# 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
*v*_{1} = *v*_{2} = *v*
- ω
_{1} · π · *D*_{1} = ω_{2} · π · *D*_{2}
- ω
_{1}ω_{2} = *D*_{2}*D*_{1}
- ω
_{2} = ω_{1} · *D*_{1}*D*_{2}
- ω
_{2} = 100(*rpm*) · 2"1" = 200*rpm*

- Meshing gears may have different torques, but when they are static (motor is stalled) they will have an equal, but opposite force
*F*_{1} = *F*_{2}
- τ
_{1}*d*_{1} = τ_{2}*d*_{2}
- τ
_{1}τ_{2} = *d*_{1}*d*_{2}
- τ
_{2} = *d*_{1}*d*_{2} · τ_{1}
- τ
_{2} = 0.5"1" · 20*in* · *lbs* = 10 *in* · *lbs*

- Gears fixed on the same shaft (2 and 3) have the same angular velocity
- Therefore
- ω
_{1} · *D*_{1}*D*_{2} = ω_{2}
- ω
_{3} = ω_{4} · *D*_{4}*D*_{3}
- ω
_{2} = ω_{3}
- ω
_{1} · *D*_{1}*D*_{2} = ω_{4} · *D*_{4}*D*_{3}
- ω
_{1}ω_{4} = *D*_{2}*D*_{1} · *D*_{4}*D*_{3}

- Gear ratios from each stage are multiplied.

- Gears fixed on the same shaft (2 and 3) also have the same torque
- Therefore
- τ
_{1} · *d*_{2}*d*_{1} = τ_{2}
- τ
_{3} = τ_{4} · *d*_{3}*d*_{4}
- τ
_{2} = τ_{3}
- τ
_{1} · *d*_{2}*d*_{1} = τ_{4} · *d*_{3}*d*_{4}
- τ
_{1}τ_{4} = *d*_{1}*d*_{2} · *d*_{3}*d*_{4}

- Again, gear ratios from each stage are multiplied

- Common mistake,
**do not** multiply gear ratios from inline gears
- τ
_{1} · *d*_{2}*d*_{1} = τ_{2}
- τ
_{2} = τ_{3} · *d*_{2}*d*_{3}
- τ
_{1} · *d*_{2}*d*_{1} = τ_{3} · *d*_{2}*d*_{3}
- τ
_{1}τ_{3} = *d*_{1}*d*_{2} · *d*_{2}*d*_{3} = *d*_{1}*d*_{3}

**Not**
- τ
_{1}τ_{3} = *d*_{1}*d*_{2} · *d*_{3}*d*_{2}

- 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,
- ω
_{1}ω_{2} = *N*_{2}*N*_{1}
- τ
_{1}τ_{2} = *N*_{1}*N*_{2}

- 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