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6.5. CONTACT 81
Figure 6.3: Illustration of Coulomb friction for a contact i. Magnitude of the tangential force fti
is restricted so that fti ≤ µi · fni and hence the tip of the contact force fci = fni + fti exerted
on Bi at pi is restricted to lie inside or on the boundary of the friction cone. Force − fci exerted
on Ai at pi and bodies Ai and Bi are not drawn. If µi = 0 then fti = 0 and the problem reduces
to non-penetration illustrated at figure (6.2).
❅❅
❅
❅
❅
fni
✻
ni
✻
✲
pi t x ✓
i
✓✓✼
✓ ✓✓✼ t y
µi · fni
i
✒
fti
✄ ✄✄✄✄✄✄✄✄✗
fci
• if vti = 0 and ati = 0 then the stable static (dry, rolling, sticking) friction is in effect at
contact i and
fti ≤ µi · fni . (6.39)
• if vti = 0 and ati = 0 then the instable static (dry, rolling, sticking) friction is in effect at
contact i and
fti = µi · fni
fti · ati ≤ 0. (6.40)
We would like to express the conditions stipulated by the Coulomb law in the form of
acceleration-level constraints (6.17). Unfortunately this is not possible because fti is a nonlinear
function and so Coulomb law has to be approximated.
Contact force fci can be imagined as if its end-tip was restricted to lie inside an infinite cone
centered at the position of the contact, the axis was pointing in the direction of the surface normal
and the “width” was determined by µi. Coulomb law says that the contact force tip should lie
inside or on the boundary of the cone in the case of stable static friction, that the tip should lie
on the boundary of the cone and the direction of the contact force should oppose the direction
of the tangential acceleration in the case of instable static friction, and that the tip should lie
on the boundary of the cone and the contact force should act in the direction opposite to the
direction of the tangential velocity in the case of dynamic friction. By setting µi to zero, the
Coulomb friction law reduces to the known acceleration-level non-penetration constraint ¨ C i p ≥ 0.
See figure (6.3) for illustration.
Note that stable and instable static friction must be handled altogether because fti affects
ati . Therefore, given a contact i, we can tell whether there is a dynamic or static friction in effect
by inspecting vti = 0, but if it is static we can not tell if it is actually a stable or instable static
friction until all constraints are solved. Depending on whether static or dynamic friction is in
effect at contact i, different constraints will be effective due to i.
We will now present how dynamic friction and static friction can be approximated. The
approximation of the static friction we are going to present will replace the friction cone by a
four-sided friction pyramid whose sides are parallel to the tangential directions t x i and t y
i and
friction along t x i will be handled independently on friction along t y
i .
t y
i
We will assume that the contact representation is augmented to hold µi and directions t x i and
, henceforward called the friction approximation directions.