Friction Formulation
This section provides information about the LuGre and Dahl formulations for friction.
LuGre Model
 The effect of the mating surfaces being pushed apart by lubricant.
 The Stribeck effect at very low speed. When partial fluid lubrication exists, contact between the surfaces decreases and thus friction decreases exponentially from stiction.
 Rate dependent friction phenomena such as varying breakaway force and frictional lag.
 Static friction between two surfaces.
 $rb$
 Ball radius
 ${F}_{N}$
 Normal force between a pair of surfaces.
 ${\mu}_{s}$
 Coefficient of static friction.
 ${\mu}_{d}$

Coefficient of dynamic friction (µd ≤ µs).
 $T$
 True friction torque between the surfaces.
 ${\sigma}_{0},\text{\hspace{0.17em}}{\sigma}_{1},\text{\hspace{0.17em}}{\sigma}_{2},\text{\hspace{0.17em}}{v}_{s}$
 Constants associated with the LuGre model.
Quantity  Formula 

Angular slip velocity  $\underset{\_}{\omega}={\left[{}_{{\omega}_{x}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\omega}_{y}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\omega}_{z}}\right]}^{T};\text{\hspace{0.17em}}{\omega}_{m}=\Vert \underset{\_}{\omega}\Vert $ 
Bristle states  $\underset{\_}{z}={\left[{}_{{z}_{x}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{z}_{y}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{z}_{z}}\right]}^{T}$ 
Stiction to dynamic friction transition function  $g\left({\omega}_{m}\right)={\mu}_{d}+\left({\mu}_{s}{\mu}_{d}\right){e}^{{\left({r}_{B}\cdot {\omega}_{m}/{v}_{s}\right)}^{2}}$ 
Coupling function  $\lambda \left({\omega}_{m}\right)=\frac{{r}_{B}{\omega}_{m}{\mu}_{d}^{2}}{g\left({\omega}_{m}\right)}$ 
Rate dependence function  ${C}_{0}\left({\omega}_{m}\right)=\frac{\lambda \left({\omega}_{m}\right){\sigma}_{0}}{{\mu}_{d}^{2}}=\frac{{r}_{B}{\omega}_{m}{\sigma}_{0}}{g\left({\omega}_{m}\right)}$ 
Model for dynamic simulation 
$\underset{\_}{z}={r}_{B}\underset{\_}{\omega}{C}_{0}\underset{\_}{z}$
$\mu =\left({\sigma}_{0}\underset{\_}{z}+{\sigma}_{1}\underset{\_}{z}+{\sigma}_{2}r\underset{\_}{\omega}\right)$ $\underset{\_}{r}={r}_{B}{\underset{\_}{F}}_{N}/\Vert {\underset{\_}{F}}_{N}\Vert $ ${\underset{\_}{\tau}}_{jforce}=\underset{\_}{\tilde{r}}\underset{\_}{\tilde{\mu}}{\underset{\_}{F}}_{N}$ ${\underset{\_}{\tau}}_{preload}=\left(\underset{\_}{\mu}/{\mu}_{s}\right){T}_{preload}$ $f\text{\hspace{0.17em}}\Vert {\underset{\_}{\tau}}_{jforce}\Vert >\Vert {\underset{\_}{\tau}}_{preload}\Vert \text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{\_}{\tau}={\underset{\_}{\tau}}_{jforce}$ $else\text{\hspace{0.17em}}\underset{\_}{\tau}={\underset{\_}{\tau}}_{preload}$ 
Model for static simulation 
Dahl Model as a Subset of the LuGre Model
Comparison Between the Dahl and LuGre Models
 Stribeck effect (also known as the stickslip effect)
 Static Friction > Dynamic Friction
 Bristle damping
 Memoryless damping (viscous damping)
 Rate dependency (such as dependency of friction force on frequency of input)!
The curves in the figure represent the hysteresis loop in the friction models that cause energy dissipation. Notice that the LuGre model (left) is sensitive to the frequency of the input. In contrast the Dahl model (right) shows no change in its response even when the frequency of the inputs change.