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    Several days ago I read an interesting article about hydraulic balance of  mechanical seals (the original article is from here, I am also putting a copy of it in the Library) . Very interesting material, definitely worth looking into.  As I was finishing reading the article, an idea of applying the same principles to piston assembly of an axial piston pump was already buzzing in my head. Many thanks to mr. Mc Nally for the idea!

   Under the Shoe.     

    Earlier I talked about hydraulic balance and its use to reduce load forces of pump components.  This article is a follow up, focused on a piston assembly of an axial piston pump (swashplate type). The pics  show a classic piston shoe design, although the hydraulic balance principle is equally applied to the "inverted" piston design, where the "ball" is part of the shoe (Linde, Komatsu,..), a design which allows a bigger swashing plate angle, thus resulting in a more compact pump design.

     Anyway, let us take a closer look at the forces acting on a piston inside a working pump. Obviously we have closing force, resulting from system pressure acting on the piston's end and opening force acting on the shoes cavity.  But besides these two obvious forces there's more. Although the clearance between the contact surfaces of the shoe and the swashplate is very small, there's still a radial oil flow and, consequently, a pressure drop in this area (pic8). Due to the very small clearance the flow is most likely laminar, and due to the fact that, as the radius increases, the flow cross section also increases,  the pressure drop is not linear. This means that the shoe balancing area is, in fact, a little bigger than the cavity area alone.

   There are two basic designs of a piston shoe, a simple design (pic6, pic7) and a complex (or labyrinth) design (pic4, pic5, just one example as there are many similar variations).  The "high" areas inside the balancing cavity of a complex piston design also contribute to the hydrostatic lift as the film of oil under them is pressurized. 

     I would really like to know why so many different shoe designs exist. The simple, single cavity no labyrinth design is the cheapest and the fastest to produce, and its efficiency and reliability is time proven on both open and closed circuit pumps by many, so if I were to design a pump I would stick to it. But, apparently, there are still plenty factories out there, who invent a new revolutionary shoe cavity design each time a new pump series is launched. Which can also be explained by the fact that creation of pump/motor components to a great extend is (A) an empirical process, and (B) a process greatly influenced by company tradition, which is no shame at all. So if someone wants to go nuts machining away all sorts of cavities and cavity-connecting labyrinths in a piston shoe, I say let them! In fact, nothing can explain better this point than this abstract from (1):

      "The general lack of basic principles for designing reliable components for hydraulic pumps and motors requires a strategy of applying experience and full scale testing. Hydraulic pump and motor components are very sparsely lubricated though they are immersed in fluid. They therefore slide in "boundary  lubrication", a regime for which there are no methods for predicting, or even estimating product life or frictional performance. The testing of sub-components in bench tests or in accelerated tests produces new uncertainties since the role of the many variables that control wear and scuffing are not well known. The best design procedure involves full scale testing of components with all of the recirculating contaminants, vibrations and misalignments included..." (highlighted by me)
    Now, to dress this theory up in numbers and to find out the balance ratio of modern piston designs (out of pure scientific curiosity, of course),  I swept shelves in the warehouse and took measurements of some random pistons (various brands, both closed and open circuit), then made some calculations and combined the results into this nice table. For a complex design piston the measurements were taken in the following manner. All measurements are in millimeters. To calculate the contact  area lift, an estimated value of  30% of the total area of the outer ring surface was used. If you ask  me where did I get the 30% figure I'll tell you it's a hunch, based on the McNally's article.

    In the "balancing area" column you have the area of the shoe cavity directly exposed to the system pressure (inner rings' surface of the complex design shoe was not taken into consideration, as the fluid film under the rings is pressurized thus equally contributing to the lifting force) plus 30% of the total area of the outer ring. Balance ratio is the shoe balancing area divided by piston head area .
Short Pump Reference Design A, mm B, mm C, mm Piston Head Area, sq.cm Balancing Area, sq. cm Balance Ratio
Sauer 90R075 simple 20.7 16.7 25.7 3.36 3.09 0.92
SPV 23 complex 20.6 18.5 22.4 3.33 3.06 0.92
SPV 21 complex 17.2 15.5 18.6 2.32 2.13 0.92
A10VO028 simple 14.5 9.3 18.7 1.65 1.3 0.79
A4VO125 complex 24.7 22.4 27 4.79 4.47 0.93
A11VO190 complex 28 25.9 30 6.15 5.81 0.94
A4V90 complex 22 20.4 23.6 3.8 3.6 0.95
A4VG125 complex 22 19.3 24 3.8 3.4 0.90
A10VSO100 simple 23 15.4 30.4 4.15 3.48 0.84
HPR100 complex 22.4 20.3 23.3 3.94 3.54 0.90
BPV100 complex 21.4 19 22.5 3.59 3.18 0.88
K3V112 complex 24 21.8 25.6 4.52 4.15 0.92
Sauer 90R100 simple 22.6 18.4 28.1 4.01 3.72 0.93
   As you can see, different brands and designs all wound up with the balance ratio of around 0.9, (the only exception being the A10VO028, with a smaller ratio of 0.8, which I am guessing can be explained by the small size).

   You may also see that complex shoe design prevails. Another advantage for the design I might think of is, probably, the fact that the narrower outer ring of the complex design, when scored by contaminant particles, affects the balance ratio to a smaller extent in comparison to the simple design with its relatively wide ring surface. But hey, check out Sauer Danfoss series 90 closed circuit pumps. Extremely reliable and more than time proven units that have used single cavity design pistons for a decade already, and seem to have done just fine.

    It is also worth mentioning  that aside direct acting forces caused by the system pressure (and the case pressure) there are other forces contributing to the piston shoe lift. There's the hydrodynamic lift, caused by the fluid trying to get under the shoe as it moves along the swashing plate. There are all sorts of vibrations and misalignments. There's the centrifugal force, friction forces between the cylinder block and the piston itself, and probably an assload of other forces I can't think of at the moment but pretty sure they exist. All of them being compensated for by the hydraulic balance.

    I can only imagine how cool must have been all the empirical work and failed attempts to design a  reliable piston. Pump designers, I salute you! Keep up the good work for us, hydraulics-junkies, to have something to back-engineer...

1. Hydraulic Failure Analysis
    fluid, components and system effects
    George E. Totten, David. K. Wills, Dierk G. Feldmann.