RVD Rolling Friction Model

The RVD Rolling Friction contact model offers an alternative friction calculation to the Standard EDEM Rolling Friction model.

In this model, the rolling friction depends on relative rotational velocity of a pair of elements in contact in a manner suggested in (Zhou, et al. 1999).

This Relative Velocity Dependent (RVD) Rolling Friction model is offered as a 3D generalization of the “Type A: Directional Constant Torque” model type discussed in the paper (Ai, et al. 2011) with one difference in the method of calculating the relative rotational velocity of two elements in contact. While in reference (Ai, et al. 2011) the relative rotational velocity is evaluated from tracking the point of contact, in the EDEM RVD Rolling Friction model, the relative rotational velocity is calculated from the values of instantaneous rotational velocities of two elements in contact as described below. This implementation detail insured proper functionality in three dimensions without affecting the computational time.

In particular, this contact model accounts for rolling friction by applying a constant torque to the contacting surfaces. The torque depends on the relative rotational velocity of two particles in contact, particle i and particle j, as follows:

τ i = − μ r F n R * ω ^ r e l τ j → = τ → i MathType@MTEF@5@5@+= feaahyart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaHep aDdaWgaaWcbaGaamyAaaqabaGccqGH9aqpcqGHsislcqaH8oqBdaWg aaWcbaGaamOCaaqabaGccaWGgbWaaSbaaSqaaiaad6gaaeqaaOGaam OuamaaCaaaleqabaGaaiOkaaaakmaaHaaabaGaeqyYdChacaGLcmaa daWgaaWcbaGaamOCaiaadwgacaWGSbaabeaaaOqaamaaFiaabaGaeq iXdq3aaSbaaSqaaiaadQgaaeqaaaGccaGLxdcacqGH9aqpdaWhcaqa aiabes8a0bGaay51GaWaaSbaaSqaaiaadMgaaeqaaaaaaa@5139@

With μr the coefficient of rolling friction, and R* is the equivalent radius of two elements in contact. The unit vector of relative rotational velocity, ωrel, is calculated as follows: 

ω ^ rel = n ^ ij  × ν → tij / v tij v → tij =  − 1 2 ( ω → i + ω → j )× r ij → ,       particle−particle  contacts          − R i ω i → × n ^ ij ,      particle−geometry​​  contacts MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaqiaa qaaiabeM8a3bGaayPadaWaaSbaaSqaaiaadkhacaWGLbGaamiBaaqa baGccqGH9aqpdaqiaaqaaiaad6gaaiaawkWaamaaBaaaleaacaWGPb GaamOAaaqabaGccaaMc8Uaey41aq7aa8HaaeaacqaH9oGBaiaawEni amaaBaaaleaacaWG0bGaamyAaiaadQgaaeqaaOGaai4lamaaemaaba GaamODamaaBaaaleaacaWG0bGaamyAaiaadQgaaeqaaaGccaGLhWUa ayjcSdaabaWaa8HaaeaacaWG2baacaGLxdcadaWgaaWcbaGaamiDai aadMgacaWGQbaabeaakiabg2da9iaaykW7daGadaabaeqabaWaaWba aSqabeaacqGHsisldaWcaaqaaiaaigdaaeaacaaIYaaaaaaakiaacI cadaWhcaqaaiabeM8a3bGaay51GaWaaSbaaSqaaiaadMgaaeqaaOGa ey4kaSYaa8HaaeaacqaHjpWDaiaawEniamaaBaaaleaacaWGQbaabe aakiaacMcacqGHxdaTdaWhcaqaaiaadkhadaWgaaWcbaGaamyAaiaa dQgaaeqaaaGccaGLxdcacaGGSaGaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaadchacaWGHbGaamOCaiaadshacaWGPbGa am4yaiaadYgacaWGLbGaeyOeI0IaamiCaiaadggacaWGYbGaamiDai aadMgacaWGJbGaamiBaiaadwgacaaMc8UaaGPaVlaadogacaWGVbGa amOBaiaadshacaWGHbGaam4yaiaadshacaWGZbaabaGaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeyOe I0IaamOuamaaBaaaleaacaWGPbaabeaakmaaFiaabaGaeqyYdC3aaS baaSqaaiaadMgaaeqaaaGccaGLxdcacqGHxdaTdaqiaaqaaiaad6ga aiaawkWaamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGSaGaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaamiCaiaadggacaWGYbGa amiDaiaadMgacaWGJbGaamiBaiaadwgacqGHsislcaWGNbGaamyzai aad+gacaWGTbGaamyzaiaadshacaWGYbGaamyEaiaaygW7caaMb8Ua aGPaVlaaykW7caWGJbGaam4Baiaad6gacaWG0bGaamyyaiaadogaca WG0bGaam4CaaaacaGL7bGaayzFaaaaaaa@DAD1@

Where nij is a unit vector pointing from particle i to the point of contact as shown in the following figure: