friction

A library containing routines for calculating the frictional response of contacting bodies.

Work-In-Progress

This is a work in progress. This library will eventually be a compilation of the friction modeling efforts I have undertaken as part of my everyday work.

Available Models

• Coulomb Model

$$F = \text{sgn} \left( v \right) \mu_{c} N$$

• Lu-Gre Model

$$F = \sigma_{0} z + \sigma_{1} \frac{dz}{dt} + \sigma_{2} v$$ $$\frac{dz}{dt} = v - \frac{\left| v \right| z}{g(v)}$$ $$g(v) = a_{1} + \frac{a_2}{1 + s^{\alpha}}$$ $$a_{1} = \frac{\mu_c N}{\sigma_{0}}, a_{2} = \frac{\mu_s N - \mu_c N}{\sigma_{0}}, s = \frac{\left| v \right|}{v_s}$$

• Maxwell Model

$$F = k \delta$$ $$\delta_{i+1} = \text{sgn} \left( x_{i+1} - x_{i} + \delta_{i} \right) \min \left( \left| x_{i+1} - x_{i} + \delta_{i} \right|, \Delta \right)$$ $$\Delta = \frac{N \mu_c}{k}$$

• Generalized Maxwell Slip Model

$$F = \sum_{i=1}^{n} \left( k_i z_i + b_i \frac{dz_i}{dt} \right) + b_v v$$ $$\begin{equation} \frac{dz_i}{dt} = \begin{cases} v & \text{if $|z_i| \le g(v)$} \\\ \text{sgn} \left( v \right) \nu_i C \left( 1 - \frac{z_i}{\nu_i g(v)} \right) & \text{otherwise} \end{cases} \end{equation}$$ $$g(v) = a_{1} + \frac{a_2}{1 + s^{\alpha}}$$ $$a_{1} = \frac{\mu_c N}{\sigma_{0}}, a_{2} = \frac{\mu_s N - \mu_c N}{\sigma_{0}}, s = \frac{\left| v \right|}{v_s}$$ $$\sum_{i=1}^{n} \nu_i = 1$$

• Stribeck Model

$$F = \text{sgn} \left( v \right) \left( \mu_c N + N \left( mu_s - mu_c \right) \exp(-|\frac{v}{v_s}|^2) \right) + b_v v$$

• Modified Stribeck Model

$$F = k \delta + b_v v$$ $$\delta_{i+1} = \text{sgn} \left( x_{i+1} - x_{i} + \delta_{i} \right) \min \left( \left| x_{i+1} - x_{i} + \delta_{i} \right|, g(v) \right)$$ $$g(v) = a_{1} + a_2 \exp(-|\frac{v}{v_s}|^2)$$ $$a_{1} = \frac{\mu_c N}{k}, a_{2} = \frac{\mu_s N - \mu_c N}{k}$$

References:

1. Al-Bender, Farid & Lampaert, Vincent & Swevers, Jan. (2004). Modeling of dry sliding friction dynamics: From heuristic models to physically motivated models and back. Chaos (Woodbury, N.Y.). 14. 446-60. 10.1063/1.1741752.
2. Rizos, Demosthenis & Fassois, Spilios. (2009). Friction Identification Based Upon the LuGre and Maxwell Slip Models. Control Systems Technology, IEEE Transactions on. 17. 153 - 160. 10.1109/TCST.2008.921809.
3. Al-Bender, Farid & Lampaert, Vincent & Swevers, Jan. (2005). The generalized Maxwell-Slip model: A novel model for friction simulation and compensation. Automatic Control, IEEE Transactions on. 50. 1883 - 1887. 10.1109/TAC.2005.858676.
4. Tjahjowidodo, Tegoeh & Al-Bender, Farid & Brussel, H.. (2005). Friction identification and compensation in a DC motor. IFAC Proceedings Volumes. 32. 10.3182/20050703-6-CZ-1902.00093.
5. Lampaert, Vincent & Al-Bender, Farid & Swevers, Jan. (2003). A generalized Maxwell-slip friction model appropriate for control purposes. 4. 1170- 1177 vol.4. 10.1109/PHYCON.2003.1237071.
6. Al-Bender, Farid & Swevers, Jan. (2009). Characterization of friction force dynamics. Control Systems, IEEE. 28. 64 - 81. 10.1109/MCS.2008.929279.
7. Al-Bender, Farid. (2010). Fundamentals of friction modeling. Proceedings - ASPE Spring Topical Meeting on Control of Precision Systems, ASPE 2010. 48.
8. Al-Bender, Farid & Moerlooze, K.. (2011). Characterization and modeling of friction and wear: an overview. International Journal Sustainable Construction & Design. 2. 19-28. 10.21825/scad.v2i1.20431.