PDF《Pure Elastic Contact Force Models》

b two bodies in the state of contact (indentation, δ)

16 2 Pure Elastic Contact Force Models Ak ? cos /k ?sin /k sin /k cos /k k ? i;

j ? ? ?2:3? A normal vector to the plane of contact, illustrated in Fig. 2.1b, can be deter- mined as n ? d d ?2:4? where the magnitude of the vector d is evaluated as d ? nT d ?2:5? The minimum distance condition given by Eq. (2.5) is not enough to ?nd the possible contact points between the contact bodies, since it does not cover all possible scenarios that may occur in the contact problem. Therefore, the contact points are de?ned as those that correspond to maximum indentation, that is, the points of maximum relative deformation, measured along the normal direction (Lopes et al. 2010;

Machado et al. 2014). Thus, three geometric conditions for contact can be de?ned as, (i) the distance between the potential contact points given by vector d corresponds to the minimum distance;

(ii) the vector d has to be collinear with the normal vector ni;

(iii) the normal vectors ni and nj at the potential contact points have to be collinear. The conditions (ii) and (iii) can be written as two cross products as (Machado et al. 2011) nj ? ni ?

0 ?2:6? d ? ni ?

0 ?2:7? The geometric conditions given by Eqs. (2.6) and (2.7) are two nonlinear equations with two unknowns, which can be solved using a Newton-Raphson iterative procedure (Atkinson 1989;

Nikravesh 1988). This system of equations provides the solutions for the location of the potential contact points. Once the potential contact points are found, the next step deals with the evaluation of the relative indentation between the contact bodies as (Flores and Ambrósio 2004) d ? ???????? dT d p ?2:8? The velocities of the contact points expressed in terms of the global coordinate system are evaluated by differentiating Eq. (2.2) with respect to time, yielding _ rP k ? _ rk ? _ Aks0P k k ? i;

j ? ? ?2:9? 2.1 Generalized Contact Kinematics

17 in which the dot denotes the derivative with respect to time. The relative normal velocity is determined by projecting the contact velocity onto the direction normal to the plane of contact, yielding (Flores et al. 2004) vN ? _ d ? nT _ rP j ? _ rP i ?2:10? This way of representing the relative normal velocity is quite convenient, in the measure that it is not necessary to deal with the derivation o........