Hydraulic remote position-controllers

HYDRAULIC REMOTE POSITION-CONTROLLERS*

By J. E. M. COOMBES, B.Sc.(Eng.), Associate Member.f

{The paper was first received 14th March, and in revised form 1th July, 1947.)

SUMMARYBrief descriptions arc given of the significant characteristics of the

more important hydraulic devices used in remote position-controllers.Such devices include valves, dashpots, pumps, approximately constant-pressure sources and hydraulic motors. Combinations of theseprimary devices are then described which, for analytical purposes,may be regarded as single "elements" in the control sequence of anyservo-mechanism of which they form a part. Amongst these com-binations are pump- and valve-controlled power drives, various typesof hydraulic relays and passive mechanical networks incorporatingdashpots. The practical applications of these hydraulic servo-elements are then illustrated by means of mathematical analyses inwhich are deduced the conditions for stability and performance to beexpected of typical hydraulic remote position-control systems.

LIST OF SYMBOLS0 --- Instantaneous angular displacement.y ••- Instantaneous linear displacement in a straight line.T — Instantaneous torque.F — Instantaneous force.P — Instantaneous pressure.Q — Instantaneous rate of liquid flow (volume per unit

time).Av - Instantaneous area of the effective port opening

of a control valve.i •• --- Instantaneous electric current./ = Moment of inertia.

M — Mass.K — Viscous friction coefficient (torque per unit angular

velocity or force per unit velocity in a straightline).

A = Stiffness coefficient (torque per unit angular dis-placement or force per unit displacement in astraight line).

r — Lever ratio of a hydraulic relay.a ~ Compressibility of oil.V Volume.y A quantity as defined in Section 4.3.

i, t2 . . . ••-• Time-constants.a, b, c, e - Miscellaneous constants as defined in the text,

a, /? = Phase angles./ = Frequency, cycles per second.7T= 3-14 . . .to — Angular velocity, radians per second.

- litf../ = V-i.p -- The differential operator d/dt.

When used as a suffix:—i = An input quantity.o = An output quantity.e — An error quantity./ --=- A feedback quantity.v --~~~ A quantity relating to a control valve.

n — A quantity relating to the natural undampedoscillation of an inertia in association with astiffness.

1, 2, 3 . . . = Suffixes to distinguish between quantities of asimilar kind.

Generalized vector quantities are denoted by[...] and their scalar values by | [ . . , ] | .

(1) INTRODUCTIONThe term "hydraulic remote position-controller" describes an

error-operated automatic positional-control system in which thefinal source of power to position the output member is hydraulic.Apart from the final power drive which is therefore alwayshydraulic, the remainder of the control system may be of anyphysical kind or combinations of kinds as may be convenient.

In general, electrical methods are usually employed to obtaina measure of the error of the system using known means suchas magslips, selsyns, etc. Mechanical methods may howeverbe used where the input and output members of the system aresufficiently close to one another physically to permit the necessarymechanical links or shafts to be accommodated conveniently as,for example, in certain machine tool applications. Measurementof the error by hydraulic methods presents great difficulties andis impracticable.

Where the error is measured electrically, the initial poweramplification of the error quantity is conveniently obtained bymeans of magnetic or electronic amplifiers which may alsoincorporate passive networks with complex transfer functions tomodify the error quantity as in all-electric r.p.c.% systems describedelsewhere.

As will appear later, a mechanical operative quantity isrequired between the penultimate element of the system and thefinal hydraulic power drive which positions the output member.This is sometimes obtained by means of an electric motor butmore usually by means of one or more of what are termed"hydraulic relays." For this purpose, hydraulic relays probablygive better response characteristics than can be obtained by anyother means.

Hydraulic viscous damping devices or dashpots are frequentlyemployed in combination with mechanical masses and springsto synthesize passive linear mechanical networks having complextransfer functions similar to passive electrical networks. Thesenetworks may be inserted directly in the main control sequenceof the system to give approximations to time derivatives and/orintegrals of the operative quantity, wherever the latter ismechanical. They may also form part of feedback and feed-forward channels and are commonly so used to impart specialcomplex transfer functions to hydraulic relays.

Hydraulic power drives for r.p.c. may be divided into twocategories:

(a) Pump-controlled.(b) Valve-controlled.

These categories may be used to classify hydraulic r.p.c.systems to some extent, but within these two broad divisions

• Measurements Section paper.t Metropolitan-Vickers Electrical Co., Ltd.

% The abbreviation "r.p.c." is used herein for "remote position-controller" and alsofor the adjectival phrase "remote position-control."

[270]

COOMBES: HYDRAULIC REMOTE POSITION-CONTROLLERS 271

systems exist in a wide variety of different forms. A generalapproach to the subject has therefore been adopted in preferenceto a discursive discussion of specific systems and theoreticalconsiderations have throughout been given precedence overpractical details.

(2) HYDRAULIC DEVICESThe hydraulic medium used is commonly oil on account of

its lubricating and anti-corrosive properties. Also, because oftheir greater chemical stability, mineral oils are preferred tovegetable and animal oils. For reasons of cost and reducedfire risk, water, rendered less corrosive and more lubricating bythe addition of suitable ingredients, is occasionally used in largeinstallations. Water has the theoretical advantage over oil thatits kinematic viscosity is less dependent upon temperature.Nevertheless, its use is rare on account of practical difficulties.

It is usually assumed, as a first approximation, that the mediumused in a hydraulic device is incompressible. In many casesthe error introduced thereby is negligible; thus the change involume of most mineral oils at 40° C is only about 0 • 3 % per1 000 lb/in2 of pressure.* The mixture of free air with oil,however, greatly increases its effective compressibility. For•example, the addition of only one per cent of free air by volumeat atmospheric pressure increases its effective compressibility atatmospheric pressure by over 200 times. Since the charac-teristics of hydraulic devices used in r.p.c. are usually modifiedin an undesirable way when the hydraulic medium is com-pressible, it is essential that such devices be designed so thatfree air may be removed and any tendency to draw it into thesystem reduced to a minimum.

Liquid flow as a result of a pressure difference may be eitherviscous (steady, streamline, laminar) or turbulent (unsteady,sinuous). In viscous flow the velocity is directly proportionalto the pressure difference and indirectly proportional to theviscosity of the liquid. In turbulent flow the velocity is directlyproportional to the square root of the pressure difference andsubstantially independent of the viscosity. In r.p.c. practiceusing oil, flow in small pipe-lines and leakage between machinedsurfaces are usually viscous, whilst flow in large pipes is frequentlyturbulent. Flow through an orifice follows the law of turbulentflow.

(2.1) Control ValvesControl valves are essentially adjustable orifices. The volume

of liquid which passes through a control valve in unit time istherefore directly proportional to the square root of the pressuredifference across it and directly proportional to its effective cross-section.

Cylinder block

I Delivery tof externalI circuit

High-pressuresupply

Low-pressure

return

• Piston member

Fig. 1.—Reciprocating-piston control valve.

Control valves exist in many different forms. A type muchused in r.p.c. technique is the reciprocating-piston arrangementshown diagrammatically in Fig. 1, in which the direction ofliquid flow in the external hydraulic circuit depends upon the

• HYDE, J. H.: "On the Viscosities and Compressibilities of Liquids at HighPressures," Proceedings of the Royal Society, A, 1920, 97, p. 240.

direction in which the piston member is displaced from its centralor closed position. The force required to move the pistonmember is mainly that necessary to overcome hydraulic reactionand stiction. In large valves working at high pressures reactionforces usually predominate, but in small valves with low pressuresstiction may be a large part of the total force required. In r.p.c.,the fact that the hydraulic reaction of a valve is not a functionof the displacement of the piston member only, greatly com-plicates the problem of operating the piston member of a largevalve by the mechanical output of the preceding element in thecontrol sequence. Where stiction is troublesome its effect isfrequently reduced by superimposing upon the operative forceproper a "dither" force of relatively high frequency having amagnitude approximately equal to the stiction.

(2.2) DashpotsDashpots are devices which, when used to connect two

mechanical members and when inertia effects are neglected,exert a force directly proportional to the first or second powerof the relative velocity between the mechanical members depend-ing upon whether the liquid flow within the dashpot is viscousor turbulent respectively. Dashpots used in r.p.c. are usuallydesigned to have viscous flow characteristics and therefore alinear relationship between force and relative velocity.

Various arrangements of dashpots are shown in Fig. 2. In

By-pass pipe

Fig. 2.—Typical hydraulic dashpots.

Fig. 2(a) the dashpot consists of a cylinder and well-fitted pistonor simple ram across which is connected a by-pass pipe. Thepipe is proportioned so that the liquid flow within it is viscousand therefore directly proportional to the pressure differenceacross the piston. In consequence, the relative velocity betweenthe piston and the cylinder is directly proportional to the forcetransmitted. In Fig. 2(b) the by-pass pipe is replaced by a holein the piston, whilst in Fig. 2(c) it is replaced by the annularspace between the cylinder and the loosely fitted piston.

An adjustable oil dashpot having several interesting practicalfeatures is shown in Fig. 2(d). The well-fitting piston-rods 1and 2 slide in a fixed cylinder block 3 into which is built anadjustable needle valve 4 designed to have viscous flow charac-teristics. The whole dashpot is submerged in a bath of oil atatmospheric or other constant pressure. The fixed cylinderblock enables the masses of the moving parts to be kept smalland*also facilitates the fitting of external means to adjust thecharacteristics of the dashpot by alteration in the setting of theneedle valve. Bimetallic or other temperature-sensitive meansare also easily fitted to alter the setting of the valve automaticallyand so render the characteristics of the dashpot independent ofchanges in viscosity of the oil.

(2.3) PumpsBoth constant-delivery and variable-delivery pumps are used

in r.p.c.As its name implies, the constant-delivery pump delivers a

272 COOMBES: HYDRAULIC REMOTE POSITION-CONTROLLERS

constant volume of oil per unit time assuming that the powersource driving the pump maintains a constant speed and thatleakage of oil within the pump from its high- to its low-pressureside is negligible. In practice, leakage is always present to someextent, and since it increases with increase in the pressuredifference across the pump, the delivery is not quite constantbut decreases slightly with increase in pressure. In r.p.c,constant-delivery pumps are used for light duties only, for whichpurpose gear-type pumps are usually employed.

In the variable-delivery pump the volume of oil delivered perunit time is adjustable. For any particular adjustment, how-ever, the pump has constant-delivery characteristics. In r.p.c,two types of variable-delivery pumps are used, that in whichthe delivery is continuously adjustable from zero to the fullcapacity of the pump and that in which the delivery is con-tinuously adjustable from full capacity in one direction ofdelivery to full capacity in the reverse direction of delivery.Variable-delivery pumps for r.p.c. are usually of the swash-plateor radial-piston types in which the delivery is adjusted by alteringthe effective stroke of the pistons.

(2.4) Approximately Constant-Pressure SourcesWhilst the hydraulic accumulator provides a source of oil at

truly constant pressure, its size and weight usually prohibits itsuse in r.p.c. Pressure-regulated pumps are therefore used which,although they do not give a truly constant pressure, are relativelymuch smaller and lighter.

In one arrangement, a constant-delivery pump is used acrosswhich is connected a spring-loaded relief valve as shown inFig. 3. Briefly, the excess delivery of the pump over the instan-

Mechaniralpower

Constant-delivery

pump

Pressure-regulatingrclu'f, valve**

I Delivery tor external

ciivuit

Fig. 3.—Constant-delivery pump with pressure regulation obtained bymeans of a relief valve.

taneous requirements of the external hydraulic circuit is by-passed through the relief valve. The energy destroyed in therelief valve by the excess oil passing through it appears as heatand may lead to rapid heating of the oil unless means are pro-vided to cool it. Whilst the arrangement is not therefore anefficient one, nevertheless, it is frequently employed with inex-pensive gear-type pumps for small-power hydraulic supplieswhere its low efficiency is of secondary importance to its sim-plicity.

The principle of a more efficient arrangement invariably usedwhere large powers are involved is shown in Fig. 4 and makes

I Delivery tor external

—J circuit

Fig< 4#—Principle of pressure-regulating arrangement for a variable-delivery pump.

use of a variable-delivery pump the stroke of which is adjustedautomatically to suit the delivery requirements of the externalcircuit. The ram 1 is connected to the stroke control lever 2 of

the pump 3 and exerts a force proportional to the pressure of thepump, the low pressure side of which is assumed to be at someconstant reference or return pressure. The force exerted by theram is opposed by a compressed spring 4. The spring and ramare arranged so that the delivery of the pump is reduced to zeroat the maximum pressure rating Px of the pump and increasedto the full capacity of the pump at some lower pressure P2. Adashpot 5 ensures that the otherwise oscillatory motion of thespring and the effective mass of the moving parts is aperiodic.The change in pressure for a change in delivery from zero to thefull capacity of the pump is therefore Px ~ P2. In r.p.c. practiceP2 is typically in the region of 0- 8 of Pv

(2.5) MotorsThe hydraulic motors used in r.p.c, apart from simple rams

used to obtain straight-line motions, are essentially reversedswash-plate or radial-piston type pumps, except that in mostcases no provision is made for varying the stroke of the pistons.The torque developed by such a motor is directly proportionalto the pressure difference across it, whilst neglecting leakage thevelocity is directly proportional to the volume of liquid passingthrough it in unit time. Since leakage increases with increasein the pressure difference and therefore with the torque developed,the velocity is not quite independent of the torque but decreasesslightly with increase in torque.

(3) HYDRAULIC SERVO-ELEMENTSThe primary hydraulic devices described in Section 2 are used

to form hydraulic combinations which have mechanical inputand output operative quantities. These combinations may betreated analytically as single "elements" in the control sequenceof any servo mechanism of which they form a part. The moreimportant of these combinations and their characteristics willnow be described.

(3.1) Pump-Controlled Power DrivesA pump-controlled power drive consists essentially of a

hydraulic motor which is driven by the controlled delivery of avariable-delivery pump as shown diagrammatically in Fig. 5.

• Header tank or otherI constant-pressure source

Mechanicalpowersource

Motoroutputshaft

Pump Motor

Fig. 5.—Basic arrangement of pump-controlled power drive.

Since the stroke of the motor is fixed, its velocity as a firstapproximation is directly proportional at all times to the deliveryof the pump and therefore to the stroke of the pump as deter-mined by the. stroke control lever. Also, since in r.p.c. speedcontrol in both directions is required, the pump must be of thereversible type mentioned in Section 2.3. For applicationsrequiring a straight-line mechanical output the motor may bereplaced by a simple double-acting ram.

The torque developed by the motor is limited only by themaximum pressure difference which it is safe to permit acrossthe pump and motor and consequently full torque is availableat all speeds. Spring-loaded relief valves 1 and 2 limit themaximum pressure difference which it is possible to build up


across the pump and motor. Any leakage of oil from thesystem is made up through non-return replenishment valves 3and 4 from a header tank or other constant pressure oil supply.Both the relief and replenishment valves are frequently built intothe casing of the pump to form a single compact unit.

In practice, compressibility of the oil modifies the simpleproportional relationship assumed above for the stroke of thepump and the resulting speed of the motor. This effect isexamined in the Appendix (Section 6) for the case in which theload driven by the motor consists of an inertia with viscousfriction. Assuming that

(a) All frictional and whirling losses due to oil flow arenegligible,

(b) The pump and motor are both 100% efficient,(c) The speed of the pump is constant,(d) No dilatation of the hydraulic circuit occurs as a result of

oil pressure,it is shown that the instantaneous angular velocity p60 of themotor in terms of the instantaneous stroke y{ of the pump isgiven by

•\d = ^T-y, • • • (1)

where ----- Moment of inertia of the load.= Viscous friction constant of the load.- "Hydraulic stiffness" of the system.- A constant of the pump.

A constant of the motor.

The velocity of the motor p9Q for a suddenly applied stroke y/\is therefore a damped oscillatory function of time. If the strokeis varied harmonically with angular velocity o>, equation (1)yields

[p0o] =b

(2)

where the square brackets denote generalized vector quantities.The ratio of the amplitude of the velocity of the motor to theamplitude of the stroke is therefore

• . . (3)

and the velocity of the motor lags the stroke by a phase angle agiven by

a - arc tan r 2-— . . . . (4)

The general way in which these amplitude and phase expres-sions vary with co are indicated in Fig. 6, where the preciseshapes of the curves depend upon the value of Ko in relationto Ao and 70. It may be noted from equation (4), however, thatthe angular velocity for which the phase lag is 90° is independentof KQ and is given by

i\

(5)

where a>n is the undamped natural angular velocity of the systemas determined by the inertia of the load in association with thehydraulic stiffness. Since it is desirable that phase lags in thefinal power drive should in general be as small as possible, itfollows from equation (5) that the hydraulic stiffness should beas large as possible.

The hydraulic stiffness (cf. Appendix) is theoretically given by

~oV(6)

where cr = Effective compressibility of the oil.2V= Total volume of oil in the hydraulic circuit.

To obtain a high value of hydraulic stiffness it is thereforenecessary that both the effective compressibility and the total

180-

Fig. 6.—Typical harmonic amplitude and phase response-curves of apump-controlled motor.

[Ao — ci-/o)- -f <o'A'^]

a - arc tan.— °—-

volume of the oil in the hydraulic circuit should be as small aspossible. As mentioned earlier, the effective compressibility ofthe oil is greatly increased by the presence in it of quite a smallamount of free air and it is therefore essential that the formationof free air be minimized. To this end a relatively high replenish-ment pressure is frequently used, since this not only discouragesthe release of dissolved air but also reduces any tendency todraw free air into the system from outside. Also, should freeair find its way into the system from outside when the latter isshut down, the subsequent application of a high replenishmentpressure compresses the air and greatly reduces the extent bywhich it increases the effective compressibility of the oil whenrunning. If the maximum working pressure is in the region of1 000 lb/in2, the replenishment pressure is typically as high as100 Ib/in2.

In practice the hydraulic stiffness is not a constant but variesfrom the value given by equation (6) to one-half of that value(cf. Appendix), depending upon the motions imparted to thestroke of the pump. Whilst the system is not therefore trulylinear, nevertheless it is usually found that for the purpose ofanalysing the stability and performance of an r.p.c. system noserious errors are introduced by assuming linear operationaccording to equation (1), provided that the hydraulic stiffnessis taken as one-half of that given by equation (6).

(3.2) Valve-Controlled Power DrivesA valve-controlled power drive consists essentially of a

hydraulic motor to which the flow of oil from an approximatelyconstant-pressure source is controlled by means of a valve.Since the speed of the motor must be variable in both directionsin r.p.c, the valve must be double-acting and is typically of thereciprocating piston type as indicated in Fig. 7. In normalpractice the oil supply is obtained from a pressure-regulatedvariable-delivery pump. Where a straight-line mechanical outputis required the motor may be replaced by a simple double-acting

ram.VOL. 94, PART IIA. 18


Control valveMotor

I!

sinusoidal velocity to the motor is readily obtained by makingthe substitutions

pd0 = [p6Q] sin cot

P29Q — [p90]co cos cot

'"loutput ôr s u c c e s s i v e values of t and solving for A,,. This has beenU shj,ft done and Fig. 8 shows the way in which Av must be varied to

100%

Fig. 7. -Basic arrangement of valve-controlled power drive.

A full analysis of the system is complicated but a simplifiedtreatment suffices to show that it is inherently non-linear. Forthis purpose it will be assumed that

(a) All frictional and whirling losses except those in thecontrol valve are negligible.

(b) The motor is 100% efficient.(r) The pressure of the oil supply is constant.(d) The oil is incompressible.(<') No dilatation of the hydraulic circuit occurs as a result of

oil pressure.If the constant pressure of the oil supply is denoted by P and

the instantaneous pressure-differences across the valve and themotor by PD and Po respectively, then

P - Pv + P*But 0o = cv'(Pv) x Av

where QQ -= Instantaneous volume of oil flowing through thevalve in unit t ime.

A., = Instantaneous effective port opening of the valve.

and

c -- A constant of the valve.

QQ - bP90

where p9Q -• Instantaneous angular velocity of the motor.b - A constant of the motor.

Hence, combining the last two equations and substituting thevalue of/1,, so obtained in the first equation gives

(7)

or, since the constant b equals the instantaneous torque of themotor per unit pressure difference across it,

(8)

where T{) is the instantaneous torque due to PQ and T is themaximum torque which would result if the whole of the availablepressure /' were applied to the motor. Since, in general, To willbe some linear function of f>0, it is evident that equation (8)represents a non-linear system, the degree of non-linearitydepending upon the magnitude of 7'0 relative to T. To carrythe analysis any further it is necessary to know the nature of theload, since this determines the relationship between To and 60.

In the important practical case where the load is essentially apure inertia /0, To equals fop

20o and equation (8) becomes

IoP%) N A, (9)

The general solution of this equation has not been found, butthe variation of the valve port opening At) necessary to impart

Fig. 8.—Theoretical variation with time of effective port opening forvarious maximum values of IQPWO expressed as a percentage ofthe maximum torque.

impart sinusoidal velocity to the motor when the inertia of theload is so adjusted that the maximum value of IOP29O rises to0, 25, 50, 90 or 100% of T. It will be observed that, whilstnon-linearity is perceptible when the maximum value of IQPZ9Q

is as small as 25 % of T, nevertheless it is not very serious upto 50 % and this value is often taken as the maximum for whicha valve-controlled power drive with predominantly inertia loadcan be regarded as a linear element in a cyclic r.p.c. system.

Valve control is essentially an inefficient arrangement due tothe energy destroyed in the valve which appears as heat. Onthe other hand, it enables a number of r.p.c. systems or otherhydraulically controlled machines to be operated from a singleconstant-pressure source and it is frequently used where itsconvenience outweighs its low efficiency.

(3.3) Passive Mechanical NetworksTypical passive mechanical networks incorporating viscous

dashpots are shown in Fig. 9 together with their operationalcharacteristics. They call for no special comment.

(3.4) Hydraulic RelaysHydraulic relays are hydraulically-opcrated power amplifiers

whose input and output operative quantities are mechanical andrelated by means of complex transfer functions. They may beused singly, in series or in parallel and are somewhat analogousto stages of d.c. electronic amplification incorporating passiveelectrical networks. Where a hydraulic relay constitutes thepenultimate element in a hydraulic r.p.c. system the outputmember of the relay varies directly the stroke of the pump orthe effective port opening of the control valve, as the case maybe, of the final hydraulic power drive.

As a general rule, hydraulic relays require to operate with asource of oil at approximately constant pressure. Since the


1

2

3

4

5

F-

n

F —•

/ -

X

Ky,-

r 1 1

l__

X

y2 fc

B x - ^K

x+/?A' yj

J :1-'~T~K+pM F

7

8

9

10

F-+

y

K

X

A'

X

«K" X

—1 * r~y

1 pWi

y\ y2

I ^x2 +p/r2

•

i «.+ pKr

X1 r

•p/C+X1

a-x2

r- \

\

\\

\\\

11

12

13

14

F-+*

F-+*

py*\

F—*

—At

H r

j-TR

XI

1

M K

K

+ PK\ v,h + pKiyi

IS

2 ^

X 4- K

x2

— ( V k-\y\\i +p\K\ +Aj)

MX *A'X F

Fig. 9.—Some passive mechanical networks incorporating viscous dashpots.K — Viscous dashpot coefficient (force per unit relative velocity). M — Mass.F •= Instantaneous force. X — Spring stiffness (force per unit displacement).y — Instantaneous displacement, p = Differential operator d/dt.

power involved is small, this is usually obtained from a gear-type constant-delivery pump fitted with relief-valve pressureregulation.

(3.4.1) Integrating Relay.An integrating relay consists fundamentally of a double-

acting ram to which the flow of oil from an approximatelyconstant-pressure source is controlled by means of a smalldouble-acting pilot valve as indicated in Fig. 10, where for sim-plicity the low-pressure return circuit from the valve to the oilsupply has been omitted.

Except in so far as a ram replaces the motor, an analysis ofthe arrangement follows closely that given in Section 3.2 andthe velocity of the ram is directly proportional to the effectiveport opening of the valve only if the pressure difference acrossthe ram is negligible compared with the supply pressure. Inmost r.p.c. calculations it is usually found, however, that directproportionality may be assumed without serious errors resulting,provided that the utilized pressure across the ram does notexceed about 50 % of the supply pressure. Hence, if the effectiveport opening of the valve is directly proportional to the displace-ment y. of the piston member from its closed position, thevelocity pyQ of the ram is effectively

y

' Constant-pressuresource

to

(10)

a

$

and

where e is a constant. For harmonic variation of y{ the complextransfer function is therefore

9090°all values of a)

Fig. 10.—Schematic arrangement and typical harmonic responserelations for hydraulic integrating ("non-corresponding,""floating," "creeping") relay.

(12)ej ytdt .[.vj].


the ratio of the output amplitude to the input amplitude is

M!>,-]

(13)

and y0 lags vf by a constant phase angle a of 90° for all valuesof OJ. The general way in which these amplitude and phaserelations vary with o) are indicated in Fig. 10 for comparisonwith the harmonic characteristics of the relays described inSections 3.4.2 and 3.4.3.

Since there is no direct proportionality between the displace-ment of the valve vf and the position of the ram y0, integratingrelays are optionally termed "floating," "creeping" or "non-corresponding" relays.

The simple integrating relay may be regarded as the funda-mental kind from which more complicated hydraulic relays arederived, and variations are usually characterized by the applica-tion to the simple integrating relay of negative feedback wherebyits characteristics are modified. It is impossible to describe hereall known variations and two typical examples only will begiven. These are described in Sections 3.4.2 and 3.4.3.

(3.4.2) Integrating Relay with Transient Displacement Feedback.An arrangement which modifies the characteristics of the

simple integrating relay in a way which is often of great advan-tage in r.p.c. is shown in Fig. 11. Here, negative feedback

and consequently the instantaneous displacement yv of the pistonmember of the valve will be given approximately by

yv = (1 - r)yt - ryf

where y. = Instantaneous displacement of end C of the lever.jy = Instantaneous displacement of end A of the lever.

BC

The dashpot-and-spring combination will be recognized aspassive mechanical network No. 2 of Fig. 9 and therefore if allinertia effects are negligible,

pKf A + pK °

where y0 = Instantaneous displacement of the ram.K = Viscous damping constant of the dashpot.A = Stiffness of the spring.?i = Time-constant of the dashpot and spring com-

bination.

Also by equation (10), writing yv for y(,

t2 „

BCAC

1 - r

h

90= - arc tan ii t /z + a>2t\t2

Fig. 11.—Schematic arrangement and typical harmonic responserelations for integrating relay with ^transient displacementfeedback.

which is a transient function of the displacement of the ram isobtained from a viscous dashpot and spring combination whilstthe proportions of a rigid floating lever ABC determine theamount of the feedback quantity utilized.

The lever is pivoted at each of the three points A, B and C,

Hence, combining the last three equations,

= (1 _)(__!:ih+ptjp-

. . (14)

where t2= (er)~i. Comparing this result with equation (11)of the simple integrating relay it will be seen that, apart froma constant factor (1 — r) giving overall attenuation, the. responsehas been modified by a factor

1 +Ptx

The use of "phase-advance" or "quick-response" terms of thisform in r.p.c. is well known and calls for no comment here.

From equation (14), the complex transfer function for har-monic variation of y. is

-r J 05)


and y0 lags yt by a phase angle a° given by

Oit\

a = 90° — arc tan

06)

. . (17)

The general way in which these amplitude and phase relationsvary with a> are indicated in Fig. 11 and are to be comparedwith the corresponding variations for the simple integrating relayshown in Fig. 10. The influence of the phase-advance factor inequation (14) on the shape of the phase-lag curve should benoticed. The angle of phase advance j8 = 90° — a has amaximum value of


which occurs when

(X) —

(3.4.3) Integrating Relay with Displacement Feedback.If the dashpot of the relay discussed i n the previous Section

is made infinitely tight, i.e. K = ao, pure displacement feedbackresults. The spring then serves no useful purpose and thearrangement simplifies to that shown in Fig. 12.

1 —r

BCAC

•yc

\zr\r

1 -r

a/—

90c

a - - arc tan co/2

Fig. 12. -Schematic arrangement and typical harmonic responserelations for integrating relay with displacement feedback ("pro-portional," "corresponding" or "follow-up" relay).

Using the symbols and methods of the previous Section, orby putting f, ---• 00 in equation (14),

1 7 1

r \\ ++ Pt2

(20)

Hence if yt is given a sudden value yj\, the final steady valueof y0 is

* - ^ , (21)

approached exponentially with time-constant t2 = (er)~l. Thefinal displacement of the ram is therefore directly proportionalto any steadily maintained displacement of the input and hencethe terms "proportional," "corresponding" and "follow-up"applied to this relay.

From equation (20), the complex transfer function for har-monic variation of yt is

l>0] . 1 ~ r 1!>,] r 1 +jcot2


1 ~ r 1

(22)

(23)

and yQ lags yt by a phase angle a given by

a — arc tan co/2 (24)

The way in which these amplitude and phase relations varywith to are indicated in Fig. 12 for comparison with Figs. 10and 11.

The response of the relay may, of course, be regarded aseffectively instantaneous and, according to equation (21), if itstime-constant is negligible in comparison with other time-con-stants of associated elements in an r.p.c. system.

Before leaving the subject of hydraulic relays one advantageof applying negative feedback to the simple integrating relayshould be mentioned which has not so far been considered.This is the well-known action of negative feedback in reducingthe effectiveness of any non-linearities in the sequence acrosswhich it is applied. Thus, in one practical design of relay withtransient displacement feedback the utilized pressure across theram may be as high as 70-80 % of the supply pressure beforethe performance and stability of any r.p.c. system of which itforms a part is noticeably affected.

(4) HYDRAULIC R.P.C. SYSTEMSHydraulic r.p.c. systems may be classified to some extent

according to the type of final power drive employed, i.e. valve-controlled or pump-controlled. Within these two broad divisionsexist a number of practical systems of which, however, it ispossible to describe here only the more typical.

(4.1) Zero-Displacement-Error SystemsWhilst not perhaps always recognized as such, nevertheless,

the simple proportional hydraulic relay described in Section 3.4.3is itself a complete valve-controlled zero-displacement-errorsystem in which the instantaneous error is measured and trans-mitted mechanically by the floating lever. Practical applicationsas an r.p.c. usually require that the displacement of the outputmember or ram should equal the displacement of the inputmember or point C of the floating lever (Fig. 12) and consequentlythat the lever ratio r = 0 • 5. With this substitution, equation (20)becomes

( 2 5 )

Hence, the instantaneous error of the system is

(26)

and (27)

These equations are typical of a system having a zero-displace-ment-error. A representative application is shown diagram-matically in Fig. 13, where the cutter of a milling machine iscaused to rise and fall automatically and thereby reproduce inthe finished work the contour of a master template or cam.Fig. 13 also shows how the use of a lever may be avoided whenthe output and input displacements of the r.p.c. are equal. Thiseliminates a possible source of errors in alignment due to back-lash in the pivots of the lever.

A valve-controlled zero-displacement-error system for theremote control of angular position in which the instantaneouserror is again measured and transmitted mechanically is showndiagrammatically in Fig. 14a and an analogous pump-controlledsystem in Fig. 146. To a first approximation, the performance


Constant-fpressurejsource I

Machine bed \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

Fig. 13.—Application of a proportional relay as a zero-displacement-error r.p.c.

Local-control Remotely-controlledshaft motor

n \ Differential /

cl-vlV-Bi-f>r%

J #

Rack'

Controlvalve

Constant-pressuresource

Loral-control R«-mot^lv-eontrolloJ sliaftshaft ' !,—i /Motor

„ \ Differential

dl

\ * V I y \>Mechanical power Variable-delivery

source pump

Fig. 14.—Zero-displacement-error r.p.c. systems.(a) Valve-controlled.(/>) Pump-controlled.

of both these arrangements may be represented by equations ofthe same form as equations (25) and (26).

Equation (27) is only approximately true for both valve andpump control and consequently equations (25) and (26) representthe actual performance of the above systems only approximately.In particular, these equations fail to show that the stabilitydecreases progressively as the inertia of the load is increasedand instability ultimately supervenes. A formal proof of thisfor valve control is difficult owing to its inherent non-linearity,but it is readily shown for pump control when the oil is takenas compressible. Thus from equation (1), equation (27) is moreaccurately replaced by

where al is a constant which includes constants a and b ofequation (1) and 9a is written for yr Hence, using the rela-tionship Bz = di 00,

+ pKQ + X0)p6i

This represents a zero-displacement-error system of which theRouth criterion of stability is*

«i V o

or

and if 70 alone is varied there is therefore a maximum valuefor 70 if stability is to be maintained. For a given value of 70,stability depends upon the value of the effective load dampingconstant Ko in relation to the value of constant a{. However,this does not mean that the system is automatically unstable ifthe actual damping of the load is negligible, since frictional andwhirling losses due to oil flow in the hydraulic circuit will alwaysgive to A'Q some effective value and therefore stability can alwaysbe achieved by reducing ax sufficiently.

Where it is inconvenient to use mechanical methods ofmeasuring and transmitting the error of the system it is possibleto substitute electrical methods using known devices such as"magslips," "selsyns," etc. Where, however, the applicationjustifies this complication it is usually desirable to effect at thesame time an improvement in the order of the error obtained.Electrical methods of error measurement and transmission aretherefore usually associated with zero-velocity-error systems andtheir use with zero-displacement-error systems will not be con-sidered here.

(4.2) Valve-Controlled Zero-Velocity-Error SystemsA typical zero-velocity-error system in which the error is

measured and transmitted electrically is shown diagrammaticallyin Fig. 15.

Local-controlshaft

Remotely-controlled/ shaft

Motor

Constant- pressure,source for power

drive

Electro-hydraulic relay^

Fig. 15.—Valve-controlled zero-velocity-error r.p.c.

The a.c. error voltage obtained from the electrical error-measuring devices 1 and 2 is fed to the input of an electronic

• ROUTH, E. J.: "Advanced Rigid Dynamics" (Macmillan and Co., Ltd., 1892),pp. 192-202.


amplifier the effective output current of which is given to a firstapproximation by

( + ptx)9z (28)i =

where / ----- Instantaneous output current of the amplifier.9. — Instantaneous error of the system.

= et-o0.t{ ~ Time-constant of the amplifier.ii[ — A constant.

This current energizes a coil situated in the air-gap of a pot-magnet so that the force resulting from the interaction of thecurrent with the magnetic flux deflects the coil axially againstthe restraint of a spring, the direction of deflection dependingupon the direction of the current in the coil. The deflection ofthe coil y2 from its position when the spring is unstressed istherefore

where M2

p2\A2 A2

Fffective mass of the coil.Damping constant of the coil assembly.Stiffness of the spring.A constant.

However, if the undamped natural frequency of the moving-coilassembly is sufficiently high the above expression may be writtenwith an acceptable degree of approximation as

1+1*2(29)

. K-,where t2 — -."

A2Referring to Fig. 15, it will be seen that the deflection of the

moving coil displaces the input member of a hydraulic relay inwhich is incorporated transient displacement feedback of thetype discussed in Section 3.4.2. By comparison with equa-tion (14), the velocity py3 of the output member of the relayis therefore directly proportional to

1+Ph

where

' 3 + '4

- Quotient of the dashpot constant K3 and thespring stiffness A3.

/4 = Reciprocal of the product of the lever ratio r andthe pilot-valve constant e.

If the difference between t3 and /4 is great enough, however,the denominator of this expression is sensibly constant and wemay therefore write with acceptable error,

py3 = a3(l + pt3)y2

and hence by equation (29),

py3

a2q3(l + pt3).

1 + 1 * 2a2a3i if t2 = t3 . . (30)

The extent to which it is possible in practice to compensate inthis way for the time delay in response of the moving coil bythe "quick response" of the hydraulic relay is illustrated by thecurves of Fig. 16. These relate to a typical "electro-hydraulic"relay unit and show the variation with frequency of the measuredphase angle between (1) the displacement of the moving coiland the current in it, (2) the velocity of the hydraulic ram and

60

3 30I

0

60

3

_-——

4 bFrequency, </s

10

Fig. 16.—Harmonic phase relations for typical "electro-hydraulic"relay.

the displacement of the moving coil and (3) the velocity of thehydraulic ram and the current in the moving coil. Since curve (3)is within ± 10° of zero phase shift from 0 to 10 c/s, little effectiveerror is introduced in most r.p.c. applications by assuming thatoperation is in accordance with equation (30).

Combining equations (28) and (30) gives

py3 =

and if it may be assumed that

• • (31)

thenPV0

r "4J3

P29Q ~ a\a2a3a^{\ +

- o0(l + pt{)9t

Hence using the relation #e = 9t— 9Q,

. . . (32)

. . . (33)

(34)(p+pol + o ) t p 9 i . . . (34)

This result represents a zero-velocity-error system which isover-, critically- or under-damped as

V(«o>(35)

whilst the steady-state error for a steady input acceleration is

- - ,¥ , (36)

Theoretically, therefore, the steady-state acceleration-error maybe made as small as desired by increasing the value of constant a0sufficiently and equation (35) may then be used to determinethe value of time constant tx for any desired degree of damping.

In practice it is found that the foregoing analysis representsactual performance fairly well, provided that the steady-stateacceleration-error attempted is not too small. When, however,a high degree of performance is required it is found:—

(a) That the stability of the system decreases as the meanvelocity of the input member is increased.

(b) That the degree of damping obtainable decreases as thesteady-state acceleration-error is decreased until finally thesystem becomes unstable for all values of fj.

The explanation for both these effects is undoubtedly to besought in the assumptions implicit in equation (32), but a formalanalysis taking into account all likely causes presents consider-able difficulties due to the inherent non-linearity of the system.With regard to (a), a major cause may be the change in effectivedamping contributed to the system by the losses in the controlvalve as the effective port area and therefore the mean velocityis increased. Also, since the hydraulic stiffness of the system


is obviously greater when the control valve is shut than whenit is open, the effective hydraulic stiffness may decrease as themean velocity is increased. It is found experimentally that thedecrease in stability with increase in velocity is greatly reducedif the effective port area of the control valve increases approxi-mately as the \/3 power of the displacementof the piston memberbut, so far as the author is aware, no really satisfactory explana-tion for this has been found. A number of causes may con-tribute to produce effect (b) above, amongst these the moreimportant are likely to be the inherent non-linearity of thesystem, the compressibility of the oil and possibly the time-lagsassociated with the pressure-regulating system of the pump.The decrease in stability which occurs when the steady-stateacceleration-error is decreased may be offset to some extent byincluding higher-order derivatives of the error in the right-handside of equation (33), but, nevertheless, it is the difficulty ofcombating this effect which ultimately limits the performanceobtainable.

Two practical features of the above system may be noticed.First, that "dither" is imparted to the pilot valve of the hydraulicrelay by the simple expedient of superimposing upon theoperative output current of the electronic amplifier a suitableconstant-amplitude alternating current of 50 c/s. Secondly, thatseveral advantages are secured by submerging the hydraulicrelay in a bath of oil. These are briefly that:—

(a) The oil bath may be the header tank for the pressure-regulated pump supplying oil under pressure to the relay.

(A) The design of the pilot valve is simplified because the low-pressure oil from it may now exhaust directly into the oil bath.

(c) A submerged temperature-compensated dashpot of thetype discussed in Section 2.2 may be used.

(d) The pivots of the floating lever are automatically lubri-cated.

Despite the relatively low efficiency of the valve-controlledpower drive, nevertheless this system of r.p.c. is widely usedover a range of 5-250 h.p. on account of its simplicity and theease with which two or more such systems and other hydraulicmachines may be operated from a single pressure-regulated oilsupply.

The zero-velocity-error system developed by the AdmiraltyGunnery Establishment shown in schematic form in Fig. 17 has

Loral-controlshaft.

Remotely-controlledshaft

^ © = A . C supply

AC. error-voltage oc0£>

L/Motor

several interesting features, of which the more importantare:—

(a) The application of negative feedback directly to the finalhydraulic power drive to reduce its non-linearity.

(b) The use of hydraulic methods to obtain the necessarystabilizing term.

As in the previous system, electrical methods are used tomeasure the error of the system and to deflect the input memberof an integrating hydraulic relay to which is applied negativetransient displacement feedback. Also as in the previoussystem, the "quick response" characteristic of the relay may beconsidered to cancel out the "slow response" characteristic ofthe electrical transmission. Unlike the previous system, however,the electrical transmission does not introduce a first derivativeof error term in the operative quantity and consequently thevelocity pyx of the ram of the relay is sensibly at all times directlyproportional to the error of the system only, or

pyx - aidt

The integrating relay is followed by a proportional relay, thesole purpose of which is power amplification. To a first approxi-mation, the response y2 °f this relay may be regarded as instan-taneous, and so

Maincontrol

200 lb/in2

1001h/m:

The ram of the proportional relay is connected to the pistonmember of a hydraulic valve through a passive mechanical net-work consisting of a compressed spring in parallel with a viscousdashpot. The deflection y3 of the piston member of the valveis negligible in comparison with the deflection y2 of the ram ofthe relay and hence, referring to Fig. 9 for the operationalcharacteristics of a viscous dashpot in parallel with a spring,the total force Fz transmitted is sensibly

F3 =F-\- (A3 -i- pK3)y2

where F— Force exerted by the spring when the ram of therelay is undeflected, i.e. y2 '- - 0 and the pressuredifference across the ram is zero.

K3 = Viscous damping constant of the dashpot.A3 = Stiffness of the spring.

Combining the last two equations gives

The hydraulic valve to which this force is applied is arrangedso that the pressure F4 on its delivery side is sensibly directlyproport ional at all times to the force applied to its pistonmember, i.e.

Pf-a4F3

and therefore

Fig. 17.—Valve-controlled zero-velocity-error r.p.c. with negativefeedback applied to the final power drive.

By suitable design, the displacement y3 of the pis ton memberof this valve required to increase or decrease the pressure P4

by an amoun t equal to the mean pressure u4F can be m a d e amat ter of only a few thousandths of an inch, which justifiesthe assumption made earlier that V3 is negligible comparedwith y2.

Referring to Fig . 17, it will be seen that the pressure P4 isapplied to one end of a floating or shutt le-type piston memberforming par t of the main control valve of the final hydraulicpower drive. The opposite end of this piston member is subjectto a pressure P^ of the form


-which therefore applies negative feedback from the outputmember of the r.p.c. system directly to the main control valve.In the arrangement shown, to a first approximation,

•where K5

a5

Viscous-damping constant of the eddy-currentbrake.

Effective moment of inertia of the eddy-currentdisc.

A constant.

If the final power drive is powerful enough and if inertia,friction and hydraulic reaction effects in the shuttle valve areneglected, it is reasonable to assume as a first approximationthat the pressure Pf always equals the pressure P4, i.e.

a^F | ^"2"4(A3 + pK2)9e = o4F + as(Ks + pI5)pd0

and hence, on rearranging and making use of the relationship

{p3a5l5 + P2asKs -}- pa0K3 + a^3)Bt - (pâ5h j- p*asK$Bt (37)

where a0 ••= aia2a4. Equation (37) represents a zero-velocity-error system of which the Routh criterion for stability (Joe. cit.) is

and which has a steady-state acceleration-error given by

0 âJkpi0

(38)

(39)

In practice, inertia, friction and hydraulic reaction effects inthe shuttle valve cannot be neglected and equation (37) istherefore only an approximate representation of the actualbehaviour of the system. Due to the disturbing effects ofhydraulic reaction forces which increase rapidly with the size ofthe motor controlled, the use of the system is at present limitedto relatively low-power applications.

(4.3) Pump-Controlled Zero-Velocity-Error System

If, in Fig. 15, the approximately constant-pressure oil supplyand the control valve are replaced by a variable-delivery pumpin which the stroke control lever is coupled to the ram of thehydraulic relay, a zero-velocity-error pump-controlled systemresults. To a first approximation, the analysis given in Sec-tion 4.2 for the valve-controlled system of Fig. 15 applies equallyto the pump-controlled version. However, since pump controlis fundamentally linear if changes in the hydraulic stiffness areignored, it is now possible to extend the formal analysis toinclude the effects of oil compressibility in association with theinertia and viscous friction of the controlled load.

The stroke imparted to the pump, for small angles of displace-ment of the stroke-control lever, is sensibly directly proportionalto the displacement y3 of the ram of the relay and hence fromequation (1) we may write

where y3 replaces yt and a4 replaces constants a and b. Com-bining this equation with equation (31) and using the relationship0e -'•= î OQ gives

This result represents a zero-velocity-error system which isstable (he. cit.) if

> # 2 + V o ' i ' o . • • .

and has a steady-state acceleration-error given by1

* a0 '. (42)

In considering equations (41) and (42) from a practical pointof view, a question of major interest is the minimum steady-state acceleration-error which it is possible to achieve whilststill maintaining stability, since this gives a measure of the"goodness" of the system. By rearranging equation (41), thecondition for stability becomes

(43)

where Y = V iFrom equation (42), the steady-state acceleration-error is a

minimum when a0 is a maximum and hence we require to knowwhen the right-hand side of (43) is a maximum. Considerationshows that this is so when y = 0-5 and substituting this valuein (43) gives the maximum value of aQ for stability as

a -

Substituting this value of a0 in equation (42) gives the minimumsteady-state acceleration-error for stability as

B

1. (44)

where/), is the undamped natural frequency in cycles per secondof the inertia of the controlled load /0 in association with thehydraulic stiffness \ . Rather surprisingly, it has therefore beenshown that the minimum steady-state acceleration-error whichit is possible to achieve with this system whilst maintainingtheoretical stability is a function only of the undamped naturalfrequency fn. It is not suggested, of course, that this representsa practical condition, since a system which is just stable haszero damping and is therefore quite useless as a practical r.p.c.The result is however of considerable value in so far as itreveals that the undamped natural frequency fn may be regardedas a criterion of the "goodness" of the system. In particular,since/w is directly proportional to the square root of the hydraulicstiffness it is evidently of the first importance that the hydraulicstiffness should be as high as possible.

In practice, the above system is invariably operated with theaddition of a term to the right-hand side of equation (31) repre-senting negative feedback of the acceleration of the controlledload and equation (31) therefore becomes

"SP28Q • • • (45)py3 =

With this modification equation (40) becomes

a4a5)

which represents a zero-velocity-error system which is stable if(loc. cit.)

KQAQIÎ -{- ^405) > Kfi + WQAQ/I/Q . . . (47)

and has a steady-state acceleration-error given by

B, (48)


The condition for stability may also be written

* o < r U +«4«s- 7)7 • • • • (49>

where 7 is defined as above. It is readily shown that, regard-ing y as the variable, the right-hand side of (49) is a maximumwhen y -- 0-5(1 -- «4a5), hence substituting this value in (49)the maximum value of aQ for stability is

Substituting this value of a0 in equation (48) gives the minimumsteady-state acceleration-error for stability as

(50)

Comparing this result with equation (44) it will be seen thatthe minimum steady-state acceleration-error for theoreticalstability has been reduced by a factor

11 4 a4a5

It is evident that theoretically there is no limit to the amountof the acceleration-feedback term, as determined by the value ofconstant as, which may be used and therefore no limit to theextent by which the steady-state acceleration-error may bereduced.

In actual practice, it is found that the system behaves sub-stantially in accordance with theory and the negative accelera-tion-feedback term enables acceleration errors to be greatlyreduced whilst still maintaining adequate damping. The limitto which this can be carried is probably associated with:—

(a) Variations in the effective hydraulic stiffness of the powerdrive (cf. Section 3.1).

(b) The degree of purity with which the acceleration-feedbackterm can be obtained.

It is not proposed to describe here the numerous ways in whichthe acceleration-feedback term may be obtained, since the methodadopted in any instance is largely a matter of practical expe-diency. It may, however, be stated that as a general rule elec-trical methods are found to be less convenient than hydraulicmethods using as the starting point the pressure difference acrossthe hydraulic motor.

(5; CONCLUSION AND ACKNOWLEDGMENTSIn conclusion, the author wishes to emphasize that the opera-

tional analyses given are only approximate and space limitationsprevent reference to more detailed treatments using harmonic-response diagrams.

The author's thanks are due to the Director of Naval Ordnanceand to Metropolitan-Vickers Electrical Co., Ltd., for permissionto publish the paper, to Dr. C. Dannatt, Director and ChiefElectrical Engineer, for help and encouragement, and to Mr.C. Ryder and other colleagues for constructive criticism.

(6) APPENDIX

Analysis of Pump-Controlled Power DriveIt will be assumed that:—(a) All frictional and whirling losses due to the oil flow are

negligible.(b) The pump and motor are both 100% efficient.(c) The speed of the pump is constant.

id) No dilatation of the hydraulic circuit occurs as a result ofoil pressure.

If ys is the instantaneous stroke of the pump then the instan-taneous rate of delivery Qt (volume per unit time) of the pumpto the high-pressure side of the system is

where a is a constant. If, however, p90 is the instantaneousangular velocity of the output shaft of the motor, then theinstantaneous rate QQ at which the motor is removing oil fromthe high-pressure side of the system is

<2o = bp90

where b is a constant. Since the compressed volume V of theoil on the high-pressure side is constant, the instantaneous ratepV at which oil is being compressed is

=-• ay( — bp90

If the instantaneous pressures on the high- and low-pressure sidesof the system are respectively (Pav + 0-5P0) and (Pav 0-5P0),where Pav is some constant pressure and PQ is the instantaneouspressure-difference across the pump and motor, then, withnegligible error,

pV^-- 0-5aVpPQ

where a is the compressibility of the oil.the last two equations,

Hence, combining

and, since the constant b is equal to the torque developed bythe motor per unit pressure difference across it, the instantaneoustorque To developed is

If the motor drives a load of inertia /0 and viscous-frictionconstant KQ, both referred to the motor shaft, then

(p2I0 + pKQ)d0 -paV

which on re-arrangement gives

+ pKQ

where

The quantity AQ is the "hydraulic stiffness" of the power driveand equals the torque which must be applied to the shaft of themotor to rotate it through unit angle against the compressionof the oil when the stroke of the pump is zero.

The mean pressures Pav are established by the action of thereplenishment valves on both sides of the system and, oncecreated, should continue to exist even when the stroke of thepump is returned permanently to zero. This follows since thesystem is theoretically a closed one to internal pressures greaterthan the replenishment pressure. In practice, of course, Pav isnot maintained indefinitely under such conditions, since inevitablemechanical imperfections permit leakage. The practical conse-quence of this is that AQ varies from the value given above toone-half of this value.

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Hydraulic remote position-controllers