Rotational Cam and Follower Simulation

Question

I have the following data of a cam profile:

camProfile = {{1.47363, 0.}, {1.48044, 0.0465248}, {1.4858, 0.0934788}, {1.48969, 
  0.140817}, {1.49207, 0.188493}, {1.49294, 0.236459}, {1.49228, 
  0.284669}, {1.49008, 0.333072}, {1.48631, 0.38162}, {1.48097, 
  0.430262}, {1.47405, 0.478947}, {1.46553, 0.527625}, {1.45542, 
  0.576242}, {1.4437, 0.624746}, {1.43038, 0.673085}, {1.41545, 
  0.721206}, {1.3989, 0.769053}, {1.38075, 0.816575}, {1.361, 
  0.863715}, {1.33964, 0.910422}, {1.3167, 0.956639}, {1.29217, 
  1.00231}, {1.26608, 1.04739}, {1.23842, 1.09182}, {1.20922, 
  1.13554}, {1.1785, 1.1785}, {1.14627, 1.22065}, {1.11254, 
  1.26193}, {1.07736, 1.3023}, {1.04073, 1.3417}, {1.00269, 
  1.38008}, {0.963257, 1.41739}, {0.922468, 1.45358}, {0.880353, 
  1.4886}, {0.836944, 1.5224}, {0.792277, 1.55493}, {0.746388, 
  1.58616}, {0.699316, 1.61602}, {0.6511, 1.64449}, {0.601782, 
  1.67151}, {0.551406, 1.69705}, {0.500016, 1.72107}, {0.447659, 
  1.74352}, {0.394382, 1.76436}, {0.340234, 1.78357}, {0.285267, 
  1.80111}, {0.229532, 1.81694}, {0.173083, 1.83103}, {0.115974, 
  1.84336}, {0.0582607, 1.85389}, {0., 1.8626}, {-0.0587502, 
  1.86946}, {-0.117931, 1.87446}, {-0.177482, 1.87757}, {-0.237344, 
  1.87877}, {-0.297453, 1.87805}, {-0.357749, 1.87538}, {-0.418167, 
  1.87077}, {-0.478644, 1.8642}, {-0.539117, 1.85565}, {-0.599519, 
  1.84513}, {-0.659787, 1.83263}, {-0.719854, 1.81814}, {-0.779655, 
  1.80168}, {-0.839124, 1.78323}, {-0.898194, 1.76281}, {-0.956801, 
  1.74041}, {-1.01488, 1.71606}, {-1.07236, 1.68976}, {-1.12917, 
  1.66153}, {-1.18526, 1.63137}, {-1.24056, 1.59932}, {-1.295, 
  1.56538}, {-1.34851, 1.52959}, {-1.40104, 1.49196}, {-1.45252, 
  1.45252}, {-1.50289, 1.41131}, {-1.55209, 1.36835}, {-1.60005, 
  1.32368}, {-1.64672, 1.27733}, {-1.69204, 1.22934}, {-1.73595, 
  1.17975}, {-1.77839, 1.1286}, {-1.81931, 1.07594}, {-1.85865, 
  1.0218}, {-1.89637, 0.966248}, {-1.93241, 0.909322}, {-1.96672, 
  0.851074}, {-1.99925, 0.791559}, {-2.02996, 0.730831}, {-2.0588, 
  0.668946}, {-2.08574, 0.605963}, {-2.11072, 0.541941}, {-2.13371, 
  0.476941}, {-2.15468, 0.411027}, {-2.17358, 0.344261}, {-2.19039, 
  0.27671}, {-2.20507, 0.20844}, {-2.21759, 0.139519}, {-2.22793, 
  0.0700156}, {-2.23607, 0.}};

If I plot it, it looks like that:

ListPlot[camProfile]

As you can see, it is the half of a cam !

Now I would like to simulate a follower, when the cam is turning and also get information on follower acceleration, speed and maybe torque.

Can someone think of a way to simulate this behaviour ? Here is a similar code: Simulation of a Camshaft

... but my cam profile is provided as single points, hence I don't know how to make it work.

EDIT: To make the cam profile a region, it was suggested by C.E. to use Line:

Graphics[Line[camProfile]]

The output:

Any help is highly appreciated.

Answer 1

Here's what I came up with. First I define the cam and the parameters for the follower. I assume the follower is a disk of radius of 0.5 which rotates about the point {3, 0} at a distance of 2.5 away.

cam = BoundaryMeshRegion[camProfile, Line[Append[Range[Length[camProfile]], 1]]]
x = 3;
R = 2.5;
r = 0.5;

We can approach this problem just like this post, where we find the rotation angle that leaves the follower center a distance 0.5 away from the rotated cam.

followerAngle[α_?NumericQ] :=
  With[{camα = TransformedRegion[cam, RotationTransform[α]]},
    θ /. Quiet[FindRoot[
      SignedRegionDistance[camα, {x + R Cos[θ], R Sin[θ]}] == r, {θ, π/2, π/2, π}]]
  ]

Now unfortunately this isn't quite right. As you can see below, the follower's shaft intersects the cam under this approach.

frames = Table[With[{θ = followerAngle[α]},
  Show[
    TransformedRegion[cam, RotationTransform[α]],
    Graphics[{Disk[{x + R Cos[θ], R Sin[θ]}, r], Thick,
      Line[{{x, 0}, {x + R Cos[θ], R Sin[θ]}}]}],
    Axes -> {True, False},
    Ticks -> None,
    PlotRange -> {{-3, 3}, {-2, 3}},
    PlotRangePadding -> Scaled[.05]
  ]],
  {α, 0, 2π, π/40}
];

Export["try1.gif", frames, "DisplayDurations" -> 0.05];

To fix this, I model the follower shaft as a line. If the corner of the rotated cam is above the shaft, choose the angle that leaves the corner lying on the shaft instead.

I leave it to you to add thickness to the shaft. It should just be a matter of offsetting from the line a constant distance.

corner = {-2.23607, 0.};

followerAngleFull[α_?NumericQ] :=
  Module[{θ, c},
    θ = followerAngle[α];
    c = corner.{{Cos[α], Sin[α]}, {-Sin[α], Cos[α]}};

    If[!RegionMember[HalfPlane[{{x + R Cos[θ], R Sin[θ]}, {x, 0}}, {x, 1}], c],
        θ,
        VectorAngle[c - {x, 0}, {1, 0}]
    ]
  ]

This looks much better.

frames = Table[With[{θ = followerAngleFull[α]},
  Show[
    TransformedRegion[cam, RotationTransform[α]],
    Graphics[{Disk[{x + R Cos[θ], R Sin[θ]}, r], Thick,
      Line[{{x, 0}, {x + R Cos[θ], R Sin[θ]}}]}],
    Axes -> {True, False},
    Ticks -> None,
    PlotRange -> {{-3, 3}, {-2, 3}},
    PlotRangePadding -> Scaled[.05]
  ]],
  {α, 0, 2π, π/40}
];

Export["try2.gif", frames, "DisplayDurations" -> 0.05];

Here's the plot of cam angle v.s. follower angle:

Plot[followerAngleFull[α], {α, 0, 2π}]

A polar plot:

PolarPlot[followerAngleFull[followerAngleFull[α], {α, 0, 2π}, PolarAxes -> {True, False}, PolarTicks -> {"Degrees", Automatic}]

A plot of the velocity:

Plot[Evaluate[DifferenceQuotient[followerAngleFull[α], {α, 0.001}]], {α, 0, 2π}, MaxRecursion -> 2, PlotRange -> All]