Literature survey of contact dynamics modelling(5)

Literature survey of contact dynamics modelling

mains in the bodies in the form of vibration.In this case the coe?cient of restitution may be signi?cantly in?uenced by this factor [2].Later in the paper,we discuss three models proposed for the coe?cient of restitution:Newton’s [12],Poisson’s [13]and Stronge’s [14,15].

1.1.

2.Tangential coe?cients

Friction modelling is another key aspect of impact and contact dynamics,since friction can stop and/or reverse the motion as well as,it contributes to energy dissipation.If friction is taken into account,the relationship traditionally used to determine the force of dry friction is the Coulomb’s law.This law states that the magnitude of the frictional force,F t ,can be related to the magnitude of the normal force,F n ,via a coe?cient and its direction is always opposite to the relative tangential motion [16].Two possible situations are distinguished:(a)sliding,with F t ¼l d F n ;(b)sticking,with F t 6l s F n ,where l d is the coe?cient of dynamic friction and l s is the coe?cient of static friction.The two coe?cients take into account the nature of the contact surfaces,mainly,the type of material and surface quality.Alternative ways to describe the be-havior in the tangential directions include the use of tangential coe?cient of restitution,based on Newton’s model,and the impulse ratio de?ned as a ratio of tangential impulse to normal impulse.The latter is a generalization of the coe?cient of friction and can take into account other tan-gential forces [3,4].

1.1.3.Impulse and impact energy

The time integral of the contact force F C acting during the impact is called the impulse,P ,and is ?nite [10]:P ¼lim D t !0

Z F C ðt Þd t ð1ÞThe energy loss E L incurred by impact can be calculated as the negative work done by F C during the collision [17],that is:E L ¼ÀZ F C Á_d

d t ¼ÀZ F C Ád d ð2Þwher

e _d

is the velocity of deformation.Alternatively,by making use of the di?erential relationship between impulse and force,one can write:E L ¼ÀZ _d Ád P ð3ÞAccording to the work-energy principle,2the work done by F C must be equal to the change in the kinetic energy,D T [18],so that:

D T þ

E L ¼0ð4ÞIt follows that the change in the kinetic energy depends on the coe?cient of restitution and im-pulse ratios.In [17],Stronge states a theorem which allows to calculate the work done by contact

2Usually,in literature on the subject,the term used is the energy-conservation principle .This terminology is not completely correct,because of the presence of friction and other energy dissipation.G.Gilardi,I.Sharf /Mechanism and Machine Theory 37(2002)1213–12391217

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