Planetary gear train with stepped planet gear set

**Library:**Simscape / Driveline / Gears

The Compound Planetary Gear block represents a planetary gear train with composite planet gears. Each composite planet gear is a pair of rigidly connected and longitudinally arranged gears of different radii. One of the two gears engages the centrally located sun gear while the other engages the outer ring gear.

**Compound Planetary Gear**

The block models the compound planetary gear as a structural component based on the Simscape™ Driveline™ Sun-Planet and Ring-Planet blocks. The figure shows the equivalent circuit for the structural component.

To increase the fidelity of the gear model, specify properties such as gear inertia,
meshing losses, and viscous losses. By default, gear inertia and viscous losses are
assumed to be negligible. The block enables you to specify the inertias of the internal
planet gears only. To model the inertias of the carrier, sun, and ring gears, connect
Simscape
Inertia blocks to ports
**C**, **S**, and **R**.

You can model
the effects of heat flow and temperature change by exposing an optional thermal port. To expose
the port, in the **Meshing Losses** tab, set the **Friction
model** parameter to ```
Temperature-dependent
efficiency
```

.

The Compound Planetary Gear block imposes two kinematic and two geometric constraints on the three connected axes and the fourth, internal wheel (planet):

$${r}_{C}{\omega}_{C}={r}_{S}{\omega}_{S}+{r}_{P1}{\omega}_{P},$$

$${r}_{C}={r}_{S}+{r}_{P1},$$

$${r}_{R}{\omega}_{R}={r}_{C}{\omega}_{C}+{r}_{P2}{\omega}_{P},$$

and

$${r}_{R}={r}_{C}+{r}_{P2},$$

where:

*r*is the radius of the carrier gear._{C}*ω*is the angular velocity of the carrier gear._{C}*r*is the radius of the sun gear._{S}*ω*is the angular velocity of the sun gear._{S}*r*is the radius of planet gear 1._{P1}*ω*is the angular velocity of the planet gears._{P}*r*is the radius of planet gear 2._{P2}*r*is the radius of the ring gear._{R}

The ring-planet and planet-sun gear ratios are:

$${g}_{RP}={r}_{R}/{r}_{P2}={N}_{R}/{N}_{P2}$$

and

$${g}_{PS}={r}_{P1}/{r}_{S}={N}_{P1}/{N}_{S},$$

where:

*g*is the ring-planet gear ratio._{RP}*N*is the number of teeth on the ring gear._{R}*N*is the number of teeth on planet gear 2._{P2}*g*is the planet-sun gear ratio._{PS}*N*is the number of teeth on planet gear 1._{P1}*N*is the number of teeth on the sun gear._{S}

In terms of the gear ratios, the key kinematic constraint is:

$$(\text{1}+{g}_{RP}{g}_{PS}){\omega}_{C}={\omega}_{S}+{g}_{RP}{g}_{PS}{\omega}_{R}.$$

The four degrees of freedom reduce to two independent degrees of freedom. The
gear pairs are (1, 2) = (*P2*, *R*) and
(*S*, *P1*).

**Warning**

The gear ratio *g _{RP}* must be
strictly greater than one.

The torque transfers are:

$${g}_{RP}{\tau}_{P2}+{\tau}_{R}\u2013{\tau}_{loss}\left(P2,R\right)\text{}=\text{}0$$

and

$${g}_{PS}{\tau}_{S}+{\tau}_{P1}\u2013{\tau}_{loss}\left(S,P1\right)=0,$$

where:

*τ*is torque transfer for planet gear 2._{P2}*τ*is torque transfer for the ring gear._{R}*τ*is torque transfer loss._{loss}*τ*is torque transfer for the sun gear._{S}*τ*is torque transfer for planet gear 1._{P1}

In the ideal
case, there is no torque loss, that is *τ _{loss}* = 0.

In the nonideal case, *τ _{loss}* ≠ 0. For more information, see Model Gears with Losses.

Gears are assumed rigid.

Coulomb friction slows down simulation. For more information, see Adjust Model Fidelity.