an investigation of hydraulic-line resonances and its attenuation - nasa - john l sewall david a...

8/13/2019 An Investigation of Hydraulic-Line Resonances and Its Attenuation - NASA - John L Sewall David a Wineman

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TECHNICALMEMORANDUM

N A S A T M X - 2 7 8 7

INVESTIGATION OFRESONANCE

ITS ATTENUATIONJohn L. Sewall, David A. Wineman, and Robert W. Herr

Research Center Va. 23665

T I O N A L A E R O N A U T I C S A N D S P A C E A D M I N I S T R A T I O N W A S H I N G T O N , D . C . D E C E M B E R 1973


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1. Report No. 2. Government Accession No.NASATMX-2787

4. Title and Subti t leAN INVESTIGATION OFHYDRAULIC-LINE RESONANCEANDITS ATTENUATION

7. Author(s)John L. Sewall, DavidA.Wineman, andRobertW.Herr9. P e r f o rm i n g O r g a n i z a t io n N a m eand Address

N A SA LangleyResearchCenterHampton, Va. 23665

12. Sponsoring Agency Name and AddressNationalAeronauticsandSpace AdministrationWashington, D.C. 20546

3. Recipient's Catalog No .5. R e p o r t DateDecember19736. Performing Organization Code

8. Performing Organization Report No .L-8738

10 . Work Unit No .501-22-05-01

11 . Contract or Grant No.

13 . Type of Report and Period CoveredTechnical Memorandum

14 . Sponsoring Agency Code

IB/Supplementary Notes

16. AbstractAninvestigation off luidresonance inhigh-pressure hydraulic lines hasbeen made withtw o

types off lu id dampers (o r filters) installed in the line. O netype involvedthe u se of one or moreclosed-end tubes branchingat right angles f ro m amain line, and theother typewas a f lu id muf f le rinstalled in-line. These devices were evaluated inforced vibrationtestswithoscillatorydistur-bances overa 1000-Hz range appliedto one end of the lineandwith oscillatory pressuresmeas-ured at various stations along the main pipe. Limited applications of acoustic-wave theory to thebranched systems are alsoincluded. Results showvarying attenuations ofpressure perturbations,depending on the numberan dlocation ofbranches and the typeofmuf f le r . Up to three brancheswere used in the branch-resonator study, and thelargest frequency range with maximum attenua-tionwasobtainedfor a three-branch configuration. Thewidest f requency rangeswith significantattenuations were obtainedwith twotypes off luidmufflers.

17. Key Words (Suggested by Author(s) )Vibrat ionHydraul ic- l ine resonanceAcous t ic -wavetheoryHydraulic muf f le rBranch-pipe damper

18.DistributionStatementUnclassified-Unlimited

19. Security Qassif.( ofthisreport)Unclassified

20. Security Classif.(of this page)Unclassified 21. No. of P a g e s77 Domestic, $3.75Foreign, $6.25Fo r saleby theNational Technical InformationService,Springfield, Virginia 22151


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A N INVESTIGATION OF HYDRAULIC-LINER E S O N A N C E AND ITS ATTENUATION

B y John L. Sewall, DavidA. Wineman,and Robert W . HerrLangley Research CenterSUMMARY

An invest igat ion offluid resonance inhigh-pressure hy draulic l ines has been madewith tw otypes off lu id dampers (or f i l ters ) installed in the line. O netype involvedthe useof one or more closed-end tubes branching at r ight angles f rom a main pipe, and the othertype was a f luid muf f le r installed in-line. Eva luations of these devices were made inforced v ibrat iontests inwhich osci l latory dis turbances over a 10 00-Hz range w ere appliedto one en d of the line and oscillatory pressure responses w ere measu red at var ious s ta-tions along th e main pipe line. Lim ited applications of acoust ic-wave theory to thebranched sys tems are also included.

Results show vary ing at tenuations ofpressure perturbat ions withf requency , depend-ing on the n u mb e r and location of branches and the type ofmu f f l e r. Up to three brancheswe re considered in the b ranch -resona tor s tudy, and thelargest f requ ency range contain ingmaximum at tenuat ion was obtained for a three-branch configurat ion. The widest f requencyranges with acceptable ands ignif icantpressure attenuations were obtained with either oftw o types ofmu f f l e rs of equivalent volumes. O ne ofthese was a commercial damper withan intr icate internal flow ar rangement which gavepressure at tenuat ions over a slightlywider freq uen cy range than the other ty pe, which was a s imple expansion chamber .

The studyalso included exploratory tests ofs tructu ral-f lu id interactions in aclosed-end U-tube conf igurat ion,andresults ofthese tests illustrated thepossible pressure reduc-tions obtainable near structural andf luid resonances as a consequen ce of adequately anchor-ing the piping system.

INTRODUCTION

Pumps transmitt ing fluid through hydraulic l ines under highpressure impart periodicpulses to the f luid which can induce undesirable vibrat ions . Thecrashof anadvancedfig hter-a ircraf t prototype was traced to a break in the hy drau lic line of the control systembecause ofsevere localizedresponses to periodic pressure pulsat ions produced by a pumpoperating at high freq uen cies . These responses were associated with s trong structural-f lu id resonances which we re notsuf f ic ient ly dampedb yf luid leakage internal to the a i rcraf t


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hydraul ic system. M oreove r , t here was noaccumula tor and , there f ore , nopressurereservoir to absorb flow disturb ances generated b y the pum p. A solution to this problemwas sought b y introd uc ing damping( or attenuating) devices into th e f luid line to reduceresonant oscillations to anacceptable level.

Fluid resonance inhy draul ic lines and its at tenuat ion have b een widely studiedandare based on the fun dam entals of wave propagat ion throug h continuou s media, as presentedin r e f e r e n c e s 1 and 2, for example . Ref erence 1 includes the developmentandapplicationof theseprinciples to the acoustics off luid inpipes with straight branches andHelmholtzresonators. Re fe rence 2contains a review ofwa te r -hammerresearch, pressure and f low-scaling relations, andrelated usefu l des ign in format ion . The work presented in re f e ren ce is also des ign-or ien ted andincludes some basic data onf lu id propert ies important to f lu idv ibra t ion .

Basic studies off luid dynamics in straight transmiss ion lines are reported inre fe ren ces 4 to 12 . Ref erence 4presents the first of aseries ofstudies cond ucted at theNASA Lewis Research Center on the dy namic behav ior ofhydraul ic f lu id in a long linesub jected to sinusoidal pulses at one end and having variab le impedance at the other endof th e l ine. Ref erenc es 5 and 6 give solutions based onva r ious fo rm sof the Navier-Stokesequat ions for f luid flow inpipes. The natural longi tudinalf requency of the pipeline itselfis also shown in r e f e r e n c e 5 to have a pronouncede ffect onoscillatory pressureresponsesRefe rence 7 is a theoretical treatise onf lu id dy namic response to periodic and nonperiodicinputs at one end of a hyd raulic l ine termina ted at the other end by nonlinear ori f ice condi-tions. Re fe ren ce 8 dem onstrates the application of f in ite -element technology applied tothe flow of blood.

Viscos i ty e f fec t s are treated invary ing depthin re f e rences 5 to 15,with r e f e r -ences 5 and 9o f fer ing somewhat more on the sub jec t thanth eothers. T he e f fect ofviscosi ty in Newtonian f luids is shown in ref eren ce 5 to depend on the parameterw her e a is the inside rad ius of the fluidline, w is the f requency offluid oscillation, andv is the kinematic (or effective) viscosi ty. Anincrease in the mag nitude of this parameteris associated with adecrease in the viscosi ty e f fect , and a value approachinginfinity corresponds to a low-viscosi ty f lu id oscillating at ve ry h igh f requenc i e s in alarge-diameter tubeFor non-Newtonian f lu ids (suchas M i l-0-5606 ando ther a i rc ra f t hydraul icoils), whereth eeffect ive viscosity decreaseswith increasing shear rate (whichvaries f rom onef lu id toanother), a//j7is shownto be the gov erning viscosity parameter. Inr e f e r e n c e 9 the appli-cation of acoust ic-wave theory toviscous Newtonian f luids is demonstrated for sinusoidaldisturbance propagation in long,straight, cyl indrical , nonvibrat ingpipelines. This work isre in forced in re fe renc e 10, inwhich the method oicharacteristics was applied to the one-d imensional modelo f t rans ien t flow and showed good agreement with experiment , part icu-larly in accurate predict ion ofdistort ion e f f ec t s characteristic ofwa te r -hamm er response.


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r e f e r e n ce 11, asmallv iscosi ty e f fect is shownfor such liquids as water, k erosene, and others having a lowvalue of v. The relation between conduit flexibility an d

viscosity isdiscussed in r e fe r ence 12 ,whic h shows thatelasticpipeline walls exerton the spatialpropagationof the modes of a viscous f luid.

Refe rences 13 to 16 are concern edwith var ious at tenuat ing devices used to reduce in f luid lines. These devices m ay b e roughly class i f ied inthree groups:

resonators, m u f f l e r s (or f i l te rs) , and internal da mpers . In the f i r s t group, the usef closed-end b ranches, or s tandpipes, perpendicular to a main hy drau lic l ine is descr ib ed reference s 13 and 14 . In re ference 13, measured pressure-per turb at ion ratios agree

withanalytical predictions oftheseratioswhicha re based onacoustic-wave theoryor nonviscousf luid in a hydraulic l ine witha branch located at the midpointof the line and

length equal to .half that of the line. Results of this study show large attenuationpressureper turbat ions at d iscre te f requenciesfor which th e branch length is an odd

ofquarter-wave length. This implies essentially complete ref lect ion ofwaves ate branch junct ionand nowave motion downstreamf rom this point. Inr e f e r e n ce 14, it is that alarge nu mb er of short b ranchresonators, closely spaced relative to the wave sinusoidal dis turbance, can eff ect ive ly at tenuate s inusoidal perturb at ions to alow level over a widef requency range. Each of the b ranc hes (so-called shuntassem- in r e f . 14) has asmall accumulator at the closed end and a flow constriction near th e

Twod i f ferent types of constriction (shunt element) are evaluated, and is shown to predict accurately th e attenuation constant and theratio

downstream-to-upstream pressures.The second groupoff luid perturbat ion at tenuators consists of muf f l e r s of var ious

ndparallelbranch configu rat ions. Considerable k nowledge about these devices is con-d in the comprehensive invest igat ion reported in ref eren ce 15, in which the at tenua-

properties ofover 75mu f f l e rs and muf f le r combinat ionsare measured and compared attenuationproperties calculated by acoust ic-wave theory . Althoughthis work was with air as thetest med ium,theresults are also applicable to liquids as long as

y and compressibi l i ty di f fe ren ces between gases and l iquids are taken into account . pressures and flows at the ends of multipleparallel branch conf igura-

ons are given in ref eren ce 6 without experimental c onf irmation .In th e third group ofattenuators, pressure amplitudes off luid oscillations ared by the introduction of f lexible tubes orbars into the main fluid line. T hepossibil-

ofthese insertsproducingsubstantial reduct ions inf luidoscillatorypressures over widef requency range are discussed in r e f e r e n ce 16 , inwhichtest results are reportedor single- or multiple-walled tubes or for a sponge-rubber bar 'contained withinthe f luid

3


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In support of an investigation of the aircraf t accident mentionedearlier, the LangleyResearch Center evaluated var ious f luid damper s and s tudiedf lu id pressure responses inhyd rau l i c l ines sub jectedto osci l latory dis turban ces . This work , reported herein , consis tedof atest program and l imited analy t ical s tudies . Vib rat iontests were conducted on straighl iquid-f i l led pipes with andwithout standpipe branches located at var ious points alongth epipe, andalso with and withoutmuf f l e r s in the pipe. Re fe renc e 13 provided the ini t ials t imulus for the branch-pipetests with one, two, or three b ranches at va r ious dis tancesf r o m one end of the main pipe and with tw o b r anchesat a common junct ion . Hy draul icf lu idat moderate- to-high staticpressure levels w assub jectedto ripples, o r osci l latory pressu re\pulses, introduced at "one end of the lineby two dif ferent shaker conf igu rat ions . M ostofth etests were conducted withthe downstream end of the main line capped, b u t someresultswere obtained with an accumulator at this end.

Muf f l e r tests were made for two dis t inct expansion chambers of the same volumeb u twith d i f f e r ing internal flow ar rangements , andresults ofthese tw otests are comparedwith each other andwithresults of the straight line without a muf f ler . Inaddition, vib ra-t iontests were made on a U-shaped closed-end pipe in order to s tudy s tructural-f luidinteractions.

Thetest program wascom plementedb y calculat ions based on the acoust ic-wave-theory presentat ions ofr e f e r e n ce s 4 and 13 for branch-pipe conf igurat ionsan dalso on afinite-element modelutilizing the NASTRANcomputer program. In thefirst oftheseanalyt ical s tudies, the procedu res d escr ibed in ref ere nc e 13 for a s ingle-b ranch config ura-tion are applied herein to obta in in ter terminal pressure andflowratios for sys t ems withmore than oneb r anch . T he f ini te-element approach allows for f lu id-s t ructural coupl ingdue to bothlateral andlong itudinal motionsof thepipes and canalso approximate th eef fec ts of f i l ters , as well as b ranc h resonators , in the system. Re fe ren ce 17 contains theessentials of this approach, alongwith interest ingresults. It shouldb e noted that bothofthese an alyt ical methods involve l inear theo ry, and direct com parison of analy t icalresultswithtestresults is notpossible becau se theampli tudesandpressures in thetest programwere suf f ic ient ly large to becons idered nonl inear .

SYMBOLSThe measu rements and calculat ions were made in the U.S. Customary Units ,which

are shown in parentheses following the International Systemof Units,S I.c speed of soundin hydraul ic f lu idd diameter ofhydraul ic f lu id in main pipe


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, dm diameter ofhydraulic f l u i d in kth and mthbranch pipes (see figs.32and 36, respectively, alsoeqs. (14) and (31))

f r e que nc y , w/277, Hz' ^m lengthof kth and mth branches (see f i g s . 32 and 36,respectively, also

eqs. (9) and(32))

I total lengthofmain pipel , 12, 1% , 4 lengths along main pipe (see figs. 34 to 50)P ( x ) pressureperturbation amplitude at station x along main pipeP Q inletpressureperturbation amplitude, P(0)PE exit pressure perturbation amplitude, P(l)P. pressure perturbation amplitude at ith station along main pipe(i = 1, 2, 3)PO , peak-to-peakpressurefluctuationo fmain pipe without branches; refer-j,r encepressurefor branched configurationsPA k, P^ m pressure perturbation amplitudeatbranch inlet (see f i g s . 32 and 36,

respectively)V - D L - >PTS pressureperturbation amplitude at closed end ofbranch (see figs. 32J I ) , K t>, m and 36,respectively)UAk, U^ velocity perturbation amplitudeatbranch inlet (see f i g s . 32 and 36,

respectively)U T D L.>Un velocity perturbation amplitude at closed end ofbranch (see figs. 32i5,K J3, m and 36,respectively)U(x) velocity perturbation amplitude at station x along main pipe

U inlet velocity perturbation amplitude, U(0)


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UE exitvelocity perturbation amplitudex longitudinal coordinate ofmain pipe (see fig.32)y branch coordinate (see fig.3 2 )Z characteristicimpedance ofhydraulic f lu id , pcZ exit impedanceHiz ( x ) dimensionless impedanceat stationx(see eq.(5))p . massdensity ofhydraulic f l u idu > f r e q u e n c y , 2rf, rad/sec

E X P E R IM E N T A L INVESTIGATIONHydraulic Line With Branch Resonators

A s is shown inreferences 13 and 14, onemethodfor reducing oscillations causedbyapumpin ahydraulic system - alsocalled pumpripple - involvestheinstallationofbranch resonatorsor dampers along and perpendicular to the main line of f lu id f low. Inthepresent test program, oscillatorypressureswere measuredtodetermine attenuationmagnitudes and frequencies for varioussingle-and multiple-branch systems. Lack ofs u f f i c i e n t instrumentationfor measuring f lowrates, andparticularly the in l e t - f lowprofile,precluded quantitative determination ofrelationshipsbetween oscillatory f l u id pressuresandvelocities, not onlyat the inletbutalsoelsewhere in the system.

Test setups.- Twoseparatetest systems were used inthis studyand are shownschematically infigures1 and 2. Longitudinalandlateralpipe motions were constrainedas m u c has possible inboth systems in an e f f o r t to isolate f l u i d resonances f r o m structuralpipe resonances. Ineach system, hydraulic f l u id (low viscosity)wasexcitedbypressurepulsations introducedat one end of ahydraulic line"thatwascapped at theotherend formost of thetests,and oscillatory pressureamplitudes were measured by means ofresistance-wire strain-gage transducers statically calibratedto6.895 MN/m^ (1000 psi)and o f f se t f rom the main linebymeansofT-jointsat the locations indicated infigures 1and2. (Pressuretransducer 2failedearlyin thetestprogram.) These pressuresweremeasuredatdiscrete frequencies ranging fromlessthan 100 Hz toover 900 Hz, andresonant frequencies were located approximately withinthe discrete-frequencyintervals.


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Thestaticpressure level ranged f rom 4 .825 to 10 .34 M N/m^ (700to 1500 psi) in the setupo f f i gu re 1 and was about 10.34 M N /m (1500 psi) in the setupoff i g u r e 2 .

In thetest setup off i g u r e1, the f luid in the system was excited b y a hydraul ic shakerwith i ts dr iving cy linder locked and the pressure pulses into the f lu id controlled by aservo-valve. The downstream end of the line was closed at the vacuumpump-downvalve . In thisapparatus, anattempt was madeto hold peak-to-peak pressure f luctu at ions (double ampli-tude) constant at the inlet throughoutthe f requency range. However , due to the power limi-tationof the servovalve, th e level of inletpressure f luc tuat ions dropped off as the f requencyincreased. This t rend is shown in f ig ure 3 by thecircular symbols whichrepresentpres-sure fluctuations in the station closest to the inlet in the line without b ranc hes. Intestswith branches , the root-mean-square ( rms) cur rent value to the servovalve w asadjus tedto the values used in thetest without b ranche s. Sincet he backpressure on the servovalvedoes affect th e flow into th e sys tem, it is evident that th e fo rc ing funct ion did not r emainconstant. .

In order to gain bet ter control of the input forcingfunct ion, th etest setup shown inf igure 2 was devised. Here a diaphragm r ipple generator was connectedto alarge electro-dynamic shaker, and it was possible to holdth e displacement of the diaphragm constant forf requencies up to 650 Hz. I t wasalsopossible to obtain constant, b u t reduced, displacementsat h igher f requenciesup to the maximumglevels of the shaker; however , it shouldb enotedthat evenwith a constant displacement, the vo lume dis turbance did not rema in constant b e-cause of the res i l ience of the diaphragm.

As shown in f i gu re 2, anaccumu lator was attached at the end ofline. Testsweremade withthe accumulator includedor excluded, dependingonwhetherthe shut-off valveat the end of line was open or closed.

Results.- Results oftests with branchresonators are presented inf i gu res 4 to 7for th e apparatus off i gu re 1 (test system I) and in f i gu res 8 to 20 for the apparatus off igure 2 (test sy stem n). Pressure fluctuations at the station closest to the downstreamend of each branch configurat ion(pressure transducer 3 in f igs . 1 and 2) are dividedbypressure f luctuat ions at the corresponding s tat ion on the pipe without b r anches ( f i g s . 3, 8,and 14), and theratio P S / P S r is shownas afunction off r equ ency . (B ecause resonantf r equenc ies were only approximately located, th e points inf igs . 3 to 20 are notconnected . )

These results indicate that for a s ingle-b ranch config urat ion witha quar t e r -wavelength,resonator located closest to the inlet ( f ig. 5) , 0 .2 5 to 0 .1 at tenuat ion (75- to9 0-percent reduc t ion in pressu re) is possible within the region of the branch resonantf requency (area indicatedb yarrows inf i gu re ) . The use of two or more dampers ofd i f -


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f e r en t wavelengths extends th e at tenuat ion ofpressure f luctuat ions over wider f requencyranges (see f igs .6 and 7) andalso reduces th e boost inpressuref luctuat ionsatf r e q u e n c i esabove andbelowth e branch resonant f requencies . T helargest f requency range with maxi-mum attenuations was obtained for the three-branch configurat ionoff i g u r e 7, inwhichbranches ofd i f ferent lengths were located at well-separated positions along the lengthofth e main pipe.

The e f f ec t s of the accumulator in the system can be seen in a comparison off i g -ures 8, 9, 10, 11, and 13with f i gu res 14, 15 , 16, 17, and 20. In the system without branches(f igs. 8 and 14), ther e appear to be fewer high-level pressure peak s with the accumulatorincluded (open) than withou t it (accu mu lator closed). M oreov er, the peakpressure levelsare lower withtheaccumulator open than withitclosed. Thesystemswithbranches showa somew hat reve rsed tende ncy , that is , more h igh-level d iscre te- f requency pressure peakswith the accumu lator than without it (compare f igs . 9, 10, 11, and 13with f ig s . 15, 20, 16,and 17,respect ively ) . M oreover, there appears to be no consistent significant ef fect of theaccumulator onpressure at tenuat ions for the systems with branches.

Other results showthe e f fec ts of twobranches at the same location and ofbranchentrance. For the f i r s t ofthese e ff ects , comparison off i gu re 13with f i gu re 9 showslessvariation inpressure peaks over the ful l frequency range for the two-branch configurat ionthan for the s ing le-branch config urat ion withlarger branch diameter; however, the s ingle-branch configurat ionresulted in highe r attenuations. Con cerning the bran ch-inlet ef fect ,f ig ure s 11 and 12 indicate oppositeresults for a 9 0 bend and a roundedinlet; that is, withthe 9 0 be nd in the bran ch near its inlet, the attenuations were hig her over u pper and lowerf requency ranges than in the middle (fig. 11), whereas withthe rounded branch inlet , th eattenuation was higher in the middle f requency range than over th e upper and lower ranges(fig. 12).

Hydraul ic Line With M u f f l e r sAnother means ofat tenuat ing pum p-ripplepressures is by use ofmu f f l e r s,or f i l ters ,

in th e hydraulic l ine. Partof the exper imental work under takenin this study involvedtheevaluat ion of a commercial pulsat ion damper which was considered for installation in theprototype aircraf t that crashed and was to be placed in the hyd raulic l ine as close to thepumpas pract icable. Acco rding to the man uf acturer ofthis f i l ter , i tconsists of aseriesof resonating chambers interconnected b y impedanceflow t ubes andcombines the salientfeatures of the Helmholtz resonator (e.g., ref . 1), the Q uincke tube (e.g., r ef . 3) and a low-pass acoust ic f i l te r .

Testapparatus.- Asmaller m odel of the f i l ter was made avai lable for the presenttests, inwhichpressure f luctuat ions were measured andcompared f or three systems: a8


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straight, closed-end hydraulic l ine with n odamper, th e same system with th e comm erc ialfil ter installed near the ripple ge nerato r, an d the closed -end pipe with a single expansionchamber havingth e same volumeas the commercia l un i tandalso located near the ripplegenera tor . Dimensionsof the th reetest sys tems are giveninf i gu r e 21 . Thehydraul ic l inewassimilar to that off igu re 2 and was 342.9 cm (135.0 in.) longwith a 0 .953-cm (3/8- in . )outside diameter and a 0 .12 45-cm (0 .049 - in . ) th ick wall . This line wasrestrained f r o mmoving b y means of rigid clamps at each end. Th reepressure transd uc ers w ere installedin th e line, as indicated inf i gu r e 21. Two ofthese trans du cers w ere located near th einlet tP]}andoutlet fa^) of the muff lers , and the th i rd t ransdu cer (P^jwas located nearthe closed end of the line. For ease of fabrication, the inside diameter of the single expan-sion chamber muf f ler was made somewhat smaller thanth e diameter of the commercia lmuff ler . This necessitated a longer cyl indrical cham ber in order to k eep i ts v olum e equalto that of the commerc ia l muf f l e r .

The ripple generator consisted of a 1.91-cm (3/4-in . ) diameter pistonwhich wasdr iven b y a 133.4-kN (30000- lb ) fo rce e lec t rody namic shaker . For thesetests, th e pistonwasd r iven th rougha constant oscil latory amplitud eof 0.0635 mm (0.0025 in . ) for f r e -quenciesup to 49 0 Hz. Above this f requency , th e force l imita t ionof the shaker combinedwith th e mass of the driving coil restr icted th e maximu m acce lera tion of the dr iving pis tonto 60g.

Resul ts ofmuf f le r tests.- The ripplepressures (s ingle ampl i tud e)of thehydrau l i cline without a muf f ler are shown as funct ions off requency inf i gu r e 22. Thereducedresponse amplitudes apparent at theh ighe r f r equ enc ie sinf igu re 2 2 ( b )are attr ibutable toth e smaller amplitudeof the r ipple-generator piston at these f requencies .

The variat ion of measured pressure f luc tua t ions with f requency for the sys tem withth e commercial damper instal led is shown in f igure 23. As may be seen, th e ripplepres-sure at the muf f ler outlet /P^) isless than that at the muf f le r inlet (P^\ throu gho utth ef requency range . At f r equ enc ie s above50 0 Hz, the pressure pulsations at the muf f le r out-let andnear the end of the closed l ine (Po)are extremely low. Resul ts obtainedwith asingle expansion chamber are givenin f igure 24, andpressure f luc tua t ionsdow nstreamf r o m th e muf f le r are alsove r y low at f r equ enc ie s above600 Hz .

In f igure 25, the rat io ofoutlet pressu re to inlet pressure for both th e commerc ia ldamper and the single expansion chamber are plotted as funct ions off r equ ency . Eitherdamper is seen to provid e appreciable attenuationo fpressure pulsa t ions over a widef requency range . M oreove r ,the frequency range is wider than that of any of theb ranchedsystems considered. T hecom merci al damper clearly provides greater attenuation overa wide r f requency range than does the expansion chamb er . T hecom mercia l damper iseffect ive down to about200 Hz and the expansion chamber to about375 Hz. The sharp


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pressureratio peak at 7 35 Hz for the expansion chamb er is at t r ib uted to the sharp drop inP. a t this f requency , as may be seen inf i gu re 24. Asomewhatsimilar behav ior accountsfor th e peak in P ^ / P - , near 400 Hz for the commercia l damper .& / i

U-TubeC onf igura t ionThe experime nts reported thu s far are concerned with f lu id resonances and their

attenuation. In an e f fo r t to de termine th e e f fec t of tube motiononpressuref luc tua t ions ,th e U-tube conf igura t ionshown in f i g u r e 26 wasassembledwith both fixed and removablepipe anchors as shown. Tests were intendedto de termine whetheror notanchor ingth ehydraul ic sys tem of the pro to type a i rcra f t prev ious ly re fe rred towoulds ign i f ican t lyreduce pressure f luc tua t ions due to pump ripple.

T he U-tube was exci ted by the electrodynamic shaker and ripple generator (testsys tem II in f i g . 2 ),wi th hydraul icf lu id supplied throughth e shut-off valve indicated inth e lowerpart off i g u r e 26. Aluminum tubingo f0.6 35 -cm (1/4-in .) outside diameter and0 .89 -mm (0 .03 5-in .) wall th ick ness was used. O sci llatingpressures were measured a tstations 1 and 2alongth e hor iz onta l legs, andosci l latory pipe def lect ions were measu redat the center of the ver t ica l leg by means of anoptical wedge (def lect ion target) . Testswere conducted withall anchor locations f ixed ( suppor t ed )andwith some anchors removedat the locat ions indicated (unsupported). A s t ruc tura l resonance ofth i s sy s tem oc cu rredat abou t1 67 Hz and a f l u id resonance at 515 Hz.

Pressure f luc tua t ionsar e shown inf i g u r e s27 and 28 as funct ions off requency forth e same input displacemen ts to the ripple generator for supported andunsu pported con-ditions. As can be seen, support ingthe t ub e at the locat ions indicatedin f i gu re 2 6 resul tedin decreased levels ofpressure f l uc tua t i onsin the v ic in i tyo fboth s t ruc tura l and f lu idresonant f req ue ncie s. This is part icularly evident in the low f requency ( s t ruc tu ra l ) r angeo f f i g u r e s 27 and 28. -

Figure 29compares response ampli tud es at the def lect ion target for twoinput rippledisplacements for the conf igurat ion shown in f i gu re 26 in the unsupported condi t ion. T hefact that doublingth e input displacements caused only sma l l d i f f e rencesin response ampli-tudesnear the st ru ctur al resonance suggests that the general level of ampli tudes (andpressures) are high enough to be considered nonlinear.

A N A L Y T I C A L INVESTIGATIONT he acous t ic -wave theo ry .p re sen t edin re f e rence 4 andapplied in r e f e r e n c e 13 to a

hydraul ic l ine with a s ing le-branch resonator is ut i l i zed here in to calculate inlet- to-exi tpe r t u rba t i onpressure andf low ratios for a pipewith more than one b ranchresonator.10


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Basic assumpt ions inc lude f lu id incompress ib i l i ty , th e ex is tence o fundam ped acoust icwaves wi th inthe system, neglect of longitudinal motionof the pipe itself, and negl ig ib leveloci ty of mean f l u id f low re la t ive to f lu id sonic speed.

Straight Pipe Relat ionsIn r e f e r e n c e 4 , d i f fe ren t ia l equat ions o fmo tion relating f l u id pressure and f low pe r -

turbations in a-straight pipe are solved to givejcu(^-x) j c o ( Z - x )

(ZF +Zn]e c + (ZF - Z n ) e cP(x)=U ( 0 ) Z 0iJi-2 >-L^-21- 1(ZE + zo) ec - (Z E -

u ( x ).u ( 0 )(ZE + zo) e - (ZE-zo ) e .

for pressure and ve l o c it y pe rtu r b a t io ns P (x ) and U(x )at any station x alongth e l ine inresponse to a sinusoidalpressure input at one end of the pipe. The posi tive and negat iveexponentials represent t ransmit ted and re f l ec ted waves in the p ipe , respect ive ly . Theinput f o r c i n gf r e que nc y is denoted by w, c is the f l u id sonic speed, I is the length of thel ine, Z E is the imped ance at the dow nstrea m end of the line, and Z o is thecharacteristicimpedance pc w h e r e p is the mass densi ty of the f l u id .

Equ ations (1) and (2) mayalso be wr i t ten asP (x )= PE cosh}g.(i-x)+ Z 0 U E sinh ^.(l - x) (3)Z 0U(x)= PE sinh^ (l - x) +Z0UE cosh&(i - x) (4)

Dividing equat ion (3 ) by equat ion (4 )gives th e d imension less impedance

ITz (x )=-

-^-5H f t _ x) + 2 . Cosh H f t _x )

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If th e pipe is capped at x = I, the f lu id velocity perturbation Ug = 0 ,whenceth e impedancep becomes in f in i te , and equation (5 )reduces toz (x) =coth (l - x) = -j cot g(l - x) . (6)

When x = 0, z(0)= 0 is satisfied b y..) ' (7)

f r o m w hi c h the well-known fundamental organ -pipe f r e q u e n c y f = is obtained w he nn= 1 for a pipe openat one end andclosedat theother. Thecorresponding pressure-perturbation distribution alongthepipe is readily obtained f r o m equation (3)w h i c h , forthe capped pipe, reduces to the simple relation

EVariationsofthisratioand its inversewithf r e q u e n c y are showninfigures 30 and 31 forx = 0, andresonances are indicatedby theuncapped peaks of PTT/PQ at odd integral mul-tiples o f theorgan-pipe frequency.

Branch-PipeEquationsEquationsfor terminal perturbationpressureand f low ratios inhydrauliclines with

straightbranches at right angles to the main pipemay bedevelopedbyapplicationofequations (1) to (4), together wi th conditions o fcontinuityo fpressure and f low at a j unc t i onb e t w e e n the main pipe and a branch. The continuity conditions require that thepressuresin all parts o f the junctionbe equal andthat the mass flow o f f lu id intothe junctionf r o mupstream beequal to the mass f low intothebranch pipeanddownstream portionof themain pipe.

Assuming the branch coordinate to be given b y y a s shown infigure 32 ,equations (3 )and (4 ) may be applied togive

= PB,k ^sh f (hk - y)+Z0UB) k sinh (hk- y) (9)

z 0u k (y ) =PB,ksinh^(hk- y) +z0u B ,k c o s h(hk - y) (10)

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for the kth (k = 1, 2, . . .)branch, wherePg ^ and Ug ^ are pressureandvelocity perturbations at the free end of the branch (pointB infig.32) and hk is the branch length. Withthe branch assumed capped, Ug ^ = 0 andequations (9) and(10) reduce to

pB,k cosh (hk -Z0Uk(y) = PB,k sinh &(hk - y) (12)

At the junctionof the branch and the main pipe, thecontinuity conditions o fpressure andf low may be written as

, , k, l - -4 A,k+ k,2where, as showninfigure32, the subscripts k, 1ident i fy conditionsin themain pipe justupstream of the junction, the subscripts k , 2 ident i fy conditions just downstream of thejunct ion , and the subscripts A ,k ident i fy conditions at thebranch entrance (y = 0 ) . Theinside pipediametersof the mainandbranch pipes are denotedby d and d^, respectively;d iv id ingthrough by thecross-sectionalareaof the main pipe gives equation (14) in some-wha t more convenientf o r m as follows:

Seriesofbranches.- For the single-branch configurationillustrated infigure33,k = 1;terminal pressureandf low ratioscan be obtained in the manner described inreference 13 and may bewritten as shownin f i g u r e 33,where equations (11), (12), (13),an d (15) have been used with k = 1 .

Equations for the terminal pressureandf low ratiosfor thetwo-branch configuration(k = 1,2) showni nfigure 34 can be developedf r o m th e single-branch equations written inth e f o r m

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cosh i. 9+ - tanhi sinhcosh.. c \ j. . (16 )

Lsinh&p + + - tanh cosh cosh

, o j rcosh -, ,

+ - ^ tanh cosh (17)w h e r e Pg,Ug for the s ingle-branch conf igura t ion are replaced b y P Qi,Uoi for thetwo-b ranch sys tem a t the main bra nch s tat ion 2 ,1 jus t ups t ream f ro m the second junc t ion .B e tween th e second junc t ionand the end of the main pipe in f i g u r e 34 , equa t ions (3) and (4)maybeappliedto givetheperturbationratios

2 ,2 .i- =cosh .+-2 sinhc Z c (.18)

Zou2,2 . , j~-3 " o ,T , = sinh + -= cosh -irTT TT ^ (19)

where subscripts 2,2 ident i fy conditions downstream f r o m the second junc t ion . At thebranch entrance station A,2, y = 0 and f r o m equations (11)and (12)for k = 2; ,

A, 2 jo>h2= cosh - T - 2 -PB,2 (20)

> = sinh (21)B,2

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Combining equat ions (13)and (15)wi th equat ions (20)and (21) g ivesP B ,2 _PB,2 P A , 2 _ PB,2P2,l_PB,2P2,2 _PE PA,2 PE PA,2PE PA,2 PE

2 ,2cosh

(22)

ZoUA,2 Zo U A ,2 PB,2E JB ,2 = tanh

2 ,2c PE (23)

0U 2 ,1 /d2f zouA,2 Zou2,2 /d2f 'u -J-= -^ ^ ' + ' = { - } tanhPEZou2,2 (24)

Dividing equat ions (16)a nd (17) each th roug hby P g for the tw o -b ranch co n f igu ra t io nandsubst i tu t ing equat ions (18) , (19) , and ( 2 4 )for the ratios P 2 i/Pgan d Z O U 2j/Pgleads tothe tw o -b ranch terminal-ratio equat ions given in f i g u r e3 4 .

Equat ions fo r te rminal pressure andf low ratios fo r the th r ee -b r a nch co n f igu ra t io n(k = 1,2,3) off i g u r e 35 can be developed in asimilar manne r f r om th e tw o -b ranch equ a -t ions wr i t ten in the f o r m

PQ=P3,l{csh(h+l2+h)+& cosh Z 3)

do tanh cosh sinh sinh sinh

sinh tanh- sinh- sinh

d o] jwhotanh sinh sinh sinh ( 2 5 )

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ZoU0=P3,lsinhH-i| tanh cosh

'd,\2# tanh c cosh tanh cosh sinh + cosh * +

0U35 coshtefa+12 + h)+ tanh cosh sinh ^ +

T /t a n h c sinh2

a.- tanh-icosh- sinh-^ +coshIgfa + 12]

(26)where Pg,Ug for thedouble-branch system are replaced by P$ 1>^31a*station 3,1justupstream f r o m the third branch junction. Betweenthe third junctionand the end of themain pipe infigure 35,equations (3) and (4) mayagain beapplied togive the ratios

3 , 2 = Cosh+ -. sinhc ZE (27)

Zou3,2__=sinh cosh (28)

wh ere t h e subscripts 3 ,2 ident i fy conditions just downstreamf r o m th ethird junction.Proceeding in the manner used for theone- andtwo-branch pipes, the f low relation f o rthe third junction analogous to eq. 24))is foundto be

^jtanh c 3 ,2 ( 2 9 )Divid ingequations 25)and 26) each by Pg for thethree-branch configuration andsub-stituting equations 27), 28),and 29)for theratios Pg jP and ZU. /P leadsto theequations given infigure 35 .

andZoU.j

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Branches at acommon junction.- The same procedure describedin theprecedingsection can be applied to the system with more thanonebranchat the same junction withth e main pipe. In the notation offigure 36, the continuityrelations ofequations 13)and 15)may bewritten as

(30)

M .2< \-f UA ,m + U D (3D=1X '

For the mthbranch

PB,mZ Q U A ,m

PB,m

(32)

=sinhq S 33)

Applyingequations (3) and (4) to main-pipe sectionsoneither sideof the junctionresultsin th efollowing:

34

Z U -35

PD u JwZ 2 Zo =cosh- - +- S

Z0Un 0 D =sinh ^+7r -cosh 37)PE c ZE

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From equations (30)to(32),PR m PR m PA rn PR m Pn 1 Pr>

'E ^A,m ^E - " "

E B ,m E E

PE -m =l

Equations (34)and (35)ma y b ewritten as

(40)

Z0Un Pr jwZ-, Z0Ur jwZ1-^2_Q = Csinh_1 o_Ccogh;L_1 42PE PE c PE c

Substitution ofequations (36), (37),and (40) into equations (41)and (42) gives th e equationsshown infigure36 .

Analyt icalResultsPerturbation terminal pressureratioswere calculatedf o r each o f thebranch-

resonator configurations tested in theexperimental program andalso fo r some conceptualconf igu rat ions. In a l l calculations, th e downstreamend of themain pipewas assumed tob eclosed, whencethe impedanceZg is inf ini te atthis point. Thus,the terminal pressureratios are given onlyby therealparts of the equationsi nfigures 33 to 36 for PQ/PE> anc*uncapped pressureratiosoccur at particular frequenciesin theplots offigures 30, 31,and 37 to 50. The speed of sound in the hydraulicf luid was chosena s 1.237 km/sec(4 8 700in./sec).

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Direct comparison between analytical and experimental pressure ratiosis not gen-erally possible becauseof the large pressure amplitudesof the test programin contrasttotheanalytical pressure amplitudes, whichare linear. However, approximate agreementis indicatedinexperimental andanalytical frequencies for resonances (|PE/PO| ~~ )anc*maximum attenuations ( PE/PQ| =OJ- In order to facilitate these comparisons, howevertenuous, figure numbersof the analytical plots are matchedwith thecorrespondingfigurenumbersof the experimental plots in table 1. -

Mainpipewithoutbranches.- Aspreviously noted,the pressure ratios infigures30and 31 for test systems I and IIwerecomputed from the simple cosine relation ofequa-tion (8) with x = 0. Within a 100-Hzfrequency range, analytical resonancesoccurred forsystem I at 203 Hz and 609 Hz and for system II at 88, 264, 440, 615, 791,and 968 Hz.Comparisonofterminal pressure ratios infigure 30withthe. nearly corresponding experi-mentalratios from figure 3 for test system Ishows fair agreementbetweenexperimentalandanalytical trends for frequenciesupthrough the first resonance andbeyond butpooragreement near andabovethesecond resonance. Asomewhatsimilar observationmay bemadefor test system IIwhichhad the same numberofexperimental andanalytical reso-nantpeaksup to 900 Hz anddeteriorating agreement between experimentaland analyticalresonant frequencies above the first resonance, as is evident by comparing figures 8and 31. These discrepancies may bepartially due to the fact that more precise valuesofresonant frequencies were not actually determined either for these testsor. for testsof thebranched configurations. Despite these deficiencies, theoverallfluid resonant responsebehavior ofthese pipeswithoutbranches seems wellenoughunderstoodtopermit someappraisalofattenuationcapabilitiesof the branched systems.

Mainpipewithbranches.- Variations ofanalytical pressure ratioswith frequencyfor the single -branchtest systems are presented infigures37 to 42. Becausethe samebranch length was used in all these cases, maximum attenuation of the pressure perturba-tion occurredat 406 Hz, as indicated by the generally discontinuousrise in PQ/PE anc*corresponding zero valueof |PE/PO| a* ^s frequency. For anequilateral system, suchas shown infigure 38, the minimumvalue of |PE/PO| *s ^/ rather than zeroowing toth e finite limit of the product ta n = s in cos --=for h^ = Zj = 1 % in the equation f or

giveninfigure 33. Moreover, the attenuation ofthis system clearly remains nearlyat a relative maximum(|PE/PO ~^/^)over a wide range offrequencies(about 200 Hz) incontrast to the other single -branch configurations forwhichpressure perturbations areattenuated over narrower frequency bandswith oneresonant frequency very close to thediscrete frequency ofmaximum attenuation (|PE/PO| = ;' e e ec *branch locationonthisresonant frequencyappears to be opposite for the twotest systems, that is, fortest system IIwiththebranch near an end of themain pipe (figs. 39, 40, and42), thisclosest resonance tendsto befarther removed from thefrequencyofmaximum attenuation

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than with the branch near the middle ( f ig. 41) , whereas areverse tendency is indicated fortest system I ( f igs . 37 and 38) .

Thee f f ec t s onanalyticalpressureperturbationsofadditional branches inseriesare shown in f i g u r e s 43 and 44(cor respondingto f igs . 6 and 7 for experimental results).Resul ts for test sys tem Iwith tw o branches appear in f igu re 43 and show f r e q u e n c i e s ofmaximum at tenuat ion at 406 Hz and 609 Hz corresponding to the longer andshorter of thetw obranc hes , respect ive ly . At tenuated pressure per turbat ions are indicated over af requency range ofabou t300 Hz interrupted by a resonance at 485 Hz. The presence of ath i rd branch extended th e r ange even f a r the r , as f i gu r e 44shows. H ere , withthe th i rdf requency ofmaxim um attenuation equal to 72 5 Hz, reduced values PE/PO occu r rednear 406 Hz andf r o m about850 to 950 Hz, and PE/PO =fromabout60 to 75 Hz-This last un in te r rup ted150-Hz f requency range ofmaximum a t tenuat ion resemb les asimilar exper imental behavior shown in f igu re 7 . These results suggest thatpressureattenuat ions can be extended over a wider f requency range b yadding more b ranchesofdi f fe ren t lengths. Apossible extrapolation ofthis idea is alarge n u m b e ro f closely spacedbranches d is t r ibu ted a longthe main pipe as suggested b yRober t J. B ladeofNASA LewisResearch Center in of f icial correspond ence. Indeed,he and Car l M .Hol land ( re f .1 4)show th e e f f ec t i venes sof aspecial dis tribu ted-branch system in reducingpressure per-turbat ions to a v ery low cont inuous level over a maximu m f requency r ange .

Results for twoequal b ranche s at a common junct ion are shown in f igure 45 . Hereth e ef fect of a smaller bran ch diameter th an main diameter is inclu ded , and the analyt icalpressure per turbat ions are ve rysimilar to those shown inf igure 41 for thes ing le -b ranchconfiguration with equal branch andmain-pipe diame ters . These results, inaddit iontoanother to be discussed, suggest that mult iple b ranch es at a common junction with th emainpipe may not b e,as e f fec t ive pressure a t tenuators as b ranches inseries.

Resul ts ofo ther b ranch edsystems.- P res su re -per tu rba t ion ratios were also cal-cu lated for a nu mb er of o ther b ranched sys tems fe atur ingarbitrary var iat ions in b ranchlength, locat ion, and cross-sect ional diam eter . Sampleresults of these calcu lat ions arepresented in f i g u r e s 46 to 50 and give some indicat ionof the sensitivity of sys tem at tenu-ation totheseproperties.

Figures 46 to 48show theoret ica l pressu re ratios for par t icu lar two- and t h r e e -branch combinat ionsi nseries. Compar isono ff igure 46with f igure 44 indicates thatholding the branch length the same in a three-b ranch conf igurat ion had the ef fect of shif t ingthe max i mu mpressure-ratio at tenuat ions to anotherf requency ban d. That is , with eachbranch length equal to the longest branch lengthin f i gu re44( 7 6 .2cm (30 .0 in . ) ), minimu m|PE/PO| values moved f rom th e 6 0 0 -to 750-Hz range inf i gu re44 to the 4 0 0 -to 600-Hzrange in f igure 46 . Variat ions ofpressure rat ios with f requency for sys t ems with b r anches

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closer to the exit (or capped)end of the main l ine (denotedb ypoint E)thanto the inlet end(denoted b y point 0) are shown in f i g u r e s 47 and 48. Inf i g u r e 47, the two-branch config-urat ion off i g u r e43 has simply been shifted towardthe capped end so that 3 is 15.2cm(6.0 in.) f r o m point E, andcom parison of the two f igures shows general ly bet ter at tenuat ioncharacteristicsb ecause ofthis change. The resonance of 591 Hz in f i g u r e 47appears tob e sharply def ined and maythe re fo re nothaveas adverse an e f fec t onpressure attenuationas resonances extending over wider f req ue ncies. Somewhatsimilar e f fec t s are evidentforthe th ree-branch sys tem of f i g u r e 48, in which a combinat ionof three closely spaced ,b ranche s of equal leng th are located close to point E and there are two sharply def inedresonances at 355 and 465 Hzwithinthe f requency bandof highest attenuation. O therresults (notshown) indicatesmaller f requency bands of at tenuatedpressureratios withthis b ranc h co mb ination close to the inlet and near the m iddle of the main pipe.

Figures 49 and 50 showth e e f fec t s ofincreas ing the n u m b e rand the d iametersofbranchresonators located at a common junc t ion on the main pipe. Thepressure responsecharacteristics in both f ig u res are clearly similar to those inf i gu re 45. Comparison off igu res 49 and 45 shows that redu cedpressure ratios |Pg /Pgl were obtained over sl ight lywider f requency ranges near the maximu m at tenuat ionf requency as the n um b e rof brancheswasincreased. Figure 50reveals asimilar e f fect o f increas ing b ranch d iameters , witheven lower valueso f over wider f requency bands interruptedbymore sharplydefined resonances. '

C O N C LU D I N G REMARKSAn investigation off lu id resonance and its at tenuat ion inh igh -p ressure hy d rau l ic

lines is reported. T he studyw as aimedat unders tand ingf lu id v ib ra tions insimpli f iedhydraul ic sys tems andev aluat ing tw od i f fe ren t types offluid pulsation dampers, or at ten-uators. O ne of these ty pes featu red one or mo re closed-end tubes, or standpipes, br anc h-in g f ro m a main pipe, and the other typewas a f lu id muf f le r installed in the main pipe line.In addi t ion, certain st ructural - f luid interact ions were explored in a closed -end U-tub e.M o stof the study involv ed fo rce d vibrat ion testing, with convent ional hy drau lic f lu id excitedby a 1000-Hz range ofoscillatory disturbances at one end of a system that was closed at theother end. Limited applications ofacoustic -wave theory to the b ranched sys tems are alsoincluded.

Results for the branched systems show varying at tenuat ions ofpressure pe r t u rba -tions, depending on the n u m b e rand location of branches alongth e main hy drau l ic line.Maximum reduct ions inpressure perturb at ions fo r the single -branch conf igu rat ion we reobtained with th e branch near th e main inlet or near th eexit. These reduct ions weresomewhatless b u tstill significant when the b ranc h was located n ear the middle of the

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main pipe, and they occurred over l imited f requency ranges. Increasing th e n u m b e rofbranches along the main pipe f ro m one to three resulted inlargepressure attenuationsover wider f req ue ncy ranges, and thelargestf requency range containing max imu m atten-uat ion was obtained for three branches ofd i f f e ren t lengths at relat ively separated posi-t ions along th e main pipe. The use of more thanoneb ranch at the same locationdid notproduce as much at tenuat ion as did b ranches inseries (i.e., at d i f f e ren t locations alongmain p ipe) . M ount ingan accumu la tor at the exit end of the main pipewithout b ranchesresulted in fewer high-level resonant peaks than withthe exit closed, bu t nosigni f icante f f ec t s of the accumula tor onpressure attenuations couldb e ob served for the branchedsystems.

T he m uf f l e r vibrat iontests showed that e i ther a com mercial d amper with an intr i -cate internal flow arrangement or a single expansion chamber of equivalent volume pro-duced attenuated pressure per turba t ions over wider f requency range s than those of thebranched sys tems. The f requency range of the comm ercia l damper w as a li t tle largerthan that of the expansion c hamb er.

Vibrat iontests of the U -tub e conf ig urat ion demon strated th e need for adequatelyanchoring the tube st ructure to reduce the levels ofpressure per turba t ions near struc-tura l and-fluid resonant f requencies .Langley Research Center,

Nat ional Aeronaut ics and Space Administrat ion,Hampton, Va., June 15,1973.

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R EF ER ENC ES1. Kinsler , L aw renc e E. ; and Frey, Au st in R. : Fu ndam entals of Acou st ics. Second ed. ,

John Wiley&Sons, Inc. , c .19 62 , pp . 186-216.2.Streeter, Victor L. , ed .: Handbook of Fluid Dy nam ics. First ed ., M cG raw -Hi l l Bo o k

Co., Inc., 1961,pp.20 -1 - 2 0 - 4 7 .3. Kel le r , George R .: Hy drau l ic Sy stem Analys is . Hy drau l ics &P neu mat ic s M ag .,

c . 1 969 .4. Regetz , John D ., Jr . : AnExper imenta l D eterminat ion of the Dy namic Respo nse o f a

Long Hydrau l ic Line . NASA T N D-576 , I 9 6 0 .5. D 'Souza , A. F.; and Oldenburger , R.: Dy namic Response of Fluid Lines. Trans.

ASME, Ser. D: J. Basic Eng., vol . 86, no. 3, Sept. 1964, pp. 589-598.6. Jarski, E. J.: D y n a m i c so fViscous Fluid O sci l lat ions in Hydrau l ic Lines .NSRDC-3201, U.S. Navy, Jan. 1970. (Available f rom DDC as A D - 86 45 95 . )7. Strunk, Richard Dean: Longitud inal Wave P ropagation in Liqu id Lines of Fini te

Length Inc l u d ingth e Ef fec t of Nonl inear Boundary Condi t ions . P h . D.Thesis,Univ. ofIllinois, 1969 .

8 . Ray, G autam: Fini te Eleme nt Analy sis M ethodsinD y n a m i c s ofPulsatile Flow.Ph . D.Thesis, Pennsylvania State Univ. , 1969.

9 . Holland, Carl M .; B lade, Ro bert J .; and Do rsch, Rob ert G. : Attenu at ion of SinusoidalPerturbations Super imposed on Laminar F lowof a Liqu id in aLong Line . NASAT N D - 3 0 9 9 , 1965.

10 . Z ie lke , W. : Freque ncy -Dependent Fr ic t ion in Transien t P ipe F low. Trans. A S M E ,Ser. D: J. Basic Eng. , vol . 90, no. 1, M ar . 1968, pp. 109-115.

11. Chang, S. S. L.: Transien t E f f e c t s of Supply an d Connect ing Condu i tsinHy d rau l icControl Systems. J. Frankl in Inst . , vol . 262,no. 6 , Dec. 1956, pp.4 3 7 - 4 5 2 .

12.G er lach , C. R.; andParker, J. D.: Wave P ropagat ion in Viscous Fluid Lines IncludingHigher M o d e E f f ec t s . Trans. ASME, Ser. D: J. Basic Eng., vol . 89 ,Dec. 1967,pp. 782-788 .

13.Lewis, W il l iam; B lade, Ro bert J.; and Dorsch, Robert G .: Studyof the E f f e c t of aClosed-End S ide Branch onSinusoidal ly P er turb ed Flow ofLiqu id in a Line .NASA T N D - 1 8 7 6 , 1 9 63.

14. Blade, Robert J. ; and Holland, Carl M .: Attenuat ion of Sinusoidal Disturban ces inNonviscous Liquid Flowing in a Long Line With Dis t r ibu ted Leakage . NASAT N D - 3 5 6 3 , 1966.

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15.Davis, Don D., Jr.; Stokes, George M;Moore, Dewey;and Stevens, GeorgeL., Jr.:Theoretical andExperimental Investigation o f M u f f l e rs Wi thCommentsonEngine-Exhaust M u f f l e r Design. NACA Rep. 1192, 1954. (Supersedes NACA TN 2 8 9 3 b yDavis, Stevens, Moore,andStokesand TN2 9 4 3by Stokes andDavis.)

16. Gebhardt, George T.: Acoustic Impedance Measurements of Liquid Filled PipesConta in ing Lossy, High-ComplianceInserts. D-15871, Boeing Airplane Co.,Oct. 15, 1954. - -

17 .Hewlett,James T .: Applicationsof NASTRANto Coupled Structural andHydrodynamicResponses inAircraft Hydraulic Systems. NASTRAN: Users'Experiences, NASAT M X -2 3 78 , 1971, pp.407-419.

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T A B L E 1. - INDEXTOPLOTS.

TestsystemI

IIII

IInIIIII

I

n

Conf igura t ionNobranches; I =152.4cm (60.0 in.) :Nobranches; I =351.8 cm (138.5 in.)Singlebranch; l = 15.2 cm (6.0 in.)Singlebranch; ^ =76.2cm (30.0 in.)Singlebranch; l =30.5 cm (12.0 in.)Singlebranch; 7i =99.4 cm (39.1 in.)Singlebranch; l =175.3 cm (69.0 in.)Singlebranch; ^ =251.1 cm (98.9 in.)T w obranches; Z j =15.2 cm (6.0 in.);

1 2 =6.1.0cm (24.0 in.)Three branches; l = 12 = 15".2 cm (6.0 in.);

. Z3 =45.7cm (18.0 in. ).T w o co mmo nbranches; Z j =175.3 cm (69.0 in.), wi th 11 = =0 . 6 5d d

FigureAnalytical

3 0 -3137383940414243

44

45-

Experimental. 3

854

a8a9

9 , a610, a20

6

7

. 13, a7

Accumulator open at downstream end; accumulator closed (i.e., downstream endclosed) for all other experimental figures intable.

:'25


28/80

J-C M -

toBF -i< Uf- 9 s ff [ ,v

pp5od>

rH 3

aH OWS HCo vo^- i-P(UwtoIUJnC M

C-ir >_*

it

i

i

_ ^ -CO

o, -3r wiTH

OOH '

^""

-12

12U

(6

6)

6

luc-hsyemesyemIAdmnoncnmes

Bracreooco

oasaoABCanD

1 J T-(m-^ P0 ^3 Q;

..J. -3 ' 1 1 S TOi rt oJJ. _ _ Ld

& "S-T O)1 ' ^ oI 1 ^ X. a *i i - ^ - ^- L-- 1- , - - ' - | r t 0 31 1 5 ^ "S0 M ci

H /' 10) S i^3 -d I 0M 01 ' tn^ J ^ 3 S 8 5 . S P^ H.S \^a x H \H +J >it) -H O^

2 6


29/80

HHHH a 3i M avo i 3-3- 1TN O- OO N oj


30/80

M W / m psi800

5.0coH >


31/80

u o

3.0

2.03,r

1.0

O

o,- Branchvl) resonant

region_ /

H P| f l100 500

v W

o00O

1000Frequency, H z

Figure 5 . -Pressure ratio versusf r equency fortest sys tem I (fig. 1 ) ;single-branch configu ration with 76.2 -cm (30.0 - in.) b ranch atstationA.

2.0 r-

1.03,r

0 OO BranchresonantQ regiono

Q0

100o

500 1000Frequency, Hz

Figure 6 .- Pressure ratio versusf r equency f or test system I ( f ig .1 ) ;two-b ranch configu ration with 76 .2-c m (30.0- in.) bran ch at s tat ion Aand 50.8-cm (20 .0- in.) branch at stationC.

2 9


32/80

2.0

ao%- po

COen0 )ft a< uft-p

a0)

3,r

00 Branch 0-

, oregion-r

Lc / -| "' < ? > i kv:v-:v-T)01 (?) i

500Frequency, Hz

1000

Figure 7.- Pressure ratio ve r sus f requency for test system I ( f ig. 1);th ree -b ranch configuration with 50 .8-cm (20 .0- in . ) b r a n c ha tstation A, 76 .2-cm (30 .0- in . ) branch at station B, and42 . 6 - c m(16.8-in.) b r a n c hat stationC.

psi1 * 0 0r

2.5

2.0

, t1

- 5

300

200

100 -

00

0

00 O0 o

o

( > 0

I100 500

Frequency, HzFigure 8 .- Peak - to-peak p re ssure f luctuat ion ve r sus f r eq u en cy f o r

test system II ( f ig .2)with nob ranches; accumu la tor c losed.

1000

30


33/80

2.0 r-

i .o _' 3 , ro

< ? oG

100

0O oooJ 150 0 1000

Frequency, HzFigure 9 . - Pressure ratio v e r s u s f requency f o rtest sys tem II ( f ig . 2 ) ;

s ingle-branch conf igurat ionwith 76 .2 -cm (30 .0 - in . ) b r anch atstat ion C; accu mu lator closed.

2.0-r-

' 3 , r 1.0 _

" '- - 00 Q 0() Q 000 ,0Q

1 { (jt\rJ/100 500

|1000

Frequency, HzFigure 10. -Pressureratioversus f requency for test system II( f ig .2 );s ingle-branch conf igurat ionwith 76 .2-cm (30 .0- in . ) b ranch at s ta-t ion D; accumulator closed.

31


34/80

2 .0

i.o- 3 , r

o0 o > o oo1

100 5 0 0F r e q u e n c y j Hz

1000

Figure 11.- Pressure ratiove r sus f requency fortest system II ( f ig .2);single-branch configuration with 76 .2 -cm (30 .0-in . ) b ranch inlet atstation C; 90bend in the branch near its inlet; accumulator closed.

2 .0 r-

l .O' 3 , roo 000

O

I100 500

F r e q u e n c y , Hz1000

Figure 12.-Pressure ra t io versus frequency for test system n (f ig. 2);s ingle-branch configuration wi th 76 .2-cm (30 .0 - ih .) b rancha t sta-tion C;rou nded b ranch in le t ; accum ula tor c losed.

32


35/80

2.0 i-

I100 500Frequency, H z

1000

Figure 13.- Pressureratio ve r s u s f requency f ortest sy s tem n ( f ig . 2) ;two-common-branch configuration with 0 .635-cm (1/4- in . ) outs ide .diameter , 76 .2-cm (30 .0- in . ) b r a n c h e sat station C; accumulatorclosed.

*onao

utn0)ft a

a< uPM

M N / m3 . 0 ,

2.0

1.0

psii t O O

2 0 0

O..,

o

0o

100 5 0 0Frequency, H z .

I1000Figure 14. - Peak- to-peak pressure f luctuation v e r s u s f requency f ortest

sys tem II ( f ig .2) with nob ranches ; accumu la toropen.

33


36/80

' 3 , r

3.0

2.0

1.0

h 0_ O

100

oo

o 000oo60I

500Frequency, H z

1000

Figure 15.- Pressure ratio versus f requency f or tes t sy stem II ( f ig . 2) ;s ingle-branch configurat ion with 76 . 2-c m (30 .0- in . ) b ranch at s ta-t ion C; s tandard T-joint; accumulator open.

3 4


37/80

" 3 , r

3.0

2.0

1.0

o_ o o

oo ooo

}0oI

100

o oo o

100000Frequency, H z

Figure 16.- Pressure ratioversus f requencyfortest system n (fig.2);single-branch conf igu rat ion with 76 .2- cm (30 .0-in .) b ranch a t sta-tion C; standard T-jo int ; 9 0bend at branch inlet; accumulator open.

3 5


38/80

3.0 T-

2.0 _

' 3 , r1.0I-

^ OOO

\

00 0 "N >- -^ y^~N. X**N v^XO d? O 00 0000

100 100000Frequency, Hz

Figure 17.- Pressureratio versus f r e q u e n c yf or test system II ( f ig . 2 ) ;two-comm on-b ranch configu ration with 0 .635-cm (1/4- in . ) outsidediameter, 76.2-cm (30.0-in.) branches at station C; accumu latoropen.

t . u

3.0

P2- 2.03,r

1.0

0

OO :o . . . - o

r ~ \ C ^ \ - '\^j ^/o . ' oo oo0 G

100 1000

36

500Frequency, H z

Figure 18.- Pressureratioversus f r equencyfortest sys tem II (fig. 2 ) ;s ing le -b ranch configuration with 76.2-cm (30.0- in.) branch a t sta-tion A; accumulator open.


39/80

I3,r

k.O

3.0

2.0

1.0

_QQ

O

100

o 0O

oo o

0 o oO

500Frequency,Hz

I1000

Figure19.- Pressure ratio versus f r equencyfortest system II ( f ig .2 ) ;s ingle-branch configuration with 76.2-cm (30.0- in.) branch a t sta-tion B ; accumu lator open.

37


40/80

3.0

' 3 , r

1.0Loo

o

100

0oo

oo

o

o

o

0

500Frequency, H z

1000

Figure 20 . - Pressureratioversus f requency fo rtest system II ( f ig . 2);single-branch conf igurat ion with 76 .2-cm (30 .0-in .) branch a t sta-tion D; accumulator open.

38


41/80

tO J

Ot

OJPH

asbC af- fa

ftftH

0a -* 2A.l 4,

--f 1

__ . B.2*~d2

7

A.2 3 If 11.1 1.2 Ld 2 , 1 - 2 , 2

=cos - tan sin cos

cos ui wn i . wti . wto , . . . /_1 tan-glsin- 1sin-J -sinf fa+

J sin - tan sin sin

do A .lan- Sln

E ^E - fel tan> d cos sin

H^iltan o sin d / tan c cos- sin , . , x.Cos Ufa+L2 )

J 1


54/80

B , 2

U. L *1 A 1rorce o . n

~

"

r~h i 7 2 i - A ;i ? . AJ ?i

f ~ -, '

^~ B.3 '-^,

h2L *.J A.3

2.2 11"

'

-*ih.3

7, . . 4 nl

12 E

- tan sin cos

tan u) h? tan 2*1sin2*1sin2*1. sin

tan- cos ... . .sin-i sin Z+ ' sln

sin cos

4) - tan l.sin sin

2 2whr t / d - i \ . .ta n-2 tan sin- sl n- -si nd o w oT tan-1 sln 2 - whi oiZi ,, ,ta n-1 sin-1 sin z +

tan sin sin

,2 I, ,2 Wh i l-l Win .. . .1 ,+l-fIt a n - j r ^ H - r l tan -^ cos -^ sin -^ - cos(, +Z , ) | l s i n(

; 2 . i , 2+ \ - f- itan,- sin | |tan cos^ sing i f t

sin- sin-

j s i n tan cos

j^j +I2 )co .

cos cos sln

sin2-a cos 2*1

Figure 35 . - Geometryandte rminal per tu rba t ion pressureandveloci tyratiosforthree-branch configuration.

52


55/80

D B . 4

sin^ - sm smM . ,2

m=ltan

ZoU0_ zoPE ZE COS %(li + - cos sm

M . x 2dmL \"d~m= l A_"* . / , \ CiV dm\ * u hm> -22- tan-.r21-

^ d - \ + to + c o sM ,2drz=l m tan

Figure 36.- Geometry and terminal perturbation pressure andvelocityratiosfor conf igura t ion with several branchesata co mmo nlocation (e.g., inabove sketch, M= 4).

53


56/80

1 0

200 400 600Freq uency , Hz

800 1000

(a) Inlet- to-exit pressure rat io.Figure 37 . - Terminal per turba t ionpressureratios f ortest sys tem I with single

b ranch close to the inlet;j = 15.2cm (6 .0 in . ) ;2 = 137.2cm (54.0 in . ) ;h =76 .2cm (30 .0 in . ) .

54


57/80

2 0 0 400 600Frequency, H z

(b ) Exi t - to- in le t pressu re ratio.Figure 37.- Concluded.

800 1000

5 5


58/80


800 1000

Figure38.- Terminal perturbation pressure ratios fortestsystem Iwithsingle -b ranchequilateral conf igurat ion; l-^ =1% = hj =76.2 .cm (30.0 in.).

56


59/80

10

4-3-

2-

200 400 600F r e q u e n c y , Hz

1000

(a)Inlet- to-exit press u re rat io.Figure 39 . - Terminal per turba t ion pressure ra t ios fortest sys tem IIwith single

branch close to the in le t ; j =30 .5cm (12 .0 in . ) ;Z2= 321 .3cm (126.5in . ) ;hj =7 6 . 2cm (30 .0 in . ).


60/80

H*

2-

200 400 600Frequency,Hz

(b ) Exi t - to- in le t pressure ratio.Figure 39 .- Concluded.

800 1000

58


61/80

0 200 400 600 800 1000Frequency, H z

(a) Inlet-to-exit pressu re rat io.Figure40. - Terminal per turbat ion pressure ratiosfortest system IIwith single

b ranch ; 11 =99.4cm (39.1 in.); 12 =252 .4 cm (99.4 in.);hj =7 6 . 2cm (30 .0in.).

59


62/80

H - < 2

I200 400 600Frequency, H z

(b ) Exi t - to- in le t pressure ra tio .Figure 40 . - Concluded.

800 1000

60


63/80

2 0 0 400 600Frequency, Hz

800 1000

(a) Inlet-to-outlet pressureratio.Figure 41. . - Terminal perturbation pressure ratios fortest sys tem IIwith

single branch abouthalfway fr om inlet to exit ;Z j = 175.3cm (69 .0 in . ) ;1 2 = 176.5c m (69 .5 in .) ;hx =7 6 . 2cm (30 .0 in . ) .

61


64/80

400 600Frequency, H z

(b ) Exit-to-inlet pressure ratio.Figure 41.- Concluded.

800 1000

6 2


65/80


800 1000

(a) Inlet-to-exit pressureratio.Figure 42. - Terminal per turba t ionpressure ratiosfortest sys temIIwith single branch;Z j =251.1cm ( 98 . 9 in .) ;Z 2= 100.6cm (39 .6 in .) ;hj =7 6 . 2cm (30.0 in . ) .

63


66/80

2-

200 400 ' 600Frequency, H z(b ) Exit- to-inlet pressure rat io.

Figure 42..-Concluded.

800 1000

64


67/80

1 0

, o 'h l

h2

E

200 400 600 800 1000Freq uency , Hz

(a) Inlet- to-outlet pressure ratio.Figure 43 . - Terminal per turb a t ion pressure ratios f o rtest sy s tem Iwithtw o

branches ;l = 15.2cm (6 .0 in . ) ;12 =61.0 cm (24 .0 in . ) ;Z3=7 6 . 2cm (30 .0 in . ) ;hj =76 .2cm (30 .0 in . ) ;h2 = 50.8cm (20 .0 in . ) .

6 5


68/80

0

h l h 2E

200 400 600F r e q u e n c y , H z(b ) Exit-to-inlet pressure ratio.

Figure 43.- Concluded.

800 1000

6 6


69/80

^o1hTh - 1112

h2

"" I "

I hI 3J 4 "

200 400 600Frequency, H z 800 1000

Figure 44. - Terminal per turba t ionpressureratiosfortest sys tem Iwiththree branches; l\ = lz= 15.2 cm (6.0 in .) ; 3 =45.7 cm (18.0 in.); 4 =7 6 . 2cm (30 .0 in . ) ;hj = 50.8cm (20 .0 in . ) ;h2 =7 6 . 2cm (30 .0 in . ) ;h3 =4 2 . 6c m (16.8 in.) .

67


70/80

400 600Frequency, H z

800 1000

(a) Inlet- to-outlet pressure ratio.Figure 4 5 . - Terminal per turba t ion pressure ratios for test system II

with tw ob ranches ; l = 175.3cm (69 .0 in . );2 = 176.5cm ( 6 9 . 5 i n . ) ;hj = h2 =7 6 . 2cm (30 .0 in . ) ;- = J. =0 .65 .

68


71/80

0

an investigation of hydraulic-line resonances and its attenuation - nasa - john l sewall david a...

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