Summary This article presents a family of variational integrators from a continuous time point of view. A general procedure for deriving symplectic integration schemes preserving an energy‐like quantity is shown, which is based on the principle of virtual work. The framework is extended to incorporate holonomic constraints without using additional regularization. In addition, it is related to well‐known partitioned Runge–Kutta methods and to other variational integration schemes. As an example, a concrete integration scheme is derived for the planar pendulum using both polar and Cartesian coordinates. Copyright © 2016 John Wiley & Sons, Ltd.

In this work, we propose some basic approaches towards a unification of the theories for deformable and rigid bodies. This unification process is based on two fundamental mechanical concepts, which are the principle of virtual work and the principle of d’Alembert–Lagrange. The basic idea is to initially look upon structural elements as general continua, and to endow them later with specific properties like rigidity, imposed by perfect bilateral constraints. It is shown by the example of a simple flexible multibody system how this unifying and systematic approach has to be carried out. The system under consideration consists of a rigid disk and a nonlinear elastic string, which may come into contact with each other. The contact is modeled as a hard unilateral geometric constraint combined with a one-dimensional Coulomb friction element. The contact interactions are formulated as set-valued force laws and impact laws, and the system is consequently treated within the framework of nonsmooth dynamics. The model of the string allows for large deformations in time and for a nonlinear elastic material response. By constraining the kinematics of the string to finite dimensions, a nonlinear finite element formulation is achieved in a very natural way.

SUMMARYIn the present work, a new director‐based finite element formulation for geometrically exact beams is proposed. The new beam finite element exhibits drastically improved numerical performance when compared with the previously developed director‐based formulations. This improvement is accomplished by adjusting the underlying variational beam formulation to the specific features of the director interpolation. In particular, the present approach does not rely on the assumption of an orthonormal director frame. The excellent performance of the new approach is illustrated with representative numerical examples. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper we use step size adjustment and extrapolation methods to improve Moreau's time-stepping scheme for the numerical integration of non-smooth mechanical systems, i.e. systems with impact and friction. The scheme yields a system of inclusions, which is transformed into a system of projective equations. These equations are solved iteratively. Switching points are time instants for which the structure of the mechanical system changes, for example, time instants for which a sticking friction element begins to slide. We show how switching points can be localized and how these points can be resolved by choosing a minimal step size. In order to improve the integration of non-smooth systems in the smooth parts, we show how the time-stepping method can be used as a base integration scheme for extrapolation methods, which allow for an increase in the integration order. Switching points are processed by a small time step, while time intervals during which the structure of the system does not change are computed with a larger step size and improved integration order. The overall algorithm, which consists of a time-stepping module, an extrapolation module and a step size adjustment module, is discussed in detail and some examples are given. Copyright (C) 2008 John Wiley & Sons, Ltd.

AbstractImpacts in rigid multibody systems cause instantaneous jumps in the generalised velocities. Naturally, the velocity after impact depends on the chosen impact law. An impact law should fulfil kinematic, kinetic as well as energy restrictions. In this paper, we study the domain of possible post‐impact velocities for arbitrary impact laws. For single‐contact collisions, this domain is at most one‐dimensional but the domain becomes higher dimensional in the multi‐contact case. The domain of possible post‐impact velocities is a compact convex subset of the tangent space to the configuration manifold. Using a complete canonical parameterisation, the post‐impact velocity of all impact laws can be addressed. For instance, the impact law corresponding to maximal dissipation as well as Newtons (extended) impact law are examples of incomplete canonical parameterisations. These impact laws are not complete as non‐local impact effects are not addressed. Here, we try to find a complete canonical parameterisation which covers non‐local impact effects. Moreover, the relationship between symmetries and conservation laws in this context will be elucidated. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Analyzing non-smooth mechanical systems requires often the solution of inclusion problems of normal cone type. These problems arise for example in the event-driven or time-stepping simulation approaches. Such inclusion problems can be written as non-linear equations, which can be solved iteratively. In this paper we discuss three different methods to derive the non-linear equations representing the inclusions arising in the event-driven simulation approach. First, we formulate inclusions describing the individual non-smooth constraints and solve them successively. Secondly, we interpret the non-linear equations as the conditions for the saddle point of the augmented Lagrangian function. As a third possibility we discuss the exact regularization of set-valued force laws. All three methods lead to the same numerical scheme, but give different insight into the problem. Especially the factor r occurring in the non-linear equations is discussed. Two iterative methods for solving the non-linear equations are presented together with some remarks on convergence.

There are three basic equations in mechanics for treating collisions: the law of impact, kinematic compatibility, and energetic consistency. In this paper, the conditions are examined under which a natural extension of the dynamics at an impact is possible without taking additional impact laws, and which additional assumptions have to be made to solve the impact for different classes of systems. It will be shown that Newton's law of impact for two colliding point masses can be derived from the concept of energy conservation and the principle of maximum dissipation, and has therefore not to be regarded as an independent equation. Moreover, it can be assigned to single-contact impacts in multibody systems as soon as the classical definition of perfect constraints is being extended to impulsive dynamics and unilateral contacts. It will further be shown that the principle of maximum dissipation leads to a unique post-impact velocity in the case of multi-contact collisions. In all other cases, however, the velocities remain undetermined, and laws of impact have to be postulated as additional and independent equations, whereas the classic definition of the restitution coefficient as a dissipation parameter can still be kept.

The aim of the present paper is to present some new results for problems when impacts occur. We prove, in the framework of linear elastic body, certain equivalent form of the d'Alembert's principle, including velocity discontinuity. We show that the theorem of stationary action still holds as an inequality. The consideration of the variation of the unknown impact time implies for the impact problem certain new variational expressions in inequality form.

The paper treats the evaluation of the accelerations in rigid multibody systems which are subjected to displacement dependent setvalued force interactions. The interaction laws are represented by nonsmooth 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.

An impact model for two-dimensional contact situations is developed which contains the main physical effects of a compliance element in the normal direction and a series of a compliance and Coulomb friction elements in the tangential direction. For systems with multiple impacts a unilateral formulation based on Poisson's hypothesis is used to describe the impulses which are transferred in the normal direction. The event of an impact is divided into two phases. The phase of compression ends with vanishing approaching velocity if normal impulses are transferred and is equivalent to a completely inelastic collision. The phase of expansion allows the bodies to separate under the action of the normal impulses whenever they are large enough. The absolute values of the tangential impulses are bounded by the magnitudes of the normal impulses, due to the Coulomb friction relationship on the impulse level. One part of the transferred tangential impulse during compression is assumed to be partly reversible which may be regarded as an application of Poisson's law. The remaining part is completely irreversible and considered friction. This formulation contains the special case of completely elastic tangential impacts as well as the situation when only Coulomb friction acts. It is proven that the presented impact model is always dissipative or energy preserving. The evaluation of the problem is done by solving one set of complementarity conditions during compression and a nearly identical set of equations during expansion. The theory is applied to some basic examples which demonstrate the difference between Newton's and Poisson's hypotheses.

In multibody dynamics, topology variations are caused by the fact that bodies, that are initially separated from one another, get into contact and slide or roll along each other under the influence of friction. These topology variant systems are characterized by the fact that, during the evolution in time, their number of degrees of freedom changes by latent constraints becoming active or passive due to and controlled by the system dynamics itself. In studying such systems, the following procedure is selected: With a system description in minimal coordinates without use of latent constraints, the constraints that are indicated as being potentially active by the evaluation of kinematic indicators, in this case being relative velocity and distance, are considered as algebraic secondary conditions and are taken into account by including Lagrange multipliers in the equation of motion. A sufficient condition for all potentially active constraints to remain active or become passive is provided by the solution of a complementarity problem that, in a planar case, is linear and argues at acceleration level by self-excluding kinetic indicators.

In the following a percussion drilling machine is examined as an example for mechanical systems with unilateral contacts. It is characteristic for such systems that the number of degrees of freedom changes during motion. To avoid a description of each possible system state using different sets of minimal coordinates, the constrained motion is taken into account by algebraic relations. This method has the advantage that the motion of the system and simultaneously the constraint forces are available, which is necessary to obtain conditions for a change in the state of the system. Furthermore different combinations of constraints can be easily taken into consideration in this way.

These lectures treat the motion of finite-dimensional mechanical systems under the influence of set-valued force laws that are derived from scalar potential functions by generalized dierentiation.