Increasing the Power Density for Axial

Increasing the Power Density for Axial-Piston Swash-Plate Type Hydrostatic Machines

Noah D. Manring, Viral S. Mehta, Bryan E. Nelson, Kevin J. Graf and Jeff L. Kuehn

[+-] Author Affiliations

Noah D. Manring

Mechanical and Aerospace Engineering,University of Missouri,Columbia, MO 65211e-mail: [email protected]

Viral S. Mehta

e-mail: [email protected]

Bryan E. Nelson


Kevin J. Graf


Jeff L. Kuehn

e-mail: [email protected] Inc.,Peoria, IL 61656

Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the Journal of Mechanical Design. Manuscript received December 19, 2012; final manuscript received February 25, 2013; published online May 10, 2013. Assoc. Editor: Alexander Slocum.

J. Mech. Des. 135(7), 071002 (May 10, 2013) (6 pages)doi:10.1115/1.4023924History: Received December 19, 2012; Revised February 25, 2013

Article

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Abstract

Abstract | Introduction | Machine Description | Analysis | Results and Discussion | Conclusion | Acknowledgements | References

Power density is an assumed attribute of an axial-piston swash-plate type hydrostatic machine. As such, very little research has been conducted to examine the nature and limit of this machine's power density and the literature is all but void of this important topic. This paper is being written to fill this void, and to provide a thorough analysis of the machine's power density. This paper is also aimed at identifying the most significant parameters that may be adjusted to increase the power density for a typical machine. As shown in this research, the power density of an axial-piston machine depends upon four dimensionless quantities that are characteristic of the machine's rotating group. As it turns out, the allowable stress for the cylinder block is the most sensitive parameter that may be adjusted for increasing the power density of this machine. It is further shown that increasing the machine's swash-plate angle, and reducing the minimum overhang length for the pistons, will have a significant impact on the power density as well. It is significant to note that altering the number of pistons in the design has essentially no impact on the power density of the machine and therefore the selection of this design parameter must be based upon other design objectives. In conclusion, it is shown in this paper that the power density of a typical machine may be increased by as much as 64% by altering a few of these parameters within a realistic realm of constraint.

FIGURES IN THIS ARTICLE

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Introduction


Background.

Axial-piston swash-plate type machinery is used to convert between rotating mechanical-energy and hydraulic fluid-energy for the purposes of controllably transmitting power in various engineering applications. Typical applications include power transmission in off-highway earth moving equipment, automotive power steering systems, industrial manufacturing and machining centers, and aerospace flight controls. These devices are typically used instead of, say, their electrical counterparts for two main reasons: (1) they provide a stiffer dynamic-response due to the fact that they exhibit higher torque-to-inertia ratios [1], and (2) they are more power dense than other power transmission devices especially in the range of high power transmission. For instance the typical power density for a diesel engine is around 250 kW/m3, for an AC electric motor the power density is near 540 kW/m3, and for an axial-piston hydraulic pump or motor the power density is near 6800 kW/m3 [2]. In other words, an axial-

piston swash-plate type hydrostatic machine exhibits a power density that is more than ten times greater than its next leading technology. In a growing number of applications, such as portable power transmission and aerospace flight control, the demand for a higher power density for hydrostatic equipment continues to be observed. In short, machine designers are being pressured to do more with smaller machines, and thus the niche attribute of power density for hydrostatic machinery continues to be an important topic for today's engineer.

Literature Review.

Power density is a well-known attribute for hydrostatic machinery. In fact, it is so well known that this attribute is simply assumed and very few resources in the literature seem to address the topic from a rigorous perspective. In a paper written by Prof. Ivantysynova [3], she acknowledges that compact design and high power density are among needed innovations for the future of hydrostatic equipment, and in her book on hydrostatic pumps and motors [4] she conducts one of the most thorough analyses of these machines to date. But nowhere among this scholarly work has the subject of power density been thoroughly analyzed for an axial-piston swash-plate type hydrostatic machine. Even works that have been published by the current author [5] have failed to address this topic thoroughly. In a literature search for papers on this topic, only a paper published by Peter Achten [6] seems to apply a rigorous analysis to the subject of power density for axial-piston machinery. In this paper, Achten presents a novel machine design called the floating cup, axial-piston design and seeks to demonstrate that lower swash-plate angles for this machine are not detrimental to its overall power density. As stated in Auchen's own article, the floating cup is a significantly different machine and it is unlikely that the same principles found in his paper will apply to the more traditional machine described in this present research.

Objectives.

Power density is an assumed attribute of an axial-piston swash-plate type hydrostatic machine. As such, very little research has been conducted to examine the nature and limit of this machine's power density and the literature is all but void of this important topic. This paper is being written to fill this void, and to provide a thorough analysis of the machine's power density. This paper is also aimed at identifying the most significant parameters that may be adjusted to increase the power density for a typical machine. As shown in this research, the power density of an axial-piston machine depends upon four dimensionless quantities that are characteristic of the machine's rotating group. As it turns out, the allowable stress for the cylinder block is the most sensitive parameter that may be adjusted for increasing the power density of this machine. It is further shown that increasing the machine's swash-plate angle, and reducing the minimum overhang length for the pistons, will have a significant impact on the power density as well. It is significant to note that altering the number of pistons in the design has essentially no impact on the power density of the machine and therefore the selection of this design parameter must be based upon other design objectives. In conclusion, it is shown in this paper that the power density of a typical machine may be increased by as much as 64% by altering a few of these parameters within a realistic realm of constraint.

Machine Description


This section describes the overall function and appearance of the axial-piston swash-plate type hydrostatic machine. The characteristics of this machine are chosen in such a way as to represent typical machine designs and therefore several details of the design have been omitted due to their lack of generality. Figure 1 shows the general configuration of the machine.

The hydrostatic machine consists of several pistons within a common cylindrical block. The pistons are nested in a circular array within the block at equal intervals about the x-axis. As shown in Fig. 1, the cylinder block is held tightly against a valve plate using the force of the compressed cylinder-block spring and a less obvious pressure force within the cylinder block itself. A thin film of oil separates the valve plate from the cylinder block which, under normal operating conditions, forms a hydrodynamic bearing between the two parts. A ball-and-socket joint connects the base of each piston to a slipper. The slippers themselves are kept in reasonable contact with the swash plate by a retainer mechanism (not fully shown in Fig. 1) and a hydrodynamic bearing surface separates the slippers from the swash plate. The swash-plate angle, α, is generally controlled by an external control mechanism.

While the valve plate is held in a fixed position, the coupled shaft and cylinder block are driven about the x-axis at an angular speed, ω. During this motion, each piston periodically passes over the discharge and intake ports on the valve plate. Furthermore, because the slippers are held against the inclined plane of the swash plate, the pistons undergo an oscillatory displacement in and out of the cylinder block. As the pistons pass over the intake port, the piston withdraws from the cylinder block and fluid is drawn into the piston bore. As the pistons pass over the discharge port, the piston advances into the cylinder block and fluid is pushed out of the piston bore. This motion repeats itself for each revolution of the machine and the basic task of displacing fluid is accomplished. If the discharge fluid is at a higher pressure than the intake fluid the machine is operating as a pump and the machine is being used to supply fluid power for a downstream circuit. If the intake fluid is at a higher pressure than the discharge fluid the machine is operating as a motor and the machine is being used to convert fluid power back into rotational mechanical power at the output shaft.

Analysis


Power Density.

The power density for the rotating group of an axial piston machine is defined as the amount of power that can be distributed by the machine, divided by the spatial volume of the envelope in which the rotating group is confined. If this quantity is divided by the operating pressure and the rotational speed of the machine, a nondimensional form for the power density may be expressed as

(1)Π≡V d V s

where Vs is the spatial volume of the machine's rotating group, and Vd is the volumetric displacement of the machine per radian. From this equation, it is clear that the power density may be increased by minimizing the spatial volume, Vs, for a given volumetric displacement of the rotating group, Vd.

Volumetric Displacement.

Equation (1) describes the power density of the axial-piston machine as a ratio of the volumetric displacement to the spatial volume of the rotating group. In this section, the volumetric displacement of the machine will be considered by analyzing the circular pattern of piston bores within the cylinder block. The following analysis assumes that the bores are spaced evenly about the centerline of the cylinder block. Three constraints are placed on the design; namely, the volumetric displacement of the machine, the maximum bore-to-bore hoop stress, and the physical space limitations which exist about the arc length of the piston-pitch circle. Using simple equations that describe these three constraints, the nominal piston pitch-radius, r, the piston-bore diameter, d, and the angular wall-thickness between bores, δ, are determined. Figure 2 shows the layout geometry of the cylinder block.

The volumetric displacement of the machine given in units of volume per revolution is generally expressed as V D =2NA p rtan(α) where N is the number of pistons, Ap is the area of a single piston, r is the piston-pitch radius, and α is the swash-plate angle [1]. Recognizing that A p =πd 2 /4 , where d is the diameter of a single piston, the volumetric displacement of the machine is given more explicitly in units of volume per radian as

(2)V d =Nd 2 rtan(α) 4

Using Fig. 2, it may be determined that δ=2π/N−ψ . Furthermore, the dimension ψ may be closely approximated as d/r; therefore,

(3)δ=2π N −d r

Equation (3) describes the space limitation of the design about the arc length of the piston-pitch circle. If a single bore is modeled as a nominal thin-wall pressure vessel, then the nondimensional hoop stress for a single bore may be calculated as

(4)σ=d 2rδ

The value for σ is a measure of the maximum allowable stress in the piston wall, and is a constant for any given design. In practice, reasonable values for σ have been around 1.3. A simultaneous solution of Eqs. (2), (3), and (4) may be used to produce an equation for the piston pitch-radius, r, the piston diameter, d, and the angular wall thickness between bores, δ. These results are given by

(5)r=(1 2π (1+2σ) σ ) 2/3 (NV d tan(α) ) 1/3

(6)d=(16πσ (1+2σ) V d N 2 tan(α) ) 1/3

and

(7)δ=2π N(1+2σ)

Equations (5) and (6) show that the piston-pitch radius and the piston diameter scale in proportion to the cube root of the volumetric displacement. This property will be used to scale other length dimensions that appear later in this paper. Equation (7) shows that the wall thickness between piston bores must decrease as the allowable maximum stress increases. These explicit forms will be used in a subsequent section to identify the most sensitive parameters that may be adjusted to increase the power density of the axial-piston machine.

Spatial Volume.

In order to calculate the power density of the axial-piston machine, Eq. (1) shows that we need to determine the spatial volume of the rotating group, Vs. As shown in Fig. 3, a number of parameters must be used to describe the envelop of the rotating group; however, if we consider this volume to be a simple cylindrical shape with a diameter D and a length L, an expression for the volume of this cylindrical envelop may be written as follows:

where D and L are shown in Fig. 3. From the geometry shown in Fig. 3 the diameter of the cylindrical envelope is given by

(9)D=2(r+h)

A parametric relationship for the dimension h may be conveniently defined as follows:

(10)h=d 2 +rδ=2πr N −d 2

where δ is shown in Fig. 2 as an angular wall-thickness between bores. The length of the cylindrical envelope may be determined from the geometry of Fig. 3. This result is given by

(11)L=t+2rtan(α)+l+m o +rtan(α)

where t is the cylinder-block face thickness, l is the engaged length of the bushing within the cylinder block, and mo is the nominal overhang length of the piston. Again, a convenient set of parametric relationships will be used to conduct the analysis of this study. These relationships are given by

(12)m o =m min +rtan(α) and l=3 2 d

where mmin is the minimum overhang length shown in Fig. 3 and the relationship between the engaged length and the piston diameter ensures that the length to diameter ratio is always 1.5. This length to diameter ratio is common and well-known throughout the industry. As a matter of convenience, the minimum overhang length and the cylinder-block face thickness will be scaled using the following relationships:

(13)t=t o (V d V d o ) 1/3 and m min =m min o (V d V d o ) 1/3

where Vd is the volumetric displacement of the machine and the subscript “o” is used to indicate nominal or baseline values associated with each parameter. The 1/3rd power comes from standard scaling practices for these machines.

By substituting Eqs. (2)–(13) into Eq. (1), it may be shown that the power density of the rotating group is given by

(14)Π=⎧ ⎩ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [N+2π N 2 (N 2 (1+2σ) σ ) 2/3 − 2π(σ N 2 (1+2σ) ) 1/3 ] 2 ×[c tan 2/3 (α) +3 2tan(α) (σ N 2 (1+2σ) ) 1/3 + 1 Nπ (N 2 (1+2σ) σ ) 2/3 ] ⎫ ⎭ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −1

where c is a dimensionless constant given by a baseline design and defined as

(15)c≡m min o +t o (16πV d o ) 1/3

See Fig. 3 and Eq. (13) for further clarification of this quantity. By examining Eq. (14) closely, it may be seen that the power density is a function of only four dimensionless parameters: (1) the swash-plate angle α, (2) the number of piston N, (3) the nondimensional hoop stress in the cylinder block σ, and (4) the nondimensional constant c. This expression for the power density of the rotating group is very nonlinear, and the only thing that can be readily discerned from this result is that a decrease in the nondimensional constant c will produce an increase in the power density. In other words, by reducing the minimum overhang length for the piston mmin, or by reducing the cylinder-block face thickness t, the power density of the rotating group will go up. This is a fairly obvious result. The following paragraphs will be used to gain insight into the effect of changing other parameters in the design by conducting perturbation analysis using Eq. (14).

Perturbation Analysis.

In order to evaluate the sensitivity of the power density to small changes in the four parameters that define Π, it will be convenient to nondimensionalize the expressions for power density and the four relevant parameters. This process of nondimensionalization is done by defining the following quantities:

(16)Π=Π o Π ∧ , α=α o α ∧ , N=N o N ∧ , σ=σ o σ ∧ , and c=c o c ∧

where the subscript “o” denotes that the parameter is evaluated at a nominal or baseline condition, and where carets indicate a nondimensional value of the parameter which is of order one. Using these definitions, a Taylor series expansion of Eq. (14) may be taken with respect to all four parameters and the leading coefficient for each term is called the sensitivity coefficient for that parameter. This Taylor series is symbolically expressed as follows:

(17)Π ∧ =1+S σ (σ ∧ −1)+S α (α ∧ −1)+S c (c ∧ −1)+S N (N ∧ −1)

where the sensitivity coefficients are given explicitly as

(18)S σ =(∂Π ∂σ σ Π )∣ ∣ o , S α =(∂Π ∂α α Π )∣ ∣ o , S c =(∂Π ∂c c Π )∣ ∣ o ,S N =(∂Π ∂N N Π )∣ ∣ o

Again, the subscript “o” denotes that the quantity in brackets is evaluated at the nominal condition of the machine design. Since these coefficients are nondimensional and scaled by the nominal design conditions of the machine, their numerical values may be compared with each other in order to determine the most sensitive design feature to change for increasing the power density of the machine. These comparisons will be made in the following section of this paper.

Results and Discussion


Numerical Evaluations.

In order to generate result from the previous analysis, a nominal or baseline machine must be specified. As it turns out, a very typical machine may be described using the following nondimensional quantities:

(19)σ=1.3556, α=0.3142, c=0.2993, N=9

Since the derivative in Eq. (18) are complicated, the sensitivity coefficients have been evaluated numerically and are presented as follows:

(20)S σ =0.5154, S α =0.4705, S c =−0.2351, S N =−0.0241

As previously mentioned, the sensitivity coefficients may be directly compared with one another since they have been nondimensionalized and scaled with respect to the baseline design.

From the results of Eq. (20), it may be seen that the hoop stress σ is the most sensitive parameter to adjust for increasing the power density of the rotating group. If the allowable hoop stress in the cylinder block goes up, the power density goes up. As shown in Eq. (7) this means that the wall thickness between bores will become smaller. It should be noted that a typical steel cylinder-block operating at a fluid pressure of 42 MPa can typically withstand a nondimensional hoop stress of up to 3.5. If the material is heat treated, allowable stress values may be increased beyond this value.

The second most sensitive parameter that may be adjusted is the swash-plate angle α. In fact, adjustments to the swash-plate angle are nearly as sensitive as adjustments in the allowable hoop stress. By increasing the swash-plate angle, the power density increases. Current industry practice is to utilize a swash-plate angle between 14 and 21 degrees (0.2443 and 0.3665 rad) with the most common swash-plate angle being 18 degrees (0.3142 rad).

The third most sensitive parameter to adjust is the dimensionless constant c. This constant may be adjusted by reducing either the minimum overhang dimension mmin, or the cylinder-block face thickness t. As will be shown later, the minimum overhang dimension may be reduced by using an inverted piston-slipper design for locating the center of the piston ball joint further inside the bore.

Finally, the least influential parameter to adjust is the number of pistons N. As shown in Eq. (20), adjustments to this parameter are ten time less sensitive than adjustments to any other parameter in

the design. Axial piston machinery typically uses either 7 or 9 pistons in the design. As will be shown in subsequent plots, the power density of the machine is so weakly dependent upon the number of piston that alterations in this design parameter may be safely neglected without losing or gaining an opportunity.

Plotted Results.

Figures 4 through 7 show plots of the nondimensional power density, as defined in Eq. (16), versus the four parameters that may be adjusted to alter the power density of the rotating group. The vertical axis in each plot is identical in scale to allow a fair comparison of each plot. The horizontal axis spans a reasonable range for each of the independent variables, and the circle on each plot identifies the nominal or baseline design configuration that is representative of Figs. 1 and 3.

The first thing to notice from Fig. 4 is that reasonable changes in the dimensionless hoop stress σ can produce fairly large changes in the power density of the machine. As noted earlier, this is the most sensitive parameter that can be adjusted and nearly a 50% increase in the power density may be obtained when coupled with an increase in the swash-plate angle α. Figure 5 illustrates this same point, but shows a bit more clearly that the power density of the machine increases almost linearly with the swash-plate angle. Figure 6 illustrates that the power density decreases almost linearly with the dimensionless constant c, and the flatness of each curve in Fig. 7 shows that the power density is only weakly dependent upon the number of pistons in the machine. In other words, there is not much power density to be gained or lost by altering the number of pistons within the design. To improve the visualization of these results the next subsection will compare two machine layouts of the same volumetric displacements.

Drawing Comparisons.

To visually illustrate the results of the research, Fig. 8 presents two drawings for an axial-piston machine. Both of the machines in this figure are designed to displace the same amount of fluid per revolution and are therefore the same size from a functional point of view. There are three primary differences between the two machines: (1) the new machine allows for more stress to be carried within the cylinder block, (2) the new machine uses a higher swash-plate angle, and (3) the new machine uses an inverted piston-slipper design that allows the minimum overhang length to vanish. Table 1 lists the design features of each machine and shows the percent change that has been realized in each feature. It is interesting to note that the piston diameter has remained essentially constant while the piston pitch radius has been reduced by almost 15%. This is accompanied with a 16% increase in the swashplate angle which essentially shows that we have linearly traded the piston pitch radius for a larger swash-plate angle in order to maintain the volumetric displacement of the machine. In summary, the new machine design has increased the power density of the baseline machine by 64%. This has been done by shrinking the spatial volume for the rotating group in both the radial and axial directions. As Fig. 8 is intended to show, there is a visual difference in the size of the new machine as the wall thickness between bores has been greatly reduced and the axial length has been reduced slightly.

Conclusion


The following conclusions are supported by the analysis, results, and discussion of this paper:

That the power density of the axial piston machine may be described as a ratio of the volumetric displacement to the spatial volume of the machine rotating group

That the cylinder-block layout is constrained by the designed volumetric displacement for the machine, the bore-to-bore hoop stress, and the physical space limitations about the arc length of the piston pitch-radius

That the piston-pitch radius and the piston diameter may be described as a function of the bore-to-bore hoop stress, the number of pistons in the machine, the volumetric displacement, and the swash-plate angle. See Eqs. (5) and (6)

That the piston-pitch radius and the piston diameter scale with the cube root of the machine's volumetric displacement

That the power density of the axial-piston machine is a function of only four dimensionless parameters: the swash-plate angle, the number of pistons, the nondimensional hoop stress in the cylinder block, and the nondimension constant shown in Eq. (15)

That the power density of the machine is most sensitive to changes in the allowable hoop stress and the swash-plate angle

That the power density of the machine is fairly sensitive to changes in the minimum overhang length of the piston and the cylinder-block face thickness as shown in Fig. 3

That the power density of the machine is essentially unaffected by the number of pistons that are used in the design

That it is possible to increase the power density of the axial-piston machine by as much as 64% by allowing the cylinder block to carry more stress, but increasing the swash-plate angle to 21 degrees, and by using an inverted piston-slipper design to eliminate the minimum overhang length of the piston

These results are insightful from a design point of view, and can be used to guide an engineer in his or her choice of design modifications that are needed for altering the power density of the axial-piston machine. As the demand for power density becomes higher and higher, the design engineer will most likely need to consider the options that have been presented in this paper during the development of a new axial-piston product.

Acknowledgements


The authors would like to thank Caterpillar Inc. for the generous support of this project, and for the many useful conversations that have taken place during the development of results.

NomenclatureAp =

cross-sectional area of a single piston

D =

diameter of the rotating-group spatial volume

d =

piston diameter

h =

distance between the outside diameter of the rotating group and the piston pitch-radius

L =

length of the rotating-group spatial volume

l =

minimum engaged length of the single piston with the cylinder block

mmin =

minimum piston overhang length

mo =

nominal piston overhang length

N =

number of pistons

r =

piston pitch-radius

Sc =

sensitivity coefficient with respect to the dimensionless constant c

SN =

sensitivity coefficient with respect to the number of pistons in the machine

Sα =

sensitivity coefficient with respect to the swash-plate angle

Sσ =

sensitivity coefficient with respect to the cylinder-block hoop stress

t =

cylinder-block face thickness

VD =

volumetric displacement per revolution

Vd =

volumetric displacement per radian

Vs =

spatial volume of the machine's rotating group

α =

swash-plate angle

δ =

angular wall-thickness between bores

Π =

power density

σ =

dimensionless hoop stress

ψ =

angular distance across a single pistion

References


1

Blackburn, J. F., Reethof, G., and Shearer., J. L., 1960, Fluid Power Control, Wiley, New York.

2

Manring, N. D., 2000, “Designing the Shaft Diameter for Acceptable Levels of Stress Within an Axial-Piston Swash-Plate Type Hydrostatic Pump,” ASME J. Mech. Des., 122, pp. 553–559. [CrossRef]

3

Ivantysynova, M., 2008, “Innovations in Pump Design: What are Future Directions?” Proceedings of the 7th JFPS International Symposium on Fluid Power, Toyama, Sept. 15–18, pp. 59–64.

4

Ivantysyn, J., and Ivantysynova., M., 2003, Hydrostatic Pumps and Motors: Principles, Design, Performance, Modelling, Analysis, Control and Testing, Tech Books International, New Delhi, India.

5

Manring, N. D., 2013, Fluid Power Pumps and Motors: Analysis, Design and Control, McGraw-Hill, Inc., New York.

6

Achten, P. A. J., 2004, “Power Density of the Floating Cup Axial Piston Pump,” ASME International Mechanical Engineering Congress and Expo, Anaheim, Nov. 13–19, pp. 1–12.

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