r is the pin radius, m; r is the bearing radius, m;

Transcription

1 International Journal of Applied Engineering Researc ISSN Volume, Number 5 (7) pp Working Out of an Analytical Model of a Radial Bearing Taking into Account Dependence of Viscous Caracteristics of Micropolar Lubrication on Pressure and Temperature Kamil Samedovic Akverdiev, Murman Aleksandrovic Mukutadze, Elena Olegovna Lagunova 3, Konstantin Sergeevic Solop 4 Rostov State Transport University (RSTU), Cair Higer Matnematics, Rostov-on-Don, 34438, Russian Federation Rostov State Transport University (RSTU), Cair Higer Matnematics, Rostov-on-Don, 34438, Russian Federation 3 Rostov State Transport University (RSTU), Cair Higer Matnematics, Rostov-on-Don, 34438, Russian Federation 4 Rostov State Transport University (RSTU), Cair Higer Matnematics, Rostov-on-Don, 34438, Russian Federation Abstract In te article we discuss te tecnique and realisation of te exact self-simulated solution of a problem on te infinite of a radial bearing operating in a ydrodynamic mode wit an incompressible micropolar liquid lubricant Te solution considers dependence of viscous caracteristics on pressure and temperature Also we consider te performance of te trust bearing wit te adapted sape of te abutment Our researc is based on te system of te equations of movement of a micropolar liquid incompressible lubrication in te operating clearance for a case of «a tin layer», a continuity equation, and te expression reflecting te law of speed cange of energy dissipation of te lubricant environment We ave also taken into account te dependence of viscous caracteristics bot from pressure and temperature Te field of speeds and pressures as been found, and analytical expressions for a bearing capacity and a friction force ave been obtained We presented te estimation of influence of te parameters caracterising dependence of viscosity on pressure and temperature, and also influence of te adapted sape on te basic performance caracteristics of te bearing Keywords: ydrodynamics, adapted sape of abutment, dependence of viscous caracteristics of a micropolar liquid lubricant on pressure and temperature Designations, is lubricant dynamic viscosity coefficient, Ns/m ; are viscosity coefficients of micropolar lubricant, Ns/m ; is caracteristic viscosity of a Newtonian lubricant, Ns/m ; κ and γ are caracteristic viscosity of a micropolar lubricant, Ns/m ; T is temperature ºC; p is ydrodynamic pressure in a lubricant layer, Pa; α, β are experimental constants; α, ω are parameters of a basic sape; α is a discarge angle of a sliding bar wit a linear circuit to axisoх r is te pin radius, m; r is te bearing radius, m; e is eccentricity; δ is radial gap, m; ω is a parameter caracterising an adapted sape of a sliding bar; Τ is caracteristics of temperature ºC; α is te parameter caracterising dependence of viscosity on pressure; β is te parameter caracterising dependence of viscosity on temperature; υ is rotation rate of microparticles, m/s; u, ν are components of lubricating material s velocity vector, m/s; Q is lubricant consumption per unit time ; C p is eat capacity at constant pressure, J/kg degrees; Ω is angle speed, s ; N, N are structural and viscous parameters of a micropolar lubricant, a micrometer 484

2 International Journal of Applied Engineering Researc ISSN Volume, Number 5 (7) pp INTRODUCTION Performance of any macines and mecanisms, teir reliability and durability, and, ence, profitability strongly depends on a design and operation of friction units Modern macine-building tendencies demand substantial increase of load and speed modes of beaviour of bearing units However, compensation of growt of specific static and dynamic loadings is impossible by increase of te area of teir abutments Te key to te solution of te problem is in te increase of bearing capacity of bearings as a result of application of an adapted sape of bearing busings tat are adjusted for conditions of ydrodynamics Besides, it is quite important to specify analytical models of bearings on te basis of te account of dependence of viscous caracteristics of a micropolar liquid lubricant bot on pressure and temperature Te last defines novelty and relevance of te obtained solution Scientific novelty of te offered solution and specification of te analytical model consists in te account of dependence of viscosity of a micropolar liquid lubricant bot on pressure and temperature; besides, usage of te adapted sape of te abutment of te bearing provides its increased bearing capacity Te simultaneous account of a complex of variable factors as allowed essential specifying of analytical model and approacing of te obtained results to te real ones Te purpose of te researc was to generate te specified analytical model of a radial bearing working in a mode of ydrodynamic lubrication It as been done on te basis of introduction in calculations of a self-simulated variable and due to te account of dependence of viscosity of a micropolar liquid lubrication bot on pressure and temperature Problem statement We consider te establised movement of a viscous micropolar liquid lubricant in te operating clearance of a radial bearing (between a sliding bar and a guide bearing) under conditions of te adiabatic process Te bearing wit te adapted profile of a basic surface isn't mobile, and te saft rotates wit an angular speedω(fig ) It is supposed tat tere is a full filling of te operating clearance wit lubrication Besides, we assume tat viscous caracteristics of a micropolar liquid lubricant depend on pressure and temperature under te following indicative law: e, e, e () p Т p Т p Т Here μ is a factor of dynamic viscosity of te lubrication; κ, γ are factors of viscosity of a micropolar lubricant; μ is caracteristic viscosity of Newtonian liquid; κ and γ are caracteristics of viscosity of micropolar lubrication; Τ is temperature; p is ydrodynamic pressure in a lubricant layer; α, β are experimental constants Figure : Simulation Sceme Te equation of te saft contours and te bearing bus in te polar coordinate system r, θ in te center of te saft are as follows: r r, r r ecosθ аsinωθ () were r is te pin radius; r is te bearing radius; e is eccentricity; е а, is small quantity of te same order δ δ δ=r r be determined ; а, ω are parameters of te contact profile to Te initial equations and boundary conditions Taking into account simultaneous dependence of values on temperature and pressure, let us study te system of te equations of movement of te lubricant wit micropolar properties as te initial one We sall consider it for a case of «a tin layer» taking into account () and te following continuity equations: u dp N, pt r r e d u u v r N N r r, (3) 484

3 International Journal of Applied Engineering Researc ISSN Volume, Number 5 (7) pp Dimensional values are connected wit corresponding dimensionless values by te following correlations: u ur ; v ; ; p p p; ; ; ; Т Т Т; r r r; T r,, ; ; l N, N, l, p r p r r Т 4 is a caracteristic temperature; α is te parameter caracterising dependence of viscosity on pressure; β is te parameter caracterising dependence of viscosity on temperature; υ is te speed of microrotation; u, ν are components of a speed vector Taking into account transition to dimensionless variables in a lubricant layer, omitting caracter strokes,we will come to te following system of te equations: u N, pt r r e d u u v r N N r r, Te boundary conditions are as follows: dp u, v, wen r cos sin ; e a u, v, wen r, ; ; p p a p p ; Let us average te second equation of system (5) over te tickness of a lubricating layer We will get te following: u dr dr dr r N N r (4) (5) (6) (7) We will obtain te solution of equation (7) as follows: A x r A x r A x (8) 3 From boundary conditions (6) it follows tat A ; A A θ x (9) 3 Taking into account (9) forυ, we will get te following expression: A r r () Substituting () in (7) accurate witin members O, O, O, we will obtain te following: N N N υ υ r r, r N r N () Taking into account (), te system of equations (5) in te approac accepted by us is as follows: u N dp r, r N e d p T N r r Precise self-similar solution u, r () We will searc for te exact self-simulated solution of system () satisfying to boundary conditions (6) as follows: ψ ψ u U r,θ, v V r,θ, r θ r V r,θ vξ, U r,θ uξ, ξ, dp C C ψ r,θ ψξ,, p βt 3 e dθ θ θ υξ ξ ξ N θ (3) Taking into account (3) from system () in our approximation, we will obtain as follows: N С, u C, N u v,, N ψ ~ dp C C pт 3 d e ~,, v v, u ψ ~ ~, u, u (4) d (5) 484

4 International Journal of Applied Engineering Researc ISSN Volume, Number 5 (7) pp We can find te solution of system (4) taking into account te boundary conditions (5), connected wit a definition of speed field, by direct integration As a result, we will obtain te following: u 3 С N 3 N С,, N N N C (6) Here С 6, and we will define С from te condition pa p pπ and p equation (4) were J k С СJ θ θ, (7) J3 θ π μθ θ dθ k Predictably, given N, v te received results completely coincide wit te result for a case of Newtonian liquid Definition of ydrodynamic pressure We will define dimensionless ydrodynamic pressure in a lubricant layer from te following equation: dp C C 3 dθ θ θ (8) For te solution of te equation (8) at first we will define as te function depending fromθ Wen determining μ θ use te expression reflecting regularity of cange of speed of dissipation of energy of te lubricant environment:: dн dθ δ θ θ μμ r θ ψ ξ u ξ dξ we (9) Ten te rise in temperature is defined by te following expression: dt dh dθ dθ C Q C Q p μμ r θ ψ ξ u ξ δ θ θ dξ () Were Q is te lubricant loss per time unit; C p is a eat capacity at constant pressure; r is te pin radius, m; is angle speed δ rc () Q r δ ψ ξ dξ Let us differentiate expression ten we will obtain te following: dμ dp dt μ θ α β dθ dθ dθ p βт e wit respect to θ, r T C δ C dp μ θ β4μ θ ψ ξ v ξ μθα dξ dθ θ θ () Taking into account equation (8), for definition μ(θ ) we come to te following differential equation: dμ αс αс 3 μ θ θ θ θ d θ θ r v 4μ β θ ψ ξ ξ T C δ C dξ Integrating tis equation, we will receive te following: μθ 3 αс J θ αс J θ K J3 θ J θ 3J θ, were 4μ βr K, ξ dξ, T C C δ θ 3 (3) (4) dθ ψ ξ vξdξ, v ξ dξ, Jк θ к Let us solve equation (4) for μ(θ ), ten we will receive te following: θ (5) μθ K J3 θ J θ 3Jθ αсj θ αсj3 θ Furter we will replace function μ(θ ) wit its average integrated value as follows: 4843

5 International Journal of Applied Engineering Researc ISSN Volume, Number 5 (7) pp π π μ μ θ dθ K J θ J θ J θ αс J θ С J θ dθ (6) Let us solve te received equations for Δ, Δ, Δ 3,, I 3 (θ ), I (θ ), I (θ )to witin te second-order term O (η ), O O,, we will get te following expressions: η C πω η 3 cosπω, πω 6 cosπω, η πω 6 cosπω, 4 N 4 (7) 3 7N Taking into account (7), finally for μ, we will receive te following expression: 4 sin N 7N K cos 4 cos, 3 6 were 4μ r K Tc sin 6 sin p δ (8) Taking into account (8) and (8), dimensionless ydrodynamic pressure is defined by te following expression: а p μ CJ θ CJ3 θ η ω η θ πω а 6μ ηsinθ cosθω cos πω (9) Findings of te researc and teir discussion Taking into account (9) and (6) for bearing capacity and friction force we will obtain te following expressions: 3 3μr μ κ Rx ηπ η ω cos πω, δ R L fr у 3 3μr μ κ η sin πω, δ π ψ u dθ π cos πω N (3) μr μ r N η θ θ ω Below is te range of te parameters in all expressions: η = η = 3 ; ω = ; K = 5; α = ; β = ; T = 3 ºC; μ = N s ; Pα = 8 35 MPa; m δ =, m; r = m; Ω = 8s ; μ = 595 N s m According to te results of te numerical analysis we ave constructed graps (Fig 7), allowing us to draw te following conclusions As a result of te carried out teoretical researces, we revealed te basic laws of interrelation of viscous caracteristics of a micropolar lubricant wit pressure and temperature in a lubrication layer, and also impact of te adapted sape of an abutment of te bearing busing and structural and viscous parameters (N and N )ofa micropolar lubricant It was establised tat viscous caracteristics of a micropolar lubricant under te conditions of ydrodynamic lubrication are intensively increased wen parameter β drops, it is caused by dependence of temperature on pressure To a lesser extent it appens during te reduction of α parameter caused by dependence of viscosity on pressure, wic similarly influences te bearing capacity of a trust bearing 3 Application of te bearing busing wit te adapted sape provides stable max of te bearing capacity in te researced range of factors, wic can be additionally increased during te values of viscous parameter of a micropolar lubricant 4844

6 International Journal of Applied Engineering Researc ISSN Volume, Number 5 (7) pp Figure : Dependence of a component of a vector of te supporting force on te following parameters: ω, caracterising an adapted sape of te abutment, and structural and viscous parameter N Figure 3: Dependence of a component of a vector of te supporting force of te bearing on te following parameters: β, caused by dependence of temperature on pressure, andα, caused by dependence of viscosity on pressure Figure 4: Dependence of a component of a vector of te supporting force on te following parameters: ω, caracterising an adapted sape of te abutment, and structural and viscous parameter N Figure 5: Dependence of a component of a vector of te supporting force of te bearing on te following parameters: β, caused by dependence of temperature on pressure, and α, caused by dependence of viscosity on pressure Figure 6: Dependence of friction force of te bearing on te following parameters: β, caused by dependence of temperature on pressure, and α, caused by dependence of viscosity on pressure Figure 7: Dependence of friction force of te bearing on te following parameters: ω, caracterising an adapted sape of te abutment, and structural and viscous parameter N 4845

7 International Journal of Applied Engineering Researc ISSN Volume, Number 5 (7) pp ACKNOWLEDGEMENTS Te publication as been issued in te scope of realization of fellowsip of OJSC «RZD» No 37/6 for development of scientific and pedagogical scools in te area of te railway transport REFERENCES [] Akverdiyev КS, Vovk AYu, Mukutadze MA, Savenkova MA, Analytical metod for prognosis of values of micropolar lubrication criteria providing stable operation of radial sliding bearing Journal of Friction and Wear 8 V 9, No P 84 9 [] Mukutadze MA, Matematical Model of a Compressible Micropolar Hydrodynamic Lubrication of a Radial Bearing wit Adapted Sape of Its Bearing Surface Bulletin of Don State Tecnical University V, No 8 (59) P 4 44 [3] Akverdiev KS, Mukutadze MA, Mukutadze AM, Radial bearing wit porous barrel Proceedings of Academic World: International Conference, 8 t of Marc, 6, San Francisco, USA IRAG Researc Forum : Institute of Researc and Journals, 6 P 8 3 [4] Akverdiev KS, Mukutadze МА, Vovk AY, Semenko IS, Hydrodynamic Calculation of a Radial Bearing Operating in a Non-Stationary Mode on a Viscoplastic Lubricant Possessing Micropolar Properties Bulletin of Rostov State Transport University 8 No 4(3) P 3 38 [5] Zadoroznaja EA, Karavaev VG, Estimation of a eat state of a eavy-loaded bearing taking into account reological properties of a lubricant Internal combustion engines All-Ukrainian researc journal Karkov: Publising ouse «Karkov Polytecnical Institute» No P [6] Zadoroznaja EA, Solution of a termoydrodynamic problem of lubrication of eavy-loaded plain bearers taking into account reological properties of a liquid lubrication Problems of mecanical engineering and reliability of macines 4 No 4 P 7 8 [7] Matveev, VA, Orlov OF, Determination of dynamic viscosity of te substance depending on pressure and temperature Te Bulletin of Bauman MSTU Series «Sciences» 9 No 3 P 6 8 [8] Albagaciev AJ, Kozemjakina VD, Cicinadze AV, Frictional, wear and temperature caracteristics of materials during ig-speed slippage in macines and devices Friction and lubrication in macines and mecanisms No 3 P 9 9 [9] Prokopyev VN, Rozdestvensky YV, Karavaev VG, Zadoroznaja EA [etc], Dynamics and lubrication of tribo-units of piston and rotary macines: A monograp Celyabinsk SUSU Publising Center, Part 36 p [] Prokopyev VN, Rozdestvensky YV, Karavaev VG, Zadoroznaja EA [etc], Dynamics and lubrication of tribo-units of piston and rotary macines: A monograp Celyabinsk SUSU Publising Center, Part p [] Prokopyev VN, Zadoroznaja EA, Karavaev VG, Levanov IG, Improvement of a calculation metod of te eavy-loaded plain bearers greased wit nonnewtonian oils Problems of mecanical engineering and macines reliability, No P [] Akverdiev KS, Mukutadze AM, Researc of Drive Factor of Damper wit Double-Layer Porous Ring wit Compound Feed of Lubricant Material International Journal of Applied Engineering Researc 7 No P