[A4-12]

LOAD-SENSING DYNAMICS OF APRESSURE-COMPENSATED VARIABLE

DISPLACEMENT VANE PUMP

Masamichi Nakagawa*, Toshio Takenaka**Ryuichiro Yamane*

*Dept . of Mechanical EngineeringTokyo Institute of Technology

Tokyo, Japan**Dept . of Mechanical Engineering

Musashi Institute of TechnologyTokyo, Japan

ABSTRACT

The present report deals with an

analysis of a pressure-compensated variable

displacement vane pump regarded as a load-

compensating control system. On the basis

of experimental data of an article on the

market, the effective transfer function of

the mechanical system can be reduced to a

form of 1st order lag system. Furthermore

including the pressure surge in actual

circuits due to the sudden change of the

load flow into the analysis, it becomes

clear that the entire transfer function of

the mechanical and hydraulic system has a

form of phase lead system. The theoretical

solution of the step response of the output

pressure is similar to the experimental

data. After the consideration of effective

parameters in the entire system based on

the above results, several guides for

parameter design are to improve the load-

sensing function and to optimize its step

response.

KEYWORDS

Load-sensing, Load-compensating control

system, Displacement varying mechanism,

Pressure surge, Step response

NOMENCLATURE

Ae: equivalent area of pressure receiving

in the cam ring, 3.66•~10-4m2

a: speed of sound, 103 m/s

B: displacement variability [=Vopix],0.028m2

C: output flow variability [=Qop/(Nen,8m2/rev

cleak: internal leakage coefficient

[= Qleak/Pop , 2•~10-11m3 /(s•EPa)],

D: diameter of the load line, 0.011 in

e: eccentricity of the cam ring, m

f comp : compensating force of the cam ring

position, N

fset: initial setting force of the cam ring

position, N

Ke: eauivalent bulk modulus of hydraulic

fl uid, 1x109 Pa

k: spring constant, 3.45•~105 N/m

L: length of the load line, 1.5 inm: total moving mass of the cam ring,

0.35kg

N: shaft rotational speed,

20, 25, 30 rev/s

Pop : output pressure, Pa

Qleak: internal leakage flow, m3/s

Qload: load flow, mi/sQop : output flow at the outlet port, m-)/s

Qpa: vane displacement flow in the pump

action, m3/s

s: Laplace operator

Vop: total volume of vane compartments

opened to the outlet port, 6x10-5 m3

w: flow velocity, m/sx: displacement of the cam ring

[=emax e in

Yset: initial displacement of the pressure

setting screw, in

ζ: damping ratio of the cam ring motion,

1.675

ρ: mass density of the hydraulic fluid,850kg/m3

ωn: natural frequency of the cam ring and

spring system, 1061 rad/s

131

132

INTRODUCTION

A pressure-compensated variable-displacement vane pump is a hydraulicsource with a built-in self-operatedmechanism to vary its output flow inresponse to its load change, and thereforeis an important module to compose load-sensing circuit. This type of pump is notsuitable for high output pressureapplications, in comparison to piston-typevariable displacement pumps, but has somemerits for the unification of thedisplacement varying mechanism into thepump; the simplicity of its structureconsidering its function and the needlessof the external control device to vary thedisplacement. In this study, the actualdynamics in load-sensing of this type ofpump is investigated with the purpose ofobtaining the knowledge available formachine designs, circuit layouts andpractical operations.

In resent papers on this type of pump,investigated subjects are the outputpressure and/or flow fluctuations due tothe finite number of vanes[5][6][7], thestability[8) and the responsibility[9] ofthe displacement varying mechanism underthe above fluctuations. The most primaryfunction of this type of pump is thecompensations of the output pressure andflow for the change of the load flow, butany reports on the actual dynamic behaviorin load-sensing such as the step responsehave apparently not been published to date.

The present report deals with ananalysis of this type of pump regarded as aload-compensating control system. On thebasis of experimental data of an article onthe market, the effective transferfunction of the mechanical system can bereduced to a form of 1st order lag system.But, it is not sufficient for thedescription of actual pressure responses toconsider only the above transfer functionof the mechanical system, because the largepressure surge can be generated in actualcircuits due to the sudden change of theload flow.

Here including the pressure surge intothe analysis, it becomes clear that the

Fig.1 A schematic of the test pump

entire transfer function of the mechanicaland hydraulic system has a form of phaselead system. The theoretical solution ofthe step response of the output pressure issimilar to the experimental data.

After the consideration of effectiveparameters in the entire system based onthe above results, several guides forparameter design are presented to improvethe load-sensing function and to optimizeits step response.

ACTUAL DYNAMICS

The test pumpThe test pump is an article sold on

the market. A schematic of the test pump isshown in Fig.1.

The experimental equipmentFig.2 Shows a schematic of the

experimental equipment. The experiment ontransient responses of the test pump wasmade for three different shaft rotationalspeeds, 1800 rpm,1500 rpm and 1200 rpm, at6.86 MPa set pressure where the static

performances are the best. The load flowcan be stepwise changed by the on/offoperation of a solenoid valve in a dummyload circuit which consists of tworestrictors in parallel, keeping theworking point within the range between thecut-off point and the dead-head point.Under these equivalent step inputs of theload flow, the responses of the outputpressure and the eccentricity of the camring were measured and recorded.

Experimental results of step responseSeveral examples in the experimental

results of step response are shown inFigs.3 and 4. Fig.3 is the sudden decreasecase of the load flow due to quick-openingof the solenoid valve. Fig.4 is in reversethe sudden increase case of the load flowdue to quick-closing of the solenoid valve.

Both cases lead that theresponsibility is better, the larger is theshaft rotational speed.

The responses of the output pressureare not stepwise but also with the largeovershoot due to the pressure surge. Thesephenomena indicate that it is necessary forthe analysis of the dynamics of this typeof pump to include the hydraulic transientin the load line.

Fig.2 A schematic of the experimental

equipment

133

LINEARIZED MODELING

The displacement varying mechanism

A self-operated mechanism to vary the

eccentricity of the cam ring is built inthe pump as the pressure-compensating and

displacement varying mechanism. Itsfunction is to vary the output flow under

the constant output pressure. It plays the

role to sense the change of the load asthat of the output pressure and to modulate

the output flow in response to the loaddemand. Consequently, its dynamics is

situated the most fundamental of all

characteristics in this type of pump.The cam ring and spring system in the

above mechanism can be regarded as aviscous-damped vibration system of one-

degree-of freedom, and then the transfer

function of the cam ring displacement vs.the output pressure is given by:

(1)

The cam ring motion varies the volumeof compartments between neighboring vanesopened to the outlet port, and the effectdue to the cam ring motion is immediatelymade feedback to the output pressure.

It is considered that the total volumeof vane compartments opened to the outletport is proportional to the eccentricity,because the output flow is proportional tothe eccentricity[1], and then

(2)

The equivalent bulk modulus of the workingfluid in the above vane compartments is

defined by:

The change of the output pressure due to

the displacement of the cam ring is given

by:

Omitting the deltas in-front of variables

to denote a minute increment, and makingthe Laplace transformation, the above

equation is transformed into:

(3)

Accordingly, the displacement varying

mechanism has a minor feedback loop and theentire transfer characteristics is

expressed by the block diagram as shown inFig.5.

Fig.5 A block diagram of the displacement

varying mechanism

(a) N= 1800 rpm (b) N= 1500 rpm (c) N= 1200 rpmFig.3 Example experimental results in the sudden decrease case of the load flo

w

(a) N = 1800 rpm (b) N = 1500 rpm (c) N = 1200 rpm

Fig.4 Example experimental results in the sudden increase case of the load flow

134

The entire block diagramConnecting blocks of the pump action,

the compressibility of hydraulic fluid, theleakage, the set spring force and the

compensation force by the output pressure

with the block diagram of the displacementvarying mechanism as shown in Fig.5, the

entire block diagram of the test pump is

obtained as shown in Fig.6.

Fig.6 The entire block diagram of the mechanical system of the test pump

The load change is the disturbance onthe load flow for the pressure-compensated

pump with the control mechanism to maintaina constant output pressure. In this case,load-sensing means the function to controlthe output flow of the pump in response tothe detected change of the load flow.Accordingly, the well fulfillment of theload-sensing function compensates theoutput flow just enough for the load flowand maintains the output pressure constant.

Therefore, the load sensibility of thistype of pump is suited for the analysis as

the control system to maintain a constant

output pressure compensating for the changeof the load flow.

Fig.7 shows the block diagram ofthe load-compensating function, which is

obtained by equivalent conversions of theblock diagram on the assumption of yset=0

in Fig.6.

Fig.7 A block diagram of the load-sensing function

EFFECTIVE TRANSFER FUNCTION

Open loop transfer functionThe open loop transfer function of the

load-compensating control system in Fig.7is reduced and rearranged in order as

follows:

(4)

Using the experimental data indicated inNOMENCLATURE:

(5)

(6)

135

In Eq.(6), the terms multiplied byreciprocals of values in Eq.(5) are almostnegligible compared with unity. Therefore,Eq.(6) is reduced to the followingequation.

(7)

where,

(8)

Eq.(7) represents that the open looptransfer function of the system in Fig.7 isreduced to a mere integral element. Afterthe above reduction of Eq.(4) into Eq.(7),the feedback block in Fig.7 is approximatedto a mere proportional element. Theelimination of Cleak in rearrangingEq.(6) into Eq.(7) seems to suggest thatthere is no effect of the internal leakageon the dynamics of load-sensing.

Closed loop transfer functionUsing Eq.(7), the closed loop transfer

function of the system in Fig.7 becomes aform of 1st order lag system as follows:

(9)

where,(10)

Comparing Fig.7 with Eq.(7), Fig.6 isreduced to Fig.8. Accordingly, it is foundout that the displacement varying mechanismis effectively regarded as a mereproportional element within the limits ofthe experiment in the preceding chapter.

Fig.8 The effective block diagram of the mechanical system of the test pump

INCLUSION OF PRESSURE SURGE

Pressure surge blockThe pressure surge in the load line

can be estimated by Joukowski's equation asfollows:

(11)

Therefore, the transfer function to express

the pressure surge is:

(12)

This transfer function in Eq.(12)expresses the inertia of hydraulic fluid inthe load line, and its block works inparallel with the block of compressibilityin Fig.6 or 8. Consequently, the surgepressure is summed into P,p, the controlledvariable of the closedloop system, ancFig.8 is redrawn into Fig.9.

Fig.9. The block diagram including the pressure surge in the load line

To investigate the function of the

pressure surge block in Fig.9, the summing

point of the surge pressure in Fig.9 istransposed from the back of the Gp blockto its fore, and the load input is unified

into the single one as shown in Fig.10.

where,

(13)

(14)Fig.10 Unification of the load inputequivalent to Fig.9

136

In Fig.10, the load input Qload is

(15)

times the case without the surge block, and

it is found that the surge block makes a

phase lead compensation for the 1st orderlag system of the main loop at the stage of

load input. Rearranging by a =•ã Ke / P,

Tsurgein Eq.(15) is as follows:

(16)

Theoretical responseThe entire transfer function of the

mechanical and hydraulic system of thistype of pump is as follows:

(17)

where,(18)

Using actually measured data presented inNOMENCLATURE, the estimation of Eqs.(16)and (18) results in:

Tguroo>T (19)

Eq.(19) means that the system expressed inEq.(17) is a phase lead system. The stepresponse of Eq.(17) is:

(20)

Because of Tsurge/T = 3.95 in the caseof the test line, the theoretical responseof the output pressure against the stepchange of the load flow is as follows:

-

Pop(t)/ RQ 61

1+2.95e-t/T

(21)

Fig.11 shows the typical solution curve ofEq.(21). As may be seen from Fig.11, thecurve has a peak. The theoretical responsein Fig.11 gives a typical curve in thesimilitude of actually measured stepresponses as shown in Figs.3 and 4.

Fig.11 Theoretical response of the output

pressure for the step change of the loadflow

PARAMETER DESIGN

Design guides of effective parametersEq.(20) means that the steady state

error of output pressure due to the samechange of the load flow is proportional toR, the closed loop gain. Therefore, thesmaller R is, the better it is. Generalways to decrease R are to decrease Ke, Band to increase ON, Vop. But, Ke is notmodifiable because of the value of thephysical property of hydraulic fluid, andit is not desirable in general to decrease'e of the hydraulic fluid as a powertransmission medium.

Since B is connected with V becauseof B = Vo/x as shown in NOE1CLATURE,x,the displacement of the cam ring, must bemore increased in order to decrease B andto increase V at the same time. It seemsto be diffictit to practice such a changeof the hardware. It would be more suitableto increase CN, the displacement flow, as aactual selection.

It is satisfactory that the concreteways are the increase of the thickness ordepth of the pump in order to increase thedisplacement flow a rotation and theincrease of the rotational speed as much as

possible. The former way results in theincrease of Voat the same time, and thelatter way isgupported by the experimentaldata as shown in Figs.3 and 4.

Optimum design of step responsePositively using the peak of pressure

surge with the effect of the phase leadcompensation, it is possible not only toeliminate the peak but also to optimize thestep response. If the following conditioncan be satisfied in Eq.(20),

Tgurgo/ T

=1 (22)

the time-dependent term can be eliminated,and Eq.(20) can be reduced as follows:

Pop(t)=-RQB1 (23)

And Eq.(17), the transfer function of load-sensing, can be also reduced to:

Pop

/ Qload=R (24)

that is, the entire system can beequivalently reduced to a mere proportionalelement. If Eq.(22) is rearranged bysubstituting Eqs.(16) and (18), thefollowing relationship equation ofeffective parameters is obtained:

4CNVop

/πD2αB=1 (25)

Considering which parameter isdesignable, the design guides on C,N,Vop,Bare given in the preceding section, and a,the speed of sound, is not modifiablebecause of the value of the physicalproperty, therefore in Eq.(25), D is leftfree as a designable parameter.

If Eq.(25) is solved for D, thefollowing formula on the optimum diameter

137

of the load line is obtained:

(26)

In the case of the test pump, the value canbe calculated easily by substitutingactually measured data as shown inNOMENCLATURE into Eq.(26), and

D= 0.0243

or 24.3 mm diameter is found out. In thepractical design, this value becausereference standard, and in order to finallydetermine the value of D, it must be made asynthetic judgment considering thecharacteristics of connected apparatus, thepressure drop in the circuit and the costetc.

CONCLUSION

The summary in this report is asfollows:(1) The displacement varying mechanism canbe effectively regarded as a proportionalelement. Therefore, the spring constant andthe parameters of the vibration system haveno effect on the load-sensing dynamics.(2) The internal leakage is not a seriousproblem on the load-sensing dynamics.(3) The effective transfer function of themechanical system is reduced to a 1st orderlag system.(4) The load-sensing function and thepressure-compensating function areequivalent in this type of pump, and theconcrete ways to improve thecharacteristics of these functions are theincrease of the thickness or depth of thepump and the increase of the rotationalspeed.(5) The pressure surge in the load linecan be estimated by Joukowski's equationand built in the block diagram of the load-sensing function, it makes a phase leadcompensation for the 1st order lag systemof the main loop at the stage of the loadinput.(6) The entire transfer function of themechanical and hydraulic system including

the pressure surge has a form of phase leadsystem.(7) The above entire system with a form ofphase lead can be made reduction to a mereproportional element by selecting theoptimum diameter of the load line.

REFERENCE

[1] Nakagawa, M., and Takenaka,T.,"Load-Sensing Characteristics of a Pressure-Compensating Variable Displacement VanePump (1st report, Static Characteristics)", Preprint for Showa 62 year's SpringLecture Meeting on Hydraulics andPneumatics, May 1987, p.1 (in japanese)[2] Nakagawa,M., and Takenaka,T., ibid.,(2nd report, Dynamics of the DisplacementVarying Mechanism ), p.5[3] Nakagawa, M., Takenaka, T.,and Yamane,R.,"Load-Sensing Characteristics of aPressure-Compensated Variable DisplacementVane Pump (3rd report, Analysis of theDynamics as a Load-Compensating ControlSystem), Preprint for Showa 63 year'sSpring Lecture Meeting on Hydraulics andPneumatics, May 1988, p.21 ( in japanese)[4] Nakagawa,M.,Takenaka,T.,and Yamane,R.,ibid.,( 4th report, Analysis of theDynamics including the Pressure Surge inthe Load Line ),p.25[5] Ueno,H., Shintani,R., and Okazima,A.," Pressure and Flow Ripples of a VariableDisplacement Vane Pump (1st report,Experiments)", Transactions of Jpn. Soc.Mech. Eng., 53-490, 1987, p.1736 ( injapanese)[6] Ueno,H., Shintani,R., and Okazima,A.,ibid.,( 2nd report, Theoretical Analysis)", p.1742[7] Karmel, A. M.," Modeling and Analysis ofthe Dynamics of a Variable-DisplacementVane-Pump With a Pivoting Cam", J. DynamicSystems, Measurement, and Control, Trans.ASME, 1988, 110-2, P.197[8] Karmel,A.N., "Stability and Regulationof a Variable-Displacement Vane-Pump",ibid., p.203[9] Pery , A.," Improving VariableDisplacement Vane Pumps for FasterResponse", Proceedings of NationalConference on Fluid Power, 1984, 38, p.173