10FTM08

AGMA Technical Paper

Calculation of LoadDistribution in PlanetaryGears for an EffectiveGear Design ProcessBy Dr.-Ing. T. Schulze andDipl.-Ing. C. Hartmann-Gerlach,DriveConcepts GmbHand Prof. Dr.-Ing. B. Schlecht,Technical University of Dresden

Calculation of Load Distribution in Planetary Gears for anEffective Gear Design Process

Dr.-Ing. Tobias Schulze and Dipl.-Ing. Christian Hartmann-Gerlach, DriveConceptsGmbH and Prof. Dr.-Ing. Berthold Schlecht, Technical University of Dresden

[The statements and opinions contained herein are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.]

AbstractThe design of gears - especially planetary gears - can just be carried out by the consideration of influences ofthe whole drive train and the analysis of all relevant machine elements. In this case the gear is more than thesum of its machine elements. Relevant interactions need to be considered under real conditions. Thestandardized calculations are decisive for the safe dimensioning of the machine elements with theconsideration of realistic load assumptions. But they need to be completed by extended analysis of loaddistribution, flank pressure, root stress, transmission error and contact temperature.

Copyright 2010

American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314

October 2010

ISBN: 978--1--55589--983--7

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Calculation of Load Distribution in Planetary Gearsfor an Effective Gear Design Process

Dr.--Ing. Tobias Schulze and Dipl.--Ing. Christian Hartmann--Gerlach, DriveConceptsGmbH and Prof. Dr.--Ing. Berthold Schlecht, Technical University of Dresden

Introduction

For the dimensioning of highly stressed toothingsthe analysis of load distribution and the definition oftooth flank modifications belongs to the principaltasks. Similar problems appear at the evaluation oftoothing damages and failure modes of wholegears. Although there are a large number ofstandards for the calculation of spur gears, ISO6336. It is necessary to have special and powerfulcalculation software which is reflecting the force--deformation--relation for every point of the contactarea more precisely. The cause is the divergencefrom the conjugated toothings at gear wheels ofspur and planetary gears. Often the flanks are mod-ified in height and width direction. With these modi-fications the load--dependent deformations of thetoothing and the surroundings as well as toothingerrors, position errors of the housing boreholes andbearing clearance can be compensated. Also thegear noise as well as the load capacity is influencedin a positive way.

For the determination of the load distribution inplanetary gear stages the deformation analysis is amore complex task as for spur gear stages. Thedeformation of the wheel body as well as the adja-cent structures and the planet carrier can’t be calcu-lated efficiency in an analytical way. They need tobe investigated with FE calculations or extendedmodel approaches.

With a detailed load distribution analyses software/7/ all necessary calculations for the load analysis ofplanetary gears are united. The relevant deforma-tions are determined with automatically generatedFE meshes of the gear wheels and planet carrierand are used for the load distribution calculationafter wards. The software MDESIGN LVRplanet

allows the load distribution calculation of planetarystages with spur, helical and double helical gearwheels. Therefore analytical functions for the con-tact area are in use. As result you can see the fastcomputing time despite of the high--resolution rough

discretization of the used model. The load,pressure, root stress distribution and width loadfactor can be interpreted, Figure 1.

Figure 1. Bad load distribution on flank

Furthermore the program is giving suggestions howto modify the flanks for a well--balanced loaddistribution, see Figure 2.

Figure 2. Good load distribution on flank

To solve all this calculation tasks in an efficient way itis necessary to install calculation software. Toreduce the necessary inputs to a minimum it is indis-pensable to have a high degree of connectionsbetween the single calculation modules (geardesign, calculation accord. Standards, loaddistribution analysis). In the background the task isefficiently solved with scientific establishedcalculation kernels and uniform interfaces. So a

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fast and secure concept, dimensioning andcalculation of the machine element, the gear andthe whole drive train are possible. There is the pos-sibility to optimize toothing, shafts, bearings andbolts with use of real load assumption. Using thiscalculation software already at an early point of theproduct life cycle, PLC, you can get secure state-ments of your finish product without manufacturingprototypes. This calculation method can’t totallyreplace the measuring campaign and test runs butunnecessary iterations can be switched off.

Applications of large gearboxes

The gearboxes described in this article are mostlyused in heavy drive trains, special purposemachinery and in wind power plants, Figure 3. Thecharacteristic of all these operating areas areturbulent and unsteady loads, uncertain boundaryconditions and in some cases very high load peaks.

Figure 3. Application – wind power plant

For wind turbines the wind, the start-- and stopprocedure are the most important dynamic inputparameter – and also the biggest unsureness forload conditions. By the mills there is thesame fact interms of flow of material, Figure 4.

A very special case study is the drive train of bigmining trucks with high power output up to 700 kWand cyclic loads for transportation in surface min-ing’s, Figure 5.

In all of these operation areas there are highrequirements to design, optimization andcalculation of drive trains and especially gearboxes.

Figure 4. Application – cement mill

Figure 5. Application – mining truck

Another example of a large gearbox with high poweroutput and cyclic loads is the foundry crane inFigure 6 for transportation of melted mass.

Figure 6. Application – foundry crane

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Basics of load distribution

The load measures in tooth contact are caused bydistribution of load on tooth pairs (εα > 1: profile loaddistribution KHα, KFα) and load distribution alongface width (lead load distribution KHβ, KFβ). The cal-culation method of load distribution is based onusing deformation influence numbers, Figure 7.

The deformation coefficients aik of influence coeffi-cients method are the basic of load distribution. Theinfluence coefficients aik are the absolute values ofdeformation in section i, which are the result of theforce in section k, in relation to the single force insection k. It applies that aik = aki.

This solver algorithm using influence coefficients isthe most effective way to calculate the loaddistribution. The quality of load distributiondependson the numbers of normal planes and accuracy ofinfluence numbers.

y1= F1 α11 + F2 α12 (1)

α11=YE1FE

(2)

α12=YE2FE

(3)

where

α11 influence coefficient at position 1 becauseof force at position (see Figure 7);

α12 influence coefficient at position 1 becauseof force at position 2 (see Figure 7);

YE1 deformation at position 1 because of unitforce at position 1 (see Figure 7);

YE2 deformation at position 1 because of unitforce at position 2 (see Figure 7).

General system of equation

The system of equation 4 must be solved to get theload distribution in form of single forces Fi.

fz= A F (4)

At the beginning of the calculation the vector ofcomplete tooth deformation fz and force vector Fare unknown. The system of equation can only besolved iteratively. Therefore the complete toothdeformation fz has to be varied until the sum ofsingle forces Fi and complete tooth normal force Fbnare equal (equation 5).

Fbn= Fi (5)

The calculation of Fbn is done with equations from[16]. In the coefficient matrix A the known deforma-tion influence numbers aij are concentrated. Thenumbers of influence are calculated on experiment-al or numeric methods. For one gear pair thecoefficients aij are substituted with the sum ofdeformation numbers bij (equation 6) of tooth andopposite tooth.

Figure 7. Principles of influence coefficients

bij= aI,ij+ aII,ij (6)

The coefficients aI,ij and aII,ij are the values for tooth1 and tooth 2. The vector fz includes the resultantcomplete tooth deformation fz,ges. Because of fz,geshas the same value in every faceplane iand ineverycontact position, for all elements of fz follows(equation 7).

f=⎪⎡⎣11⋮1⎪⎤⎦

fz,ges (7)

The complete tooth deformation fz of one gear pairresults from a load and leads to the total transmis-sion error of booth gear wheels. In the simplest way-- only contact on the contact line -- the fz,ges is the

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result of elastic tooth deformation fzI,i and fzII,i frompinion and gear (equation 8).

fz,i= fzI,i+ fzII,i= fz,ges= const. (8)

In the simplest way the contact line deviationconsists of the existing deviation of the designedflank and eventually added modifications. Thecontact line deviation is calculated for one positionof contact and unencumbered load case, so it is aconstant initial value for solving the completesystem of equation. The current values arecombined in vector fk, the contact line deviation. Inthe advanced form all existing deformations anddisplacements, i.e. deflection of shaft, bearingclearance, excepting deformation of numbers ofinfluences, are considered in vector fk. Thesedeformations and displacements can be calculatedindependently from load distribution. If anadditional contact line deviation fk is superposed onelastic deformations, equation 8 must be changedto equation 9.

fz,i= fzI,i+ fzII,i+ fk,i− gi= fz,ges= const.(9)

The resultant deviation of the gear pair after elasticdeformation is expressed with the part gi. The elast-ic tooth deformations fzI, i and fzII, i are replaced in thesystem with force vector F and coefficient matrix A.So an enlarged system of equation 10 is built.

fz,ges= A F+ fk− g (10)

When the result of the system of equation is anegative single force Fi, it means that this sectiondoesn’t contact under complete tooth force and aresultant flank deviation gi exists. The system ofequation is compressed and solved again afterdeletion of row i and column for every negative sec-tion force. The operation is repeated until no negat-ive single forces exist. After deletion the system ofequation 11 remains. One row of the system ofequation for one gear pair can be expressed(equation 12) according to Figure 7.

A F= fz,ges− fk (11)

nk=1

aik Fk = fz,ges− fki (12)

Enlarged system of equation: If more results ofHohrein/Senf [14, 15] are used, the system ofequation 13 for F and fz can be solved directly.

⎪⎪⎪

⎡

⎣

⎪⎡

⎣A ⎪⎤

⎦

− 1− 1⋮− 1

1 1 ⋅⋅⋅ 0

⎪⎪⎪

⎤

⎦

⎪⎪⎪⎪

⎡

⎣

F1F2

3.

Fnfz,ges

⎪⎪⎪⎪

⎤

⎦=⎪⎪⎡

⎣⎪⎡⎣− fk⎪⎤⎦Fbn

⎪⎪⎤

⎦(13)

The matrix of deformation numbers of influence Aconsists of the matrix of Hertzian deformation AHand the matrix of the resultant tooth deformation AZ(equation 14).

A= AH+ AZ (14)

A directly solving of the system of equation 13,respectively the solving algorithm, is only possibleby linear interrelations between load and deforma-tion. Non linear variables (i.e., Hertzian pressure)are linearized or considered in the system ofequation by iterative operations. To include the loaddepending Hertzian influence on deformationexactly, the algorithm of load distribution has to bedone in one position of contact several times. Afterevery step of iteration the Hertzian stiffness isadapted to the actual load distribution. The iterativecalculation is also necessary to involve deformationof surrounding from resilience of surrounding toachieve a correct load distribution. As aforemen-tioned, the negative single forces Fi because offlank deviation are deleted by compressing thesystem of equation.

If at the same time more tooth pairs n are in contact,the system of equation 13 is enlarged and the coeffi-cient matrix Ap (equation 15) of one tooth pair p isseen as matrix of discontinuity. If in this equationalso the null terminated matrices are reserved theopposite influence (cross influence) of neighboringtooth pairs can be shown. When using solid wheelsthe cross influence is disregarded and the teeth areuncoupled.

A=⎪⎪⎡

⎣

A1 0 0⋮ ⋮ ⋮ ⋮0 Ap 00 0 An

⎪⎪⎤

⎦(15)

Flank modifications

The general way to use flank modifications isdescribed by the following 3 steps:

1. Load depending deformations should beinfluenced by design arrangements in this waythat they are reduced or maybe compensated.

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2. The residual linear variable part of contact linedeviation depending on load, thermic and centri-fugal force has to be compensated by helicalflank modification.

3. The contact line deviation, which balancesaround expected value 0, caused by measure-ment deviation of gears, deviation of gearwobbling and other raisings of load on the faceside has to be reduced by an additional leadcrowning.

Figure 8 and Figure 9 show some possiblemodifications on the tooth flank.

Figure 8. Selection of flank modifications(left: helix angle modification --

right: lead crowning)

Figure 9. Selection of flank modifications(left: tip-- and root relief --

right: topological modification)

Calculation of planetary gear sets

The advantage of planetary gears is the division ofpower to the planets so that a high density of powercan be achieved. Due to manufacturing deviationsthe load distribution on the planets is not exactly,which is expressed with the factor Kγ. An additionalproblem is the load distribution along face width, asmentioned above.

The calculation of load distribution in a planetarygear system essentially depends on the helix angeldeviation between the contact flanks of the gearpairs. It can be understood as the sum of differentinfluences. It is assumed that the effects are

overlying independently, the sum of contact linedeviation can be calculated with the singledeviations. The calculation of single displacementsand deformations of all gear box bodies – especiallythe planet carrier, the coupling of ring gear and gearwheel bodies and the deformation of teeth – is inplanetary gearboxes more complex than in spurgearboxes.

To determine the load distribution the flank deviationfor the tooth contact sun/planet and the toothcontact planet/ring gear is calculated by the newsoftware MDESIGN LVRplanet. The result of thecalculation is the excessive of the line load, which isexpressed by the factor KHβ. In general theexcessive of the line load is on the flank sideopposite to the deviated flank side.

Next to the calculation of the ratio of maximum andmiddle line load the software gives detailed informa-tion about tooth flank pressure and tooth root stressdistribution. See Figure 10.

Figure 10. Verification of planetary gearstages

The flank deviation (FLKM) consists of followingparts:

S elastic deformation of gear body (veRK);

S elastic tilting difference of roller bearings [13](veWL);

S torsion deformation of planet carrier (vePT);

S tilting of planet because of sliding bearing(verkippPL);

S effective helix angle modification (fHβeff);

S elastic deformation of tooth flank;

S elastic deformation difference of planet carrierbearing;

S deformation of housing.

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The helix angle deviation for tooth contact sun/planet is calculated by equation 16.

FLKM1∕2= ve1+ ve21∕2+ veWL1∕2+ vePT1∕2

+ verkippPL1∕2+ fHβeff1∕2(16)

The helix angle deviation for tooth contact planet/ring gear is calculated by equation 17.

FLKM2∕3= ve22∕3+ ve3+ veWL2∕3+ vePT2∕3

+ verkippPL2∕3+ fHβeff2∕3(17)

where

ve1 deformation difference of sun;

ve2 deformation difference of planet;

ve3 deformation difference of ring gear.

The deformation is calculated by the FE--methodand after wards it is added to the flank deviation. Allparts of the helix angle deviation have to be addedas values normal to the flank.

The database of the calculation is saved in XML--Format. With this a structured depositing ofdesign--, modifications--, deviation--, load-- andcontrol data is possible. Furthermore the programhas a project management which is based on adatabase to save projects, for standard examplesand more calculation guidelines.

After input of all necessary parameters all data arechecked, the design models are generated and theFE--models for the gears with coupling design andthe planet carrier are created. Figure 11 showssome design variants of planet carriers.

Figure 11. Design variants of planet carrier

For an efficient calculation it is necessary andreasonable to use software. DriveConcepts GmbHdevelops software solutions for drive technology,which is characterized by clear and intuitivehandling of all data.

In the background academic establishedcalculation kernels and consistent structuredinterfaces help to solve the actual task efficiently.

Gearbox design

Especially for design concepts of planetary andspur gearboxes the newest development of Drive-Concepts GmbH the product MDESIGN gearbox isestablished, see Figure 12. This calculationsoftware gives complete product information in theearly phase of product life cycle, PLC. Thecalculation can’t replace measurements and testdrives, but iteration steps can be reducedeconomically.

Figure 12. Design and optimization ofplanetary gear boxes

The software allows an intuitive and easy handlingin the design process of whole gearboxes from thedimensioning of the machine elements – shafts,bearings and toothings -- according to the actualstandards. See Figure 13.

Figure 13. Gearbox model of a multi stagespur gear in the software MDESIGN gearbox

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In a second step the consideration of CADgeometry data of housings will be possible. There-fore the software imports a standard geometryformat, generate Finite Element models, calculatestiffness matrices for the housing and deliver thisinformation to the design process of MDESIGNgearbox. See Figure 14.

Figure 14. Selection of interfaces betweenbearing and housing

Case study

The example of a wind turbine with 2000 kW outputpower should show the consequences of differentflank modifications with constant load. SeeFigure 15.

Figure 15. Case study – wind power plant

The main gearbox consists of three gear stages(helical gearing). The detailed parameters of the 1st

planetary gear stage are listed in Table 1. Additionalparameters (e.g., the bearing characteristics,carrier pin tolerances) are not shown, but they had

to consider for realistic determination of loaddistribution. See Table 1.

Table 1. Parameters of planetary gear stage

Module m 16 mm

Number of teeth z1/2/3 20/36/--91

Center of distance a 463 mm

Pressure angle α 205

Helix angle β 85

Face width sun b1/2/3 310 mm

Add. modification sun x1 0,4

Add. modification planet x2 0,3156

Add. modification ringgear

x3 --1,6429

The next figures show for one special time situationwith a typical load the possibilities of load condi-tions. In Figure 16 the initial state of unmodifiedgearing under nominal load is shown (axis “L”means “line of contact”). In this case the loadintensity ratio of line load is 1,67.

Figure 16. 1st gear stage -- original state

In the first step with a helix flank modification thefactor can be reduced to KHβ =1,23, see Figure 17.

Figure 17. 1st gear stage – 1st optimization

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The rest of unbalanced distribution along the facewidth, which comes from planet carrier torsiondeformation, can be compensated with an optimallead crowning. The load intensity ratio can bereduced to KHβ =1,16, Figure 18.

Figure 18. 1st gear stage – final design

Figure 19 shows that an oversized lead crowningcan also lead to poor conditions. In this case thelead load distribution changes to KHβ =1,98.

Figure 19. 1st gear stage – bad solution withtoo much lead crowing

The example shows the must oft the right dimensionof used modifications. If these are right the leadload distribution KHβ can be reduced from 1,67 to1,16. But with unfavorable modifications theopposite will be achieved.

The case study shows many advantages ofintegrated software, like MDESIGN LVRplanet, todevelop gearboxes.

References

1. Börner, J. Senf, M.: Verzahnungsbeans-pruchung im Eingriffsfeld – effektiv berechnet.Antriebstechnik 1, 1995

2. Börner, J.: Very efficient calculation of the loaddistribution on external gear sets – the methodand application of the program LVR. Interna-tional ASME Conference, San Diego, 1996

3. Hartmann--Gerlach, Christian: Verformungsan-alyse von Planetenträgern unter Verwendungder Finiten Elemente Methode. Internal draft,TU Dresden 2008

4. ISO 6336:2006 Calculation of load capacity ofspur and helical gears

5. Linke, H.: Beitrag zur Ermittlung der Zahn-flanken-- und Zahnfuβtragfähigkeit unter Ber-ücksichtigung der Abweichungen geomet-rischer Gröβen, Deformation der Getriebeteileund der Werkstoffkennwerte. Habilitationss-chrift TU Dresden, 1978

6. MDESIGN LVR 2010, software for loaddistribu-tion of multi stage spur-- and helical gears.DriveConcepts GmbH, 2010

7. MDESIGN LVRplanet 2010, software for loaddistribution of planetary gear stages. Drive-Concepts GmbH, 2010

8. MDESIGN gearbox 2010, design and calcula-tion software for multi stage gearboxes. Drive-Concepts GmbH, 2010

9. Schlecht, B. Senf, M.; Schulze, T.: Beans-pruchungsanalyse bei Stirnradgetrieben. An-triebsstränge in Windenergieanlagen -- Hausder Technik e.V., Essen, o9./1o. März 2010

10. Schulze, Tobias: Ganzheitliche dynamischeAntriebsstrangsbetrachtung von Windener-gieanlagen. Sierke Verlag 2008, PhD thesis TUDresden

11. Schulze, Tobias: Getriebeberechnung nach ak-tuellen wissenschaftlichen Erkenntnissen, Vor-trag anlässlich des Dresdner Maschinenele-mente DMK2007 in Dresden, DriveConceptsGmbH, 2007

12. Schulze, Tobias: Load distribution in planetarygears under consideration all relevant influ-ences. JSME International Conference on Mo-tion and Power Transmissions May 13--15,2009, Matsushima Isles Resort, Japan

13. Schulze, Tobias: Load distribution in planetarygears. Danish gear society “Gearteknisk In-teresseGruppe” 11th february 2010 at SDU inOdense, Denmark

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14. Wiche, E.: Radiale Federung von Wälzlagernbei beliebiger Lagerluft. Konstruktion, Berlin19(1967)5

15. Hohrein, A.; Senf, M.: Untersuchungen zurLast-- und Spannungsverteilung an

schrägverzahnten Stirnrädern. Diss. TUDresden, 1978

16. Linke, H.: Stirnradverzahnung – Berechnung,Werkstoffe, Fertigung. München, Wien:Hanser, 1996