Literature survey of contact dynamics modelling
G.Gilardi,I.Sharf/Mechanism and Machine Theory37(2002)1213–12391219 Multiple frictional contacts in multi-body systems have been studied by several authors. Hurmuzlu and Marghitu[22]examine the problem where a planar rigid-body kinematic chain undergoes an external impact and an arbitrary number of internal impacts.The latter was de?ned as the situation when two bodies are simply in contact while impact occurs elsewhere in the system [22].They developed a di?erential–integral approach,extending Keller’s work[21]and using all three models for the coe?cient of restitution,and an algebraic approach,based on Newton’s model of restitution.Han and Gilmore[23]proposed a similar approach,using an algebraic formulation of motion equations,Poisson’s model of restitution and Coulomb’s law to de?ne the tangential motion.Di?erent conditions that characterize the motion(slipping,sticking,reverse motion)are detected by analyzing velocities and accelerations at the contact points[24],similarly to Hurmuzlu and Marghitu[22].Han and Gilmore[23]veri?ed their simulation results with experiments for a two-body and three-body impact.
Haug et al.[25]solve directly the di?erential equations of motion by using the Lagrange multiplier technique.For impact,Newton’s model is used,while Coulomb’s law is used for friction.Wang and Kumar[26],Anitescu et al.[27]reduce the problem to a quadratic program-ming problem.
1.2.2.Continuous models
Application of the impulse–momentum methods to model the impact dynamics of rigid bodies leads to several problems.First,in the presence of Coulomb friction,cases arise in which no solution or multiple solutions exist.Examples and analysis of these inconsistencies can be found in Wang and Kumar[26]and Mason and Wang[28].These ambiguities have been attributed to the approximate nature of Coulomb’s model and to the inadequacy of rigid body model,but no clear explanation has been found.The second problem is that energy conservation principles may be violated during frictional impacts,as shown by Stronge[14],as a consequence of the de?nition of the coe?cient of restitution.Finally,the discrete approach is not easily extendible to generic multi-body systems.The use of compliance or continuous contact models where the impact force is a function of local indentation can overcome the problems encountered in the discrete formula-tion[26,29].
Di?erent models have been postulated to represent the interaction force at the surfaces of two contacting bodies[2,3].The?rst model was developed by Hertz[30],in which an elastostatic theory was used to calculate local indentation without the use of damping.The corresponding relationship between the impact force and the indentation is allowed to be non-linear.In the?rst and simplest model of damping,referred to as spring-dashpot model[9],the contact force is represented by a linear spring-damper element.Dubowsky and Freudenstein[31]presented an extension of this model called the impact-pair model,where they assumed a linear viscous damping law and a Hertzian spring for modeling the behavior of the impact surfaces.Hunt and Crossley[32]showed that a linear damping model does not truthfully represent the physical nature of the energy transfer process.Thus,they proposed a model based on Hertz’s theory of contact with a non-linear damping force de?ned in terms of local penetration and the corre-sponding rate.Lee and Wang[33]proposed a similar model,but with a di?erent function spec-ifying the non-linear damping term.Other damping models have been proposed to describe totally or partially plastic impacts[2,3,15].