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\title{A Computational Approach on the position of Load Centre of a Slipper Bearing}
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\begin{document}

             \author[1]{Dr. Kishan  Choudhuri}

             \affil[1]{  National Institute of Technology, Agartala.}

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\date{\small \em Received: 8 December 2011 Accepted: 3 January 2012 Published: 15 January 2012}

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\begin{abstract}
        


The slipper bearing is an integrated part of an axial piston pump. Proper lubrication is important for successful operation of the slipper bearing. This type of bearing is a type of hydrostatic thrust bearing. Many works have been done in the recent past on this type of bearing. This current work is based on the theoretical investigation of the locus of the load centre of this type of bearing. The leakage losses and the slipper drag are two important factors on which this type of bearing is designed. The load carrying capacity has a direct impact on those two factors. On the other hand the stability of the bearing depends upon the position of the load centre. There are many input factors on which the position of load centre is varied. These input factors thus play an important role on the smooth operation of such bearing. Such input variable are slipper tilt, applied pressure on the slipper, slipper speed, slipper non flatness angle or slipper land size. On the variation of these input variables, the nature of the position of load centre is plotted in this work. Based on the results the reasonable conclusions are made.

\end{abstract}


\keywords{Slipper, position of load centre, slipper tilt, lubrication.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
xial piston swash-plate type hydrostatic pumps are being used extensively in aircraft, industrial and agricultural systems since they can transmit large specific power and the flow rate from them can be varied. A basic difference in the design of various models of axial piston pumps is how the pistons contact the swash plate. Many design use a bronze slipper positioned between the piston and the swash plate. With this design, hydraulic fluid is fed through internal passages to the piston/slipper and slipper/swash plate interfaces to supply lubrication at these surfaces. Some axial do not use a slipper, but rather finish each piston with a case-hardened spherical dome. The spherical dome contacts the swash plate in such a fashion, much like the contact that occurs in ball bearings. Elimination of the slipper reduces costs and eliminates the disadvantages of the slipper design, but unfortunately, it creates other problems. One of these is wear at the spherical dome/swash plate interface. The fig. \hyperref[fig_2]{1} shows a typical slipper-piston assembly. The slipper is pivoted on the ball at the end of the piston to allow it to adjust to the swash plate angle and to rotate relative to the piston. High pressure fluid from the piston is connected via the control orifices in the piston and slipper to the central slipper pool allowing covered the influence of the orifice size on the performance of the bearing. Koc and Hooke \hyperref[b0]{[1]} examined the effect of the tilting couples on the behavior of the slippers experimentally. Wang and Yamaguchi \hyperref[b1]{[2]}, \hyperref[b3]{[3]} clarified experimentally and theoretically the effects of nozzle and thermoplastic materials on the characteristics of hydrostatic bearing/seal parts in water hydraulic axial pumps and motors. Manring \hyperref[b4]{[4]} investigated the effects of pressureinduced deformations on the characteristics of hydrostatic thrust bearing. Manring \hyperref[b5]{[5]} investigated experimentally, the effect of different socket geometry in the performance of slipper bearing. They found the effect on the leakage flow, load carrying capacity and the film thickness of the slipper bearing. In the work of Nie S. L. \hyperref[b6]{[6]}, the characteristic equation of the hydrostatic slipper bearing with an annular orifice damper is formulated, where the effects of various geometric parameters (e.g. damping length, supporting length, and clearance between the piston and the cylinder bore) are reflected. S. Kumar, J.M. Bergada, J. Watton \hyperref[b7]{[7]} presented static and dynamic characteristics of a piston pump slipper with a groove. Three dimensional Navier Stokes equations in cylindrical coordinates have been applied to the slipper/plate gap, including the groove. In the work of M. Borghi, E. Specchia and B. Zardin \hyperref[b8]{[8]} a numerical procedure is used to solve the Reynolds equation, written here with respect to the slipper-swash plate gap, whose height is considered variable in a two dimensional field and with time. In the work of Hong Liu, Zeng Xiong Peng, Chu Jing Shen \hyperref[b9]{[9]} the calculation of film shape is simplified as a single objective optimization problem with two decision variables. A genetic algorithm is used to investigate about the film shape of the entire slipper bearing In the work of Fazil Canbulut, Erdem Koç, Cem Sinanoglu \hyperref[b10]{[10]}, the slipper geometry and working conditions affected on the slipper performance have been analyzed experimentally. The model of the slipper system has been established by original neural network (NN) method. The objective of the present work is theoretical investigation of the position of the load centre of the slipper bearing which is not studied extensively by the previous authors. ? The flow is laminar.\par
? Surface velocities are considered to be constant in direction.\par
? The lands is approximately conical.\par
Slipper is having a circular pocket, which is surrounded by a land as shown in figure  {\ref .} 1. The orifice connects the slipper pocket to the piston bore which feeds it with oil thus establishing a pressure in the slipper pocket which is approximately equal to piston pressure. The oil inside the slipper pocket lubricates the total slipper area. Referring to the Figure \hyperref[fig_4]{2} the clearance between the land of an untilted slipper and the swash plate can be expressed as . When slipper is tilted (Figure.  
\section[{Mathematical Model}]{Mathematical Model}\par
The following assumptions are taken into account to derive all the equations:\par
? Body forces acting on the lubricant such as gravitational, magnetic or electrical are neglected\par
? The pressure induced flow in the circumferential direction is neglected\par
? The pressure is assumed to constant through the thickness of the lubricating fluid.\par
? The lubricant is Newtonian.\par
? Viscosity is considered to be constant through the thickness of the lubricating film. This leads to a pressure distribution:\par
()\textbf{1}\par
where, is the non-dimensional width of the slipper land.\par
The first group of the equation ( \hyperref[formula_0]{1}) corresponds the hydrostatic pressure distribution for flat and untilted slipper and the second group corresponds the hydrostatic pressure distribution produced by coning of the land and the tilt of the slipper. The final group represents the hydrodynamic effects due to conical shape of the land, slipper tilt and the slipper velocity.\par
The first group of the equation ( \hyperref[formula_0]{1}) can be replaced by an analytical solution of pressure for flat and untilt slipper which is derived in Appendix A. That analytical solution can be found out as .\par
Thus ultimately the equation of pressure distribution over the land of the slipper can be written as Non-dimensional load is given as   {\ref 2}) the first group is load carrying capacity for flat and untilted slipper. The second group is its modification for conical shape of the land and slipper tilt. The third group is the hydrodynamic effect on the load carrying capacity.\par
Finding out the appropriate G value is the main issue to solve the slipper equations. This is done from ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? = ? ? ? ? Volume XII Issue V Version I ? ? = sin j j y ? ? = cos j j x\par
which is dependent on and\par
The moment of the load with respect to x and y axis can be found out by integrating the load of small strip abut x and y axis. These moments are given by and From these two moments the load centre of the load can be found out by dividing the moment with the load carrying capacity. The derivations of the moments are given in Appendix C. Now the dimensional moments are given by  
\section[{Results}]{Results}\par
As the solution is directly got from the analytical method, the plotting can be done using MATLAB programming window. All the equation are set in the programming environment of the MATLAB software and the solutions are plotted. Generally the maximum clearance occurs very near to the leading edge. In the analysis of theoretical slipper drag, programming is developed with the fact that in the leading edge maximum clearance occurs. The position of the load centre is plotted against the slipper tilt. the slipper are considered as perfectly flat. For the figure \hyperref[fig_6]{3}, the slipper speed is kept constant at 1500 rpm and the plot is drawn for different pressure. It can be seen from the graph that after some point of tilt angle the position of load centre goes out of the slipper land area. This prove that slipper runs with very small tilt angle. For figure \hyperref[fig_7]{4} the pressure is kept constant at 120 bar and the plot is drawn for different speeds. It is observed for a higher theoretical tilt the slipper does not work. Apparently the fluid film breaks after a maximum angle of tilt.\par
It can be observed from figure \hyperref[fig_6]{3} through figure 6, that the slipper center stays in the center line of motion or X axis for small tilt angles. For small tilt angle the center of load moves firstly towards the negative X axis. More tilt brings the center of load on the positive side of X axis. A little more tilt bring the centre of load to the actual centre of slipper. But for more tilt angle the centre of tilt angle the load goes outside the area of slipper which is not possible. Therefore a higher tilt is not possible to occur in actual slipper operation. Moreover the variation of pressure, speed, slipper area and non flatness angle put different response to the position of load centre. It can be observed that higher pressure in the slipper pocket tries to keep the slipper load centre in the actual centre of slipper and lower pressure tries to deflect the load centre away from actual slipper centre. In the same way higher speed of slipper tries to keep the slipper load centre in the actual centre of slipper and lower speed tries to deflect the load centre away from actual slipper centre. From figure 5, it can be observed that a particular amount of nonflatness angle tries to keep the load centre in the actual slipper centre.    \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Op??}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-3.png}
\caption{\label{fig_1}A}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-4.png}
\caption{\label{fig_2}Figure 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2}\includegraphics[]{image-5.png}
\caption{\label{fig_4}Figure 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-6.png}
\caption{\label{fig_5}M}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3}\includegraphics[]{image-7.png}
\caption{\label{fig_6}AFigure 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{4}\includegraphics[]{image-8.png}
\caption{\label{fig_7}Figure 4 :}\end{figure}
   		 		\backmatter  			 \par
From figure  {\ref 6}, it can be observed that a more amount of slipper are tries to keep the load centre in the actual slipper centre and hence stabilize the bearing fast.\par
IV. 
\subsection[{CONCLUSION}]{CONCLUSION}\par
From the results of the computation of the position of load centre, it can observed that higher tilt angle gives irrelevant results. It may happen the fluid film breaks at higher tilt angles. The stability of slipper increases with increase of slipper pressure (vide figure  {\ref 3}). The stability increases with increase of slipper speed (vide figure  {\ref 4}). The stability of slipper increases with increase of slipper non flatness angle (vide figure  {\ref 5}). The stability increases with increase of slipper land size (vide figure  {\ref 6}). For more practicality of this tilt angle more computation is required.			 			  				\begin{bibitemlist}{1}
\bibitem[ Tribology International ()]{b13}\label{b13} 	 		\textit{},  	 	 		\textit{Tribology International}  		1997. 30  (11)  p. .  	 
\bibitem[ Tribology International ()]{b2}\label{b2} 	 		\textit{},  	 	 		\textit{Tribology International}  		2002. 35 p. .  	 
\bibitem[Fisher ()]{b21}\label{b21} 	 		\textit{A theoretical determination of some characteristics of tilted hydrostatic slipper pad},  		 			M J Fisher 		.  		 RR728.  		1962.  	 	 (BHRA Report) 
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\end{bibitemlist}
 			 		 	 
\end{document}
