Gears, belts, chains and power transmission

Define which speed is divided by which

For two compatible external gears in maintained mesh, the magnitude of output speed divided by input speed equals the input tooth count divided by output tooth count. Direct external mesh reverses rotation. A gear pair’s pitch circles represent the kinematic contact geometry; they are not the outer tooth surfaces or a manufacturing drawing.

Some people quote a reduction ratio as input speed/output speed instead. State the convention to avoid calling the same pair both one-third and three without explanation. Tooth counts and compatible pitch geometry determine the nominal ratio. Backlash, elastic deformation and motion errors can affect instantaneous position transfer while the intended average engagement ratio remains fixed.

Engagement and friction require different models

A smooth belt transfers tangential force through contact friction with pulleys. Its nominal no-slip speed relation uses pulley pitch radii, but slip and stretch can affect the actual record. A toothed belt or chain provides positive engagement with suitable pulleys or sprockets; it still has stiffness, alignment, wear and operating limits.

Belts, chains and gears differ in packaging, mass, noise, maintenance and attainable positioning behaviour. A general label cannot select one. Identify the required speed, torque, duty, environment and access. Supporting bearings also carry forces generated by transmission elements; transferring torque does not make those transverse or axial loads disappear.

Use torque and angular speed at matching ports

Mechanical power magnitude for steady rotation is P=τω, where τ is torque and ω is angular speed in rad/s. For an ideal lossless reduction, a lower output speed permits proportionally greater output torque. With losses, output power is smaller than input power over a consistent boundary and interval.

Efficiency here is useful output power divided by input power under a stated steady condition with no net storage change. It need not stay constant with load or speed. During acceleration, some input can increase stored kinetic energy. A quoted ratio and an assumed efficiency therefore do not establish startup torque, thermal behaviour, tooth strength or permissible continuous duty.

Calculate speed and actual torque from supplied power

Use 18 input teeth and 54 output teeth. With an input 180 rpm clockwise, the output is 180×18/54=60 rpm counterclockwise. The reduction ratio input/output is 3. Converting to radians gives ωin=6π rad/s and ωout=2π rad/s.

Suppose the input power is 6 W and the constructed steady efficiency 0.80. Output power is 4.8 W and other transfer 1.2 W. Input torque magnitude is 6/(6π)≈0.318 N·m; output torque is 4.8/(2π)≈0.764 N·m. The torque ratio is 2.4 rather than the lossless 3. Those values describe a supplied operating account, not recommended ratings for a gear or motor.

Try the reasoning

1. Why does the output rotate oppositely for this pair?

   Show the answer and reasoning

   The two external gears contact directly; their tangential motions at engagement require opposite rotation.

2. Does a 3: 1 reduction triple the available power?

   Show the answer and reasoning

   No. Ideally power is conserved; actual output power also reflects losses and storage changes.

3. Why should a smooth belt’s measured ratio be checked?

   Show the answer and reasoning

   The no-slip geometric ratio is an assumption; slip, stretch and operating conditions can alter actual transfer.

What to carry forward

Keep ratio, rotation direction, power balance and component suitability as connected but separate checks.