Friction Models 

In[5]:=

X GraphicsRow[{Plot[{pos1[t]}, {t, 0, 1}, AxesLabel -> {"t", "x"}, PlotRange -> All], Plot[{vel1[t], beltv[t]}, {t, 0, 1}, AxesLabel -> {"t", "v"}, PlotRange -> All]}, ImageSize -> 500]

In[8]:=

X GraphicsRow[{Plot[{pos2[t]}, {t, 0, 1}, AxesLabel -> {"t", "x"}, PlotRange -> All], Plot[{vel2[t], beltv[t]}, {t, 0, 1}, AxesLabel -> {"t", "v"}, PlotRange -> All]}, ImageSize -> 500]

In[11]:=

X GraphicsRow[{Plot[{pos3[t]}, {t, 0, 1}, AxesLabel -> {"t", "x"}, PlotRange -> All], Plot[{vel3[t], beltv[t]}, {t, 0, 1}, AxesLabel -> {"t", "v"}, PlotRange -> All]}, ImageSize -> 500]

In[16]:=

X GraphicsRow[{Plot[{pos4[t]}, {t, 0, 8}, AxesLabel -> {"t", "x"}, PlotRange -> All], Plot[{vel4[t], beltv[t]}, {t, 0, 8}, AxesLabel -> {"t", "v"}, PlotRange -> All]}, ImageSize -> 500]

In[17]:=

X Plot[{pos1[t], pos2[t], pos3[t], pos4[t]}, {t, 0, 2}, AxesLabel -> {"t", "x"}, PlotRange -> {0.9, 1.1}, PlotLegends -> {"viscous", "viscous+Coulomb", "viscous+Coulomb+Stribeck", "viscous+Coulomb+Stribeck+static"}, ImageSize -> 400]

In[18]:=

X coilSpring[loc_: {0, 0}, force_: 0, segments_: 4, height_: .1, endLength_: .1, thickness_: .015, ghost_: None] := Module[{f, s, e}, { f = force + 1; s = (f - 2 endLength)/(segments + .5); e = endLength + s/4; GraphicsGroup[{ Black, Flatten@{ Thickness[thickness], JoinForm["Round"], CapForm["Butt"], If[ghost === "Ghost", GrayLevel[0, .2], Black], Line[Flatten[{{loc + {0, 0}, loc + { endLength, 0}}, Flatten[ Table[{loc + { e + a s, -height}, loc + {e + a s + .5 s, height}}, {a, 0, segments - 1}], 1], {loc + {f - e, -height}, loc + {f - endLength, 0}, loc + {f - endLength, 0}, loc + {f, 0}}}, 1]], If[ ghost =!= "Ghost", {White, CapForm["Round"], Thickness[.5 thickness], Line[{loc + {0, 0}, loc + { endLength, 0}}], Table[Line[{loc + { e + a s, -height}, loc + {e + a s + .5 s, height}}], {a, 0, segments - 1}], Line[{loc + {f - e, -height}, loc + {f - endLength, 0}}], Line[{loc + {f - endLength, 0}, loc + {f, 0}}]}]} }] }];

In[19]:=

X conveyorBelt[loc_: {0, 0}, speed_: 0, length_: 2, height_: .4] := Module[{p, l}, { GraphicsGroup[{ LightGray, EdgeForm[{Black}], Disk[loc + {0, -height/2}, height/2], Disk[loc + {length, -height/2}, height/2], Black, AbsoluteThickness[2], Circle[loc + {0, -height/2}, height/2, {Pi/2, 3 Pi/2}], Circle[loc + {length, -height/2}, height/2, {-Pi/2, Pi/2}], Line[{loc + {0, 0}, loc + {length, 0}}], Line[{loc + {0, -height}, loc + {length, -height}}], AbsolutePointSize[10], Point[loc + {0, -height/2}], Point[loc + {length, -height/2}], (* movement *) l = ((2 Pi height/2) + 2 length)/990;(* last number is path divisions >>> (0,#) *) p = Partition[ Flatten@{ Table[loc + {a, 0}, {a, 0, length, l}], Table[loc + {length, -height/2} + height/2 {Sin[a], Cos[a]}, {a, 0, Pi, (Pi/(2 Pi height/4)) l}], Table[loc + {a, -height}, {a, length, 0, -l}], Table[loc + {0, -height/2} + height/2 {Sin[a], Cos[a]}, {a, Pi, 2 Pi, (Pi/(2 Pi height/4)) l}] }, 2, 2, 1, {}]; White, AbsolutePointSize[3], Array[Point[p[[speed + #]]] &, 5] }] }];

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Model a block on a moving conveyor belt anchored to a wall by a spring, using different models for the friction force . Compare positions and velocities for the different models. The preceding animation shows the final refinement of the model, which incorporates four types of friction: viscous, Coulomb, Stribeck, and static.

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In[1]:=

X m = l = 1.; beltv[t_] = .1; spring[x_] = 1000. (l - x);

Viscous friction is proportional to the relative velocity .

In[2]:=

X sys := {m x''[t] == spring[x[t]] + friction[x'[t]], x[0] == 1, x'[0] == 0};

In[3]:=

X viscous[v_] := -30. (v - beltv[t]); friction[v_] := viscous[v];

In[4]:=

X {pos1, vel1} = NDSolveValue[sys, {x, x'}, {t, 0, 2}];

The block quickly settles in a stable position.

Out[5]=

Coulomb friction is proportional to the sign of relative velocity .

In[6]:=

X coulomb[v_] := -25. Sign[v - beltv[t]]; friction[v_] := viscous[v] + coulomb[v];

In[7]:=

X {pos2, vel2} = NDSolveValue[sys, {x, x'}, {t, 0, 20}];

With the addition of Coulomb friction, the block initially sticks to the belt before settling down.

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Stribeck friction is a refined Coulomb friction .

In[9]:=

X stribeck[v_] := -.3 Sign[v] Exp[-2 Abs[v]]; friction[v_] := viscous[v] + coulomb[v] + stribeck[v];

In[10]:=

X {pos3, vel3} = NDSolveValue[sys, {x, x'}, {t, 0, 20}];

The relative effect of Stribeck friction in the model is small.

Out[11]=

Static friction holds the block in place until the spring force exceeds some value , depending on the roughness of the surfaces.

In[12]:=

X friction[v_] := viscous[v] + coulomb[v] + stribeck[v];

In[13]:=

X staticsys := {x''[t] == If[stuck[t] == 1, beltv'[t], spring[x[t]] + friction[x'[t]]], x[0] == 1, x'[0] == 0};

In[14]:=

X \[Mu] = 100; stick = WhenEvent[x'[t] == beltv[t], stuck[t] -> Boole[spring[x[t]]^2 < \[Mu]^2]]; slip = WhenEvent[spring[x[t]]^2 > \[Mu]^2, stuck[t] -> 0];

In[15]:=

X {pos4, vel4} = NDSolveValue[{staticsys, stick, slip, stuck[0] == 0}, {x, x'}, {t, 0, 8}, DiscreteVariables -> stuck[t]];

The block repeatedly sticks to the belt due to frictional forces and then slips back due to the spring force.

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Compare the successively refined friction models.

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