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