Type C Rolling Friction Model

The Type C Rolling Friction contact model, proposed by Ai et al. (2011), is an extension of the Standard Rolling Friction model.

This model includes a non-viscous term in the damping torque equation.

M r = M r k + M r d MathType@MTEF@5@5@+= feaahyart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGYbaabeaakiabg2da9iaad2eadaWgaaWcbaGaamOCaaqa baGcdaahaaWcbeqaaiaadUgaaaGccqGHRaWkcaWGnbWaaSbaaSqaai aadkhaaeqaaOWaaWbaaSqabeaacaWGKbaaaaaa@4017@

Where Mr is the total damping torque vector, Mrk is the non-viscous damping torque vector and Mrd is the viscous damping torque vector. The non-viscous damping torque vector is a function of the relative particle rotation angle and the rolling stiffness as defined in Equations 2 and 3.

M r k = − K r θ r MathType@MTEF@5@5@+= feaahyart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGYbaabeaakmaaCaaaleqabaGaam4Aaaaakiabg2da9iab gkHiTiaadUeadaWgaaWcbaGaamOCaaqabaGccqaH4oqCdaWgaaWcba GaamOCaaqabaaaaa@3FE4@
k r = λ k n μ r 2 R r 2 MathType@MTEF@5@5@+= feaahyart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaBa aaleaacaWGYbaabeaakiabg2da9iabeU7aSjaadUgadaWgaaWcbaGa amOBaaqabaGccqaH8oqBdaWgaaWcbaGaamOCaaqabaGcdaahaaWcbe qaaiaaikdaaaGccaWGsbWaaSbaaSqaaiaadkhaaeqaaOWaaWbaaSqa beaacaaIYaaaaaaa@43A8@
M r k ≤ μ r R r F n MathType@MTEF@5@5@+= feaahyart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaaca WGnbWaaSbaaSqaaiaadkhaaeqaaOWaaWbaaSqabeaacaWGRbaaaaGc caGLhWUaayjcSdGaeyizImQaeqiVd02aaSbaaSqaaiaadkhaaeqaaO GaamOuamaaBaaaleaacaWGYbaabeaakiaadAeadaWgaaWcbaGaamOB aaqabaaaaa@44C3@

Where kr is the rolling stiffness, θr is the relative rotation angle, Rr is the equivalent rolling radius, kn is the normal contact stiffness and μr is the Coefficient of Rolling Friction.
Note: The non-viscous torque magnitude is limited according to Equation 4, where Fn is the normal contact force.

The viscous damping torque vector is defined in Equation 5 and is a function of the relative rotational velocity vector at the contact point and the rolling viscous damping ratio.

m r d = − 2 η I r k r ω MathType@MTEF@5@5@+= feaahyart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaWGYbaabeaakmaaCaaaleqabaGaamizaaaakiabg2da9iab gkHiTiaaikdacqaH3oaAdaGcaaqaaiaadMeadaWgaaWcbaGaamOCaa qabaGccaWGRbWaaSbaaSqaaiaadkhaaeqaaaqabaGccqaHjpWDaaa@4384@

Where η is the rolling viscous damping ratio, Ir is the equivalent Moment of Inertia for the rotational vibration mode about the contact point, and ω is the relative rotational velocity vector at the contact point.

The viscous damping torque is only applied if the non-viscous torque magnitude is below the limit as defined in Equation 4. In cases where significant non-viscous rolling resistance is expected at particle contacts, such as when contact forces are high or the modeled material consists of predominantly angular particles, the use of Type C Rolling Friction may result in a more physically accurate behavior and greater stability of the EDEM model.