Army Motei-iei Development · ~ /0 Gary V. Amith 9. PERFORMING ORGANIZATION NAME AND ADDRESS 10....

10YN

- ~Ai r" ~ f

sOASTEANAXlATICAL AND) EYYERIaENAL StUDIES OF

LANIW4'TAR 'PiOPORTIONAL AMT'LIFXERS -

- PRhfL 1976

F-ren'red by - -

'Pm- Pcrit-na S1rate fUiversity'$v-t.~!13 'in6Cotrols La borarory

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!ThivcQ. 'Y Ti rUPcmyva 160902Z

-Under Cnrc

-~ ~-.U.S. Army Motei-iei Development

7.orsd Readiness Commanrd

W4ARRZY DIAMOND L ASO. RATC0R!V

Adlelptd, Mcryknd 207S3

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-4 'Some Analytical and Experimental Studies of . T OP REPORT & PERIOD COVERED

'(' aminr Prportona Amplifiers ______________

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S. CONTRACT OR GRANT NUMBER(s)

I DAAG-34-73-C-0213J. Lowe hearer DRCMS Code: 611102.11.71200

~ /0 Gary V. Amith9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASK

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It. KEY WORDS (Continue on reerse side it necessary and Identify by block number)

Proportional Amplifier (Laminar) Pressure Noise CharacteristicsSupply CharacteristicsInput Characteristics

N' Transfer Characteristicstput CharacteristicsI. TA.T (Continue on reverse side It noes~eaary and Idenilly by block number)-his report presents the results of an analytical and experimental investi-

gation of a pressure-controlled proportional amplifier which operates with a sup.ply jet in the laminar flow regime. A mathematical model is developed which may

deIecope isicbe used for the prediction of the supply characteristics, input characteristics

for three types of input signals, transfer characteristics, and output characteristics of the amplifier. Acceptable agreement is found to exist between theoryand experimental data although theory predicts a slightly greater aspect ratiodependency than the data indicate. (More complete abstract found on page 3).

.. N.P I 14OV #. .i s L ,0NC... . ....

I'

ABSTRACT

This report presents the results of an analytical ard experimentalinvestigation of a pressure-controlled proportional amplifier which operateswith a supply jet in the laminar flow regime. A mathematical model isdeveloped which may be used for the prediction of the supply characteristics,inpUL characteristics for three types of input signals, transfer character-istics, and output characteristics of the amplifier. Acceptable agreementis found to exist between theory and experimental data although theory pre-dicts a slightly greater aspect ratio dependency than the data indicate.

In general, the data indicate that the normalized operating character-istics are relatively weak functions of the aspect ratio and Reynolds numberover the investigated range, although a strong dependency of the operating

characteristics on the mean control pressure was found to exist. Both thedeflected-jet input resistance and pressure gain of the amplifier were foundto decrease with increasing mean control pressure. It was also determinedthat a reduction in the setback of the control knife-edges results in anincrease in the null input resistance as predicted by theory. A setbackreduction of one-half to one-fourth the supply nozzle width was found tonot decrease the linear range of operation of the amplifier. An experimentalinvestigation of the differential output noise characteristics of the ampli-fier indicates a linear increase of noise with increasing aspect ratio andexponential growth of noise with increasing Reynolds number for a fixedbandwidth of noise measurement.

I

I

3 3

I

TABLE OF CONTENTS

A.BSTR.ACT .. .. ........................... 3

1. INTRODUCTION .. .. ......................... 9

j2. MATHEMATICAL MODEL. .. ....................... 12

2.1 Supply Characteristics .. ... . ..... ..... . ... 1212.2 Input Characteristics. .. ... ..... ...... .... 16

2.3 Transfer Characteristics .. ... ...... ........ 22

2.4 Output Characteristics .. ... . ..... ..... . ... 24

3. EXPERIMENTAL RESUJLTS. .. ... ..... ...... ....... 26

3.1 Supply Characteristics .. ... . ..... ..... . ... 26

3.2 Input Characteristics. .. ... . .... ...... . ... 29

3.3 Transfer Characteristics .. ... ...... ........ 31

13.4 Output Characteristics .. ... ...... ..... .... 35

3.5 Differential Output Noise. .. .. . ..... ........ 39

34. CONCLUSIONS. .. .. ...... ..... ...... ...... 39

*REFERENCES .. .. .. ..... . ..... ...... ...... 42

3DISTRIBUTION .. .. . ...... ...... ..... ...... 43

LIST OF ILLUSTRATIONS

Figure

1 Schematic of LPA used in this study. .. .. ...... .... 112 Control port region fluid circuit diagram for supply jet in

null position. .. .. ..... ............... 173 Control port region fluid circuit diagram for'deflected'

supply jet .. .. .. ..... . ..... ..... . ..... 174 Supply characteristics for a - 0.56. .. .. ...... . ... 275 Supply characteristics for a - 0.56; a-1.12; a -1.56 . . . 286 Null and push-pull input characteristics for a -1.12;

Nh- 110.... .. .. .. .. ........ . .. .. .. .. .. .... 30#17 Null and push-pull input characteristics for B SB 0.25 andB 3 -B 0.5. .. .. ..... . ..... ...... ...... 32

38 Push-pull transfer characteristics for a - 1.12; N -1100 339 Blocked-load pressure gain variation with mencon rolpressure .. .. . ................ ................ 34

10 Variation of pressu~re gain under blocked-load conditions withIReynolds number. .. .. ...... ...... ........ 3611 Normalized pressure gain versus output loading. .. ... . ... 3712 Output characteristics for a - 1.12; N~h - 1000; Pm - 0.15 . 38

13 Differential output noise as a function of Reynolds number .. 40

ii5

'77 IV

NOMENCLATURE

B' Control channel widthcc

Br Average receiver channel widthra

' Receiver inlet widthri

B' Supply nozzle widths

B;B Setback of control knife-edge

Cd Discharge coefficient

Cfd Coefficient of fully developed friction factor

C Vena contracta coefficient -,vc

Ce Momentum flux discharge coefficient

D' Diameter of cylinderical leading edge of splittercyl

G Pressure gain

H' Amplifier height

it Control momentum flux incident upon receiverscr

if Supply momentum flux incident upon receiverssr

kd Coefficient of Yd

K, Pressure drop factor

L' Axial distance from supply nozzle exit to control knife-edgesc

L' Control channel lengthcc

L' Receiver channel length

L;' Axial distance from supply nozzle exit to receiver entrance

L' Axial distance from control knife-edges to receiver entrance jv

L' Supply nozzle throat lengththL' Virtual origin lengthvoN Reynolds number, V'sH'/v'

Nb Reynolds number, V' B'/v'Rb saa

q Bias pressure6

IP'bm Mean bias pressure [P'bl + P tb )/2 ]

P1 Control pressure

P: Mean control pressure (P' + P' )/2]cm cl c2

Pt Kinetic (i.e. dynamic) pressure of entrained flow

P' Output pressure0

P Null output pressure

5 Pt Receiver pressurer

Null receiver pressureP rn

P Supply pressure

AP' Pressure drop due to boundary layer growth

of Fluctuating component of the differential output pressure

Qc Control volumetric flow rate

Control volumetric flow rate for null operation

Id Volumetric flow rate through orifice formed by lateraldisplacement of supply jet

SQ' Output volumetric flow rate

Qon' Null output volumetric flow rate

IQ Supply volumetric flow rate

R ' Incremental fluid resistance of null supply jet-controlknife-edge orifice

Apparent fluid resistance of null supply jet-control knife-S'PPedge orifice

R' Incremental fluid resistance of control channel| cc(Ra p Apparent fluid resistance of control channel

R' Incremental fluid resistance of orifice formed by lateraldeflection of supply jet

R' Input resistance

RIncremental load resistance

3 (R)app Apparent load resistance

7

R' Incremental fluid resistance of receiver channelr

(Rw) app Apparent fluid resistance of receiver channel

vt Velocity in boundary layer

V' Potential core velocity of supply flowp

V' Average exit supply velocitysa

x' Coordinate parallel to axis of LPA

y' Transverse coordinate of LPA

!Yb Incremental fluid admittance of null supply jet-controlknife-edge orifice

y Lateral jet displacement at control knife-edge

Incremental fluid admittance of orifice formed by laterald deflection of supply jet

Yr Lateral jet displacement at inlet to receiver

Greek Symbols

Incremental change

6*' Displacement thickness

Stt Boundary layer thickness

V1 Kinematic viscosity

p' Fluid density

0 Aspect ratio, H'/B'8

Superscripts

Denotes dimensional quantity

Subscripts JBL Blocked-load conditions

cv converging section of supply nozzle

e exit

i inlet

1,2 Denotes opposing sides of inputs and outputs of LPA

8-

1" . .. .... . .... ..... ........ . .

i. INTRODUCTION

a One of the primary components used in the design of fluidic control sys-Items is the proportional amplifier. The type of proportional amplifier whichhas received the most attention is the jet-deflection amplifier (also known asthe beam deflection amplifier) originally conceived by B. M. Horton in 1959.Early work on the jet-deflection amplifier was centered on a turbulentsupply jet being deflected by the momentum of turbulent control jets.

The primary disadvantages of jet-deflection amplifiers which operate withturbulent supply jets are low gain, low signal-to-noise ratios and stabilityproblems which primarily arise when large output loads are applied to theoutput ports of the amplifier. The low gain and low signal-to-noise ratiosof the amplifier are primarily due to the nature of the turbulent supply jet.Both the gain and normalized pressure recovery are limited because of therapid spread of a turbulent jet. In addition, the noise output of theamplifier is strongly related to the turbulent nature of the supply jet.

One approach to circumvent these problems associated with the jet-de-flection amplifier is to operate with a laminar supply jet. If the supplyjet Reynolds number at the exit of the supply nozzle is not too low, thespread of the laminar jet will be much less than a turbulent jet--resultingin an increase in both gain and normalized pressure recovery. Signal-to--noise ratios for jet-deflection amplifiers designed for operation withlaminar supply jets would also be expected to be much greater than amplifiersdesigned for operation with turbulent jets.

The development of a jet-deflection amplifier which operates in thelaminar flow regime is described by Manion and Mon,

1 Smith and Shearer, 2

Manion and Drzewiecki, 3 and Smith. Since this is the final summary reportrequired under this contract it will briefly review aspects of earlier re-ports and referenced literature which are pertinent and essential to thedevelopment of ideas, concepts and analysis employed in this research.

1Manion, F. M., and Mon, G., Fluerics: 33. Design and Staging of LaminarProportional Amplifiers, Harry Diamond Laboratories-TR-1608, Adelphi, MD, 1972.

2Smith, G. V., and Shearer, J. L., Research Investigation on Laminar Pro-portional Amplifiers, Final Report Contract DAAG-39-72-C-0190, The PennsylvaniaState University, University Park, PA., 1974.

I 3Manion, F. M., and Drzewiecki, T. M., Analytic Design of Laminar Pro-portional Amplifiers, Proceedings HDL Fluidics State-of-the Art Symposium,Adelphi, MD, 1974.

4Smith, G. V., An Analysis of Laminar Proportional Amplifiers, Ph.D.Thesis, The Pennsylvania State University, University Park, PA., 1975.

* 9

The work of Manion and Mon1 describes the development of a laminar pro-portional amplifier (LPA) very similar to that shown in Fig. 1. This design,which has a setback of one half of the supply nozzle width, uses a pressurefield to deflect the laminar supply jet. They report that a pressure fieldeffectively deflects the laminar jet without disturbing the flow field

whereas a momentum exchange design introduces noise.

The Reynolds number range in which the LPA should operate is limited byexcessive jet spreading on the one hand and transition to turbulence on theother. Manion and Mon,1 Smith and Shearer,2 and Smith4 report on the resultsof flow visualization studies and report that the device should operate pro-perly in a Reynolds number range of approximately 700 to 1500 where theReynolds number is based on the amplifier height and the average supplyvelocity. Smith and Shearer2 and Smith4 also report on the feasibility ofoperating the LPA with reduced setback.

Manion and Mon present a control volume analysis for the static anddynamic pressure gain of the LPA. Their analysis, which is restricted to apush-pull input signal, relies in part on experimental data and does notinclude all of the effects of the bias pressure on the performance of theLPA. Their report also includes a method to properly stage the LPA.

Manion and Drzewiecki3 present a fairly comprehensive analysis of theoperating characteristics of an LPA. Their analytical investigation of the inputcharacteristics, transfer characteristics and output characteristics are com-pared to data and, in general, acceptable agreement is found to exist. It isfelt, however, that their analysis of the control port-supply jet interactionregion does not include all of the bias pressure effects. Their analysis ofthe transfer characteristics is limited to blocked-load conditions at theoutput ports while their analysis of the output characteristics is limited tonull supply jet operation. In addition, both the input characteristics andtransfer characteristics are limited to push-pull input signals. Their paperalso includes a dynamic analysis of the LPA as well as geometric configurationand contamination sensitivity problems.

iManion, F. M., and Mon. G., Fluerics: 33. Design and Staging of LaminarProportional Amplifiers, Harry Diamond Laboratories-TR-1608, Adelphi, MD, 1972.

Smith, G. V., and Shearer, J. L., Research Investigation on Laminar Pro-portional Amplifiers, Final Report Contract DAAG-39-72-C-0190, The PennsylvaniaState University, University Park, PA., 1974.


3Manion, F. M., and Drzewiecki, T. M., Analytic Design of Laminar Pro-

portional Amplifiers, Proceedings HDL Fluidics State-of-the Art Symposium,Adelphi, MD, 1974.

10

o (n

1 -

I 2C4

002

C: 0. 0 2

0 f0

The purpose of this paper is to report the results of an analytical andexperimental investigation of the operating characteristics of the LPA shownin Fig. 1

2. MATHEMATICAL MODEL

In this section a mathematical model for the prediction of the operatingcharacteristics of laminar proportional amplifiers is presented. In theformulation of this model primary consideration was given to developing atheory which, in addition to accurately predicting the operating character-istics of an LPA, would easily indicate the effects of various operatingparameters and important geometric parameters. In addition it was deemedimportant to develop a mathematical model which could easily be used fordifferent types of input signals commonly encountered in actual applications.

In this analysis primes are used to denote dimensional quantities whileunprimed quantities denote dimensionless quantities. All pressures, flowrates, and length dimensions are normalized with respect to the supply pres-sure, supply flow rate and supply nozzle width respectively.

2.1 Supply Characteristics

The basic approach to this analysis of the pressure-flow character-istics of laminar flow in the supply nozzle of the LPA shown in Fig. 1 willbe the determination of the pressure drop across the nozzle for a givenvolumetric flow rate.

The supply nozzle is composed of three separate geometric regimes:a parallel wall plenum chamber; a converging section; and a parallel wallthroat section. Calculations indicate that the pressure drop across therelatively large cross sectional area plenum chamber is sufficiently small, ascompared to the presssure drop across the converging section and the throatsection, that it may be neglected without appreciable error.

For the converging section the inlet flow will be assumed to have auniform velocity profile and the boundary layer growth will be determinedthrough the use of von Karmen's momentum integral equation for two-dimension-al incompressible boundary layers with an axial pressure gradient. Although

the flow in the converging section is three-dimensional due to the presenceof the bounding plates of the LPA, the growth of the boundary layer is assumedto be the same on the vertical walls as on the bounding plates and smallenough so that two-dimensional flow may be assumed. The assumption of asuitable velocity distribution in the boundary layer (v'=V' sin [wy'/26'])enables the momentum thickness of the boundary layer and t~e wall shear

t

stress to be related to the displacement thickness of the boundary layer sothat von Karmen's momentum integral equation may be expressed in non-dimen-sional form as

* d6* * 2dV_

6 V - + 4.65 (6) 2dV 1.51 (1)p dx dx N

Rb

where V and both 6 and x are normalized with respect to V' and B'p sa srespectively.

12

........... ....... ....ll ... .. .. .....il ... ....... ... ........ .. . ...... -~ 1.. .......... .... ... ... .......

IThe variation of the core velocity with axial distance is estimated

from continuity for this converging nozzle as

v __ __ (2)p (5 - 0.4x)

Substitution of Eqn. (2) and its derivative into Eqn. (1) yields

6 d6* 1.86 6*2 1.51)1 (5 0.4x) dx +(5 -0.4x)2 1Rb

If it is assumed that the displacement thickness increases propor-

tionally to the square root of the ratio of the axial distance to the Rey-nolds number based on nozzle width (an assumption justified by Smith

4 ), the

displacement thickness at the end of the converging section is determined

from Eqn. (3) to be

6* 0.89 (4)

(N Rb

The core velocity at the end of the converging section is obtained

from continuity as

pi (1 - 25) (1 - 26./a) (5)

The pressure drop in the converging section for the case with bound-

ary layer growth is greater than that would occur with ideal hydrodynazaic flow.The increase in the pressure drop is found by applying Bernoulli's equation toa centerline stream tube which yeilds

A 'cv =(Vpi)2 (6)

I ~pV' 2 p

The pressure drop in the throat section is obtained from an analysis

of laminar flow development in rectangular channels with a uniform velocity

profile at the inlet. Schlicting5 has shown that the difference between thecore velocity and the average velocity in the entry region of a parallel

plate channel is proportional to the square root of the axial distance. This

relationship continues to apply in the entry region of a rectangular channelwhere the interaction between boundary layers is negligible. Applying

Bernoulli's equation to the centerline stream tube enables this relationship

4 Smith, G. V., An Analysis of Laminar Proportional Amplifiers, Ph.D.

Thesis, The Pennsylvania State University, University Park, PA. 1975.

5 Schlicting,H. Boundary Layer Theory, McGraw-Hill Book Co., Inc., N.Y.,

I 1968.

13I

to be expressed as

____) [12 F) N. Rh (7)

The constant of proportionality, Cl, is determined from the experimental dataof Sparrow et al. 6 as approximately equal to 14. It is noted that this valueis close to the value of 13.74 obtained by Shapiro et al.7 for developingflow in tubes.

The analysis of the pressure drop in the throat section is madeslightly more complicated due to the fact that the velocity profile enteringthe throat section is not uniform but is partially developed due to boundarylayer growth in the converging section. Thus a "virtual origin length",defined as the entrance length of a rectangular channel over which the pres-sure drop due to boundary layer growth is the same as in the convergingsection, is obtained by substitution of Eqn. (6) into Eqn. (7) which yields

L NRh (2(V) ) 2 (8)vo 196 1(Vi) - ]

It should be noted that for Reynolds numbers (based on amplifier height) from600 to 1500 and for aspect ratios of 0.5 to 2.0, Eqn. (8) varies by less than10 per cent from a "virtual origin length" of 0.36.

The total pressure drop in the supply nozzle is that due to theboundary layer growth plus that required to accelerate the fluid under idealhydrodynamic flow conditions to the average supply velocity. Thus the totalpressure drop is given as

p , p ,( A P ' )s bp+. 1+ W ?) 2 (9)

sa sa

The supply nozzle discharges coefficient is defined from the relation=' [2(Ps PI) 1

Q I = CdH'B' b (10)qs d s P'(o

6Sparrow, E. M., Hixon, C. W., and Shavit, G., Experiments on Laminar

Flow Development in Rectangular Ducts, Trans. ASME, Journal of Basic Engine-ering, Vol. 89, 1967.

7Shapiro, A. H., Siegel, R., and Kline, S. J., Friction Factors in theLaminar Entry Region of a Smooth Tube, Proc. U. S. Nat. Congr. Appl. Mech.,1954.

14,

-4-,i

I I II I I I I I I II I I II I I I I I I I I IiI -

Substituting Eqns. (7) and (9) into Eqn. (10) and solving for thedischarge coefficient yields

C Cd + a 1LoI + 14 2o J) N (11)

It is interesting to note that the term, (1+ a)/2(a) varies by amaximum of 6 per cent from a value of unity for aspect ratios of 0.5 to 2.0.Thus it is noted that the discharge coefficient is primarily determined by the

Reynolds number based on amplifier height.

It shguld be noted that Eqn.(l) is based on the experimental data ofSparrow et al. and although strictly valid only near the entrance of thechannel, may be applied without appreciable error for

12

NRh >65(L ° + Lth) "(12) '1

For Reynolds numbers less than that indicated by Eqn.'(12) an expres-

sion for the pressure drop for fully developed flow in channels with anentrance length correction should be used. Smith4 giv- .s such an expressionfor low Reynolds numbers as well as an expression-for the discharge coeffici-ent when the Reynolds number is based on the pacit potential core velocity asopposed to the average exit velocity. -'

If it is assumed that t oundary layer thickness is approximatelythe same on all sides at the it of the supply nozzle, the displacementthickness at the nozzle t is given as

*_ + a I [ 1 Cd] (13)

The momentum flux discharge coefficient, is obtained by reducingthe nozzle dimensions by the sum of the displacemen2 thickness and the momen-tum thickness. If it is assumed that the pressure gradient at the exit ofthe nozzle is negligible, the ratio of the displacement thickness to themomentum thickness is 2.59. The momentum flux discharge coefficient maytherefore be expressed as

6Sparrow, E. M., Hixon, C. W., and Shavit, G., Experiments on LaminarFlow Development in Rectangular Ducts, Trans. ASME, Journal of Basic Engine-ering, Vol. 89, 1967.

4Smith, G. V., An Analysis of Laminar Proportional Amplifiers, Ph.D.Thesis, The Pennsylvania State University, University Park, PA. 1975.

15

* *

C = [1 - 2.77 6e ] [1 - 2.77 6 /c] (14)

2.2 Input Characteristics

The mathematical model for the determination of the input character-istics of the LPA is based on a lumped parameter analysis of the input flowfield when the supply jet is in the centered (i.e. null) position. The re-sulting flow equations are then linearized for the input characteristics whenthe jet is deflected due to a lateral pressure gradient.

Figures 2 and 3 present the fluid circuit diagrams for the supplyjet in the null position and in the deflected position respectively. Thesubscripts 1 and 2 denote opposing control ports.

In Fig. 2 P is the control pressure applied to the control port,Q is the control flow rate, P is the bias pressure existing at the supplyc extadbnozzle exit and P is the vent pressure. All pressures in this and succeed-

Ving sections will be gage pressures measured relative to the vent pressure.

In Fig. 3 the 6's are used to denote small changes of the variablesfrom their mean or null operating value. The change in control flow rate,5Q , is due to the difference between the change of control pressure, 6Pan& the change of bias pressure, 6P . The change in the null control flowrate, 6Q n' is due to a change in tke bias pressure while 6Q is the incre-mental fiow rate through the area formed by the lateral displacement of thesupply jet. It should be noted that 6Qd may be either plus or minus depend-ing on the direction of the lateral displacement of the supply jet.

The apparent resistance of the control channel, (R )a , is obtainedfrom the pressure-flow characteristics of laminar flow in rectanRular chan-nels which includes both fully developed losses and entrance losses associatedwith developing flow. In normalized form this may be expressed as

Pc - Pb =(R) Q (15)

where

CfdLccCd ] 2 Q (16)(R)ap = +-K 1 -- Q (16)cc app 4B cNa B cc Bcc j

Typical experimental values of Cfd and K from Sparrow et al.6 aregiven as

6Sparrow, E. M., Hixon, C. W., and Shavit, G., Experiments on LaminarFlow Development in Rectangular Ducts, Trans. ASME, Journal of Basic Engine-ering, Vol. 89, 1967.

16

L[

clcc app bi

(Rb)appI

Figure 2 Control port region fluid circuit diagramfor supply jet in null position

5C1 cclbl

R b1 R dl

5 6Qcnl 6Q dI

Figure 3. Control port region fluid circuit diagramI for deflected supply jet.

II 17

ca - 0.5; C = 62.2, K1 = 0.99fd1

- 0.2; Cfd 76.3, K1 = 0.89,

8while analytical values of Cfd may be obtained from Han.

The incremental resistance of the control channel, R , defined asthe slope of the pressure-flow characteristic at a given operating conditionis obtained by differentiation of Eqn. (15) (using Eqn. (16) for (R cdapp)with respect to Q Cwhich yeilds c p

fdcc d2 Cd 12Rcc 4B B + 2K, [ Bc (17)

ccB cc cc

The pressure-flow characteristics for the region at the end of thecontrol port channel when the supply jet is in the null position is obtainedby modeling the area bounded by the control knife-edge, supply jet and thetop and bottom plates as an orifice. The analysis of this orifice area iscomplicated because of the necessity of defining an effective jet width atthe axial location of the control knife-edges and the dependency of the jetwidth on both the bias pressure and the spreading of the laminar jet. It isalso noted that the flow rate through this orifice is not zero when the pres-sure drop across the orifice is zero but is equal to the rate of control flowentrained by that side of the supply let.

Smith4 discusses the effect of the bias pressure on the width ofthe supply jet and uses a simplified approach which employs an effectivepressure drop to account for the complex nonlinear relationship that existsbetween the total flow and the entrainment and pressure drop factors. Hestates that the flow rate through the orifice may be expressed as

FcSB + 0.25 P 1Qc d P b + Pkin (18)

where L r [C2 12

Pkin 2N CBS (19)

8Han, L. S., Hydrodynamic Entrance Lengths for Incompressible Laminar

Flow in Rectangular Ducts, Trans. ASME, Journal of Applied Mechanics, Vol. 82,1960.

4Smith, G. V., An Analysis of Laminar Proportional Amplifiers, Ph.D.

Thesis, The Pennsylvania State University, University Park, PA., 1975. .1

18

The first term in brackets in Eqn. (18) is the effective area of theorifice normalized with respect to the effective area of the supply nozzleexit. The term C is a coefficient to account for the vena contracta effectin this orifice wtich is taken as equal to 0.75 from an inviscid analysis byEhrich 9 for vena contracta coefficients in two-dimensional needle valves withan input angle of 450. The increase in this orifice area with increasingbias pressure as indicated by Eqn. (18) was observed in flow visualizationstudies carried out by Smith 4.

The second term in brackets is the effective pressure drop acrossthe orifice which consists of the bias pressure, Ph' and the dynamic pressureof the entrained flow rate, P . An estimate of he entrained flow rate isnobtained from Eqn. (18) when te bias pressure is zero.

Defining the apparent resistance of the control knife-edge orifice,(1)a, as the ratio of the pressure drop across the orifice to the flow[ ra eaRrough the orifice yields

C=db (20) .

[[app [CBSB + 0.25 Pb] [Pb + Pk

The incremental resistance of the control knife-edge orifice isobtained by differentiation of Eqn. (18) with respect to Pb which yields,

I 8Cd [eb + P kin ] (21)'b 4 C B + 3P + 2P

Yb vc SB b kin

In Eqn. (21) Yb is the incremental admittance of the control knife-edgeorifice. Both the apparent resistance and incremental resistance of thecontrol knife-edge orifice are noted to be strong functions of the biaspressure.

The input characteristics for the supply jet in the null positionis given as

PC[R + (Rb) QC (22)

where (Rcc)app and (Rb)app are given by Eqns. (16) and (20) respectively.L9Ehrich, F. F., Some Hydrodynamic Aspects of Valves, ASME Paper No. 55--

A-114, 1955.


19

It is noted, however, that the apparent resistance of the control

channel is a function of the control flow rate while the apparent resistanceof the control knife-edge orifice is a function of the bias pressure. Themost straight forward procedure for the determination of the null inputcharacteristics for given operating conditions is to determine the controlflow rate for a given bias pressure from Eqn. (18). The corresponding con-trol pressure is then easily obtained from Eqn. (15) for the given biaspressure and the calculated flow rate.

Before the deflected-jet input characteristics can be obtained, itis necessary to determine the relationship between the supply jet deflectionand the differential pressure gradient existing across the supply jet. Afluid particle, upon exiting from the supply nozzle, undergoes a transverseacceleration due to the transverse pressure gradient acting upon it. Assumingthat each fluid particle in the supply jet at a given axial location has under-gone the same transverse displacement, that the axial jet velocity isessentially constant over the control port region and that the transversepressure gradient is linear, Smith 4 has shown that the lateral jet displace-ment, yc, at the end of the control port region is given by

L 2 (Pbl - b2(Yc 4 C@ (i Pbm (23)

The flow rate through the additional orifice formed by the lateraldisplacement of the supply jet, Qd' is given by

Yc Pbm i(24

d Cd (24)

Adopting the sign convention that a positive jet displacement resultsin a flow rate out of the control port region yields

2SLc Pbm (25)

dl = 4CdCe(I - Pbm) [bl - Pb2(

Perturbation of Eqn. (25) for small changes in the bias pressureyields

6Q Y SP (26)dl Ydl bl

Smith, G. V., An Analysis of Laminar Proportional Amplifiers, Ph.D.Thesis, The Pennsylvania State University, University Park, PA., 1975.

20K ,

wherePb2

y__ 1 k b2 (27)Ydl Rdl d bi

and Lc2 P bm

kd 4Cd C(-cm) (28)

The term k is a deflection coefficient of the admittance, Y' and,although it is the game for each control port, it is a strong function of thebias pressure. The admittance is also observed to depend on the variation ofP with P (i.e. 6P /6P ) indicating a dependency on the type of inputbi2 b2 bignal applied to the symmetrically opposed control ports.

Solving the fluid circuit shown in Fig. 3 for the deflected-jetinput resistance yields

6P 1 + R (Y + Ydl (R - + (29) 4

i Qcl b dl

As previously noted the admittance, Y1I and thus the deflected-jetinput resistance, Ri, of control port 1 depends on the nature of the sourcesused to provide the Input signals applied to the sy-metrically opposed con-trol ports. This is due to the cross-coupling existing between the controlports provided by the supply jet. Smith4 has analyzed three different typesof input signals and has determined the resulting deflected-jet input resist-ance for control port I when it is driven by an ideal pressure or flow source.The resulting expressions are given as:

Case I, 6Pc2 - - 6Pl; (Ideal push-pull pressure sources)

1 + Rc(Yb + 2 kd) (0R Ril" Yh + (30)3 ii b + 2 d

Case II, -Pc2 0 0; (Control port 2 driven by ideal pressuresource held constant)

1 + R c[2(Yb + kd ) + Rc Y b(Yb + 2 kd)

R = ~ b+ - c bd (31)Ril " Yb + kd + RccYb(Yb + 2 k ) (3),.3b d cceb b d


21

Case III, 6Qc2 = 0; (Control port 2 driven by ideal flowsource held constant)

R b d cc b (Yb + 2 kd)il Yb (Yb + 2 kd) (32)

Smith 4 has also analyzed and developed an expression for the degreeof cross-coupling existing between the control ports.

2.3 Transfer Characteristics

The transfer characteristics are analyzed by determining the pressuregain of the LPA under blocked-load conditions. The receiver pressure, P , isdefined as the pressure existing at the entrance to the receivers while theoutput pressure, P is defined as the pressure existing at the output ports0of the LPA. The pressure gain, G, of the LPA is defined as the change ofdifferential output pressure divided by the change of differential inputcontrol pressure. For a small-signal analysis this is expressed as

G P ol - SPo2S cl - Pc2(33)

The notation of control ports 1 and 2 and receivers 1 and 2 is suchthat a positive differential input signal results in a positive differentialoutput signal.

Under blocked-load conditions the flow through each receiver is zeroso that the receiver pressure is equal to the output pressure. The blocked-load pressure gain, GBL, is then given as

(aPol - 6Po2)BL (6Prl - 6 r2)BLG BL 6P - 6P 6P - 6P (34)BL cPP2 SPcl 6Pc2

Equation (34) may be expressed as

F (Prl - 6Pr2)BL 6 r 6Pbl - 6P b2r6GL j bl 2 (35)

r bl b2 cl c2

The first term in brackets in Eqn. (35) is the change in differential ioutput pressure for a change in the lateral Jet displacement at the entranceof the receivers. Smith4 analyzes this term on the basis that the pressure at


22

the entrance of the receiver is determined by the net x-momentum flux incidentupon the receiver entrance. Noting that the total incident momentum flux isequal to the sum of the supply momentum flux and the control momentum fluxminus momentum flux losses due to shear effects, he perturbs the resultingequations to obtain

(6P - 6 2 L F ~ Lrl- Pr2)BL 4 1 Jcr c 1 (6

6Yr r B r Jsr 2LT-L c

where the momentum influx at the receiver entrance due to the supply flow isgiven by ;

Pbm 8 CdLT 1.34 [(37sr e 2 ] N Rho NRh I37

and the momentum influx at the receiver entrance due to the control flow isgiven by Pbm 8CdPbm

Jcr = [2C vcB + -] [Pbm N Rhm (38)

Both J and J are normalized with respect to the momentum effluxfrom the supply nozzle ir ideal hydrodynamic flow.

The decrease in the blocked-load receiver pressure gain given byEqn. (36) with increasing mean bias pressure is due to the reduction of supplymomentum flux with increasing bias pressure and asymmetry of the flow pastthe control knife-edges when the supply jet is deflected.

The second term in brackets in Eqn. (35) is the change in lateral jetdisplacement at the receiver entrance for a change in lateral pressure differ-ence existing across the supply jet. Assuming that the supply jet travels ina straight line after being deflected an amount yc over the control portregion, Smith 4 evaluates this term as 2I6Y_ r 2Lc - Lc

P - 6Pb 2 4 -l_ (49)

hi ~b2 4C bm

The third term in brackets in Eqn. (35) is the change in differentialbias pressure for a change in differential control pressure and is obtainedfrom the fluid circuit diagram in Fig. 3 as

6? -6?bl b2 1(40)

6P - c2 1 - Rc(Yb + 2kd)

4 Smith, G. V., An Analysis of Laminar Proportional Amplifiers, Ph.D.Thesis, The Pennsylvania State University, University Park, PA., 1975.

23

Substitution of Eqns. (36), (39) and (40) into Eqn. (35) yields anexpression for the blocked-load pressure gain as

SLc Jcr 2LTLc -Lc2Jsr 2LT -Lc B (1 - Pbm)

GBL 1 + Rcc[Yb + 2 k d (41)

As previously noted, the first term in brackets in the numeratordecreases with increasing bias pressure while the second term in bracketsis noted to increase with increasing bias pressure. Calculations indicatethat the bias pressure effects on these two terms tend to cancel out so thata reasonable estimate of the numerator can be obtained by evaluating it forzero bias pressure. However, the denominator of Eqn. (41) is noted to be astrong function of the bias pressure--increasing with increasing bias pres-sure. Thus Eqn. (41) indicates that the pressure gain of the LPA underblocked-load conditions should decrease with increasing bias pressure.Because of the reduction in normalized shear losses with increasing Reynoldsnumber, the gain should increase with increasing Reynolds number for a givenaspect ratio. For a given Reynolds number (based on amplifier height) thesum of the normalized shear losses decreases with increasing aspect ratio-indicating larger pressure gains with higher aspect ratios.

2.4 Output Characteristics

The output characteristics of the LPA are analyzed by determiningthe output pressure-flow characteristics for one of the receivers as theapplied load on the output port is varied for a particular value of differ-ential control pressure.

The pressure at the entrance of the receiver, P , is the "driving"pressure or "source" pressure for the output flow under nu[Y conditions,Q - When the load resistance applied to the LPA is infinity, the flowtrough the receiver channel is zero and the pressure at the entrance to thereceivers is determined by the fact that all of the incident x-momentum fluxis converted into y-momentum flux. As the loading on the LPA is reduced, flowis allowed to enter the receiver channel reducing the amount of x-momentumflux that is converted into y-momentum flux. Thus the "source" pressuredecreases as the load resistance is decreased.

The relationship between the pressure at the receiver entrance fora given output flow and the pressure at the receiver entrance for blocked-load conditions may be obtained by making the assumption that the momentumflux incident upon the receivers is evenly distributed. The amount ofx-momentum flux converted into y-momentum flux is then given by the differ- Pence between the x-momentum flux incident upon the receiver and the momentum 1.flux entering the receiver channel. This is expressed as

JrI24

j + Ii

I

I The relationship between the null receiver pressure and the nulloutput pressure is obtained from the pressure drop for laminar flow inrectangular channels as

Prnl -Ponl (Rrl) appQonl (43)

where

CodirRs ( + ] 2 +K Q n1(44)Rrlaap p 1 N + K rai

Solving Eqns. (42) and (43) yields the desired null output pressure-flow characteristics as

2J +J C

onI B - r Qonl + (Rrl)app Qonl (45)

The deflected-jet output characteristics under blocked-load condi-tions may be easily obtained from the definition of the blocked-load pressuregain which results in

p = p + GBL [6Pcl - Pc2 (46)ol onl 2I 4Smith also presents a linearized analysis of the variation of the

pressure gain with equal resistive loads applied to the output ports. His

equation for the pressure gain is then given as

[o] 1 (47)

4B Qon r

I whereC L 2

+CfdL r Cd ] + 12 F~d]2raN~ Rh BJ l ri~o

4 Smith, G. V., An Analysis of Laminar Proportional Amplifiers, Ph.D.3Thesis, The Pennsylvania State University, University Park, PA., 1975.

S25

As would be expected, the pressure gain of the LPA approaches the

blocked-load pressure gain as the applied resistive loads approach infinity.

It is important to note that the normalized pressure gain givenby Eqn. (47) is a strong function of the null output flow rate (note that Ris also a function of Qo ). It is therefore necessary to determine the nulloutput flow rate for a given resistive load applied to the output ports asdiscussed by Smith.4 As would be expected it is therefore necessary to knowthe complete pressure-flow characteristics of the load to be driven by theLPA before the pressure gain can be determined.

Smith4 has also presented an analysis of the dynamic input imped-ance of the LPA for three types of input signals as well as an analysis ofthe dynamic pressure gain of the LPA.

3. EXPERIMENTAL RESULTS

In this section the results of an experimental investigation of thestatic operating characteristics of an LPA are presented and compared totheory. The dimensions of the LPA used in this study are shown In Fig. 1.Unless otherwise noted all data were taken using a model with a setback ofone-half of the supply nozzle width. In order to investigate the effects ofreduced setback on amplifier performance, some data were taken using a modelwith a setback of one-fourth of the supply nozzle width.

In order to obtain easily measurable pressures and flow rates, a hydraulicfluid (MIL-H-5606) was used as the working fluid. Flow rates were measuredwith calibrated variable area flow meters while pressures were measured

manometrically.

3.1 Supply Characteristics

Figure 4 presents the supply pressure-flow characteristics for an

aspect ratio of 0.56 and for nozzle exit pressures of up to 30 per cent ofthe supply pressure. Also shown on this figure is the predicted pressure-flow characteristic from Eqn. (10). Good agreement between theory and experi-mental data is noted to exist.

As previously noted, the discharge coefficient is primarily deter-mined by the Reynolds number based on amplifier height. Since the supplyflow rate is directly proportional to this Reynolds number, one would expectthat if the supply flow rate were plotted versus the difference between thesupply pressure and the nozzle exit pressure times the square of the aspectratio, the data would collapse into a single curve. This data is presentedin Fig. 5 and it is observed that the data essentially does collapse into asingle curve. Also shown on this figure are the predicted pressure-flowcharacteristics for an aspect ratio of 1. The predicted curves for each of


Thesis, The Pennsylvania State University, University Park, PA., 1975. *

26

0 0 0 0

*0 E:

o oi

-4-

0 00 C100

* 227

C4

-1 C4 uo

0 0 4

d -4

0

0 -4

1-1I

4 0Cd 11- w

00

Ln

00

CDU

CN X

od4 28I

tthe aspect ratios are essentially the same as that shown. This figure isespecially useful in the design of systems utilizing laminar amplifierssince a single curve is obtained for the relationship between the supplyReynolds number (or flow), supply pressure, and aspect ratio. Thusspecification of two of these parameters easily yields the third parameter.

3.2 Input Characteristics

Figure 6 presents data on the input characteristics for both thenull condition and the push-pull (i.e. 6P - 6P ) deflected-jet conditionfor an aspect ratio of 1.12 and a Reynolds numberof 1100. Data are pre-sented for the deflected-jet pressure-flow characteristics for mean controlpressures of 5, 10, 15, 20 and 30 per cent of the supply pressure. Alsopresented on this figure are the predicted pressure-flow characteristicsfor the null condition given by Eqn. (22) and for the push-pull condition

given by Eqn. (30).

During this data acquisition and for all data to be presented, the

supply pressure was held constant when investigating the effects of meancontrol pressure variation. Thus for a given supply pressure the supplyflow rate and Reynolds number decreased as the mean control pressure wasincreased. The supply flow rate used in all flow rate normalization isthat supply flow rate occurring for a zero mean control pressure. TheReynolds number appearing on all figures are the Reynolds numbers for zeromean control pressure.

The agreement between theory and data shown in Fig. 6 is noted tobe quite satisfactory. As theory predicts the push-pull input resistancedecreases with increasing mean control pressure. The push-pull data areobserved to be essentially linear over the entire operating range. Thisobservation indicates that the predicted push-pull characteristics, obtainedfrom a linearized analysis, need not be restricted to the case of extremely5 small signal changes.

4Data presented and discussed by Smith on the input characteristics3 for aspect ratios of 0.56, 1.12 and 1.56 and for Reynolds numbers of 730 to

1400 indicate that the input characteristics do not demonstrate a strong de-pendendy on the aspect ratio or Reynolds number. The data for an aspectratio of 0.56 indicated a lower deflected-jet input resistance than theory

I predicted.

Smith4 also presents data on the deflected-jet input characteristicsfor low mean control pressures and for single-sided pressure (i.e. 6P -0)input signals as well as an analysis of the degree of cross-coupling exhibitedby the LPA. He states that anomalies may occur in the deflected-jet inputcharacteristics at low mean control pressures and that the supply jet mayexhibit bistable characteristics at low mean control pressures where the null


29

I 1 .... ..

C',14

-4 N

-4

C%4-a

1-4 1-4

00

03

J

input resistance becomes very large. The LPA also indicated a high degreeof cross-coupling existing between the control ports as the mean controlpressure is increased above zero.

Since Smith and Shearer discussed the possibility of satisfactoryoperation for setbacks less than one-half the supply nozzle width, data were

taken on the input characteristics for a setback of one-fourth the supplynozzle width. These data are presented in Fig. 7 for an aspect ratio of1.07 and a Reynolds number of 750. Data are also shown for a setback ofone-half the supply nozzle width to facilitate interpretation of the effectsof reduced setback. As is evident by this figure, the reduction in setbackyields a greater null input resistance as theory predicts. However, both3 data and theory indicate that the push-pull input resistances are essentiallythe same for both values of setback at a given mean control pressure.

3.3 Transfer Characteristics

Figure 8 presents a typical set of transfer characteristics for anaspect ratio of 1.12, Reynolds number of 1100 and mean control pressure of 5,

10,20 and 30 per cent of the supply pressure for a push-pull input signalunder blocked-load conditions. From this figure and data for other aspectratios and Reynolds numbers as presented by Smith4 it is apparent that thepressure gain of the LPA decreases with increasing mean control pressure.For a given aspect ratio the decrease in pressure gain with increasing meancontrol pressure was found to be slightly more pronounced for the lowerReynolds numbers as theory predicts. The LPA is observed to exhibit relative-

ly flat saturation characteristics. It was determined, however, that for eachaspect ratio and Reynolds number a mean control pressure existed at whichpoint the saturation characteristic would begin to droop. The linear range

of operation was also found to increase slightly with increasing Reynoldsnumber.

As might be expected the LPA model with a reduced setback exhibiteda smaller operating range than the original model although the gain wasessentially the same. It is important to note, however, that the range ofoperation of the reduced setback model included the original linear range of

operation.

Figure 9 is a plot of the pressure gain versus the mean controlpressure for aspect ratios of 0.56, 1.12 and 1.56. Also shown on this figureare the predicted gains for each aspect ratio. The data and theory indicatea decrease in the pressure gain as the mean control pressure is increased.

At lower mean control pressures both data and theory predict a higher pressure

I 2Smith, G. V., and Shearer, J. L., Research Investigation on LaminarProportional Amplifiers, Final Report Contract DAAG-39-72-C-0190, Theg Pennsylvania State University, University Park, PA., 1974.

4Smith, G. V., An Analysis of Laminar Proportional Amplifiers, Ph.D.Thesis, The Pennsylvania State University, University Park,PA., 1975.

I 31

cat

040

u-

P-4- c

.6J -4

1.-

10

4 04-4

onC 0

o 0 o

32 Li

--.

I,

I0.8

i ol - o2 I <r " -oP -P.-o---

a | - 1.12 0.6 , 'o

NRh 1100 ,/ -

I P Cm- 0.05 o .

0 0- o0.1 0.4

] PC , - 0.2

I A Pcm 0.3 0.2I

l"I

-0.06 -0.04 -0.02 0 02 0 04 0.06P1 - P2"I

ci c2

S / 0.4

C1 /- / /

_ , -0.6O-I-O - -

I0 -0.8

IFigure 8. Push-pull transfer characteristics for

C - 1.12; NRh - 1100.

33

EEquation (41) .

200

G BL

N Rh 740

10 0 0 0.56

0 1=.12

ZI a 1.56

0 0.1 0.2 0.3

Pcm

I Figure 9. Biocked-load pressure gain variation withmean control pressure.

34 11

gain for the higher aspect ratios. It is noted, however, that the decrease

in the pressure gain with increasing mean control pressure for an aspectratio of 1.56 is greater than theory predicts. The theory is observed topredict a stronger aspect ratio dependency than the data indicates.

Figure 10 is a plot of the blocked-load pressure gain versus Rey-nolds number for a mean control pressure of 15 per cent of the supply pres-sure. Data for each of the aspect ratios are presented. The predicted pres-sure gain for an aspect ratio of unity is also presented in this figure.

Theory predicts a pressure gain increase with increasing Reynolds number thatis greater than the data indicate. It appears that the pressure gain beginsto level off as the Reynolds number is increased. This, perhaps, is 0 wouldbe expected since the pressure gain should reach a maximum at some value ofReynolds number and then decrease.

Although the transfer characteristics of proportional amplifiersare normally presented for blocked-load conditions, they are not, in general,operated under blocked-load conditions in actual applications. Thus datawere taken on the pressure gain of the LPA for various equal resistive loadsapplied to the output ports.

-- The effect of applied output load resistance on the pressure gain ofthe LPA is more clearly understood if the pressure gain is normalized with re-I spect to the blocked-load pressure gain and is plotted versus the incrementalload resistance applied to the output ports for a particular aspect ratio andReynolds number. Figure 11 presents this data for an aspect ratio of 1.12,IReynolds number of 1000 and for mean control pressures of 10, 15 and 20 per

cent of the supply pressure. Although the blocked-load pressure gain has beenshown to be a strong function of the mean control pressure, the effect of meancontrol pressure on the normalized gain is found to be small. The data pre-I sented in this figure indicate that for output resistive loads greater thanapproximately three times the supply resistance, the pressure gain of the LPAis essentially the same as for the blocked-load condition. Also presented onthis figure is the predicted normalized gain calculated from Eqn. (47). Goodagreement is noted to exist between the experimental data and the mathematicalmodel predictions.

The information presented in Fig. 11 is especially useful in thedesign of systems utilizing laminar amplifiers since it may be possible todesign the LPA so that the pressure gain is essentially independent of theoutput load. However, the fact is emphasized that both the incremental loadresistance and apparent load resistance must be known before the pressure gain

can be determined for given operating conditions. This is to say that thecomplete pressure-flow characteristics of the load to be driven by the LPAmust be known. The data and theory presented in Fig. 11 are for a particulartype of load (primarily of an orifice type) and care must be taken in applyingthe information presented in this figure to other types of applied loads.

3.4 Output Characteristics

A typical set of output characteristics are presented in Fig. 12for equal resistive loads applied to the output ports and for various valuesof push-pull input signals. On this figure considerably more data are

35

20 Equation (41).

G BL

P =0.15cm

0 y - 0. 5610

U 0=1.12

9- 1.56

0 400 800 1200

N Rh

Figure 10. Variation of pressure gain under blocked-load conditions with Reynolds number.

36

.5'..

LIII 0

Ir~ 01 -4 -4 ~N

0 0 0

0 II I~ II0

.- 40 ~

<1 C) C) U

It

It In

Ir.-7 0

I 0

U

'7 0U

000

II -4

I I.-

0

I *0

N-4-4

ii:0

-~ z

31 -4x

0

I 08-4

1 0 '0 -~

0

0 0 0 0

I -~I1 37

Pci c2

00.6 t00

± 0.02

Qol Equation (46)

0.4

0.2

0 0.2 0.4 0.6

Figure 12. output characteristics .for =1.12; N Rh 1000;

P - 0.15.,

38

I

presented for the null condition since during repeatability tests it wasobserved that the output characteristic data points did not fall along assmooth a curve as was originally thought. Thus, the data were repeated withconsiderably closer spacing of data points for the null condition. Alsoshown on this figure are the predicted output characteristics calculated fromEqn. (46). Reasonable agreement is noted to exist between theory and dataalthough the data do not fall along as smooth a curve as theory predicts.

The deflected-jet data follow the same trend as the null condition data. Thisanomaly in the output characteristics was also ound to exist for other valuesof mean control pressure as discussed by Smith. Smith 4 also presents datafor other aspect ratios, Reynolds numbers, mean control pressures and outputloading conditions. He reports that a considerable amount of cross-couplingexists between the output ports and that anomalies can occur in the output

Icharacteristics.

3.5 Differential Output Noise

The true &MS value of the fluctuating component of the differentialoutput pressure was measured for a bandwidth of 0 to 200 Hz. A portion ofthis data for a mean control pressure of 15 per cent of the supply pressure~is presented in Fig. 13 which indicates that the differential output pressure

noise increases proportionally with aspect ratio and exponentially withReynolds number. Smith4 also presents data on both the true RMS value andthe power spectral density of the fluctuating component of the differentialoutput pressure for various aspect ratios, Reynolds number and mean controlpressures. Generally it was observed that the differential output pressurenoise was a minimum for mean control pressures around 15 per cent of thesupply pressure and that the major components of the pressure noise were con-centrated in the low frequency spectrum.

4. CONCLUSIONS

Based on the results of this study it is concluded that the use of pres-sure-controlled amplifiers operating with laminar supply jets can offercertain advantages over amplifiers operating in the turbulent supply jet

regime. Some of these advantages include: 1) significant increase in pres-sure gain; 2) significant increase in signal-to-noise ratios; 3) reducedpower consumption; 4) stable operation under blocked-load conditions; 5)decreased output resistance.

Some of the disadvantages of the LPA as compared to typical amplifiersoperating with turbulent supply jets include: 1) significant decrease in both

the null input resistance and deflected-jet input resistance; 2) strong de-pendency of the amplifiers performance on the mean control pressure; and 3)high degree of cross-coupling between control ports.


Thesis, The Pennsylvania State University, Utiiversity Park, PA., 1975.

39

M.

10-2

of 000

P Cm 02.1500 a 0.56

o ) 1.12?a 1.56

400 800 1200 1600

N Rhj

Figure 13. Differential output noise as a function ofReynolds number.

464

I

It was shown, however, that the null input resistance could be signifi-cantly increased by a reduction in the setback of the control knife-edgesfrom 0.5 B' to 0.25 B' . It was determined that the reduction in operatingrange resulting from This reduced setback did not affect appreciably thelinear range of operation.1 3In general, acceptable agreement between the experimental data and thepredicted characteristics was observed--verifying, to a certain degree, therange over which the model is reasonably valid. However, the mathematicalmodel was found to predict a stronger dependency on the aspect ratio than wasobserved experimentally.

An analysis of the differential output noise indicated an exponentialI growth of the noise with increasing Reynolds number. Thus, low Reynolds

number operation may be desirable for applications where low output noise isimportant.

I

I

iI

I

I

I

K'

REFERENCES

1. Manion, F. M., and Mon, G., Fluerics: 33. Design and Staging of

Laminar Proportional Amplifiers, Harry Diamond Laboratories-TR-1608,

Adelphi, MD, 1972.

2. Smith, G. V., and Shearer, J. L., Research Investigation on LaminarProportional Amplifiers, Final Report Contract DAAG-39-72-C-0190, ThePennsylvania State University, University Park, PA., 1974.

3. Manton, F. M., and Drzewiecki, T. M., Analytic Design of Laminar Pro-portional Amplifiers, Proceedings HDL Fluidics State-of-the ArtSymposium, Adelphi, MD, 1974.

4. Smith, G. V., An Analysis of Laminar Proportional Amplifiers, Ph.D.Thesis, The Pennsylvania State University, University Park, PA., 1975.

5. Schlicting, H. Boundary Layer Theory, McGraw-Hill Book Co., Inc., N. Y.1968.

6. Sparrow, E. M., Hixon, C. W., and Shavit, G., Experiments on LaminarFlow Development in Rectangular Ducts, Trans. ASME, Journal of Basic

Engineering, Vol. 89, 1967.

7. Shapiro, A. H., Siegel, R., and Kline, S. J., Friction Factors in theLaminar Entry Region of a Smooth Tube, Proc. U. S. Nat. Congr. Appl.Mech., 1954.

8. Han, L. S., Hydrodynamic Entrance Lengths for Incompressible LaminarFlow in Rectangular Ducts, Trans. ASME, Journal of Applied Mechanics,Vol. 82, 1960.

9. Ehrich, F. F., Some Hydrodynamic Aspects of Valves, ASME Paper No. 55-A-114, 1955.

4I

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Air Force Flight Test Center

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Duke UniversityCollege of EngineeringDurham, NC 27006

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Washington UniversitySchool of EngineeringP.O. Box 1185St. Louis, MO 63130ATTN: W. M. Swanson

West Virginia University S

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51

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I

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Controls

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I 53

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Laboratory

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Officer/Carter, W. W./Sommer, H.

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ATTN: Kimmel, S., PIOATTN: Lhief, 0021 -

ATTN: Chief, 0022ATTN: Chief, Lab 100

ATTN: Chief, Lab 200

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54 4;;

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ATTN: Chief, Lab 400ATTN: Chief, Lab 500ATTN: Chief, Lab 600ATTN: Chief, Lab 700ATTN: Chief, Lab 800ATTN: Chief, Lab 900ATTN: Chief, Lab 1000ATTN: Chief, 041ATTN: HDL Library 3 copiesATTN: Chiarman, Editorial Committee 4 copiesATTN: Chief, 047ATTN: Tech Reports, 013ATTN: Patent Law Branch, 071

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