This article shows how to assemble a two-stage compound planetary gear set in Engrenarium. The example uses a complete planetary gear train in the first stage and a second stage without sun gear, connected to the first by couplings between carriers and between planet gears.

The mechanism can be mounted in Engrenarium desktop software or Engrenarium Web. The idea is to use the first stage to drive the second through couplings, rather than treating each planetary gear set as an isolated system.

First stage

Open a new project in Engrenarium. In the basic parameters panel, configure the first planetary with the following tooth counts:

Planetary 1 Tooth count
Sun gear 20
Planet 28
Ring 76

Then click add planetary gear set. Engrenarium stops the simulation and reports that the kinematic system is underdetermined. This is to be expected: when creating a second stage, it still lacks known speeds or sufficient couplings to define the motion.

Second stage without sun gear

In planetary 2, delete sun gear and configure only the planet and the ring gear:

Planetary 2 Tooth count
Planet 30
Ring 78

This detail is important: a planetary stage does not need to contain all possible parts to participate in the mechanism. In this example, the second planetary works without sun gear and receives movement through the couplings.

Speeds and couplings

Now impose the known angular velocities on planetary 1:

Element Angular velocity
Planetary 1 - Sun gear \(10.0\ \mathrm{rpm}\)
Planetary 1 - Carrier \(5.0\ \mathrm{rpm}\)

Then, in couplings, connect the corresponding elements between the two stages:

In relation to speeds, select Planetary 1 - Sun gear as input and Planetary 2 - Ring as an output.

Assembly of two-stage compound planetary gear set in Engrenarium
Example with two stages: planetary 1 has sun gear, planet, ring gear and carrier; planetary 2 works without sun gear and is driven by couplings.

First stage equations

In planetary 1, the relationship between sun gear, ring gear and carrier can be written by the speeds relative to carrier:

\[ Z_{S1}(\omega_{S1}-\omega_{B1}) + Z_{A1}(\omega_{A1}-\omega_{B1}) = 0 \] \[ 20(10.0-5.0) + 76(\omega_{A1}-5.0) = 0 \]

It follows:

\[ \omega_{A1} = 5.0 - \frac{20}{76}(10.0-5.0) \approx 3.7\ \mathrm{rpm} \]

The speed of the first stage planet comes from the contact relationship with sun gear:

\[ \omega_{P1} = \omega_{B1} - \frac{Z_{S1}}{Z_{P1}}(\omega_{S1}-\omega_{B1}) \] \[ \omega_{P1} = 5.0 - \frac{20}{28}(10.0-5.0) \approx 1.4\ \mathrm{rpm} \]

Second stage equations

The two couplings transfer speeds to the second stage:

\[ \omega_{B2} = \omega_{B1} = 5.0\ \mathrm{rpm} \] \[ \omega_{P2} = \omega_{P1} \approx 1.4\ \mathrm{rpm} \]

As planetary 2 does not have sun gear, the speed of the ring gear can be obtained by the relationship between planet, ring gear and carrier:

\[ \omega_{P2} = \omega_{B2} + \frac{Z_{A2}}{Z_{P2}}(\omega_{A2}-\omega_{B2}) \] \[ \omega_{A2} = \omega_{B2} + \frac{Z_{P2}}{Z_{A2}}(\omega_{P2}-\omega_{B2}) \] \[ \omega_{A2} = 5.0 + \frac{30}{78}(1.4-5.0) \approx 3.6\ \mathrm{rpm} \]

With input at the sun gear of planetary gear set 1 and exit at the ring of planetary 2, the relationship indicated by Engrenarium is:

\[ i = \frac{\omega_{S1}}{\omega_{A2}} = \frac{10.0}{3.6} \approx 2.8 \]

Expected results

The results panel should show the relationship Planetary 1 - Sun gear / Planetary 2 - Ring equal to approximately \(2.8\).

Element Angular velocity
Planetary 1 - Sun gear \(10.0\ \mathrm{rpm}\)
Planetary 1 - Planet \(1.4\ \mathrm{rpm}\)
Planetary 1 - Ring \(3.7\ \mathrm{rpm}\)
Planetary 1 - Carrier \(5.0\ \mathrm{rpm}\)
Planetary 2 - Planet \(1.4\ \mathrm{rpm}\)
Planetary 2 - Ring \(3.6\ \mathrm{rpm}\)
Planetary 2 - Carrier \(5.0\ \mathrm{rpm}\)

Why are both couplings needed

Coupling just the carriers is not enough. Physically, the carriers of the two stages could rotate together while the planets of each stage would rotate at different speeds. Therefore, the coupling between planet gears is also essential to transform the two stages into a compound planetary gear set with determined movement.

Carrier of planetary 1 with extended shafts to couple the two stages
With the coupling between carriers, Engrenarium extends the axes of the carrier of planetary gear set 1 to fit the planet gears of the two stages.

This behavior also explains why the rendering of the second carrier disappears. The model now represents a common carrier, adjusted to support the planets of both stages and maintain the correct kinematic constraints.