User:100kWhr/sandbox

Derivation


Two independent gears; Sun and Planet; angular velocities:



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

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



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:

which must always be true, regardless of Carrier motion.



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:

To immediately solve with planet angular velocity eliminated:

($$) into ($$) :

($$) into ($$) :

which completes the analysis.

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

To solve for equations containing planet angular velocities :

($$) :

($$) into ($$) :

Which when rearranged, respectively are:

Note : Equation ($$) + Equation ($$) = Equation ($$)

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.





A sample reference method.

Variable Names: