conf 2000b flucom

8/13/2019 Conf 2000B Flucom

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ON THE DYNAMICS OF SWASH PLATE AXIAL PISTON

PUMPS WITH CONICAL CYLINDER BLOCKS

KASSEM, S.A.

Professor, Mechanical Design and ProductionDepartment, Faculty of Engineering, Cairo University,

Giza 12316 Egypt.

E-mail: [email protected]

BAHR, M.K.

Graduate student,Mechanical Design and ProductionDepartment, Faculty of Engineering, Cairo University,

Giza 12316 Egypt.

E-mail: [email protected]

ABSTRACT

In this paper a mathematical model for swash plate

axial piston pumps with conical cylinder blocks ispresented. Simulation runs are carried out using this

model to evaluate the performance of a pump with

certain design parameters under different operating

conditions. The results show that both the moment

acting on the swash plate in the direction perpendicular

to its inclination and the torque acting on the driving

shaft are nearly constants under certain operatingconditions, and increase linearly with the increase of

the delivery pressure and/or increase of the swash plate

inclination angle. The other component of the moment,

which tends to change the swash plate inclination

angle, is found to be periodic, and can be fairlyrepresented by four harmonics and an average value

which acts in the direction that decreases the inclination

angle. The average value increases nearly linearly with

the delivery pressure increase and/or decrease of the

swash plate inclination angle. Pump leakage was found

not to affect the moment components. Results show

also that the force tending to detach the piston from the

swash plate decreases nearly linearly with the increaseof the cylinder block cone angle, and that the

percentage reduction of this force is independent of the

pump rotational speed.

KEY WORDS

Piston pump, swash plate, moments, piston forces,

conical cylinder block.

1. INTRODUCTION

The static and dynamic characteristics of swashplate axial piston pumps with cylindrical cylinder

blocks were extensively investigated during the last

three decades. The variation of cylinder pressure during

one complete revolution of the pump was studied in [ 1,

2, 5, 6, 7 and 8] with the effects of sloping and non-sloping, square or semi-circular, and triangular

silencing grooves on the variation of cylinder pressure

investigated. When the pump suction pressure was

atmospheric, cavitation in the piston chamber was

recorded [7]. Kaliafetis and Costopoulos [3] studied

both theoretically and experimentally the static anddynamic characteristics of a variable displacement

swash plate pump with constant pressure regulator.Modeling and designing a variable geometric volume

axial piston pump was carried out in [4]. Effect of the

suction pressure, rotational speed, shape and sealingcondition of the valve plate on cavitation in an axial

piston pump was studied experimentally by Atsushi [5].

The average moment acting on the swash plate of an

axial piston pump was studied experimentally in [6],

and was shown to depend upon the swash plate

inclination angle, rotational speed, entrapment angle,and system pressure. It decreases as the swash plate

angle increases. Cylinder pressure and the moment

acting on the swash plate were calculated in [7] for two

different types of valve plate. The calculated values

coincided well with the measured ones when a valveplate with wide, short and deep notches was used.

Pumps with conical cylinder blocks are now widely

used in both industrial and mobile applications. In these

pumps the piston line of stroke is inclined to the pump

axis of rotation in order to reduce the piston inertia

force effect that tends to detach either the pistons fromthe slipper pads or the slipper pads from the swash

plate. This technique allows driving the pumps at

higher speeds, which increases the pump specific

power. Kassem and Bahr [8] recently carried out a

comprehensive theoretical study to investigate the

effect of the triangular silencing groove dimensions onthe piston chamber pressure and pump flow rate

fluctuation for this type of pumps. They showed how to

determine the port plate configuration that causes

gradual rise and drop of the piston chamber pressure,

with cavitation avoided, and low delivery flow rate

fluctuation.

In the present work, the moment acting on the swash

plate and the pump driving torque are investigated for

pumps with conical cylinder blocks under different

operating conditions. Also the effect of the cylinder

block cone angle on the force tending to separate the

piston from the swash plate is investigated.

2. PUMP MODELING

Figure 1 shows the studied pumping mechanism.The displacement s of the k

tpiston as function of its

angular position kis given by [8]

,0.5DR,tansincoscos

LtancosRLwhere

)1()cos/L(Ls

22k

1k2

1k

=

+=

=
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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2

okcpck

ksks

skdsk

dkdk

dkddk

kck

L

kdkkpsk

V)s(0.5LAVand

),p-p(sgnp-p2

ACQ

),p-p(sgnp-p2

ACQwhere

)2(B

pV

R

pQsAQ

+=

=

=

++=+

Fig. 1. Layout of the pumping mechanism

Assuming that the pump speed, the suction and delivery

pressures ps and pd are constants, the inertia effect of

the oil column inside the piston chamber is negligible

and that the leakage flow rate out of the piston chamber

is proportional to the piston chamber pressure, the

continuity equation as applied to the control volume

(C.V.) shown in Fig.2 takes the form [8]:

Knowing the dimensions of the valve plate, the values

of Askand Adkcan be evaluated at each , and thus thepiston chamber instantaneous pressure p and delivery

flow rate Qdkcan be determined by solving the forgoing

equations numerically. The pump instantaneous

delivery flow rate is given by:

Neglecting friction, the force F acting axially on the kth

piston and its slipper pad is given by

The normal force acting on the swash plate due to the

force Fkcan be shown to be given by:

In the shown coordinate system, the moment acting on

the swash plate due to the kth

piston normal force has

the three components:

The total moment acting on the swash plate has thus the

components:

The moment, M tends to change the swash plateinclination angle. The moment Mz equals nearly the

pump driving torque. The resultant of the two

components Mx and Mz; namely M , acts on the swash

plate bearing system and is given at any instant by:

Fig. 2. Piston chamber control volume and forces acting on the swash plate.

angle.ninclinatioplateswashtheisand,0.5DR

,)/LR-(Rtan,N

1)-(k2t

11

2211-

k

=

=+

=

)3(QQ

N

1k

dk

==

sin)sLp5.0(Rsin)sx(Rrwhere

)(4FsinrmampAF

k1kc1ck

sck2

pkpkpk

+=+=

+++=

(5))cossinsincos/(sinFF knk =

.LLcos-zand

cosRsinsinLy

,sinRsincosLxwhere

yFxFMand

zFxFM,zFyFM

1k

k2kk

k2kk

knxkk

nykzk

knxkk

nzkykk

nykk

nzkxk

+=

+=

+=

=

==

===

===N

1k

zkz

N

1k

yky

N

1k

.,kxxMMandMMMM

MMM2

z2

xb +=


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found to be as shown in table 2. The frequency of the

first harmonic is seen to be nine times the pump

rotational speed, and the frequencies of the subsequent

harmonics are multiples of the first one.

Harmonic Myo 1 2 3 4 5 6

Amplitude

[ Nm ]

-62 104 27 28 33 12 11

Frequency

[ Hz ]

0 217 434 651 868 1119 1336

3. PUMP SIMULATION

A software package based on Matlab was developedand used in [8] to evaluate the pump kinematics. This

package has been extended to calculate the forces and

moments acting on the swash plate of the pump. For a

nine piston pump with the dimensions (in mm) shown

in table 1, running at 1450 rpm and having a leakageresistance RL equal to 107 MPa/(m

3/s), the

recommended valve plate dimensions were found in [8]

to be as shown in Fig.3.

D1 D2 D3 dp L171.75 60.20 54.70 17.00 76.60

L2 Lc Lp 66.10 57.30 59.10 15

Table.1. Pump dimensions

Fig. 3. Recommended valve plate configuration

Simulation of the pump performance with the

recommended valve plate was carried out. The obtained

results are shown in Fig.4, which depicts the gradual

rise and drop of the piston chamber pressure at three

different delivery pressures. It shows also that the pump

delivery flow rate fluctuation increase with the increase

of the delivery pressure. Each of the moments M z and

M is seen to be nearly constant at each deliverypressure, and increases linearly with the increase of pd.The other component of the moment; namely M , is

seen to depend on the pump angle of rotation . Itvaries between positive and negative values, with

higher peak negative values. The mean value of this

moment is always negative, i.e. it tends to decrease .For the investigated three delivery pressures, the

moment mean values were found to be 10, 35 and 62

Nm respectively. At any delivery pressure, the moment

Mycan be analyzed into harmonics, such that

For a delivery pressure of 30 MPa, the amplitude and

frequency of each harmonic, up to the sixth one were

Table.2. Harmonic analysis of the moment My

Figure 5 shows a comparison between the moment

M as predicted from pump simulation, and that

calculated taking into account the first four harmonics

only. The figure shows reasonable agreement betweenthe two curves. The same conclusion has been reached

after extensive study of the moment M at different

delivery pressures, running speeds and swash plate

inclination angles (results are not presented). Thisverifies that, for design purposes, the moment M can

be represented by a negative average value and fourharmonics only. The average value of the moment M ,

has been found to increase with the increase of the

delivery pressure and decrease of the swash plate

inclination angle as shown in Fig. 6. The average value

of the moment Mzwhich nearly equals the pump drive

shaft mean torque ( since is generally small ) is seento increase with the increase of the pump delivery

pressure and/or flow rate, as shown in Fig. 7. The

moment Mbacting on the swash plate bearing system is

seen not to be influenced much by the swash plate

inclination angle as depicted in Fig. 8.

4. EFFECT OF CONE ANGLE ON

PISTON DETACHING FORCE

During the suction stroke the piston chamberpressure is nearly atmospheric, and due to piston inertia

it might be detached from its slipper pad or separated

from the swash plate. When the cylinder block is

conical, the centrifugal force acting on the piston due to

its rotation has a component acting in the direction of

the piston line of stroke, which always pushes the

piston towards the swash plate. This componentdecreases the piston detaching force. To investigate the

effect of the cone angle on the piston detaching force,

the maximum value of this force is computed for

various cone angles and rotational speeds. The obtained

results are depicted in Fig.9. The piston maximum

detaching force is seen to decrease nearly linearly with

the increase of at any rotational speed, while itincreases with the increase of the speed. The percentage

reduction of the piston detaching force is seen to

increase nearly linearly with , while it does not dependon the pump rotational speed as depicted in Fig. 10..

Thus it is recommended to increase to the maximum

value allowed by design considerations.

=

++=n

1i

iiyiyoy )tisin(MMM


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4

pd= 10 MPa pd= 20 MPa pd= 30 MPa

Fig. 4. Dynamics of a pump with a recommended valve plate


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5

Fig. 5. Harmonic analysis for moment My.

Fig. 6 Effect of swash plate inclination angle on

average My.

Fig. 7. Effect of swash plate inclination angle on the

shaft average torque.

Fig. 8. Effect of swash plate inclination angle onaverage Mb.

Fig. 9. Effect of cylinder block cone angle on piston

maximum detaching force

Fig. 10. Effect of cylinder block cone angle on

percentage reduction of the detaching force

5. CONCLUSION

Simulation of performance of swash plate axial piston

pumps show that the component of the moment which

affect the swash plate in its swinging direction isperiodic, and can be fairly represented by an average

value and four harmonics. This average value tends to

drive the swash plate towards the pump minimum

geometric volume position and increases with the

increase of the pump delivery pressure and/or decrease

of the swash plate inclination angle. Fluctuation in the

driving shaft torque is found to be small, and the

average torque increases with the increase of the pump

delivery pressure and/or geometric volume. The

component of the moment acting on the swash plate

bearing system is found to be nearly constant at

constant delivery pressure and increases linearly with

the delivery pressure. It is not affected by the swashplate inclination angle. Simulation results show the

advantage of using conical cylinder blocks. Using such

a block reduces the force that tends to detach the piston

from the swash plate during the suction stroke. It is

verified that the percentage reduction in the detaching

force increases nearly linearly with the increase of thecylinder block cone angle.

REFERENCES

[1] K. A. Edge, and J. Darling, Cylinder Pressure

Transients in Oil Hydraulic Pumps with Sliding Plate

Valves, Proc. Instn. Mech. Engrs., Vol. 200, No. B1,

1986, 45-54.[2] N.D. Marring, The Torque on the Shaft of an Axial

Piston Swash Plate Type Hydrostatic Pump, Journal of


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Dynamic Systems, Measurement, and Control, Vol120, 1998, 57-62.

[3] P. Kaliafetis, and T. Costopoulos, Modeling and

Simulation of an Axial Piston Variable Displacement

Pump with Pressure Control, Mech. Mach. Theory,

Vol. 30, No.4, 1995, 599-612.

[4] N.D. Marring, and R.E. Johnson, Modeling andDesigning a Variable-Displacement Open-Loop

Pump, Journal of Dynamic Systems, Measurement,

and Control, Vol. 118, 1996, 267-272.

[5] Y. Atsushi, Cavitation in an Axial Piston Pump,

Bulletin of the JSME, Vol. 26, No. 211, 1983, 72-78.[6] S.J.Lin, A. Akers and G. Zeiger, The Effect of Oil

Entrapment in Axial Piston Pumps, ASME

Puplication G00282, 1984, 127-134.

[7] I. Kiyoshi, and N. Masakasu, Study of the

Operating Moment of a Swash Plate Type Axial

Piston Pump, The Journal of Fluid Control, Vol. 22,Issue 1, 1994, 30-46.

[8] S.A. Kassem and M.K. Bahr, Effect of Port PlateSilencing Grooves on Performance of Swash Plate

Axial Piston Pumps, Current Advances in Mechanical

Design and Production, Proc. of 7th

MDP Conf., Cairo

Pergamon press, 2000, 139-148.

NOMENCLATURE

a Piston acceleration m/s2

Ad,As Delivery, suction porting area m2

Ap Piston cross-section area m2

B Effective bulk modulus, 1.3x103 MPaCd Coefficient of discharge, 0.611 -

D1 Cylinders pitch circle diameter at

cylinder block front surface

m

D2 Cylinders pitch circle diameter at

cylinder block back surface

m

D3 Diameter of pitch circle of valve

plate

m

Fk Resultant axial force acting on the

kth

piston

N

Fn

k Normal force acting on the swash

plate

N

Fs Cylinder block spring force axial

component

N

Fn

x,y,z Components of the force acting on

the swash plate

N

k Piston number in the arrangement

of the piston group

-

L1, L2 Lengths m

Lc Cylinder length m

Lp Piston length m

Mb Moment acting on the swash platesupporting bearings

Nm

mp Piston mass kg

Mx,y,z Components of the moment acting

on the swash plate

Nm

n Pump rotational speed rpm

N Number of pistons

pk Time rate of change of chamber

pressure

Mpa/s

pd Pump delivery pressure MPa

pk Piston chamber pressure MPa

ps Pump suction pressure MPa

Q Pump total output flow rate m3/s

Qd Delivery flow rate from one

cylinder

m3/s

Qs Suction flow rate into one cylinder m3/s

rck Radius of piston center of gravity m

RL Resistance to leakage out of

cylinder

MPa s/ m3

sk Piston velocity m/s

sk Piston displacement m

t Time s

V Additional volume m3

Vc Cylinder volume m3

xc Distance between piston spherical

head center and piston center of

gravity

m

xk, yk

& zk

Cartesian coordinates of piston

spherical head center

m

Swash plate inclination angle

Cylinder block cone angle

Phase shift

k Angular position of the kth

piston

Oil density, 850 kg/m3

Angular velocity rad/s

ACKNOWLEDGMENT

The authors would like to thank Eng. K. T. Hamza for

his help.

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