Effect of surface roughness and slip velocity on the performance of a hydrostatic circular footstep bearing S.D. Shukla 1)
a*
, G.M. Deheri b,
Department of Mathematics, Shri R.K.Parikh Arts and Science College, Petlad, Gujarat, India- 388450 2) Department of Mathematics, S.P.University, Vallabha Vidyanagar, Gujarat, India- 388450 * Corresponding author:
[email protected]
Abstract An endeavour has been made to study and analyze the performance of a hydrostatics circular rough footstep bearing considering the slip velocity. The velocity slip is modelled by the method of Beavers and Joseph. The stochastic model of Christensen and Tonder has been adopted to evaluate the effect of transverse surface roughness. With the aid of suitable boundary conditions, the governing equation for the radial flow rate is solved to obtain the pressure distribution, leading to the calculation of load carrying capacity. It is found that the combined effect of slip velocity and surface roughness is to decrease the load carrying capacity significantly, in general. However, the situation remains far improved when negatively skewed roughness is involved and the slip velocity is at minimum. This effect further enhances when negative variance occurs. Keywords: circular footstep bearing, roughness, slip velocity, load carrying capacity
1.
Introduction the different recess depth and rotating velocity using finite volume method and the simulation results indicated that an improved characteristic would be affected by recess depth and rotating velocity. Less attention has been paid to study of effects of velocity-slip at the surface, although it may be of importance in the flow behavior of gases and liquids, particularly, when the film is thin. Singh et al. [11] obtained the steady state solution for aerostatic porous thrust bearings incorporating the effect of velocity slip. Chattopadhyay and Majumdar [12] theoretically investigated the static characteristics of hydrostatic porous oil bearings with tangential velocity slip at the porous interface. Rao et al. [13] investigated the effects of velocity slip and viscosity variation on squeeze film lubrication of two circular plates. Here, it was observed that load capacity decreased due to slip velocity. After having some run-in and wear, the bearing surfaces tend to develop roughness. When the gap between two mating surfaces becomes smaller, the effects of roughness become more important. In most of the applications, the smooth bearing surfaces would not be valid for the accurate prediction of the performance and life of the bearings. Thus, surface roughness has been studied with much interest in the recent years because all bearing surfaces are rough to some extent. Also, to increase the performance of different bearings, it is essential to study the influence of surface roughness. The random character of the roughness was recognized by Tzeng and Seibel [14] who employed a stochastic approach to study the roughness. To investigate the effects of surface roughness, many methods have been proposed and implemented in the context of surface lubrication. The efforts were made by Christensen and Tonder ([15], [16] and [17) and Chow and Cheng [18] within the frame work of the stochastic theory . Lin [19] theoretically investigated
The basic idea of a hydrostatic bearing is to pressurize a fluid to produce a fluid film between two surfaces which move relative to each other. The fluid film thickness is larger than the surface roughness, so the two surfaces never contact during motion. Additionally, because of the external fluid pressurization, the supporting force is independent of surface speed. This insensitivity to relative surface speed differentiates hydrostatic bearing from a hydrodynamic bearing. Hydrodynamic bearings have great use in Mount Palomar telescope, and in many radar installations. Hydrostatic bearings appear in the literature as early as 1851Girard [1]. The performance of a thrust bearing was analyzed by Elwell and Sternlicth [2] theoretically as well as experimentally. Dowson [3] studied the inertia effects in hydrostatic thrust bearing. Stiffness and damping characteristic of compensated thrust bearings were considered by Ghosh and Majumdar [4]. Another hydrostatic development of note was invented by Wasson and Slocum [5] which involved placement of all fluid routings on the surface of the bearing. Kotilainen and Slocum [6] expanded upon Wasson’s work by casting the surface features in to the inner diameter of a bearing sleeve. Lu et al. [7] developed fluid-structure interaction model and calculated temperature rise and pressure distribution by making use of computational fluid dynamics. Srinavasan [8] analyzed the static and dynamic load on hydrostatics bearing with variable viscosity and pressure. Sharma and Yadav [9] studied the Rabinowitsch fluid model for the analysis of the performance of hydrostatic circular thrust bearing. Yu et al. [10] theoretically studied the dynamic pressure of hydrostatic thrust bearing under Corresponding author:
[email protected]
336
ISBN - 978-93-5107-261-4
where is the mean film thickness, while is a randomly varying portion measured from the mean level characterizing the random roughness. Here is assumed to be governed by the probability density function
$ # % " & & " , !" "$
the effect of surface roughness on the dynamic stiffness and damping characteristics of compensated hydrostatic circular step thrust bearing Here, it has been deemed appropriate to make an investigation in the performance of a rough hydrostatic circular footstep bearing taking the slip velocity in to account.
2.
' ()*(+ ((
Analysis
The details regarding the mean α, standard deviation σ, and skewness ε associated with the characterization of roughness, can be seen from Christensen and Tonder [15], [16] and [17]). Stochastically averaging equation (1) by the method of Christensen and Tonder, and integrating with the the associated boundary condition ', expression for pressure distribution is found to be
The geometry and orientation of the bearing structure is given in Fig. 1(a) and 1(b). A shaft of outer radius is located co-axially over a plane pad with a circular recess of radius . The chamber is supplied with oil at pressure
-. )01 2 4 / 3
where 5 6 $ 7 $ 6 $ 8 7 $ 6 6 The other boundary condition is ,
Using the following dimensionless scheme 8 7 6 8 9 7 9 6 9 9 9
5 > (Cameron [20]). The load carried is the pressure integral over the entire bearing surface,
Fig.1 (b) Flow from a footstep bearing
Under usual assumptions of hydrostatic lubrication theory the governing equation for the radial flow rate Cameron [20], by employing the Beavers and Joseph [21] model for smooth bearing system takes the form
$E 3D
?
