Planet Planet Gear

Planet Planet Gear component     

The Planet Planet Gear component models a set of carrier, inner planet, and outer planet gear wheels with a specified gear ratio without inertia, elasticity, or backlash. The inertia of the gears and carrier may be included by attaching Inertia components to the ‘inplanet’, ‘outplanet’, or ‘carrier’ flanges respectively. The damping in the bearing connecting the planet(s) to carrier can be included via the component options. Bearing friction on the ‘inplanet’, ‘outplanet’, and ‘carrier’ shafts may be included by attaching Bearing Friction component(s) to these flanges.

Note 1: Since the outer planet’s mass is rotating at a distance from the Planet Planet gear axis, ensure that when adding inertia to the ‘outer planet’, proper inertia is also added to the ‘carrier’.

Note 2: When attaching a bearing friction component to the outer planet shaft to represent outplanet/carrier bearing friction, the configuration shown in the figure below should be used to correctly account for the relative velocity of the outer planet with respect to the carrier.

Including Outer Planet/Carrier Bearing Friction

Kinematic Equation

$\left(1\+r_{\mathrm{O}\/I} \right){\mathrm{\ϕ}}_c \= r_{\mathrm{O}\/I}·{\mathrm{\ϕ}}_{\mathrm{O}} \+{\mathrm{\varphi }}_I$ 

where $r_{\mathrm{O}\/I}$  is the gear ratio and is defined as:

  $r_{\mathrm{O}\/I}\=\frac{N_{\mathrm{O}}}{N_I}$ 

where   $N_{\mathrm{O}}$ is the number of teeth of the outer planet gear and   $N_I$  is the number of teeth of the inner planet gear.
 Also ${\mathrm{\ϕ}}_C \,$${\mathrm{\varphi }}_{\mathrm{O}}$ and ${\mathrm{\varphi }}_{\mathrm{I}}$ are defined as the rotation angles of the carrier, outer planet, and inner planet, respectively.

Torque Balance Equation (No Inertia)

$n_{\mathrm{pl}} ·r_{\mathrm{O}} \= r_{\mathrm{O}\/I}·\left({\mathrm{\τ}}_I-{\mathrm{\τ}}_{\mathrm{loss}}\right)$

${\mathrm{\tau }}_C \+n_{\mathrm{pl}}· {\mathrm{\tau }}_{\mathrm{O}} \+ {\mathrm{\tau }}_I\= 0$

where${\mathrm{\τ}}_C \, {\mathrm{\τ}}_{\mathrm{O}} \,{\mathrm{\τ}}_I$  are defined as the rotation angles of the carrier, outer planet, and inner planet, respectively.$n_{\mathrm{pl}}$ is the number of identical outer planets meshing with the inner planet.

Also${\mathrm{\τ}}_{\mathrm{loss}}$ is the loss torque and is defined as:

  ${\mathrm{\tau }}_{\mathrm{loss}}\&equals; -\frac{{\mathrm{n}}_{\mathrm{pl}}}{r_{\mathrm{O}\&sol;I}} · d·{\mathrm{\&omega;}}_{0\&sol;C} \&plus; \left\{\begin{array}{cc}\&lpar;1-{\mathrm{\&eta;}}_2\&lpar;{\mathrm{\&omega;}}_{I\&sol;C}\&rpar;\&rpar;· {\mathrm{\&tau;}}_1& {\mathrm{\&tau;}}_1\ge 0 \\ \&lpar;1-\frac{1}{{\mathrm{\eta }}_1\&lpar;{\mathrm{\omega }}_{I\&sol;C}\&rpar;} \&rpar;·{\mathrm{\&tau;}}_1& {\mathrm{\&tau;}}_1<0\end{array}\right.$        

Also

${\mathrm{\omega }}_{I\&sol;C}$ = ${\mathrm{\omega }}_I$ - ${\mathrm{\omega }}_C$

${\mathrm{\omega }}_{\mathrm{O}\&sol;C}$ = ${\mathrm{\omega }}_{\mathrm{O}}$ - ${\mathrm{\omega }}_C$

where

$\mathrm{\&omega;\_\_x} \&equals; \mathrm{\&varphi;\&middot;\_\_x}$  ,      $x \in \left\{I\&comma;\mathrm{O}\&comma;C\right\}$

Power Loss

The power loss ($P_{\mathrm{loss}}$) is calculated as:

$P_{\mathrm{loss} }\&equals; \left\{\begin{array}{cc}0& \mathrm{ideal}\&equals;\mathbf{true} \\ n_{\mathrm{pl}}·d·{{\mathrm{\omega }}^{}}_{0\&sol;C}^{}{ }^2\&plus; \left(1-{\mathrm{\eta }}_2\right){\mathrm{\tau }}_1·{\mathrm{\omega }}_{I\&sol;C}& {\mathrm{\&tau;}}_1· {\mathrm{\omega }}_{I\&sol;C}\ge 0 \\ n_{\mathrm{pl}}·d·{{\mathrm{\omega }}^{}}_{0\&sol;C}^{}{ }^2\&plus; \left(1-\frac{1}{{\mathrm{\eta }}_1}\right){\mathrm{\tau }}_1·{\mathrm{\omega }}_{I\&sol;C}& {\mathrm{\&tau;}}_1· {\mathrm{\omega }}_{I\&sol;C}\ge 0 \end{array}\right.$

Connections 

Parameters

Note:  Gear ratio $r_{\mathrm{O}\&sol;I}$  must be strictly greater than zero.

See Also

Driveline Library Overview

MapleSim Library Overview

1-D Mechanical Overview

Basic Gear Sets

References

Pelchen C., Schweiger C., and Otter M., “Modeling and Simulating the Efficiency of Gearboxes and Planetary Gearboxes,” 2nd International Modelica Conference, Proceedings, pp. 257-266.