Compound planetary gear#

The Compound planetary gear block models a planetary gear transmission with composite planetary gears. Each composite planetary gear is a pair of rigidly connected gears of different radii located along a common axis. One of the two gears meshes with the central sun gear, and the other meshes with the outer ring gear.

This block is a structural component consisting of the blocks Sun-planet and Ring-planet Fig. 1 :

The model imposes two kinematic and two geometric constraints on the connected transmission shafts: CompoundPlanetaryGear :

$r_{вод.} \cdot \omega_{вод.} = r_{сол.}\cdot \omega_{сол.} +r_{сат.1} \cdot \omega_{сат.}\\r_{корон.} \cdot ω_{корон.}= r_{вод.} \cdot ω_{вод.} + r_{сат.2}\cdot ω_{сат.}\\r_{вод.}= r_{сол.}+ r_{сат.1} \\r_{корон.}= r_{вод.} + r_{сат.2}$$(CompoundPlanetaryGear)

where:

$r_{car.}$ – distance from the transmission axis to the satellite axis (satellite installation radius);

$\omega_{car.}$ – angular velocity of the carrier;

$r_{crown.}$ – radius of the crown gear;

$\omega_{crown.}$ – angular velocity of the crown gear;

$r_{sun}$ – radius of the sun gear;

$\omega_{sun}$ – angular velocity of the sun gear;

$r_{sat.1}$ – radius of the first stage of the satellite;

$r_{sat.2}$ – radius of the second stage of the satellite;

$ω_{sat.}$ – angular velocity of the satellite.

The gear ratio between the ring gear and the second stage of the planetary gear (1) :

(1)#\[\begin{split}g_{PS} = \frac{N_{сат.1}}{N_{\text{сол.}}}= \frac{r_{сат.1}}{r_{\text{сол.}}}\\g_{RP} = \frac{N_{\text{корон.}}}{N_{сат.2}}= \frac{r_{\text{корон.}}}{r_{сат.2}}\end{split}\]

where:

$g_{PS}$ – gear ratio between the sun gear and the first stage of the satellite;

$g_{RP}$ – gear ratio between the ring gear and the second stage of the satellite;

$N$ is the number of teeth of the corresponding gear.

Note

For the transmission to operate correctly, the gear ratio g_RP must be strictly greater than one.

The main kinematic constraint in terms of gear ratios relates the angular velocities of the shafts of the transmission (2):

(2)#\[(1 + g_{RP} \cdot g_{PS} ) \,\cdot \omega_{\text{вод.}}= \omega_{\text{сол.}} +g_{RP} \cdot g_{PS} \, \cdot \omega_{\text{корон.}}\]

The four degrees of freedom of the system are reduced to two independent degrees of freedom. The model uses two gear pairs: “second stage of satellites – crown gear” and “sun gear – first stage of satellites”, for which the transmission agreement (1,2) = (P2, crow.) and (sun, P1) is adopted.

The transmission of torque in the block is described by the equations:

for engagement solar – first stage of satellites (3) :

(3)#\[g_{PS} \cdot \,M_{\text{сол.}} + M_{P1} - M_{\text{потери}} = 0\]

for engagement the second stage of the satellites is the crown gear (4) :

(4)#\[g_{RP} \cdot \,M_{P2} + M_{\text{корон.}} - M_{\text{потери}} = 0\]

where:

$M_{P2}$ – transmission moment for the second stage of satellites;

$M_{crow.}$ – transmission moment for the crown gear;

$M_{sun}$ – transmission moment for the sun gear;

$M_{losses}$ – the moment of loss (equal to zero in the ideal case).

1. Properties#

Property	Default Value	Description	Units of measurement
Ring (R) to planet (P) teeth ratio (N_R/N_P)	2.0	-	-
Planet (P) to sun (S) teeth ratio (N_P/N_S)	1.0	-	-

2. Inputs and outputs#

Parameters	Input/Output	Description	Units of measurement	Port type
Carrier (C)	Input	-	-	Rotational motion
Ring (R)	Output	-	-	Rotational motion
Sun (S)	Output	-	-	Rotational motion

In order to see the connection capabilities of the block, you need to go to page Description of block ports