UDC/UDK 621.8 ISSN 1580-2949 Original scientific article/Izvirni znanstveni članek MTAEC9, 41(2)73(2007) NUMERICAL DETERMINATION OF THE CARRYING CAPACITY OF ROLLING ROTATIONAL CONNECTIONS NUMERIČNA DOLOČITEV NOSILNOSTI VTRLJIVIH KOTALNIH ZVEZ Robert Kunc, Andrej Žerovnik, Matej Žvokelj, Ivan Prebil Faculty of Mechanical Engineering, University of Ljubljana, Aškerčeva ul. 6 1000 Ljubljana, Slovenia robert.kuncŽfs.uni-lj.si Prejem rokopisa – received: 2006-05-17; sprejem za objavo - accepted for publication: 2007-02-16 The exploitation of large axial bearings includes load peaks, which cause permanent deformation of the rolling contact. The plastic strain of the base material under the hardened rolling layer starts to grow and micro-cracks on the edge of the hardened layer and the peeling of this layer may occur. In our computation we have used an elasto-plastic model that combines isotropic and kinematic hardening and the growth of material damage. The damage model allows us to follow the variation of the elastic and plastic stress and deformation as a function of the number of cycles. In the article, an experimental verification of the described numerical model is shown, which can be used to determine the actual carrying capacity of the rolling contact for low-speed axial bearings. Key words: rotational connections, low cycle fatigue, cyclic plasticity, damage Pri uporabi velikih aksialnih ležajev se pojavljajo preobremenitve, ki povzročijo trajno deformacijo ležajnega kontakta. Plastična deformacija osnovnega materiala pod utrjeno ležajno površino začne rasti in lahko nastanejo mikrorazpoke na robu kaljene plasti ter njeno luščenje. Pri našem izračunu smo uporabili elastoplastični model, ki kombinira izotropno in kinematsko utrjevanje ter rast poškodovanosti materiala. Ta model omogoča sledenje spremembe elastoplastične napetosti in deformacije kot funkcijo števila ciklov. V članku je opisana eksperimentalna verifikacija opisanega numeričnega modela, ki ga lahko uporabimo za določitev prave nosilnosti ležajnega spoja za aksialne ležaje, ki delujejo pri majhni hitrosti. Ključne besede: vrtljiva zveza, malociklična utrujenost, ciklična plastičnost, poškodbe 1 INTRODUCTION Bearings are among the most frequently used elements in machine engineering. Owing to their widespread use, the requirements that a bearing must meet are highly diverse. Bearing use ranges from applications in which a bearing collapse does not constitute a major problem, to applications where the collapse of a bearing could lead to enormous economic losses and potentially disastrous consequences for people. The large bearings used in rolling rotational connections are an example of such an application. The basic element of a rotational connection is a large rolling bearing, with attachment holes in the bearing rings, and a gear wheel (Figure 1). Alloyed steels such as 42 CrMo 4 and C 45 - ISO 683/1 are normally used in the production of bearing rings since the process requires thermal and mechanical treatments. The raceway is surface-hardened to attain the minimum hardness necessary to prevent the pressing-in of the rolling element. The actual external load and the elasticity of the upper and support structures cause an uneven distribution of the external loads over the rolling bearing’s diameter 13. This leads to local plastic deformations of the bearing raceway and to the initiation of damage, largely conditioned by the load size and material fatigue 45. The existing models for determining the carrying capacity of large rolling-bearing conne- ctions with surface-hardened raceways do not fully consider or even greatly simplify the actual state of the bearing ring 312. In analyzing the actual carrying capacity of rolling contacts in large rolling bearings with surface-hardened raceways we have decided to use a combined elasto-plastic constitutive model, which links the material C7|-|gT - Max. Hertz stress aH Figure 1: Elements of rotational connections Sllika 1: Elementi rotacijskih povezav Materiali in tehnologije / Materials and technology 41 (2007) 2, 73-76 73 R. KUNC ET AL.: NUMERICAL DETERMINATION OF THE CARRYING CAPACITY ... damage mechanics with the isotropic and kinematic hardening/softening 13-15. The material model considers 17 material parameters, selected from a series of monotone and cyclic experimental tests. The model, which was built in the finite-element code, was used to investigate the development of elasto-plastic deformations, stress areas and damage occurrence for low-cycle loads that occur in the raceways of large rolling bearings at a low rotational speed. In the article an experimental verification of the described numerical model is shown, which can be used for the determination of the actual carrying capacity of the rolling contact in low-speed axial bearings. 2 NUMERICAL MODEL A developed and built-in finite element is used in the application of a numerical model focusing on the rolling contact between a raceway model and a rolling element, and is carried out in the ELFEN© commercial software environment. The discretisation of a bearing raceway’s geometric model applies a developed finite element that covers isotropic hardening/softening, kinematic hardening and material damage 13-15. This makes it possible to monitor the actual low-cycle development of the strain deformation or the hardening/softening and material damage growth. The end code of a developed element is written in the FORTRAN programming language. The geometry of the contact between the rolling element and the raceway is determined with square surface-contact elements (Figure 2). The numerical model of a rolling contact considers the actual geometrical and material characteristics of bearing raceway models. The characteristics of a bearing ring model change in line with the raceway depth. After having measured the changes in hardness along the cross-section of a bearing raceway model, we entered various material parameters into the damage model. 0.02 0.01 0.00 Ó 5 10 Ě5 20 25 30 35 40 NunlaerofCycles.N Figure 3: Development of the stress ovm and strain Łvm paths on the edge of the raceway-model hardened layer Slika 3: Razvoj napetosti ovm in deformacije Łvm na robu utrjene plasti ležajne tečine These parameters were derived from the results obtained from single-axis monotone and cyclic tear tests 13. In describing the material characteristics of raceway models, we also took into account the remaining material stresses that occurred during the production of the test raceways 14. 3 NUMERICAL RESULTS The numerical rolling-contact model was used to simulate a contact of the rolling element and a surface-hardened bearing raceway at a varying contact force from Fmax = 35.2 kN to Fmin = 1 kN. The model enables every element to be assigned its deformation strain state and growth of damage relative to the number of contact load cycles. Figure 3 shows the evolution of the equivalent von Mises stresses sVM and strains eVM at the location of maximum-damage accumulation for the first 40 loading cycles, and Figure 4 shows a distribution Figure 2: Numerical model Slika 2: Numerični model Figure 4: Damage distribution in 3D model after 100 load cycles Slika 4: Porazdelitev poškodbe v 3D-modelu po 100 nihajih 74 Materiali in tehnologije / Materials and technology 41 (2007) 2, 73–76 R. KUNC ET AL.: NUMERICAL DETERMINATION OF THE CARRYING CAPACITY ... 0.008 0.006 0.000 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Depth, D/mm 3.5 4.0 4.5 Figure 5: Damage distribution in ring depth Slika 5: Porazdelitev poškodbe v globino ležajnega obroča of the damage after 100 contact force cycles. In contrast, Figure 5 shows the damage distribution along the ring depth that determines the damage accumulations in the base material on the edge of a hardened layer. The carrying capacity of the raceway model for rolling elements of different diameters was determined on the basis of a diagram showing the growth of material damage relative to the number of load cycles (Figure 5)14. 4 VERIFICATION OF THE MODEL A comparison of the results of the experimental work and the numerical calculations of the contact between the raceway model and the rolling element can only be made by comparing the directions of the maximum total and permanent displacements of the bearing-raceway model. 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