# 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 