknown accelerations is stated, known in classical mechanics as the Principle of Least Constraints. The theory is applied to unilaterally constrained systems

Formulation of rigid body systems with nonsmooth and multivalued interactions

International audienceThis paper treats the formulation of interaction laws in rigid multibody systems. The interaction laws may be represented by a certain class of convex C 0 potential functions and then derived through subdifferentiation. The resulting multifunctions contain the cases of smooth force characteristics, bilateral constraints, as well as combinations of them like unilateral constraints, dry friction, or prestressed springs or damper combinations. Impacts are excluded. A generalization of the principles of d’Alembert, Jourdain, and Gauss in terms of variational inequalities will be given

The Principles of d'Alembert, Jourdain, and Gauss in Nonsmooth Dynamics Part I: Scleronomic Multibody Systems

International audienceThe paper treats the evaluation of the accelerations in rigid multibody systems which are subjected to set-valued forceinteractions. The interaction laws may be represented by non-smooth potential functions, and then derived through gen-eralized differentiation. The resulting multifunctions contain the cases of smooth force characteristics, bilateral con-straints, as well as combinations of them like unilateral constraints, dry friction, or prestressed springs with play. Im-pacts are excluded. A generalization of the classical principles of d’Alembert, Jourdain, and Gauss in terms ofvariational inequalities will be given. A strictly convex minimization problem depending on the unknown accelerations ofthe system will be stated, known in classical mechanics as the Principle of Least Constraints

Rigid body dynamics with a scalable body, quaternions and perfect constraints

In this paper, we present a formulation of the quaternion constraint for rigid body rotations in the form of a standard perfect bilateral mechanical constraint, for which the associated Lagrangian multiplier has the meaning of a constraint force. First, the equations of motion of a scalable body are derived. A scalable body has three translational, three rotational, and one uniform scaling degree of freedom. As generalized coordinates, an unconstrained quaternion and a displacement vector are used. To the scalable body, a perfect bilateral constraint is added, restricting the quaternion to unit length and making the body rigid. This way a quaternion based differential algebraic equation (DAE) formulation for the dynamics of a rigid body is obtained, where the7×7 mass matrix is regular and the unit length restriction of the quaternion is enforced by a mechanical constraint. Finally, the equations of motion in the form of a DAE are linked to the Newton-Euler equations of motion of a rigid body. The rigid body DAE formulation is useful for the construction of (energy) consistent integrator

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 flow

Non-smooth modelling of electrical systems using the flux approach

The non-smooth modelling of electrical systems, which allows for idealised switching components, is described using the flux approach. The formulations and assumptions used for non-smooth mechanical systems are adopted for electrical systems using the position-flux analogy. For the most important non-smooth electrical elements, like diodes and switches, set-valued branch relations are formulated and related to analogous mechanical elements. With the set-valued branch relations, the dynamics of electrical circuits are described as measure differential inclusions. For the numerical solution, the measure differential inclusions are formulated as a measure complementarity system and discretised with a difference scheme, known in mechanics as time-stepping. For every time-step a linear complementarity problem is obtained. Using the example of the DC-DC buck converter, the formulation of the measure differential inclusions, state reduction and their numerical solution using the time-stepping method is shown for the flux approac

A time-stepping method for non-smooth mechanical systems

International audienceTime-stepping algorithms allow a robust simulation of dynamical problems with many unilateral constraints and friction. Its idea is to calculate velocity updates instead of accelerations. As a consequence, contact behaviour and impact can be treated by the same equations

Formulation and Preparation for Numerical Evaluation of Linear Complementarity Systems in Dynamics

In this paper, we provide a full instruction on how to formulate and evaluate planar frictional contact problems in the spirit of non-smooth dynamics. By stating the equations of motion as an equality of measures, frictional contact reactions are taken into account by Lagrangian multipliers. Contact kinematics is formulated in terms of gap functions, and normal and tangential relative velocities. Associated frictional contact laws are stated as inclusions, incorporating impact behavior in form of Newtonian kinematic impacts. Based on this inequality formulation, a linear complementarity problem in standard form is presented, combined with Moreau's time stepping method for numerical integration. This approach has been applied to the woodpecker toy, of which a complete parameter list and numerical results are given in the pape

Application of the nonsmooth dynamics approach to model and analyze the contact-impact events in cam-follower systems

The dynamic modeling and analysis of planar rigid multibody systems that experience contact-impact events is presented and discussed throughout this work. The methodology is based on the nonsmooth dynamics approach, in which the interaction of the colliding bodies is modeled with multiple frictional unilateral constraints. 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 forces and impulses in 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 an augmented Lagrangian approach, which is embedded in the Moreau time-stepping method. The effectiveness of the methodologies presented in this work is demonstrated throughout the dynamic simulation of a cam-follower system of an industrial cutting file machine.This work is supported by the Portuguese Foundation for the Science and Technology under the research project BIOJOINTS (PTDC/EME-PME/099764/2008). The first author expresses his gratitude to the Portuguese Foundation for the Science and Technology for the postdoctoral scholarship (SFRH/BPD/40067/2007). This research was conducted during a post-doctoral stay of the first author with Professor Christoph Glocker at the Center of Mechanics, ETH Zurich