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
- 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,
- Combining these, we verify how speed (angular velocity) and torque are inversely proportional in gear train
- These equations can also be applied to sprockets & chain and pulleys & belts