A Breadboard Fluidic-Controlled Pneumatic Stepper Motor

7/30/2019 A Breadboard Fluidic-Controlled Pneumatic Stepper Motor

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S A T E C H N IC A L N O T E N A S A T N- Ae. 1

D-4495-

LOAN COPY: RETURN TOKIRTLAND AFB, N ME&AFWL (WLIL-2)

y WiZZium S. Grz@nReseurch CenterCZeveZund, Ohio

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A E R O N A U T IC S A N D S PA 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 . A P R I L 1968


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TECH LIBRARY KAFB,NMI111111111111111111111IIHIllllllln111Il1

A BREADBOARD FLUER IC-CO NTRO LLED PNEUMATICSTE PPI NG- MOTOR SYSTEM

By Wil l iam S. Grif f inL e w is R e s e a r c h C e n t e r

Cleve land , Ohio

NATIONAL AERONAUTICS AND SPACE ADMINISTRATIONFo r s a l e by the C lea r inghouse fo r Fede r a l Sc ien t i f i c ond Techn i ca l ln fo rmo t ion

Spr ing f ie ld , V i rg in ia 22151 - CFSTI p r i c e $3.00


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A B REA DBOAR D FLUERIC CONTROLLED PNEUMATIC STEP PING -MOTOR SYSTEMby William S. Griffin

Lewis Research CenterSUMMARY

This repor t descr ibes NASA Lewis developed flueric' cir cui try used to drive a novelpneumatic stepping motor. The design and breadboard implementation of the circui tryare presented along with so me of the techniques used for interconnection of digi tal fluidjet ampli fiers. The experimental performance of a breadboard flueric-drive-circuitry -stepping-motor actuator system is evaluated. Finally, a comparison is made between theresult ant pneumatic stepping-motor system and the mo re conventional pneumaticpiston-in-cylinder actuator.

The principal conclusions of the work ar e that(1)The NASA flueric-drive-circuitry - stepping-motor combination constitutes a

reliable, fast, open-loop digi tal stepping actua tor sys tem which has high resolution andoutput stiffness. The breadboard flueric-drive-circuitry - stepping-motor systemcould be stepped at 173 steps per second in either direction and cyclicly reversed,without missing steps, at 115 steps pe r second. The step size of the shaft output motionwa s 0.25' and a maximum sta tic output torque of 70 inch-pounds force (788.9 cm-N) wasobtained.flow than a conventional elect ropneumat ic piston-in-cylinder actua tor designed to do thesa me job.

(2) The flueri c-drive-circuitry - stepping-motor actuator system consumes more

INTRODUCTIONThe advent of the nuclear engine fo r rocket vehicle application (NERVA) created a

demand for actuator sys te ms which are simple and reliable, have a minimum of moving'Fluerics is the te rm adopted in July 1965 by the Government Fluid Amplifier

Coordination Group to describe a general class of fluid devices, such as fluid jet ampli-fi er s, vortex amplifiers, turbulence amplifiers, etc.rep or t to denote suc h devices.

Fluer ics is used throughout thi s


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parts and sliding surf ace s, and can withstand a high surroundin g nuclear radiation field.As a result of these requirements, a novel pneumatic stepping motor was developed bythe Bendix Corporation under NASA contract NAS 3-5214. The motor, which operates onthe nutating gear principle, can be operated either digitally or in analog fashion. Devel-opment of a complete actuat or system , however, require d synthesis of means fo r ad-mitting the various working pre ss ur es to the stepping m otor. Preferably, the devicesfo r introducing working pre ss ur es should be sim ple and at least as reliable as thestepping motor itsel f. Because of the radiat ion level at the actuat or location, the ad-vantages of fluid am pl if ie rs fo r this purpose appe ared obvious.

Several approaches have been taken to develop a fluid-amplifier drive circuit forthe stepping mot or. Under NASA contrac t NAS 3-5214, a n analog ci rcui t composed ofvortex amplifiers w a s developed. Although this ci rc ui t had the advantage of being pro-portional, thus enabling the motor to be re ve rs e driven under excessive load torques,it proved much slower than specified (ref. 1). An alternative method for driving themotor w as proposed by Blaiklock (ref. 2). Blaiklock's circuit is digital and open loop indesign and u ses fluid jet amplifiers rat he r than vortex amplifiers. Thus, it is expectedto be faster than the analog cir cu it of refe renc e 1. However, Blaiklock's ci rcuit req ui ressuccessful development of a n axisym metri c, trista ble fluid jet amplifier for its success-fu l implementation.

The approach described i n this rep ort was to design and develop a high-speed, open-loop, digital stepping cir cu it which could be eas ily syn thesized fro m fluid jet amplifiersof conventional design.motor drive circuitry (ref. 3), its perform ance, and the perf ormance of the completedrive-circuitry - stepping-motor actuator system are descri bed herein. The nomen-clature, som e of the techniques used fo r interconnection of the elements, and details ofthe design of cer tain su bassem blies are presented in appendixes A to C. Finally, acomparison is made between the fluer ic-driv e -circui try - stepping-motor actuatorsystem and the mo re conventional pneumatic piston-in-cylinder actuator.

The design and breadboard implementation of thi s stepping-

STEPPING MOTORAs shown schematically in figure 1, the actuator h as only two moving parts: a

gimbal-supported driving gear f r ee to nutate (wobble) but not to ro tate and an output gearfr e e to rotate but not to nutate. By unequal pressurization of eight bellows attached toits periphery, the driving gear is made to tilt and contact the output ge ar . As the bellowspressurization pattern is sequenced, the point of contac t between the tw o gears travelsaround the circumference of the output gear. Since the output ge ar h as 180 teeth and thenutating gear 181, the output gea r advances by 1 tooth, o r 2O, f or ever y complete revolu-2


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- - - rForce vec torsI , f r o m b e ll o w s---

--2- S u p p o r t b e a r i n g s

O u t p u t shaft-

F i g u r e 1. - S t e p p i n g - m o t o r o p e r a t i o n .

C-66-3984

EC-66-3983

F i g u r e 2. - S t e p p i n g m o t o r .3


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l l l l l I

tion made by the point of contact. If the bellows pressuriz ation pattern, and hence thepoint of contact, is advanced only by a frac tion of a revolution, the output shaft will beadvanced by the sa m e frac tion of 2 Since eight bellows are used for manipulation ofthe driving gear, the pressuriz ation pattern can be cyclicly advanced in eight steps andthe output-shaft posit ion in inc rem ents of 0.25'. If the bellows pressuri zation patt ernis fixed at one position, the output-shaft position is al so fixed for any load torques lessthan those which would cause disengagement of the ge ar s. When load torques areapplied, a sm al l output-shaft deflection occur s accompanied by movement of the nutatinggear . However, fo r torques less than those which cause disengagement, the maximumoutput-shaft deflection is less than one step size (0.25'). Thus, in the absence of ex-cessive load torques, the actuator motor can accurately position a load at aqy shaftposition specified by sequencing of the bellows pres sur iza tion patt ern.

tively. A mo re complete descript ion of the motor and the procedure s used in its designis given in reference 1 .

The motor and its cross-sectional drawing are shown in figur es 2 and 3 , respec-

Pressure

-Gi mba lr i ng

. Inlet portto bellows

CD-8813Figure 3. - Cross section of pneumatic stepping motor.

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BELLOWS DRIVE CIRCUITRY DESIGNA key performance re quire ment affecting the design of the bellows dri ve ci rc ui t was

that it sequence the bellows pr es su re s in ex ce ss of 160 s teps per second. An open-looptal counting cir cu it was used to achieve this sequencing speed. With no external

osition feedback, the ci rcu it speed of operation was limi ted only by the speed of itslements and the del ays of its int ernal logic feedback paths. Fluid jet amplifiers wer e

used as the operational elemen ts since they tend to be of the or de r of 3 to 10 t imesthan other flueric components (vortex ampl ifiers , turbulence ampl ifier s, or

mpact modulators).r Command pulse, forwardIr l n p u t I rT iming pulses r C ou nte r ou tput ,,-Bellows pre ss ure s

I! I I / II

I condit ioningI SteppingPowernitI Counter\ :Command pulse, backwa rd motorcondit ioningunit

Figure 4. - Block diagram of breadboard actuator system.

Figure 4 is a block diag ram of the drive- cir cui try and stepping-moto r actuationThe drive circ uitry consi sts of thre e main parts: (1) two pulse conditioning

s which acce pt forward and backward d irecting command pulses and convert themwell-defined timing puls es Tf and Tb7 (2) a counter ci rcuit which accepts the

a pressurization patt ern on its eight outputs,nd (3) the power ampl ifi ers which take the low-power signals del ivered by the counter

t them into high pr es su re and flow signals fo r driving the bellows of themotor. The following sections discu ss design as pe ct s of the various ci rcuit

Counting C ircu itThe counting cir cu it is a ring-type counter on which is s tored a pressurization

shifted forwa rd o r backward by either of two timing pulse trains. As shownthe darkened circ les in figure 5(a), the required pressurization pattern

four pre ssu ri ze d bellows next to each other s o that maximum force is exertedthe point of contact between the nutating and output ge ar s. To be avoided is a

pattern, such as that shown in figure 5(b), which shif ts the center of5


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I I1 I I I1 I I . . .

-A(a) C orrect pressur izat ionpattern.

A(b) ncorrect pressur iza-tio n pattern.

A- E;Q;- c c;QgJDE0;

-A A A AStar t Pu l se 1 P u l s e 2 P u l s e 3

B - - - -A A A AP u l s e 4 P u l s e 5 P u l s e 6 P u l s e 7

(c) Sequencing of pressure pattern by forward counting inpu t pulses .F igure 5. - Bellows pressurization patterns.

TABLE I. - SEQUENCING O F PRESSURIZATION PATTERNForward timing

Puke,Tf

012345678000000000

Backward timiniPuke,

Tb000000000012345678

~

-A

100001111111111111

-B

110000111111000011

Bellows-C

1110000111100DD111

-I:

1111000011000D1111

~-A

011110000000011110

~-B00111100DDD3111L1I

-C000111100001111333

00001111001111DD3D

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for ce toward the cen ter of the nutating g ea r and deliv ers a higher percentage of thebellows output force to the gi mbal s which supp ort the nutating gear rat her than to thepoint of contact between the gear s. Thus, the counting cir cui t is required to maintainthe press uriza tion p atter n shown in figure 5(a) and to index this pa ttern sequentiallyaround the circumfe rence of the nutating gear , as shown in figur e 5(c).pressurization by the logical state 0, then the following se t of logical equations may beused to r ep res en t the sequencing of the pressurization pattern:

If bellows pressurization is denoted by the logical state 1 and absence of bellows

1A =T b - B+Tf-" +B.f i =RA- -RA =T b * B +T f - D + B - D =Sx

- _RC =Tb*D+T f - B+B - D =%

J,, =Tb-A +Tf.C =Rjj

Ft,, =Tb.A + Tf .C =SDSetting A (SA) denotes t ransit ion of bellows A from logical state 0 to logical state 1whileresett ing A (RA) denotes transiti on from 1 to 0.surization pattern, SA- A while SA- A.AND and logical OR, respect ively. Table I represe nts, in tabular form, the sequenc-ing of the bellows pr es su re s as a function of for ward o r backward timing pulse s.tions were derived by using a synthesis procedure described in reference 4, and makeuse of the fact that only eig ht of the poss ible combinat ions of the four Boolean variab le sA, B, C, and D rep rese nt acceptable pressurization patterns. The resultant Set andRe se t equations (minus the error correcting term s) contain a minimum number ofte rm s. These equations enable the use of a simp le c irc uit design which has low flowconsumption. During application of the timing pulse s, four Se t and Res et equations aresatis fied by any given counter state. However, thr ee of these are redundant and simplyinstruct a bistable element to r ema in in the state it is alr eady in. As long as a cor rec t

Because of the sym me try of the p re s -The symbols * and + denote logical

Two comments should be made regarding logical equations (1). First, the equa-

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pressurization pattern has been initially stor ed on the cou nter, this redundancy does notaffect the de si re d counting sequence, and no need exists for er ro r correction.

Secondly, four terms in the logical equations do not contain either forward Tf orbackward Tb pulses. These te rm s are er ro r correcting te rms , which detect an in-co rre ct pressurization pattern, such as shown in figure 5(b), and issue the appropriateSet or Reset signal to eliminate it. Without these t er ms , an incorre ct pressurizationpattern would remain inc orre ct as it was propagated around the outputs of the counte r.At times, the err or correcting te rm s issue a signal when a n e rr o r does not exist. How-ev er , the generation of a n additional signal under n orm al operation is again a redun-dancy which ins tructs the affected fluid je t amplifier to re mai n in the state it is alreadyin. Thus, the erro r-co rrec tin g ter m for Sc would issue a Set signal for C during theinitial state of the counter in table I even though no e r r o r was present . Once the counterhad been advanced two step s for war d (from Tf - 0 to Tf - 2 in table I), B would belogical 0 and would cause the er ro r correction term B - D in Sc to al so be logical 0(i .e. , o vanish).The counter cir cui t which was developed to satisfy the logical equations (eqs. (1)) sshown schematically in figur e 6 . The pressurization pattern is sto red and advanced onfour cent ral bistable fluid je t ampl ifiers , designated I, II, ID, and IV. Thei r outputsare designated A,x, B, B, C , c , D , and 15, respectively, corresponding to simi larlydesignated bellows in figur e 5. An active two-input OR unit is connected to each co ntrolpor t and furnishes its con tro l signal s. To the input of each act ive OR unit is connectedthe output of an OR unit acting as a passive AND. These units accept the output of oneof the ce nt ra l bistable un its as a power supply and a forward Tf or backward Tb timingpulse as their control signal. Thei r output is thus the logical product (AND) f theoutputs of the ce nt ra l bistable units and the timing pulse. For example, the input de-livered to bistable amplifier I fro m the passive AND unit A 1 s a rese t for I and is thelogical product of the forward pulse Tf and the output of D (i.e . , Tf.D). Examinationof equation (1) shows that this is indeed the condition by which A is rese t by a forwardpulse. Car efu l examination of the circui t shows that the logical equations (1) ncludingthe er ro r correcting te rm s are satis fied. The connections which implement the latterare shown as dotted lines in figure 6 .

Operation of the circuit is as follows: Shor t timing pu lses of one type only (eitherTf or Tb) are applied simultaneously to all contr ol po rt s of the passive AND units.Those AND units whose power nozzles a r e pre ssu rized by the outputs of the cent ral bi-stable amplifiers (I, 11, 111, and IV) genera te output signal s of durat ion approximatelyequal to the length of the timing pulse. These output sig nal s tr av el to the cont rol por tsof the activ e OR units and cause them in turn to gener ate s ho rt output pulses. Thesepulses switch the ce nt ra l bistable amplifi ers. The outputs of the cen tra l bistable unitsare delayed a n amount of t ime T before they reach the nozzles of the passive AND units.Thus, if the timing pulse is sh or t and has vanished before changes in outputs from the8


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Passive

units1AN D

Activeun i t sCentra lbistables whichdrive power

Active

Passiveun i t s1AN D

Figure 6. - Schematic diagram of counting circuit for nutator motor

bistable amp lifi ers re ac h the power nozzles of the passive AND units, nothing fu rt he rwill happen and the counter will be se t fo r the next timing pulse.repeated with the appl ication of ea ch timing pulse Tf or Tb.ce nt ral bistable units to the passive AND units is important.the c a r ry signal will rea ch the power nozzles of the passive AND units before the timingpulse Tf has disappeared. If the delay is too long, the carry signals will not reach thepower nozzles of the passive AND units by the time the next timing pulse occurs. Fig-ure 7 schematically illu strate s the time history of the various pre ssu res in the countercircuit which are necessary to "set" A (i .e . , cause the output A to appear).

If the counter is initially i n the state Tf-3, as shown in table I, and if a forwardtiming pulse Tf is applied to the control ports of all the passive AND units, amplifier IVwill be reset, and the output will appear. The counter will thus have advanced tostate Tf-4 of table I. Since the respons es of the passive AND unit, the active OR unit,and the ce nt ra l bistable unit are not instantaneous, the output 6 will appear a time tsw

The sequence isChoice of the proper delay -r with which to retard the c a r r y signals from the

If the delay is too short,

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IIIIIIII Il l I 1 I I 1

I Io u t p u t E

I I I -I 0.005 0.010IIL r t-

-Ou t p u t d e l a y e d byt r a n s i t t i m e q

I =0.005 0.010

O u t p u t A I' -0 0.005 0.010Time, t , sec

F i g u r e 7. - S e q u e n c e o f p r e s s u r e s n e c e s s a r y t o s e t A .

after initiation of the timing pulse. The total switching time tsw, pulse dispersion inthe signal lines in the counter, and the rise and decay ti me s of the timing pulse determinewhether it is necessa ry to delay the ca r r y signal by an additional amount of time T~ toensure that the timing signal has time to vanish. These time s T~ +tsw constitute thetotal delay tim e T , as is shown in figure 6. The resul tant delayed output Ddelayed isshown in the thi rd line of figure 7 and sets the counter fo r the next timing pulse Tf. Anamount of time tand the earliest possible occurrence of the next timing pulse Tf rep res en ts an idlecounter state. A timing pulse may be again applied at any time during t to index theqcounter in its normal manner. Hence, the only inherent limi t to the operating speed ofthe counter is the application of timing pul ses so rapidly that the period t goes to zero.In the breadboard counter circui try to be described in the section, BREADBOARDACTUATOR SYSTEM, the width of the timing pulse was 1 millisecond, the total combinedelement switching time (including pulse propagation ti me s within the elements) tsw wasapproximately 0.0015 second, and the delay tim e T~ was approximately 0.002 second.

qSince the actuator was designed to operate at 160 steps per second, the quiet period tis approximately 0.00275 second fo r the breadboard counter cir cui t which was im-plemented. Thus, the breadboard counter should be capable of operating at approximately

which exists between the last change of the ca rr y signal Ddelayedq

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286 s tep s per second. If the m easur ed delay in element switching time tsw could beconsidered dependable, it could be use d as the total delay time 7. With a total delay Tof only 0.0015 second, the circu it could, in theory, be stepped at 667 st eps pe r second.

Signasource

Pulse Conditioning Unit

I Pu l seLong t ransmiss ion l ine unit Timing pulse

II ~ condit ioning I

As illustrated in figur e 8(a), the command pulses delivered to the actuator systemcan be highly distorted as a resul t of propagating down a long transmission line. Theirrounded leading edges, low ampl itude , and long tails make them unsuitable for directuse as timing pulses. The function of the pulse conditioning unit, shown schema ticallyin figure 8(b), is to convert the command pulses into properly shaped timing pulses.singlesided and of variable amplitude, into a push-pull signal of fixed ampli tude . Sincethe output of the OR-NOR unit does not have fast ri se and decay times, a bistable unit(unit 2 in fig. 8) is used to give the command pulse s ha rp leading and trailin g edges. The

The first element, a n OR-NOR unit, is used to conver t the command pulse, which is

(a) Command pulse wave forms.P ulse width fixation

Ampl i f ier F igure

(b) Sc hematic diagram.F igu re 8. - P ulse condi t ion ing unit.

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output of unit 2 is fed into a unit which fixes the width of the pulse at 1 millisecond. Thiswidth was considered to be the minimum value compatible with the switching resp onse ofthe pass ive AND units of the counter and the dispe rsive effec ts of the lines used to dis-tribute the timing pulse. Pulse width fixation is accomplished by introducing the commandpulse simultaneously to a n orific e which feeds control po rt C1 f amplifier 3 and to a1-millisecond delay line connected to control port C2 f the sa me amplifier. Thus, thereshaped command pulse initially causes flow to go through control por t C1 o set the am -plifier, and to produce an output on receive r R2. The s am e command pulse, however,propagates along the transmi ssion line and, 1 millisecond later, pres suri zes control portC2.set and the flow from its receiver R2 hut off, thus termina ting the timing pulse. Designof the pulse width fixation portion is tre at ed in appendix C. The remaining bistable a m -plifier, unit 4, ser ve s to amplify the pulse in pr es su re and flow and send it to a manifoldfo r distribution to the control por ts of the passiv e AND units of the counter.

If the pulse del ivered by the delay line is of pr op er magnitude, ampl ifier 3 will be re-

P ower A mplifierThe Lewis SB1 power fluid je t amplifier was used to drive the bellows of the stepping

motor. Shown in outline form in figure 9 , the amplifier has a supersonic nozzle tocre at e the main power je t and use s a conventional, inclined-wall interaction region fo rdeflection of the jet. The receiv ers a r e of the Y type and follow the design philosophyuse d on the Lewis B1 subsonic bistable amplifier (ref. 5). Rev erse discharge and dis-placement flows deli vered by the bellows are thus dumped to atmosphere through thevent V3 nstead of into the interact ion region. A s pointed out in referenc e 5 , this featureperm its mor e rapid switching at lower control-port pr es su re s and flows than would be

b \~. -1 3

~' 3.'5" E>* - 1.22 1. IS.-0. 111.6- = 1 --- - ~. .._-I2025.6 - -I

Figure 9. - Lewis S B - 1 power amplifier. ( A l l l inear dimens ions to be multipl ied by Dn.)12


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mm ( a ) Number of s t eps pe r second, 173.NmE, .-2....z

L0 '025 .050Time, sec(b l Number of steps per s e c o n d , 28.7.

Figure 10. - B e l l o w s pressure as functionof time.obtained with a fluid jet amplifier which used conventional rece iv ers ai med at the inter-action region. A penalty in pre ss ur e recovery associated with amplifiers using the Ytype of receivers is believed to be offset by th eir incre ased switching spee ds and lowercontrol-port pressures.follows:

The salient chara cte rist ics of the SB1 amplifier when driving the bellows a r e as

Power nozzle throat size, in. (cm)Width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0.040 (0.101)Depth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0.060 (0.153)

Design supply to exhaust press ure ratio . . . . . . . . . . . . . . . . . . . . . . . . 4 . 0Maximum normalized receiver pressure (nominal),

(PR - Pe)/(Ps - Pe), percent of supply . . . . . . . . . . . . . . . . . . . . . . . . . 40Maximum normalized re ce ive r flow (nominal), mR/ms, percen t of supply. . . . . . . 0Mach number ofpower nozzle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . -1.63Design load volume, in. (cm ). . . . . . . . . . . . . . . . . . . . . . . . 0.576 (9.43)Charging time constant of load volume, sec . . . . . . . . . . . . . . . . . . . . . 0.008Control-port switching pressure (nominal), (Pc - Pe)/(Ps - Pe) . . . . . . . . . . 0.07Cont rol-port switching flow (nominal), mc/ms . . . . . . . . . . . . . . . . . . . 0.07

3

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Figure 11 - Control-port pressure-flow characteristics of Lewis S B - 1 fluid jet amplifier.


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To be noted are the short be l lows charging time constant of 0.008 second and the ra therla rg e internal volume which the amplifier mu st drive . Figure 10 shows oscilloscopetraces of the bellows pr es su re as a function of time. As can be seen, the experimentaltime constant is close to the theo retic al, thus indicating validity of the proced ure s usedto predic t the bellows charging times and, hence, the requi red amplifier size. Fig-ures 11and 12 show typical pressure-flow characteristics of the amplifier control por tsand receive rs.

0 . I .1 0 .15 .20

xNormalized pressure, (PR - P e t / ( P s - P el

Figure 12. - Receiver pressure-flow characterist ics of Lewis S B - 1 f lu id jet amplif ier,

BREADBOARD ACTUATOR SYSTEMTo establish the practicality of the previously discussed c ircui ts, the bellows drive

circu itry was implemented, in breadboard fo rm , and connected to the stepping motor toform a complete breadboard actuator system. The resultant system is shown in figu re 13,and its design performance specifications are listed in table II.

Breadboard Bellows Drive C ircui tryExcept fo r the power ampl ifi ers , the breadboard implementation of the bellowsdrive cir cuit ry used standard, commerciall y available fluid je t amplifiers.

merci al f luid jet ampl ifiers we re made of photoetched ce ra mi c and had barbed hosefittings for input-output connections. A l l OR units (including those actin g as passive

The com-

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I I I I I I I I 1 I 1

TStepping motor

Pulse condition-

1iti

Command pulsef, $ ;A i - 3982

Figure 13. - Breadboard actuator system.

TABLE II. - DESIGN PERFORMANCE SPECIFICATIONS FOR BREADBOARDACTUATOR SYSTEM

Working fluidAmbient tempe ratu re, OR (%)Supply pre ssu res , psia (N/cm )

Power valvesCounter circuitPulse conditioning unit

Amplifier 3Amplifiers 1 , 2, and 4 2Exhaust pressur e (all components), psia (N/cm )

Maximum output torque, in. -1b force (cm-N)Load ine rti a, lb mass -in. (kg-cm2)Maximum stepping rate, steps/secPeak-to-peak ( 2 9 amplitude response, cpsBellows internal volume, in. (cm3)Bellows stroke , in. (cm)Bellows charging time constant, secBellows actuating pressu re , psid (N/cm differential)Total manifolding volume per bellows (calculated),Stepsize, deg

2( i n . 3 , (cm3

Air530 (2951

59. 5 (41 .1)2 6 . 7 ( 1 8 . 4 )1 7 . 7 ( 1 2 . 2 )20 . 7 (1 4 .4)14 . 7 ( 1 0 . 2 )103 (1165)

2 8 ( 8 2 . 5 )160

70 . 2 3 7 ( 3 . 8 9 )

0 070 (0. 178)0.008

2 0 . 5 ( 1 4 .1 )0. 39 (5 .56)

0 . 2 5

16

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C-66-3978

(a) Bistable amplifier; power nozzle, 0.020 by 0.080in c h (0.0508 by 0.204 cm). (b ) OR-NOR unit; power nozzle, 0.010 by0.040 in c h (0.0254 by0.102 cm). (c) Small bistable amplifier; power nozzle,0.010 by 0.040 in c h (0.0254 by 0.102 cm).F igure 14. - F luid jet amplifiers used in breadboard actuator system and silhouettes.

,Forward co mma nd pulse , Tf/'

I'Ampl i f ier 2,

I r Delay

,Ampl i f ier 40\

\\b \ ';Amplifier 3I

L O u t p t i t Tf ,C -66-3979Figure 15. - Pulse condit ioning unit.

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AND units) wer e of th e asym metri cal wall-attachment type and had a 0.010- by0.040-inch power nozzle. The cen tra l bistable amplifiers (I, 11, III, and IV) of thecoun ter and the bis tab le ampl if ie rs (2, 3, and 4) of the pul se conditioning unit weresym met ric al wall-attachment units. The centra l bistable units and amplifier 4 of thepulse conditioning unit had a 0.020- by 0.080-inch (0.0508- by 0.203-cm) power nozzle.All other amplifiers had a 0.010- by 0.040-inch (0.0254- by 0.101-cm) power nozzle.Figu re 14 shows the units as delivered and the manufacturer's silhouettes fo r theelements.

line of th e pulse width fixat ion portion is somewhat involved and is discussed in detail inappendix C.

Fig ure 15 shows the pulse conditioning unit. Design of the input orifi ce and delay

Ce ntral ampl i f iersfI,II, 111 IN\ T iming pu l sef manifold

\- 1 J u n c ti o nmanifolds c-66-3980*?~ --.A-**

Figure 16. - Breadboard counter c ircuit.The counter circuit is sh jwn in figure 16. Each cen tra l bistable UL, (I, 11, III, and

N) ith its associated OR units forms a single layer of seven elements. Four suchlayers are stacked on top of each other to for m the complete counter circuit. A manifoldis used to distr ibute the timing pulses, Tf and Tb, as received fr om the outputs of thepulse generator. The output of the centr al bistable ampl ifi ers is fed to junction mani-folds which ar e located on and pre ssu riz e the control por ts of the power amplifi ers.Fr om the control ports of the power amplifi ers, the output signa ls from ampli fiers I,11, 111, and IV are fed back to the supply nozzles of the var ious p assiv e AND units in thecounter circu it proper.

18


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Time, secFigure 17. -T imin g pulse delivered tocontrol port of passive AND unit.Supply pressure to pulse ondit ioningunit, 6.0 psig (4.14 N/cmi age).

(a) C i r uit pressure, 12.0 psig (8.35DVIm ._ Nlcm gage)..aiEz...LVII3iL2

I I I I I I0 .02 .04 . C 6 .08 .1 0Time, s ec

(b ) C ircui t pressure, 6.0 psig (4.15F igure 18. - Counter outputs D an d

N / C J age).

Dd.

19


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Of interest is a n interconnection technique used to reduce signal attenuation to thetransmission lines of the counter circuit. If a driving sourc e characteri stic typical ofmos t fluid jet amplifiers is used, it is often possible to find an oversized transmissionline in which the pulse reflected by the load is completely cancelled by the driving sourceimpedance. By this method, it is possible to deliver a pulse of pr ope r waveform to theload with a n oversized line and thus incur much lower frictional l oss es than if anacoustically matched line wer e used. Thi s technique, which might be termed "singlereflection termination, 'I is described in appendix B and was used to s iz e both the linesused to distribu te the timing pulse and those used to c ar ry the counter outputs to thepower nozzles of the passive AND units.the timing pulse and the ca r r y signal D, respective ly. The height and width of thetiming pulse are close to the theoretical v a l u e s . The small steps before and aft er thepulse resu lt fro m the pre ss ur e transducer being mounted about 1 inch (2.54 cm) awayfr om the interac tion region of the AND unit. The delayed ca r r y signal Ddelayed, hassharp rise and decay tim es and is al so of proper height and width. To be noted is thepresence of the reflected wave in the so ur ce (output D). A detailed ana lysis of thesewaves is given in appendix B.

Figures 17 and 18 show oscilloscope traces of

Exper men aI Performa nce of Breadboa rd Actuator Syste mA series of performance tests was conducted on the complete breadboard actuation

system to determine the ability of the bellows dri ve c irc uit ry to drive the stepping motor.As shown in figure 19, the output shaft could be stepped in ei th er direction at 173 st epspe r second. Rather smal l, periodic waviness occurred on the otherwise smooth outputtrace. To investigate the so urce of waviness, the actu ato r was stepped at slower speedsand photographs were made of oscilloscope t ra ce s of the shaf t position. As shown infigure 20, the step sizes are nonuniform. Once ev er y eight steps, a larg e ste p ofapproximately 0.4' occurs. As the stepping rate is increased, the smaller steps smoothout, but the large step remains . The actuator was disassem bled and inspected fo robvious flaws, such as dirt between the gear teeth or damaged bellows, which mightcause such a change in step size. None were found, and the cause of the uneven ste psre ma in s unexplained.

Analogous to the frequency response test performed on proportional actuators, thestepping motor can be commanded to cyclicly r ev er se direction aft er a fixed number ofsteps. A s shown in figu re 21, the inertially loaded actuato r system can be cycliclystepped eight ste ps (2') in each direct ion without missing, at a stepping rate of 115 stepsper second. This performance is roughly comparable to a bandwidth of 7.2 cps at *loamplitude fo r a conventional piston actuator .20


23/67

u.2 .4 .6 . 8 1.0(a) Steps per second, 10.mc-0c-B

(a) Forward direction.

I 1 1 -0 .W .08 .12 .16 .20(b ) Steps per second, 28.7.

I I I I0 .1 .2 . 3 . d .!Time, sec(b ) Backward direction.

Figure 19. - Output shaft posit ion as func -tion of time at 173 steps per second. 1 -0 .01 .02 .03 .04 .05Time, sec(c) Steps per second, 115.

Figure 20. - Output shaft position asfunc tion of time. Steady steppingrate.Maximum output torque was measu red by incre asing the load torque on the actuator

output shaft while the actuator was being stepped until the driving g ea r would disengage.The res ult s shown in figure 22, indicate a maximum s t a t i c output torque of 70 inch-pounds force (789 cm-N) and a maximum slewing rate of approximately 37' per second.The reduction in output torqu e from the design value of 103 inch-pounds fo rce (1165 cm-N)can be attributed to (1) the inherent stiffness of the bellows an d driving gear gimbalflexures and (2) he fact that the differential output p res su res of the power amplifiers

21


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Ill IIIII I I I I

900

700

600

Va-3E

300

200

100

I I I 1 I I0 .01 .02 .03 .M .0 5-0rY).-

(a) Steps p er second, 57.5.n

-800-

--

a025 0 0 - z

4m-gE

c.-aiL

---

0-

I l l1.0- 0 .OM ,008 ,012 .016 .020Time, sec

(b) Steps per second, 115.Figu re 21. - Smdll amplitude responseof breadboard actuator system. Eightsteps each direction.

0 10 30

\\-

Speed, deglsecF igure 22. - Output torque as function of speed 01breadboard actuator system.

2we re somewhat lower than the 2 0 . 5 psi ( 1 4 . 1 N/cm ) design value. A differentialbellows pre ss ur e of 4 psi ( 2 . 7 6 N/cm ) w as required to force the nutating gear into contactwith the output gea r. Subtracting this pre ss ur e from the 18 psi ( 1 2 . 4 N/cm ) differentialpr es su re experimentally delivered by the drive c ir cui try power ampl ifiers and thendividing by the 2 0 . 5 design value output p re ss ur e for the breadboard circ uit (table II)yields the following predicted reduction in output torque:

22

This reduction indicates a n experim ental output torque of 0 . 6 8 3 times 1 03 or 7 0 . 3 inch-pounds fo rc e (800 cm-N), which is approximately that observed. Since the steppingmotor is designed fo r a bellows differential operating p re s su re of 70 psi ( 4 8 . 3 N/cm ),the torque efficiency can be increased by raisin g the bellows actuating pre ss ur es t o thisvalue. This in cr ea se may be accomplished by increasing both the supply and exhaustpr es su re s of the bellows driv e circuitry. A prototype bellows drive circuit operating atNERVA supply and exhaust pr es su re s of 2 0 0 psia and 50 psia (138 and 3 3 . 5 N/cm abs),respectively, would deliver a differential output pr es su re of approximately 6 1 . 3 psi

2

2

2 2

I I I I1111 I 1111


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2(42.5N/cm ) to the bellows. The output torque, under such conditions, would probablyrise to 82 perce nt of its design value.

The reduction in maximum, unloaded slewing velocity observed in the torque-speedhas not been explained but probably re su lt s from interactions between the actuator

large load pulley used f o r applying torque s. When the rated load inertia of228 pounds mass-inch squared (82.5 kg)(cm )) was replaced on the actuator outputthe actuator could again be operated i n both direc tions at 43.3' per second or

173 steps per second (shown by the solid symbol in fig. 22).C O M P A R I S O N O F B R E A D B O A R D N UT A TO R A C TU A TO R S Y S T E M

WITH CONVENTIONAL P ISTON ACTUATORAlthough the nutator actuator system has a unique set of ch ar ac te ri st ic s which make

t inherently well adapted to some applications and less well adapted to othe rs, it isit with the mo re conventional piston actuator. The pistonis considered to be friction less, drive n by a flapper valve, and to have a

at le as t equa l to that of the ideal perform ance of the breadboardII).

put torques, and their maximum slewing velocities a r e used as the cri te ri a of com-Two ca se s are considered for the stepping-motor system: the first is with a

that calculated fo r the breadboard actuator system(0.576 n. 3; 9.4 cm ), the other is with a power ampl ifi er load volume equal to the

0.237 cubic inch (3.88 cm ) plus the int ern al volume of a channel1/16 inch square by 7.5 inches long (0.159by 19.1 cm). The latter volume is considered

a prototypecomputing the flow requ ire d by the power valves and then scaling the rest of the

The flows required by the actuator sy st em s, their maximum

33

The flow requ ired by the prototype actuator syste m was dete rmined by

The resul t s of comparing the two systems are as follows:

23


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Design performance specifications :Working fluid . . . . . . . . . . . . . . . . . . . . . . . . . . Air at 530' R (294' K)Supply pressure, psia (N/cm ) . . . . . . . . . . . . . . . . . . . . . . . 59.7 (41.2)Exhaust pr es su re , psia (N/cm ). . . . . . . . . . . . . . . . . . . . . . . 14.7 (10.3)Resolution, deg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kO.25Maximum slewing rate, deg/secLoad ine rti a, lb mass-in . (kg-cm ) . . . . . . . . . . . . . . . . . . . . 28.0 (82.5)Frequency response , 2 peak-to-peak amplitude, cps . . . . . . . . . . . . . lat to 7

22

Maximum output torque, in. -1b force (cm-N) . . . . . . . . . . . . . . . . 103 (1165). . . . . . . . . . . . . . . . . . . . . . . . . . . 402

Actual stepping-motor syst em performance:Maximum output torque, in. -1bforce (cm-N) . . . . . . . . . . . . . . . . . . 0 (789)

. . . . . . . . . . . . . . . . . . . . . . . . . . 43.3Frequency response , 2 peak-to-peak amplitude, cps . . . . . . . . . . . . . . . 7.2Resolution, deg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *0.25Maximum slewing rate, deg/sec

Hypothetical piston actuator performance:Maximum output torque, in. -1b force (cm-N) . . . . . . . . . . . . . . . . . 03 (1165)Maximum slewing rate, deg/sec . . . . . . . . . . . . . . . . . . . . . . . . . 99.5Frequency response, 2 peak-to-peak amplitude, cps . . . . . . . . . . . . . . . .

Total system flow consumption:Hypothetical piston ac tuat or

7

lb mass/se c (kg/sec). . . . . . . . . . . . . . . . . . . . . . . . 0.00071 (0.000321)3 3standard f t /min (standard m /min . . . . . . . . . . . . . . . . . 0.945 (0.0268)0.576 in.lb mass/ sec (kg/sec) . . . . . . . . . . . . . . . . . . . . . . . . 0.0211 (0.00955)

Breadboard stepping-motor system (Load volume per bellows,3or 9.4 cm ) - 16.9 (0.479)standard f t3 min (standard m3/min) . . . . . . . . . . . . . . . . . .

Prototype stepping-motor system (Load volume per bellows,3or 4.36 cm ).266 in.standar d ft3/sec(standard m3/sec) . . . . . . . . . . . . . . . . . . . . 8.3 (0.235)lb ma ss /s ec (kg/sec) . . . . . . . . . . . . . . . . . . . . . . . 0.01035 (0.00469)

A s can be seen, the pr im ar y disadvantages of the stepping-motor syst em a r e flow con-sumption and maximum slewing speed. The breadboard sys te m consumes 29.7 time s asmuch flow as a n equivalent piston actuator, while the prototype system consumes14.6 times as much. It is possible to furt her reduce flows requi red by the prototypesystem (1) if higher -pre ssur e re cove ry power valves are used, thus permitting smallerbellows, (2) if the bellows st if fne ss is reduced, thus permitting the sa me output torquesfor a lower power-valve supply pr es su re , (3) if a la rg er bellows charging time constantcan be allowed, thus permitting sma ll er power amplifiers, and (4) if either the bellowsvolume or line manifolding volume is reduced. The last approach could yield large24


27/67

decreases in power amplifier flow consumption, since the bellows charging flows out-weigh displacement flows by a factor of 8.2. Reduction of the bellows volume, however,wil l probably be a difEicult task s inc e approximately half of thei r int ernal volume alr eadyis eliminated by an insert.

CONCLUSIONSFr om a n investigation of a breadboard fluer ic -controlled pneumatic stepping-motor

sys tem , the following conclusions wer e drawn:1. The fluid je t amplifier bellows drive circuitry is sufficiently fa st that bellows

charging time is the pri ma ry limitation to the maximum stepping-motor-system speed.Although it has not been test ed at higher rates, we believe that the cu rre nt bellows drivecircuitry can operate satisfactorily at wel l above 173 steps per second.

2 . The improved bellows drive circuitry and the stepping motor constitute a reliable,.open-loop stepping actuat or sys tem with inherently high output stiff ness, reason ableslewing speeds, and sm al l step size. Fur the r improvements in both the actuator and thepower valves of the bellows dr iv e cir cui try should inc rea se syst em perfo rmance beyondthat reported herein.

3. The stepping-motor-system flow consumption will be much higher than that of a nequivalent flapper-valve-driven piston actuator designed to do the sa me job. However,fo r many aerosp ace applications, this disadvantage in flow consumption may not belarge in comparison to the advantages in simplicity, apparent reliability, and high outputstiffness that are offered by the pneumatic stepping motor in combinat ion with a fluericdigital drive system.Lewis Research Cen ter,

National Aeronautics and Space Administ ration,Cleveland, Ohio, October 6, 1967,

122-29-03-09-22.

25


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A P P E N D I X AS Y M B O L S

A

DDng0k1mmnnPRR P R 2di

%otrS

line areaequivalent choked nozzle areaoutput of counteramplifier control portsspeed of sound, in. /sec; cm/secline diameter, in. ; cmwidth of power nozzle exit, in. ; cmacce ler ation of gravity, (lb ma ss )(in. )/(lb force)(sec ); (kg) (cm)/(N)(sec2)ratio of specific heatslength of line, in. ; cmma ss flow rate, lb mass/se c; kg/sec2 n+ 1, eq. (E19)dummy indexpressure , lb force/in. 2; N/cmReset

2

2

amplifier receiversline resistance per unit length, (lb force)(sec)/(lb mass)(in. );

total line resista nce, (lb force)(sec)/(lb mass)(in. '1; (N)(sec)/(kg)(cm2)reflection coefficientSet

3(N)sec)/&g) (cm3,

S Lap hcia n operator, sec-l ; supplyT timing pulse; torque , in. -1b fo rc e; cm-Nt time, secV ventX distance down line, in. ; cm

26


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line shunt admi ttance per unit length, (lb mass) (in. )/(lb force) (sec);

impedance, (lb force)(sec)/(lb mass)(in. '); (N)(sec)/(kg)(cm2)3line s er ie s iner tne ss pe r unit length, (lb force)(sec)/(lb mass)(in. )

total line surg e impedance, (lb force ) (sec)/(lb mass)(in. );

(kg) (cm)/" (set)

22(N) (sec)/(kg) (cm 1(N)(sec)/(kg) (cm2)

2line surge impedance in absence of friction, (lb force)(sec)/(lb mass)(in. );

attenuation te rm in transmission-line equationstorque efficiencypropagation operato rviscosity, (lb force)(sec)/(in. '); (N)(sec)/(cm )density, lb mass/ in. 3; kg/cmtime delay, se ccentral bistable amplifiers

23

Subscripts:A, B, C, Da start of lineb backward, end of line1 7 27 3C control, cha rac ter isti c valued delayede exhaustf forwardj jet2 line0 reference conditionsq quietR receiver

Boolean va riable s

amp lifi ers in counter circ uit (identified in app. B)

amplifier receiversR 1 ' R 227


30/67

s supplysw switchingt transmission linexySuperscripts:(-) logical complement of quantity(')('*)

conditions before wave in lineconditions after wave in line

dimensionless impedance or resistance, compressib le flow in amplifierdimensionless impedance or resistance incompressible flow in amplifier

28


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A P P E N D I X BS IMPLlF lED INTERCONNECTION TECHNIQUES FOR

DIGITAL FLUID JET A MPL IF IERSThis discussion of s ome simplified techniques for interconnec tion of digital fluid je t

amplifiers is included as a n aid for understanding both appendix C and the pu ke reflec-tion technique used to minimize steady-state viscous flow lo ss es in a transmission line.

T r a n s m i s s i o n l i n e

S

nL1A m p l i f i e r A

L1A m p l i f i e r B

F i g u r e 2 3. - Typical i n t e r c o n n e c t i o n of t w o b i s t a b l e f l u i d j e ta m p l i f i e r s .

The interconnection prob lem is illustrated schematically in figure 23. The output ofa bistable fluid jet amplifier is connected to a trans missio n line which, in turn, feeds thecontrol po rt of a second bistable ampl ifier. Amplifier A initially has its output directe dthrough its rec eiver R1, while ampli fie r B has its output directed to its receiver R2.If amplifier A is switched to its receiver R2, a pr es su re pulse will be s en t down theline. If the cont rol-port impedance of ampli fier B does not completely abs or b thispulse, reflected waves will be sen t up and down the line. A change in the control-portimpedance of amplifier B during this proces s, as a re su lt of switching, will fur the rcomplicate the situation.coupled with the nonlinear input-output cha rac ter isti cs and dynamics of the fluid jetampli fiers, this task becomes formidable. Fortunately, however, the fluer ic tr ans-missi on lines used in many pra ctical engineering situations are sufficiently sh or t thatfriction and pulse disper sion may be ei ther neglected or considered to be small. Also,for well-designed fluid je t amp lifi ers , the switching dynamics of the ampli fier are fastin comparison to the transit times of the pulses se nt down the lines. This fast switch-ing perm its the ampli fiers to be regarded as two state, nonlinear so ur ce s and loads.analy sis of digital fluid je t amp lifi er interconnection may be considerably simplified. Inthe case of zero line friction or pulse dispersion, the tra nsm iss ion line may be modelledas a pure, bilateral delay unit with equa l input and output cha ra ct er is ti c impedances ofmagnitude:

A rigor ous analy sis of the transm ission line, by itself, is involved and, when

With the foregoing assumptions of low line friction and fast amplifier switching, the

29


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zco=(:)(&)=ewhere p / ptube, respectively. Since the input-output pr es su re -flow ch ar ac te ri st ic s of fluid je tamplifiers are often plotted i n normalized for m, it is convenient to normal ize equa-tion (Bl) with respect to the power nozzle pr es su re s and flows of the fluid jet amplifierdriving it. Equation (Bl) may be rewritten as follows:

is the ra ti o of the fluid densities after and before the wave se nt down theY X

-zo -

For air,0.813(2\

2For approximately inco mpress ible power nozzle flow (


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1.6

1.4

1. 2

1.0

.8.E"-.Ea-c' 6-EF . 4E

II

Nm.-0 . i

C

-. i

-.

-_

i

ItJco

Forwardpoint-,,4-/

i\is.I\./ "L-%i t ch ing/-

\iii

uI

Ii,IC-.15 -.10 -.05 0 .05 .10 .15 . 2 0Normalized pressure, (P - Pe)/ (Ps - P

' Ou t p u t o f amplifier' A- L ine si-- ontroof am le impedance -iort characterist icfier B

5

Figure 24. - Load-l ine solution to t ransmiss ion- l ine pressures.

conditions aft er all transi ents have died out. The su rg e impedance of the line has noeffect on the location of the se two steady-state operating points.

Between these two steady-state conditions, however, the transmission-line dynmicscome into play. During the transi ent i n which amplifier A is switched into the line, thefrictionless line initially appea rs as a resi sti ve impedance Z c o which is slightly non-linear as a re su lt of the p / pengineering cases, this nonlinearity is smal l and will be neglected. Thus, as shown infigure 24, a pulse of pre ss ur e and flow indicated by intersection 2 is sen t down the line.A t the end of the line, the pr es su re pulse me ets the terminating load impedance. If thepr es su re and flow c ar ri ed by the initial pulse are not compatible with the pressure-fl ow

I

t e r m in equations (Bl) to (B3). For most practicalY X

31


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cha rac te ri st ic s of the terminating impedance, a readjustmen t of flow conditions at theend of the line must occu r. To the load, the line behaves as a source w i t h impedance

(since positive flow is now defined as being out of the line) which has initial pres-su re s and flows specified by intersection 2 of figure 24 . Thus, the pre ssu re at the endof the line will incre ase and flow will decrease until they reac h a value compatible withthe pressur e-flow chara cteri st ic s of the load (intersection 3, fig. 24). These new flowconditions, however, differ fro m the pulse initially sent down the line. Thus, a pres-su re pulse is sent up the line to the receiv er of the driving ampli fier A. Again, the newconditions represented by intersection 3 a r e not acceptable to the rec eiv er of amplifier Aand another readjustment takes place, at intersection 4. Reflections continue to occuruntil the f inal steady state is reached.the sa me points as do the load characteristics (dashed line, impedance Z i , fig. 25), no

-%0

If a transmission line is used which intersects the receiver output characteristics at

L

I/////I-

i i. 0 -.15 -.10 -.05 0 .05 ,10 .15 . 2 0 .2 5 .30 .35 dN o r m a l i z e d p r e s s u r e , ( P - Pe)/ (Ps - P elF i g u r e 25. - S i n g l e- r ef l ec t i o n l i n e t e r m i n a t i o n t e c h n i q u e .

32


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reflect ed pulse will be nec essa ry to re adj ust flow conditions at the end of the line.Steady-state conditions will exist after pass age of the initi al switching pulse. A secondcase can be considered in which the tra nsm iss ion line is oversized (impedance ZB,fig. 25) but intersects the receiv er output characteristi cs at a point (intersection 2,fig. 25) such that the refl ected pulse rea djust s flow conditions in the line to the final,stea dy- stat e value (intersection 3, fig. 25). In the absenc e of wave-form dis tort ion dueto the effects of friction, the pulse delivered to the load will be indistinguishable fromthat delivered by an acoustic ally matched line since, to the load, the reflected wave isnot a separate pulse. Thi s technique enables properly shaped pulses to be delivered toloads by oversiz ed tra nsm issi on lines. The oversiz ed lines, because of the ir la rg erintern al diame ter, will have much lower frictional loss es and pulse dispersi on thanlines acoustically matched to load impedances.ation. Steady-state viscous flow lo ss es may be approxi mated by the equation fo r Hagen-Poiseuille flow in a pipe:

The presence of friction in the line, even in sma ll amounts, complicates the situ-

128 bpi!Pa - PbPl n-D4

which, when normalized to the power nozzle pr es su re s and flows of the driving amplif ieris expressed as

o r , fo r approximately incompr essi ble power nozzle flow as

Although equations (B4), (B5), nd (B6) redict steady-state losses, they do notaccurat ely account for the effects of fr ict ion when the pulse is travelling in the line. Asimplified anal ysi s of these effects is given in appendix D for the case of an acoust ica llyterminated line of low intern al friction (10 perc ent of its surg e impedance). The resu ltsindicate that i f a step change in pressure is applied to one end of the line, attenuation of

33


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111 I 111 , . . ,.

ine wi.,e pu,se at the time it reaches the other end of the be half the final, steady-state value. After a period of time equal to two tr ans it tim es of the line has elapsed,the los ses will have r ise n to their fin al steady-state value. If these resu lts can begeneralized to the nonacoustically terminated line, then, fo r the following ca ses , itwould appear possible to treat the effects of fri cti on approximately by constructing anequivalent source.alent so urc e would be equal to the actua l driving sou rc e with one-half the ste ady-sta telo sse s subtracted.

equivalent source would be the driving source minus the steady-state fricti onal lo sse s.In both cas es , the transmi ssion line would be treate d as if it wer e frictionless and thesource - transmission line - load inte ractio ns would be analyzed in the s ame manner aswas done in figure 3 .

The ana lysis in appendix D is based on el ec tri ca l transmission-line theory and onlyapproximates conditions in a pneumatic trans missi on line. The resu lts , therefore, arenot highly accu rate and the ana lys is should only be used to indicate whether or not atransm issio n line has acceptable o r unacceptable l oss es. For many engineering applica-tions, however, suc h information is sufficient. The correct ions used to reduce lo ss esusually reduce them by fac to rs of tw o or more. In comparison with the e r r o r thatmight occur in the predicted losses , the corrective changes are large.

The technique of subtracting steady-state lo sse s from the driving sou rce and con-structi ng an equivalent source was used to pred ict the behavior of the timing pulse andthe ca rr y signal. As mentioned in the te st , eight lines (1/16 in. (0.159 cm) i.d. and8 in. (20.3 cm) long) were used to ca rr y the timing pulse from amplifier 4 of the pulseconditioning unit to the control ports of the passive AND units. The combined normalizedimpedance of these lines was

(1) If the pulse duration is shor t in comparison to one line t ransit t ime, the equiv-

(2)If the pulse duration was equal to or greater than tw o line tran sit times, the

Zko = 0.167and their combined fricti onal resistance was

R' =0.0169

The normalized receive r pressure-flow ch ara ct eri st ics of amplifier 4 of the pulse con-ditioning unit a r e plotted in figure 26. The dashed lines a r e the rece iv er cha racteri sticsminus the steady-state frictional pres sur e drop in the line. On the sa me plot is shownthe combined control-por t impedance of the eight passi ve AND units.

Starting at location 1 in figure 26, the combined timing pulse line impedance is seen

34


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Receiver output characterist ic-- odified receiver ou tput1.2~ Control-port pressure-flow curve

Normalized transmission-l inee

-Final steadystate ' ,

.0 5 20Figure 26. -Tra jectory of t iming pulse.

to intersect the recei ver output characteristics at location 2 .

\ '!

.3 0 .35 .40 .45

Thus, a pr es su re pulse of- 2approximately 0.19 (Ps - Pe) or 1.14 psi (0.788 N/cm ) will be sen t down the line. Atthe load, it is reflected as a pressure pulse with a f i n a l value of 0.247 (Ps - Pe) or1. 48 psig (1.025 N/cm2 gage) (location 3).viently handled with the aid of the wave diagram shown in figure 27. The diagram is thesame as normally used f or computations of one-dimensional, unsteady flow (for a nexample, see Shapiro, ref. 6, vol. 11, ch. 23). Posit ion of the wave is plotted as afunction of time (vertical axis) and distance along the transmis sion line (horizontal axis).The different areas in the diagram correspond to conditions in the line at different loca-tions and times . Thus, the initial steady-state conditions in the line are denoted as land occupy the lower righ t cor ne r of the diagram . As the wave advances down the tube(from left to right i n the diagra m), conditions change to 2 (the sam e as intersection 2

Determination of the rest of the pr es su re s in the line as a function of time is con-

3 5


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.... . . ...

4. c3.5

3. c

2.5r....w.-4-4 2.0Nm.-E0z 1.5

1.0

.5

1 1 I I I I I I ICo ndition Dimensionless Dimensionless masspres sure, flow. rate,P' m'

1 -0.012 0.05- 2 .192 1.273 .247 .9 6- 4 -. 12 .055 .M3 -. 6- 6 .025 -. 67 -. 10 .05- 8 -.OM -. 59 -. 20 -. 75- 10 -. 12 .05

--

Compression waveR arefaction waveLocation of Dress ure trans ducer- ---

0 .2 . 4 .6 .aRatio of distance down line to length of line, x /ZF igure 27. - Wave diagram of timin g pulse. Length ofl ine, 8.0 inches (20.3 cm); time de lay , 0 .588~10 '~second.

36

. ....


39/67

in fig. 26). Reflection from the right end of the tube (load end) causes a pre ssu re waveto be sen t back down the tube with conditions 3 behind it. The output ampli fie r of thepulse conditioning unit is switched away from the transm issio n line before the reflectedwave has a chance to a r r i v e at its receiver. Thus, the rec eiv er will immediately assu meconditions 4 (intersection 4 n fig. 26) since it has no way of knowing that conditions inthe line are any different than those specified by intersection 2. Region 5 denotes acondition which is internal to the line and exists only momentaril y at the driving ampli-f ie r and the control ports of the passive AND units. Condition 5 can be deter mine danalytically by the linearized theory of characteristics (ref. 6)o r graphically by use offigure 26. If the theory of cha rac te ri st ic s is used, the magnitude of the wave betweenconditions 2 and 3 is calculated

A p t =0.247 - 0.192 = +o. 055A m ' =0.96 - 1.27 =-0.31

and its pr es su re and flow changes are added to those existing in region 6. Thus, thenormalized pr es su re i n region 5 would be

P'= -0.012 +0.055 =0.043and the normalized flow would be

m' =0.05 - 0.31 = -0.26Alternat ively, lines of slope Z' and -ZLo may be drawn from intersec tions 3 and 4,respectively, in figure 27. The intersec tion of these two lines, intersection 5, representsthe same conditions in the line as condition 5 in figur e 28. The rea son for this canbe se en by considering a cut in the transmissi on line at station x where the waves ofintersections 3 and 4 irst meet.portion of the line to the ri ght of the cut would appear to have the pr es su re s and flowsspecified by intersec tion 3 (fig. 26) and a n output impedance of ZLo. The port ion of theline to the left of the cu t would appear to have the pr es su re s and flows of in tersec tion 4(fig. 26) and a n output impedance of -ZL0. The pr ess ures and flows that re sul t from themeeting of the two waves must be compatible with these two equivalent sou rce s. Thus,the intersection of the ZLo and -ZL0 lines drawn from them r epr ese nt the only set ofconditions mutually acceptable to the two equivalent so urces.and source are nonlinear. Conditions 6 at the driving amplifier are specified by a n ex-

At the time the waves arrive at this station, the

Use of figure 26 is most convenient to determine intersections 6 and 7 since the load

37


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tension of the line 3-5 to the unswitched rece ive r ch ar ac te ri st ic s, while conditions 7 atthe load are a n extension of a line at slope -Zko from intersection 5 to the control-portpressure-flow curve. The remaining intersections in figures 26 and 27 may be dete r-mined in sim il ar manner. The resultant theoretical time histor ies of the timing pulseat the control po rt of the AND unit and the pre ssu re transducer used to meas ure thetiming pulse are plotted in figu re 28.

1IL

a- (a) Observed by con trol port.

I l l11 I III I Ihl IIn3Normalized time, tlr(b) Observed by press ure transduce r.F igure 28. - Theoretical timing pulse waveforms.

The experimental timing pulse, shown in figure 17, has a peak amplitude of 1. 2 to21 . 3 psig (0.829 to 0.896 N/cm gage), a duration of the peak of approximately 0.6 to0 .7 millisecond, and steps at the beginning and end of the pulse of approximately0.6-psig (0.414-N/cm -gage) magnitude. Comparison with the theoret ical timing pulsepr es su re (fig. 28) indicates that the analysis underpredicts steady-state lo sse s and pre-dicts a higher initial wave in the line than was observed. The e r r o r in initi al waveheight suggests that the experimental transmission line had eith er a lar ger diameter thanassumed or that its walls were highly flexible. A check of the line size indicated that the

2

38


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actual diam eter was close to theoretical. Line elastance was al so checked, but contri-butes only approximately 3.5 perc ent to the compressibility of the air in the tube. Thus,in this example, the analy sis c or re ct ly predicted the qualitative shape of the timing pulsewaveform and the o rd er of magnitude of the los ses , but subs tanti al quantitative e r r o r sexist between experiment and theory. However, the predicted pulse attenuation was sm al lenough to be re latively unimportant. If the analysi s had predicted a lar ge r pulse attenu-ation, the lines would have been either shortened o r incre ased in diame ter sufficiently toreduce lo sses to a minimum, thus, again their absolute magnitude would be unimportant.

The carry signal from the ce nt ra l bistable units to the power nozzles of the passiveAND units was analyzed in the s am e manner. The output of each c ent ral bistable unit isloaded with one control po rt of a n SB-1 power ampl ifier a nd two 1/16-inch- (0.159-cm-)inside dia met er rubber lines which feed the power nozzles of two passive AND units.The combined normalized res ist anc e and impedance of the rubber l ines are

Zbo =0.469R'= 0.0515

The control por t of the SB-1 am plifie r, although located an appreciable distance fr om theoutput of the ce ntr al bistable amp lif ier s, was considered to be in close contac t with them.Thus, the scaled SB-1 control-port cha rac ter isti cs (see appendix E and fig. 11)weresubtracted f rom the outputs of the c en tra l bistable units t o give a n equivalent output(long-dash line, fig. 29). The pres su re drops due to line frictional res ista nce were thensubtracted fro m the new cu rve to give a n equivalent source output (short-dash line,fig. 29) which was used for computation. The first six reflections a r e shown in the fig-ure.observed on the experimental model (fig. 18, p . 19). The predicted f inal pre ssu re of4.0 psig (2.76 N/cm gage) (33 pe rcen t of gage supply pr ess ur e) is lower than that mea-su re d, and the experimental pulse had no observable overshoot. Thus, it would againappear that the line used in the tests had a larg er internal diameter than had beenassumed in the calculations or that its walls were sufficiently flexible to appreciablydecrease its effective acoustica l impedance. However, the l ine used for the c ar ry sig-nals was the same as that used to distribute the timing pulse. There fore, this suggestionmust be regarded as incorrect. In this example, the analysis overpredicted the los ses ,whereas, in the cas e of the timing pulse, it underpredicted lo sse s.

Figur e 29, which shows mo re than six reflections, does not ag re e with the waveforms2

39


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A0


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A P P E N D I X CDESIGN OF THE PUL SE CONDI T I ONI NG UN I T

This appendix illustrates the techniques used to size the delay line and input orificeof the pulse width fixation portion of the pulse conditioning unit (fig. 8 (p. 11)). Typicalcalculations and g raphs are included. The approaches to be des cribe d should be validfo r us e with any fluid jet amplifiers whose switching characteristics are fast in com-par iso n with the duration of the delivered output pulse.

Choice of the proper input-orifice and delay-line di am et ers is made difficult bycross flow between the contro l po rt s of the fluid je t ampl ifier . When one control po rtof a bistable fluid je t amplifier is pressur ized, the pressure-flow charac terist ics of theopposite control port are changed. Thus, a n input orifi ce can be sized to set amplifier 3(switch it to receiv er R2) when control po rt C2 is left open to atmosphere. A delay-line dia mete r ca n be calculated which will reset it (switch it to re ce iv er R1) when contr olport C 1 is vented. However, because of control-port crossflow , the orifi ce and delayline, in combination, probably will not provide the signal to reset amplifier 3, once ithas been set.

To account fo r the effects of cont rol-port crossflow, it is ne ce ss ar y to obtain botha plot of the rece iv er output pressure-flow ch ara ct eri st ic s and a set of control-portpressure-flow characteristics as a function of opposite control-port pr es su re . Theplots used are shown in figur e 30 and were taken on the amplifie r shown in figure 14(a)(p. 17) . With the ai d of th ese gr aph s, the following iterative procedure can be used tosize both the input-orifice and delay-line diameters:

(1)An initial amplifier 2 output pressure is assumed. With this pressu re , a n inputorifice is calculated which will furnish sufficient p re ss ur es and flows to set amplifier 3(switch it to its receiver R2).equivalent source fo r feeding control po rt C1 may be constructed. By plotting thepressure-flow cha rac ter ist ics of this sou rce on the control-port cha rac ter ist ics of

(2) With the calculated input orifice and as su me d amplifier 2 output pressure, a n

- P~)/ (P -mplifier 3 (figs. 30(a) and (b)), a set of valu es of Pc(P c - Pe)/(Ps - Pe) may be determined. as a function ofS( 13- 0(3) With the or ific e calculated in s tep ( l ) ,a n equivalent input-pressure-flow

characteristic fo r the input orifice - con tro l port C1 combination may be constructed(Pc2 - Pe is assumed to be zero). Thi s pressure-flow characteristic is subtractedfro m the amplif ier 2 output pressure-flow characteristic to give a n equivalent sourc ewhich is available fo r driving the delay line.

41


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. 5r ward swi tching pointII I-.1 0 .1 . 2 . 3 .4 . 5 .6 .7No rmalize d press ure, (Pc - Pe)/ (Ps - P el

(a) Uns witched control port, amplif ier 3; supply pressure m inus exhaust pressure, 3.0 psig (2.07 N/cm Z gage).F igure 30. - Pressure-f low characteristics.

1

- Pe)/(Ps - P ) as a function of P - Pe)/(Ps - P,), fi gu re s 30(a)e ( c2(4)Given 1and (b) can be used to determine the pressure-flow ch ara ct eri st ic s of control port C2of ampl if ie r 3.plotted on the equivalent output char acte ris tic determined i n (3). This plot shows whetheror not amplifier 2 can deliver sufficient pr es su re to reset the ampli fier 3.

(6) If amplifier 2 can deliver sufficient pressure to reset amplifier 3, then a normal-izedde lay-li ne impedance can be selected. Starting from the inters ection of the controlport C2 characteristics of ampli fier 3 with the unswitched output characte rist ics ofamplifier 2 , the impedance will int ersect the switched output cha ra ct er is ti cs of ampli-f i e r 2 at a pre ssu re equal to that assumed when calculating the orifice diameter in ste p (1).The result ant waves and reflections a r e then calculated by use of the techniques ofappendix B to see whether a n acceptable reset signal is delivered to control port C2. I

(5) The pressure-flow ch ara ct eri sti cs of control por t C2 determined in (4) are then

42


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-.1 0 .1 .2 . 3 . 4 .5 .6 .7 .9 1.0 1.1 1.2P (P, - P el(pel - e)/Normalized pressure,

(b ) Switched control port, a mplifier 3; supply pressure minus exhaust pressure, 3.0 psig 12.07 N/cm2ga ge).Figure 30. -Continued.


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AA

\. .- . 6 . - ~ ~ ~ ~ ~-. 5 -.10 -. 5 0 .05 .10 .15 .a .25 .30 .35 .40 .45 .50Normalized pressure, (PR - Pel / (Ps - P

(c) Receiver, amplifier 2 supply pressure minus exhaust pressure, 6.0 psig (4.15 N/cm2gage).Figu re 30. - Concluded.


47/67

the signal delivered is satisfactory, a delay-line diameter is calculated direc tly, by useof equations (B2) or (B3) . If the quality of the reset signal is not satisfactory, then anew output pr es su re f rom amplifier 2 is assurred, a new orifice diameter is calculated in( l ) , and the iteration procedure is repeated.proce dure. The experimentally mea sur ed values of orifice diameter and timing pulsepressure are used in step (1) as initial values so that a comparison may be made betweenthe predicted results and those actually obtained.

A typical iteration is given subsequently in illustrat ion of the previously desc ribed

The supply pr es su re s and power nozzle areas of amplifiers 1 to 3, were as follows:2Ps - Pe = 6.0 psig ( 4 . 1 4 N/cm gage) (supply to amp lif ier 2)2Ps - Pe = 3 . 0 psig (2.07 N/cm gage) (supply to amp lif ier 3)

Since the power nozzle supply pr es su re s, and hence flows, for the two amplifiers aredifferent, scaling fa ct or s must be used when plotting the normalized chara cteris tics ofone on the normalized c ha ra ct er is tic s plot of the other. For these supply pr es su re s,the corresponding pr es su re and flow scaling facto rs are:

(s - Pelamp(p s - Pel =2.0amp 3(ms)(hS)

amp = 1 . 4 1 5amp 3

Thus, if the control-port ch ar ac te ris tic s of amplifier 3 are to be plotted on the receivercha rac teri stic s of amplifier 2, its normalized pr es su re s must be divided by a factor oftwo and its normalized flows by a factor of 1 . 4 1 5 .

Step 1An inlet orifice diame ter of 0 . 0 1 3 5 inch ( 0 . 3 4 3 mm) and a n initial pulse height of21 . 9 psig ( 1 . 3 1 N/cm gage) are assumed to be the actual, f i n a l values. Plotting the

4 5


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I Forward switching pointExtr,I --- lation of iswitcht

/i/

No

mtrol -port character is tic

/I

.45

. 4 .6ized pressure,

-Normali;(pcz - f

7

1.0port C1 o! amplif ler 3 and no rmal ized pressurePe). A mplif ie r switched to control port C1

output pressure-flow characteristics of this sou rce on the unswitched amp lifier 3 control-port characteristics indicates that a switching, or se t press ure, of 0 . 1 7 5 (Ps - Pe)will be delivered to the control port (fig. 31) . Since this pr ess ure is higher than the actualswitching press ure, the control-port characteristic is extrapolated past its triggeringpoint and is shown as a dashed line.

Step 2The orifice pressure-f low c hara cteri stic is also plotted on the switched control-port

characteristic (fig. 3 2 ) . The values that Pc - P,)/(P, - pe) assu mes for variousval ues of (Pc2 - Pe)/(Ps - Pe) is specified by the inte rse ctions of the constant(Pc2 - Pe)/(Ps - Pe) line s with the orifice output pressure-flo w chara cteri stic. Fig-

( 1

46


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lp -

.-4V.Ea-ce *22 .-IVII

FIN.-

-. 0 .1 . 2 . 3 . 4 .5 .8 .9 1.0 1.1 1.2Normalized pressure,Figure 32. - Determination of values of normalized pressure of normalized pressureAmplifier 3 switched to control port C 2


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-orward switching point-pC1 - P,)(P, - pel as function0 Orifice flowsof (Pc2 - Pe)/(Ps - P el -+p212 Normalized ressure,(pC2 - p e ) j p s - pe l1


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.9

Figure 34. - Determination of orifice inpu t pressure-flow characteristics. Amplifier 3 switched to control port C p


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cn0

1.4+

1 . 2 L

1. 0

0

-. 2

-.4-

chara

\,Normal receiver outputEquivalent receiver output \- (orifice flows subtracted)-Orifice input pressure-flow

eristic \

~ __-_------.--------------- ----

___- .0 5 .10 . I 5 .2 0 .25 .30 .35 .4 0Normalized pressure, (P R - Pe)/ (Ps - P elFigure 35. - Equivalent amplifier 2 receiver output characteristic,

.50


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ure 3 1 is used when the power jet is switched to rec eiv er R1, and figure 32 is used whenthe jet is directed toward receiver R2.control-port pressure of Ppressure to rise to Pc - Pe)/(Ps - Pe) =0 . 2 5 . Thi s value is shown as a vertical

Thus, fo r example, figure 32 shows that a C2- P,)/(P, - Pe) =0.15 will cause the C 1 control-port( c2( 1

dashed line in the figurg.

Step 3To determine the input pressure-flow ch ar ac te ri sti cs , the source curve of the input

orifice was shifted sideways on the two control-port characteristics plots. The in te r-sect ion of the shifted orifice sourc e curve with the horizon tal (press ure) axis indicatesthe amplifier 2 pre ssu re which is driving the orifi ce. The inte rsect ion of the orifi ce

(P s - Pe) =0 determinesurve with the zero opposite control pressure curvethe flow which control p or t C1, nd hence the orifi ce, will consume. Shifting the orifi ceoutput curve sideways without changing its fo rm may be justified on the bas is that the flowthrough the orifice is approximate ly incompressibl e and the upstr eam density of the fluiddoes not change appreciably for the amplifier 2 output pr es su re s that are encountered.The (Pc2 - Pe)/(Ps - Pe) = 0 line wa s used to determine control-port C 1 flows,sinc e control-port C1 flow consumption is of in te re st only to dete rmine the pressuresand flows available for driving the delay line. During the t ime when the in itial portion ofthe delay-line pulse is being created, no pre ss ur es will be applied to control po rt C 2 .

The resultant constructions are shown i n figur es 33 and 34. The intersections ofthe orifice output cha ra ct er is tic s with the P (Ps - Pe) =0 line a r e shown asdiamonds. The resu ltant pressure-flow curve determined by the inter sections of theorifice curves with the horizontal pressure axis and the ( c2 - pe)/(ps - =0line ar e plotted on the amplifier 2 recei ver ch aracte ristic s in figure 3 5 . The equivalentamplifier 2 output char acter istic when the orifice flows are subtracted from it is shownas a dashed line in figure 35.

( p c 2 - p e Y

( c 2 - "dl

Step 4Because (Pc - Pe)/(Ps - Pe) is uniquely related to - Pe)/(Ps - Pe) as

- Pe)/(Ps - Pe). Since the1determined by s tep s 1 and 2, the control-port cha rac ter ist ic plots may be used again to

determine the variation of mcJms as a function of

51


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I I IFo rward swi tching point-pC2 - P,)(P, - P as funct iono f (PC1 - Pe)/(Ps - P el -

!cont ro liaracter i~

..

)rt C2 pressure-ic s

I ISA@;-

R 1

.4 5.5 5p8

-.1 0 .1 . 2 .3 . 4 5 . 6 . 7 .a . 9Normalized pressure, (Pc, - Pe) / (Ps - P

p

1.0

F igure 36. -E ffective pressure-f low cha racteristics of control port C2, ampli fier 3. Am plifie r switche d to control port C2

amplifier is assumed to be symmetrical, the control-port charact eristic plots shownin figures 31(a) and (b) can be used to repr esen t contro l port C2 as well as the controlpor t C1. The subscri pt s 1 and 2 on the plots are merel y interchanged. By use ofthe orifice pressure-flow plots in figure s 31 and 32, cu rv es ofI;. may be obtained. These curv es are shown in fi gu re s 36 and 37function of mas dashed lines. Fig ure 31 wa s used to obtain the cu rve in figure 37. Figure 32 w a sused to obtain the curve in figu re 36. Values of (Pc - Pe)/(Ps - Pe), as determinedfrom step 2, fo r vari ous val ues of pC - P,)/(P, - Pe) are plotted as diamonds. Thedashed line drawn through these points r ep re se nt s the effective input press ure- f lowcharact eristi cs for control port C2.

P - p e p s Pel as( c2c2

1( 2

Step 5The effective control-port C2 characteristics determined in step 4 are multiplied

by the previously determined scaling fa ct or s and plotted on the equivalent ampl ifer 252


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.8-

.l---

.6-e,V.Ea,-ce .5-z .4-52 . 3 - - -

cv)I

V,.--m

--+- quivalent control-port C2 press ure--pC 2 - P,)(P, - P as functionflow characteristics Iof (PC1 - Pe)/(Ps - P el -.100 .1 . 2 . 3 . 4 . 5 , . b . , . l .8 .9 1.0 11 12Normalized pressure, (P - Pe)/(Ps - Pc 2Figure 31 . -Effective pressure-flow cha racteristics of control port C2 a nplifier 3. Amplifier switched to control port C1.


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1.6

\.4

~- -~Receiver output characteristic, amplifier 2--- eceiver output characteristic minus input

1.2- orific e flow, a mplifier 2 -+-- Ideal delay-line su rge impedance Experimental valu; of amplif ier 2output pressure (use d as as-,umed o ri fic e in pu t p re ss ur e ini tep (a))----k-elay-line surge impedance actually usedControl-port C 2 pressure-flow characteristics,1.0- amplifier 3-- eceiver output minu s inpu t orifice flow andYI.E.Ea,-e3

-c

0-cYIYIEna,m.-E0z

- -.15 -. 0 -.05 0 .0 5 .I O . I 5 .20 .2 5 .3 0 .35 .40 .45 .50Normalized pressure, (P - Pe)/(Ps - P elF igure 38. - Determination of value of reset pulse.


57/67

output chara cteris tics. Thus, the dashed lines in figures 36 and 37 are scaled and plot-ted on the amplifier 2 output cha rac ter ist ics in figure 3 8 .

As shown in figure 38, amplifier 2 can provide sufficient pre ss ur e and flow to resetamplifier 3 . The delay-line sur ge impedance c an be determined by drawing a linebetween point A, the in itia l condition of the delay line, to point B, the inter sec tion of theinitial assumed orifice pressure with the equivalent amplifie r 2 output charac terist ic(shown as a short-dash line in fig. 38). The delay-line sur ge impedance cannot bearb itra rily chosen, as was done in appendix B. Its sur ge impedance is effectivelyspecified by the initial pr es su re s and flows in it before amplifier 2 is switched (point A)and the pressures and flows that re su lt when amplifier 2 is switched into it (point B).Thes e points were chosen at the beginning of the i terat ion proc edure to s iz e the inputorifice. They mu st be used to calculate the delay-line impedance in ord er to maintainconsistency in the calculations since both the delay line and the input orifice are drivenin parallel by the sa me amplifier 2 receiver.points A and B was not available and a slightly sma ll er one was used instead. Thus, asshown in figure 38, this sm all er line should cause a slightly higher amplifier 2 outputpressure to exist (point C) han was actually observed.

To compute the reset pulse delivered to control por t C2, he delay-line staticpressure losses are first subtracted from the amplifier 2 output characteristics (asdesc ribed in appendix B). The resu ltant equivalent output so ur ce is shown in figu re 3 8 .A sma ll reflection o ccu rs which is al mo st completely absorbed by the equivalent outputso ur ce impedance. The pred icted value of the reset pulse is close to the experimentaland thus indicates the validity of the techniques used to si ze the delay line and to predi ctits performance. In parti cular , single reflection line termination w a s used to goodadvantage. A sm al le r line acoustically matched to the control-port C2 impedancewould have prohibitive fri cti ona l los ses , and a pulse of insuffic ient magnitude to resetthe amplifier would be delivered to the control port.

The line size corresponding to the impedance obtained by drawing a line between

I l l 10 .@I2 .OM .006 .008 .010Time, secFigure 39. - Experimental traces of amplif ier 2 output pressureand reset pulse delivered to contro l port C2, amplif ier 3.

55


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Experimental pr es su re pro files of the pulse as delivered to the beginning of the delayline and as measured by a transducer approximately 1 nch (2 .54 cm) from control portC2 are shown in figure 39. The initial pulse rise time is fast and has a small step in it.This step, which is the reflected wave, appe ars since the transducer is a finite distancefrom the junction of the control port and the interaction region. If the transducer couldhave been mounted directly at the junction between the control port and the interactionregion, thi s reflected wave would not have appeared as a separate pulse.the procedur es outlined in appendix B. The result ant output pulse delivered by ampli-fier 4 into an acousti cally terminated 1/8-inch- (0.318-cm-) inside- diameter line isshown in figure 40. A s is shown, a pulse with fast rise and decay tim es, constantamplitude, and 1-millisecond duration could be delivered by the pulse conditioning unit.

The lines between amplif iers 3 and 4 were sized for no-reflection termination by

:Ew oLm- m.-

F2cVI_CL (a) C i rc ui t pressure, 12.0 psig. (8.26N

E E- Nlcm' gage. 13u-z VI2-3VInEa :E

0E

VI

aE t0*I

-1- I. 1 ~ - 10 .002 .OM .OM .008 .010Time, sec

(b) C ircu i t pressure, 6.0 psig. (4.13Nlcm' gage. 1Figure 40 - Output of pulse co ndi t ioning unit.

56


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APPENDIX DSIMPLIFIED TRANSMISSION LINE

This appendix pr es en ts a brief t rea tment of an acoustically terminated pneumatic(1) To illust rate some of the effects resulting from transmission-line friction(2) To provide justification for use of a n equivalent source and frictio nless tra ns -Several sophisticated models of pneumatic transmis sion lines have been made which

transmission line:

mission line to repre sen t a transmission line with s ma ll but finite frictiontake into account the effec ts of time va rient heat transfer and changing velocity profi les(refs. 7 o 9). Although the analyse s accura tely predict the performance of se mi -infinite lines, they are hard to apply in a situation where numerous interactions ca noccur, by means of waves, between a nonlinear source and load. Thus, some autho rshave chosen to modify the s imple r equations fo r a n electromagnetic transmiss ion line tore pr es en t the physically m or e complicated pneumatic line (refs. 10 to 12). If highaccuracy is not de si re d and if approp riate co rrecti ons a r e made to account for changingvelocity p rofiles and heat transfer, this latter approach can, on occasion, yield reason-able results (refs. 10 to 12).than the exact analysis, the elec tr ic al transmission-line analogy is used in thi s appendixto point out the effects of line friction.

Because it can yield reasonable re su lt s with fa r less effort

Figure 41. - Simplif ied m o d e l of t ransmiss ion l ine.Borrowing the analogy from electromagnetic transmission-line theory, the

pneumatic trans missi on line may be regar ded as composed of a n infinite number ofseries impedances and shunt admittances, as shown in figu re 41 (refs. 9, 11, and 13).The following equations fo r continuity and momentum can be written f or a n elementallength d x of the line (refs. 9, 11, and 13):Continuity:

ddx-m (x, ))=- Y s)P(x,s)


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Momentum:ddx- P(x,s)) = - Z(S)l;l(X,s)

Combining equations (Dl) nd (D2) o solve for the pressure P(x, s) and solving theresultant differential equation yield the following classical result fo r waves in a trans-mission line (refs. 9 and 14):

where

is usually ref er re d to as the propagation operator.Pa(s). Fr om equation @2)

The m as s flow m ( s ) may be solved fo r in te r m s of the initially applied pr es su re

or

where

is usually refe rre d to as the surge impedance of the line. For sma ll friction and s hor tt imes (s - a), t can be assumed that

@ * F O-


61/67

which permits equations (D4) and (D7) to be rewritten

z - zcop+ ; )Cwhere

XC

T = -

@xcy=-2zco

If , f o r the moment, only right traveling waves (Pb(s) =0) are considered,

Thus, the term, xr(s)

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A Breadboard Fluidic-Controlled Pneumatic Stepper Motor

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