
60 Friction-Induced Vibration and Self-Organization
For a given penetration depth d of two surfaces in contact, only asperities
with the height h > d participate in contact (Figure4.2a). Furthermore, for an
asperity of height h, the penetration is z = h – d, the contact radius is propor-
tional to (h – d)
1/2
, or just z
1/2
, while the contact area is proportional to z. The
elastic force, P, is proportional to the power 3/2 of the contact area; therefore,
P ~ z
3/2
. The total contact area and the total normal force are found by inte-
gration of all asperities participating in contact:
∫∫∫
−= =
∞
−
−
∞
−
∞
Apzdzdzzedze ze dz~( )
r
zd
h
d
h
z
h
000
(4.7)
∫∫∫
−= ...