Planet Ring Gear

Planet Ring Gear component

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

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

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

Including Planet/Carrier Bearing Friction

Kinematic Equation

$\left( r_{R\/P}-1\right){\mathrm{\ϕ}}_c \= r_{R\/P}·{\mathrm{\ϕ}}_R-{\mathrm{\ϕ}}_P$ 

where $r_{R\/P}$  is the gear ratio and is defined as:

  $r_{R\/P}\=\frac{N_R}{N_P}$ 

where $N_R$ is the number of teeth of the outer planet gear and   $N_P$  is the number of teeth of the inner planet gear.

Also ${\mathrm{\varphi }}_C \,$${\mathrm{\varphi }}_R$ and ${\mathrm{\varphi }}_P$ are defined as the rotation angles of the carrier, outer planet, and inner planet, respectively.

Torque Balance Equation (No Inertia)

$r_{R\/P}·n_{\mathrm{pl}} · {\mathrm{\tau }}_p \= {\mathrm{\τ}}_{\mathrm{loss}} - {\mathrm{\tau }}_R$

${\mathrm{\tau }}_c\+n_{\mathrm{pl}} · {\mathrm{\tau }}_p \+ {\mathrm{\tau }}_R \= 0$

where ${\mathrm{\tau }}_C$ , ${\mathrm{\tau }}_R$ and ${\mathrm{\tau }}_p$ are the torques applied to the carrier, ring, and planet, respectively and $n_{\mathrm{pl}}$ is the number of identical planets meshing with the ring.

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

  ${\mathrm{\tau }}_{\mathrm{loss}}\= {\mathrm{n}}_{\mathrm{pl}} · r_{R\/P} · d·{\mathrm{\ω}}_{P\/C} \+ \left\{\begin{array}{cc}\(1-{\mathrm{\η}}_1\({\mathrm{\ω}}_{R\/C}\)\)· {\mathrm{\τ}}_R& {\mathrm{\omega }}_{R\/C} · {\mathrm{\τ}}_R \>0 \\ \(1-\frac{1}{{\mathrm{\eta }}_2\({\mathrm{\omega }}_{R\/C}\)} \)·{\mathrm{\τ}}_R& {\mathrm{\omega }}_{R\/C} · {\mathrm{\tau }}_R \le 0\end{array}\right.$        

Also

${\mathrm{\omega }}_{\mathrm{P}\/C}$ = ${\mathrm{\omega }}_P$ - ${\mathrm{\omega }}_C$

${\mathrm{\omega }}_{R\/C}$ = ${\mathrm{\omega }}_R$ - ${\mathrm{\omega }}_C$

where

$\mathrm{\ω\_\_x} \= \mathrm{\ϕ\·\_\_x}$  ,      $x \in \left\{P\,R\,C\right\}$

Power Loss:

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

$P_{\mathrm{loss} }\= \left\{\begin{array}{cc}0& \mathrm{ideal}\=\mathbf{true} \\ n_{\mathrm{pl}}·d·{\mathrm{\omega }}_{P\/C}^{}{ }^2\+ \left(1-{\mathrm{\eta }}_1\right){\mathrm{\tau }}_R·{\mathrm{\omega }}_{R\/C}& {\mathrm{\τ}}_R· {\mathrm{\omega }}_{R\/C}\ge 0\\ n_{\mathrm{pl}}·d·{{\mathrm{\omega }}^{}}_{P\/C}{ }^2\+ \left(1-\frac{1}{{\mathrm{\eta }}_2}\right){\mathrm{\tau }}_R·{\mathrm{\omega }}_{R\/C}& {\mathrm{\τ}}_R· {\mathrm{\omega }}_{R\/C}\ge 0\end{array}\right.$

Connections 

Parameters

Note:  Gear ratio $r_{R\/P}$  must be strictly greater than one.

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.