Pressure, Flow, Force, and Torque Between the Barrel and Port Plate in an Axial Piston Pump

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J. M. BergadaFluid Mechanics Department,

ETSEIAT UPC,Colon 11,

Terrassa 08222, Spaine-mail: [email protected]

J. WattonCardiff School of Engineering,

Cardiff University,Queen’s Buildings,

P.O. Box 925,Cardiff, CF24 0YF Wales, UK

e-mail: [email protected]

S. KumarFluid Mechanics Department,

ETSEIAT UPC,Colon 11,

Terrassa 08222, Spaine-mail: [email protected]

Pressure, Flow, Force, andTorque Between the Barrel andPort Plate in an Axial PistonPumpThis paper analyzes the pressure distribution, leakage, force, and torque between thebarrel and the port plate of an axial piston pump. A detailed set of new equations isdeveloped, which takes into account important parameters such as tilt, clearance androtational speed, and timing groove. The pressure distribution is derived for differentoperating conditions, together with a complementary numerical analysis of the originaldifferential equations, specifically written for this application and used to validate thetheoretical solutions. An excellent agreement between the two approaches is shown,allowing an explicit analytical insight into barrel/port plate operating characteristics,including consideration of cavitation. The overall mean force and torques over the barrelare evaluated and show that the torque over the XX axis is much smaller than the torqueover the YY axis, as deduced from other nonexplicit simulation approaches. A detaileddynamic analysis is then studied, and it is shown that the torque fluctuation over the YYaxis is typically 8% of the torque total magnitude. Of particular novelty is the predictionof a double peak in each torque fluctuation resulting from the more exact modeling of thepiston/port plate/timing groove pressure distribution characteristic during motion. Acomparison between the temporal torque fluctuation pattern and another work shows agood qualitative agreement. Experimental and analytical results for the present studydemonstrate that barrel dynamics do contain a component primarily directed by thetorque dynamics. �DOI: 10.1115/1.2807183�

Keywords: axial piston pump, barrel to port plate leakage, torque dynamics

IntroductionIt is known that an axial piston barrel experiences small oscil-

ations due to the forces acting over it. Cavitation also occurs inany cases, sometimes damaging the plate and barrel sliding sur-

aces and therefore reducing the volumetric and overall efficiencyf the pump. More importantly, the resulting failure of the pumps often a critical issue in modern industrial applications. Pistonumps and motors are not fully understood in analytical detailince problems related to cavitation, mixed friction, and barrelynamics, among others, are yet to be resolved via explicit meth-ds. This paper attempts to bridge this gap by bringing togetherurely analytical solutions, with numerical validation, in connec-ion with an important area of barrel/port plate leakage flow andssociated torque dynamics.

Helgestad et al. �1� studied theoretically and experimentally theffect of using silencing grooves on the temporal pressure andeakage fluctuation in one piston cycle. Triangular and rectangularilencing grooves versus port plate “ideal timing” and standardort plate were compared. For a range of operating conditions, thehoice of triangular entry grooves was deduced to be the mostppropriate. Martin and Taylor �2� analyzed in detail the start andnish angles for the pressure and tank grooves to have ideal tim-

ng. As in Ref. �1�, graphs are presented to understand the tempo-al pressure and flow in a single piston, but leakage flow was notonsidered. The results showed that triangular silencing groovesere more appropriate in all cases, except when the pump param-

ters are fixed; in such case, ideal timing main grooves were pref-

Contributed by the Dynamic Systems, Measurement, and Control Division ofSME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CON-

ROL. Manuscript received February 21, 2006; final manuscript received June 8,
007; published online January 8, 2008. Review conducted by Suhada Jayasuriya.
ournal of Dynamic Systems, Measurement, and ControlCopyright © 20

om: http://dynamicsystems.asmedigitalcollection.asme.org/ on 05/30/2014

erable. Edge and Darling �3� presented an improved analysis ableto evaluate piston temporal pressure and flow, the improvementbeing based on taking into account the rate of change of momen-tum of the fluid during port opening. As in previously reportedwork, triangular silencing grooves were shown to be most appro-priate for a piston pump operating over a wide range of workingconditions. With regard to cavitation erosion, they defined themost severe region to be at the end of the inlet port and at the startof the delivery port.

Jacazio and Vatta �4� studied the pressure, hydrodynamic force,and leakage between the barrel and plate. The study used Rey-nolds equation of lubrication, integrating it when consideringpressure decay in the radial direction and including rotationalspeed. They found equations for the pressure distribution and liftforce, which showed the dependency of these parameters withrotational speed. Yamaguchi �5� demonstrated that a port platewith hydrostatic pads allows fluid film lubrication over a widerange of operating conditions. When analyzing the barrel dynam-ics, he took into account the spring effect of the shaft, and bychanging some physical parameters he determined the most likelycases for metal to metal contact between barrel and valve plate tooccur. Yamaguchi �6� experimentally studied the barrel and platedynamics, using position transducers, and used four differentplates for experimentation, three of them with a groove, one with-out a groove, and no outer pad. He found that the gap between thebarrel and plate oscillates, the oscillation having a large peak andan intermediate smaller peak. For any kind of fluid used, it wasfound that the film thickness and amplitude increased with in-creasing inlet pressure.

JANUARY 2008, Vol. 130 / 011011-108 by ASME

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Matsumoto and Ikeya �7� experimentally studied the friction,eakage, and oil film thickness between the port plate and cylinderor low speeds. They found that the friction force was almostonstant with rotational speed, but strongly depended on supplyressure and static force balance. In a further paper, Matsumotond Ikeya �8� focused more carefully on the leakage characteris-ics between the cylinder block and plate, again for low speedonditions. The results showed that the fluctuation of the tilt anglef the barrel and the azimuth of minimum oil film thickness de-ended mainly on the high pressure side number of pistons. Koba-ashi and Matsumoto �9� studied the leakage and oil film thick-ess fluctuation between a port plate and a barrel. They integratedumerically the Reynolds equation of lubrication, taking into ac-ount the pressure distribution in both the radial and the tangentialirection. The flow, barrel tilt, and barrel/port plate clearance ver-us angular position were determined at very low rotationalpeeds. Weidong and Zhanlin �10� studied the temporal leakageow between a barrel and a plate and between a piston and aarrel, and considered separately the leakage from each barrelroove and the effect of the inlet groove. The barrel tilt was notaken into consideration. Yamaguchi �11� gave an overview of theifferent problems found when considering tribological aspects ofumps. When assessing the plate and cylinder block performance,e pointed out the effect of the leakage for different fluid viscosi-ies when the port plate has or has not a hydrodynamic groove. Itas found that the use of a groove stabilizes the leakage for dif-

erent fluid viscosities �12�.Manring �13� evaluated the forces acting on a cylinder block

nd its torque over the cylinder main axis. He considered theressure distribution at the pump outlet as constant and the decaylong the barrel lands as logarithmic, independent of the barrel tiltnd turning speed. In a further study �14�, he also investigatedarious port plate timing geometries within an axial piston pump.t was found that a constant area timing groove design had thedvantage of minimizing the required discharge area of the timingroove, the linearly varying timing groove design having the ad-antage of utilizing the shortest timing groove length, and theuadratic timing groove design had no particular advantages overhe other two. Zeiger and Akers �15� considered the dynamicquations of the swash plate, which were linked with piston cham-er pressures. They defined first the temporal piston chamberressure, taking into account the area variation at the inlet andutlet groove entrances. The torque over the swash plate was dy-amically and statically evaluated, finding that the torque averagehanged mainly with the swash plate angle, turning speed, andutlet pressure. They compared simulation and experimental re-ults, finding a good correlation, although leakage was not evalu-ted. In a further study �16�, they presented a model consisting ofsecond order differential equation of the swash plate motion and

wo first-order equations describing the flow continuity into theump discharge chamber and into the swash plate control actuator.

One of the first studies focusing on the understanding of theperating torques on a pump swash plate was undertaken by Inouend Nakasato �17,18�. They found theoretically that the excitingorque acting on the swash plate had a sawtooth shape. They also

easured the torque on the swash plate, finding that it had twoeaks while the exciting one had a single peak. They defined theecond peak as the one appearing when the system reached itsatural frequency. Manring and Johnson �19� defined the dynamicquations of the swash plate in an axial piston pump, such equa-ions having regard to the effect of the two actuators that maintainhe swash plate in position. Wicke et al. �20� simulated the dy-amic behavior of an axial piston pump using the programATHFP. They focused the study on understanding the influence ofhe swash plate angle variation on the piston forces and the yoke

oment around the turning axis They found that an increase of thewash plate angle increased the risk of cavitation in the cylinderhamber at the beginning of the suction port and also decreased
he time averaged yoke moment and increased peak to peak varia-
11011-2 / Vol. 130, JANUARY 2008


tions. In the paper by Manring �21�, he further analyzed the dy-namic torque acting on the swash plate. As in a previous study, hedid not consider the swash plate inertia and damping. He noticedthat piston and slipper inertia tend to destabilize the swash plateposition, although the most important term that created torqueonto the swash plate was due to piston pressure. Gilardino et al.�22� defined the dynamic equations that give the torque onto theswash plate, including the torque created by the displacement con-trol cylinders.

In Ivantysynova et al. �23�, a new method of prediction of theswash plate torque based on the software CASPAR is presented andcalculates the nonisothermal gap flow and pressure distributionacross all piston pump gaps. The study defined a direct link be-tween the dynamic torque acting on the swash plate and the smallgroove dimension located at the entrance and exit of the valveplate main groove. Manring �24� studied the forces acting on theswash plate in an axial piston pump and took into account the“secondary swash-plate angle” as well as the primary swash-plateangle. He demonstrated that the use of a secondary swash-plateangle will require a control and containment device that is capableof exerting a thrust load in the swash-plate horizontal axis direc-tion. In a further study �25�, he examined the control and contain-ment forces for a cradle-mounted, axial-actuated swash plate,showing that an axial-actuated swash plate tends to keep theswash plate well seated within the cradle during all operatingconditions. Bahr et al. �26� used the swash-plate dynamic equa-tions, found in previous papers, to create a dynamic model of apressure compensated swash-plate axial piston pump with a coni-cal cylinder block. They implemented the equations of the com-pensating unit to create a full model of the pump. The equationswere integrated using MATLAB SIMULINK, finding that the lateralmoment acting on the swash plate fluctuates in a periodic fashionand contains nine harmonics and a negative mean value.

Despite the amount of papers published, evidence has not beenfound regarding a full analytical investigation of the describedeffects, and this paper considers this task in a methodical way byconsidering pressure distribution, leakage, force, and torque dis-tribution and then by adding barrel dynamics.

2 Mathematical Analysis

2.1 Pressure Distribution and Leakage Between the Barreland Port Plate: Main Groove Effect. The fundamental equationsare now presented together with the appropriate solutions, theintermediate details of each analysis being presented in theAppendix.

Figure 1 represents the barrel and port plate face of an axialpiston pump, one of the pistons being drawn for clarification. Theport plate transfers the flow rate from the external connectingports via two large kidney-shaped slots machined in the port plateinner face, one at the pump inlet and the other at the pump outlet.These port plate slots, often called grooves, are shown in Fig. 1,where the main dimensions and the central axes are also shown.Notice that a timing groove is placed at the entrance of the maingroove on the pressurized side. The entrance to each piston in thebarrel is via an associated small kidney-shaped port referred tolater as the “piston groove,” that is, there are nine piston groovesmachined on the face of the barrel. The sign convention is that thepositive side of the X axis is toward the left side of the Y axis, andthe barrel is slightly tilted with respect to the port plate with a tiltangle �. The port plate is secured to the main body of the pumpwith four bolts, and the barrel is pushed toward the port plate bya spring located at the bottom of the barrel �not shown in Fig. 1�.This fixing mechanism therefore carries an additional load in-duced by the torque created by the pressure differential across thepump when in operation. Since laminar flow exists, then taking tiltand rotation into account, assuming that the flow moves in a radialdirection, then Reynolds equation takes the following polar coor-
dinate form:
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�

�r�rh3�p

�r� = 6��r

�h

��1�

his equation must be integrated along each particular land, ex-ernal and internal, associated with the main port plate groove andhe timing groove, four lands in total. Each land has its particularoundary conditions, which will be used to find the constants ofntegration.

The pressure distribution for the external land rextr�r�rexter2ill then be

pext land = pint�1 +ln�r/rext�

ln�rext/rext 2�� + pexterln�rexr/r�

ln�rext/rext 2�

+3��rm ext sin �

�h0 + �rm ext cos ��32��rext

2 − r2�

+�rext 2

2 − rext2 �ln�rext/r�

ln�rext/rext 2� � �2�

nd for the internal land rint2�r�rint,

pint land = pint�1 +ln�r/rint�

ln�rint/rint 2�� + pextln�rint/r�

ln�rint/rint 2�

+3��rm int sin �

�h0 + �rm int cos ��32��rint

2 − r2�

+�rint 2

2 − rint2 �ln�rint/r�

ln�rint/rint 2� � �3�

or these equations, the relation between � and � has the condi-ions that when �=0, then �=�max and the main port plate groovexists between −�i�� j.

Once the pressure distribution across the lands on both sides ofhe main port plate grooves has been found, the next step is toetermine the leakage, where the leakage due to the main portlate groove can be expressed as

Qleakage =−�i

�j 0

h

verdyd� +−�i

�j 0

h

virdyd� = Qext + Qint �4�

Once the integrations have been performed, the flows across thexternal and internal lands associated with the main port plate

Fig. 1 Barrel/port plate configuration

roove are given by

ournal of Dynamic Systems, Measurement, and Control


Qext =�pext − pint�

12� ln�rext/rexr2��h03��−�i

�j + 3h02�rmext�sin ��−�i

�j

+ 3h0�2rmext2 �1

4sin�2�� +

�

2�

−�i

�j

+ �3rmext3 � 1

12sin�3�� +

3

4sin ��

−�i

�j � �5�

Qint = −�pext − pint�

12� ln�rint/rint2��h03��−�i

�j + 3h02�rmint�sin ��−�i

�j

+ 3h0�2rmint2 �1

4sin�2�� +

�

2�

−�i

�j

+ �3rmint3 � 1

12sin�3�� +

3

4sin ��

−�i

�j � �6�

The leakage due to the main groove effect will be the addition ofthe two leakage flows. According to Eqs. �5� and �6�, the leakageis now explicitly expressed as a function of geometry, internal andexternal pressures.

2.2 Force and Torque on the Barrel Due to the PressureDistribution Along the Main Groove. The force between thecylinder block and the port plate due to the main port plate grooveand the associated two lands is given as

F =−�i

�j rint

rext

Pintrd�dr +−�i

�j rint2

rint

Pint landrd�dr

+−�i

�j rext

rext2

Pext landrd�dr �7�

The solution then gives

F = Pint„� j − �− �i�…

4� �rext

2 − rext22 �

ln�rext/rext2�−

�rint2 − rinr2

2 �ln�rint/rint2��

+ Pext„� j − �− �i�…

2�rext2

2 − rint22 � + Pext

„� j − �− �i�…4

�� rint2 − rint2

2 �ln�rint/rint2�

−�rext

2 − rext22 �

ln�rext/rext2�� 8�

According to Eq. �8�, a linear variation between force and internalpressure is expected. When studying the torque created by thenonuniform pressure distribution, two torques need to be consid-ered, and they have the following general forms:

MXX =−�i

�j rint

rext

Pintr2 sin �drd� +

−�i

�j rint2

rint

Pint landr2 sin �drd�

+−�i

�j rext

rext2

Pext landr2 sin �drd� �9�

MYY =−�i

�j rint

rext

Pintr2 cos �drd� +

−�i

�j rint2

rint

Pint landr2 cos �drd�

+−�i

�j rext

rext2

Pext landr2 cos �drd� �10�

The internal and external land pressures will be given by Eqs.�2� and �3�. Since the case studied is for a symmetrical groove, thefirst integral of Eq. �9�, amongst others, have a zero value. There-
fore, after integration and rearrangement, it is found that
JANUARY 2008, Vol. 130 / 011011-3


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MXX =−�i

�j 3��rmint sin2 �d�

�h0 + �rmint cos ��3 � rint2

6�rint

3 − rint23 � −

�rint5 − rint2

5 �10

+�rint2

2 − rint2 �

6�rint

3 − rint23 �

ln�rint�ln�rint/rint2�

−�rint2

2 − rint2 �

2 ln�rint/rint2�� rint23 − rint

3 �9

+rint

3 ln�rint� − rint23 ln�rint2�

3��

+−�i

�j 3��rmext sin2 �d�

�h0 + �rmext cos ��3 � rext2

6�rext2

3 − rext3 � −

�rext25 − rext

5 �10

+�rext2

2 − rext2 �

6�rext2

3 − rext3 �

ln�rext�ln�rext/rext2�

−�rext2

2 − rext2 �

2 ln�rext/rext2�� rext3 − rext2

3 �9

+rext2

3 ln�rext2� − rext3 ln�rext�

3�� 11�

he torque over the Y axis, when considering a symmetric groove,ill be

MYY =pint�sin ��−�i

�j

9� �rint2

3 − rint3 �

ln�rint/rint2�+

�rext3 − rext2

3 �ln�rext/rext2��

+pext�sin ��−�i

�j

9�−

�rint23 − rint


−�rext

3 − rext23 �

ln�rext/rext2��+

pext�sin ��−�i

�j

3�rext2

3 − rint23 � �12�

It should be noticed that the torque about the XX axis dependsn the pump turning speed and tilt, which means that for theymmetrical groove case studied here, such a torque will be zerof any of these parameters is zero. On the other hand, the torquebout the YY axis is independent of the tilt and pump turningpeed, and depends only on the geometry and the internal pres-ure.

2.3 Effect of the Entrance Timing Groove. In order to re-uce pump noise and minimize sudden pressure changes and as-ociated backflow losses, it is common to add a small groove,sually called timing groove, at the entrance of the main outletroove. Therefore, the next step is to implement the equationslready developed to include this timing groove effect. As for theain groove, the equations for the timing groove will be based on

he Reynolds equations of lubrication �Eq. �1��. The equationsiving the pressure distribution along the internal and externalands next to the timing groove are similar to the ones alreadyound for the main groove, the main differences when solving theifferential equation being the boundary conditions and the limitsf integration. Using the appropriate boundary conditions shownn the Appendix, and considering Fig. 1, the solution becomes asollows:

For the external land, Rext�r�rext2,

pext landsg= pint�1 +

ln�r/Rext�ln�Rext/rext2�� + pext

ln�Rext/r�ln�Rext/rext2�

+3��Rmext sin �

�h0 + �Rmext cos ��32��Rext

2 − r2�

+�rext2

2 − Rext2 �ln�Rext/r�

ln�Rext/rext2� � �13�

or the internal land: rint2�r�Rint,

11011-4 / Vol. 130, JANUARY 2008


pint landsg= pint�1 +

ln�r/Rint�ln�Rint/rint2�� + pext

ln�Rint/r�ln�Rint/rint2�

+3��Rmint sin �

�h0 + �Rmint cos ��32

��Rint2 − r2� +

�rint22 − Rint

2 �ln�Rint/r�ln�Rint/rint2� � �14�

The leakage associated with the timing groove, bearing in mindthe boundary conditions, is now given as

Qleakagesg=

−��i+��

−�i 0

h

verdyd� +−��i+��

−�i 0

h

virdyd�

= Qextsg+ Qintsg

�15�

where the velocities ve and vi and the gap depth h will have thesame generic form as in the main groove case. Since the limits ofintegration are now nonsymmetrical, some of the terms for themain groove that were previously zero now exist. Therefore, fol-lowing the procedure established in the main groove case, theinternal and external land leakages will be

Qintsg=

3��RmintRint2

12�cos�− �i� − cos„− ��i + ��…�

+3��Rmint

12 ln�Rint/rint2��rint2

2 − Rint2 �

2�cos�− �i� − cos„− ��i + ��…�

−pext − pint

12� ln�Rint/rint2��h03�− �i − „− ��i + ��…�

+ 3h02�Rmint�sin�− �i� − sin„− ��i + ��…� + 3h0�2Rmint

2

��1

4sin„2*�− �i�… +

− �i

2−

1

4sin�2*

„− ��i + ��…�

−− ��i + ��

2�

+ �3Rmint3 � 1

12sin„3*�− �i�… +

3

4sin�− �i�

−1

12sin�3*

„− ��i + ��…� −3

4sin„− ��i + ��…�� 16�

Qextsg=

3��RmextRext2

12�cos�− �i� − cos„− ��i + ��…�

−3��Rmext

12 ln� Rext

rext2�

�rext22 − Rext

2 �2

�cos�− �i� − cos„− ��i + ��…�

+pext − pint

12� ln�Rext/rext2��h03�− �i − „− ��i + ��…�

+ 3h02�Rmext�sin�− �i� − sin„− ��i + ��…�

+ 3h0�2Rmext2 �1

4sin„2*�− �i�… +

− �i

2

−1

4sin�2*

„− ��i + ��…� −− ��i + ��

2�

+ �3Rmext3 � 1

12sin„3*�− �i�… +

3

4sin�− �i�

−1

12sin�3*

„− ��i + ��…� −3

4sin„− ��i + ��…�� 17�

It is interesting to note that the leakage due to the timing groove



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epends on the barrel rotational speed, this effect appearing due tohe groove asymmetry.

The force caused by the action of the timing groove is

Fsg =−��i+��

−�i Rint

Rext

Pintrd�dr +−��i+��

−�i rint2

Rint

Pint landsgrd�dr

+−��i+��

−�i Rext

rext2

Pext landsgrd�dr �18�

hen substituting the pressure distribution derived earlier, Eqs.13� and �14�, into Eq. �18� and after integration and rearrange-ent it is found that

Fsg = Pint�− �i − „− ��i + ��…�

4� �Rext

2 − rext22 �

ln�Rext/rext2�−

�Rint2 − rinr2

2 �ln�Rint/rint2��

+ Pext�− �i − „− ��i + ��…�

2�rext2

2 − rint22 �

+ Pext�− �i − „− ��i + ��…�

4� �Rint

2 − rint22 �

ln�Rint/rint2�−

�Rext2 − rext2

2 �ln�Rext/rext2��

+−��i+��

−�i 3��Rmint sin �

�h0 + �Rmint cos ��3d�

� �Rint2 − rint2

2 �2�1

8−

ln�Rint�

4 ln� Rint

rint2� −

1

8

1

ln� Rint

rint2��

+rint2

2 − Rint2

4

1

ln� Rint

rint2� �rint2

2 ln�rint2� − Rint2 ln�Rint��

+−��i+��

−�i 3��Rmext sin �

�h0 + �Rmext cos ��3d�

� �rext22 − Rext

2 �2�−1

8+

ln�Rext�

4 ln� Rext

rext2� +

1

8

1

ln� Rext

rext2��

−rext2

2 − Rext2

4

1

ln�Rext/rext2�„rext2

2 ln�rext2� − Rext2 ln�Rext�…

�19�

he remaining integrals in Eq. �19� must be solved numerically.he torque over the X and Y axes created by the timing groove at

he entrance will be given by

MXXsg=

−��i+��

−�i Rint

Rext

Pintr2 sin �drd�

+−��i+��

−�i rint2

Rint

Pint landsgr2 sin �drd�

+−�� +��

−�i R

rext2

Pext landsgr2 sin �drd� �20�

i ext



MYYsg=

−��i+��

−�i Rint

Rext

Pintr2 cos �drd�

+−��i+��

−�i rint2

Rint

Pint landsgr2 cos �drd�

+−��i+��

−�i Rext

rext2

Pext landsgr2 cos �drd� �21�

When substituting the pressure distribution in each land, Eqs.�13� and �14�, into the torque equations and after some develop-ment, it is found that

MXXsg=

pext

3�rext2

3 − rint23 ��− cos ��−��i+��

−�i +�pint − pext�

9� rint2

3 − Rint3

ln�Rint/rint2�

+Rext

3 − rext23

ln�Rext/rext2��− cos ��−��i+��−�i

+−��i+��

−�i 3��Rmint sin2 �d�

�h0 + �Rmint cos ��3

�� Rint5 − rint2

5 �15

−�rint2

2 − Rint2 �

18

�rint23 − Rint


+−��i+��

−�i 3��Rmext sin2 �d�

�h0 + �Rmext cos ��3

�� rext25 − Rext

5 �15

−�rext2

2 − Rext2 �

18

�Rext3 − rext2

3 �ln�Rext/rext2�� 22�

MYYsg=

pext

3�rext2

3 − rint23 ��sin ��−��i+��

−�i +�pint − pext�

9

�� rint23 − Rint

3

ln�Rint/rint2�+

Rext3 − rext2

3

ln�Rext/rext2��sin ��−��i+��−�i +

3��Rmint

h03

�� Rint5 − rint2

5 �15

−�rint2

2 − Rint2 �

18

�rint23 − Rint


�� 1

��Rmint/h0�2� 1

1 + ��Rmint/h0�cos �

−1

2�1 + ��Rmint/h0�cos ��2�−��i+��

−�i �+

3��Rmext

h03 � �rext2

5 − Rext5 �

15−

�rext22 − Rext

2 �18

�Rext3 − rext2

3 �ln�Rext/rext2��

�� 1

��Rmext/h0�2� 1

1 + ��Rmext/h0�cos �

−1

2�1 + ��Rmext/h0�cos ��2�−��i+��

−�i � �23�

Again, the remaining integrals in Eq. �22� must be solved nu-merically. The overall leakage, pressure distribution, force, andtorque versus the barrel main axis will be the addition of theequations of the main groove and the timing groove for each case.Special attention has to be made when regarding each torque signconvention due to each direction of action.

2.4 Effect of Cylinder Pressures. To complete the analysis, itis now necessary to include the effect of the pressure inside eachcylinder chamber. Such a summation of pressure forces will createboth forces and torques that will act in an opposite direction thanthe ones already presented. The first thing to be noticed is that
pressure inside the cylinder chamber changes with time. Follow-
JANUARY 2008, Vol. 130 / 011011-5


ist

apZbl

F=

caamhstwoaj

3

tftcapgifcammfmagtctm

4

tgcrmin2mTdc

0

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ng previous work and a consideration of fluid volumes in thistudy, compressibility effects are negligible; therefore, to find theemporal pressure, the following equation is sufficient:

Pint cylinder = Pint +�

2� Q

CdAflow�2

= Pint +�

2�VpistonAcylinder

CdAflow�2

�24�It has been assumed that since the leakage flow between piston

nd cylinder is of a much lower order of magnitude than theiston flow, its effect can be neglected when calculating pressure.ero reference time is defined when one of the pistons is at itsottom dead center. The piston velocity can be calculated as fol-ows:

V = �R tan sin��t� �25�

or the case under study, max=20 deg, �=1440 rpm, R0.03434 m, and �=875 kg /m3.There are few useful data regarding detailed discharge coeffi-

ients inside piston pump cylinders, and it will be assumed that anverage discharge coefficient value of 0.75 is reasonable. Whenll values are substituted into Eq. �24�, it is noticed that the maxi-um pressure inside the cylinder is only around 0.003 MPa

igher than the pressure outside, taking into account that the out-ide pressure can be typically 10–35 MPa. As the barrel rotateshe force and torque created by the pressure inside, the cylindersill vary with rotation angle, hence time. It is necessary to pointut that for 40 deg of rotation, 28 deg embraces five pistons thatre connected to pump pressure, while during the other 12 degust four pistons are connected to pump pressure.

Additional Numerical SolutionThe developed solutions are complex, some requiring addi-

ional numerical integrations, and are human error prone. There-ore, to validate the solutions developed, a numerical solution ofhe original differential equations has also been undertaken spe-ifically for this study of flow through the gap between the barrelnd the port plate. Regarding the mesh selection, it is important tooint out that the mesh created in the � direction along the mainroove lands has a step size much bigger than the step size neededn the � direction along the timing groove lands. This is due to theact that the land for the barrel plate in the radial direction asso-iated with the main groove is much smaller than the one associ-ted with the timing groove. Therefore, although the flow will beostly radial in both cases, the effect of barrel rotation will beore intense along the small groove lands. Nevertheless, it was

ound that the radial direction pressure drop in both cases wasuch bigger than the tangential pressure drop. To increase the

ccuracy of the results at any cell, the average pressure of all fourrid points was used. The program was written and executed usinghe MATLAB software. In fact, the program consists of a code thatalculates the pressure distribution between barrel and plate; fromhe pressure distribution, leakage, forces, and torques are deter-

ined.

Analytical Results

4.1 Pressure Distribution. The theoretical pressure distribu-ion according to Eqs. �2�, �3�, �13�, and �14� along the mainroove and timing groove lands is represented in Fig. 2 for a set oflearances and turning speeds. At this point, it is interesting toeflect on the work by Edge and Darling �3�, who found that theost severe region for cavitation erosion was at the end of the

nlet port and at the start of the delivery port, as predicted by theew theoretical solution shown in Fig. 2. From Figs. 2�a� and�b�, it is shown that cavitation at the entrance of the timing andain groove is more likely to appear as the clearance is reduced.he pressure asymmetry is accentuated as the central clearanceecreases, and it is also accentuated as the inlet pressure de-
reases. It is important to notice that as the turning speed increases
11011-6 / Vol. 130, JANUARY 2008


�Figs. 2�c� and 2�d��, the pressure distribution asymmetry in-creases as well and will tend to create a low pressure area at bothsides of the groove entrance. Therefore, it can be said that accord-ing to the theory developed, cavitation at the entrance of the tim-ing and main grooves is more likely to appear when workingunder low pressures, small clearances, and high turning speeds.

In Fig. 3, some analytical and numerical pressure distributionsare compared, and it will be noticed that for the particular casespresented, the pressure distribution comparisons are very similareven when cavitation appears at the timing groove entrance. Thisgives further support to the analytical predictions being presentedhere.

4.2 Leakage in the Main Groove and the Timing Groove.Figure 4 represents the leakage for different central clearances h0,where for each clearance the maximum angle � possible has beenused. A comparison between the numerical solution and the theo-retical equations is also shown. Figure 4�a� shows the leakagefrom the main groove, and the timing groove, and an excellentagreement between the theoretical equations and the numericalsolution is evident.

Although not presented, the leakage variation with pressure dif-ferential is linear, as expected for both grooves. From thesegraphs, it is clearly noticed that leakage is much higher on theexternal land than on the internal one. Leakage also has the ten-dency to reach an asymptotic value as the barrel-plate distanceincreases. When comparing Figs. 4�a� and 4�b�, it can be said thatthe leakage due to the timing groove is less than 5% of the maingroove leakage. One of the reasons why the timing groove leak-age is so small is because the land between the groove and theexterior is much bigger than in the main groove.

4.3 Force Acting on the Barrel. When evaluating the forceon the barrel due to the main groove, according to Eq. �8�, it isexpected that a linear variation between the force and the internalpressure will exist. It is also noted from Eq. �8� that such a forcewill not depend on the rotational speed or tilt, but just on theremaining geometry. This lack of significant force variation is dueto the groove symmetry, and when the symmetry disappears, it isexpected that some further force variation will exist. In fact, theforce variation with angular velocity and clearance exists whenstudying the force created by the timing groove. The force for thebarrel-plate versus pressure differential, due to the timing groove,for a constant turning speed and a given clearance is representedby Eq. �19�, from where it can be noticed that the force slightlyincreases with the central clearance. Although not presented, theforce decreases with the increase of angular velocity. It must betaken into account that for pressures less than 15 MPa, cavitationis much likely to appear for this design with a 1 �m clearance, butonly for a very small zone around the small groove entrance. Tofind out the net force over the barrel, it is necessary to consider theopposing force due to the pressure inside each cylinder chamber,described in Sec. 2.4. This force will depend on the number ofpistons pressurized; therefore, in Fig. 5, the overall force is pre-sented when four or five pistons are under pressure. The reality isthat values will fluctuate between the two total limits.

When computing the overall force due to the main groove,timing groove, and cylinder pressure in Fig. 5, it is shown that thetiming groove plays a very small role. Therefore, all changes inforce due to parameters other than pressure will be negligible. Itcan be said that the inclusion of the small groove would bring anincrease of force of typically 10% for the case studied. Again, thenumerical analysis results and analytical results show an excellentagreement, the two approaches producing indiscernible predic-tions. It is very important to point out that when considering theforce due to the cylinders, it is noticed that when the fifth piston isunder pressure the total force over the barrel is much more bal-anced than when just four cylinders operate. In fact, the fluctua-tion of the total force when acting on four or five pistons is almost
40% of the total force acting over the barrel.


msn

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4.4 Mean Torques About the XX and YY Axes. Perhaps theost interesting aspect of this study regarding average steady

tate conditions is the fact that torque over the X and Y axes canow be clearly and explicitly defined.

Consider first the torque about the XX axis (Eqs. (11) and (22)).ue to the main groove and timing groove effects the torque is

epresented in Fig. 6, where it is noticed that the maximum torquever the X axis will be found for the highest turning speeds andmallest clearances. Notice that the torque found via the numericalpproach is very close to the theoretical predictions.

To be analytically appropriate, the minimum clearance has beenet to 3 �m since cavitation exists in the small groove below thislearance, even though the area of cavitation is very small, asreviously discussed. It has to be pointed out that the torque in-reases with the central clearance for the timing groove and wille numerically greater than the effect due to the main groove.ote also that the torque due to the main groove is not zero due to

he effect of angular velocity, which distorts the pressure distribu-ion around the groove. It is also noticed that the main grooveorque acts in the opposite direction to the timing groove torque;he main groove torque is mainly created by the effect of theotational speed, but the timing groove torque is mainly created byhe static pressure.

Consider next the torque about the YY axis (Eqs. (12) and23)). It is important to point out that the torque due to the main

Fig. 2 Theoretical pressure distribution alongparameters. Maximum tilt.

ort plate groove �Eq. �12�� is independent of the central clear-



ance, the turning speed, and the tilt. Therefore, it is expected thata linear behavior versus the pressure differential will exist. On theother hand, the torque over the Y axis created by the timinggroove makes it clear that it will depend on tilt angle, turningspeed, central clearance, inlet pressure, and geometrical dimen-sions �Eq. �23��. Figure 7 shows the torque Myy for the maingroove and the timing groove versus tilt angle and for a set ofcentral clearances. These have been represented in a consistentway to those for Mxx, but it is clear that the contribution from themain groove is substantially constant for a particular pump pres-sure. Theory also suggests that there is no speed effect, and Fig.7�a� shows that the effect of central clearance change isnegligible.

It is important to note when considering the main groove thatthe contribution from the pistons varies with angular position.Therefore, Fig. 7�a� shows the average torque evaluated over arotation of 40 deg. It will be demonstrated later that the cylindertorque fluctuation is about 7.5%. The timing groove effect is, ofcourse, substantially lower than the main groove effect althoughMyy does increase with increasing central clearance and decreaseswith increasing speed. This effect is perfectly understandable onceit is understood what happens with the pressure distribution onboth lands of the timing groove when modifying the barrel turningspeed and or the central clearance. As the rotational speed de-creases and/or the clearance increases, the pressure distribution

e main and small grooves for a set of different
th
across and along the timing groove lands tends to be higher and

JANUARY 2008, Vol. 130 / 011011-7


mbmags

5

mmtupaIptRbs

sol

0

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ore symmetrical, giving a higher value for the torques aboutoth axes. Figures 6�b� and 7�b� clearly indicate that the maxi-um torques due to the timing groove occur at the lowest speed

nd highest central clearance. As in the previous cases, the resultsiven by the equations presented in this paper and the numericalolution have a very good agreement.

Torque DynamicsUntil this point, pressure, leakage, force, and torque versus theain barrel axis have been evaluated in the absence of pistonotion effects by just considering average values. However, as

he barrel turns around the pump main axis, then, in fact, the areander pressure will depend on the number and position of theistons appropriate to the geometry. Therefore, the leakage, force,nd torque over the two main axes will vary with barrel position.n this next step, it will be assumed that once a timing groove andiston entrance groove �piston groove� overlap is established, thenhe same static pressure will be found along the entire groove.egarding the effect of the piston leaving the main groove, it wille assumed that when a piston leaves the main groove, its pres-

Fig. 3 Pressure distribution along the maineters. Maximum tilt, numerical and analytical

ure will remain until the piston reaches the top dead center. In

11011-8 / Vol. 130, JANUARY 2008


order to clarify the different acting torques at each time, Table 1has been created and it represents the different grooves andgroove angles, which are active during one cycle. It also repre-sents the beginning and ending of its particular cycle, the durationof a particular cycle may be of 4 deg or 8 deg of the total cycleduration of 40 deg equivalent to 0.00463 s. Since the pump hasnine pistons, nine cycles of 40 deg will be appearing for eachrevolution and Table 1 starts when one of the pistons is at itsbottom dead center.

In the third row of Table 1 and for the time interval between0.00139 s and 0.00231 s, the effect of the piston exiting pressureis highlighted. For each case defined in Table 1, Eqs. �22� and �23�are used. A crucial modification to these equations is necessary interms of the integration limits appropriate due to the changingeffective cross-sectional area as the piston groove passes over andthen leaves the timing groove. Similar considerations must bemade for a piston groove as it leaves the main port plate groove.The angle range is different at the average external and internalradii due to the relative widths of the timing groove and the pistongroove seen by the piston. It has to be recalled that numerical

small grooves for a set of different param-utions.

and

integrations have to be performed when finding the Mxx torque.



Btiocpgs

Fd

Ftt

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efore introducing the overall dynamic torques, Fig. 8 shows theemporal torques over the barrel main axis due to the pressurenside the cylinders. As stated previously, such a torque dependsn the cylinders position, fluctuating considerably when one of theylinders is pressurized and depressurized. Notice that the graphsresented are given for pressures of 25 MPa and 18 MPa. Theraphs at 18 MPa will be used to compare with experimental re-

ig. 5 Mean force over the pump barrel for a set of pressureifferentials, numerical solution, and theory

ig. 4 Leakage between the barrel and plate for different cen-ral clearances and for maximum � at each clearance. The in-ernal pressure is 25 MPa. „a… Main groove. „b… Timing groove.

ults.



Fig. 6 Mxx torque due to the main and timing groove effects,maximum tilt, 25 MPa. „a… Main groove. „b… Timing groove.

Fig. 7 Myy torque due to the „a… main and „b… timing groove
effects, maximum tilt, 25 MPa.
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Faoetth

1waob

IPdTd

Td

Td

Td

Td

Td

Fi22

0

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The temporal net dynamic torque over the barrel is presented inig. 9. At every step, the effect of the torques due to the timingnd main grooves, piston entrance and exiting effects, and thepposite dynamic torques due to the cylinder pressure is consid-red. It is noticed that the torque Myy is about ten times greaterhan the torque Mxx. The value of Myy, however, has a variation ofypically 8% of the total torque for both inlet pressures presentedere.

On the other hand, the torque Mxx at 25 MPa changes from4 N m to −38 N m every 40 deg and from 12 N m to −27 N mhen the inlet pressure is 18 MPa. Therefore, the oscillations cre-

ted about the XX axis are expected to be much higher than thenes created about the YY axis. It is important to point out thatoth torques have two peaks every 40 turning degrees, such peaks

Table 1 Groove and groove angle

nitial position.iston in loweread center �LDP�

Initial time=0 s Central groove In

Final time �0.000926 s

Small groove

otalisplacement=8 deg

Piston entrance


Initial time �0.000926 s

Central groove In




Central groove Inpo


Piston exiting



Central groove In




Central groove In


Small groove



Central groove InSmall groove


Piston entrance

ig. 8 Temporal changes in the torques generated due to cyl-nder pressure effects during motion, 1440 rpm, 18 MPa, and5 MPa. „a… XX axis, 25 MPa. „b… XX axis, 18 MPa. „c… YY axis,
5 MPa. „d… YY axis, 18 MPa.
11011-10 / Vol. 130, JANUARY 2008


being created every time a piston reaches the top dead center andevery time a piston groove is just in physical proximity with thetiming groove.

At this point, it is recalled that the work done by Inoue andNakasato �17,18� focused on the dynamics of a swash plate, andduring measurements they found that the torque over the XX axishad two peaks of different amplitudes every cycle. They thoughtthat the first peak was due to the excitation torque, while thesecond peak was created when the excitation torque frequencywas near the system natural frequency. From Fig. 9, it can be seenthat in reality the excitation torque is composed of two peaks ofdifferent amplitudes, but results from a different mechanism pro-posed by the authors. Ivantysynova et al. �23�, using the numerical

hich are active during one cycle

position 86�−66 Final position 94�−66

−66�−74 −66�−66

−74�−106 −66�−98


n66�−94 Final

position66�−86

98�66 106�74



−82�−82 −74�−82

position 78�−74 Final position 86�−66−74�−82 −66�−74−82�−114 −74�−106

Fig. 9 Dynamic torque variation about each axis, 1440 rpm,18 MPa, and 25 MPa, h0=4 �m, �=0.006104 deg „a… YY axis,25 MPa. „b… YY axis, 18 MPa. „c… XX axis, 25 MPa. „d… XX axis,

s w

itial

itial

itialsitio

itial

itial

itial

18 MPa.



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nalysis program CASPAR, also reported the two peaks character-stic of the torque acting on the swash plate. They found a clearnteraction between the groove dimensions on the port plate andhe torque fluctuation acting on the swash plate. In this work, itas clearly seen that modifying the small groove length producescomplete different torque variation.Due to the lack of direct torque measurements on the barrel, for

his study, the theoretical exciting torque on the barrel Mxx isualitatively compared with the computer model performed byvantysynova �23� and shown in Fig. 10, albeit using a differentxial piston pump. Such comparisons can be done when realizinghat both dynamic torques in fact have the same origin, that is, theiston pressure fluctuation when turning around the port plate.

Although the two pumps compared are different, a good quali-ative agreement can be found since in both cases the input torqueresents a large peak and a smaller peak. As Ivantysynova pointedut, the shape and dimensions of the grooves cut on the port plateill be decisive on the torque dynamics acting on the barrel and

he swash plate. Therefore, the differences in peak shape are easilynderstood when considering the timing groove dimensions andosition between the present pump and the one used by Ivan-ysynova. It is clearly stated, on the other hand, that both torqueynamics have the same origin.

Experimental Test RigTo measure the barrel dynamic displacement, the test rig pre-

ented in Fig. 11 was created. Two Micro-Epsilon inductive posi-ion transducers, capable of measuring to an accuracy of 0.1 �m,ere used and located at one end of the Y axis and at 45 deg. The

ransducer calibration showed an excellent linearity and producedalibration factors of 47.75 �m /V. The barrel dynamic displace-ent was measured, and Fig. 12 presents the dynamic displace-

ig. 10 Qualitative comparison of torques acting on the barrelnd on the swash plate. „a… Present study, analytical, 25 MPa.b… Ivantysynova †23‡, computer model.

ig. 11 Scheme of the test rig, showing the transducer’s
osition


ment measured at 18 MPa. Notice that at least two waves aresuperposed; the main wave has a frequency of around 24 Hz,which is the pump rotational frequency, the frequency of the sec-ond wave being nine times the pump frequency. The low fre-quency wave is due to the barrel face runout, while the highestfrequency is due to each piston torque effect when entering andleaving the pressure groove. Therefore, the superimposed sinu-soidal wave is easily extracted from the measurement.

7 Barrel DynamicsOnce the torques MXX and MYY acting over the barrel swash-

plate system are known, the theoretical dynamic barrel positionversus the barrel main axis can be calculated. The independentdynamic equations over the XX and YY axes may be approximatedas follows:

Ix�̈ + Bx�̇ + Kx�� − �0� + Fforcesxsgn��̇� = MXX �26�

Iy�̈ + By�̇ + Ky�� − �0� + Fforcesysgn��̇� = MYY �27�

The damping coefficient for the barrel piston assembly may beconsidered to be very small. Friction forces exist on the barrelface and the barrel circumference, and act mainly against the turn-ing movement of the barrel, and can be considered small whenstudying the barrel XX and YY axis acceleration. Also, the tor-sional spring constant, although unknown, has to play an impor-tant role when studying the barrel dynamics, and inertia effectswill need to be considered. Since the moment of inertia changes asthe barrel rotates, a first approximation using an average momentwill be considered. The calculated average moments of inertiawhen pistons are not considered are

Ixx = 0.00909066 kg m2 Iyy = 0.00909066 kg m2

When the piston inertia is considered, the barrel plus pistonmoment of inertia will be time dependent but negligible in thisexample. The peak to peak variation is around 0.5% of the meanvalue, the mean value being Ixx= Iyy =0.0127 kg m2. The twotorques of 18 MPa presented in Fig. 9 therefore act as inputs andare generated as data files to allow a dynamic solution via theMATLAB SIMULINK package. The theoretical barrel dynamic gapposition is presented in Fig. 13 and shows the small effect of thedamping coefficient on the dynamic behavior. Notice that, as ex-pected, an increase in the damping coefficient will reduce thefluctuation, therefore reducing the magnitudes of both peaks. Fig-ure 14 shows the measured temporal barrel position obtained us-ing the test rig defined in Sec. 6 and after the runout signal hasbeen removed.

It is noticed that the experimental result is very noisy and isattributed to high frequency fluid dynamic effects not included in

Fig. 12 Barrel dynamic gap measurements obtained from theposition transducer located over the YY axis. Outlet pressureof 18 MPa.

the mathematical model �20�. This frequency of 1250 Hz is pos-

JANUARY 2008, Vol. 130 / 011011-11


sateigepadm

8

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taAlgca

Fs

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ibly created by the fluid momentum change during port opening,s reported by Edge et al. �3�, and may very well explain some ofhe peaks of the experimental results obtained. Nevertheless, thexperimental trends show clearly the peak created by the torquencrease as each piston groove faces the timing groove, having aood agreement with the theory presented. It is to be noticed thatxperimental and analytical results have the same frequency, theumping frequency. Results by Yamaguchi �6�, using a differentxial piston pump, show a good qualitative agreement with theisplacements found, although the amplitudes of vibration wereuch higher and no fluid dynamic effects were evident �Fig. 15�.

ConclusionsA set of new equations has been developed and allows progress

o be made on the analytical solution for the pressure distribution,eakage, force, and both torques between the barrel and port platef an axial piston pump. The Reynolds equation of lubrication haslso been compared using a numerical method specifically appli-able to the gap between the barrel and port plate. Results fromhe numerical model and the theoretical model were compared,iving a very good agreement in all cases.

Leakage was found to be greater across the external land thanhe internal land, for the same operating pressure, and typically byfactor of 2 whether it be related to pressure or central clearance.s expected, the small timing groove produced a significantly

ower leakage than the main groove and could probably be ne-lected from a total flow loss point of view. It was found thatavitation in pumps is more likely to appear for smaller clear-nces, smaller output pressures, and bigger turning speeds.

Fig. 13 Computed gap position dynamics, Xefficient, 25 N m/rad s−1. „b… Damping coeffici

Fig. 14 Experimental position dynamics, XX axis, 18 MPa

ig. 15 Experimental barrel position dynamics XX axis, mea-
ured by Yamaguchi †6‡
11011-12 / Vol. 130, JANUARY 2008


The effect of precisely defining piston position was shown to becrucial when calculating the total force on the barrel, the totalforce reducing as the number of active piston changes from 4 to 5during one cycle. This was shown to be due to the balance be-tween piston pressure effects and groove effects, the former be-coming more dominant as the number of piston active changesfrom 4 to 5.

The total static torque is dominated by the contribution aboutthe YY axis as anticipated from other work and simple consider-ations of the pumping mechanism. Individual torque contributionswere shown to increase with pressure in almost a linear manner,the torque effect of the combined grooves being larger than theopposing torque due to piston pressure.

Both dynamic torques acting over the barrel XX and YY axeswere carefully studied and were introduced into a Simulink modelto evaluate the barrel temporal position. Comparisons betweennumerically analyzed torques presented by another researcher andthe ones presented in this paper show a good qualitative agree-ment, and clarify the role of the small grooves cut on the portplate regarding torque dynamics. In particular, the importance of aprecise modeling of the flow continuity mechanism was shown tobe crucial in predicting the correct wave form shapes.

Temporal torque calculations showed a marked difference inshape for each axis considered due to piston pressure effects, thepeak to peak values being much greater across the XX axis thanthe YY axis, although the average torque over the YY axis is muchhigher than the one over the XX axis. This extends to the overalltorque fluctuation when groove effects are also taken into account.

A test rig has been created to measure the barrel dynamic dis-placement. The results clearly show the torque increase as eachpiston groove is connected to the timing groove, as established bythe theory presented. This was in spite of the requirement to ex-tract the required data from a noisy signal due to barrel runout. Agood correlation between analytical and experimental results wasfound, and it can be concluded that the barrel dynamic displace-ment above the runout displacement is created by torque dynam-ics. The torque increase, as the piston groove faces the timinggroove, is responsible for the small peak fluctuation presented.

A direct analytical foundation has been therefore established forfuture design possibilities. Further work needs to be addressed tostudy in detail the barrel dynamics for different inlet pressures andswash-plate angles, and the aim would be to fully understand theorigin of the higher frequencies appearing in the experiments.

NomenclatureAcylin � cylinder area �m2�Aflow � flow cross section at the end of the cylinder

�m2�B � barrel damping coefficient �N m / rad s−1�

c1, c3 � constants �N m�c2, c4 � constants �N /m2�

Cd � discharge coefficientF � force due to the main groove �N�

xis. Frictional torque 3 N m. „a… Damping co-30 N m/rad s−1.

X aent

Fsg � force created by the timing groove �N�



A

flcpowgt

w

J

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Fforce � barrel frictional torque �N m�h � barrel general height �m�

h0 � barrel central height �m�I � barrel moment of inertia versus a generic angle

�kg m2�K � spring torsional constant acting over the barrel

�N m�MXX � torque versus the barrel X axis �N m�MYY � torque versus the barrel Y axis �N m�

pi � general pressure �N /m2�Pint cyl � pressure inside the cylinder �N /m2�

pext � pump inlet �tank� pressure �N /m2�pint � pump outlet pressure �N /m2�

pext land � pressure distribution across the external land,main port plate groove �N /m2�

pint land � pressure distribution across the internal land,main port plate groove �N /m2�

pext land sg � pressure distribution across the internal land,timing groove �N /m2�

pint land sg � pressure distribution across the internal land,timing groove �N /m2�

Qi � generalized leakage flow �m3 /s�Qext � leakage across the external land, main port

plate groove �m3 /s�Qint � leakage across the internal land, main port

plate groove �m3 /s�Qext sg � leakage across the external land, timing groove

�m3 /s�Qint sg � leakage across the internal land, timing groove

�m3 /s�r � barrel generic radius �m�

rint � internal radius of the main groove �m�rext � external radius of the main groove �m�Rint � internal radius of the timing groove �m�Rext � external radius of the timing groove �m�

rm � average radius between land borders �m�r0 � groove central radius �m�t � time �s�

ve � flow generic velocity across an external land�m/s�

vi � flow generic velocity across an internal land�m/s�

V � piston velocity �m/s�� barrel tilt angle perpendicular to the Y axis

�rad� � swash-plate angle �rad�� barrel tilt angle perpendicular to the X axis

�rad�� small groove angle �rad�� fluid dynamic viscosity �K g/m s�� barrel angle �rad�� barrel angular velocity �rad/s�

ppendix: Further Details of the Analytical SolutionWhen taking tilt and rotation into account, and assuming the

ow moves in a radial direction, the Reynolds equation in polaroordinates takes the form of Eq. �1�. This equation will be ap-lied to four different lands, called the internal and external landsn the main port plate groove and the timing groove �see Fig. 1here the internal and external lands on both sides of the mainroove are clearly stated�. For any generic land, it will be assumedhat

h = h0 + �rm cos � �A1�

here rm is the average radius of each particular land.
Derivation of Eq. �A1� versus � will give


�h

��= − �rm sin � �A2�

Substituting Eqs. �A1� and �A2� in Eq. �1� and after the first inte-gration it is found that

dp

dr= −

3��rm sin �r

�h0 + �rm cos ��3 +c1

r�h0 + �rm cos ��3 �A3�

After the second integration,

p = −3��rm sin �

�h0 + �rm cos ��3

r2

2+

c1

�h0 + �rm cos ��3 ln r + c2 �A4�

This equation can be applied to any generic land. For each case,the constants of integration will be found via boundary conditions.

1 Pressure Distribution and Leakage Between Barreland Port Plate: Main Groove Effect

From Fig. 1, the boundary conditions for the external or internalland will be as follows: external land,

r = rext p = pint

r = rext2 p = pext = ptank �A5�

rmext =rext + rext2

2

internal land,

r = rint p = pint

r = rint2 p = pext = ptank �A6�

rmint =rint + rint2

2

When applying the boundary conditions for the external land itis found that

pint = −3��rmext sin �

�h0 + �rmext cos ��3

rext2

2+

c1

�h0 + �rmext cos ��3 ln rext + c2

�A7�

pext = −3��rmext sin �


rext22

2+

c1

�h0 + �rmext cos ��3 ln rext2 + c2

�A8�

and for the internal land,

pint = −3��rmint sin �

�h0 + �rmint cos ��3

rint2

2+

c3

�h0 + �rmint cos ��3 ln rint + c4

�A9�

pext = −3��rmint sin �


rint22

2+

c3

�h0 + �rmint cos ��3 ln rint2 + c4

�A10�

From Eqs. �A7� and �A8�, the value of the constants C1 and C2can be found. Eqs. �A9� and �A10� will be used to find the con-stants C3 and C4, obtaining

c1 = �pint − pext −3��rmext sin �


�rext22 − rext

2 �2

��


�A11�
ln�rext/rext2�
JANUARY 2008, Vol. 130 / 011011-13


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w

a

w

TE

f

Ipt

it

0

Downloaded Fr

c3 = �pint − pext −3��rmint sin �


�rint22 − rint

2 �2

��


ln�rint/rint2��A12�

c2 = pint�1 −ln rext

ln�rext/rext2�� + pextln rext

ln�rext/rext2�

+3��rmext sin �

�h0 + �rmext cos ��32�rext

2 +ln rext

ln�rext/rext2��rext2

2 − rext2 ��

�A13�

c4 = pint�1 −ln rint

ln�rint/rint2�� + pextln rint

ln�rint/rint2�

+3��rmint sin �

�h0 + �rmint cos ��32�rint

2 +ln rint

ln�rint/rint2��rint2

2 − rint2 ��

�A14�

he pressure distribution for the external land rext�r�rext2 afterubstituting the constants C1 and C2 in Eq. �A4� will be given byq. �2�. For the internal land rint2�r�rint, when substituting C3nd C4 in an equation homologous to Eq. �A4�, the pressure dis-ribution is to be given by Eq. �3�.

Once the pressure distribution has been found, a logical nexttep would be to determine the leakage. The equation representinghe total leakage due to the main groove is Eq. �4�, where it isoticed that the velocity distribution across the internal and exter-al lands is required. The velocity distribution according to Poi-eulle’s law can be given as follows: for the external land,

ve =1

�

dp

dr

y

2�y − h� �A15�

here

h = h0 + �rmext cos � �A16�

nd for the internal land,

vi =1

�

dp

dr

y

2�y − h� �A17�

here

h = h0 + �rmint cos � �A18�

he pressure distribution versus radius from the first integration ofq. �1� will be as follows: for the external land,

dp

dr= −

3��rmext sin �r

�h0 + �rmext cos ��3 +c1

r�h0 + �rmext cos ��3 �A19�

or the internal land,

dp

dr= �−

3��rmint sin �r

�h0 + �rmint cos ��3 +c3

r�h0 + �rmint cos ��3��− 1�

�A20�

t is necessary at this point to state that for the internal land theressure decreases as the radius decreases; therefore, its sign haso be changed to produce the required effect.

When substituting Eqs. �A15�, �A16�, and �A19� into the firstntegral of Eq. �4� and after performing one of the two integra-
ions, the leakage at the external land will be given as
11011-14 / Vol. 130, JANUARY 2008


Qext =−�i

�j

−�h0 + �rmext cos ��3r

12��−

3��rmext sin �r


+c1

r�h0 + �rmext cos ��3�d� �A21�

For a symmetrical groove or, in other words, when �� j�= �−�i� andafter some integrations, the external flow is given as

Qext =−�i

�j �pext − pint�12� ln�rext/rext2�

�h0 + �rmext cos ��3d� �A22�

Once this integration is performed, it is obtained by Eq. �5�.Operating similarly, when Eqs. �A17�, �A18�, and �A20� are

substituted in the second integral of Eq. �4�, the leakage across theinternal land will be given as

Qint =−�i

�j �h0 + �rmint cos ��3r

12��−

3��rmint sin �r


+c3

r�h0 + �rmint cos ��3�d� �A23�

when �� j�= �−�i�; after some minor integrations, the internal flowwill be

Qint =−�i

�j

−�pext − pint�

12� ln�rint/rint2��h0 + �rmint cos ��3d� �A24�

After integration, the internal flow due to the main groove will begiven by Eq. �6�.

The total leakage for the barrel plate will be the addition of theleakage due to the main port plate grove and the leakage due tothe timing groove. For the main groove, the leakage will be theaddition of leakages given by Eqs. �5� and �6�. The leakage willdepend on the geometry, internal and external pressures, tilt, andcentral clearance.

2 Pressure Distribution and Leakage: Effect of the En-trance Timing Groove

As for the main port plate groove, the equations for the timinggroove will be based on the Reynolds equations of lubrication�Eq. �1��. The equations giving the pressure distribution along theinternal and external lands next to the timing groove are similar tothe ones already found for the main groove, the main differenceswhen solving the differential equation in this case being theboundary conditions and the limits of integration. The boundaryconditions when focusing on the small groove �see Fig. 1� will beas follows: for the external land,

r = Rext p = pint

r = rext2 p = pext = ptank �A25�

Rmext =Rext + rext2

2

for the internal land,

r = Rint p = pint

r = rint2 p = pext = ptank �A26�

Rmint =Rint + rint2

2

The limits of integration would be from −� to −��+��. Follow-ing the same procedure as in the main groove and taking intoaccount that the constants C1, C2, C3, and C4 will be having thesame form although it is necessary for this case to change rint by
Rint and rext by Rext �see Fig. 1�, it is found that for the external


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at

T

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J

Downloaded Fr

and, Rext�r�rext2, the resulting equation is Eq �13�, and for thenternal land, rint2�r�Rint, the resulting equation is Eq. �14�.

The equation representing the leakage associated with the smallroove was Eq. �15�, where the velocities ve and vi, the genericap depth h, and the pressure variation with radius will have theame generic form as in the main groove case �Eqs. �A15� toA20��, being in this case necessary to substitute in these equa-ions, rm ext by Rm ext and rm int by Rm int. Since the limits of inte-ration are now nonsymmetrical, some of the terms for the mainroove that were zero now do exist. Therefore, following the pro-edure established in the main groove case, the internal and ex-ernal land leakage will be given by Eqs. �16� and �17�. It isnteresting to note that the leakage due to the small entranceroove depends on the barrel rotational speed, this effect appear-ng due to the groove asymmetry.

Force and Torque on the Barrel Due to the Pressureistribution: Main Groove EffectThe force between the cylinder block and the pump port plate

ue to the main groove was given as Eq. �7�. Assuming that theressure inside the groove is constant �although it is time depen-ent�, the first term of the integration will be

F = Pintrext

2 − rint2

2„� j − �− �i�… +

−�i

�j rint2

rint

Pint landrd�dr

+−�i

�j rext

rext2

Pext landrd�dr �A27�

The external and internal land pressures are given by Eqs. �2�nd �3�, and after some integrations and rearrangement it is foundhat

F = Pint„� j − �− �i�…

4� �rext

2 − rext22 �

ln�rext/rext2�−

�rint2 − rinr2

2 �ln�rint/rint2��

+ Pext„� j − �− �i�…

2�rext2

2 − rint22 � + Pext

„� j − �− �i�…4

�� rint2 − rint2


−�rext

2 − rext22 �

ln�rext/rext2�� +−�i

�j 3��rmint sin �

�h0 + �rmint cos ��3d�

� ��rint2 − rint2

2 �2�1

8−

ln�rint�4 ln�rint/rint2�

−1

8

1

ln�rint/rint2��+

rint22 − rint

2

4

1

ln�rint/rint2��rint2

2 ln�rint2� − rint2 ln�rint��

+−�i

�j 3��rmext sin �

�h0 + �rmext cos ��3d�

� ��rext22 − rext

2 �2�−1

8+

ln�rext�4 ln�rext/rext2�

+1

8

1

ln�rext/rext2��−

rext22 − rext

2

4

1

ln�rext/rext2��rext2

2 ln�rext2� − rext2 ln�rext�� A28�

he remaining integrals need to be solved numerically.Although, as reported by Jacazio and Vatta �4�, the force de-

ends on the turning speed �, in reality the terms given by thentegrals are much smaller than the first terms of Eq. �A28�. Suchntegration terms would play a much bigger role when using non-ymmetrical slots. In the case under study, the groove is sym-etrical and integrations of the type �−�i

� j �sin � / �h0

�rm cos ��3�d� will be equal to zero. Therefore, the force on the
arrel due to the main port plate groove effect will just depend on


the geometry and the internal pressure; the resulting equation isEq. �8�, which represents a linear variation between force andinternal pressure.

The torque over both axes created by the nonuniform pressuredistribution along the main port plate groove and the lands asso-ciated will have the general form given in Eqs. �9� and �10�. Aswhen studying the force, the external and internal land pressuredistributions are given by Eqs. �2� and �3�. Since the case studiedis for a symmetrical groove, ��i�= �� j�, the first integral of Eq. �9�amongst others has a zero value; therefore, Eq. �9�, after substi-tuting the equations for the pressure distribution, will look like

Mxx =−�i

�j rint2

rint �pint + pintln�r/rint�

ln�rint/rint2�+ pext

ln�rint/r�ln�rint/rint2�

+3��rmint sin �


�� rint2 − r2

2+

rint22 − rint

2

2

ln�rint/r�ln�rint/rint2��r2 sin �drd�

+−�i

�j rext

rext2 �pint + pintln�r/rext�

ln�rext/rext2�+ pext

ln�rext/r�ln�rext/rext2�



�� rext2 − r2

2+

rext22 − rext

2

2

ln�rext/r�ln�rext/rext2��r2 sin �drd�

�A29�

Once integrated versus the radius and after some rearrangement,the torque over the X axis due to the main port plate groove willtake the form of Eq. �11�, where the remaining integrals need to besolved numerically.

The torque over the Y axis when substituting the equationsgiving the pressure distributions �2� and �3� in Eq. �10� is given by

MYY = pintrext

3 − rint3

3�sin ��−�i

�j +−�i

�j rint2

rint �pint + pintln�r/rint�

ln�rint/rint2�

+ pextln�rint/r�

ln�rint/rint2�+

3��rmint sin �


�� rint2 − r2

2+

rint22 − rint

2

2

ln�rint/r�ln�rint/rint2��r2 cos �drd�

+−�i

�j rext

rext2 �pint + pintln�r/rext�

ln�rext/rext2�+ pext

ln�rext/r�ln�rext/rext2�



�� rext2 − r2

2+

rext22 − rext

2

2

ln�rext/r�ln�rext/rext2��r2 cos �drd�

�A30�

When performing the rest of the integrations, taking into accountthat the groove under consideration is symmetrical, after rear-rangement Eq. �12� is obtained.

It is noticed when checking the torque equations that the torqueover the X axis depends on the pump turning speed and plate tilt,which means that for the symmetrical groove case studied here,such a torque will be zero if any of these parameters is zero. Onthe other hand, the torque over the Y axis is independent of tilt andpump turning speed, and just depends on the geometry and the
internal pressure.
JANUARY 2008, Vol. 130 / 011011-15


4i

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wptpa

gfocsao

R

0

Downloaded Fr

Force and Torque Caused by the Action of the Tim-ng Groove

The force over the barrel due to the timing groove can be givenccording to the general equation �Eq. �18��. When substitutingqs. �13� and �14� into Eq. �18�, after integration and rearrange-ent, Eq. �19� was found, where the remaining integrals need to

e solved numerically.The torque over the X and Y axes created by the timing groove

ill be given by Eqs. �20� and �21� and, when substituting theressure distribution for each land, by Eqs. �13� and �14�. In theorque equations, following the procedure established for the mainort plate groove and after some development, Eqs. �22� and �23�re to be found.

The remaining integrals in all the equations need to be inte-rated numerically. The overall leakage, pressure distribution,orce, and torque versus the barrel main axis will be the additionf the equations of the main groove and the small groove for eachase. Special attention has to be made when regarding the torqueince different signs will mean different torque directions. Thelready developed equations need to be implemented by the effectf the pressure inside the cylinders, as explained in Sec. 2.4.

eferences�1� Helgestad, B. O., Foster, K., and Bannister, F. K., 1974, “Pressure Transients

in an Axial Piston Hydraulic Pump,” Proc. Inst. Mech. Eng., 188�17/74�, pp.189–199.

�2� Martin, M. J., and Taylor, B., 1978, “Optimised Port Plate Timing for an AxialPiston Pump,” Fifth International Fluid Power Symposium, Cranfield, UK,Sept. 13–15, Vol. B5, pp. 51–66.

�3� Edge, K. A., and Darling, J., 1989, “The Pumping Dynamics of Swash PlatePiston Pumps,” ASME J. Dyn. Syst., Meas., Control, 111, pp. 307–312.

�4� Jacazio, G., and Vatta, F., 1981, “The Block-Lift in Axial Piston HydraulicMotors,” The ASME/ASCE Bioengineering, Fluids Engineering and AppliedMechanics Conference, Boulder, CO, June 22–24, pp. 1–7.

�5� Yamaguchi, A., 1987, “Formation of a Fluid Film Between a Valve Plate anda Cylinder Block of Piston Pumps and Motors �Second Report, a Valve PlateWith Hydrostatic Pads�,” International Journal of the Japan Society of Me-chanical Engineers, 30�259�, pp. 87–92.

�6� Yamaguchi, A., 1990, “Bearing/Seal Characteristics of the Film Between aValve Plate and a Cylinder Block of Axial Piston Pumps: Effects of FluidTypes and Theoretical Discussion,” J. Fluid Control, 20�4�, pp. 7–29.

�7� Matsumoto, K., and Ikeya, M., 1991, “Friction and Leakage CharacteristicsBetween the Valve Plate and Cylinder for Starting and Low Speed Conditionsin a Swashplate Type Axial Piston Motor,” Trans. Jpn. Soc. Mech. Eng., Ser.C, 57, pp. 2023–2028.

�8� Matsumoto, K., and Ikeya, M., 1991, “Leakage Characteristics Between theValve Plate and Cylinder for Low Speed Conditions in a Swashplate-TypeAxial Piston Motor,” Trans. Jpn. Soc. Mech. Eng., Ser. C, 57, pp. 3008–3012.

�9� Kobayashi, S., and Matsumoto, K., 1993, “Lubrication Between the Valve

11011-16 / Vol. 130, JANUARY 2008


Plate and Cylinder Block for Low Speed Conditions in a Swashplate-TypeAxial Piston Motor,” Trans. Jpn. Soc. Mech. Eng., Ser. C, 59, pp. 182–187.

�10� Weidong, G., and Zhanlin, W., 1996, “Analysis for the Real Flow Rate of aSwash Plate Axial Piston Pump,” Journal of Beijing University of Aeronauticsand Astronautics, 22�2�, pp. 223–227.

�11� Yamaguchi, A., 1997, “Tribology of Hydraulic Pumps,” ASTM Special Tech-nical Publication No. 1310, pp. 49–61.

�12� Yamaguchi, A., Sekine, H., Shimizu, S., and Ishida, S., 1987, “Bearing/SealCharacteristics of the Oil Film Between a Valve Plate and a Cylinderblock ofAxial Pumps,” J. Jpn. Hydraul. Pneum. Soc., 18�7�, pp. 543–550.

�13� Manring, N. D., 2000, “Tipping the Cylinder Block of an Axial-Piston Swash-Plate Type Hydrostatic Machine,” ASME J. Dyn. Syst., Meas., Control, 122,pp. 216–221.

�14� Manring, N. D., 2003, “Valve-Plate Design for an Axial Piston Pump Operat-ing at Low Displacements,” ASME J. Mech. Des., 125, pp. 200–205.

�15� Zeiger, G., and Akers, A., 1985, “Torque on the Swashplate of an Axial PistonPump,” ASME J. Dyn. Syst., Meas., Control, 107, pp. 220–226.

�16� Zeiger, G., and Akers, A., 1986, “Dynamic Analysis of an Axial Piston PumpSwashplate Control,” Proc. Inst. Mech. Eng., Part C: Mech. Eng. Sci., 200�1�,pp. 49–58.

�17� Inoue, K., and Nakasato, M., 1994, “Study of the Operating Moment of aSwash Plate Type Axial Piston Pump. First Report: Effects of Dynamic Char-acteristics of a Swash Plate Angle Supporting Element on the Operating Mo-ment,” J. Fluid Control, 22�1�, pp. 30–46.

�18� Inoue, K., and Nakasato, M., 1994, “Study of the Operating Moment of aSwash Plate Type Axial Piston Pump. Second Report: Effects of DynamicCharacteristics of a Swash Plate Angle Supporting Element on the CylinderPressure,” J. Fluid Control, 22�1�, pp. 7–29.

�19� Manring, N. D., and Johnson, R. E., 1996, “Modeling and Designing aVariable-Displacement Open-Loop Pump,” ASME J. Dyn. Syst., Meas., Con-trol, 118, pp. 267–271.

�20� Wicke, V., Edge, K. A., and Vaughan, D., 1998, “Investigation of the Effects ofSwash Plate Angle and Suction Timing on the Noise Generation Potential of anAxial Piston Pump,” Fluid Power Systems and Technology, ASME, 5, pp.77–82.

�21� Manring, N. D., 1999, “The Control and Containment Forces on the SwashPlate of an Axial Piston Pump,” ASME J. Dyn. Syst., Meas., Control, 121, pp.599–605.

�22� Gilardino, L., Mancò, S., Nervegna, N., and Viotto, F., 1999, “An Experiencein Simulation the Case of a Variable Displacement Axial Piston Pump,” FourthJHPS International Symposium, Paper No. 109, pp. 85–91.

�23� Ivantysynova, M., Grabbel, J., and Ossyra, J. C., 2002, “Prediction of SwashPlate Moment Using the Simulation Tool CASPAR,” Proceedings of IMECE2002, ASME International Mechanical Engineering Congress and Exposition,New Orleans, LA, Nov. 17–22, Paper No. IMECE2002-39322, pp. 1–9.

�24� Manring, N. D., 2002, “The Control and Containment Forces on the SwashPlate of an Axial Piston Pump Utilizing a Secondary Swash-Plate Angle,”Proceedings of the American Control Conference, Anchorage, AK, May 8–10,pp. 4837–4842.

�25� Manring, N. D., 2002, “Designing a Control and Containment Device for aCradle-Mounted, Axial-Actuated Swash Plates,” ASME J. Mech. Des., 124,pp. 456–464.

�26� Bahr, M. K., Svoboda, J., and Bhat, R. B., 2003, “Vibration Analysis of Con-stant Power Regulated Swash Plate Axial Piston Pumps,” J. Sound Vib.,
259�5�, pp. 1225–1236.


Documents

Pressure, Flow, Force, and Torque Between the Barrel and Port Plate in an Axial Piston Pump