1、10FTM08AGMA Technical PaperCalculation 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 DresdenCalculation of Load Distribution in Planetary Gears
2、 for anEffective Gear Design ProcessDr.-Ing. Tobias Schulze and Dipl.-Ing. Christian Hartmann-Gerlach, DriveConceptsGmbH and Prof. Dr.-Ing. Berthold Schlecht, Technical University of DresdenThe statements and opinions contained herein are those of the author and should not be construed as anofficial
3、 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
4、 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 loaddist
5、ribution, flank pressure, root stress, transmission error and contact temperature.Copyright 2010American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314October 2010ISBN: 978-1-55589-983-73Calculation of Load Distribution in Planetary Gearsfor an Effective Ge
6、ar Design ProcessDr.-Ing. Tobias Schulze and Dipl.-Ing. Christian Hartmann-Gerlach, DriveConceptsGmbH and Prof. Dr.-Ing. Berthold Schlecht, Technical University of DresdenIntroductionFor the dimensioning of highly stressed toothingsthe analysis of load distribution and the definition oftooth flank m
7、odifications 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 i
8、s 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-dependen
9、t 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 ge
10、ar 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 cant be calcu-lated efficiency in an analytical way. They need tobe investigated with FE calculations or extendedmodel approac
11、hes.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 calcul
12、ationafter wards. The software MDESIGN LVRplanetallows 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 r
13、oughdiscretization of the used model. The load,pressure, root stress distribution and width loadfactor can be interpreted, Figure 1.Figure 1. Bad load distribution on flankFurthermore the program is giving suggestions howto modify the flanks for a well-balanced loaddistribution, see Figure 2.Figure
14、2. Good load distribution on flankTo 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, calculatio
15、n accord. Standards, loaddistribution analysis). In the background the task isefficiently solved with scientific establishedcalculation kernels and uniform interfaces. So a4fast and secure concept, dimensioning andcalculation of the machine element, the gear andthe whole drive train are possible. Th
16、ere 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 me
17、thod cant totallyreplace the measuring campaign and test runs butunnecessary iterations can be switched off.Applications of large gearboxesThe gearboxes described in this article are mostlyused in heavy drive trains, special purposemachinery and in wind power plants, Figure 3. Thecharacteristic of a
18、ll these operating areas areturbulent and unsteady loads, uncertain boundaryconditions and in some cases very high load peaks.Figure 3. Application wind power plantFor wind turbines the wind, the start- and stopprocedure are the most important dynamic inputparameter and also the biggest unsureness f
19、orload conditions. By the mills there is the same 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-ings, Figure 5.In all of these operation areas there are
20、 highrequirements to design, optimization andcalculation of drive trains and especially gearboxes.Figure 4. Application cement millFigure 5. Application mining truckAnother example of a large gearbox with high poweroutput and cyclic loads is the foundry crane inFigure 6 for transportation of melted
21、mass.Figure 6. Application foundry crane5Basics of load distributionThe load measures in tooth contact are caused bydistribution of load on tooth pairs ( 1: profile loaddistribution KH, KF) and load distribution alongface width (lead load distribution KH, KF). The cal-culation method of load distrib
22、ution is based onusing deformation influence numbers, Figure 7.The deformation coefficients aikof influence coeffi-cients method are the basic of load distribution. Theinfluence coefficients aikare the absolute values ofdeformation in section i, which are the result of theforceinsectionk, in relatio
23、n 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 distribution dependson the numbers of normal planes and accuracy ofinfluence numbers.y1= F111+ F212(1)11=YE1F
24、E(2)12=YE2FE(3)where11influence coefficient at position 1 becauseof force at position (see Figure 7);12influence coefficient at position 1 becauseofforceatposition2(seeFigure7);YE1deformation at position 1 because of unitforceatposition1(seeFigure7);YE2deformation at position 1 because of unitforcea
25、tposition2(seeFigure7).General system of equationThe system of equation 4 must be solved to get theload distribution in form of single forces Fi.fz= AF (4)At the beginning of the calculation the vector ofcomplete tooth deformation fzand force vector Fare unknown. The system of equation can only beso
26、lved iteratively. Therefore the complete toothdeformation fzhas to be varied until the sum ofsingle forces Fiand complete tooth normal force Fbnare equal (equation 5).Fbn=Fi(5)The calculation of Fbnis done with equations from16. In the coefficient matrix A the known deforma-tion influence numbers ai
27、jare concentrated. Thenumbers of influence are calculated on experiment-al or numeric methods. For one gear pair thecoefficients aijare substituted with the sum ofdeformation numbers bij(equation 6) of tooth andopposite tooth.Figure 7. Principles of influence coefficientsbij= aI,ij+ aII,ij(6)The coe
28、fficients aI,ijand aII,ijare the values for tooth1 and tooth 2. The vector fzincludes the resultantcomplete tooth deformation fz,ges. Because of fz,geshas the same value in every face plane iand in everycontact position, for all elements of fzfollows(equation 7).f =111fz,ges(7)The complete tooth def
29、ormation fzof 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,gesis the6result of elastic tooth deformation fzI,iand fzII,ifrompinion and gear (equation 8).fz,i= fzI,i+ fzII,i= fz,ges= co
30、nst. (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 completesy
31、stem 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 dis
32、placements can be calculatedindependently from load distribution. If anadditional contact line deviation fkis 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
33、 expressed with the part gi. The elast-ic tooth deformations fzI,iand fzII,iare replaced in thesystem with force vector F and coefficient matrix A.So an enlarged system of equation 10 is built.fz,ges= AF+ fk g (10)When the result of the system of equation is anegative single force Fi, it means that
34、this sectiondoesnt contact under complete tooth force and aresultant flank deviation giexists. 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 th
35、e system ofequation 11 remains. One row of the system ofequation for one gear pair can be expressed(equation 12) according to Figure 7.AF= fz,ges fk(11)nk=1aikFk= fz,ges fki(12)Enlarged system of equation: If more results ofHohrein/Senf 14, 15 are used, the system ofequation 13 for F and fzcanbesolv
36、eddirectly.A 1 1 111 0F1F23.Fnfz,ges= fkFbn(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 alg
37、orithm, 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 di
38、stribution 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 di
39、stribution. As aforemen-tioned, the negative single forces Fibecause 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 a
40、s 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 00 0 Ap00 0 An(15)Flank modification
41、sThe 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.72. The residual linear variable part of contact linedeviation depending on load, thermic a
42、nd 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 leadcrowni
43、ng.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
44、gear setsThe 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 K. An additionalproblem is the load distribution alon
45、g 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 areoverlying independentl
46、y, 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 comp
47、lex 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 K
48、H. 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. Veri
49、fication of planetary gearstagesThe 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 (fHeff);S elastic deformation of tooth flank;S elastic deformation difference of planet carrierbearing;S deformation of housi
