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the integral of the contact forces and not with the forces itself, the methods can handle both motion and impulsive events like impacts. As a drawback, the accuracy of time-stepping integrators is low. This can be fixed by using a step-size refinement at switching points. Smooth parts of the motion are processed by larger step sizes, and higher order integration methods can be used to increase the integration order.
98:, which uses set-valued force laws to model mechanical systems with unilateral contacts and friction. Consider again the block which slides or sticks on the table. The associated set-valued friction law of type Sgn is depicted in figure 3. Regarding the sliding case, the friction force is given. Regarding the sticking case, the friction force is set-valued and determined according to an additional algebraic
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transition occurs. At these switching points, the set-valued force (and additional impact) laws are evaluated in order to obtain a new underlying mathematical structure on which the integration can be continued. Event-driven integrators are very accurate but are not suitable for systems with many contacts.
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The woodpecker toy is a well known benchmark problem in contact dynamics. The toy consists of a pole, a sleeve with a hole that is slightly larger than the diameter of the pole, a spring and the woodpecker body. In operation, the woodpecker moves down the pole performing some kind of pitching motion,
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Event-driven integrators distinguish between smooth parts of the motion in which the underlying structure of the differential equations does not change, and in events or so-called switching points at which this structure changes, i.e. time instants at which a unilateral contact closes or a stick slip
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The two main approaches for modeling mechanical systems with unilateral contacts and friction are the regularized and the non-smooth approach. In the following, the two approaches are introduced using a simple example. Consider a block which can slide or stick on a table (see figure 1a). The motion
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Time-stepping integrators are dedicated numerical schemes for mechanical systems with many contacts. The first time-stepping integrator was introduced by J.J. Moreau. The integrators do not aim at resolving switching points and are therefore very robust in application. As the integrators work with
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The integration of regularized models can be done by standard stiff solvers for ordinary differential equations. However, oscillations induced by the regularization can occur. Considering non-smooth models of mechanical systems with unilateral contacts and friction, two main classes of integrators
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techniques. The non-smooth approach provides a new modeling approach for mechanical systems with unilateral contacts and friction, which incorporates also the whole classical mechanics subjected to bilateral constraints. The approach is associated to the classical
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of the block is described by the equation of motion, whereas the friction force is unknown (see figure 1b). In order to obtain the friction force, a separate force law must be specified which links the friction force to the associated velocity of the block.
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To conclude, the non-smooth approach changes the underlying mathematical structure if required and leads to a proper description of mechanical systems with unilateral contacts and friction. As a consequence of the changing mathematical structure,
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anymore. As a consequence, additional impact equations and impact laws have to be defined. In order to handle the changing mathematical structure, the set-valued force laws are commonly written as
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This section gives some examples of mechanical systems with unilateral contacts and friction. The results have been obtained by a non-smooth approach using time-stepping integrators.
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software. An open-source software dedicated to the modeling and the simulation or nonsmooth dynamical systems, especially mechanical systems with contact and
Coulomb's friction
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In the following it is discussed how such mechanical systems with unilateral contacts and friction can be modeled and how the time evolution of such systems can be obtained by
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Potra F.A., Anitescu M., Gavrea B. and
Trinkle J. A linearly implicit trapezoidal method for integrating stiff multibody dynamics with contacts, joints and friction.
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Consider two colliding spheres in a billiard play. Figure 5a shows some snapshots of two colliding spheres, figure 5b depicts the associated trajectories.
275:: Applications with unilateral contacts and friction. Static applications (contact between deformable bodies) and dynamic applications (Contact dynamics).
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Drumwright, E. and Shell, D. Modeling
Contact Friction and Joint Friction in Dynamic Robotic Simulation Using the Principle of Maximum Dissipation.
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Anatomic tissues (skin, iris/lens, eyelids/anterior ocular surface, joint cartilages, vascular endothelium/blood cells, muscles/tendons, et cetera)
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Time-stepping methods are especially well suited for the simulation of granular materials. Figure 4 depicts the simulation of mixing 1000 disks.
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Figure 1: Block which can slide or stick on a table. Figure a) depicts the model, figure b) the equation of motion with unknown friction force
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Stewart D.E. and
Trinkle J.C. An Implicit Time-Stepping Scheme for Rigid Body Dynamics with Inelastic Collisions and Coulomb Friction.
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Brogliato, B. Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics.
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Glocker Ch. and Studer C. Formulation and preparation for
Numerical Evaluation of Linear Complementarity Systems.
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or by transforming the inequality/inclusion problems into projective equations which can be solved iteratively by
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problems. The evaluation of these inequalities/inclusions is commonly done by solving linear (or nonlinear)
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Numerics of
Unilateral Contacts and Friction—Modeling and Numerical Time Integration in Non-Smooth Dynamics
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If a motorbike is accelerated too fast, it does a wheelie. Figure 6 shows some snapshots of a simulation.
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407:, Lecture Notes in Applied and Computational Mechanics, Volume 47, Springer, Berlin, Heidelberg, 2009
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can occur, and the time evolutions of the positions and the velocities can not be assumed to be
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Motion of many particles, spheres which fall in a funnel, mixing processes (granular media)
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Pfeiffer F., Foerg M. and
Ulbrich H. Numerical aspects of non-smooth multibody dynamics.
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which is controlled by the sleeve. Figure 7 shows some snapshots of a simulation.
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Advanced Robotics: Algorithmic Foundations of Robotics IX
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exist, the event-driven and the so-called time-stepping integrators.
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480:, an open source multi-physics simulation engine, see also project
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in Ch. Glocker's group (High-Performance
Computing with MPI), 2016
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Augmented time-stepping integration of non-smooth dynamical systems
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Non-smooth Mechanics and Applications, CISM Courses and Lectures
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Unilateral Contact and Dry Friction in Finite Freedom Dynamics,
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Brogliato B. Nonsmooth Mechanics. Models, Dynamics and Control
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of simulating compression of large assemblies of hard particles
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400:, PhD Thesis ETH Zurich, ETH E-Collection, to appear 2008
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Dynamik von Starrkoerpersystemen mit Reibung und Stoessen
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Figure 5: a) Snapshot. b) Trajectories of the two spheres
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Squealing of brakes due to friction induced oscillations
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Granular Rigid Body Simulation Framework developed at
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Computer Methods in Applied mechanics and Engineering
357:Jean M. The non-smooth contact dynamics method.
345:VDI Fortschrittsberichte Mechanik/Bruchmechanik.
151:theory and leads to robust integration schemes.
254:A simulation and visualization can be found at
58:Arbitrary machines with limit stops, friction.
327:Communications and Control Engineering Series
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321:Springer Verlag, LNACM 35, Heidelberg, 2008.
109:Figure 3: Set-valued force law for friction
250:Figure 7: Simulation of the woodpecker toy
329:Springer-Verlag, London, 2016 (third Ed.)
69:. In addition, some examples are given.
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38:Contacts between wheels and ground in
94:A more sophisticated approach is the
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444:, Rensselaer Polytechnic Institute.
256:https://github.com/gabyx/Woodpecker
420:, Center of Mechanics, ETH Zurich.
391:Int. J. Numer. Methods Engineering
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424:Lehrstuhl fĂĽr angewandte Mechanik
377:Comput. Methods Appl. Mech. Engrg
201:Figure 4: Mixing a thousand disks
279:Lubachevsky-Stillinger algorithm
233:Figure 6: Wheely of a motorbike
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300:"Contact in multibody system"
432:, INRIA Rhone-Alpes, France,
347:VDI Verlag, DĂĽsseldorf, 1995
238:Motion of the woodpecker toy
16:Motion of multibody systems
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384:Int. J. Numer. Meth. Engng
379:195(50-51):6891-6908, 2006
352:Multibody System Dynamics
173:Time-stepping integrators
22:deals with the motion of
418:Multibody research group
164:Event-driven integrators
132:complementarity problems
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155:Numerical integration
136:quadratic programming
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67:numerical integration
354:13(4):447-463, 2005
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90:Non-smooth approach
28:unilateral contacts
442:Multibody dynamics
268:Multibody dynamics
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190:Granular materials
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503:Dynamical systems
273:Contact mechanics
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286:References
124:inequality
100:constraint
51:Clockworks
498:Mechanics
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262:See also
206:Billiard
182:Examples
73:Modeling
32:friction
482:website
436:Siconos
116:impacts
478:Chrono
336:, 2010
140:Jacobi
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