Literature survey of contact dynamics modelling(10)

Literature survey of contact dynamics modelling

2.1.2.Newton’s model

In Newton’s model the coe?cient of restitution is de?ned as[12]:

e¼ÀCðt fÞÁn

Cðt0ÞÁn

¼À

C f

C0

ð9Þ

This model is based on a kinematic point of view and only the initial and?nal values for the relative normal velocity are taken into account.The line of termination is given by[10]ð1þeÞC0þC1P nþC2P t¼0;t¼t fð10Þ

2.1.

3.Stronge’s model

This model is based on the internal energy dissipation hypothesis[14].The coe?cient of resti-tution is de?ned as the square root of the ratio of energy released during restitution to the energy absorbed during compression.In terms of the work done by the normal force during the two phases,the coe?cient of restitution can be calculated from:

e2¼

W r

ÀW c

ð11Þ

It can be shown that the energy hypothesis leads to the only model which ensures that the energy loss from sources other than friction is non-negative,and is zero when e¼1.In[47],Stronge applies the above de?nition to derive a theoretical expression for e in terms of W c and the work required to initiate yield.

In a recent work[15],Stronge considers the problem of oblique impact of a rigid cylinder on a deformable half-space.It is noted that in this and similar cases of collisions between objects of very di?erent sizes,the energy loss to stress waves,W w,is substantial and can be accounted for with the following de?nition of the coe?cient of restitution:

e2¼W rÀW w

ÀW c

:ð12Þ

2.2.Additional relations

To obtain additional equations,one can pursue either of two possibilities.The?rst is to de?ne coe?cients of restitution for each of the other directions.For instance during slipping,one can obtain the following relationships by applying Newton’s model in the tangential plane[3]:

e t¼ÀCðt fÞÁt

Cðt0ÞÁt

;e b¼À

Cðt fÞÁb

Cðt0ÞÁb

ð13Þ

In the same manner,it is possible to de?ne three coe?cients to model the rotational e?ects of the impact[3,4].The second possibility is to de?ne a relationship between quantities in the normal and tangential directions.Brach[3,4]proposed to relate the tangential and normal impulses with equations analogous to Coulomb’s law:

P t¼l t P n;P b¼l b P nð14Þwhere coe?cients l are called impulse ratios.Smith[20]extended the use of Coulomb’s law to allow for a change in the direction of tangential velocity.If sticking occurs before the end of the impact,relationships(13)and(14)must be replaced with a condition of zero slip,i.e.,CÁt¼0 and CÁb¼0,referred to as lines of sticking[23].

1222G.Gilardi,I.Sharf/Mechanism and Machine Theory37(2002)1213–1239

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