Literature survey of contact dynamics modelling(9)

Literature survey of contact dynamics modelling

m1v1

ðÀv10Þ¼P

m2v2

ðÀv20Þ¼ÀP

h1Àh10¼d1ÂPþM

h2Àh20¼d2ÂPÀM

ð5Þ

In the above,indexes1and2specify the body,while0denotes the initial conditions.The un-knowns are the linear and angular velocities of the two bodies and the impulses P and M.The angular impulse M is neglected in the majority of formulations since,consequently to the basic assumptions,the contact region must be small.

Additional relations are required to solve for the unknown impact variables.For the normal direction,one relation is provided by the coe?cient of restitution.In the tangential direction,the relational laws may have to be replaced with kinematic constraints(for instance,during sticking, zero tangential velocity is imposed).

2.1.Coe?cient of restitution models

Let the vector triadðn;t;bÞde?ne a coordinate system with origin at the contact point,where n is the normal to the two bodies at that point,and vectors t and b de?ne the tangent plane[3,4]. Then the linear impulse can be written as

P¼P n nþP t tþP b bð6ÞThe relative linear velocity at the contact point,denoted by C,has a component along the normal direction,called the compression velocity,and a component along the bi-tangential direction, called the sliding velocity[10].The principle models of restitution are introduced below.

2.1.1.Poisson’s model

In Poisson’s model[13],the total normal impulse,P f,is divided in two parts,P c and P r,cor-responding to compression and restitution phases,respectively.The coe?cient of restitution is de?ned as[10]

e¼P r

P c

;P f¼P cþP rð7Þ

The condition for the end of compression phase is zero relative velocity along the normal di-rection,that is CÁn¼0.In the P nÀP t space,this represents the line of maximum compression[10]. Using this de?nition and Eqs.(5)–(7),it is possible to de?ne the line of termination as:

C0þ

C1

1þe

P nþC2P t¼0;t¼t fð8Þ

where C0is the approach velocity,C1and C2are parameters depending on initial conditions, geometry and inertia[10].

G.Gilardi,I.Sharf/Mechanism and Machine Theory37(2002)1213–12391221

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