MODELLING OF THERMOELASTIC TRANSIENT CONTACT INTERACTION FOR BINARY BEARING TAKING INTO ACCOUNT CONVECTION

Serviceability of metal-polymeric "dry-friction" sliding bearings depends on many parameters, including the rotational speed, friction coefficient, thermal and mechanical properties of the bearing system and, as a result, the value of contact temperature. The objective of this study is to develop a computational model for the metallic-polymer bearing, determination on the basis of this model temperature distribution, equivalent and contact stresses for elements of the bearing arrangement and selection of the optimal parameters for the bearing system to achieve thermal balance. Static problem for the combined sliding bearing with the account of heat generation due to friction has been studied in [1]; the dynamic thermoelastic problem of the shaft rotation in a single and double layer bronze bearings were investigated in [2, 3].


INTRODUCTION
Clarification for the behavior of surface layers in metal-polymeric thermoelastic tribocontact -one of the central problems in tribotechnology.Therefore, for a deeper knowledge of the processes on the contact surface, it is necessary to develop not only the experimental methods of diagnosis but adequate theoretical models.At present, there is a single point of view that the determining factor in the operational mode of the metal-polymer interface is a thermal stress at friction node.
In recent years, in domestic and foreign scientific literature much attention is paid to the theoretical (numerical, analytical) and experimental studies of the bearing performance of polymeric "dry friction" bearings [4][5][6][7][8][9][10][11][12].The widely used polymer in the manufacture of bearing is PTFE-4.The main reason that caused the interest to this material is the fact that during "dry friction" of metals and PTFE-4 at low sliding speed the friction coefficient is very small and does not usually exceed normal coefficients of friction in metal bearings in the presence of lubricants.Pure PTFE has good chemical resistance, low friction and a wide operating temperature range, but it is subject to deformation under load and heavy wear.Fillers introduced into PTFE improve wear resistance about a thousand times, the resistance of the load pressure 2-5 times, and thermal expansion is reduced by 2-3 times.The investigation of the influence of sliding speed, bearing pressure and temperature on the friction and wear in sliding bearings made of PTFE with fillers is subject of [13].
In contrast to the above-mentioned works, the present article examines metallic-polymer plain bearing with PTFE inclusions in unsteady interaction of bearing components, taking into account the heat generation from friction and convective heat transfer, based on a 3D model.
Bearing type with anti-friction PTFE-4 inclusions is shown in Fig. 1.

THE MATHEMATICAL FORMULATION OF THE PROBLEM
Let us describe the nonstationary dynamic coupled contact problem of thermoelastic interaction of elastic homogeneous cylinder (hereinafter -the shaft) with the inner surface of the cylindrical layer of finite length with inserts (hereinafter -the bearing).The geometry of bearing is in a cylindrical coordinate system and 2a) shows a three-dimensional examined object (Figure 1b) -statement of the problem in a front plane.(Fig. 2 (a), this system corresponds to the shaft) on the surfaces ) of the shaft, there are applied loads, and each is 2 / P and acts vertically down.
Bearing rotates counterclockwise with an angular velocity ω (rad/sec) during int T seconds.Between the shaft and the bearing, there are Coulomb friction forces with the coefficient k .The corresponding values of friction coefficients for stainless steel/PTFE (Teflon) bearings could be found in [14].Let shaft S ± be surfaces of shaft ends, lying in planes , respectively.It is assumed that at the shaft and bearing surfaces that make contact with the environment there is defined convective heat exchange with a convective heat transfer coefficient a .Let the initial state of the shaft temperature and the bearing be the same as the environment and equal to С °0 .
As a result, we come to the solution of non-stationary coupled contact thermoelastic problem with the classical equations of motion for the thermoelastic medium [15] (index Constitutive relations for coupled thermoelasticity have the form: stresses and strain tensors.
The boundary and initial conditions have the form: -is the contact surface between the shaft and the bearing, cont q -heat flow in the contact region, cont k -contact thermal conductivity coefficient,

PROBLEM STUDY
To solve this problem the finite element method and the special software ABAQUS were used.To improve the convergence of the algorithm for solving the problem and reducing calculation time, the solution was carried out in two stages.The first stage is static contact problem of elasticity for the shaft pressing on the surface of the bearing, the second is coupled transient thermoelastic contact problem of the shaft rotation.
The finite element mesh is built using 8-node coupled thermoelastic C3D8T element.The resulting finite element model of the bearing system is shown in Figure 2.For the simulation of contact interaction between the inner surface of the bearing and the outer surface of the shaft, both were covered by the contact pairs of elements.To solve the non-stationary problem, minimal and maximal time steps are set, and the settings that allow the package to choose the optimal time step during the calculations are defined.

RESULTS
In the numerical experiments, the following values of the geometrical and material parameters were set as in Tab.The bearing is assumed to be made of bronze and the inserts contacting with the shaft of PTFE-4.During the calculations shaft is assumed to be made of steel.Corresponding material properties for the materials mentioned could be found in Tabs.1-2.The convective heat transfer coefficient for the metal-air pair on the surface of the shaft and bearing bordering with surroundings is indicated as a .It was chosen to be in the calculations equal to N and rotation speed of π 10 rad/sec after 5 seconds.For all other stated cases, the picture of distribution for the temperature field and contact stresses are typically the same, the difference is only in numerical values.Taking this into account, the presentation of corresponding figures is not sufficient and only plots for contact stresses and temperatures with time will be given.As expected, in the zones of PTFE inserts, the temperatures are less than those for other zones.It should be mentioned that temperature distribution is unsymmetrical.
Elements of a bearing system are significantly heated during rotation of the shaft, and the increase in the speed or rotation time leads to a rise in temperature.Figure 5 shows the character of the temperature change of the maximum values for the shaft and the bearing.Here and after all results correspond to point of shaft with coordinates ( .It can be seen that the bearing temperature is considerably lower than the temperature of the shaft, due to the greater heat transfer surface area.Ramp character for bearing temperature is due to the presence of the gap between the contacting bodies.Bearing point, entering the contact zone, receives a certain amount of heat due to friction, then moving in the area of the gap between the shaft and the bearing, some part of the heat is redistributed inside the bearing volume and is emitted into the environment, which leads to a small decrease in temperature for the corresponding point.

CONCLUSIONS
Studies have shown that of great significance in achieving thermal balance for the demonstrated system is the value of convection for the shaft and bearing free surfaces, while increasing parameters such as load value, the coefficient of friction, the achieving of thermal balance in the bearing system is slowed down.It has been observed that during some initial period of time the temperature rises significantly, then the growth pattern shape becomes flatter.Note that the finite element method using Abaqus package for this problem turned out to be quite effective and allows one to explore the effectiveness of such problems for different values of the input geometrical and mechanical parameters.
Statement of the problem, experimental characterization of PTFE-4 polymer and investigation of the influence of parameters of thermoelastic contact interaction in binary bearing on its stressdeformed state and temperature fields are made by I.V. Kolesnikov

Fig. 2 .
Fig. 2. Statement of the problem Рис. 2. Постановка задачи The outer surface of the bearing ext S ( The influence of different values of the coefficient of friction k and the load P on the maximum temperature max t of the bearing (marked with index bear ) and shaft (marked index shaft ) for the problem after the rotation for 5 int = T seconds was studied.Figs.3-4show the distribution of temperature and contact pressure for the value of pressing force equals

Fig. 6 .Fig. 7 .
Fig. 6.The character of the maximum temperature change for the various pressing force values Рис. 6. Характер изменения максимального значения температуры для различных значений силы вдавливания and S.A. Danilchenko and supported by the Russian Science Foundation under grant 14-29-00116 and performed in Rostov State Transport University.Mathematical formulation of the problem, CAD geometry preparation and finite element modeling are made by M.I.Chebakov, A.A. Lyapin and E.M. Kolosova and supported by the Ministry of Education and Science of the Russian Federation under project no.213.01 11/2014 28, the Russian Foundation for Basic Research under project no.16-08-00852 and performed at the Southern Federal University.

Table 2
Material properties for components of system

Table 3
The dependence of the thermal expansion coefficient for PTFE on temperature