MeccanicaDOI 10.1007/s11012-013-9830-8

Reliability for design of planetary gear drive units

Milosav Ognjanovic · Miloš Ristic ·Predrag Živkovic

Received: 26 January 2013 / Accepted: 22 October 2013© Springer Science+Business Media Dordrecht 2013

Abstract Gear drive units are important componentsof technical systems (TS) and need to be of high qual-ity. Planetary gear units are very compact and efficientmechanical power transformers, but further increase ofoperating quality level requires the application and de-velopment of the new design methodology. The sub-ject of this contribution is presentation of Reliabilityfor design as the new approach of reliability mod-elling suitable for the new design methodology appli-cation, especially for planetary gear units using vari-ous kinds of experimental and exploitation data. Themethodology follows V-model for TS design whichis in this work adapted for gear units design and forpresentation of the new methodology based on prop-erty based design, axiomatic design and robust designmethodology. To this end, the procedure for total reli-ability of TS decomposition, and methodology for ele-mentary reliability for design of structure componentscalculation is developed and presented. The reliabil-ity for design is established in reverse form of reli-ability for maintenance which presents common per-ception of the “reliability” term. This approach is in-tended to provide further increase of planetary gear

M. Ognjanovic (B) · M. RisticFaculty of Mechanical Engineering, Universityof Belgrade, Belgrade, Serbiae-mail: mognjanovic@mas.bg.ac.rs

P. ŽivkovicFaculty of Technical Sciences, Kosovska Mitrovica, Serbia

unit’s quality and efficient usability of gear unit com-ponent resources. The design directions are oriented toproviding equal level of elementary reliability of com-ponents.

Keywords Planetary gear units · Reliability ·Experiments · Design

1 Introduction

Current research trends in the field of technical sys-tems design offer a set of new methods that enableefficient design process and technical systems devel-opment of high quality level. For the purpose of cre-ating the arrangement of these methods application,by VDI-2206 standard, a V-model for technical sys-tems (TS) design is proposed. The model provides thepossibility for applying robust design methodology inorder to design parameters (DP) definition insensitiveto service conditions variation and to provide satis-factory results at first attempt. Axiomatic methods arealso compatible with robust approach and when com-bined provide efficient DP definition. Design for Xmethodology is oriented to a certain design propertyor requirement and can be involved in V-model of de-sign process. The application of those various meth-ods implies decomposition of design structure and de-sign properties and also integration in reverse direc-tion. The procedure of TS and its property decompo-sition and integration is developed and presented in

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the article [1]. This is the base for the property-baseddesign methodology application. Numerous examplescan be cited to support this approach, such as [2–7]. Inthis process, the reliability of TS and its componentscan be used as a functional requirement (FR) of TS,then as a design property of TS components and asan indicator of TS quality. The common meaning ofreliability term is broad and connected to the neces-sity of TS maintenance. These are the predictions ofTS malfunction, causes of malfunctions, arrangementof a certain component replacement and fixing com-ponents, or its relation. For the purpose of TS design,reliability modelling and application has to be orientedto the TS creation. Various approaches are developedin order to create the relation between TS or compo-nent reliability and some of a TS parameter or propertyin order to establish its relation that can support the de-sign process of TS [8–16]. Suggested approaches arepredominantly oriented to solving a certain type of de-sign problem without possibilities to extend generali-sation to all problems and to all types of TS’s. Also,it is important to notice that Reliability for TS mainte-nance has a long tradition and for the purpose of its re-liability monitoring in exploitation, modelling and ap-plication various methods are developed. All of themare intended for identification, estimation or predic-tion of TS malfunctions or reliability level, using ana-lytic, probabilistic or numerical simulation methodol-ogy. They do not show clearly the interaction betweenthe main causes that produce exploitation conditions(load probability) and failure probability of TS com-ponents. Also, it is not possible to extract them, usingthese models, by decomposition of the desired relia-bility of components. These probabilities are the basefor DP definition for TS design with a desired relia-bility level. The term “Design for reliability” is con-tained in the field of TS design as a part of a specificgroup of design methodologies “Design for X”, how-ever this area also treats TS reliability during exploita-tion segment of the TS life cycle, so that informationimportant for design can be collected. In general, re-liability is not directly related to DP of TS definition.Also, the load probability and failure probability arenot clearly divided and related in the form to extractthese probability components in a reverse approach.When gear transmission units are not considered inparticular, they are usually treated in reliability modelsas a component of wider TS. In this field, there muchmore research works about gear teeth mesh, teeth fail-ures, bearing’s inside relations and failures, gear unit

applications etc. [17–30]. This is the reason why it isnecessary to establish a new specific approach in theform of Reliability for design which eliminates men-tioned deficiencies. This paper defines it as a separatesub-field of reliability inverse oriented to reliability formaintenance.

Methodology for gear transmission unit’s devel-opment and design requires provision and use of alarge amount of experimental data. However, the scopeof the volume of this data is smaller compared withthe empirical approach to the TS design. Also, suchmethodology allows avoiding a lengthy and often im-practicable testing of the reliability of complete geardrive unit. Using the suggested Reliability for designapproach, reliability of design components and over-all reliability is calculated and tests can be carried outin order to prove the established models only. Theobjective of this article is to establish the procedurefor methodology of Reliability for design definitionof gear drive units, especially of planetary gear driveunits, which will be applicable in the form of designproperty and design constraint, functional requirementand gear drive unit quality indicator. Developed designmethodology of gear drive unit components with ap-plication of these elementary reliabilities is partly pre-sented in articles [6] and [11], but complete presenta-tion is the subject of a separate paper. The numericexample presented in this work is intended to ver-ify the calculation of elementary reliability for designand effects of reliability level on design parameters ofgears and bearings. The support to this approach canbe found in the V-design model. The general form ofthis model had to be transformed and adapted. For-mation of V-design model, suitable for use in the de-sign of gear transmission units, and then decomposi-tion of planetary gear drive unit structure and overallreliability to the level of gear unit components (com-ponent property), and then its integration are all givenin the first part of this paper. Elementary reliability ofTS components will support the application of prop-erty based design, axiomatic design and robust designmethodology application. The complex planetary geardrive unit structure is used as a case study that can in-spire discussions about various design problems of theTS. This is a planetary gear drive unit of bucket wheelexcavator applied at open pit coal mine. The secondpart of the article contains an example of elementaryreliability for design calculation and analysis of the ef-fects of this reliability varying in the design parame-ters.

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Fig. 1 V-model of the design process for gear transmissionunits design

2 V-model for gear drive units design

The general V-design model shows that the processstarts from necessary functional requirements of TSand finishes with the identification of TS quality andbehaviour indicators. The design procedure in thismodel, established by VDI-2206, contains a blockabout TS architecture, then a block about design of TScomponents, and finally a block about TS integration.By [1], the V-model is adapted to property-based de-sign procedure. Starting from functional requirementsand to end up with indicators of the system behaviour,20 steps in the design of TS are defined including thedefinition of properties and characteristics of TS com-ponents. This form of the V-model represents generalproperty based design methodology of TS. Variousproperties of TS and its components can be the subjectof processing, using this procedure. For the purpose ofgear transmission units design, in Fig. 1 is presentedthe solution of V-model which includes the three maingroups (blocks) of these activities. By this solution ofa model, the authors intend to bring in relation the ex-perience and results of various gear drive units designwith actual new methodology in TS design and devel-opment.

The first block of standardised V-model, accordingto VDI-2206, contains TS architecture which involvesTS structure creation and structure decomposition. Inthe case of gear transmission units the term of TS ar-chitecture contains various combinations of gear pairs

or planetary gear sets, shafts, bearings etc. In orderto define design parameters (DP) of structure compo-nents, the first step is decomposition (Fig. 1). Struc-ture decomposition means disassembly of conceptualdesign to the level of components suitable for DP def-inition or to the level of components which have tobe replaced in complete in maintenance (reparation)process. Then, decomposition contains analytic hier-archy processing of gear unit functional requirementsand service properties to the level of structural com-ponents. This means the hierarchy processing of gearunit load capacity, speed of rotation, desired overallreliability, limited overall vibration and noise, limitedlevel of temperature etc. The next block in V-model(Fig. 1) is the design of structure components i.e. de-sign parameters (DP) definition, according to decom-posed functional requirements (FR) and desired oper-ation properties which can be used as limitations i.e.design constraints. Axiomatic design methodology issuitable to bring in relation DP with FR. The axiomsof independence and axiom of information minimi-sation are extremely important in creating calculationmodel efficient. Standardised gear calculation method-ology is suitable for adapting for these axioms appli-cation. Also, it is possible to apply this approach inshafts and bearing calculation. Axiomatic methodol-ogy is also suitable for inclusion of stochastic vari-ation of exploitation conditions and stochastic vari-ation of structural components characteristics. Theseparameters basically relate reliability of components.Using elementary reliability as a design constraint inaxiomatic relations makes identified design parame-ters get robust i.e. insensitive to stochastic parametersvariation. This combination leads to the developmentof integrated methodology of the gear transmissionunits design. The thread block of V-model in Fig. 1contains the gear unit component and properties in-tegration. The direction is opposite to decomposition.The process contains the creation of design modelsand drawings including DP harmonisation in order toobtain a higher level of compatibility and maximumof operation properties. Using the same models uti-lized for decomposition, integration in opposite direc-tion of overall reliability, vibration, noise etc., usingcomponent properties, is performed. The result is gearunit behaviour indicated by corresponding indicatorsof behaviour such as overall reliability level, vibrationlevel, noise level, etc.

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3 Reliability for design of gear drive units

Reliability is a very broad term and implies the prob-ability of any malfunction of the TS, over the wholeperiod of exploitation. Reliability defined in this formis suitable for use in TS maintenance process. How-ever, this is a very broad term and is unsuitable formodeling, and model results deviate significantly fromthe actual results obtained in exploitation. For the pur-pose of design requirements this term and its mean-ing has to focus on certain failures and conditions im-portant for realization of DP definition methodology,i.e. axiomatic and robust design methodology appli-cation. To this end, the next several problems in thiswork have to be solved: (a) to identify the features ofelementary probability suitable to be used as designconstraints in axiomatic and robust design approach,(b) to define procedure and rules of design structureand overall (total) reliability of gear transmission unit(TS) decomposition, and (c) to define methodology forelementary reliability application in axiomatic and ro-bust design procedure i.e. integrated methodology forgear transmission units design.

3.1 Features of elementary reliability

The main feature of elementary reliability is that it hasto be competent to determine the component DP by ap-plying axiomatic and robust methodology. This meansthat elementary reliability has to be focused on a cer-tain possible failure in the component. Furthermore, ithas to be combined of influences and its probabilitiesin order to bring in relation failure causes and DP’s.Relations have to be clear and simple not for academicpresentation and understanding only, but for axiomaticapproach rules satisfaction. These are the reasons thiselementary reliability is defined as the complex prob-ability composed of the two probabilities, operatingstress (load) probability and failure probability causedby this stress. Since in the course of operating life ofa component, the operating stress varies, unreliabilitycan be calculated as

Fp =k∑

i=1

piPFi; pi = nΣi

nΣ

; PFi = 1− e−(

σHiη

)β

(1)

where i = 1, . . . , k the stress level, pi—probabilityof the operating stress appear, PFi—failure proba-

Fig. 2 Relation between gear operating stress spectrum and therange of the teeth flanks failure probability distribution

bility under certain operating stress σHi , η and β—parameters of Weibull’s distribution function. In Fig. 2graphically is presented the calculation process of el-ementary reliability of a certain gear according toEq. (1). This is the relation between gear flank stressspectrum and the range of pitting failure probabilitydistribution range.

Equation (1) is based on Weibull’s function of fail-ure probability distribution with the two parameterswithout the thread, the time parameter. The reasonsare the next. This function does not refer to reliabilityand presents failure probability only, which is timelyindependent. Failure probability is associated with thestress cycles number until failure, which is the resultof laboratory testing independent of time duration. Ex-ploitation conditions and time duration are involvedby pi that includes stress cycles number nΣi result-ing from operating (exploitation) life. In this way, cal-culated unreliability Fp includes operating time butwithout the time parameter in Weibull’s function. Pre-sentation of elementary reliability or unreliability bythree parametric Weibull’s functions, without separa-tion in pi and FPi produces the problem in creatingindependent members in axiomatic matrix.

3.2 Decomposition of the total reliability of planetarygear drive unit

According to previous discussion, two approaches toTS, especially to gear transmission units’ reliabilitymodeling and use are possible to perform. The first,more common, is for the maintenance requirementsand the other one is to design the requirements. Thetotal value of overall reliability in both approaches has

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Fig. 3 Structure of planetary gear unit for bucket wheel exca-vator drive

to be the same. The objective of decomposition is toprovide the elementary reliabilities of components inorder to apply axiomatic and robust design methodol-ogy, and in the process of TS integration (in V-model)to get realistic value of total reliability as an indica-tor of TS quality. In order to achieve these goals, itis necessary that the decomposition process includesthe following: (a) decomposition of TS structure intocomponents which have to be replaced in complete inthe maintenance process, (b) creation of decomposi-tion model of the total desired reliability that will con-tain the elementary reliabilities of components, relia-bilities of component relations and reliabilities of sec-ondary functions not in directly related with the TScomponents, and (c) creation of analytic model whichcan provide calculation (processing) of TS desired to-tal reliability. The main goal of decomposition processis to extract elementary reliabilities of TS componentsfrom the total desired reliability.

The case study presented here for reliability baseddesign is a planetary gear drive unit (planetary re-ducer) applied for bucket wheel drive at the minebucket wheel excavator with power 375 kW, trans-

mission ratio 182, input speed of rotation 1480 rpmand output speed of 8.15 rpm (Fig. 3a). The gear unitconsists of three sections. The first one (I) is an inputsection that contains one planetary gear set and bevelgear pair (Fig. 3b). Sections II and III are two plane-tary gear sets i.e. planetary stages in power transmis-sion for torque and speed transformation. One plane-tary gear set (Fig. 3c) consists of the central pinion,three gear satellites and one inside toothed gear ring.This ring is fixed and by rotation of the central pinion,planetary gears rotate together with the satellite car-rier. Input shaft is connected to the central pinion andoutput shaft with the satellite carrier. In section I thedesign structure is different and presents the combina-tion of the bevel gear pair and planetary gear set withthe input shafts. The main power input (375 kW) pro-vides bevel gear pair with a corresponding (side) shaft.During operation the main drive motor, the another-alternative input shaft (planetary central pinion—sungear) is fixed by the brake. In these conditions the gearring rotates together with the bevel gear. The outputtorque of section I carries out the satellite carrier con-nected to stage II by the spline joint. The rule is thatbucket wheel excavators have to be provided with aux-iliary drive of lesser power. During auxiliary drive op-eration, the bevel gear pair is fixed by the brake atthe input shaft and power transmission is going onvia planetary gear set settled inside the bevel gear.The role of planetary gear set in these conditions ischanged and transmission ratio is also changed. Thering together with the bevel gear is fixed and sun gearrotates. The output torque also carries out satellite car-rier connected to stage II by the spline joint.

Decomposition of design structure, functional re-quirements and operating properties, according toFig. 1 produces models which are in close relation andspecific for every certain technical system. This meansthat it is not possible to create a model of TS reliabil-ity decomposition or a model of total load capacitydecomposition independently of the design structuremodel. Also, it is important to mention that the objec-tive of total reliability decomposition is to determinethe elementary reliability of a certain component thatwill form the base of robust design parameters of thecomponent definition. For this purpose, it is neces-sary to extract elementary reliabilities of componentsfrom other various influences on the overall reliabil-ity of TS. In this sense, the model of total reliabil-ity decomposition has to include reliabilities against

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appearance (a) malfunctions of design components,(b) malfunctions of connections and relations betweendesign components and, finally, (c) appearance of mal-functions of secondary technical processes such as lu-brication, cooling, control etc. In the case of plane-tary gear unit presented in Fig. 3, the three sectionsin design structure are identified. Each of them con-tains a corresponding gear set, bearings, gaskets, satel-lite carriers and various connecting fits. The model,in the form of reliability tree, obtained by total re-liability decomposition contains total reliabilities ofdesign components Ra , connection fits (spline joints,bolted joints, interference-clarence joints etc.) Rb andsecondary processes in the unit (lubrication, cooling,. . .) Rc (Fig. 4). Total reliability of design compo-nents Ra is structured in three branches of reliabilitytree, each for every structure sections I, II and III. Theblue leaves present elementary reliabilities of the gearsets, the green leaves elementary reliabilities of bear-ings and the red leaves elementary reliabilities of gas-kets at two input and one output shaft. Each of theseelementary reliabilities is specific and with specificfeatures which have to be respected in the presentedmodel. These features are related with the componentbehaviour in the exploitation process, design charac-teristics, failure process etc. and will be discussed inthe next section. All of them have to be included in themodel for analytic hierarchy processing of total reli-ability which has to be developed based on presentedmodel in Fig. 4 and which is not the subject of thisarticle.

3.3 Elementary reliability of gear sets

Planetary gear units are very compact structures witha very high specific load capacity. The main disad-vantage of these structures is non-uniform failure ofplanetary gear sets, small space for bearings and heat-ing of gear unit caused by small dimensions, whichreduces heat radiation. The subject of further discus-sion is non-uniform failure of gears in a planetary gearset. Figure 5 shows the distribution of teeth flanks fail-ures at gears in planetary gear set. This is the result oflaboratory testing of planetary gear units at back-to-back testing rig with a permanent torque during test-ing process. In this structure the central pinion makesalmost three meshes (1) in one revolution. Also, theteeth flank of this pinion is exposed to a very high sur-face stress, which is the result of the small pinion di-ameter. The result of these conditions is progressive

Fig. 4 Reliability decomposition of planetary gear transmis-sion unit (Fig. 3)

damage of the teeth active flank. Surface layer failurein this progressive process is a combination of pittingand scuffing caused by material particles released bypitting. This progressive failure process has taken outa hardened (carbonised) layer, and in further operationprocess the failure continues with acceleration. This isuntypical process according to standardised models ofgear teeth failure. Satellite gears with one teeth flankare meshing with the central pinion and with anotherone with the inside toothed ring. The first one is alsoexposed to the same high surface stress as the centralpinion, but teeth mesh frequency (stress cycles num-ber) is much smaller. The picture in Fig. 5 shows thatat this flank, pitting started in the middle region of theflank. The opposite teeth flank of satellite gear in themesh (2) with inside toothed ring is exposed to lessersurface stress, because concave and convex shapes offlanks are in contact. Stress cycles number is less com-pared to opposite flank. The result is the beginning ofmicro-pitting of this flank, as shown in Fig. 5. Further-more, the teeth flanks of inside toothed ring are ex-posed to less stress cycle’s number and the wear pro-cess of the teeth flanks was not so serious at first. It ispossible to notice small surface damages that are pro-duced by plastic penetrations of particles from othergears taken by the oil.

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Fig. 5 Teeth wear distribution in planetary gear set

The situation is that a complete planetary set hasto be removed after central pinion failure occurs. Forthe gear pairs, the rule is that when teeth flanks of onegear get significant, it is not possible to replace justthis gear, on the contrary, it is necessary to replace acomplete gear pair. In the case of a planetary gear set(Fig. 5), after the teeth flanks failure of central pinion,it is necessary to replace the complete gear set. In gen-eral, various damages can occur in one TS component.The damage that occurs first causes the replacementof a complete component. This is the reason why thereliability for design of a component is equal to ele-mentary reliability in relation to failure with minimalelementary reliability. In the case of a planetary gearset, it is elementary reliability against teeth flanks fail-ure of planetary central pinion. In the model in Fig. 4these are RPGS-II and RPGS-III for sections II and III. Inthe case of section I, the bevel gear pair and the plan-etary gear set (Fig. 3b) create one gear set complete,i.e. one component. Failure of teeth flanks of the bevelgear pair or planetary gear set will cause replacementof the complete set. In the model in Fig. 4, reliabil-ity of input gear set RIGS is equal to the lower valueof elementary reliability of teeth flanks of the bevel orplanetary set.

In general, gear teeth can be broken due to bendingand similar stresses. In the case of involutes’ gears thiscan be accidental or the result of errors in the produc-tion. This is the reason why this possibility has to be

included in the part of total reliability Rc in Fig. 4, butnot in the reliability for design of a gear set.

3.4 Elementary reliability of bearings and gaskets

In planetary gear transmission unit bearings are theexecutors of auxiliary functions, but they are alsovery important components from reliability aspect.The main function of power transmission is carriedout by gear sets and this process is supported by aux-iliary functions of bearings. Every planetary gear setcontains two groups of bearings: one group is appliedto support satellite carrier (marked by BC in Fig. 3a)and the second one to support satellite gears, markedby BG. Bearings of satellite carriers BC operate sim-ilarly to all bearings of common shafts in the gearunits. Additionally, as the forces of planetary gear setsare in a relative balance, these bearings are not sig-nificantly loaded. Unlike the previous group, the bear-ings of satellite gears BG are highly loaded and set-tled in a small space inside the satellite gears. Highload and small space (of relatively small bearing di-mensions) indicate that reliability of these bearings isnot enough. With increase of gear load capacity andreduction of gear dimensions, the problem with thisgroup of bearings becomes bigger. In some gear driveunits, operating life of these bearings has to be shorterthan of other components and replaced when its liferesource expires. This fact has to be included in the

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analytic hierarchy processing model of total reliabilityof gear drive unit. It is also important to mention thatall three bearings of satellite gears (BG) are exposedto the same operating conditions. From the point ofview of maintenance expenses, in the case of damageof one of them, all three have to be replaced. That isthe reason why in the reliability tree the elementaryreliability RBG of just one bearing represents all threebearings for every satellite gear set.

The reliability model in Fig. 4, for the first section(I) is more complex and specific because the designstructure is specific. Complete rotating mass whichconsists of bevel gear together with planetary gear issupported by gears BV-3 and BV-4, and input shaft ofbevel gear pair by bearings BV-1 and BV-2. Exceptfor these bearings, the section also contains additionalfour bearings, the two BC1 and BC2, for planetarycarrier supporting, and another two, BC3 and BC4,for auxiliary input shaft supporting. As Fig. 4 shows,the branch of reliability for design model for the sec-tion I contains elementary reliabilities RBV 1 · · ·RBV 4,then elementary reliabilities RBG1-I · · ·RBG4-I, andone RBG-I which represents elementary reliability ofall three bearings in satellite gears. Elementary relia-bilities of bearings in satellite gears in all three sec-tions have great influence on the total reliability ofplanetary gear transmission unit.

The components with short exploitation life andthose that have to be frequently replaced are gasketswhich seal the contact between shafts (input and out-put) and housing. Damage of a gasket can producedamage of gears and bearings. This probability has tobe included in an analytic model for decomposition.In Fig. 3, the two input shafts contain two seals in sec-tion I of the planetary gear unit, marked by SE. In sec-tion III there is one seal (SE) included in the reliabilitysystem (Fig. 4). These elementary reliabilities RSE arelinked to the number of shaft revolutions in the courseof service life only.

4 Reliability based design of gear unit components

The objective of reliability based design is to providethe design parameters (DP) of components which willsatisfy desired elementary reliability obtained by totalreliability decomposition. In this way, a set of addi-tional objectives can be also satisfied. Values of DP’s

will be harmonised with stochastic operating condi-tions and robustness of design structure will be pro-vided i.e. DP’s will be insensitive to operating con-ditions variation. Also, DP will ensure minimisationof design structure volume and desired total reliabilityin relation to operating life i.e. the time of operation,and desired level of system quality. For this purpose,total reliability of the system is involved as a func-tional requirement and an indicator of system qualityand system behaviour. In this process, elementary reli-ability is the design property of a component and con-straint in applied axiomatic design methodology. Reli-ability based design is a part of general methodologycalled Property based design and reliability is one ofthe properties.

According to axiomatic design methodology, twoaxioms have to be satisfied. The first one is provisionof information minimum. In the case of planetary geartransmission units this means that of all DP’s of com-plete design structure, in the first step it is necessary toidentify the main DP’s of gears. This implies the defi-nition of the central pinion diameter and width for ev-ery planetary set (transmission stage or section). Theother DP’s of gears result from the relation betweenDP’s in a planetary gear set. The DP’s of satellite car-riers are defined in relation to the DP’s of gears. Thismeans that the axiom of information minimum is com-pleted by calculating central pinions DP’s only with-out influence of other DP’s. The second is the axiomof independence. DP’s have to be calculated, if it ispossible, independently of each other. In this case, thisaxiom is fulfilled by successive calculation of DP’s ofgears and DP’s of bearings. Both groups of DP’s haveto be in harmony with each other and with elemen-tary reliabilities. Harmonisation is an iterative processwhich follows DP’s axiomatic calculation of gears andbearings. By satisfying these two axioms, the maincondition of axiomatic design methodology applica-tion is provided that matrix of functional requirements(FR) transformation into DP’s is uncoupled [6] and[11].

4.1 Reliability based design of gears

In planetary gear transmission units, the main compo-nents are gears i.e. planetary gear sets. According toprevious discussion, the central pinion in the planetarygear set is exposed to higher teeth stress with higherstress cycles number, compared to other gears in the

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Fig. 6 Load spectrums of central pinion in section III for bucketwheel excavator operating conditions

set. This is the reason why elementary reliability, asso-ciated with the central pinion flank failure, presents el-ementary reliability of the planetary gear set assembly.As a case for this elementary reliability calculation,the central pinion of planetary gear set in section III ofgear drive unit in Fig. 3a, is selected. In Fig. 2 and byEq. (1) the calculation process based on operation loadspectrum (load probability) and gear failure probabil-ity, is presented.

Operation load spectrum of bucket wheel excava-tor, where gear drive unit is applied, is presented inFig. 6. This machine operates in a regime, where re-sistance torque at the bucket wheel is not uniform andvaries its value. The level of this torque depends of thematerial resistance, operating intensity, handling wayetc. The load spectrum presented in Fig. 6 is the re-sult of extensive monitoring, torque measurement forselected samples of conditions and statistical analysisand assessments. The spectrum presents an arrangedcollection of loads of various values, which shows par-ticipation of all of them in the chosen number of cycles(revolutions). The value of selected stress (or load) cy-cles for load structure presentation is, as a rule, onemillion, 106 cycles. Heavy service regime implies highparticipation of high load values. The light regimemeans high participation of low or lower torque val-ues, but bucket wheel excavators, as a rule, are notused in the light regime. The light regime of exploita-tion corresponds to vehicles and similar technical sys-tems.

Failure probability distribution range of planetarycentral pinion is presented in Fig. 7. This results fromtesting methodology presented in [6], and presents

Fig. 7 Failure probability distribution of gears with carbonisedteeth flanks

the relationship between teeth flank stress (Hertzianstress) σH and stress cycles number until flank fail-ure occurs. The range of testing results distribution isbounded by the lines with failure probability PF0.1 andPF0.9. For various gear materials and surface thermaltreatment in DIN 3990 and ISO 6336, data for theseranges creation are presented. The range in Fig. 7 isalso defined using those data and corresponds to al-loyed steel with carbonised and grinded gear teeth.For certain selected stress cycles number nΣi , failureprobability is defined by two parametric Weibull’s dis-tribution functions

PF (σH ) = 1 − e−(

σHη

)β (2)

Weibull’s function parameters η and β are the resultof Eq. (3) obtained by transformation of Eq. (2) forvalues σH = [σH ]0.1 from the line PF (σH ) = PF0.1 =0.1, and for the value σH = [σH ]0.9 from the linePF (σH ) = PF0.9 = 0.9 in Fig. 7. Total stress cyclesnumber nΣi during the entire operating life corre-sponding to every stress level σHi , Fig. 7, is definedby vertical line and values of [σH ]0.1 and [σH ]0.9 areidentified, and then these parameters are calculated byapplying Eq. (3)

β = log(lnPF0.1/ lnPF0.9)

log([σH ]0.9/[σH ]0.1)

η = [σH ]0.1β√− ln(1 − PF0.1)

= [σH ]0.9β√− ln(1 − PF0.9)

(3)

For every load (operating stress) level i = 1,2,3 and 4identified in the load spectrum (Fig. 6), the total stress

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Table 1 Elementary unreliability calculation of central planetary pinion (Heavy operating regime, duration of 10 years)

I 1 2 3 4

Ti-III, kN m 117.6 111.3 96.9 68.2

σHi 1511.2 1470.1 1371.9 1150.9

nΣi 2.2 × 107 1.98 × 108 1.32 × 108 8.8 × 107

[σHi ]0.1 1532 1315 1315 1318

[σHi ]0.9 1952 1687 1733 1779

βi 12.73 12.38 11.17 10.28

ηi 1828.2 1577.1 1608.4 1640.4

PFi 0.08473 0.34229 0.15561 0.02580

pi = nΣi/nΣ 0.05 0.45 0.3 0.2

pi · PFi 0.0042365 0.1540305 0.046683 0.00516

cycles number in the course of operating life nΣi isidentified and presented in Table 1. Using the diagramin Fig. 7, for every nΣi endurance boundaries, [σH ]0.1

and [σH ]0.9 are identified, included in the table, andusing Eqs. (3) the parameters of Weibull’s function ηi

and βI are calculated. These parameters make possi-ble to use Eq. (1) to calculate failure probability PFi

i.e. PF (σHi) for every flank stress level σHi calculatedusing the corresponding central pinion torque Ti fromload spectrum for heavy service regime (Fig. 6). Re-sults of the stresses σHi calculation given in Table 1correspond to central planetary pinion of section IIIwhere the gear module is mn = 18 mm, and gear widthb = 200 mm, number of teeth of pinion is 14 with off-set factor x = 0.2309, teeth number of satellite gearsare 25 and teeth number of ring is 64, both withoutoffset factors of involute profiles. Output speed of ro-tation, as already mentioned, is 8.15 rpm. In the ta-ble, the sum of products Fpi = piPFi presents the to-tal unreliability of central pinion of the planetary setin section III, Fp = 0.206 and reliability R = 0.794,after a 10-year gear unit operation. This is also the el-ementary reliability of planetary gear set in section III,RPGS-III = 0.794. As a rule, this reliability is small andpresents the main problem in planetary gear units de-sign. Also, it should be noted that service regime ofbucket wheel excavators is not too heavy, as presentedin Fig. 6, and this reliability can be higher. Addition-ally, an important fact is that very often central pinionoperates in conditions of progressive teeth flank dam-age.

The presented procedure of elementary reliabil-ity calculation, by DP’s definition of the planetary

Fig. 8 Effect of load spectrum variation at gear design allow-able stress σHdes

gear set, is used for design allowable stress defini-tion (calculation) for the desired elementary reliabilityobtained by decomposition of the total reliability ofgear transmission unit. Design allowable stress σHdes

is equal to the maximal operating stress in the spec-trum σH1 which provides the desired elementary reli-ability (Fig. 8). By variation of the stress levels in thestress spectrum (all of them in the same proportion)and by iterative calculation of elementary reliability,we obtain the value of σH1,which provides the desiredelementary reliability, as a corresponding design al-lowable stress σHdes. The lower σ ′

Hdes corresponds tothe higher elementary reliability and provides biggerdimensions (diameter or withed) of the central pinion.In the same relation, the other gears (satellite and ring)have to change dimensions.

Figure 8 shows clearly the effect of stress level andstress cycles number variation on elementary reliabil-ity value and also the effect of desired elementary reli-ability on design allowable stress and gear dimensions.

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Fig. 9 Effect of bearing revolutions at its elementary reliability

This variation process via σHdes is included in transfor-mation matrix in the axiomatic procedure which de-fines DP’s in relation of functional requirements FR.Also, σHdes includes possible variations of operatingstress and failure probability of gear material and pro-vides robustness of DP’s i.e. independence of condi-tion variation.

4.2 Reliability based design of bearings and gaskets

Planetary principle of power transformation is char-acterised by the use of the number of bearings. Bear-ings damage can result from a large number of effectssuch as operating load and loads caused by assemblycomponents interaction, bad or lack of lubrication andcooling, metal particles produced by the gear flankfailure, effect of other dirties etc. As mentioned inSect. 3, all of these effects in reliability decompositionprocedure have to be separated from elementary relia-bility in relation to possible failure caused by operat-ing load. In the case when the dimensions of the plan-etary gear sets are significantly reduced, the problemarises with the significantly reduced space for bearingsof satellite gears, which leads to their probability andoperating life reduction.

Reliability based design procedure for bearings, es-pecially bearings of satellite gears is very similar toreliability based design procedure for gears. Startingfrom load spectrum presented in Fig. 6, we calculatethe loads of bearings i.e. forces acting to the bearing.For every torque level Ti , a corresponding force is act-ing on bearing Fi . The shape of bearing load spectrum(Fig. 9) is the same but numbers of bearing revolutionsnΣi are recalculated according to gear satellite revo-lutions. Standards and bearing producers define bear-ing load capacity C which corresponds to NC = 106

revolutions until failure occurs with failure probabil-ity PF = 0.1. Inclination of the boundary line withPF = 0.1 is also defined. Position of the boundary linewith PF = 0.9 varies depending of bearing producers.Both lines for PF = 0.1 and for PF = 0.9 can be theresult of extensive failure testing of a certain type ofbearing.

Variation of load levels in the load spectrum ofbearings (Fig. 9) can be the result of variation of geardimensions. Lower levels of bearing loads correspondto larger gear dimensions. For a defined level (spec-trum) of bearing load, the number of bearing revolu-tions nΣi has to be harmonised with the desired el-ementary reliability of the bearing. As these revolu-tions in the load spectrum increase from nΣi to n′

Σi ,elementary reliability decreases (see Fig. 9). For thedesired level of bearing elementary reliability for acertain load spectrum and required operating life, thenecessary load capacity C is obtained by iterative cal-culation. This load capacity corresponds to PF = 0.1,but it is the bearing characteristic only, while bearingoperating properties result from all service conditionsrelations. This is incorporated into axiomatic and ro-bust design procedure for the bearings selection. Loadcapacity has to satisfy the desired elementary reliabil-ity with operating life equal to gear operating life or toone half of gear operating life when it is necessary toreplace bearings at the half of gear operating life.

Gaskets seal input and output rotating shaft in re-lation to housing. The gasket load is not the result ofoperating load of gear transmission unit. It is the re-sult the gasket design, production and mounting. Thisis why the gasket elementary reliability is related tothe number of revolutions during operating life:

Fp = PF (nΣ) = 1 − e−(

nΣη

)β (4)

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where η and β are parameters of Weibull’s distributionof gasket failures. For the gasket with small speed ofrotation, operating life until unreliability Fp becomesunacceptable is much longer compared to slow speedof the shaft rotation. Since the speeds of input and out-put shaft are very different, elementary reliabilities andoperating life of corresponding gaskets are also differ-ent. Also, it is necessary to take into account that pa-rameters of function (4) are related to gaskets’ dimen-sions for the same type of gasket. The difference inoperating life has to be included in the analytic modelfor the desired total reliability processing.

5 Conclusion

This paper is oriented to the impact of modern designmethods, experimental results and quality increase ofplanetary gear transmission units. Suggested method-ology belongs to the group of property based design.The reliability for design lies in the role of planetaryunit quality indicator and functional requirement (inthe total form) and in the role of design property anddesign constraint in the form of component’s elemen-tary reliability. The major innovative contributions ofthe article are as follows:

• Reliability system of planetary gear unit modellingand decomposition based on elementary reliabilityof components connected to the most probable fail-ure.

• Reliability for design definition, in the form of ele-mentary reliability of components (gears and bear-ings) based on service regime and failure probabil-ity distribution.

• Design parameters definition of planetary gear unitcomponents (gears and bearings) based on ax-iomatic and robust design approach aimed at ob-taining the components with uniform reliability re-sources.

• An adapted V-model (according to VDI-2206) forgear drive units design.

The suggested approach does not include an an-alytic model for desired total reliability of planetarygear transmission units’ decomposition which willprovide corresponding values of elementary reliabil-ity of structural components. This can be the part ofseparate paper together with complete calculation pro-cess.

Acknowledgements This work is a contribution to the Min-istry of Education and Science of Serbia funded project TR035006.

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