Spinning Friction Contact Model

The Spinning Friction contact model is used to account for the friction that would occur if a particle face is rotating against another particle or Geometry.

Friction involves calculating the contact area and then using this value to obtain the torque on the particles as follows:

M = 2 3 μ F n R d i s k MathType@MTEF@5@5@+= feaahyart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGnb Gaeyypa0ZaaSaaaeaacaaIYaaabaGaaG4maaaacqaH8oqBcaWGgbWa aSbaaSqaaiaad6gaaeqaaOGaamOuamaaBaaaleaacaWGKbGaamyAai aadohacaWGRbaabeaaaOqaaaaaaa@41D3@

Where Fn is the normal force, μ is the coefficient of friction, and Rdisk is the effective disk radius given by approximating the contact as a disk with area equal to the normal area Acontact of the overlap region:

R d i s k = A c o n t a c t π MathType@MTEF@5@5@+= feaahyart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGKbGaamyAaiaadohacaWGRbaabeaakiabg2da9maakaaa baWaaSaaaeaacaWGbbWaaSbaaSqaaiaadogacaWGVbGaamOBaiaads hacaWGHbGaam4yaiaadshaaeqaaaGcbaGaeqiWdahaaaWcbeaaaaa@453A@

The torque acts in the opposite direction to the normal part of the relative angular velocity.

A Limit is applied to this torque to avoid oscillating behavior when the angular velocity is small. This is done by finding the torque which would completely eliminate the angular velocity in one Time Step and is described as:

M t a n g , t i m s t e p = C s a f e t y ω r e l , n min ( I x , y , z ) δ t MathType@MTEF@5@5@+= feaahyart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWG0bGaamyyaiaad6gacaWGNbGaaiilaiaadshacaWGPbGa amyBaiaadohacaWG0bGaamyzaiaadchaaeqaaOGaeyypa0ZaaSaaae aacaWGdbWaaSbaaSqaaiaadohacaWGHbGaamOzaiaadwgacaWG0bGa amyEaaqabaGcdaabdaqaaiabeM8a3naaBaaaleaacaWGYbGaamyzai aadYgacaGGSaGaamOBaaqabaaakiaawEa7caGLiWoaciGGTbGaaiyA aiaac6gacaGGOaGaamysamaaBaaaleaacaWG4bGaaiilaiaadMhaca GGSaGaamOEaaqabaGccaGGPaaabaGaeqiTdq2aaSbaaSqaaiaadsha aeqaaaaaaaa@5FBF@

The ωrel,n is the normal component of the relative angular velocity, Δt is the Time Step, and Ix,y,z is the minimum component of the moment of inertia of either of the two contacting objects excluding geometries. The extra Csafety factor is to try to avoid possible instability in cases with multiple contacts and at the moment has a value of 0.125.

The damping to damp rocking is defined as:

M normaldamping = γ n 4 R disk 2 ω rel,t MathType@MTEF@5@5@+= feaahyart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGUbGaam4BaiaadkhacaWGTbGaamyyaiaadYgacaaMc8Ua amizaiaadggacaWGTbGaamiCaiaadMgacaWGUbGaam4zaaqabaGccq GH9aqpcaaMc8UaaGPaVlabgkHiTmaalaaabaGaeq4SdC2aaSbaaSqa aiaad6gaaeqaaaGcbaGaaGinaaaacaWGsbWaa0baaSqaaiaadsgaca WGPbGaam4CaiaadUgaaeaacaaIYaaaaOGaeqyYdC3aaSbaaSqaaiaa dkhacaWGLbGaamiBaiaacYcacaWG0baabeaaaaa@5956@

Where γn is the linear damping coefficient in the normal direction from the base model, and ωrel,t is the tangential part of the relative angular velocity.