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Planetary Gear

Planetary Gear component

The Planetary Gear component models a gearbox without inertia, elasticity, or backlash. It consists of an inner sun wheel, an outer ring wheel, and a planet wheel located between sun and ring wheel. The bearing of the planet wheel shaft is fixed in the planet carrier.

The component can be connected to other elements at the sun, ring and/or carrier flanges. An option is provided to activate a planet flange. If inertia is taken into consideration, the sun, ring, and carrier inertias can be added by attaching the Inertia component to the corresponding connectors. The inertia of the planet wheels are included by attaching an Inertia component to the planet flange. The damping in the bearing connecting the planet(s) to carrier can be included via the component options. Bearing friction of the ‘ring’ and ‘carrier’ shafts may be included by attaching the Bearing Friction component(s) to these flanges.The icon of the planetary gear signals that the sun and carrier flanges are on the left side and the ring flange is on the right side of the gear box. However, this component is generic and is valid independently to how the flanges are actually placed (for example, the sun wheel may be placed on the right side instead on the left side). According to the overall convention, the positive direction of all relevant vectors, namely, the absolute angular velocities and cut-torques in the flanges are along the axis vector shown in the icon.

Note 1: Since the planet’s mass is rotating at a distance from the planetary gear axis, ensure that when adding inertia to the ‘planet' that the 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, use the configuration shown in the figure below to correctly account for the relative velocity of the planet with respect to the carrier.

Including Planet/Carrier Bearing Friction

Kinematic Equation

The gear ratio of the planetary gear is $r_{R / S}$ and is defined by:

$r_{R / S} = \frac{N_{R}}{N_{S}}$

Where $N_{R}$ is the number of ring teeth and $N_{S}$ is the number of sun teeth.

The number of planet teeth $N_{P}$ has to fulfill the following relationship:

$N_{P} = \frac{1}{2} (N_{R} - N_{S})$

There are two types of kinematic equations depending whether the Planet flange is disabled or enabled:

Note: When the Planet flange is enabled ( $planet port = true$ ) an extra equation is added.

Planet flange is disabled

$(1 + r_{R / S}) ϕ_{c} = ϕ_{S} + r_{R / S} \cdot ϕ_{R}$

Planet flange is enabled

$(1 + r_{R / S}) ϕ_{c} = ϕ_{S} + r_{R / S} \cdot ϕ_{R}$

$(r_{R / S} - 1) ϕ_{p} = r_{R / S} \cdot ϕ_{R} - ϕ_{S}$

Where $ϕ_{C},$ $ϕ_{O}$ , and $ϕ_{I}$ are defined as the rotation angles of the carrier, outer planet, and inner planet, respectively

Internal Structure

Torque Balance Equation (No Inertia)

There are two sets of kinematic torque balance equations depending on whether the Planet flange is disabled or enabled:

Planet flange is disabled

$- \frac{1}{r_{R / S}} τ_{R} + τ_{S} + τ_{loss} = 0$

$τ_{R} + τ_{S} + τ_{C} = 0$

Planet flange is enabled

$- \frac{1}{r_{R / S}} τ_{R} + τ_{S} - \frac{2 n_{pl}}{r_{R / S} - 1} τ_{P} + τ_{loss} = 0$

$τ_{R} + τ_{S} + τ_{C} + n_{pl} \cdot τ_{O} + τ_{P} = 0$

Where $τ_{C}, τ_{R}, τ_{S}, τ_{O}, τ_{P}$ are defined as the rotation angles of the carrier, ring, sun, and planet respectively. $n_{pl}$ is the number of identical planets meshing with the ring and sun gears.

Also $τ_{loss}$ is the loss torque and is defined as:

$τ_{loss} = \{\begin{array}{c} 0 & ideal = true \\ n_{pl} \cdot \frac{2}{r_{R / S} - 1} \cdot ω_{P / C} + τ_{mR} - τ_{mS} & ideal = false \end{array}$

Where $n_{pl}$ is the number of planets meshing with the Sun and the Ring gears, and

$τ_{mR} = \frac{1}{r_{R / S}} \{\begin{array}{c} (1 - η_{11} (ω_{R / C})) \cdot τ_{R} & ω_{R / C} \cdot τ_{R} \geq 0 \\ (1 - \frac{1}{η_{12} (ω_{R / C})}) \cdot τ_{R} & ω_{R / C} \cdot τ_{R} < 0 \end{array}$

$τ_{mS} = \{\begin{array}{c} (1 - η_{22} (ω_{S / C})) \cdot τ_{S} & ω_{S / C} \cdot τ_{S} \geq 0 \\ (1 - \frac{1}{η_{21} (ω_{S / C})}) \cdot τ_{S} & ω_{S / C} \cdot τ_{S} < 0 \end{array}$

Where $η_{11} (ω_{R / C})$ and $η_{12} (ω_{R / C})$ are the forward and backward Ring/Planet meshing efficiency, respectively and $η_{22} (ω_{S / C})$ and $η_{21} (ω_{S / C})$ are the forward and backward Sun/Planet meshing efficiency, respectively.

Also

$ω_{R / C}$ = $ω_{R}$ - $ω_{C}$

$ω_{S / C}$ = $ω_{S}$ - $ω_{C}$

$ω_{S / R}$ = $ω_{S}$ - $ω_{R}$

$ω_{P / C}$ = $- 2 \frac{r_{R / S}}{{(r_{R / S})}^{2} - 1} ω_{S / R}$

Where

$ω__x = ϕ·__x$ , $x \in \{P, S, R, C\}$

Power Loss

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

$P_{loss} = P_{loss1} + P_{loss2} + P_{loss3}$

Where

$P_{loss1} = \{\begin{array}{c} 0 & ideal = true \\ (1 - η_{11}) τ_{R} \cdot ω_{R / C} & τ_{R} \cdot ω_{R / C} \geq 0 \\ (1 - \frac{1}{η_{12}}) τ_{R} \cdot ω_{R / C} & τ_{R} \cdot ω_{R / C} < 0 \end{array}$

$P_{loss2} = \{\begin{array}{c} 0 & ideal = true \\ (1 - η_{22}) τ_{S} \cdot ω_{S / C} & τ_{S} \cdot ω_{S / C} \geq 0 \\ (1 - \frac{1}{η_{21}}) τ_{S} \cdot ω_{S / C} & τ_{S} \cdot ω_{S / C} < 0 \end{array}$

$P_{loss3} = {\begin{array}{c} 0 & ideal = true \\ η_{pl} \cdot d_{} \cdot {(ω_{P / C})}^{2} & ω_{P / C} \geq 0 \end{array}$

Connections

Name	Condition	Description	ID
$Carrier$	$-$	Carrier flange	carrier
$Planet$	$planet port = true$	Planet flange	planet
Ring	$-$	Ring flange	ring
$Sun$	$-$	Sun flange	sun
$Loss Power$	$ideal = false$	Conditional real output port for power loss	lossPower

Parameters

Symbol

Condition

Default

Units

Description

$ideal$

$true$

Defines whether the component is:

true - ideal or

false - non-ideal

ideal

data source

$ideal = false$

GUI

Defines the source for the loss data:

•	entered via GUI [GUI]

•	by an attachment [attachment]

•	by an external file [file]

datasourcemode

$same loss data$

$ideal = false$

true

Defines whether one efficiency data table is used for all meshing loss calculations [ $same loss data = true$ ] or the efficiency of each meshing gear pair is given by a separate data table [ $same loss data$ $= false$ ].

SameMeshingEfficiency

$r_{R / S}$

$2$

Gear ratio

ratio

$n_{pl}$

$ideal = false$

$1$

Number of planet gears

numberofPlanets

$η (ω)$

$ideal = false$

same loss data = true

data source = GUI

$[0, 1, 1]$

$[\frac{rad}{s}, -, -]$

Defines all velocity dependant meshing efficiencies.

The columns:

[ $ω_{}$ $η_{1}$ ( $ω$ ) $η_{2}$ ( $ω$ )]

Five options are available:

•	1 by 1 array: entered value is taken as the constant efficiency for forward and backward cases

$η_{1}$ ( $ω$ ) = $η_{2}$ ( $ω$ ) = $η_{}$

•	1 by 2 array: first entered value is taken as the constant efficiency for forward case and the second for backward cases

$η_{1}$ ( $ω$ ) = $η_{1}, η_{2}$ ( $ω$ ) = $η_{2}$

•	1 by 3 array: first column is ignored and the second and third values are taken as constant efficiencies for forward and backward cases, respectively

•	n by 2 array: Second column is forward and backward efficiency

$η$ ( $ω$ ) = $η_{1}$ ( $ω$ ) = $η_{2}$ ( $ω$ )

•	n by 3 array:

Second column is forward efficiency

$η_{1}$ ( $ω$ )

Third column is backward efficiency

$η_{2}$ ( $ω$ )

meshinglossTable3

$ideal = false$

same loss data = true

data source = attachment

Defines velocity dependant meshing efficiency

First column is angular velocity ( $ω_{}$ )

(See $col$ $[η]$ below)

data3

$ideal = false$

same loss data = true

data source = file

fileName3

$col$ $[η]$

same loss data = true

data source = attachment or file

$[2, 3]$

Defines the corresponding data columns used for forward efficiency ( $η_{1}$ ) and backward efficiency ( $η_{2}$ )

Two options are available:

•	1 by 1 array:

Data column corresponding to the column number is used for both forward and backward efficiency ( $η = η_{1} = η_{2}$ )

•	1 by 2 array:

Data column corresponding to the first column number is used for forward efficiency ( $η_{1}$ ) and data column corresponding to the second column number is used for backward efficiency ( $η_{2}$ )

columns3

$η_{R / P} (ω_{R / C})$

$ideal = false$

same loss data = false

data source = GUI

$[0, 1, 1]$

$[\frac{rad}{s}, -, -]$

Defines Ring/Planet velocity dependant meshing efficiency as a function of $ω_{R / C}$ .

The columns are:

[ $ω_{R / C}$ $η_{1}$ ( $ω_{R / C}$ ) $η_{2}$ ( $ω_{R / C}$ )]

First column is angular velocity of the ring gear w.r.t. the carrier ( $ω_{R / C}$ )

Five options are available:

•	1 by 1 array: entered value is taken as the constant efficiency for forward and backward cases

$η_{1}$ ( $ω_{R / C}$ ) = $η_{2}$ $(ω_{R / C})$ = $η_{}$

•	1 by 2 array: first entered value is taken as the constant efficiency for forward case and the second for backward cases

$η_{1}$ ( $ω_{R / C}$ ) = $η_{1}, η_{2}$ ( $ω_{R / C}$ ) = $η_{2}$

•	1 by 3 array: first column is ignored and the second and third values are taken as constant efficiencies for forward and backward cases, respectively.

•	n by 2 array: Second column is forward and backward efficiency

$η_{}$ ( $ω_{R / C}$ ) = $η_{1}$ ( $ω_{R / C}$ ) = $η_{2}$ ( $ω_{R / C}$ )

•	n by 3 array:

Second column is forward efficiency

$η_{1}$ ( $ω_{R / C}$ )

Third column is backward efficiency

$η_{2}$ ( $ω_{R / C}$ )

meshinglossTable1

$ideal = false$

same loss data = false

data source = attachment

Defines the velocity dependent meshing efficiency

First column is angular velocity ( $ω_{R / C}$ )

(See $col$ $[η_{R / P}]$ below)

data1

$ideal = false$

same loss data = false

data source = file

fileName1

$col$ $[η_{R / P}]$

$ideal = false$

same loss data = false

data source = attachment or file

$[2, 3]$

Defines the corresponding data columns used for forward ( $η_{1}$ ) and backward ( $η_{2}$ ) friction torques.

Two options are available:

•	1 by 1 array:

Data column corresponding to the column number is used for both forward and backward efficiency ( $η_{1} = η_{2} = η)$

•	1 by 2 array:

Data column corresponding to the first column number is used for forward efficiency ( $η_{1}$ ) and

Data column corresponding to the second column number is used for backward efficiency ( $η_{2}$ )

columns1

$η_{P / S} (ω_{S / C})$

$ideal = false$

same loss data = false

data source = GUI

$[0, 1, 1]$

$[\frac{rad}{s}, -, -]$

Defines Planet/Sun velocity dependant meshing efficiency as a function of $ω_{S / C}$ .

The columns are:

[ $ω_{S / C}$ $η_{1}$ ( $ω_{S / C}$ ) $η_{2}$ ( $ω_{S / C}$ )

First column is angular velocity of the sun gear w.r.t. carrier ( $ω_{S / C}$ )

Five options are available:

•	1 by 1 array: entered value is taken as the constant efficiency for forward and backward cases

$η_{1}$ ( $ω_{S / C}$ ) = $η_{2}$ $(ω_{S / C})$ = $η_{}$

•	1 by 2 array: first entered value is taken as the constant efficiency for forward case and the second for backward cases

$η_{1}$ ( $ω_{S / C}$ ) = $η_{1}, η_{2}$ ( $ω_{S / C}$ ) = $η_{2}$

•	1 by 3 array: first column is ignored and the second and third values are taken as constant efficiencies for forward and backward cases, respectively.

•	n by 2 array: Second column is forward and backward efficiency

$η_{}$ ( $ω_{S / C}$ ) = $η_{1}$ ( $ω_{S / C}$ ) = $η_{2}$ ( $ω_{S / C}$ )

•	n by 3 array:

Second column is forward efficiency

$η_{1}$ ( $ω_{S / C}$ )

Third column is backward efficiency

$η_{2}$ ( $ω_{S / C}$ )

meshinglossTable2

$ideal = false$

same loss data = false

data source = attachment

Defines the velocity dependent meshing efficiency

First column is angular velocity ( $ω_{S / C}$ )

(See $col$ $[η_{P / S}]$ below)

data2

$ideal = false$

same loss data = false

data source = file

fileName2

$col$ $[η_{P / S}]$

$ideal = false$

same loss data = false

data source = attachment or file

$[2, 3]$

Defines the corresponding data columns used for forward ( $η_{1}$ ) and backward ( $η_{2}$ ) efficiency

Two options are available:

•	1 by 1 array:

Data column corresponding to the column number is used for both forward and backward efficiency ( $η_{1} = η_{2} = η)$

•	1 by 2 array:

Data column corresponding to the first column number is used for forward efficiency ( $η_{1}$ ) and

Data column corresponding to the second column number is used for backward efficiency ( $η_{2}$ )

columns2

$ideal = false$

$[\frac{N \cdot m}{\frac{rad}{s}}]$

linear damping in planet/carrier bearing

smoothness

$ideal = false$

Table points are linearly interpolated

Defines the smoothness of table interpolation

There are two options:

•	Table points are linearly interpolated

•	Table points are interpolated such that the first derivative is continuous

smoothness

Note: Gear ratio $r_{R / S}$ must be strictly greater than zero.

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Machine Design & Industrial Automation

Other

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Examples & Applications

Community

About Maplesoft

Media Center

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