Gear Trains

Overview

Engineering reference data for Gear Trains in mechanics.

Key Formulas

Newton's Second Law

$F = ma$

Force = mass × acceleration.

Work

$W = Fd\cos\theta$

Work = force × displacement × cos(angle).

Kinetic Energy

$E_k = \frac{1}{2}mv^2$

Energy of motion.

Potential Energy

$E_p = mgh$

Gravitational potential energy.

Variables

Typical Gear Ratios

The following table provides typical gear ratios for common gear set types.

Gear Arrangements with Idler Wheels

When the driver and follower must rotate in the same direction, an idler gear is inserted between them. The movement ratio for this arrangement is:

$$i_M = \frac{n_D}{n_F} = \frac{t_I}{t_D} \cdot \frac{t_F}{t_I} = \frac{t_F}{t_D}$$

where:

• $t_I$ = number of teeth on the idler gear.

This shows the idler gear does not change the overall gear ratio ($t_F / t_D$), but it reverses the rotation direction back to the original.

Bicycle Gearing

A bicycle's drivetrain is a practical application of gear trains. The distance traveled by the wheel per pedal revolution is calculated using:

$$l = \pi d \frac{n_D}{n_F} = \pi d \frac{t_D}{t_F}$$

where:

• $l$ = distance traveled by the wheel
• $d$ = outer diameter of the wheel
• $t_D$ = number of teeth on the pedal sprocket (chainring)
• $t_F$ = number of teeth on the wheel sprocket (cog)

Example Calculation

For a bike with a 26-inch wheel diameter, a 24-tooth front chainring, and a 34-tooth rear cog:

• Wheel circumference $c = \pi d \approx 81.7$ in.
• Distance per pedal stroke $l = (81.7 \text{ in}) \times (24/34) \approx 57.7$ inches.

For the highest gear (42T front, 14T rear):

• Distance per pedal stroke $l = (81.7 \text{ in}) \times (42/14) \approx 244.8$ inches.

References

• Original Source