Oynak Lokmalı Eksenel Hidrodinamik Kaymalı Yataklarının Davranışının Sayısal Olarak Belirlenmesi İçin Yeni Bir Algoritma

Abstract

Bilgisayarların devreye girmesi, bellek kapasitelerinin ve işlem hızlarının giderek artması çeşitli nümerik metodların hemen hemen her türlü yağlama probleminin çözümü için kullanılmalarına imkan vermiştir. Bu metodlardan "sonlu farklar" metodu günümüzde yağlama problemlerinin nümerik olarak çözümünde en fazla kullanılan bir yöntem olarak bilinmektedir. Yapılan bu çalışmada herhangi bir lokma geometrisine ve akışkan film şekli geometrisine bağlı kalınmaksızın istenilen geometride çalışma imkanı sunulmuştur. Yani belli geometriler için çalışma sınırlaması ortadan kaldırılarak daha geniş bir geometrik serbestlik getirilerek geometrisi literatürde olmayan yataklar için de performans değerleri hesaplanmaktadır. Bu çalışmada daire sektörü şekline sahip oynak lokmalı eksenel hidrodinamik kaymalı yatakların performans karekterlerini hesaplamak için Genelleştirilmiş Reynolds denklemi kutupsal koordinatlara göre elde edilmiş ve sonlu farklar metodu kullanılarak nümerik şekilde çözülmüştür. Hazırlanan bilgisayar programı istenilen yatak geometrisinin, ve yatak büyüklüklerinin boyutlu olarak tanımlanmasına imkan vermekte ve böylelikle sonuçlar doğrudan, herhangi bir dönüşüm yapmadan boyutlu olarak elde edilebilmektedir. Böylelikle farklı geometrilere sahip yatakların belli geometrik yataklar için hazırlanmış çeşitli boyutsuz eğrilere bağlı kalmaksızın performans değerlerinin doğrudan boyutlu olarak elde edilmesi imkanı sağlanmıştır. Yatak performans değerleri olarak yük taşıma kapasitesi, lokmanın istenilen herhangi bir yönündeki basınç değişimi, lokma açılan, minumum yağ film kalınlığı ve yatak üzerindeki yeri, ortalama film kalınlığı, basınç merkezi gibi değerler hesaplanmaktadır.

The study of hydrodynamic lubrication is, from a mathematical standpoint, the study of a particular form of Navier-Stokes equations. This particular differential equation was formulated by Osborne Reynolds in 1886 in the wake of a classical experiment by Beauchamp Tower in which the formation of a thin fluid film was for the first time observed and understood to be the basic mechanism of hydrodynamic lubricatin. This Reynolds equation can be deduced either from the Navier-Stokes equations or from first principles, provided of course, that the same assumptions are used in both methods. The Reynolds equation contains viscosity, dencity, and film thicness as parameters. These parameters both determine and depend on the tempereture and pressure fields and on the elastic behavior of the bearing surface. Thus, to get a complete and accurate representation of the hydrodynamics of the lubricatimg film, it is offentimes necessaty to consider simultaneously the Reynods equations, the energy equation, the elasticity equation, and the equation of state [1]. The differential equation originaly derived by Reynolds is restricted to incompressible fluids. This, however, is an unneccessary restriction ; for the equation can be formulated broadly enough to include effects of compressibility and dynamic loading. We have called this the generalized Reynolds equation. To get this equation we make this assumptions (see figure 1.3.). 1. The heith of the fluid film y is very small compared to the span and length x, This permits us to ignore the curvature of the fluid film. 2. No variation of pressure accros the fluid film. Thus, cp/dy = 0 3. The flow is laminar; no vortex flow and no turbulence occur anywhere in the film. 4. No external forces act on the film. IX 5. Fluid inertia is small compared to the viscous shear. These inertia forces consist of acceleration of the fluid, centrifugal forces acting in curved films, and fluid gravity. y t U2 ui z.*- Figure 1. The Fluid Film. 6. No slip at the bearing surfaces. 7. Compared with the two velocity gradients du I dy and dw I dy, all oyher velocity gradients are considered negligible. Since « and, to a lesser degree,^ are the predominant velocities and y is a dimension much smaller man either x or z, the above assumption is valid. Thus any derivatives of terms other than du I dy and dw By will be of a much higher order and negligible. We can thus omit all derivatives with the exeption of cP-u/dy^ and d^u/dy^ With all above assumptions and setting up a force balance of an element of fluid in the lubricant film and continuity equation and other mathematical computation we get this equation, ax r"u3 a_ > ph^_ap r| dx. f~Ui X~\ + dz ph v n dz) !-<».-«j%W This equation is called as generalized Reynolds equation [1]. ' This equation is valid for both incompressible and compressible lubricants. But in this study we deal on the incompressible lubricants so we can set p = constant in the equation. This time we get this equation, d_ dx ph dp d h dp = 6(U1-U2)£+12V In cylindrical coordinates using the substitutions x = r Cos© and z = r Sin6 U = wr =3inr/30 the generalized Reynolds equation becomes *frh^]+IJL(h.|^ dx \ dx) x 8QK. BQJ öh = 6wr^+12rV method. In this study this equation is solved numerically with using mite difference L*& Figure 2. Mesh ofThe Pad XI A computer programme have been developed for determining the thrust bearing Performance characteristics. For determining the pressure distribution in the lubricant film is obtained by solving the established steady-state Reynolds equation for incompressible lubricant by finite difference method [2]. The parameters in this equation are solved numerically a 40x40 mesh size and using central difference expression for the Reynolds equation. The film thickness at any point on the mesh can be written as both the function of r and 9, so ve have h (r,0) = h0 - ( Rjj. Cos0 - Rq ) tanp + Ry SinG. tana And we can compute partial differencial of this according to r and 0, 6h dr = HARETU = -Cos0- tanp + Sin0- tana dh 50 HATETU = Rt (j Sin0- tanp + R; tj CosG tana On the mesh we define i,j points. With an assumed film thickness at the center of the pad, the film thicness at every mash point is determined. Firstly an assumed pressure assigned every point. Then the second set of pressures are determined and this proces of iteration continues till the pressures converge to a limit of error of 0.001 Now final pressures are determined. d? d? c:P 1 cP 1 d:? h3 - + 3h:r. HARETU. - + rh3 -r^r + - 3h:. HATETU ?- - +- h3 -r = 6wrn.. HATETU er er dr- r cQ r 36" >-j P -P -^ ri-l,j ri-l,j ^ 2Ar + 3h-J-rij fP -P ^ ri-l.j fi-l.j, 2Ar HARETU+ r, j. h30 ^P +P -^P ^ Ar2 i f P -P J_ 2 ri,j^l ri.j-l ry '< 2A0 J HATETU+- ?.j '.j p + p _ op fi,J-l ^ fi.J-l - ri.J A02 6wrM. HATETU Xll A: = Ki fp -P ^ ^ 2Ar A2= 3hr,rrio p -p ri^l,j ri-l.j 2Ar.HARETU A3= wK P.-i.+P,-,, Ar: A4 = "T hf.j i.j P -P ri,J-l ri-J-I 2A6 HATETU A,= h3/p..,+P ^ i.J ?i,j-l "i-J-l A92 D = owr^HATETU B = P r-2-r-hf; 2-hf, '.j 1.J l.J '.J V Ar- rl0. A9-, A, + A2 + A3 + A4 + As + ^ j. B = D pi.= D -( A1+A2+A3+A4+A5) B In this study performance characteristic of thilting-pad thrust bearing is computed in dimensional form. In the thilting-pad thrust bearing, the pads are free to tilt on spherical pivots, and can adapt to any equilibrum position for a given lubricant film thicknes at pivot point. In studying dimensinlees form we make some set of graphical configurations and then for a given bearing we choose some values at the configuration and then we do some compution to transform this values to dimensional form. Besides this we only can determine the same shape of bearings as in the given at graphic. Xlll Whith presented computer programe in this study we can define any shape of thrust bearing in the characteristic of dimensional form and we directly get the balance characteristic of the bearing such as pressure distribution both circumfrential and longitudinal direction, pad balance angles, minumum film thicknes and its position on the pad, pressure centre and its position on the pad. And we get a set of solution by setting different bearing velocities, different viscosities, different mean film thicknes and different pivot positions.