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Epicyclic Gearing

Gear Ratios

Algebraic Method

Derivation

Two independent external involute spur gears.

Two independent gears; Sun and Planet; angular velocities:

$\omega _{\text{s}}$ (1)

$\omega _{\text{p}}$ (2)

Two external involute spur gears linked by a carrier arm but not in mesh.

Constrain pivot centers to a Carrier arm; gears not in mesh:

$\omega _{\text{c}}$ (3)

The angular velocities apparent to an observer on the Carrier are:

$\omega _{{\text{s}}_{\text{c}}}=\omega _{\text{s}}-\omega _{\text{c}}$ (4)

$\omega _{{\text{p}}_{\text{c}}}=\omega _{\text{p}}-\omega _{\text{c}}$ (5)

Two external involute spur gears linked by a carrier arm and in mesh.

When brought into mesh, gear pitch circle velocities must equate. Assuming equal tooth pitch throughout, it is apparent to an observer on the Carrier that:

${\frac {\omega _{{\text{s}}_{\text{c}}}}{\omega _{{\text{p}}_{\text{c}}}}}=-{\frac {N_{\text{p}}}{N_{\text{s}}}}={\frac {\omega _{\text{s}}-\omega _{\text{c}}}{\omega _{\text{p}}-\omega _{\text{c}}}}$ (6)

which must always be true, regardless of Carrier motion.

Epicyclic Gear Train: Sun, Planet, Planet Carrier Arm, and Internal Ring Gear.

An internal Ring gear is acted upon by the Planet gear. The relationship between Ring and Planet gear angular velocities and gear teeth counts, as apparent to an observer on the Carrier is:

$(\omega _{\text{r}}-\omega _{\text{c}})={\frac {N_{\text{p}}}{N_{\text{r}}}}\omega _{{\text{p}}_{\text{c}}}$ (7)

To immediately solve with planet angular velocity eliminated:

(5) into (6) :

$\omega _{{\text{p}}_{\text{c}}}={\frac {N_{\text{s}}}{N_{\text{p}}}}(\omega _{\text{c}}-\omega _{\text{s}})$ (8)

(8) into (7) :

$N_{\text{r}}\omega _{\text{r}}+N_{\text{s}}\omega _{\text{s}}-(N_{\text{r}}+N_{\text{s}})\omega _{\text{c}}=0$ (9)

which completes the analysis.

Equation (9) is directly useful for gear ratios, while equation (8) is in effect, the planet bearing speed.

To solve for equations containing planet angular velocities :

(6) :

${\frac {\omega _{\text{s}}-\omega _{\text{c}}}{\omega _{\text{p}}-\omega _{\text{c}}}}=-{\frac {N_{\text{p}}}{N_{\text{s}}}}$ (6')

(5) into (7) :

${\frac {\omega _{\text{r}}-\omega _{\text{c}}}{\omega _{\text{p}}-\omega _{\text{c}}}}={\frac {N_{\text{p}}}{N_{\text{r}}}}$ (10)

Which when rearranged, respectively are:

$N_{\text{s}}\omega _{\text{s}}+N_{\text{p}}\omega _{\text{p}}-(N_{\text{s}}+N_{\text{p}})\omega _{\text{c}}=0$ (11)

$N_{\text{r}}\omega _{\text{r}}-N_{\text{p}}\omega _{\text{p}}-(N_{\text{r}}-N_{\text{p}})\omega _{\text{c}}=0$ (12)

Note : Equation (11) + Equation (12) = Equation (9)

General Applicability

The only assumption made throughout, was that of equal tooth pitch, ie, equal angular displacement per tooth. The results are otherwise general. One may for example, equate sun and ring tooth counts to analyze a differential; add an extra tooth to the ring gear for more play; etc, and the mathematical results (angular velocity and displacement ratios) will be correct.

Where

$\omega _{\text{s}},\omega _{\text{c}},\omega _{\text{p}},\omega _{\text{r}}$ are the absolute angular velocities of the Sun gear, Planet Carrier arm, Planet gear, and Ring gear respectively, and

$\omega _{{\text{s}}_{\text{c}}},\omega _{{\text{p}}_{\text{c}}},\omega _{{\text{r}}_{\text{c}}}$ are the apparent angular velocities of the Sun, Planet, and Ring gears with respect to the planet Carrier arm respectively, and

$N_{\text{s}},N_{\text{p}},N_{\text{r}}$ are the number of teeth on the Sun, Planet, and Ring gears respectively.

Epicyclic Gear Train: Sun, Planet and Ring Gears (Planet Carrier Arm not shown).

Epicyclic Gear Train comprised of three external gears linked by a common Carrier arm.

A samplereference method.

Variable Names:

$\omega _{\text{s}}$	absolute angular velocity of the sun gear
$\omega _{\text{p}}$	absolute angular velocity of the planet gear
$\omega _{\text{r}}$	absolute angular velocity of the ring gear
$\omega _{\text{c}}$	absolute angular velocity of the carrier arm
$\omega _{{\text{s}}_{\text{c}}}$	apparent angular velocity of the sun gear wrt the carrier arm
$\omega _{{\text{p}}_{\text{c}}}$	apparent angular velocity of the planet gear wrt the carrier arm
$\omega _{{\text{r}}_{\text{c}}}$	apparent angular velocity of the ring gear wrt the carrier arm
$N_{\text{s}}$	number of teeth on the sun gear
$N_{\text{p}}$	number of teeth on the planet gear
$N_{\text{r}}$	number of teeth on the ring gear

References

https://en.wikipedia.org/wiki/Epicyclic_gearing#Fixed_carrier_train_ratio

https://en.wikipedia.org/wiki/Epicyclic_gearing#Gear_ratio_of_standard_epicyclic_gearing

https://en.wikipedia.org/wiki/Gear_train

https://en.wikipedia.org/wiki/Angular_velocity