Literature survey of contact dynamics modelling(6)

Literature survey of contact dynamics modelling

1218G.Gilardi,I.Sharf/Mechanism and Machine Theory37(2002)1213–1239

forces during periods of unidirectional slip as a simple function of the corresponding impulse and the average relative velocity at the contact point.

1.2.Literature overview

1.2.1.Discrete models

The impact between two rigid bodies was analysed initially by Sir Isaac Newton[19],and expanded by Whittaker[12]to account for frictional impulse.In that model the coe?cient of restitution is a kinematic quantity that de?nes a relationship between the normal components of the velocities before and after the impact at the contact point(referred to as Newton’s model). Routh[13]presented a graphic method based on a kinetic hypothesis to de?ne the coe?cient of restitution(referred to as Poisson’s model).The coe?cient of restitution is de?ned as a kinetic quantity that relates the normal impulses that occur during the compression and restitution phases.The two approaches are also di?erent in the treatment of the motion in tangential di-rection during the impact.In Routh’s study,the possibility of changes in slip direction during contact is taken into account,while in Whittaker’s study it is not.In many simple cases,the two approaches lead to the same result,as shown by Wang and Mason[10],while in other cases,they can produce inconsistent results,as shown by Kane and Levinson[18]and Stronge[14].This is a consequence of the possible changes in the slip direction.Ignoring these can lead to the overes-timation of the?nal velocity after the impact,as illustrated with the Newtonian approach. Poisson’s model,instead,can result in an increase of energy in some con?gurations for a perfectly elastic impact[14].

Brach[3,4]proposed an algebraic solution scheme,revising Newton’s model and introducing impulse ratios to describe the behavior in the tangential directions.He de?ned the tangential impulse as a constant fraction of the normal impulse––the constant ratio of the two being the impulse ratio.It is equivalent to the friction coe?cient in many cases.Brach also demonstrated that work-energy and kinematic constraints impose an upper bound on the impulse ratio.He also expanded this approach to include the impulse moments.Alternatively,the motion in the tan-gential direction was described by using the tangential coe?cient of restitution.Smith[20]pro-posed another purely algebraic approach to the problem using the Newtonian de?nition for the coe?cient of restitution.Impulse ratio is determined using as velocity an average value of di?erent slipping velocities.Keller[21]developed an approach which involves the integration of the contact impulse variables.Thus,the system is treated as an evolving process parameterized by cumulative normal impulse.Also,by using a revised Poisson’s model,Keller concluded that during impact, no increase in energy is http://www.77cn.com.cning Routh’s graphical method to analyse the contact models, Wang and Mason[10]identi?ed the impact conditions under which Newton’s and Poisson’s models give the same solution.Stronge[14]demonstrated the energy inconsistencies in some solutions obtained with Poisson’s model when the coe?cient of restitution is assumed to be in-dependent of the coe?cient of friction.In that case Poisson’s model does not lead to vanishing dissipation for a perfectly elastic impact.As a result,Stronge proposed to de?ne the coe?cient of restitution as the square root of the ratio of the elastic strain energy released during restitution to the energy absorbed by deformation during compression.With this de?nition no energetic in-consistencies are present[14].

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