Prediction Method of Static Torque Sharing Distribution of Planetary

Gear Sets Generated by Manufacturing Error Distribution

Hiroki Kakiuchi, Honda R&D Co., Ltd.

Romax Technology European Summit 2013

2

Overview of Romax application in Honda

Misalignment

Shaft

Gear

Carrier

Bearing

Housing

Original tool Damage

Contact Stress

Damage

Durability : Bearing

Contact Stress

Contact Stress

Damage

Durability : External Gear

Load

T.E

.

Transmission

Error Micro Geometry

Misalignment

Mis

alig

nm

ent.

Load

NV : Gear Whine

Durability : Internal Gear

Contents

・Background and objective

・Generation mechanism of uneven torque sharing

・Relational expression between manufacturing errors and torque sharing

・Verification of relational expression

・Verification of accuracy of parameter estimation

・Summary

4

Prediction of Strength of planetary Gears

Prediction of torque distribution necessary

for strength design of planetary gear sets.

The gear with the greatest torque break first.

・・・・・ × ×

×

Pro

bab

ilit

y

Statistics

Planet gear torque

Statistics of torque of each planet

Torque when torque sharing is equal

Problematic region at strength design

5

Method for Estimating Torque Distribution

Probability Density

Manufacturing error

Error 1

Error 2

…

Error N

Method①

Histogram

Planet gear torque

Num

ber

of tim

es

Simulation Sharing 1

Sharing 2

…

Sharing N

Input

Torque Sharing Error Pattern

output

N times calc.

Method②

Parameter

Estimation

1 time calc.

←Existing method, feasible, accuracy unknown, large amount of computation.

←Distribution formula, parameter estimation method, accuracy unknown, less computation.

Probability Density Function

;PTfp

Planet gear torque

Probability Density

Estimate torque distribution with the distribution formula

6

Objective & Issues

●Objective： To establish a method for estimating torque distribution using a distribution formula.

① Derivation of a relational expression between manufacturing error and torque sharing

（*n-planet）

② To establish parameter computation method

●Research question：

Manufacturing error distribution： Normal distribution

Relational expression between manufacturing error and torque sharing

Torque distribution formula

Precise & quick estimation of torque distribution

7

Equivalent transmission Error

Planet 1

x

y

Planet 2

Planet 3

Planet 4

Root circle of internal gear

Base circle of internal gear

Base circle of sun gear

Base circle of planet gear

Nominal position of planet 1

・Ex：pinion axis displacement

Planet 1

Planet 2

Planet 3

Planet 4

IPjSPjPjPj iid

IPji

SPji

・Equivalent transmission error：

Expressing the effect of manufacturing errors on the line of gear action

Projection

Displacement

Displacement : Pjd

Internal gear

Sun gear

Contact line

8

Stiffness Model

Ground

x

yPlanet 1

Planet 2

Planet 3

Planet 4

IPk

SPk

SPk

SPk

SPk

IPk

IPk

IPk

P1 P2 P3 P4 P1

P2

P3

P4

Clearance

Serial connection of

two synthetic springs

IPSP

IPSPP

kk

kkK

＊2D model of transverse plane

In contact Not in contact

Synthetic spring

Synthetic spring

Synthetic spring

SPk

IPk

Consideration of

clearance and

synthesis Sun gear

Internal gear

9

Relation between Manufacturing Error and Torque Sharing (1/3)

P1 P2 P3 P4

IPjSPjPjPj iid

Pj

Planet 1

x

y

Planet 2

Planet 3

Planet 4

Root circle of internal gear

Base circle of internal gear

Base circle of sun gear

Base circle of planet gear

Nominal position of planet 1

Displacement

Sun

Difference in clearance

IPji

SPji

Projection

Displacement : Pjd

Contact line

P1

P2

P3

P4

P1

Internal

Clearance differences due to pinion axis displacement

・Transmission error：

10

Relation between Manufacturing Error and Torque Sharing (2/3)

Planet 1

x

y

Planet 2

Planet 3

Planet 4

Planet gear position without carrier translation

In contact In contact

Not in contact

Not in contact

In contact In contact

Not in contact

Not in contact

P1 P2 P3 P4

IPjSPjPjPj iicc

Pjc Self-centering action

Sun

P1

P2

P3

P4

Internal

After self-centering

Effect of self-centering action

also expressed on the

transmission action line

11

Relation between Manufacturing Error and Torque Sharing (3/3)

P1 P2 P3 P4 P1 P2 P3 P4

F

Difference when meshing starts is difference in torque distribution

Px̂

PPP

PPP

PPP

PPP

xKf

xKf

xKf

xKf

ˆ

ˆ

ˆ

ˆ

4

3

2

1

IPSP

IPSPP

kk

kkK

Two planet gears begin to mesh

F

The other two begin to mesh

Gear-mesh

force：

Serial connection of

2 synthetic springs

Synthetic spring Synthetic spring

constant：

12

Equation of Equilibrium of Forces

Ground

x

yPlanet 1

Planet 2

Planet 3

Planet 4

IPk

SPk

SPk

SPk

SPk

IPk

IPk

IPk

0ˆ1

n

j

PjPPjPjPP iacxK

S

n

j

BSPjPjPP TrcxK 1

ˆ

PK

Equilibrium of forces：

Balance of input ant output torques：

IPjSPjP iia

PIPjSPjPj aiii

Equilibrium of force

P1 P2 P3 P4

Px̂

F

Equilibrium of input and output

Direction of synthesized

gear mesh forces:

（*n-planet）

13

Relational Expression between Manufacturing Error and Torque

Projected error： PjBPPBSBPSPj ErKnrrTT

nVUEn

h

PhPjPjPjPj

1

njuUPj 12cos2 π

njvVPj 12sin2 π

n

h

Ph nhu1

12cos π

n

h

Ph nhv1

12sin π

Torque of each planet gear is in proportion with projected error

Ideal

torque

Product of stiffness

& base circle radius

Y-intercept Slope

Projected error

Y-intercept Slope

Pin

ion

gear

torq

ue

Relation

Pinion

gear

torque

Linear sum of equivalent

transmission error

14

Projected error

Pin

ion

gear

torq

ue

Verification of Relational Expression

Equation for 4-Pinion：

Experimentally verified equation for 4-pinion

PjBPPBSBPSPj ErKrrTT 4 44321 PPPPPjE

44321 PPPPPjE （j = 1,3）

（j = 2,4）

Y-intercept Slope

・Verification method

① Experiments using

gears with different

manufacturing errors

③ Confirmation of

linearity and

Y-intercept

②Data plot

Projected error

Pinion gear torque

・・・

15

Sun gear ： input

Planetary carrier ：output

Internal gear ： fixed

Testing Rig

Test piece gear set Jig gear set

Motor

16

Test piece

Tested 6 carriers with pin hole intentionally displaced

・Pin hole displacement of test carriers [μm]

Carrier Planet 1 Planet 2 Planet 3 Planet 4

dx dy dx dy dx dy dx dy

A -1.5 40.2 0.5 -9.2 1.7 -3.3 -2.5 -5.4

B -13.4 -1.4 -6.9 5.7 -13.4 -17.1 -20.3 5.9

C -25.9 -2.0 -30.5 -8.6 -24.6 -16.5 -17.9 91.1

D -11.6 2.5 -23.6 89.9 89.5 -20.9 1.0 -108.8

E -24.7 1.1 -37.1 -12.3 -23.7 6.1 -11.8 -14.6

F -20.4 3.0 -19.8 3.1 -19.1 54.4 -20.1 4.2

P1

P2

P3

P4

x

y

Test piece

17

Method for Torque Prediction (1/2)

Torque on each planet gear estimated from internal gear strain

Strain of internal gear

Torque on

each planet gear

Estimate

18

Method for Torque Prediction (2/2)

Strain of internal gear

-200

-100

0

100

200

300

400

500

Time [sec]

Str

ain [

μst

]

118 Nm 235 Nm

Torque on each planet gear estimated

from the proportion of peak values and ratio

Phase [degree]

Str

ain

[μ

st]

Strain of internal gear for test carrier A

P2 P1 P3 P4

43214321 :::::: PPPPPPPP TTTT

Rate of strain peak Rate of planet gear torque

19

0

10

20

30

40

50

60

-30 -20 -10 0 10 20 30

Projected error ： EP1 [μm]

To

rqu

e ：

TP

1 [

Nm

]

118 Nm 157 Nm 196 Nm 235 NmL 118 Nm L 157 Nm L 196 Nm L 235 Nm

Verification of Relational Expressions

Validity of relational expression verified

by the experimental results.

Projected error [μm]

To

rqu

e [

Nm

]

Torque of planet 1

PjBPPBSBPSPj ErKrrTT 4

Input torque

@Sun

Equally distributed torque on each planet gear

118 22.6

157 30.1

196 37.6

235 45.0

Relational expression：

・Equally distributed planet gear torque [Nm] (Theoretical value)

20

Mean：

SD：

Torque distribution formula Torque distribution formula（With displaced pin holes）

Stiffness Kp needs to be computed for parameter estimation

222exp21 mTTf PjPj

BSBPS nrrTm

nnrKa PHBPPP 3

6bPH

Variables other than Kp can be obtained

from existing data.

←Normal distribution

PjBPPBSBPSPj ErKnrrTT Relational

expression：

b: Pin hole position accuracy φb

21

Projected error

Slope P

inio

n g

ear

torq

ue

Estimation

Method for Stiffness Estimation

PjBPPBSBPSPj ErKnrrTT

Stiffness Kp estimated by plotting the results of structural analysis

Relational expression：

Base circle

diameter

of planet gear

◆Procedure for stiffness Kp estimation

① Structural analysis

with RomaxDesigner

② Result plot

Manufacturing error input

③ Calculation of KP

t

t

Slope

BP

Pr

tK

Slope

Sun: IN

Carrier: OUT

Ring: FIXED

22

Torque of planet 1

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Measurement value [Nm]

An

aly

sis

valu

e [

Nm

]

□ 118 Nm

◇ 157 Nm

△ 196 Nm

○ 235 Nm

-5%

+5%

Confirmation of precision of estimated stiffness

Precise estimation of stiffness Kp by numerical analysis confirmed.

Calc

ula

tio

n v

alu

e [

Nm

]

Torque of planet 1

Measurement value [Nm]

Plot analytical results against experimental results

Input measured manufacturing error for structural analysis

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Summary 1. Derived a relational expression between

manufacturing error and the torque on each planet gear in a normal n-planet planetary gear set.

2. Verified the relational expression using experimental results.

3. With RomaxDesigner, Stiffness parameter Kp for distribution can be precisely estimated using existing data.

In conclusion, the distribution of torque on each planet gear in a planetary gear set originating from manufacturing errors can be estimated precisely and efficiently.

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