# What is the pressure, temperature and slip speed dependent friction coefficient model for two lubricated contacts?

To specify my problem properly:

I have two lubricated surfaces. One of them is static and the other one is rotating with velocity ω and is pressed onto the first one with force F:

I need to find model of the friction coefficient. From the literature, i know the function should be

$μ=f(ω,P,T)$

(citation 1 - Prediction of torque response) where P is applied pressure and T is temperature.

I know that the friction coefficient is dependent on velocity via Stribeck equation (citation 2 - Friction models):

$F=(F_C+(F_S- F_C ) e^{(-(|ω|/ω_s)^i )})tanh⁡(k_{tanh}ω)+k_vω$

Where $F_C$ is Coulomb sliding friction force, $F_S$ is maximal static friction force and $ω_s$, $i$ are Stribeck coefficients and $k_{tanh}$ is transition coefficient and $k_v$ viscous friction coefficient.

Somehow, they turned it into this (citation 3 - Wet clutch modelling):

$μ=(μ_C+(μ_S- μ_C ) e^{(-(|ω|/ω_s)^i)}) tanh⁡(k_{tanh}ω)+k_v ω$

My guess on how they did it is following:

$μN=(μ_C N+(μ_S N- μ_C N) e^{(-(|ω|/ω_s)^i)}) tanh⁡(k_{tanh}ω)+k_v ω$

$μ\frac{P}{S}=(μ_C\frac{P}{S}+(μ_S\frac{P}{S} - μ_C\frac{P}{S})e^{(-(|ω|/ω_s)^i)})tanh⁡(k_{tanh}ω)+k_vω$

$μ=(μ_C+(μ_S - μ_C )e^{(-(|ω|/ω_s)^i)}))tanh⁡(k_{tanh}ω)+k_vω\frac{S}{P}$

where N is normal force, P is pressure and S is area to which the pressure is applied. Then they probably assumed that S/P is constant or they omitted it.

Is my guess right? How do I incorporate temperature into this model? Could you give me some references to articles/books?