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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 rR/S and is defined by:

 rR/S=NRNS 

Where   NR is the number of ring teeth and    NS  is the number of sun teeth.

The number of planet teeth   NP has to fulfill the following relationship:

 

NP = 12NR  NS 

 

 

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+rR/S ϕc = ϕS + rR/S·ϕR 

 

Planet flange is enabled

 

1+rR/S ϕc = ϕS + rR/S·ϕR 

 

 rR/S  1 ϕp = rR/S·ϕR  ϕS

 

Where ϕC , ϕO , and ϕI 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

 

1rR/SτR  + τS+τloss = 0 

 

τR  + τS+τC = 0

 

Planet flange is enabled

 

 

 

1rR/SτR + τS  2nplrR/S  1τP +τloss = 0 

 

 

τR + τS + τC +npl· τO + τP=  0

 

Where τC , τR , τS , τO ,τP  are defined as the rotation angles of the carrier, ring, sun, and planet respectively. npl is the number of identical planets meshing with the ring and sun gears.

Also τloss is the loss torque and is defined as:

 

τloss = 0                  ideal=true npl · 2rR/S  1· ωP/C+ τmR  τmS                  ideal=false 

 

Where npl is the number of planets meshing with the Sun and the Ring gears, and

 

τmR&equals;  1rR&sol;S &lpar;1η11&lpar;ωR&sol;C&rpar;&rpar;· &tau;R       ωR&sol;C · &tau;R0  &lpar;1 1η12&lpar;ωR&sol;C&rpar;&rpar;· &tau;R    ωR&sol;C · τR<0

 

τmS&equals;  &lpar;1η22&lpar;ωS&sol;C&rpar;&rpar;·&tau;S       ωS&sol;C · τS0  &lpar;1 1η21&lpar;ωS&sol;C&rpar;&rpar;·&tau;S    ωS&sol;C · τS<0

 

Where η11&lpar;ωR&sol;C&rpar; and η12&lpar;ωR&sol;C&rpar; are the forward and backward Ring/Planet meshing efficiency, respectively and  η22&lpar;ωS&sol;C&rpar; and η21&lpar;ωS&sol;C&rpar; are the forward and backward Sun/Planet meshing efficiency, respectively.

Also

ωR&sol;C = ωR  - ωC

ωS&sol;C = ωS  - ωC

ωS&sol;R = ωS  - ωR

ωP&sol;C =  2 rR&sol;SrR&sol;S2 1ωS&sol;R

Where

&omega;x  &equals; &varphi;·x  ,      x  P&comma;S&comma; R&comma;C

 

Power Loss

 

The power loss (Ploss) is calculated as:

 

Ploss &equals;  Ploss1 &plus; Ploss2&plus; Ploss3

 

Where

 

Ploss1  &equals; 0ideal&equals;true 1η11τR·ωR&sol;CτR · ωR&sol;C0  11η12τR·ωR&sol;CτR · ωR&sol;C<0  

 

Ploss2  &equals; 0ideal&equals;true 1η22τS·ωS&sol;CτS · ωS&sol;C011η21τS·ωS&sol;CτS · ωS&sol;C<0

Ploss3  &equals; &lcub;0ideal&equals;true  &eta;pl·d·ωP&sol;C2ωP&sol;C0  

 

 

Connections 

Name

Condition

Description

ID

Carrier

Carrier flange

carrier

Planet

planet port&equals;true

Planet flange

planet

Ring

Ring flange

ring

Sun

Sun flange

sun

Loss Power

ideal&equals;false

Conditional real output port for power loss

lossPower

 

Parameters

Symbol

Condition

Default

Units

Description

ID

ideal

-

true

-

Defines whether the component is:

true - ideal or

false - non-ideal

ideal

data source

ideal&equals;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&equals;false

true

-

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

SameMeshingEfficiency

rR&sol;S

-

2

-

Gear ratio

ratio

npl

ideal&equals;false

1

Number of planet gears

numberofPlanets

η&omega;

ideal&equals;false

 

same loss data = true

 

data source = GUI

  0&comma;1&comma;1 

rads&comma;&comma;

Defines all velocity dependant meshing efficiencies.

The columns:

[ω     η1 (&omega; )     η2 (&omega; )]

Five options are available:

• 

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

η1 (&omega; ) = η2 (&omega; ) = η

• 

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

η1 (&omega;) = η1 &comma; &eta;2 (&omega; ) = η2

• 

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

&eta; (&omega;) = η1 (ω ) = η2(&omega; )

• 

n by 3 array:

Second column is forward efficiency

η1 (&omega;)

Third column is backward efficiency

η2 (ω )

meshinglossTable3

ideal&equals;false

 

same loss data = true

 

data source = attachment

- 

 

 

 

 

 

 

 

Defines velocity dependant meshing efficiency

First column is angular velocity (ω)

(See col &eta; below)

data3

ideal&equals;false

 

same loss data = true

 

data source = file 

- 

fileName3

col &eta;

same loss data = true

 

data source = attachment or file

 

  2&comma;3

 

-

Defines the corresponding data columns used for forward efficiency (η1) and backward efficiency (η2 )

Two options are available:

• 

1 by 1 array:

Data column corresponding to the column number is used for both forward and backward efficiency (&eta;&equals;&eta;1 &equals; η2  )

• 

1 by 2 array:

Data column corresponding to the first column number is used for forward efficiency ( &eta;1) and data column corresponding to the second column number is used for backward efficiency (η2)

columns3

&eta;R&sol;PωR&sol;C

ideal&equals;false

 

same loss data = false

 

data source = GUI

  0&comma;1&comma;1 

rads&comma;&comma; 

Defines Ring/Planet velocity dependant meshing efficiency as a function of ωR&sol;C .

The columns are:

[ωR&sol;C     &eta;1(ωR&sol;C )     &eta;2(ωR&sol;C )]

First column is angular velocity of the ring gear w.r.t. the carrier (ωR&sol;C)

Five options are available:

• 

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

η1(ωR&sol;C ) = &eta;2&omega;R&sol;C = &eta;

• 

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

&eta;1(ωR&sol;C ) = &eta;1 &comma; η2(ωR&sol;C ) = η2

• 

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 (ωR&sol;C ) = η1 (ωR&sol;C ) = η2 (ωR&sol;C )

• 

n by 3 array:

Second column is forward efficiency

η1 (ωR&sol;C )

Third column is backward efficiency

η2 (ωR&sol;C )

meshinglossTable1

ideal&equals;false

 

same loss data = false

 

data source = attachment

 

 

 

- 

 

 

 

 

Defines the velocity dependent meshing efficiency

First column is angular velocity (ωR&sol;C )

(See col ηR&sol;P below)

 

 

 

data1

ideal&equals;false

 

same loss data = false

 

data source = file

 

 

-

 

 

fileName1

col &eta;R&sol;P

ideal&equals;false

 

same loss data = false

 

data source = attachment or file

  2&comma;3

-

Defines the corresponding data columns used for forward (η1) and backward (η2) 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 (η1&equals;η2 &equals; &eta;&rpar; 

• 

1 by 2 array:

Data column corresponding to the first column number is used for forward efficiency (η1) and

Data column corresponding to the second column number is used for backward efficiency (η2)

columns1

&eta;P&sol;SωS&sol;C

ideal&equals;false

 

same loss data = false

 

data source = GUI

  0&comma;1&comma;1 

rads&comma;&comma; 

Defines Planet/Sun velocity dependant meshing efficiency as a function of ωS&sol;C .

The columns are:

[ωS&sol;C     η1(ωS&sol;C )     η2(ωS&sol;C )

First column is angular velocity of the sun gear w.r.t. carrier (ωS&sol;C)

Five options are available:

• 

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

η1(ωS&sol;C ) = &eta;2ωS&sol;C = &eta;

• 

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

&eta;1(ωS&sol;C ) = &eta;1 &comma; η2(ωS&sol;C ) = η2

• 

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 (ωS&sol;C ) = η1 (ωS&sol;C ) = η2 (ωS&sol;C )

• 

n by 3 array:

Second column is forward efficiency

η1 (ωS&sol;C )

Third column is backward efficiency

η2 (ωS&sol;C )

meshinglossTable2

ideal&equals;false

 

same loss data = false 

 

data source = attachment

 

 

 

 

 

- 

 

 

 

 

 

Defines the velocity dependent meshing efficiency

First column is angular velocity (ωS&sol;C )

(See col ηP&sol;S below)

 

 

 

data2

ideal&equals;false

 

same loss data = false

 

data source = file

 

 

 

 

-

 

 

 

fileName2

col &eta;P&sol;S

ideal&equals;false

 

same loss data = false

 

data source = attachment or file

  2&comma;3

-

Defines the corresponding data columns used for forward (η1) and backward (η2) efficiency

Two options are available:

• 

1 by 1 array:

Data column corresponding to the column number is used for both forward and backward efficiency (η1&equals;η2 &equals; &eta;&rpar; 

• 

1 by 2 array:

Data column corresponding to the first column number is used for forward efficiency (η1) and

Data column corresponding to the second column number is used for backward efficiency (η2)

columns2

d

ideal&equals;false

0

Nmrads

linear damping in planet/carrier bearing

d

smoothness

ideal&equals;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 rR&sol;S must be strictly greater than zero.

 

See Also

MapleSim Driveline Library Overview

MapleSim Library Overview

1-D Mechanical Overview

Basic Gear Sets

Planet Planet Gear

Planet Ring Gear

 

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.

 

 


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