e eccentricity; δ radial clearance, m;

Transcription

1 International Journal of Alied Engineering Research ISSN Volume, Number 9 (7) Research India Publications. htt:// Radial Plain Bearings Oerating on Viscoelastic Lubricant Caused by the Melt, Taking into Account the Deendence of the Viscosity of the Lubricant and the Shear Modulus on the Pressure Elena Olegovna Lagunova, Rostov State Transort University (RSTU), Chair Higher Matnematics, Rostov-on-Don, 34438, Russian Federation. Orcid ID: Abstract The method of forming an exact self-similar solution for the hydrodynamic calculation of a radial lain bearing oerating on a viscoelastic liquid lubricant due to the melt of the guide is given in the aer, taking into account the deendence of the viscosity of the lubricant and the shear modulus on ressure. Based on the system of equations for the motion of the incomressible fluid with viscoelastic roerties, the analytical deendence for the rofile of the molten surface of the guide is obtained for the case of a "thin layer", taking into account the deendence of the viscosity of a viscoelastic lubricant and the shear modulus on ressure, and the continuity equation, exressions for the energy dissiation rate. In addition, the main oerating characteristics of the friction air under consideration are determined. The effect of the arameter due to the melt of the guide and the Deborah number on the comonents of the suorting force vector and the friction force are estimated. Keywords: hydrodynamics, radial lain bearing, viscous incomressible liquid viscoelastic lubricant, molten surface of a guide. INTRODUCTION Tribosystems of modern machines oerate under high loadseed regimes. In these conditions, the use of liquid friction rovided by liquid lubricants is the most romising. And the absolute advantage in this mode of lubrication belongs to the hydrodynamic rocess. The wide alication of hydrodynamic lubrication in a wide variety of machines and mechanisms required the develoment of a significant range of the new highly effective lubricating fluids, such as micro-olar, viscoelastic, viscolastic comressible and incomressible. The ractice of using these liquid lubricants significantly outstris the develoment of models and theoretical calculations necessary for their wide alication in ractice. The attemt to develo a general methodology for calculating the hydrodynamic lubrication regime for different bearing designs, lubricated with liquids with different hysical and mechanical roerties is of articular imortance and scientific interest. Signs μ characteristic viscosity, Ns/m ; hydrodynamic ressure, Pa; α constant exerimental value, μ coefficient of dynamic viscosity of the lubricant, Ns/m ; r shaft radius, m; r bearing radius, m; e eccentricity; δ radial clearance, m; ω, a arameters characterizing the adated bearing rofile; uv, comonents of the velocity vector of the lubricating medium; α arameter characterizing the deendence of viscosity on ressure, β Deborah number, G characteristic value of the shear modulus, Pa; G shear modulus, Pa. h oil film thickness, m; angular rate, с ; η relative eccentricity of the bearing bush. It should also be noted that currently there are a number of romising directions for new research in the field of 98

2 International Journal of Alied Engineering Research ISSN Volume, Number 9 (7) Research India Publications. htt:// hydrodynamics, the further develoment of which will substantially accelerate their bringing to industrial alication. Such areas include smearing of low-melting metal alloys with a melt. TASK SETTING The steady flow of an incomressible viscoelastic lubricant in the ga of an infinite radial lain bearing of a melt-coated lowmelting coating is under consideration. The shaft is rotating with the angular seed (Fig. ), and the bearing bushing is stationary. It is assumed that the sace between the eccentrically located shaft and the bearing is comletely filled with a viscoelastic lubricant, and the bearing bush is made of a material with a low melting oint. In the olar coordinate system r, θ, ole of which is located in the center of the bearing bush, the equation of the contour of the shaft of the molten surface of the bearing bush and the surface of the bearing bush covered with a metallic melt will be written as follows: r r H, r r, r r f, () e where H cos sin..., ; r is the shaft r radius; r is radius of the bearing bush covered with a metal melt; е is eccentricity; ε is relative eccentricity; f is bounded function with is subject to defining.. The radial comonent v r the seed is slightly less than its circumferential comonent v θ. 3. The ressure is constant throughout the thickness of the lubricating film. The deendence of the viscosity of the lubricant and the shear modulus on the ressure is exressed by the deendences: α р е, μ = μ G α = Gе р, () where μ is characteristic viscosity, μ is coefficient of dynamic viscosity of the lubricant, р is hydrodynamic ressure in the lubricating layer, α is constant, G is characteristic value of the shear modulus, G is the shear modulus. INITIAL EQUATIONS AND BOUNDARY CONDITIONS The system of dimensionless equations of motion of a lubricant having viscoelastic roerties (the Maxwell liquid) is taken as initial equations for the case of a "thin layer", and also the continuity equation with allowance for (): v d d, u v e e, r d d r is the Deborah number,, where G (3) uvare comonents of the velocity vector of the lubricating medium; is hydrodynamic ressure in the lubricating layer. In order to define the function f, caused by the molten surface of the bearing, we use the formula for the rate of energy dissiation. d K d h v dr, r (4) Figure : Design Scheme The conditions for the motion of the infinite radial lain bearing are considered under the following assumtions:. All the heat released in the lubricating film goes to the melting surface of the material of the bearing bush. r K L where dissiation of mechanical energy, is arameter characterizing the rate of secific heat of fusion er unit volume. h θ ηcosθ, L is The boundary conditions for the system of equations (3) and (4) within the accuracy of Оε will be written as: 99

3 International Journal of Alied Engineering Research ISSN Volume, Number 9 (7) Research India Publications. htt:// u, v sin at r cos h ; v, u at r ;, (5) a e ; ;. where f When forming the analytical exression for hydrodynamic ressure, we transform the boundary conditions, assuming that the lubricant undergoes a shift at the entrance to the loaded zone. We believe that the lubricant enters the zone of hydrodynamic flow with comlete relaxation. Then the corresonding boundary conditions for the hydrodynamic ressure can be written in the form: dc β d, at θ. dθ μ dθ (6) If the shift occurs quickly enough comared to relaxation, the lubricant will be in a relaxed state and its shift will occur at the time it enters the loaded bearing zone. Suose that this is the case. This is equivalent to assuming: at θ. (7) or, alying this condition to the entire liquid at the time it enters the bearing, d с, at θ. dθ (8) The corresonding boundary conditions are defined as: d, с dθ at р θ ; р π. (9) р Relations between dimensionless and dimensional variables are given in the form: р а r r r, r r ; v r v; θ μr vr δ u; ; ; δ c μ ; G GG; c c c; μ r δ ;. () Let s introduce the signs, let Z = e α. Having differentiated both sides of the equation within the accuracy of the members O αβ the first equation of the system (3) takes the form: d Z dz C C. 3 d d h h Then the system of equations (3) and (4) takes the following form: d Z dz C C u v ; ; 3 d d h h r θ cos d v Z K dr d r With corresondent boundary conditions () u, v sin at r cos ; v, u at a ; ; r Z Z e Z ; c. () Taking K as a small arameter due to the melt and the rate of energy dissiation, we seek the function as: 3 3 K K θ K θ H, (3) 3 where H K K K θ θ θ. 3 Boundary conditions for dimensionless velocity comonents u and v on the contour r θ can be written as follows: v v v H v H H ; r r r r u u u H u H H. r r r r (4) We seek the asymtotic solution of the system of differential equations () with allowance for the boundary conditions () and (4) in the form of series under degrees of the small arameter К: v v ( r, ) Kv ( r, ) K v ( r, )...; 93

4 International Journal of Alied Engineering Research ISSN Volume, Number 9 (7) Research India Publications. htt:// u u ( r, ) Ku ( r, ) K u ( r, )...; 3 K K K θ θ ; 3 3 Z Z KZ K Z K Z (5) 3... Performing the substitution (5) into the system of differential equations (), taking into account the boundary conditions () and (4), we obtain the following equations: - for the zeroth aroximation: v d Z dz v u, r d d r With boundary conditions: u v, v sin at r cos ;, u at r ; K Kg ; a Z Z ; c ; Z. (7) for first aroximation: v d Z dz v u ; ; r d d r cos d v Z K dr d With boundary conditions: r r r r r v u v ; u ; c ; Z ; v ; u at r cos ; (6) (8) Z Z ; K K,. (9) EXACT SELF-SIMILAR SOLUTION The exact self-similar solution of the roblem for the zeroth aroximation will be sought in the form: v V r, ; u U r, ; r r r, ; ; cos V r, v ; U r, u h ; d Z dz C C ; 3 h cos. d d h h () Substituting () into the system of differential equations (6), taking into account the boundary conditions (7), we obtain the following system of differential equations: C v C u v ; ; ; () And boundary conditions: u v,, sin, ; Z a ; Z u, v, v d. () By the direct integration we get: C C, v C, 3η С 6, С 6α. (3) 6η 5. Determination of hydrodynamic ressure Solving the equation for hydrodynamic ressure d Z dz C C taking into account the ; 3 d d h h boundary condition get: Z Z e a a Z we 3η βcos θ+ sinθ ηβcosθ+sinθ 3η θ Z 6α θ+ β 4β + 3η βcosθ+ sinθ 3η 3ηβcosθ+sinθ 6α θ+ 3η θ 6η β 4β + 93

5 International Journal of Alied Engineering Research ISSN Volume, Number 9 (7) Research India Publications. htt:// 6η β β 4ηβ 4β 6α. 6η 4β β α a e In order to define taking into account the equation (3), we come at the following equation: (4) d v h d. d h h (5) Integrating the equation (5), we obtain: 3 3 h h h d d d. where (6) C d ; v d C ; v d 4. 3 Solving equations (6) taking into account (7) and condition K K, we obtain: (7) C 3 3 sin 3 sin C sin sin 4 sin sin. Then for the first aroximation we get: v V r, ; u U r, ; r r r, ; ; cos V r, v ; U r, u h ; d Z dz C C ; h 3 cos. d d h h Substituting (9) into the system of differential equations (8), taking into account the boundary conditions (9), we obtain the following system of differential equations: С ; v C ; u v ; (3) And boundary conditions: u v,,, ; (9) (8) 93

6 Z International Journal of Alied Engineering Research ISSN Volume, Number 9 (7) Research India Publications. htt:// a Z ; Z ;,,. (3) u v M v d a From conditions Z Z ; Z, we obtain Z β 4β + 3η β cos θ+ sinθ η β cosθ+sinθ 3η θ 6αМ θ+ 3η β cos θ+ sinθ 3η 3ηβ cosθ+sinθ θ+ 3η θ 6η β 4β + 6η β β 4ηβ 4β, 6η 4β β (33) where С 3η 6α М, 6η v sin M su su 3 cos sin cos [: ] r r [: ] cos 3 sin cos cos sin cos sin cos 4 3 cos sin cos sin cos sin cos sin sin 7 C sin 4 sin sin. 4 4 (34) 933

7 International Journal of Alied Engineering Research ISSN Volume, Number 9 (7) Research India Publications. htt:// Then for Z Z KZ we get the following exression Φ α a Z 6α KM x e, (35) where 3η βcosθ+ sinθ ηβcosθ+sinθ 3η θ Φ x θ+ β 4β + 3η βcosθ+ sinθ 3η 3ηβcosθ+sinθ θ+ 3η θ 6η β 4β + 6η β β 4ηβ 4β. 6η 4β β Then alying Taylor series exansions α e and α a e, we get a α α e 6α KM x e Φ (36) a a 6 KM x. (37) Solving the equation (37) within the accuracy of O(α 3 ), O a 3 a a a 6 KM x. for hydrodynamic ressure we obtain (38) RESULTS OF THE RESEARCH AND THEIR DISCUSSION Let us now turn to the determination of the basic oerating characteristics of the bearing. Taking into account (6), (8), and (38) for the comonent of the suorting force vector and the frictional force, we obtain: 3 r a Rx cosd 3 6 r a a 3 3 KM, 6 934

8 International Journal of Alied Engineering Research ISSN Volume, Number 9 (7) Research India Publications. htt:// 3 r a Ry sind (38) 3 6 r a a 3 3 KM, 6 r e v v Lтр K d r r r r r KM 6 M arctg. For the verification calculations, the following values are used based on the theoretical models obtained: μ =,85 Нc/м ; η =,3 м; r =,9995,493 м; Ω = 8 с ; δ =,5 3,7 3 ; К =,,5;,8,35 МПа; α =...; L = 35,33 38, N/m ; М = 6,5. а Based on the results of numerical calculations, the grahs shown in Fig. -5 were built. Figure : Deendence of the comonent of the suorting force R ( x ) on the arameter α, which characterizes the deendence of viscosity on ressure, and on the arameter M, which characterizes the thickness of the molten film. Figure 3: Deendence of the comonent of the suorting R force ( x ) on the arameters: M, characterizing the thickness of the melted film, and K, due to the melt and the rate of energy dissiation. 935

9 International Journal of Alied Engineering Research ISSN Volume, Number 9 (7) Research India Publications. htt:// Figure 4: Deendence of the comonent of the suorting force R ( x ) on the arameter α, characterizing the deendence of viscosity on ressure, and on the arameter K, due to the melt and the rate of energy dissiation. Figure 5: The deendence of the frictional force on the arameter α, which characterizes the deendence of viscosity on ressure, and on the arameter K, due to the melt and the energy dissiation rate. CONCLUSIONS The analysis of the calculated models and grahs allows us to draw a number of the following conclusions:. A refined design model of a radial lain bearing oerating under conditions of hydrodynamic lubrication with a melt of a low-melting coating is obtained, taking into account the deendence of the viscosity of the lubricant and the shear modulus on ressure.. A significant contribution of the constructive arameter K due to the melt is shown. As the design arameter K increases, the frictional force decreases, and the bearing caacity increases. ACKNOWLEDGEMENTS The ublication has been issued in the scoe of realization of fellowshi of OJSC «RZD» No. 37/..6 for develoment of scientific and edagogical schools in the area of the railway transort. REFERENCES [] Kroachev D.Y. Methods For the Raid Measurement of the Temerature of Metal Melt for the Needs of Machine-Building Enterrises / D.Y. Kroachev, А.А. Grishin, A.D. Maslo // Casting and Metallurgy.. 3(66). Pages 6 7. [] Wilson. Lubricant with Melt / Wilson // Problems of Friction and Lubricant Page 9. [3] Beretta. Sliding Bearings, Lubricated by Their Own Melt or a Product of Subli-mation / Beretta, Niro, Silvestri // Works of American Engineering Society С [4] Physical Values. Directory. М. : Energoatomizdat, 99. [5] Khavin, V.Y. Brief Chemical Directory / V.Y. Khavin. L. : Khimiya, 99. [6] Perelman V.I. Brief Reference Book of the Chemist / V.I. Perelman. M.-L.: Kimiya, 964. [7] Soldering Directory. nd Edition, Modified and Added. М. : Mashinostroenie, 984. [8] Kotelnitskaya L.I. Calculation of Radial Bearings with Efficient Work on Lubrication with Melt in Turbulent Mode / L.I. Kotelnitskaya, N.N. Demidova // Bulletin of Rostov State Transort University... ages 8 3. [9] Prikhodko V.M. Mathematical Model of Hydrodynamic Lubrication During Melting of Bearing Surface of Radial Bearing / V.М. Prikhodko, L.I. Kotelnitskaya // Friction and Wear.. V., 6. Pages [] Akhverdiev K.S. On a Solution of the Problem of Hydrodynamic Lubrication with a Liquid Formed During the Melting of a Guide, in the Presence of Forced Lubrication / K.S. Akhverdiev, L.I. Kotelnitskaya, N.S. Voronin // Bulletin of the Don 936

10 International Journal of Alied Engineering Research ISSN Volume, Number 9 (7) Research India Publications. htt:// State Technical University.. V., (). Pages [] Akhverdiev K.S. The Steady Motion of a Viscoelastic Fluid Between an Inclined Slider and a Guide Taking Into Account the Inertial Forces of the Lubricating Comosition/ K.S. Akhverdiev, I.A. Zhurba // Friction and Wear. 4. V. 5, 6. Pages [] Mukutadze М.А. Develoment of the System of Design Models of Sliding Bearings Based on the Develoment of the Hydrodynamic and Rheodynamic Theory of Lubrication: Thesis work дof PhD in Technical Sciences / М.А. Mukutadze. Rostov n/d, 5. Pages [3] Akhverdiev, K.S. Radial Bearing with Porous Barrel / K.S. Akhverdiev, M.A. Mukutadze, A.M. Mukutadze // Proceedings of Academic World : International Conference, 8 th of March, 6, San Francisco, USA. IRAG Research Forum : Institute of Research and Journals, 6. P [4] Akhverdiyev КS, Vovk AYu, Mukutadze MA, Savenkova MA, Analytical method for rognosis of values of microolar lubrication criteria roviding stable oeration of radial sliding bearing. Journal of Friction and Wear. 8. Т. 9,. P [5] Lagunova, E.O. Simulation Model of Radial Bearing, Taking into Account the Deendence of Viscosity Characteristics of Micro-Polar Lubricant Material on Temerature // International Journal of Alied Engineering Research. 7. Vol