Gears, Pulleys, and Sprockets

These three powertrain elements transfer energy through rotary motion.

Change the speed of rotation

Change the direction of rotation 

Change the amount of torque available to do work

Gears

A gear train is a mechanism used for transmitting rotary motion and torque through interlocking teeth.

A gear train is made when two or more gears are meshed

Driver gear causes motion

Motion is transferred to the driven gear

Mating gears always turn in opposite directions.

An Idler Gear allows the driver and driven gears to rotate in the same direction.

Mating gears always have the same size teeth (diametric pitch)

The rpm of the larger gear is always slower than the rpm of the smaller gear.

RPM is Revolutions Per Minute

Gears locked together on the same shaft will always turn in the same direction and at the same rpm (rotations per minute)

Gear Ratios

Variables to know 

n= number of teeth

D = diameter

W = angular velocity (speed)

t = torque

Subscripts in and out are used to distinguish between gears.

n_in = 6                          n_out = 12

d_in = 2 in.                    d_out = 4 in.

w_in = 40 rpm             w_out = 20 rpm

t_in = 40 ft-lb              t_out = 80 ft-lb

GR/1 = N_out/N_in = d_out/d_in = ⍵_in / ⍵_out = 𝝉 _out  / 𝝉_in

? / 1 = 12/ 6     4 in / 2 in      40 rpm / 20 rpm       80 ft - lb / 40 ft - lb

GR / 1 = n_out / n_in = 12 / 20 = 0.6 / 1

GR / 1 = n_out / n_in = 5 / 12 = 0.42 / 1

Idler gears don’t affect GR!

What is the TOTAL gear train gear ratio?

0.6 / 1 0.42 / 1 4 / 1 = 1/1 

If gears A and D were directly connected to each other, what would the resulting gear ratio be?

If the last gear had 40 teeth, what would be the total gear ratio?

0.6 / 1 0.42 / 1 8/1  = 2/1 or GR / 1 = n_out / n_in = 40 / 20 = 2 / 1 

Gear Ratios - Compound Machines

• Are used similarly to MA

• Apply to torque instead of force

F_out = MA * F_in

T_out = GR * T_in

• In a compound machine, total MA and GR are products of components

MA_total = MA₁ * MA₂ …. MA_n

GR_total = GR₁ * GR₂ … GR_n

• MA is used only to calculate forces, not torques. 

• GR is used only to calculate torques, not forces.

Example Compound Machine

Here are three mechanisms in series:

1. Wheel-axle

2. Gear train

3. Wheel-axle

Gear 1: 60 tooth radius 1.5 in (F_R)

Gear 2: 24 tooth radius 0.6 in (F_E)

D = 4.0 in

Mechanism #1: Wheel-axle

MA = D_E / D_R = 4.0 in / 1.5 in = 2.67

Mechanism #2: Gear train

Gr = n_out / n_in = 24/60 = 0.4

Mechanism #3: Wheel-axle

MA = D_E / D_R = 0.6 in./4.0 in. = 0.15

Total Mechanical Advantage:

MA_total = MA₁ MA₃ = 2.67 0.15 = 0.4

Simulation:

The two middle gears share a common axle, so they rotate at the same speed. 

This allows the final gear to rotate slower and produce more torque than if it were connected only to the driver gear.

Compound Gear Ratios:

What is the gear ratio between gears A and B

4/ 1 * 2.5 / 1 = 10/ 1 

D_out / d_in = ω_in / ω_out = 𝝉_out / 𝝉_in

6 in. / 2 in.         30 rpm / 10 rpm        55 ft-lb / 18.33 ft - lb

D = diameter 

ω = angular velocity (speed)

𝝉 = torque

n _out / n_in = d_out / d_in =  ⍵_in / ⍵_out = 𝝉 _out  / 𝝉_in

22/11              3in/1.5 in 90 rpm/45rpm 120 ft - lb / 60 ft - lb 

N = number of teeth 

D = diameter 

w=angular velocity

t= torque