Literature survey of contact dynamics modelling(12)

Literature survey of contact dynamics modelling

m1ðv1Àv10Þ¼PþP U

1þP F

1

m2ðv2Àv20Þ¼ÀPþP U

2þP F

2

h1Àh10¼d1ÂPþMþd U

1ÂP U

1

þd F

1

ÂP F

1

h2Àh20¼d2ÂPÀMþd U

2ÂP U

2

þd F

2

ÂP F

2

ð15Þ

where P U is the vector of unknown impulses associated with geometric constraints on the bodies, P F is the vector of external known impulses,d U and d F are the position vectors from the center of mass to the point of application of the respective impulses[3].To solve Eq.(15),we need to resort to additional relations for the impact parameters.Di?erent models for these parameters lead to di?erent forms of the?nal equations and possibly di?erent results,depending on the particulars of the impact.

2.3.2.1.Algebraic http://www.77cn.com.cning Newton’s or Poisson’s models to de?ne the coe?cients of restitution in any direction(or about any axes),purely algebraic equations are obtained.Together with the impulse ratios,these equations can be written in the form:

e¼eðP;M;vÞ;l¼lðP;MÞð16ÞExamples of analytical solutions of Eqs.(15)and(16)can be found in Brach[3,4],Smith[20]and Mac Sithigh[48].Lagrange’s equations describing impact between two rigid bodies are presented in[16,25].These formulations solve for unknown generalized coordinates,the Lagrange multi-pliers associated with impact forces(or normal impulses)and the friction forces due to stiction. This approach has also been applied by some authors to?exible-body systems(see,for example, Kulief and Shabana[49],Yigit et al.[50]).In this case,the coe?cient of restitution value for relatively compact bodies must be used with care as it may be a?ected by the?exibility.Ref.[50] demonstrates that the‘‘rigid body’’concept of the coe?cient of restitution can be used for the ?exible beam considered by these authors.

2.3.2.2.Integral–di?erential equations.Another approach to solving the impact problem is to think of the impact as an evolving process parameterized by cumulative normal impulse[21,48].An application of this approach is reported by Keller[21],where Poisson’s model of restitution is used.Stronge[51]employes a similar analytical method to investigate changes in relative velocity, but with the use of the energetic coe?cient of restitution.The linear impulse P is divided in the usual two components,normal and tangential,given by:

P nðtÞ¼

Z t

0F CðsÞÁn d s¼

Z t

F n d s;P t¼À

Z t f

l F n t d t¼À

Z P f

l t d P n¼À

Zð1þeÞP m

l t d P n

ð17Þ

The normal component P n is used as an independent variable.The solution of the impact problem is reduced to determining P m,as well as the variation in the slip direction speci?ed by the tan-gential unit vector t.To this end,di?erentiating the relative linear velocity with respect to the normal impulse P n gives

1224G.Gilardi,I.Sharf/Mechanism and Machine Theory37(2002)1213–1239

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