Gearing

Gearing

  • Gear to increase or decrease speed (angular velocity) and torque
  • Speed is inversely dependant on torque
    • As speed increases, torque decreases (and vice versa)
  • Meshing gears may have different angular velocities, but they will always have the same linear velocity
    • v1 = v2 = v
    • ω1 · π · D1 = ω2 · π · D2
    • ω1ω2 = D2D1
    • ω2 = ω1 · D1D2
    • ω2 = 100(rpm) · 2"1" = 200rpm
  • Meshing gears may have different torques, but when they are static (motor is stalled) they will have an equal, but opposite force
    • F1 = F2
    • τ1d1 = τ2d2
    • τ1τ2 = d1d2
    • τ2 = d1d2 · τ1
    • τ2 = 0.5"1" · 20in · lbs = 10 in · lbs
  • Gears fixed on the same shaft (2 and 3) have the same angular velocity
  • Therefore
    • ω1 · D1D2 = ω2
    • ω3 = ω4 · D4D3
    • ω2 = ω3
    • ω1 · D1D2 = ω4 · D4D3
    • ω1ω4 = D2D1 · D4D3
  • Gear ratios from each stage are multiplied.
  • Gears fixed on the same shaft (2 and 3) also have the same torque
  • Therefore
    • τ1 · d2d1 = τ2
    • τ3 = τ4 · d3d4
    • τ2 = τ3
    • τ1 · d2d1 = τ4 · d3d4
    • τ1τ4 = d1d2 · d3d4
  • Again, gear ratios from each stage are multiplied
  • Common mistake, do not multiply gear ratios from inline gears
    • τ1 · d2d1 = τ2
    • τ2 = τ3 · d2d3
    • τ1 · d2d1 = τ3 · d2d3
    • τ1τ3 = d1d2 · d2d3 = d1d3
  • Not
    • τ1τ3 = d1d2 · d3d2
  • Middle gear is "idle", it does not effect the overall gear ratio
  • In gears, the diameter used to calculate speed and torque ratios is called the pitch diameter
  • It isn't possible to measure this value with callipers
  • Instead, we can use the number of teeth, which can easily be counted
  • So we get,
    • ω1ω2 = N2N1
    • τ1τ2 = N1N2
  • Combining these, we verify how speed (angular velocity) and torque are inversely proportional in gear train
    • τ1τ2 = ω2ω1
  • These equations can also be applied to sprockets & chain and pulleys & belts