Elastomere Friction - The Best Friend international

4 L.Busse,A.LeGal,andM.Klüppel
2.2 Hysteresis Friction Simulation
Friction μges can be divided into two parts: adhesion friction μAdh and hysteresis
friction μHys.
μges = μAdh + μHys. (8)
The latter is caused by the energy dissipations where the rubber sample
is deformed at local asperities. The excited frequencies follow a spectrum
as given in Equations (5) and (6) and thus can be used for the hysteresis
integral [6–9] for the friction force FHys under normal force FN and thus
friction coefficient μHys is according to our model
μHys(ν) ≡ FHys
FN
= 〈δ〉
2σ0ν
�� ω2
ωmin
ω · E ′′ � ωmax
(ω) · S1(ω)dω +
ω2
(9)
ω · E ′′ �
(ω) · S2(ω)dω .
Again we assume a bifractal approach to suit best the nature of the HDC.
Microstructures influence mainly to the lower velocities [6]. E ′′ is the loss
modulus of the elastomer, σ0 is the applied pressure and 〈δ〉 is the mean
excitation depth inside the rubber
〈δ〉 = b ·〈zp〉 (10)
with the mean penetration depth zp of the asperities into the rubber, scaled
by the factor b. Further, only wave lengths above λmin =2πν/ωmax contribute
to Equation (9). It holds that [8]
λmin∼=
ξ� � �λ2
ξ �
� 3(D−2−D1) 0.09πξ⊥ ·|E(2πν/λmin)|·F0(t) · 6π · √ 3λ 2 c · ns
s 2/3 · ξ � ·|E(2πν/ξ �)|·F 3/2(ts)
where ns ∼ (3 − β2)/(5 − β2) isthesummitdensityand
Fn =
� ∞
t
� 1
3D 2 −6 ,
(11)
(x − t) n · φ(x)dx (12)
with n =0, 1, 3/2 are the Greenwood–Williams functions [10] with the normalized
distances t = d/σHD and ts = d/σSHD, where the gap distance d
indicates the rubber distance from the mean substrate level; σHD and σSHD
are the standard deviations of the height distribution or summit height distribution,
respectively.
For pure hysteresis friction, it becomes [7, 9]
λmin ∼ =
�
|E(λmin) · Cz|(λmin)
. (13)
σ(λmin)