Modeling supply and return line dynamics for an electrohydraulic

ulic

stream ofons at theain in a

ating theo directmodel-

on

ISATRANSACTIONS®

ISA Transactions 44~2005! 329–343

Modeling supply and return line dynamics for an electrohydraactuation system

Beshahwired Ayalew* Bohdan T. Kulakowski†

The Pennsylvania Transportation Institute, 201 Transportation Research Building, University Park, PA 16802, USA

~Received 30 June 2004; accepted 6 December 2004!

Abstract

This paper presents a model of an electrohydraulic fatigue testing system that emphasizes components upthe servovalve and actuator. Experiments showed that there are significant supply and return pressure fluctuatirespective ports of the servovalve. The model presented allows prediction of these fluctuations in the time dommodular manner. An assessment of design changes was done to improve test system bandwidth by eliminpressure dynamics due to the flexibility and inertia in hydraulic hoses. The model offers a simpler alternative tnumerical solutions of the governing equations and is particularly suited for control-oriented transmission lineing in the time domain. © 2005 ISA—The Instrumentation, Systems, and Automation Society.

Keywords: Hydraulic system modeling; Supply and return line dynamics; Accumulator model; Hydraulic hoses; Modal approximati

ntnes

-arhee

fi-thestor

tso-

theou-ofex-eanly

aep-

,terrome

andatInse

1. Introduction

A very common assumption in the developmeof models for valve-controlled hydraulic actuatiosystems is that of constant supply and return prsures at the servovalve@1–4#. On the other hand, asurvey of work on fluid transmission line dynamics suggests that significant pressure dynamicsintroduced in hydraulic systems as a result of tcompressibility and inertia of the oil as well as thflexibility of the oil and the walls of pipelines@5–8#. Transmission line dynamics can be signicant on the supply and return lines betweenhydraulic power unit~pump! and the servovalve awell as between the servovalve and the actuamanifold.

*Corresponding author. Tel.:~814! 863-8057; fax:~814!865-3039.E-mail address: [email protected]

†Tel.: ~814! 863-1893; fax:~814! 865-3039.E-mailaddress: [email protected]

0019-0578/2005/$ - see front matter © 2005 ISA—The Instru

-

e

Close-coupling~i.e., mounting the servovalvedirectly on the actuator manifold! is often used asa solution to the problem of minimizing the effecof transmission line dynamics between the servvalve and the ports of short-stroke actuators. Incase of long-stroke actuators, where close cpling may not be physically feasible, the effecttransmission line dynamics can be analyzed byplicitly including a transmission line model in thmodel of the servosystem, as shown by VSchothorst@9#. However, in the case of the suppline to the servovalve, close coupling may not beconvenient solution for either short- or long-strokactuators, since usually the hydraulic power suply ~HPS! unit, including the hydraulic pumpdrive unit, heat exchangers, and cooling wapumps, needs to be housed separately, away fthe work station of the actuator or the load framsupporting the actuator. In such cases, supplyreturn lines from the HPS to the servovalve thare of significant length may be unavoidable.addition, from installation considerations, the

mentation, Systems, and Automation Society.

330 Beshahwired Ayalew, Bohdan T. Kulakowski / ISA Transactions 44 (2005) 329–343

331Beshahwired Ayalew, Bohdan T. Kulakowski / ISA Transactions 44 (2005) 329–343

Fig. 1. Schematic of test system~HP5high pressure, LP5low pressure!.

es

r-ly-lya’stheca-er-

leso-xi-er-ns

dac-plylvendtheicknd

ina-

enoo

ithr,

ns-the

f-

ain-a

e

v-

ld8-erron-2gu-M

at-w

un-at

for

tharely

supply and return lines are usually flexible hosrated for the appropriate working pressure.

Aside from the extensive presentation by Viesma@7#, not much has been reported on the anasis of a hydraulic servosystem including supppressure variations at the servovalve. Viersmanalysis was done in the frequency domain andemphasis was to provide design rules for the lotion and sizing of the components of hydraulic svosystems. Yang and Tobler@10# developed amodal approximation technique that enabanalysis of fluid transmission lines in the time dmain via state space formulations. Modal appromation results provide modular and simpler altnatives to direct numerical time-domain solutioof the flow equations.

In this paper, the authors present and use moapproximation results within a model of an eletrohydraulic actuation system to investigate supand return pressure variations at the servovadue to transmission line dynamics. Open-loop aclosed-loop tests were conducted to validatemodel. The model was then used to make a quevaluation of alternative layouts of the supply areturn lines.

2. Description of test system

The hydraulic system shown schematicallyFig. 1 was designed for fatigue testing applictions. The servovalve is a 5-gpm~19 lpm! two-stage servovalve employing a torque motor drivdouble nozzle-flapper first stage and a main sp

l

l

output stage. The servovalve is close coupled wa 10-kN, 102-mm-stroke symmetric actuatowhich is mounted on a load frame. Pressure traducers are used for sensing the pressures atfour ports of the servovalve. A linear variable diferential transformer~LVDT ! is mounted on theactuator piston for position measurement.

Control and signal processing is done withdSpace® 1104 single processor board, whichcludes onboard A/D and D/A converters andslave digital signal processor~DSP!. An amplifiercircuit converts a 0–10-V control output from thdSpace® D/A to a high-impedance650-mA cur-rent input to the torque motor coils of the servoalve.

The unit labeled hydraulic services manifo~HSM! is connected to the servovalve using 3.04m-long SAE-100R2 hoses. The hydraulic powsupply ~HPS! unit, including its heat exchangeand drive units, is housed separately and is cnected to the HSM via 3.048-m-long SAE-100Rhoses. The HSM provides basic line pressure relation via the accumulators. In addition, the HSis equipped with a control manifold circuitry topermit selection of high- and low-pressure opering modes, low-pressure level adjustment, slopressure turn-on and turn-off, and fast pressureloading. The drain line provides a path for oil thseeps past the seals in the actuator and alsodraining oil from the HSM pressure gage.

During a normal fatigue testing operation, bothe low-pressure and high-pressure solenoidsenergized, the main control valve is complete


Fig. 2. Simplified system.

s

n-i-

hetheenatatoaw

ofain

angh

s atedin,. Itpedla-

ro-ly-ethefi-

p-

re,endS

theic-re-am

or-

c-ec-thesesel

eo-

ionidiones-p-

ey,itys-

orebyl,d

wide open, and the circuitry of the HSM allowflow at full system pressure@11#. We thereforemodel the HSM by considering the lumped nolinear resistance arising from change of flow drections, flow cross sections, as well as flow in tfilter element. The total pressure drop betweenpressure inlet and outlet ports of the HSM is givin manufacturer specifications. The available dsatisfy a nonlinear expression relating flow ratepressure drop. While the HSM unit is rated forwide range of flow rate capacities, the rated florate through the servovalve is within 10%nominal flow rate of the HSM. We therefore uselocal linear approximation to account for lossesthe HSM,

DpHSM5RHSMq. ~1!

We further assume that the check valve isideal one, so that its own dynamics are fast enouto be neglected and its backflow restriction halarge enough parallel resistance that the permitbackflow is very small. We also consider the draflow to be negligible. With these simplificationsthe system reduces to the one shown in Fig. 2should be noted that the resistances were lumin the HSM excluding the gas-charged accumutors.

We make one more assumption before we pceed. Through extensive frequency domain anasis, Viersma@7# has shown that, provided that thaccumulator and the pressure relief valve onhydraulic power supply unit are located sufciently close to the pump outlet~within 0.3 m!, thepump flow pulsation frequencies can be su

pressed from the pump output pressure. Therefoin the following analysis, it is assumed that thoutput pressure just after the accumulator apressure relief valve connection points in the HPunit can be set as a known pressure input torest of the system. In fact, this is not a very restrtive assumption, since the modeling approach psented here is modular and models of the upstrecomponents of the whole HPS unit can be incporated if needed.

For the simplified system, we now have two setions of transmission hoses to model. The first stion is for the supply and return hoses betweenHPS and the HSM and the second is for the hobetween the HSM and the servovalve. A modapplicable for each section is discussed next.

3. Model of transmission lines

To model the hydraulic hoses for our system, wrefer to previous rigorous results on validated slutions of the mass and momentum conservatequations governing flow in one-dimensional flutransmission lines having a circular cross sect@5,6,8#. In general, some assumptions are necsary for the basic results to hold. These assumtions include laminar flow in the lines, negligiblgravitational effects, negligible tangential velocitand negligible variations of pressure and densin the radial and tangential directions. We also asume constant and uniform temperature and ignheat transfer effects in the fluid line. We therelimit the discussion to the linear friction modewhich does not include distributed viscosity an

is-e

lls

se

sfs

ies

inx-

ofin

i-m

lein-ll-x-nd

lly

forre-ithAy-

atn-

ion

-rm

edhebeusaleput

m-ratew

andntc-a-c-

n-ffi-a-ilof

-ely,


heat transfer effects@6,10#. Corrections are appliedto account for the frequency dependence of vcosity on the linear friction model, following thwork of Yang and Tobler@10#.

The flow lines are assumed to have rigid wain some derivations@1,7#. However, Blackburnetal. @12# and McCloy and Martin@13# arrive at thesame governing equations as the rigid wall ca~for a frictionless flow! by allowing for wall flex-ibility and defining an effective bulk modulucombining the flexibility of the wall and that othe oil. Their definition of effective bulk moduluis the same as that derived by Merritt@4#, wherethe effective bulk modulus is viewed as a serinterconnection of the ‘‘stiffness’’ of the oil, of acontainer wall and even of entrapped air volumethe oil. Following this approach, we consider fleibility effects via the effective bulk modulusbe .The model parameters required for any sectionthe transmission line reduce to the ones shownFig. 3. For the hydraulic hoses in this work, nomnal values of the bulk modulus were taken froRef. @13#.

Using the above assumptions for the singtransmission line, the conservation laws can betegrated in the Laplace domain to yield a weknown distributed parameter model commonly epressed as a two-port matrix equation asometimes known as the four-pole equation@1,7#.The four-pole equations can take four physicarealizable causal forms@6,9,12#. Two of these fourare readily relevant to the problem at hand: onethe supply line hoses and another one for theturn line hoses. The third one also finds use waccumulator connection lines, as discussed byalew and Kulakowski@14#.

Taking the supply line case first, we notice thin most hydraulic servosystem applications, a cotrol signal modulates the servovalve consumptflow rate downstream of the supply line,qd(t),following the excursions of the~loaded! actuatorpiston. Thenqd(t) is a preferred input to the transmission line model, and a realizable causality forequires that eitherpu(t) or qu(t) should be the

Fig. 3. A fluid transmission line.

other input@12#. Since we have already assumthe pressure just after the connection point of tpressure relief valve and first accumulator totaken as an input to the system, the desired caform of the four-pole equations for the supply linis the so-called pressure-input/pressure-outcausality form@15#. It can be derived by definingthe boundary conditions for the distributed paraeter model as the upstream pressure and flow(pu ,qd) and the downstream pressure and florate(pd ,qu) at the opposite ends of the line,

F Pd~s!Qu~s!G

5F 1

coshG~s!2

Zc~s!sinhG~s!

coshG~s!

sinhG~s!

Zc~s!coshG~s!

1

coshG~s!

G3F Pu~s!

Qd~s!G . ~2!

The definitions of the propagation operatorG(s)and the line characteristic impedanceZc(s) de-pend on the friction model chosen@6,9#. In thispaper, the authors use the approach of YangTobler @10# that incorporated frequency-dependedamping and natural frequency modification fators into analytically derived modal representtions of the four-pole equations for the linear frition model.G(s) andZc(s) are defined by

G~s!5Dn

d2s

4n Aa2132abn

sd2, ~3!

Zc~s!5Z0A32abn

sd21a2. ~4!

The frequency-dependent correction factorsaand b are obtained by comparing the modal udamped natural frequencies and damping coecients of the modal approximations of the dissiptive ~‘‘exact’’ ! model, which is described in detain Ref. @16#, against the modal representationthe linear friction model@10#. Corrected kinematicviscosity ~n! values are used@8#. The dimension-less numbersDn andZ0 are the dissipation number and the line impedance constant, respectivand are given by

sis

these

ate

n-l

iesal

teer-

re-thatq.

ehede

-tedhe-ye-is

sis-eirof

the

tyteeslts

optp-,ate


Dn54ln

cd2, ~5!

Z054rc

pd2, ~6!

wherec is the speed of sound in the oil,

c5Abe

r. ~7!

The three causal functions1/coshG(s),Zc(s)sinhG(s)/coshG(s), and sinhG(s)/Zc(s)coshG(s) can be represented as infinite sumof quadratic modal transfer functions. The goalto use a finite number of modes to approximateotherwise infinite sum of the modal contributionfor the outputs. Of particular interest for the timdomain models sought in this paper is the stspace formulation@10,15#,

F pdi

quiG5F 0 ~21! i 11Z0vci

2~21! i 11vci

Z0a22

32nb

d2aG Fpdi

quiG

1F 0 28nZ0

d2Dn

8n

d2Z0Dna20

G Fpu

qdG ,

i 51,2,3...n. ~8!

The vci are the modal undamped natural frequecies of blocked line for the linear friction modegiven by

vci5

4npS i 21

2Dd2Dn

, i 51,2,3,...n. ~9!

The modification factorsa and b are given asfunctions of the dimensionless modal frequencd2vci/4n @10#. The output is the sum of the modcontributions and is expressed as

Fpd

quG5F (

i 51

n

pdi

(i 51

n

qui

G5@ I 2 I 2¯I 2#

3@pd1 qu1 pd2 qu2¯pdn qun#.

~10!

It should be noted that the truncation to fininumber of modes will introduce steady-staterors. Some methods have been suggested tocover the steady-state output based on the factat steady state the original four-pole equation, E~2!, reduces to

Fpd

quGss

5F1 28Dn

0 1 G Fpu

qdGss

. ~11!

Hsue and Hullender@16# discussed rescaling thtruncated sum of the modal approximation for tdissipative model by its zero-frequency magnituto bring about Eq.~11!. Van Schothorst@9# andHullender et al. @15# described an additive approach where the steady-state error is eliminaby adding a corrective feed-through term on toutput equation~10!. However, the transfer functions so implemented will no longer be strictlproper. This may entail the need for off-line algbraic manipulations when the transmission lineconnected to static source and/or load linear retances or other transmission line models with thown direct feed-through gains. The eigenvaluesthe coupled system may then be altered bysteady-state correction@15#.

Yang and Tobler@10# introduced methods thamodify the input-matrix or use a state similarittransformation matrix to affect the steady-stacorrection while preserving the modal eigenvaluof the truncated model. Since comparable resuare obtained by the use of either method, we adthe input-matrix modification method here. Supose matricesAi and Bi represent, respectivelythe feedback and input matrices in the modal stspace Eq.~8!. Introducing the input-matrix modi-fier G,

F pdi

quiG5AiFpdi

quiG1BiGFpu

qdG . ~12!

The steady-state value of then-mode approxima-tion is then

lue

nn.-v-

eion

n-e

Eq.wth

th-dri-plyingesr-

tohetedforebeut,d

tedionseeriaap

Ingi-re

isnd

in-s ise

eti-

rs,

ionaslso

nlity

toion


Fpd

quGss

5(i 51

n Fpdi

quiGss

52(i 51

n

~Ai21Bi !GFpu

qdGss

.

~13!

Comparing with the desired steady-state vagiven by Eq.~11! and solving forG,

G52S (i 51

n

Ai21Bi D 21F1 28Z0Dn

0 1 G . ~14!

The number of modesn to be chosen depends othe frequency range of interest for the applicatio

Similarly, for the return line from the servovalve, we can define the flow rate at the servoalve end and the pressure at the downstream~to-ward the tank! end as inputs to the model of threturn line based on the other four-pole equatof causality dual to Eq.~2! @6#. Equally, we canuse the observation that switching the sign covention of the flow direction for just the return linand using the four-pole equation dual to Eq.~2!yields the same set of four-pole equations as~2!, provided the inputs to the model remain florate toward the servovalve end and pressure atother end. This fact can easily be shown maematically, but we omit it here for brevity anstate that for the return line model all of the devations presented above for modeling the supline hold. The caveat is to exercise care in usthe proper signs for the input and output flow ratat both ends of the return line when forming inteconnections with other system components.

For step response simulations, it is desirablehave good estimates of the initial conditions of tmodal states, especially when the interconnecsystem model contains nonlinearities. Usually,a single pipeline section, the derivative of thmodal output can be assumed to be zero justfore the application of the step change in the inpand modal initial conditions can be computefrom

Fpdi~02!qui~02! G52Ai

21BiGFpu~02!qd~02! G , ~15!

where the@pu(02),qd(0

2)#T are the inputs justbefore the step change. For an interconnecpipeline system, the inputs to one pipeline sectmay be outputs of another section, in which cathe determination of proper initial values for thmodal states of each section can be done by tand error. The step disturbances can also be

e

-

l-

plied after initial transients have died down.general ‘‘steady’’ simulations, like those involvinsinusoidal fatigue test wave forms, the modal intial conditions of the interconnected system aless important.

As shown in Fig. 3, the model described in thsection requires few parameters to set up asimulate each transmission line section using lear state space models in the time domain. Thiparticularly more convenient than finite differencapproaches, which may require rigorous descrzation methods.

4. Modeling accumulators

For the nitrogen gas-charged accumulatowhich are of piston type for our system~Fig. 4!,we assume that the piston mass and seal frictare negligible. Under this assumption, the gpressure and the oil pressure are equal. We aneglect the compressibility of the oil volume ithe accumulator compared to the compressibiof the gas. Furthermore, we consider the gasundergo a polytropic expansion and compressprocess with polytropic exponentm.

pgVgm5const. ~16!

Fig. 4. Piston accumulator.


Fig. 5. Model interconnections for the supply line~Version I!.

oflns-ua-ro-rkb

,

r-bleopsal

c--

the

--

di-chec-

eq.

c-intion

a

e-

torndede-plyina-h-

rn-s-

ledthisnor

The exponentm approaches 1 for a slow~nearlyisothermal! process and the specific heat ratiothe gas for a rapid~adiabatic! process on an ideagas. However, it should be noted that if heat trafer effects were to be included, then real gas eqtions of state should be used together with apppriate energy conservation laws. For the wopresented here, the present model was found tosufficiently accurate.

Given initial gas pressurepg0 and gas volumeVg0 , the gas pressure is computed from Eq.~16!,which is equivalent to

pg5mpg

Vg02*0t qadt

qa , pg~0!5pg0 , ~17!

whereqa is the flow rate of the hydraulic oil to theaccumulator. For simulations involving distubances applied at the servovalve, it is reasonato assume that the accumulator already develan initial gas pressure through a slow isothermprocess1 (m51). The initial gas volumeVg0 canbe estimated by applying Eq.~16! between thepre-charge state~the gas pre-charge pressure at acumulator capacity! and the initial state at the onset of the disturbance. The initial gas pressurepg0

1The present system has an additional slow turn-on/tuoff accumulator that enables the HSM to come to full sytem pressure from an off state in a slow and controlmanner. The servovalve disturbances considered instudy are applied after the whole system has reachedmal operating conditions.

e

can be estimated as the HSM pressure minuspressure drop in the connection lines.

5. Model interconnections for the supply line

The components on the supply line of the simplified model shown in Fig. 2 can be interconnected as shown in Figs. 5 or 6. The arrows incate the input-output causality assigned for easubsystem. Each of the blocks section I and stion II implement Eqs.~8!–~14! for the corre-sponding sections of the supply line.

Integration causality is the desired form for thmodel of the accumulator, which is given by E~17!. It was pointed out by Viersma@7# that theflow dynamics in the short branch-away connetion lines to the accumulators are significantmost cases. Under the linear resistance assumpgiven by Eq. ~1!, the subsystem ‘‘manifold andcheck valve loss’’ can be configured either aspressure-input/flow rate-output subsystem~in ver-sion I, Fig. 5! or as a pressure-input/pressuroutput subsystem~in version II, Fig. 6!. As a con-sequence, the model of the short accumulaconnection line changes between version I aversion II. The model of the short connection linin version I has the same structure as the onescribed above for the sections of the main supline. The model for the short connection lineversion II is derived using the modal approximtion for the relevant four-pole equations wit(pu ,pd) as input and(qu ,qd) as output, as de

-


Fig. 6. Model interconnections for the supply line~Version II!.

ec-rmau-ongdelelsoit

fies

aeatn-tem

a-eedd

ton

-isr

ce,

e-onsooilvo-in

theare

ht

theof

ngthe

ve.

scribed by Ayalew and Kulakowski@14#. It wasshown there that the dynamics of the short conntion line can be approximated by a first-order tethat reduces to a series interconnection of hydrlic resistance and inertance. This result goes alwith the convenient assumption that the oil sicompressibility in the accumulator is negligibcompared to that of the gas side. This result amakes version II preferable to version I, sincereduces the order of the overall system and veria physically supported model order reduction.

The model for the return line is developed insimilar way noting the reverse direction of thflow, as mentioned earlier. It should be noted ththe modularity of the subsystem model interconections allows changes to be made to the sysmodel with ease.

Fig. 7. Schematic of a rectilinear actuator and servoval

6. Model for servovalve and actuator

Physical models of electrohydraulic servoactutors are quite widely available in the literatur@1–4,7,9,17#. The model presented here is adaptto apply to a four-way servovalve close couplewith a double-ended piston actuator.

Fig. 7 shows a double-ended translational pisactuator with hydraulic flow ratesqt from the topchamber andqb to the bottom chamber of the cylinder. Leakage flow between the two chamberseither internal(qi) between the two chambers oexternal from the top chamber(qe,t) and from thebottom chamber(qe,b). At and Ab represent theeffective piston areas of the top and bottom farespectively.Vt and Vb are the volumes of oil inthe top and bottom chamber of the cylinder, rspectively, corresponding to the center positi(xp50) of the piston. These volumes are alconsidered to include the respective volumes ofin the pipelines between the close-coupled servalve and actuator as well as the small volumesthe servovalve itself.

We assume that the pressure dynamics inlines between the servovalve and the actuatornegligible due to the close coupling.2 Furthermore,even for a long-stroke actuator used in a fligsimulator application, Van Schothorst@9# has

2This is to say that any resonances introduced byshort-length lines are well above the frequency rangeinterest for our system. In fact, this can be verified usithe model development presented earlier or referring toresults of Van Schothorst@9#.

totedthendrt o

-lk

beby

ci-twohe

etors,

g.

inileco-

ta

e

c-

y-orex-

rec-r isas

tois

sged

s-


shown that the pressure dynamics in the actuachambers need not be modeled using distribuparameter models. It is therefore assumed thatpressure is uniform in each cylinder chamber ais the same as the pressure at the respective pothe servovalve.

Starting with the continuity equation and introducing the state equation with the effective bumodulus for the cylinder chambers, it canshown that the pressure dynamics are given~see, for example Ref.@2#!

dpb

dt5

bc

Vb1Abxp~qb2Abxp1qi2qe,b!,

~17a!

dpt

dt5

bc

Vt2Atxp~2qt1Atxp2qi2qe,t!.

~17b!

These equations show that the hydraulic capatance, and hence the pressure evolution in thechambers, depends on the piston position. Tleakage flowsqi , qe,b , and qe,t are considerednegligible.

The predominantly turbulent flows through thsharp-edged control orifices of a spool valve,and from the two sides of the cylinder chambeare modeled by nonlinear expressions@1,2,4#. As-suming positive flow directions as shown in Fi7, we have

qb5Kv,1sg~xv1u1!sgn~pS2pb!AupS2pbu

2Kv,2sg~2xv1u2!sgn~pb2pR!Aupb2pRu,

~18a!

qt5Kv,3sg~xv1u3!sgn~pt2pR!Aupt2pRu

2Kv,4sg~2xv1u4!sgn~pS2pt!AupS2ptu,

~18b!

where thesg(x) function is defined by

sg~x!5H x, x>0

0, x,0. ~19!

The parametersu1 , u2 , u3 , u4 are included toaccount for valve spool lap conditions as shownFig. 7. Negative values represent overlap whpositive values represent underlap. The valveefficientsKv,i are given by

r

f

Kv,i5cd,iwiA2/r, i 51,2,3,4. ~20!

These coefficients could be computed from dafor the discharge coefficientcd,i , port widthswi ,and oil densityr. If we assume that all orifices aridentical with the same coefficientKv , then thevalue ofKv can also be estimated from manufaturer data for the rated valve pressure drop(DpN),rated flow (QN), and maximum valve stroke(xv max) using the following equation@1,4#:

Kv,i5Kv5QN

xv maxA1/2DpN

, i 51,2,3,4.

~21!

As an approximation of the servovalve spool dnamics, a second-order transfer functionequivalently a second-order state space wastracted from manufacturer specifications,

Xv~s!

I v~s!5

Gvvn,v2

s212zvvn,vs1vn,v2

. ~22!

The state equations governing piston motion aderived considering the loading model for the atuator. For the test system, the actuator cylinderigidly mounted on a load frame, which we usean inertial frame as shown in Fig. 8.

The upward force on the actuator piston duethe oil pressure in the two cylinder chambersgiven by

Fp5Abpb2Atpt . ~23!

The friction force on the piston in the cylinder idenoted byF f and the external loading includinspecimen damping and stiffness forces are lumptogether inFext. The equations of motion are eaily derived by applying Newton’s second law:

xp5vp , ~24!

Fig. 8. Actuator with a simple load model.

uaua

hear

on-

,

y-hen-onuf-

c--

toralrsnionen

stsnonheur-erav

ed-e

m-edngofst.tic

en

cta-nt

is

ns

re, aonheree-rals

heicese

fora


vp51

mp@Abpb2Atpt2Fext2F f2mpg#.

~25!

Eqs. ~17a!, ~17b!, ~24!, and ~25! with qb and qt

given by Eqs.~18a! and ~18b! constitute the statespace model for the servovalve and loaded acttor subsystem under consideration. These eqtions also contain the major nonlinearities in tsystem: the variable capacitance and the squroot flow rate versus pressure drop relations. Nlinearity is also introduced in Eq.~25! by the non-linear friction force, which includes Coulombstatic, and viscous components@1,18# as discussednext.

7. Modeling and identification of friction

Friction affects the dynamics of the electrohdraulic servovalve as well as the dynamics of tactuator piston. Friction in the servovalve is geerally considered to be of Coulomb type actingthe spool of the valve and can in practice be sficiently eliminated by using dither signals@9#.The particular friction effect of interest in this setion is the model of the frictional forces that appear in the equations of motion of the actuapiston. The literature offers various empiricmodels applied to specific hydraulic actuato@1,3,18,19#. In the most general case, friction ithe actuator cylinder is considered to be a functof the position and velocity of the piston, thchamber pressures~the differential pressures whethe piston is sticking near zero velocity!, the localoil temperature and also running time.

In this study, open-loop and closed-loop tewere performed to identify the friction force othe actuator piston by assuming it to be a functiof velocity. Open-loop tests involved changing tset current input to the servovalve while measing the steady-state piston velocity and cylindchamber pressure responses. The system behas a velocity source in the open loop. The closloop test involved tracking a 2-Hz 35-mm sinwave piston position command underP controlwhile measuring acceleration, velocity, and chaber pressures. In both the open-loop and closloop tests, the friction is then estimated usiNewton’s second law. Fig. 9 shows the resultmultiple open-loop tests and a closed-loop teThe closed-loop test clearly shows the hystere

--

e

es

-

behavior of friction. For simplicity, we use thcommon memory-less analytical model of frictio~without hysteresis! as a function of velocity,given by Eq.~26! and included in Fig. 9,

F f5Fv6xp1sgn~ xp!~Fc

61~Fs62Fc

6!e~ xp /Cs!!.~26!

The asymmetry of the friction force with respeto the sign of the velocity observed in the mesured data is taken into account by taking differecoefficients in Eq.~26! for the up and down mo-tions. The strong scatter in the estimation datatypical of friction phenomena.

8. Experimental validation of the model

The model described in the previous sectiowas implemented inMATLAB/SIMULINK , and base-line open-loop and closed-loop experiments weconducted to validate it. In these experimentssimple load mass is rigidly attached to the pistrod. For the models of each of the sections of tsupply and return line hoses, only six modes aretained in the modal approximation. This was dcided considering the actuator hydraulic natufrequency of 172 Hz computed using formulafrom linear models~see Merritt@4#! and selectingthe natural frequency of the highest mode of tapproximation for each section to be close to twthis value. From manufacturer frequency respondata for the servovalve, the natural frequencythe servovalve was estimated to be 140 Hz withdamping ratio of 1.

Fig. 9. Piston friction force.

uc-tha

er--

mnsnden-se40

seis

rningneea-n-at-edellthe

bertone. Itre-re-dy

eser

ny

ele-ainela

m,he

nge es.


To measure the supply and return pressure fltuation, pressure transducers were mounted onsupply and return ports at the servovalve and50-mA rated step current was supplied to the svovalve in the open-loop. Fig. 10 shows a comparison of the supply and return pressure fromeasurement and simulation. Two observatiocan be made from this figure. First, the supply areturn pressure fluctuations contain the fundamtal periods of 25 and 32 ms, respectively. Thecorrespond to fundamental frequencies of aboutand 31 Hz, respectively. The implication of thefluctuations is that the bandwidth of the actuatorlimited by the dynamics of the supply and retuhoses, since the other dynamic elements includthe servovalve and the actuator have higher corfrequencies. Second, the model follows the msurement well. In particular, the frequency cotents match nicely. The discrepancies can betributed to errors in the estimation of effectivbulk moduli for the different hose sections, anthe estimation of manifold pressure drops as was unknown but estimated parameters inadopted simplified model of the servovalve.

The open-loop response of the system caninvestigated further by looking at the cylindechamber pressures shown in Fig. 11 and the pisvelocity response shown in Fig. 12 following thsame step-rated current input to the servovalvecan be seen from Fig. 11 that the simulation pdictions of chamber pressures follow the measuments and that the supply and return pressure

Fig. 10. Supply and return pressure following a step chain servovalve current.

e

r

-

namics introduced by the long sections of hosare reflected in the individual cylinder chambpressures.

In the open-loop piston velocity response showin Fig. 12, the velocity signal was obtained blow-pass filtering and differentiating the LVDTposition signal. The figure shows that the moddoes a fairly good job of predicting the piston vlocity response as well. The differences are agattributed to uncertainties in the servovalve modand also errors in friction estimation, which hasconsiderable scatter, as shown in Fig. 9.

9. Predicting system performance

It should be recalled that, for the present systethe fundamental frequencies of oscillation of t

Fig. 11. Open-loop cylinder chamber pressure respons

Fig. 12. Open-loop piston velocity response.

thetheheeasplyn-ste-ly

nethssi-

outheassef-

reAtillc-13theed

talndrnen-he

bygi-eden

a-ac-ots

arebya-ntto

hub-

ng

led


supply and return line pressures are lower thannatural frequencies of both the actuator andservovalve. This implies that the bandwidth of ttest system is limited by supply and return lindynamics. The model described in this paper wused to predict pressure fluctuations in the supand return lines by running simulations for cotemplated modifications to the layout of the tesystem. In particular, it was desired to see if rducing the lengths of sections II of both the suppand return hoses in the layout~Fig. 2! producessignificant changes in the supply and return lipressure dynamics at the servovalve. The lengof sections I were kept unchanged since the phycal constraints of housing the HPS required ab3.048 m of hose lengths for sections I of both tsupply and return lines. The goal of the study wto see if actually there will be significant gainfrom these measures in terms of increasing thefective bandwidth for the actuator.

As a first case, the hoses of sections II wereduced to half the original length of 3.048 m.length of 1.524 m was considered sufficient to splace the HSM unit on the ground while the atuator was mounted on the load frame. Fig.shows the supply and return line pressures atservovalve following the open-loop test describpreviously with the lengths now at 1.524 m.

It can be seen from Fig. 13 that the fundamenperiod of pressure fluctuation shifts to 13.3 a18.5 ms, respectively, for the supply and retulines. These correspond to fundamental frequcies of 75 Hz on the supply line and 54 Hz on t

Fig. 13. Supply and return pressures with 1.524-m-lolines.
return line. The settling times are reduced
about 20 ms or about 22% compared to the orinal setup with 3.048 m lengths of sections II of thsupply and return line hoses. The peak amplituof the oscillation is also slightly reduced, as showin Table 1. The bandwidth of the hydraulic actutor can therefore be expected to be improvedcordingly. However, the accumulators were nshown to be effective in filtering the fluctuationcompletely.

As a second case, the sections II of the hosesconsidered to be removed from the system andso doing the HSM is close-coupled with the actutor. This consideration would need significachanges to the physical design of the load frameallow the HSM to be mounted on it together witthe actuator. In the model, the corresponding s

Fig. 14. Supply and return pressures with close-coupHSM and shorter lines.

Table 1Summary of step response simulation results.

Lengths ofsections II

~m!

Fundamental fre-quency of oscillation

instep response~Hz!

Peak amplitude ofoscillation

~Mpa!

Supplypressure

Returnpressure

Supplypressure

Returnpressure

3.048 40 31 1 0.942.286 49 39 0.97 0.911.524 76 57 0.94 0.880.7512 147 115 0.87 0.710 N/A N/A 0.0 0.0

e-ltsform

s inmnthere

the

524theor.neendre-rte

a-the

hledi-elyns

c-neliciondes.licion

neerslkre-indel

sls.or

s

re-

mbe

theedof

byll

e-re-

el-n.

hes-if-ent.edtedofe-

eesvo-ngthevo-or.glyneithla-ethe

lichefor

csup-e intheess-rn-hers-


system of section II was removed from the schmatic in Fig. 6. Fig. 14 shows simulation resufor this case as well as a hypothetical casewhich the hoses are shortened further to 0.7512It can be seen that the effect of the accumulatorfiltering out the dynamics of the sections I frothe supply and return lines is particularly evidein the close-coupled HSM case. In addition, tsteady-state pressure drops in the supply andturn line are smaller in this case compared tooriginal setup ~Fig. 10! and to the case with1.524-m lengths for sections II~Fig. 13!. The im-provements predicted by the case of lengths 1.and 0.7512 m seem rather small compared tocase of close coupling the HSM with the actuat

Table 1 summarizes these results, including omore additional length for sections II. It can bseen that the peak amplitudes of the supply areturn pressure oscillations are progressivelyduced as the length of the hoses becomes showhile the fundamental frequency of the oscilltions for the step responses becomes larger. Inlimit case of 0-m length for sections II, whicmeans the HSM accumulators are close coupwith the servovalve, the oscillations are elimnated. In this case, the accumulators effectivsuppress the oscillations arising out of the sectioI of the supply and return line hoses.

10. Conclusions

In this paper, a model of an electrohydraulic atuator system focusing on supply and return lidynamics was presented. A model of hydrautransmission lines based on modal approximattechniques was adopted yielding state spacescriptions suitable for time domain simulationThe interconnection model of the electrohydrausystem developed here is an attractive simulattool for the following reasons.

1. The distributed parameter transmission limodel requires only aggregate line parametsuch as line length, diameter, and effective bumodulus in state space dynamic models. It thefore enables a quick investigation of line effectsa time domain analysis. It was used here to mohydraulic hoses with experimental validation.

2. The time domain simulation, in turn, makeit possible to include nonlinear actuator modeThis is particularly useful for the study of actuatmodels with linear and nonlinear control system

.

-

r

-

without ignoring supply and return line pressuvariations introduced by transmission line dynamics.

3. The model of the overall hydraulic systewas modular. Subsystem models can easilychanged for desired emphasis observing onlyinput-output causality. For example, a detailmodel of the servovalve can be used in placethe simplified model used in this paper, thereimproving the predictive power of the overamodel.

Simulation results for some open-loop step rsponses were compared against experimentalsults for the test system. It is felt that the modperforms well, given the minimal amount of information it uses to enable a time domain simulatioThe use of constant bulk modulus values for those material, which in reality depends on presure and frequency, may explain some of the dference between the simulation and measuremA more detailed servovalve model with measuroperating characteristics, instead of the raspecifications, could also improve the accuracythe overall electrohydraulic system model prsented here.

A simulation analysis of the effects of thlengths of sections of the supply and return linbetween the HSM accumulators and the servalve was done. It was concluded that reducithe lengths of these lines progressively reducessupply and return pressure fluctuation at the servalve during dynamic excursions by the actuatThe test system bandwidth was correspondinimproved. The best scenario was shown to be owhere the accumulators were close coupled wthe servovalve, thereby employing the accumutors effectively in filtering out the effects of thsections of the supply and return lines betweenpump ~HPS! and the accumulators.

In conclusion, it is recommended that hydraucontrol system design and analysis include tmodeling approach presented here, to accountthe effects of supply and return line dynamirather than assuming constant values for the sply and return pressure as has been usually donthe literature. It was also demonstrated thatapproach could be used to make a quick assment of alternative layouts for supply and retulines in terms of minimizing transmission line dynamics. The method can easily be adopted to otapplications of hydraulic machinery and test sy

of

:

-i-

u-r-

y,

e-

d-g.

y-e--

ri-e-ns-T.

u-lof

p-r

X,,

uid

ted

ent

e

gis-3,

i-u-y-i.

nd,

eg.

ofe-e-

as,

e

,-

e.e,

n,-.

ss,.

invi-


tems where hydraulic oil supply and return linessignificant length are involved.

References

@1# Jelali, M. and Kroll, A., Hydraulic Servo-systemsModelling, Identification and Control~Advances in In-dustrial Control!. Springer-Verlag, London, 2003.

@2# Kugi, A., Non-linear Control Based on Physical Models, Lecture Notes in Control and Information Scences No. 260. Springer-Verlag, London, 2001.

@3# Sohl, G. A. and Bobrow, J. E., Experiments and Simlations on the Nonlinear Control of a Hydraulic Sevosystem. IEEE Trans. Control Syst. Technol.7~2!,238–247~1999!.

@4# Merritt, Herbert E., Hydraulic Control Systems. WileNew York, 1967.

@5# D’Souza, A. F. and Oldenburger, R., Dynamic rsponse of fluid lines. J. Basic Eng.86~3!, 589–598~1964!.

@6# Goodson, R. E. and Leonard, R. G., A survey of moeling techniques for fluid line transients. J. Basic En94~2!, 474–482~1972!.

@7# Viersma, T. J., Analysis, Synthesis and Design of Hdraulic Servosystems and Pipelines, Studies in Mchanical Engineering I. Elsevier Publishing Co., Amsterdam, the Netherlands, 1980.

@8# Woods, R. L., Hsu, C. H., and Chung, C. H., Compason of theoretical and experimental fluid line rsponses with source and load impedance. Fluid Tramission Line Dynamics, edited by M. E. Franke andM. Drzewiecki. ASME Special Publications, NewYork, 1983, pp. 1–36.

@9# Van Schothorst, G., Modeling of Long-Stroke Hydralic Servo-Systems for Flight Simulator Motion Controand System Design. Ph.D. thesis, Delft UniversityTechnology, Delft, The Netherlands 1997.

@10# Yang, W. C. and Tobler, W. E., Dissipative modal aproximation of fluid transmission lines using lineafriction model. J. Dyn. Syst., Meas., Control113,152–162~March 1991!.

@11# MTS Corporation, Hardware Product Manual, 290.1290.2X, and 290.3X Hydraulic Service Manifold1985.

@12# Blackburn, J. F., Reethof, G., and Shearer, J. L., FlPower Control. MIT Press, Cambridge, MA, 1960.

@13# McCloy, D. and Martin, H. R., Control of FluidPower: Analysis and Design, second edition. HalsPress, Chichester, UK, 1980.

@14# Ayalew, B. and Kulakowski, B. T., Transactions of thASME, Journal of Dynamic Systems, Measuremeand Control~in press!.

@15# Hullender, D. A., Woods, R. L., and Hsu, C.-H., Tim

domain simulation of fluid transmission lines usinminimum order state variable models. Fluid Transmsion Line Dynamics. ASME special publication, 198pp. 78–97.

@16# Hsue, C. Y.-Y. and Hullender, D. A., Modal approxmations for the fluid dynamics of hydraulic and pnematic transmission lines, Fluid Transmission Line Dnamics, edited by M. E. Franke and T. M. DrzewieckASME special publication, 1983, pp. 51–77.

@17# Dransfield, P., Hydraulic Control Systems: Design aAnalysis of their Dynamics, Springer-Verlag, Berlin1981.

@18# Alleyne, A. and Liu, R., A simplified approach to forccontrol for electro-hydraulic systems. Control EnPract.8, 1347–1356~2000!.

@19# Manhartsgruber, B., Singular perturbation analysisan electrohydraulic servo-drive with discontinuous rduced dynamics. Proceedings of the 1999 ASME Dsign Engineering Technical Conference, Las VegNevada, 1999, pp. 2001–2011.

Beshahwired Ayalew wasborn in Goba, Ethiopia in1975. He earned his BS degrein mechanical engineeringfrom Addis Ababa University,Ethiopia and his MS degreealso in mechanical engineering, from the PennsylvaniaState University, USA. He iscurrently a Ph.D. candidate inmechanical engineering at thPennsylvania State UniversityHis research interests includdynamic systems and control

energy systems, mechanical design, and vehicle dynamics.

Bohdan T. Kulakowski ob-tained his Ph.D. from the Pol-ish Academy of Sciences in1972. He has been with PenState University since 1979where he is currently a professor of mechanical engineeringDr. Kulakowski teaches under-graduate and graduate courseon engineering measurementsystem dynamics, and controlsHe is a co-author of the book‘‘Dynamic Modeling and Con-trol of Engineering Systems.’’

Dr. Kulakowski is a Fellow of ASME. His research interests arevehicle testing, vehicle dynamics, and vehicle interaction with enronment.

Documents

Modeling supply and return line dynamics for an electrohydraulic