Literature survey of contact dynamics modelling(8)

Literature survey of contact dynamics modelling

1220G.Gilardi,I.Sharf/Mechanism and Machine Theory37(2002)1213–1239

Contact sti?ness and damping forces are dependent,at the minimum,on two parameters––the coe?cient of sti?ness and the coe?cient of damping.For simple contact between two bodies,the former is determined by the geometry and the material of the contacting objects,while the co-e?cient of damping can be related to the coe?cient of restitution[32,34,35].An important ad-vantage of continuous contact dynamics analysis is the possibility of using one of many friction models available in literature.Di?erent models have been developed to permit a smooth transi-tion from sticking to sliding friction[29,36–38].Non-linear models,as well as non-local models have been used to represent the real behavior of the surface irregularities that cause the friction. The use of continuous models for contact forces allows to generalize the contact dynamics methodology to multi-body/multi-contact scenarios,as well as contact involving?exible bodies [5,38].

Two solution approaches can be distinguished in the context of continuous impact models.In the?rst,the contact model is expressed as an explicit functional relationship between the contact force and the generalized coordinates and their rates,with dependencies on certain geometric and material parameters.Application of this approach has been studied by several authors,including Ma[38],Kraus and Kumar[39],Deguet et al.[40],Vukobratovic and Potkonjak[41].The contact condition is a geometric state,and involves determination of the minimum distance or interference between surfaces[5,38].Several friction models have been used with the explicit contact model,for example,Coulomb’s model and its variations[39,41–43],or the bristle model[38].

The second approach for solving impact/contact dynamics within the continuous framework takes into account the deformation due to contact directly via the?exibility of the contacting bodies.No explicit relationship is employed between the normal contact forces and the inden-tation,however,the condition of impenetrability at the contact point must be enforced.This approach has been used by Kim[5],Bathe and Bouzinov[44],Farahani et al.[45],Heinstein et al.

[46].Impact can still be detected by checking the minimum distance between the bodies,similarly to the explicit solution.By imposing the geometric condition of impenetrability,it is possible to calculate the contact force,using the Lagrange multipliers method[5,44,46]or with other mathematical techniques[45].This method is typically used in conjunction with the?nite element discretization of the contacting bodies(or contacting regions).It is the closest to reality and makes no assumptions nor approximations on the fundamental nature of contact dynamics.

2.Discrete contact dynamics models

The discrete formulation is based on the assumptions that[10]:the impact process is instan-taneous and impact forces are impulsive;kinetic variables have discontinuous changes while no displacements occur during the impact,and that other?nite forces are negligible.This model is used mainly if the impact involves rigid or very hard and compact bodies,while the e?ects of deformation at the contact point are taken into account through coe?cients.The impact problem is solved by using the linear impulse–momentum principle,the angular impulse–momentum principle,and the relations between the variables before and after impact[3,4].If m is the mass,v the center of mass velocity,P the linear impulse due to impact,h the angular momentum,d the distance from the center of mass to the point of impact and M the angular impulse due to impact, the impact dynamics equations are:

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