Abstract:
A machine and method for a hydrostatic axial sliding bearing that has bearing and sealing functions and is capable of exhibiting reduced power losses. The hydrostatic axial sliding bearing of the machine is defined by axial sliding bearing surfaces that are separated by a fluid film and adapted for movement relative to each other during operation of the machine. The machine has first and second elements that define a first and a second of the axial sliding bearing surfaces. In combination, the first and second axial sliding bearing surfaces function as bearing and sealing surfaces for the hydrostatic axial sliding bearing. The machine is adapted so that the first and second axial sliding bearing surfaces move relative to each other in a first direction of motion, and at least the second axial sliding bearing surface has a surface profile with an oscillatory waveform in the first direction of motion.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims the benefit of U.S. Provisional Application No. 61/041,257, filed Apr. 1, 2008, the contents of which are incorporated herein by reference. 
     
    
       [0002]    This invention was made with Government support under Award EEC-0540834 awarded by the National Science Foundation. The Government has certain rights in this invention. 
     
    
     BACKGROUND OF THE INVENTION 
       [0003]    The present invention generally relates to hydrostatic sliding bearings, including hydrostatic sliding bearings suitable for use in positive displacement machines. 
         [0004]    Positive displacement pumps and motors, such as axial and radial piston machines, generally comprise an array of pistons that reciprocate within a cylinder block. In axial piston machines, the piston-cylinder combinations are parallel and arranged in a circular array within the cylinder block. An inlet/outlet port is defined at one end the cylinder block for each individual piston-cylinder combination, such that a fluid can be drawn into and expelled from each cylinder through the port as the piston within the cylinder is reciprocated. The end of the cylinder block containing the inlet/outlet ports defines an axial sliding bearing surface that abuts a surface of a valve plate, while the opposite end of the cylinder block is connected to a drive shaft for rotation of the cylinder block. The valve plate defines an inlet opening and an outlet opening that are sequentially aligned with the inlet/outlet of each cylinder, so that fluid is drawn into each cylinder through the cylinder&#39;s inlet/outlet port when aligned with the valve plate inlet opening and expelled from each cylinder through the cylinder&#39;s inlet/outlet port when aligned with the valve plate outlet opening. The mating surfaces of the cylinder block and valve plate are axial sliding bearing surfaces separated by a film of the fluid being worked on by the machine, defining a hydrostatic axial sliding bearing that is subjected to an axial load applied by the cylinder block to the valve plate. In addition to carrying this axial load, the axial sliding bearing must also minimize fluid leakage between the block and valve plate. Consequently, the axial sliding bearing has both a bearing function and a sealing function, which differentiates the hydrostatic axial sliding bearing from typical bearing applications that have only a load-bearing function. 
         [0005]    While the axial sliding bearing of a positive displacement machine is often referred to as a hydrostatic bearing, it is well known that there are hydrodynamic effects present that influence the load-bearing and sealing functions of the bearing as the cylinder block rotates relative to the valve plate. Nonetheless, for convenience sliding bearings of positive displacement machines and similar applications will be referred to herein simply as hydrostatic bearings. 
         [0006]    One end of each piston protrudes from the cylinder block and is coupled with a stationary swash plate inclined to the axis of the cylinder block, causing the pistons to reciprocate within the cylinder block as the block is rotated relative to the swash plate. The stroke length of each piston, and therefore displacement of the piston-cylinder combinations, can be made variable by changing the inclination (cam angle) of the swash plate. To provide this capability, the protruding end of each piston may be configured to have a ball-and-socket arrangement. The socket portion of this arrangement, or slipper, may have a planar surface that bears against the swash plate. The spherical mating surfaces of each piston-slipper combination and the planar mating surfaces of the swash plate and each slipper define axial sliding bearing surfaces, which are separated by a fluid film formed with, for example, the fluid being worked on. The resulting hydrostatic axial sliding bearings transfer the piston force to the swash plate during relative motion between the slipper and swash plate. 
         [0007]    Axial sliding bearing surfaces similar to those described above can also be found in other machines, including other positive displacement machines such as bent axis piston machines, radial piston machines, vane type machines, gear machines, screw-type machines, etc. 
         [0008]    The efficiencies of machines with sliding bearing surfaces are dependent on the torque losses attributable to each sliding bearing surface. For positive displacement machines, efficiencies are also dependent on power losses attributable to fluid leakage at the axial sliding bearing surfaces defined by the cylinder block and valve plate. Designs for axial sliding bearings are widely known and described in the literature. For example, descriptions of axial sliding bearings for different positive displacement machines can be found in Ivantysyn J. and Ivantysynova, M., Hydrostatic pumps and motors, New Delhi. Akademia Books International, ISBN-81-85522-16-2 (2001). Design principles and calculation methods typically assume that the gap height between the sliding bearing surfaces is uniform. Using the axial piston machine described above as an example, the sliding bearing surfaces of the planar mating surfaces of the cylinder block and valve plate and the planar mating surfaces of the slippers and swash plate are assumed to be parallel, and the sliding bearing surfaces of the spherical ball-and-socket mating surfaces of the pistons and slipper are assumed to be perfectly spherical and concentric. For manufacturing purposes, absolute deviations of flatness are defined in ranges of micrometers. It is also common practice to assume ideally smooth surfaces within the sizing process, and to allow a minimum surface roughness for manufacturing purposes, typically less than a one micrometer Ra and more typically in a range of thousandths to tenths of a micrometer, requiring an abrasive finishing operation such as lapping. Common design principles and calculation methods further assume that the fluid film between the sliding bearing surfaces is of constant thickness. 
         [0009]    A disadvantage of the above commonly-used design approach is that, in the event of an asymmetrical bearing load during relative motion between the sliding surfaces, the surfaces will incline with respect to each other and form a gap of variable height, leading to hydrodynamic effects. In the case of axial sliding bearings used in positive displacement machines, inclination of the surfaces can lead to conditions with very low gap heights on one side and very high gap heights on the opposite side. Such conditions increase friction in areas of relatively small gap heights and increase leakage in locations of relatively large gap heights, resulting in increased power losses of the machine and reduced machine efficiency. This problem is common for all asymmetrically-loaded axial sliding bearings that have a sealing function in addition to a sliding bearing function. 
         [0010]    In view of the above, there is a desire to minimize power losses, resulting from friction and/or fluid leakage, in machines with hydrostatic axial sliding bearing surfaces. 
       BRIEF DESCRIPTION OF THE INVENTION 
       [0011]    The present invention provides a machine having one or more hydrostatic axial sliding bearings capable of exhibiting reduced power losses, and to a method for configuring a machine having one or more hydrostatic axial sliding bearings so that the machine exhibits reduced power losses. 
         [0012]    According to a first aspect of the invention, the machine has at least two axial sliding bearing surfaces that are adapted for movement relative to each other during operation of the machine and are separated by a fluid film to define at least one hydrostatic axial sliding bearing having bearing and sealing functions. The machine further comprises a first element that defines a first of the axial sliding bearing surfaces and a second element that defines a second of the axial sliding bearing surfaces. In combination, the first and second axial sliding bearing surfaces function as bearing and sealing surfaces for the hydrostatic axial sliding bearing. The machine is adapted so that the first and second axial sliding bearing surfaces move relative to each other in a first direction of motion, and at least the second axial sliding bearing surface has a surface profile comprising an oscillatory waveform in the first direction of motion. 
         [0013]    According to a second aspect of the invention, the method involves reducing power losses of a machine having at least two axial sliding bearing surfaces separated by a fluid film to define at least one hydrostatic axial sliding bearing. The machine comprises first and second elements that define a first and a second of the axial sliding bearing surfaces, respectively, that move relative to each other in a first direction of motion. The method then entails forming at least the second axial sliding bearing surface to have a surface profile comprising an oscillatory waveform in the first direction of motion. 
         [0014]    The invention is applicable to a wide variety of machines, including positive displacement machines, particular examples of which include axial and radial piston pumps and motors. In the example of an axial piston machine, the axial sliding bearings may be formed by mating surfaces of a rotating cylinder block and a valve body, with one of the mating surfaces being modified to be a structured sliding surface comprising the oscillatory waveform. The oscillatory waveform can have various dimensions and a variety of micro-profiles, including symmetrical and asymmetrical oscillatory waveforms capable of generating additional hydrodynamic effects that can reduce power losses when there is relative motion between the mating surfaces. The profile and dimensions of the oscillatory waveform can be tailored to achieve an increased load-carrying capability, which can permit reduced power losses and/or allow for reductions in the surface areas of the axial sliding bearing surfaces. The profile and dimensions of the oscillatory waveform can also be tailored to achieve reduced volumetric losses. The benefits are particularly significant if the axial sliding bearing is asymmetrically loaded. 
         [0015]    Other aspects and advantages of this invention will be better appreciated from the following detailed description. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0016]      FIG. 1  shows perspective views of a cylinder block and a valve plate representative of components of an axial piston machine. 
           [0017]      FIG. 2  is a cross-sectional view of one-half of an assembly comprising a cylinder block, a piston, a slipper, and a swash plate as representative components of an axial piston machine. 
           [0018]      FIG. 3  shows a plan view of the valve plate of  FIG. 1  and three nonlimiting examples of structured surfaces that can be formed on an axial sliding bearing surface of the valve plate. 
           [0019]      FIG. 4  is a graph plotting a profile for a structured surface of the valve plate, wherein the profile has a sinusoidal waveform. 
           [0020]      FIG. 5  is a graph plotting the calculated gap height between a prior art cylinder block and valve plate predicted by a computer model as the cylinder block rotates relative to the valve plate. 
           [0021]      FIG. 6  is a graph plotting the calculated gap height between a cylinder block and valve plate predicted by the computer model as the cylinder block rotates relative to the valve plate, wherein the bearing surface of the valve plate has a structured surface having a profile with a sinusoidal waveform in accordance with an embodiment of this invention. 
           [0022]      FIG. 7  is a graph plotting the calculated pressure field distribution between the modeled prior art cylinder block and valve plate of  FIG. 5  as predicted by the computer model. 
           [0023]      FIG. 8  is a graph plotting the calculated pressure field distribution between the modeled cylinder block and valve plate of  FIG. 6  as predicted by the computer model. 
           [0024]      FIG. 9  is a graph plotting the calculated leakage between the modeled prior art cylinder block and valve plate of  FIG. 5  and between the modeled cylinder block and valve plate of  FIG. 6  as predicted by the computer model for one full revolution of their respective cylinder blocks. 
           [0025]      FIG. 10  is a graph plotting the torque losses for the modeled prior art cylinder block and valve plate of  FIG. 5  and the modeled cylinder block and valve plate of  FIG. 6  as predicted by the computer model for one full revolution of their respective cylinder blocks. 
           [0026]      FIG. 11  is a bar graph comparing the maximum and minimum gap heights predicted by the computer model under selected operating conditions for the prior art cylinder block and valve plate of  FIG. 5 , the cylinder block and valve plate of  FIG. 6 , and two additional cylinder block and valve plate pairs in accordance with additional embodiments of the invention. 
           [0027]      FIG. 12  is a bar graph comparing the power losses predicted by the computer model for each of the cylinder block and valve plate pairs of  FIG. 11  and under the same selected operating conditions of  FIG. 11 . 
           [0028]      FIGS. 13 through 18  are bar graphs in which the power losses of  FIG. 12  are separated into power losses attributable to fluid losses (leakage) and power losses attributable to torque losses for each of the cylinder block and valve plate pairs and each of the selected operating conditions of  FIG. 12 . 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0029]    The invention provides a hydrostatic axial sliding bearing that has both bearing and sealing functions and is characterized by reduced power losses, corresponding to reduced friction, when subjected to either a symmetrical or nonsymmetrical load.  FIG. 1  is representative of a cylinder block  10  and valve plate  12  suitable for use in an axial piston machine.  FIG. 2  schematically represents a cross-sectional view of one-half of the cylinder block  10 , and shows a piston  14  received within a cylinder bore  16  of the cylinder block  10 , a slipper  18  coupled to one end of the piston  14 , and a swash plate  20  abutting the slipper  18 . Consistent with conventional axial piston machines, the piston  14  and bore  16  define an axis  22  that is parallel to the axis  24  of the cylinder block  10 . While the invention will be described in reference to an axial piston machine and the components represented in  FIGS. 1 and 2 , it should be understood that the invention is applicable to a variety of other machines capable of utilizing a hydrostatic axial sliding bearing, including other positive displacement pumps and motors such as bent axis piston machines, radial piston machines, vane type machines, gear machines, screw-type machines, etc. All such applications are within the scope of this invention. 
         [0030]    The cylinder block  10  represented in  FIG. 1  comprises a circular array of parallel cylinder bores  16 , each of which receives a piston  14  in a manner similar to that represented in  FIG. 2 . The cylinder block  10  is formed to have an inlet/outlet port  26  for each of the cylinder bores  16 , such that a fluid can be drawn into and expelled from each cylinder bore  16  through the port  26  as the piston  14  within the bore  16  is reciprocated. The end of the cylinder block  10  containing the inlet/outlet ports  26  defines an axial sliding bearing surface  28  that abuts an axial sliding bearing surface  30  of the valve plate  12 , represented in  FIG. 2  as having an axis that coincides with the axis  24  of the cylinder block  10 . Though not shown, it is well known in the art to configure the opposite end of the cylinder block  10  for connection to a drive shaft for rotation of the block  10  relative to the stationary valve plate  12 . The valve plate  12  defines a pair of arcuate inlet and outlet slots  32 , which are depicted as having the same radius of curvature from the axis  24  so as to axially align with the inlet/outlet ports  26  of the cylinder bore  16  as the cylinder block  10  rotates relative to the valve plate  12 . However, it is foreseeable that the slots  32  could have different radii of curvature, and that the placement and shape of the slots  32  could differ from what is depicted in  FIG. 1 . Which of the slots  32  will serve as the inlet and which will serve as the outlet will depend on the directional rotation of the cylinder block  10  relative to the valve plate  12 . Regardless of rotational direction, the slots  32  are sequentially aligned with the inlet/outlet port  26  of each cylinder bore  16  so that, as evident from  FIG. 2 , fluid is drawn into each bore  16  through its inlet/outlet port  26  while aligned with the inlet slot  32  of the valve plate  12  and expelled through its inlet/outlet port  26  while aligned with the outlet slot  32  of the valve plate  12 . 
         [0031]    When mated, the axial sliding bearing surfaces  28  and  30  of the cylinder block  10  and valve plate  12  are separated by a film (not shown) of the fluid being worked on, defining a hydrostatic axial sliding bearing that exhibits hydrodynamic effects as the block  10  rotates relative to the valve plate  12 . To minimize fluid leakage, the block  10  and valve plate  12  are held together or otherwise subjected to an axial load that limits the gap distance (height) between the block  10  and plate  12 . In the configuration shown in  FIG. 1 , the bearing surface  28  of the cylinder block  10  is represented as having a ring groove  28 A that is coaxial with the axis  24  of the block  10  and fluidically connected (vented) with slots  28 D to the perimeter of the bearing surface  28 . The groove  28 A delineates two distinct lands  28 B and  28 C on the bearing surface  28 . The land  28 B circumscribed by the groove  28 A will be referred to as a sealing land  28 B in view of the inlet/outlet ports  26  of the cylinder block  10  being located within the sealing land  28 B, such that the sealing function required of the hydrostatic axial sliding bearing is likely to be primarily performed by the land  28 B. The remaining land  28 C circumscribing the groove  28 A will be referred to as a bearing land  28 C, as its primary function is likely to be the load-bearing function required of the hydrostatic axial sliding bearing as a result of the groove  28 A being vented to whatever pressure (likely atmospheric) that surrounds the cylinder block  10 . However, it should be noted that both lands  28 B and  28 C are likely to share in the load-bearing function. In addition, it is foreseeable that the bearing surface  28  of the cylinder block  10  could be modified to enable the bearing land  28 C to contribute to the sealing function of the hydrostatic axial sliding bearing. 
         [0032]    As presented in  FIG. 2 , one end of each piston  14  protrudes from its bore  16  in the cylinder block  10  and engages the slipper  18 . The slipper  18  engages the swash plate  20 , which is stationary and inclined to the axis  24  of the cylinder block  10  to cause the pistons  14  to reciprocate within the cylinder block  10  as the block  10  is rotated relative to the swash plate  20 . To provide a variable stroke/displacement capability, the assembly represented in  FIG. 2  is configured to allow the inclination (cam angle) of the swash plate  20  to be altered relative to the cylinder block axis  24 . In particular, the protruding end  34  of the piston  14  has a spherical surface  36  that engages a complementary spherical-shaped socket  38  formed in the slipper  18 , providing a ball-and-socket coupling that allows the end  34  of the piston  14  to rotate and pivot within the socket  38  as the cylinder block  10  rotates and the slipper  20  follows a circular path on the facing surface  40  of the swash plate  20 . The slipper  18  has a planar surface  42  that bears against the surface  40  of the swash plate  20 . The planar mating surfaces  40  and  42  of the swash plate  20  and each slipper  18  define axial sliding bearing surfaces. Each pair of bearing surfaces  40  and  42  is separated by a lubricating fluid film. For example, the film may be supplied by fluid drawn from the cylinder bore  16 , through the piston  14 , and through a passage  46  in the slipper  18 . The resulting hydrostatic axial sliding bearings provide a load-bearing function that transfers the piston forces to the swash plate  20  as the slippers  18  orbit the surface  40  of the swash plate  20 . The bearing surfaces  40  and  42  also provide a sealing function that limits fluid leakage from the interface between the slipper  16  and swash plate  20 . As with the bearing surface  28  of the cylinder block  10 , separate sealing and load-bearing regions could be delineated on the bearing surface  42  of the slipper  18 , for example, with a groove that may or may not be vented to the surrounding atmosphere. 
         [0033]    For purposes of discussing the present invention, other relevant structural and functional aspects of the axial piston machine and its axial sliding bearings represented in  FIGS. 1 and 2  will be well understood by those skilled in the art, and therefore will not be discussed in further detail here. 
         [0034]      FIG. 3  contains an axial view of the axial sliding bearing surface  30  of the valve plate  12 , and represents a cross-section line “A” that is the basis for three unwrapped cross-sectional views representing three of the various surface profiles  44  envisioned by this invention. As evident from  FIG. 3 , the cross-section line A defines a circular path on the bearing surface  30  of the valve plate  12  and has an axis that, after mating the cylinder block  10  to the valve plate  12 , will coincide with the axis  24  of the cylinder block  10  such that the instantaneous direction of motion of a point on the profile  44  lying on the circular path is tangential to the circular path. As evident from  FIG. 4 , because the oscillatory waveforms of the profiles  44  lie on a circular path, the peaks and valleys of the waveforms lie on radials of the path axis. 
         [0035]    The surface profile  44  is preferably present on at least the portion of the valve plate bearing surface  30  that surrounds the valve plate inlet/outlet slots  32  and will face the sealing land  28 B of the cylinder block  10  when the block  10  and valve plate  12  are mated. The profile  44  may also extend toward the perimeter of the valve plate bearing surface  30 , so as to face the bearing land  28 C of the cylinder block  10  when the block  10  and valve plate  12  are mated. The bearing surface  28  of the cylinder block  10  is represented in  FIG. 3  as being smooth and lacking a profile  44 , though it should be understood that a profile  44  could be provided at the bearing surface  28  instead of or in addition to the profile  44  on the valve plate bearing surface  30 . Consistent with the aforementioned sealing function of the sealing land  28 B, a profile  44  provided on the bearing surface  28  would preferably be present on at least the sealing land  28 B of the cylinder block  10 , though may also extend into the bearing land  28 C at the perimeter of the bearing surface  28 . 
         [0036]    The profiles  44  represented in Examples 1, 2 and 3 of  FIG. 3  will be referred to as triangular, sinusoidal, and sawtooth, consistent with the ordinary use and meaning of these terms in reference to oscillatory waveforms. The triangular and sinusoidal profiles  44  are understood to be symmetrical, whereas the sawtooth profile  44  can be seen to be asymmetrical. Other variations of symmetrical and asymmetrical oscillatory waveforms are foreseeable and therefore also within the scope. The oscillatory waveforms may have a wide range of peak-to-peak amplitudes, with a suitable maximum amplitude believed to be about one hundred micrometers. A minimum peak-to-peak amplitude is believed to be at least 0.1 micrometer. In the absence of a profile  44  (for example, the bearing surface  28  in  FIG. 3 ), the bearing surfaces  28 / 30  may have a surface roughness that is an order of magnitude lower than the peak-to-peak amplitude of the oscillatory waveform. 
         [0037]    The oscillatory waveforms of the profiles  44  represented in  FIG. 3  are intended to generate additional hydrodynamic effects for the axial sliding bearing surfaces of the assemblies shown in  FIGS. 1 and 2 , which are often asymmetrically loaded, particularly in view of the common practice of using an odd number of pistons  14 . The asymmetric loading of the bearing surfaces incurs power losses in conventional axial sliding bearings that can be reduced by hydrodynamic effects provided by the invention. It is believed an oscillatory waveform for the surface profile  44  and its dimensions can be defined to minimize power losses over a range of operating conditions. A surface profile  44  also has the ability to increase the load-carrying capability of the axial sliding bearing and/or allow for reductions in the surface areas of the axial sliding bearing surfaces. 
         [0038]    In investigations leading to the present invention, sinusoidal waveforms (Example 2 of  FIG. 3 ) were chosen to be modeled and analyzed using a proprietary computer model. The waveforms were analyzed as present on an otherwise conventional axial piston pump design with nine pistons. For comparison, the identical axial piston pump (minus a surface profile) was also modeled and analyzed in the investigation as a baseline (“standard”) model. Three simulation models were assessed, identified as DM. 7 , DM. 8  and DM. 9 . For the simulation and standard models, the displacement volume (V i ) of the pump per rotation was set at 75 cc. For the simulation model identified as DM. 7 , the surface profile consisted of ten full sinusoidal waves along the circumferential direction of the cylinder block/valve plate surface, as represented in  FIG. 4 . The amplitude of the DM. 7  waveform was +/−2 micrometers. The simulation model identified as DM. 8  also had a surface profile consisting of ten full sinusoidal waves ( FIG. 4 ), but with an amplitude of +/−1 micrometer. The simulation model identified as DM. 9  had a surface profile consisting of fifteen full sinusoidal waves with an amplitude of +/−1 micrometer. Finally, the standard model was simulated to have a smooth planar bearing surface. Because the models were otherwise identical, other structural and dimensional aspects of the models are not deemed to be necessary for an understanding of the investigation and its results, and therefore are not reported here. 
         [0039]    For each model, variations in six operating conditions were simulated. The approximate operating conditions are summarized in Table I below. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Simulation Parameters 
               
             
          
           
               
                 Operating 
                 Δp 
                 n 
                 p HP   
                 p LP   
                 T HP   
                 T LP   
                 T CASE   
                 β 
               
               
                 Condition 
                 [bar] 
                 [rpm] 
                 [bar] 
                 [bar] 
                 [° C.] 
                 [° C.] 
                 [° C.] 
                 [%] 
               
               
                   
               
             
          
           
               
                 #1 
                 100 
                 1000 
                 120 
                 20 
                 48 
                 47 
                 64 
                 100 
               
               
                 #2 
                 300 
                 1000 
                 320 
                 20 
                 56 
                 54 
                 67 
                 100 
               
               
                 #3 
                 300 
                 3000 
                 320 
                 20 
                 54 
                 48 
                 94 
                 100 
               
               
                 #4 
                 100 
                 1000 
                 120 
                 20 
                 54 
                 51 
                 62 
                 17 
               
               
                 #5 
                 300 
                 1000 
                 320 
                 20 
                 70 
                 55 
                 73 
                 17 
               
               
                 #6 
                 300 
                 3000 
                 320 
                 20 
                 61 
                 52 
                 91 
                 17 
               
               
                   
               
             
          
         
       
     
         [0040]    With reference to the components as identified in  FIGS. 1 through 3 , Δp is the system pressure differential between the inlet and outlet slots  32  of the valve plate  12 , n is the rotational speed of the cylinder block  10 , p HP , is the pressure at the high pressure (outlet) port, p LP , is the pressure at the low pressure (inlet) port, T HP  is the fluid temperature at the high pressure port, T LP  is the fluid temperature at the low pressure port, T CASE  is the temperature of the cylinder block  10 , and β is the inclination of the swash plate  20  relative to the axis  24  of the cylinder block  10 . A swash plate angle of 100% refers to a maximum inclination angle, corresponding to a maximum displacement for the machine. 
         [0041]    Simulations were performed with the computer model to determine three-dimensional gap heights, three-dimensional pressure fields, leakage losses, and torque losses for the standard model and the three simulation models DM. 7 , DM. 8  and DM. 9 . 
         [0042]    In the simulation, as the cylinder block rotates the resulting external force pressing the cylinder block  10  against the valve plate  12  varies and causes a change of gap height between the cylinder block  10  during the course of one revolution.  FIGS. 5 and 6  are graphs comparing the three-dimensional gap heights for the standard model and the DM. 9  simulation model, respectively, at operating condition #1 (Table I) while the cylindrical block was at a specified rotational position (φ) designated as 0 degrees for the simulation. As evident from  FIGS. 5 and 6 , the gap height around the circumference of the standard model is highly asymmetric, while the gap height around the circumference of the simulation model is sinusoidal but highly symmetric. The results depicted in  FIGS. 5 and 6  are representative of additional results that were obtained at rotational positions (φ) corresponding to angles of 45, 90, 135, 180, 225, 270, and 315 degrees. 
         [0043]      FIGS. 7 and 8  are graphs comparing the three-dimensional pressure field for the standard model and the DM. 9  simulation model, respectively, at operating condition #1 while the cylindrical block was at the rotational position (φ) of 0 degrees. As would be expected from the different fluid pressures at the inlet and outlet slots  32  of the valve plate  12 ,  FIGS. 7 and 8  evidence that the pressure fields are asymmetric for both the standard and DM. 9  simulation models. The results depicted in  FIGS. 7 and 8  are representative of additional results obtained at rotational positions (φ) corresponding to angles of 45, 90, 135, 180, 225, 270, and 315 degrees. 
         [0044]      FIGS. 9 and 10  are graphs comparing the calculated leakage and torque losses for the standard model and the DM. 9  simulation model over an entire revolution at operating condition #1. As would be expected, leakage is cyclic for the DM. 9  simulation model, whereas the leakage calculated for the standard model is relatively lower and the cyclic effect is much less pronounced. However,  FIG. 10  evidences that torque losses associated with the DM. 9  simulation model are less than half of the torque losses associated with the standard model. Consequently, at least under the conditions of operating condition #1, the DM. 9  simulation model caused a relative increase in fluid leakage and a significant decrease in torque losses. 
         [0045]      FIGS. 11 through 18  are bar graphs comparing the average maximum and minimum gap heights and average power losses for the standard model and all three simulation models (DM. 7 , DM. 8  and DM. 9 ) over one full revolution of the cylinder block  10 . For each angular position, φ, the computer model calculated maximum and minimum gap heights and power loss due to leakage (P q ) and friction at the interface of the cylinder block  10  and valve plate  12 . As evident from  FIGS. 11 through 18 , the operation parameters used to compute the maximum/minimum gap heights and average power losses for the standard and simulation models were those identified as operating conditions #1 through #6 in Table I, which include a system pressure differential (Δp) of 100 or 300 bar, a rotational speed (n) 1000 or 3000 rpm, and either a low (17%) or high (100%) displacement (β) of the swash plate  20 . 
         [0046]    The results plotted in  FIGS. 11 through 18  evidence that the performances of the simulation models relative to the standard model were far better in terms of lower torque losses under most conditions. While the data appear to suggest that performance was dependent on the system pressure differential (Δp), it was concluded that this effect was attributable to the 300 bar pressure being near the capacity of the modeled pump. Subsequent simulations have shown that performance improvements can be achieved at pressures much higher than 300 bar. Experimental results have also verified the improved performance of hydrostatic axial bearing surfaces modified in accordance with the present invention, with actual reductions in power loss of about 10% being attained for axial piston machines. 
         [0047]    While the invention has been described in terms of specific embodiments, it is apparent that other forms could be adopted by one skilled in the art. Therefore, the scope of the invention is to be limited only by the following claims.