Gear trains and belt drives

The idea

Gears, belts and cords pass motion from one body to another. The key is what’s shared at the contact: where two gears mesh (or a belt grips a pulley without slipping), the touching points move together, so they have the same speed along the contact.

The rules

Meshing gears, and belts or cords with no slip: $r_A{\omega }_A=r_B{\omega }_B{r}_A{\alpha }_A=r_B{\alpha }_B$ Things on the same shaft share $\omega$ and $\alpha$.

• A small gear driving a big one slows the output and (ideally) multiplies the torque by the same ratio.
• Meshing gears turn opposite ways. A belt keeps the same direction (unless it’s crossed).
• To go from rotation to a load on a cord or drum: $v=r\omega$, $a=r\alpha$.

Mechanism vocabulary

• A link is a rigid body with at least two attachment points (nodes). Two nodes = binary, three = ternary.
• Full joints allow one degree of freedom: a pin (revolute) joint allows rotation only; a slider (prismatic) joint allows translation only.

Worked examples

1. A motor’s 20 mm pinion turns at 1500 rpm and drives an 80 mm gear. Output speed and ideal torque ratio?

Answer

${\omega }_B=1500\times \frac{20}{80}=375$ rpm, turning the opposite way. The torque is multiplied by $80/20=4$ (ignoring losses).

2. Gear A (r = 50 mm) starts from rest with ${\alpha }_A=3$ rad/s² and drives gear B (r = 150 mm). A 100 mm drum on B’s shaft winds up a cable lifting a load. Find the load’s acceleration and speed after 4 s.

Answer

Mesh: ${\alpha }_B=\frac{50\times 3}{150}=1$ rad/s². The drum shares B’s shaft, so ${\alpha }_D=1$ rad/s². $a_{load}=r_D{\alpha }_D=0.1\times 1=0.1$ m/s² $v=0+0.1\times 4=0.4$ m/s

The pattern for any drive problem

1. Find what’s shared at each contact: the same tangential $v$ and $a$.
2. Convert between linear and angular with $r$ at each step.
3. Use the constant-α equations if α is constant.

Common mistakes

• Using diameters on one side and radii on the other. Either works, but be consistent.
• Assuming meshing gears have the same ω. They have the same rim speed.
• Forgetting gears on a shared shaft have the same ω even if their radii differ.

Check yourself

1. A 0.15 m pulley at 7.1 rad/s drives a 0.4 m pulley by belt. Speed of the second pulley's rim?

Same as the first: $0.15\times 7.1=1.07$ m/s. Its ω is $1.07/0.4=2.66$ rad/s.