Literature survey of contact dynamics modelling(11)

Literature survey of contact dynamics modelling

2.3.Solution of impact problem

2.3.1.Planar impact of two bodies

Planar impact analysis is conveniently carried out with a graphical approach [10,23].The im-pact is represented in the P n -P t plane with the following lines as illustrated in Fig.4:

?

Line of limiting friction L LF ,de?ned by Coulomb’s friction model.?

Line of sticking L S ,de?ned by zero velocity along the tangential direction.?

Line of maximum compression L M ,de?ned by zero relative velocity in the normal direction.

?Line of termination L T ,de?ned according to the restitution model used.

The graphical construction as in Fig.4is completely speci?ed by the following parameters:the geometric conditions of the impact,the inertia properties of the colliding bodies,the velocities before impact,the coe?cients of friction and restitution.It is also noted that the use of graphical approach described here assumes Poisson’s or Newton’s models of restitution.

In Fig.4,Point A is the intersection between lines L S and L T ,and point B is the intersection between lines L S and L M .The resulting lines OA and OB de?ne three regions that allow to dis-tinguish the di?erent contact modes.In particular,if L LF lies in region #1,slipping without sticking is present;if it lies in region #2,sticking is reached after the maximum compression;if it lies in region #3,sticking is reached before the maximum compression and it may be followed by reverse motion [23].A detailed analysis of contact processes and their dependence on parameters de?ning the planar rigid-body collisions is presented in [47].The main disadvantage of the graphical (and analytical)approaches is that they are not easily extendible to three-dimensional impact,primarily because of the di?culty in describing the limiting friction [10,23].

2.3.2.General formulation and solution

Several formulations of the motion equations have been applied to solve the impact problem between two (or more)rigid bodies with a discrete approach.The more general form of these equations,previously stated in Eq.(5),

is:Fig.4.Graphical impact analysis.

G.Gilardi,I.Sharf /Mechanism and Machine Theory 37(2002)1213–12391223

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