 Contact dynamics

Contact dynamics deals with the motion of multibody systems subjected to unilateral contacts and friction. Such systems are omnipresent in many multibody dynamics applications. Consider for example
 Contacts between wheels and ground in vehicle dynamics
 Squealing of brakes due to friction induced oscillations
 Motion of many particles, spheres which fall in a funnel, mixing processes (granular media)
 Clockworks
 Walking machines
 Arbitrary machines with limit stops, friction.
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 numerical integration. In addition, some examples are given.
Contents
Modeling
The two main approaches for modeling mechanical systems with unilateral contacts and friction are the regularized and the nonsmooth 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 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.
Regularized approach
A regularized force law for friction writes the friction force as function of the velocity, see figure 2. Doing so, one can eliminate the friction force to obtain a system of ordinary differential equations. A regularized force law for a unilateral contact corresponds to a spring whose stiffness vanishes for an open contact. The regularized approach is easy to understand but has numerical drawbacks in application. The resulting ordinary differential equations are stiff and require therefore special attention. In addition, oscillations may occur which are induced by the regularization. Also the choice of suitable regularization parameters is a problem. Considering a unilateral contact, the regularization parameter can be interpreted as contact stiffness. The regularization parameter of a friction element lacks such a physical interpretation. Considering a regularized friction law, also the sticking case is associated with small velocities, which does not correspond to the physical nature of friction. The regularized approach is related to the concept of regularization.Normally one would use lube to reduce friction, because if there is too much kinetic friction it will turn in to static enabling you to move in
Nonsmooth approach
A more sophisticated approach is the nonsmooth approach, which uses setvalued force laws to model mechanical systems with unilateral contacts and friction. Consider again the block which slides or sticks on the table. The associated setvalued 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 setvalued and determined according to an additional algebraic constraint.
To conclude, the nonsmooth 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, impacts can occur, and the time evolutions of the positions and the velocities can not be assumed to be smooth anymore. As a consequence, additional impact equations and impact laws have to be defined. In order to handle the changing mathematical structure, the setvalued force laws are commonly written as inequality or inclusion problems. The evaluation of these inequalities/inclusions is commonly done by solving linear (or nonlinear) complementarity problems, by quadratic programming or by transforming the inequality/inclusion problems into projective equations which can be solved iteratively by Jacobi or Gauss–Seidel techniques. The nonsmooth 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 DAE theory and leads to robust integration schemes.
Numerical Integration
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 nonsmooth models of mechanical systems with unilateral contacts and friction, two main classes of integrators exist, the eventdriven and the socalled timestepping integrators.
Eventdriven integrators
Eventdriven integrators distinguish between smooth parts of the motion in which the underlying structure of the differential equations does not change, and in events or socalled switching points at which this structure changes, i.e. time instants at which a unilateral contact closes or a stick slip transition occurs. A these switching points, the setvalued force (and additional impact) laws are evaluated in order to obtain a new underlying mathematical structure on which the integration can be continued. Eventdriven integrators are very accurate but are not suitable for systems with many contacts.
Timestepping integrators
Socalled timestepping integrators are dedicated numerical schemes for mechanical systems with many contacts. The first timestepping 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 do work with the integral of the contact forces and not with the forces itself, the methods can handle both nonimpulsive motion and impulsive events like impacts. As a drawback, the accuracy of timestepping integrators is low. This lack can be fixed by using a stepsize 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.
Examples
This section gives some examples of mechanical systems with unilateral contacts and friction. The results have been obtained by a nonsmooth approach using timestepping integrators.
Granular matters
Timestepping methods are especially well suited for the simulation of granular materials. Figure 4 depicts the simulation of 1000 disks which are mixed.
Billiard
Consider two colliding spheres in a billiard play. Figure 5a shows some snapshots of two colliding spheres, figure 5b depicts the associated trajectories.
Wheely of a motorbike
If a motorbike is accelerated too fast, it makes a wheely. Figure 6 shows some snapshots of a simulation.
Motion of the woodpecker toy
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, which is controlled by the sleeve. Figure 7 shows some snapshots of a simulation.
See also
 Multibody dynamics
 Contact mechanics: Applications with unilateral contacts and friction. Static applications (contact between deformable bodies) and dynamic applications (Contact dynamics).
 LubachevskyStillinger algorithm of simulating compression of large assemblies of hard particles
References
Further reading
 Acary V. and Brogliato, B. Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics. Springer Verlag, LNACM 35, Heidelberg, 2008.
 Brogliato B. Nonsmooth Mechanics. Communications and Control Engineering Series SpringerVerlag, London, 1999 (2nd Ed.)
 Glocker, Ch. Dynamik von Starrkoerpersystemen mit Reibung und Stoessen, volume 18/182 of VDI Fortschrittsberichte Mechanik/Bruchmechanik. VDI Verlag, Düsseldorf, 1995
 Glocker Ch. and Studer C. Formulation and preparation for Numerical Evaluation of Linear Complementarity Systems. Multibody System Dynamics 13(4):447463, 2005
 Jean M. The nonsmooth contact dynamics method. Computer Methods in Applied mechanics and Engineering 177(34):235257, 1999
 Moreau J.J. Unilateral Contact and Dry Friction in Finite Freedom Dynamics, volume 302 of Nonsmooth Mechanics and Applications, CISM Courses and Lectures. Springer, Wien, 1988
 Pfeiffer F., Foerg M. and Ulbrich H. Numerical aspects of nonsmooth multibody dynamics. Comput. Methods Appl. Mech. Engrg 195(5051):68916908, 2006
 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. Int. J. Numer. Meth. Engng 66(7):10791124, 2006
 Stewart D.E. and Trinkle J.C. An Implicit TimeStepping Scheme for Rigid Body Dynamics with Inelastic Collisions and Coulomb Friction. Int. J. Numer. Methods Engineering 39(15):26732691, 1996
 Studer C. Augmented timestepping integration of nonsmooth dynamical systems, PhD Thesis ETH Zurich, ETH ECollection, to appear 2008
 Studer C. Numerics of Unilateral Contacts and Friction—Modeling and Numerical Time Integration in NonSmooth Dynamics, Lecture Notes in Applied and Computational Mechanics, Volume 47, Springer, Berlin, Heidelberg, 2009
External links
 Multibody research group, Center of Mechanics, ETH Zurich.
 Lehrstuhl für angewandte Mechanik TU Munich.
 BiPoP Team, INRIA RhoneAlpes, France,
 Multibody dynamics, Rensselaer Polytechnic Institute.
 Siconos software
 dynamY software
 LMGC90 software
 Solfec software
Categories: Mechanics
 Dynamical systems
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