Investigations into the accuracy of hydraulic servomotors

OF HYDRAULIC SERVOMOTORS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP AAN DE TECHNISCHE HOGESCHOOL TE DELFT OP GEZAG VAN DE RECTOR MAGNIFICUS Dr. R. KRONlG, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP WOENSDAG 19 APRIL

1961 DES NAMIDDAGS TE 2 UUR

DOOR

TACO JAN VIERSMA WERKTUIGKUNDIG INGENIEUR GEBOREN TE WESTSTELLINGWERF

Geoffry Fisher.

Aan de Directie van het Natuurkundig Laboratorium der N.V. Philips' Gloeilampenfabrieken te Eindhoven ben ik veel dank verschuldigd voor de welwillende medewerking bij het onderzoek en het verschijnen van dit proef-schrift. Voor de ondervonden medewerking van collega's, assistenten en ver-taler ben ik evenzeer bijzonder erkentelijk.

INVESTIGA TIONS INTO THE ACCURACY OF HYDRAULIC SERVO-MOTORS

1. GENERAL INTRODUCTION . . . .

1.1. The importance of the accuracy in hydraulic servomotors .

1.2. Analysis of the control accuracy. Measuring results . . . . 2 1.3. Principle of operation of a number of hydraulic servomotors. 6 1.4. Feedback methods . . . 9

l.s. Scope of the investigations under description 12 2. PORT FLOWS . . . 14 2.1. General. . . 14 2.2. The velocity-time constant Lv and the flow rate V_o 14 2.3. Flowequations . . . 17 3. DIFFERENTTAL EQUATIONS FOR HYDRAULIC

SERVO-MOTORS. . . 24 3.1. General . . . 24 3.2. Fundamental equations . 24 3.2.1. Equilibrium equation. 24 3.2.2. Flow equations . . . 26 3.2.3. Equations of continuity 28

3.3. Turbulent port flow . . . . 29

3.3.1. Hydraulic servomotors with four controlled ports. 29 3.3.2. Hydraulic servomotors with one and two controlled ports 37

3.4. Laminar port flow 41

3.5. Summary. . . 44 4. METHODS FOR INTRODUCING LOAD COMPENSATION 47 4.1. Genera!. . . 47 4.2. First method of 10ad compensation . . . 47 4.3. Second method of load compensation . . . 51 4.4. General considerations on methods for load compensation 53 5. METHODS FOR SOLVING THE NON-LINEAR

DIFFEREN-TIAL EQUATION 5.1. General. . . . 5.2. Exact solution . . . 5.3. Numerical solution . 5.4. Electrical analogue .

5.5. The method of the harmonic balance

56 56 59 61 61 62

MOTORS IN THE ABSENCE OF COULOMB FRICTION 66

6.1. General . . . 66

6.2. Servomotor behaviour . . . 68 6.3. Regulator behaviour . . . 75 6.4. More detailed considerations and practical data. 77

6.4.1. General . . . 77

6.4.2. The effect of the servovalve 78

6.4.3. Damping . . . 80

6.4.4. Time constants Tv and Tr 83

7. ANALYSIS OF THE DYNAMIC BEHAVIOUR OF HYDRAU-LIC SERVOMOTORS WITH COULOMB FRICTION, USING THE DESCRIBING-FUNCTION METHOD . . . . . 87 7.1. General. . . . . 87 7.2. Open-loop serv'omotor behaviour. Representation in the complex

plane. . . 87 7.3. Closed-loop servomotor behaviour . . . 90 7.4. Analytical study ofthe servomotor behaviour . 93 7.5. Stability . . . 95 7.6. Frequency-response curves for servomotor behaviour 98 7.7. Regulator behaviour . . . 100 7.8. Critical examination ofthe results obtained in th is chapter 105 7.8.1. Genera! . . . . .'. . . 105 7.8.2. Stability. . . . . . 106

7.8.3. Amplitude describing functions 106

8. ELECTRICAL ANALOGUE. . . 111

8.1. Lay-out. . . 111 8.2. Simulation of the case of hydraulic servomotors with

"DF-dry-friction" . . . 1 13 8.3. Simulation of the case of hydraulic servomotors with idealized

dry friction . . . 115 9. APPLICATION OF A HYDRAULIC SERVOMOTOR FOR A

NUMERICALLY CONTROLLED CAM-MILLING MACHINE 120 Appendix A. HYDRAULIC SERVOMOTORS WITH FOUR

CON-TROLLED PORTS (asymmetrical piston and regulating spo ol) . . . 131 Appendix B. DIRECTIVES FOR THE DESIGN OF HYDRAULIC

SERVOMOTORS. List of symbols Samenvatting. . . . 139 142 144

GENERAL INTRODUCTION 1.1. The importance of the accuracy in hydraulic servomotors

In the present thesis attention will be given to the problem of controlling displacements to very high accuracies. Displacement control of machine parts is of ten etfected by hydraulic servomotors in which a ram can move back and forth in a cylinder. Wh en considering the use of hydraulic servo-motors in the field of machine tools for instance, we see a slide fixed to the moving ram (or to the cylinder) so that the cutting tooi can be moved with respect to the workpiece. The accuracy of the finished workpiece depends, amongst other factors, on accuracy of control of the slide motions.

By way of illustration Fig. I.l gives a sectional diagram of a hydraulically controlled copying lathe. Here the support moves at uniform speed over the bed, carrying the slide with it. The transverse movements of the cross-slide are such th at the cutting tooi, which is fixed to it, follows every movement of the stylus while this is scanning the con tours of the template. This folIowing-up can be explained as follows:

workpiece 4568 E 11 -l~ 1 stylus Dil streams

/

Ip

~

~s

t

E

E

Fig. 1.1. Sectional diagram of hydraulically controlled copying lathe.

With the regulating spool in the neutral position as drawn, the cross-slide - when under no load - will not move as the oil streams out of and into the two cylinder compartments are all equal. IT, however, the spool is forced away from the neutra I position, the oil streams are throttled in such a way that the cylinder - and hence the cutting tooi - is made to move until the spool is back in the neutral position.

It can be seen from the diagram that the deflection of the spool from its neutra I position equals the control error EO of the hydraulic follow-up system. Obviously the total workpiece error will be larger, because the forces acting on the system re sult in elastic deformation of components and in the back-lash being forced to one side. The modern copying lathe is accurate to within 20 to 50 microns. It is evident th at the demands imposed on the hydraulic servomotors under these conditions must be very stringent.

In the present thesis special attention will be given to this problem of accuracy, which is not only of importance in the case of copying lathes but also for numerically-controlled machine tools, i.e. machine tools th at require one, two or more motions to be controlled simultaneously. These machine tools are particularly weil suited to produce workpieces with curved surfaces, which have to be manufactured to close tolerances, for instance airplane wings, scale models of ships, turbine vanes, dies and cams. Hydraulic servo-motors have proved their worth for this purpose and are therefore used in ever-increasing numbers.

Beyond the range of machine tools, requirements concerning accuracy also apply, but as a rule they are less stringent. For this reason application of hydraulic servomotors in the field of machine tooI con trol will be treated more thoroughly in the following pages, but the approach will be such that, with certain restrictions, the results are of general interest.

1.2. Analysis of tbe control accuracy. Measuring results

We shall now analyze the problem of control accuracy by consuIting Fig. 1.2, which gives the block diagram of the hydraulic follow-up system depicted in Fig. 1.1. We see that the spool, when moving away from the neutraI position, provides the input signal for the servomotor. We have al ready seen that the deflection of the spooI equals the difference EO between the intended and the actuaI displacement of the ram:

EO

=

ft-fo.

(1,1)

~ ~

reference input I-+-'''-r--ou''tp-ut-:--m-'-otion to

feed back path 4569

Fig. 1.2. Bloek diagram of hydraulie follow-up system.

The difference EO is converted by the hydrauIic servomotor into a dispIacement constituting the output signalfo of the system. The servomotor working as an integrator, we find the relationship between difference signal EO and displacement

velocity jo

=

d/o/dt to be within certain limits a linear one. This relationship is expressed by the velocity-time constant Tv

=

E/

jo. Non-linearities taken into account, this time constant can be defined by:

(1.2)

It is only natural that we want E to be as small as possible, which we can achieve by taking the time constant TV as low as possible. We shall see later on that stability conditions limit the fre~dom of choosing the value of TV'

In addition to the displacement velocity, also the load on the system affects the control er~or. In this respect we may here introduce the conception of "hydraulic rigidity" of a hydraulic follow-up system. This hydraulic rigidity defines the relation between axial load and ram displacement :

(1,3) where J;P is the sum of external, inertia and frictional forces. Wh en we assume, for the time being, th at there are no relations between the velocity jo and the hydraulic rigidity Ch, nor between the load J;P and the time constant TV, we can obtain by superposition of the "velocity error" and the total "load error" of the equationsAI,2) and (1,3):

dE

=

Tv.djo- dJ;P/Ch. We then have for the region around the neutral position:

E

=

TV.jo - J;P/Ch. (1,4)

The total control error expressed by this formula is shown in Fig. 1.3a; plotted against the displacement velocity jo and with the laad J;P as a param-eter. This graph directly shows the relationship between the three quantities when the infl.uence ofthe Coulomb friction is not given explicitly. This Coulomb friction together with the external load P constitutes the total load J;P, wh en we neglect inertia forces. The Coulomb friction is assumed to represent a

P..o arctan1~ I I ---.. E I I : I

~~/~=2Ew

5064

Fig. 1.3. Control error e plotted against displacement velocity io, with the. load P as a parameter; quasi-steady-state conditions.

constant load, but its sign reverses with the direction of the displacement, thus causing the con trol error to be a discountinous function of the displace-ment velocity

jo.

At

jo

=

0 th is function has two break points and the region between these is ca lied the dead zone. Let the Coulomb friction be expressed by Wo, then the dead zone is:

(l,5)

Fig. l.3b shows this dead zone graphicaIly. From the point of view of control engineering the de ad zone is very unfavourable because it irnp lies that the regulating member of the con trol system can be moved over a distance corresponding to the dead zone without resulting in any displacement of the ram or the cylinder.

Summarizing, we can state that the total control error is built up of three components:

(a) frictional error: Ew

=

WO/Ch;

(b) velocity error : ES

=

TV·

jo

;

(c) load error : Ep

=

P/Ch.

(1,6)

It should be noted that eq. (1,6) and Fig. 1.3b are only accurate under quasi-steady-state conditions, if load P is constant and inertia forces may be ignored.

By way of i1lustration let us take a practical example. For most workpieces to be copied on a latbe the diameter of tbe cylindrical parts is of highest importance. The velocity error is, of course, absent; the load error is repro-ducible and may tberefore be eliminated in ,an easy manner (viz. by adjusting tbe position of the cross-slide). The frictional error, however, may manifest itself in variations of the workpiece diameter. The maximum variation is twice as large as tbe dead zone (see Fig. 1.1):

Dmax - Dmin

=

4Ew

=

4WO/Ch.

If the de ad zone is, say, 15 microns, a spread in the diameter may occur as large as 30 microns! For hydraulic copying lathes a dead zone of 15 microns is not at all unusual, as may be concluded from the following measuring results, which were obtained at a machine-tooI exhibition *). The measuring procedure was very simpje: a displacement indicator was fixed to the moving slide by means of a magnetic base; the indicator pin was put against either the oil-regulating spool or the gauge head. Tbe relative deflection of the spool could thus be measured, see Fig. IA. By making the slide move back and forth at extremely low speeds the dead zone could be read direct from the indicator. Finally, by using a conical temp late for giving the slide a uniform speed and

then measuring the deflections of the spool, the time constant Tv could be quickly determined by varying the velocity of the slide.

With this set-up twelve copying lathes in all were checked and it was found that the dead zone was on the average 15 to 20 microns. The time constant of six of the lathes averaged 40 milliseconds, a value which is very high when we realize how this works out when we want to obtain a slide speed of, say, 1 millimetre per second; the con trol error then has to be as large as 40 microns.

It must be admitted th at the general conditions for measuring at the exhibition

were far from ideal. The machine-tooI manufacturers did apparently not know

Fig. 1.4. Sectional diagram of measuring set-up.

this measuring procedure. As a rule they examine the workpiece itself and fr om it draw their conclusions as to the accuracy of the hydraulic follow-up system. However, the finished article will also show the effects of other defi-ciences of the machine tooI as a whole and the cutter in particular, namely backlash, wear, non-parallelism of guide ways, etc., so th at this method is a

very unreliable one. The method outlined above, on the contrary, gives direct

information on the quality of the hydraulic follow-up system.

For more data the reader is referred to two publications by Backé 1) 2)

(see end of this chapter). The dead-zone values mentioned in these publications agree reasonably with those obtained by the author of the present thesis. The values given for the velocity-time constant TV are lower; the hydraulic

rigidity is said to vary between 20 and 200 N/ fL (~2-20 kgf/ fL). The publications are also interesting because of the measuring methods described in them and because they take into account the influence of the measured quantities on various workpieces.

1.3. Principle of operation of a number of hydraulic servomotors

Fig. 1.5 shows a general functional diagram, which holds for all hydraulic servomotors to be discussed in these chapters. A ram can move back and forth in a cylinder *). The cylinder is split up into two compartments by the piston. The ram movements are governed by the amounts of oil that flow into and out of the compartments. These oil streams are controlled by four variabie orifices or ports, included into two oil circuits between supply and return lines of an oil pump. It will be clear that the oil pressures at either side of the piston are derived from the pump pressure and the flow resistances of the respective ports. The oil in the supply line is at the pump pressure which is kept constant within narrow limits by means of arelief valve. The return line discharges the oil into a reservoir; hence, when we ignore the pressure drop in this line for a moment, we may take the pressure in the return line to be equal to the atmospheric pressure. We shall discuss the line-pressure drop in a later chapter.

return lIne_ t - - - + - - - - ' ~sUPPly line (ps)

(p = 0 ) , n rel/et valve

'<\'---t---"

I

pump

J

-4572

Fig. 1.5. Functional diagram of hydraulic servomotor.

Evidently the ports are of essential importance in the con trol of the ram movements. They represent restricted passages in the lines, and the oB flow th at passes each port depends on the oil-pressure drop across it, as weil as on its size. So long as the unloaded ram is stationary, the oi1 streams in the two circuits, passing through inlet and outIet ports, are the same, i.e. Ql

=

Q2 and Q3

=

Q4. When the inlet and outlet ports, and their respective supply lines, are identical, th en the pressure drops across the ports will all be the same. Since the sum of the pressure drops across two ports connected in series equals the pump pressure, it follows that the pressure drop across each port *) In the ensuing considerations the cylinder will be assumed stationary; of course the ram may be stationary and the cylinder moving, but this makes no difference to the mathe-matical derivations to be made.

equals half the pump pressure. Hence the pressures PI and P2, at either side of the piston, may be expressed as:

PI

=

P2

=

0.5 ps . (1,7)

The ram can be forced to move backward and forward by varying the port openings through the medium of the spool. When ports 2 and 3 become larger, and 1 and 4 smaller, Q2 and Q3 wil! increase, and QI and Q4 decrease. As a consequence the oil volume in the left-hand cylinder compartment will decrease and th at in the right-hand compartment will increase. This has the result th at the ram is pushed to the left at uniform speed. In the reverse case ports 1 and 4 become larger, ports 2 and 3 smaller, so the ram is pushed to the right at uniform speed.

Since a uniform speed can be maintained without a force being exerted, the pressures PI and P2 at either side of the symmetrical piston, remain equal. That they remain also equal to half the pump pressure (according to eq. (1,7)) can easily be proved by indirect demonstration. Let the unloaded ram move at a speed dy as a result of a deftection dx of the regulating spool from the neutra 1 position, and let the corresponding oil pressures be PI

=

P2

=

0.5 ps

+

dp (see Fig. 1.6a). Then a deftection -dx of the spo ol will result in the ram moving at a speed - dy, with the oil pressures becoming: PI

=

P2

=

0.5 Ps - dp

(see Fig. 1.6b). Now, when we rotate one of the figures, 1.6a or 1.6b, about a vertical axis over 180°, we see that they contradict one another, as they are identical with the exception of the oil pressures PI and P2. This contradiction can only be eliminated by the assumption that the oil pressures are equal and remain equal to half the pump pressure, provided the ram is unloaded.

J

@

control valve

Fig. 1.6. The pressures at either side of the piston are independent of the uniform ram speed.

It is often wrongly assumed that the pressures PI and P2 are altered by a change in the port openings, and that the movement of the ram is caused by the resulting difference between these pressures. Ignoring the total load I;P

pressures PI and P2 can only depend on the total laad, EP. And, because friction is part of this totalload, we may conclude that there is an indirect relationship between ram speed and oil pressures, friction being a function of direction and magnitude of ram speed.

ConsuIting Fig. 1.5 again we shall examine the influence of the load on the ram movements. When the load is constant and directed towards the left, P2 must rise andfor PI drop, as compared with the neutral value, in order to

reguTatiflg -~.--~~~;:::..:~~~:....=~~~ spool

4574

Fig. 1.7. Hydraulic servomotor with 4 controlled ports.

make equilibrium of forces possible. This means that the pressure drop across ports 2 and 3 will increase and that across ports 1 and 4 decrease, so th at Q2 and Q3 will rise and QI and Q4 drop. Hence, under the influence of the load, the ram will move to the left at uniform speed. Conversely, a load directed towards the right will result in a ram movement to the right.

Summarizing, we cao say that the ram movements are governed by the position of the regulatiog spool, which determines the port openings, and by the total axial load on the ram. In the absence of an axial load we find that, within certain limits, there is a linear relationship between the ram speed and the position of the regulating spoot.

Some typical examples of servomotors with controlled ports are shown in Figs 1.7, 1.8 and 1.9. Each port is cylindrical and has an axial length hand a diameter d. The position of the spool fixes the axial length and hence the

port size. The spool can be so constructed that a number of ports can be varied at the same time. The number of versions of hydraulic servomotors is very large, but all belong to three main types:

(a) a type with 4 controlled ports, see Fig. 1.7; (b) a type with 2 controlled ports, see Fig. 1.8; (c) a type with 1 controlled port, see Fig. 1.9.

reguTating spooT

4576 Fig. 1.9. Hydraulic servomotor with one controlled and one fixed port.

These th ree main types can easily be reduced to the arrangement of Fig. 1.5; in the case of type (a) th is is very simpIe; in order to obtain types (b) and (c), ports 3 and 4 of Fig. 1.5 will have to be omitted. The full (constant) pump pressure, Ps, then prevails in the right-hand cylinder compartment. Remarkable in these two types of servomotors are the thick ram rods, which are necessary because the pressure in the left-hand cylinder compartment is about equal to half the pump pressure, when the spool is in the neutral position. If an equi-librium of forces is to be possible, the surface area of the piston must be twice that of the rod. Hydraulic servomotors with one or two controlled ports are easily recognizable by the thick rods. In the case of the servomotor with one controlled port it should also be noted that only one of the ports (as required by the user) is variabie, the other being fixed.

There are numerous variations of the three main types outlined above. Thus Fig. 1.1 represents a hydraulic servomotor with 4 controlled ports and Fig. 1.4 one with 2 controlled ports. In motors with only one controlled port the variations are mostly connected with the arrangement of the fixed port, which may even find a place in the ram itself. It is beyond the scope of this thesis to discuss all variations, interesting though they may beo In most cases the differences lie in the constructional details; from the point of view of the control engineer the variations have no or only very limited influence on the servomotor properties.

1.4. Feedback methods

To obtain a closed-Ioop control system it is necessary that the output signa I of the servomotor is fed back to the input. Next the measured difference

between reference input and output, i.e. E

=

ft -

/0,

can be used for the deri-vation of the deflection x of the regulating member in the con trol valve.

Fig. 1.10 gives the block diagram of a hydraulic follow-up system with the transfer function of the servovalve,

X/E,

expressed by K1Gl, and the transfer function of the servomotor by K2G2.

fi

4577

Fig. 1.10. Block diagram of hydraulic follow-up system.

There are many methods of feedback, the simplest being the all-mechanical one, as depicted in Figs 1.1 and 1.4. When the gauge head or stylus follows the contour of the temp late properly, the deflection of the spool will be equal to (or at least proportional to) the error E, so that K1Gl

=

1 (or at least a constant). In reality the problem is more complex. Due to the pressure with which the gauge he ad is kept pressed against the template, both the template and the gauge lever will be deformed slightly, and at the same time there is the danger of resonance vibrations occurring 3). Tbe lever is used in most cases for constructional reasons ; tbe transmission ratio is seldom higher tban 1 :2, because it is difficult to magnify mechanically the very small signals (corresponding to deflections of the order of some microns) by a larger factor.

-When the lever is rigid and dimensioned properly, the gauge-head pressure is low, and the template is fixed rigidly, th en all problems may find a satisfactory solution, so that we may take K1Gl to be constant with a high degree of approximation.

Besides the mechanical feedback we have since 1945 a large number of arrangements for obtaining electrical feedback. The electrohydraulic actuator developed at the Massachusets Institute of Technology may be looked upon as the forerunner of these electrical arrangements. In th is piece of equipment an electric current which is proportional to the measured difference E is converted into a displacement of a miniature ram which can be Iinked with the regulating spool of the servomotor. Fig. 1.11 gives the sectional diagram of the actuator and the following short description is taken from literature reference 4), mentioned at the end of this chapter.

"A small 4-way control valve with a hole-slot-and-plug construction is used to moduh,lte the flow of hydraulic fluid from a constant-pressure source to the chambers of a double-acting ram which delivers the power required to produce output motion. The con trol valve moves in response to various forces generated by the electric current through the torque-motor coils, and the force exerted

by the feedback spring, the lower end of which is attached to the ram. Initially, the torque-motor current is steady, the valve is centered, and the ram is motion-less at a position such that the net force due to deflection of the feedback

spring just balances the electromagnetic force generated by the flow of current in the torque-motor coils. A change in torque-motor current may cause left-ward motion which produces a flow tending to move the ram to the right, thereby increasing the spring deflection and creating a force opposite to that generated by the torque motor; hence, if the system is stabIe, the valve even-tually will be returned to its center position, and the ram will assume a new position which is proportional to the magnitude of the torque-motor current". This actuator has a bandwidth of 200 cis and weighs less than half a

kilogram.

Fig. l.1l. Sectional diagram of miniature electrohydraulic actuator.

The actuator was soon followed by other systems of which the spool was a regular part. Electrohydraulic servovalves made by several manufacturers compete for compactness and con trol power. They were originally intended for use in airplanes and rockets, where the ratio between weight and control power is of prime importance. Since 1956, however, similar servovalves are used for numerical control of machine tools incorporating hydraulic folIow-up systems. Some caution is, however, needed in view of the fact that the servovalves were initially developed for purposes in which the lifetime of the instrument was of secondary importance. The bandwidth, too, appears to have been more or less neglected in some of the designs. It should be remembered that servovalves to be used with machine tools should be reliable for a con-siderably long time, and of large bandwidth, rather than light and economical as regards oil consumption.

1.5. Scope of the investigations onder description

In the preceding pages we have discussed the necessity for using

high-precision hydraulic follow-up systems in certain applications. It is only natural

that the reproducibility of control should also co me up to severe requirements.

The quasi-steady-state behaviour of hydraulic servomotors (see Fig. 1.3b)

prompts us to aim at a low-value velocity-time constant and a high hydraulic

rigidity Ch. In other words the servomotor should respond with a high sensitivity

to deflections of the spool, whereas its sensitivity to axial loads should be a minimum. As far as the dynamic behaviour is concerned, the equipment must possess a large bandwidth and good stability. This means that the control error must be small even when the input signal varies rapidly and a dynamic load (cutting force) is present.

To find arrangements which meet the requirements, the differential equations of three different types of hydraulic servomotors will be drawn up, in which many factors, including non-linearities, will be accounted for in the best possible

way. Among the non-linear factors th ere are the port flow and the Coulomb

friction. For tbis reason the study of the differential equations will be preceded by an extensive examination of the port flows, which wW be found to have a strong influence on the velocity-time constant TV' Experiments have shown that a very low value of TV can only be obtained with turbulent port flow, and th at

only then the behaviour of the servomotor is reproducible. This latter fact is very plausible, because in the case of laminar port flow the amount of oil which passes through the port is strongly dependent on the viscosity and thus on the temperature of the oiI. Other factors accounted for in tbe differential equations are the compressibility of the oil, the viscous friction and the external axial load.

Viscous friction is neglected in many publications. In discussions on stability

this sometimes leads to one-sided conclusions. Thus, Campbell 5) and

Cbestnut 6) have drawn up certain requirements for the leakage flow around

the servovalve, to be fulfiIIed in order to obtain adequate damping. However,

tbe leakage flowaffects the hydraulic rigidity Ch very unfavourably. Damping

is, however, also influenced by viscous friction; therefore, under certain con-ditions it is not necessary that the requirements of adequate damping and high hydraulic rigidity contradict one another. It will also be demonstrated that acceleration feedback gives the same effects as viscous friction.

The external axial load is another factor which is of ten omitted in publications. Without this factor it is hard to understand the interaction between regulator

behaviour and servomotor behaviour. It will be shown that an unexpected

improvement in the hydraulic rigidity is possible without serious detriment to the stability. With the method of load compensation as appUed for the first time in the Philips Research Labs the rigidity of the hydraulic system is made extre-mely high which implies, inter aUa, that under quasi-steady-state conditions no

dead zone is present. Obviously it is then much easier to achieve high control accuracies.

The influence of the Coulomb friction in the steady state (sinusoidal input) wil! be analyzed in three of the following chapters. At the very low frequencies the highly unfavourable effect of the friction can be almost eliminated by the load-compensation method mentioned above. Remarkable enough the dead zone re-appears at higher frequencies, and its magnitude is found to be pro-portional to frequency. Evidently the bandwidth is adversely affected by Coulomb friction. A theoretica I analysis of the effect of friction on the servo and regulator behaviour will be given in a separate chapter, using the method of the describing function. It is found that, especially at low frequencies, very remarkable phenomena occur. An e1ectrical analogue obeying the non-linear differential equation was built, and the results obtained with it were recorded on film strips. In general these we re found to confirm the theoretical results of the special analysis.

The last chapter contains a description of a numerically controlled cam-milling machine, which illustrates that the hydraulic follow-up system can be applied with success. The velocity-time constant Tv of the follow-up system was 4 milliseconds, the resonance-time constant Tr 1 millisecond; thanks to

the load compensation the dead zone could be ignored (0.1 to 0.3 micron). The actual control accuracy obtained with this system during the cutting process was about 2 microns.

REFERENCES

1) W. Backé, .Das Verhalten hydraulischer Kopier Systeme, 7. Forschungsbericht des Laboratoriums für Werkzeugmaschinen und Betriebslehre der Rhein.-Westf. Technischen Hochschule Aachen, 1957.

2) W. Backé, Einflüsse auf die Genauigkeit beim Hydraulischen Nachformdrehen, 12.

Forschungsbericht des .... T. H. Aachen, 1958.

3) W. Backé, Untersuchungen über die Stabilität Hydraulischer Kopier Systeme, 12. Forschungsbericht des .... T.H. Aachen, 1958.

4) S. Y. Lee and J. L. Shearer, Development of a miniature electrohydraulic actuator,

Trans. ASME 77, p. 1077, 1955.

5) D. P. Campbell, Automatische regel inrichtingen, Leergang van 24 colleges te Delft, p. 65, 1950.

6) H. Chestnut and R. Mayer, Servomechanisms and regulating system design, Mc Graw-Hill, New Vork, 1959, vol. 1, 2nd ed., p. 210.

2.1. General

PORT FLOWS

In the operation of hydraulic servomotors an important role is played by the ports, restricted passages in the pipes to and from the cylinder compartment(s). These ports offer astrong resistance to the flow of oil, resulting in a pressure difference or drop across them; this pressure drop is derived from the pump pressure. The oil pressure in the supply line between pump and cylinder is kept constant within close limits by arelief valve. The pressure in the return line from the cylinder to the reservoir is usually about equal to the atmospheric pres su re ; sometimes a slight back-pressure is maintained in the reservoir to keep the oil from foaming.

When the ram is at rest, the oil flows through the two ports will be the same; when we assume the dimensions of the ports to be equal, then the pressure drops across' them will also be equal. The sum of these pressure drops equals the pump pressure ps, consequently either pressure drop equals half the pump pressure, or

L1Pl

=

L1P2

=

0.5ps . (2,1)

At a pump pressure of 20-30 atmospheres, until recently a much used value, the pressure drop across a port equals 10-15 atm. Hydraulic servo-motors have come into use lately, however, in which the pump pressure is as high as 200 atm, so that the pressure drop across each of the ports is loo atm. Due to this high pressure drop the oil-flow rate is very high and may easily reach a value of 50 metres per second or 180 kilometres per hour.

The oil flows can be controlled by changing the size of the port openings; in this way it is possible to control the movements of the ram. It wiU be c1ear that the shape, dimensions, and finish of the ports have astrong influence on the nature of the oil flow and thereby on the properties of the hydraulic servo-motors. The ports are, in more than one sense, the heart ofthe matter. Therefore it will be worthwhile ·to analyze the flows.

2.2. The velocity-time constant 1'v and the flow rate Vo

In Chapter 1 we have seen that under no-Ioad conditions a hydraulic servo-motor works as an integrator, th at is to say: there is a linear relation between

the ram velocity

y

and the deflection x of the spool from the neutral position.

The spool is in this neutra I position when the ram is at rest and when no external load P, no friction Wand no inert ia forces m

y

are present. It can be demonstrated that there is a direct relationship between the velocity-time constant 'Tv and the velocity Vo at which oil flows through the ports wh en the spool is in the neutral position. The velocity-time constant, as defined in eq. (1 ,2)

can also be written as

The differential quotient b

y

/bx will now be determined for the region around the neutra I position for the case of the hydraulic servomotor depicted in Fig. 2.1, which has two controlled ports. In doing this, we ignore the total axial 10ad, EP = P - W - m

y,

as weIl as the compressibility of the oil, for the time being; when drawing up the differential equations in Chapter 3 we shall take both factors into account.

I .. LP-O

4579

Fig. 2.1. Servomotor with 2 controlled ports.

In Fig. 2.1 we can see that the ram velocity

y

follows directly from the difference Q2 - Ql, i.e. from the difference between the amounts of oil let into and removed from the left cylinder compartment:

y

=

(Q2- QÜ/F. Substitution into eq. (2,2), yields

1 F Cl _

E

-TV = - ' - = . by/bx bQ2/bx - bQl/bx (2,3) (2,4) Hence we have to find the partial differential quotients bQ2/bx and bQl/bx when we have to determine the velocity-time constant. To this end we first'

determine the quantities Ql and Q2, which are equal when the control valve is in the neutral position :

(2,5) According to eqs (2,1) and (2,5), the widths of the two ports will also be equal under those conditions :

(2,6) In the case of a hydraulic servomotor with two controlled ports a deftection of the spool of magnitude x will cause the width of one port to become as much larger as that of the other port becomes smaller:

h2

=

ho

+

x,

l

(2,7)

Hence we find for the quantities Qi and Q2:

Q2

=

7Tdn2v2

=

7Td(ho

+

X)V2,

Qi

=

7Tdhi vi

=

7Td(ho - X)Vi.

Partial differentiation with respect to x now yields

àQ2 [ àV2] Q2 [ ho

+

X àV2]

-

=

7Td V2

+

(ho

+

x) -

= - -

1

+ - -

.

-

,

àx àx ho

+

X V2 àx

àQi [ àVi] Qi [ ho- X àVi]

-

=

7Td - Vi

+

(ho - x) -

=

- -

-

1

+

- -

.

-

.

àx àx ho - X Vi àx

(2,8)

(2,9)

The last equation shows that the magnitude of the differential quotients depends on the nature of the port flow. As will be explained further on in this chapter, either a laminar or a turbulent flow through the port openings may occur, depending on the magnitude of the Reynolds number. According to the theory of hydrodynamics, flow rate V is independent of port width h, in

the case of turbulent flow. With h

=

ho

±

x we have àV2/àx

=

àVi/àx ~O and thus we find for the differential quotients àQ2/àx and àQi/àx, with the

control valve in the neutra I position:

(2,10) Finally, substitution into eq. (2,4) yields the velocity-time constant TV for turbulent port flow:

F TV

=

0.5 - -.

7Tdvo (2,11)

Vnder conditions of laminar port flow the flow rate Vo is proportional to port width h. With

we now have V2

=

bh2

=

b(ho

+

x),

I

Vi

=

bhi

=

b(ho - x), ) àV2/àx

=

b

=

v2/(h o

+

x), àVi/àx

=

- b

=

- vi/(ho - x). (2,12) (2,13) With the aid of eqs (2,9) and (2,13) we again find the differential quotients for the neutral position of the control valve:

(2,14) Finally substitution into eq. (2,4) yields the time constant TV for laminar port flow:

F TV = 0.25 - -.

The expressions (2,11) and (2,15) have been derived for a hydraulic servomotor with 2 controlled ports. It is, however, easy to see that the expressions are also valid for a servomotor with 4 controlled ports, on the condition that it has a symmetrical construction. We may not use the expressions in the case of a servomotor with only one controlled port, because then the port width of the so-called fixed port is constant, with the result th at either oQl/'öx

=

0 or

o

Q2/0X

=

O. Eqs (2,4), (2,10) and (2,14) show that in that case time constant Tv

is twice that for the other servomotors. SummalÎzing we may say that

F

TV = a - -,

TTdvo

where Tv = velocity-time constant of the hydraulic servomotor;

F

=

piston area;

(2,16)

Vo

=

oil-flow rate with the control valve in the neutral position ;

a = a constant depending on the type of servomotor and the nature

of the port flow, see the Table bel ow.

Type of hydr. servom. Turbulent flow Laminar flow

4 controlled ports a

=

0.5 a

=

0.25

2 controlIed ports a

=

0.5 a

=

0.25

1 controlled port a

=

1 a

=

0.5

Equation (2,16) shows that, given the piston area and the port diameter, a smalI value of the velocity-time constant can be obtained by ensuring that the oil flows through the ports at a high rate. This means th at we should not choose the port width too smalI,. because otherwise the boundary layer of oil formed in the port reg ion will slow down the oil flow. TV will then become relatively

large and, as we have seen, the control açcuracy will be affected. The turbulent flow of oil arising when the port width is large enough will be favourable to obtain a small time constant.

2.3. Flowequations

Many versions of ports for hydraulic servomotors are known. They all have this in common that they form restricted passages in the oil circuits. In literature only a few theoretical and experimental data will be found of the oil flow through the annular ports which have been discussed in the preceding sections. We shall, therefore, turn our attention fust to other types of ports, of which such data do appear in literature. The port depicted in Fig. 2.2 is formed by a diaphragm, that shown in Fig. 2.3 by a long slit between two

straight knife edges. The flow through these ports is given by

v

=

a Y2Llp/p, (2,17)

where v = the ave rage flow rate, referred to the full cross-section of the

. port;

Llp

=

the pressure drop across the port;

p

=

the specific mass of the oil;

a

=

hydraulic coefficient.

"

... / \ I \

1

1---/~', \ 2r r ( ~

I

l \ I I -\---~-'" I \ ,I \ ... _, / I ~ v' _I / I

Fig. 2,2. Port consisting of a diaphragm.

The hydraulic coefficient a depends on the Reynolds number. At very low values of-the Reynolds number Wuest 1) calculated for a:

a

=

aYRe.

"

-.. 581

Fig. 2.3. Port formed by slit between two knife edges.

A simple analysis shows us that under these conditions the oil flow is a laminar one; the constant

a

is therefore called the laminar-flow coefficient. At the higher va lues of the Reynolds number the coefficient a will be constant; already in 1917, von Mises 3) calculated its value:

,TT

a

=

ao

=

- -

=

0.611.

TT + 2 (2,19)

In this case the oil flow is turbulent and hence the constant ao is called the turbulent-flow coefficient. Fig. 2.4 shows the relationship between the coeffi-cient a and the Reynolds number for the diaphragm-type of port. The asymp-totes of the curve for the higher and lower Reynolds numbers (i.e. for turbulent

and laminar flow) are also shown. Naturally th ere is a transition region in the neighbourhood of the point of intersection of the asymptotes, this point being defined by:

Re

=

(ao/8)2 . (2,20)

Wh en the diaphragm is infinitely thin, the laminar-flow coefficient is,

according to Wuest 1), 8

=

0.2. With ao

=

0.611 we find that Re I::::! 9. The

theoretica I results obtained by Wuest conform roughly to the experimental ones of Johansen 2), who found a value of 0.17 for the laminar-flow

coeffi-cient 8. This means that the transition region lies around Re

=

12. The

discrepancy can be explained by the finite thickness of the diaphragm used in the experiments. This is very plausible, transition taking place at

Re I::::! 2300 in the case of an infinitely thick diaphragm. So, the thinner the

diaphragm, the closer the approximation to the theoretical values found by Wuest. According to Johansen 2) and Dali 4) the value of the coefficient a in the transition region depends strongly on the area ratio: when this ratio is high, the curve approaches closer to the asymptotes and it extends from

Re

=

5 to Re

=

20 (see Fig. 2.4). 0.6 05 a

jO

.

4

0.3 4582

lam/nar flow turbulent flow

10 ---. Re

Fig. 2.4. Hydraulic coefficient a plotted against the Reynolds number.

It can be proved, from the amount of oil flowing through a diaphragm-type of

port, that in practical cases the flow must be turbulent. Using eq. (2,17) and Re

=

2 vr/v, we find the flow volume to be:

Q

=

7Tr2v

=

~

(VRe)2 v

=

~

(vRe) 2 •

4 v 4 a-Y2L1p/p (2,21)

We now choose the Reynolds number to have a value at which we just expect'

LJp

=

1.5 MN/m2 _{(~15 kgf/cm2), for the kinematic viscosity y}

₌

_{80 mm}2_/sec

(= 80 eSt) and for the specific mass p = 900 kg/m3 . We th en find: Q = 57x 10-9 m3_/sec

₌

_{57 mm}3_/sec.

Evidently reasonable ram speeds can only be obtained at much larger va lues of Q and Re. At these higher va lues of Re (above 20) we may be quite sure that the port flow is turbulent.

Fig. 2.S. Annular port.

Experimental data on the oil flow through ports formed by knife edges are lacking. However, the way in which the calculations made by Wuest on dia-phragm-type ports have been confirmed by experiments, is sufficient proof that we may use eqs (2,17), (2,18) and (2,19) as the starting point for further considerations, which wiU be valuable because there is a close relationship between the knife-edge type of port and the much-used annular port. The latter may be considered as to be formed by wedge edges. See Figs 2.5 and 2.6.

~

IL'-

I I I I I h I 4584 v

Fig. 2.6. Wedge-type port.

The angle at the wedge edge is 90° and as aresuIt there will be a stronger tendency for a boundary layer to be formed (i.e. for the flow to be laminar) than in the case of the knife-edge port, where the angle at the edge is 0°.

There is reason to assume that, at lower va lues of the coefficient 0, the transition from laminar to turbulent flow occurs at higher Reynolds numbers, . see eq. (2,20); publications by Zahor 5) and Chaimowitsch 6) point in that direction. The data given by Zahor yield, after some caIculations, 0

=

0.033.

Chaimowitsch mentions a value of 0

=

0.05 but he does not give a detailed description of the type of port. The infiuence of the laminar-fiow coefficient 0

on the hydraulic coefficient a (and hence on TV) becomes evident from Fig. 2.7 where a is plotted against Re at different va lues of

o.

Two other important factors are the clearance of the fIow-regulating com-ponent in its housing and any rounding off of the edges. It wiII be clear that rounded-off edges must overlap if they are to close the port completely. The rounding-off will also considerably strengthen the tendency of forming a boundary layer, with the result that transition from laminar to turbulent flow will happen at still higher values of the Reynolds number. This was proved,

a o=0.61 06

,

,

_.

_"

.-"

,

,

_"

"

,

,

I

"

, / .-05 _I I

"

.-,

,

'"

.-.- flowa=6VRe a _I I

_"

_" lamin.ar

t

4 I I _Ç;"',,'"

"

/ " _turbulen.t flow a =a o

:J'

. j _Ç;'''' 0'3.- .-ó~O;'"/ _Re=(~o)2 ,,/ ,'}''! _t>'*'" limit value 0.3 '0, 'Ol

'"

, / , 1

'"

, / , 1

'"

" " , 1

_"

, / 02 1

_"

_" / 1

'"

, /

'"

"

"

1 1 '" / 1 '" .-,/ ","',,/ ,'(,.,/ I I I I I 4585 10 25 50 100 200 300 Re _ _

Fig. 2.7. Coefficient a versus Reynolds number at different values of laminar-flow coefficient c5. inter alia, by experiments carried out at the Philips Research Labs, the coefficient a being measured indirectly in accordance with:

F/a b

TV

=

7TdY PS/p

=

~'

(2,22) which equation has been obtained by substitution of eqs (2,1) and (2,17) into eq. (2,16) with a = 1.

InitiaUy the measured time constant agreed weU with the calculated value for which eq. (2,22) was used, with a = 0.61 and b = 2.4 milliseconds. When, however, the equipment was kept in operation for a long time, TV was found to increase from 4 milJiseconds to more than 12 milliseconds. To the control engineer this is a very unsatisfactory state of affairs. It was clear that the in-crease in TV was caused by a drop in the hydraulic coefficient a from 0.61 to about 0.2. This, in turn, was the consequence of a rounding-off of the edges due to wear (see Fig. 2.8). The radius of the worn edges was found to be of the order of 30 microns; the effective port width, h, was about 50 microns and

the Reynolds number about 40. Af ter a new valve with sharp port edges had been installed (r<10 microns), the original values: Tv

=

4 milJiseconds and a

=

0.61, were found again. The explanation is that the laminar-flow coeffi-cient 0 dropped from about 0.1 to about 0.03 which, in accordance with Fig. 2.7, causes the centre of the transition region between laminar and tur-bulent flow to shift from Re ~ 40 to Re ~ 4oo! Actually the whole problem is even more complicated due to the intricate geometry of the port with its rounded-off edges and clearance aspects; the simple equations (2,17) to (2,20)

v

4566

Fig. 2.8. Port formed by 2 rounded-off wedges.

incl. no longer suffice. That is why Chaimowitsch and Zahor give very elaborate empirie formulae on port flows, but these are only valid in special cases and for a limited region. American publications usually start from a turbulent flow obeying eqs (2,17) and (2,19). As we have seen above, however, th is is only aIlowed ifthe Reynolds number is above a certain limit value. Our experiments have shown that a turbulent flow is certain to occur through ports with sharp edges, if

Re ~ Re

=

20. (2,23)

It is now easy to calculate the port width necessary for obeying this turbulent-flow condition. With eq. (2,17) we find the limit value of the Reynolds number:

v h aoh

y2

-1p

-Re

=

2 -

=

2 - - - ~ Re.

v v · P (2,24)

Equating the pressure drop across the port to half the pump pressure, we find for the minimum port width at which the oil flow is still just turbulent:

vRe

ht

=

.

2aoY Ps/p

(2,25) By way of iIIustration we shaIl take a practical example: Let v

=

80 eSt

=

80x 10-6 _{m2/see; Re}

₌

_{20; ao}

₌

_{0.61; Ps}

₌

_{10 MN/m2 (~ 100 kgf/cm2)}

v

=

64 m/sec; with the port diameter being 12 mm, the flow volume is

Q

=

30 cm3_/sec

₌

_{1.8 dm}3_/min.

When the turbulent-flow condition, Re ~ Rë, is not met, it is very

difficult to calculate a. In that case we can determine a roughly with

a

=

0 -V Re, (2,18)

and with the aid of eqs (2,19) and (2,20) we find for 0:

0=0.13. (2,26)

Next the flow volume can be determined with Q

=

7rd h v, viz.

1/2,1p ,1

-for turbulent flow: Q

=

a7rdh

V

-p-

=

a07rdh

r

2,1 p/ p , (2,27)

1/2,1p

for laminar flow: Q

=

a7rdh

V

-p-

=

4021YJ.7rdh2,1p. (2,28)

It should be stressed again that with laminar flow only rough estimates of the coefficients 0 and a can be made. We then have the additional circumstance that a is strongly dependent on tempera tu re, the Reynolds number being inversely proportional to the dynamic viscosity 'YJ. Therefore a hydraulic servo-motor in which laminar port flow occurs is extremely sensitive to temperature variations. Some time after putting into operation, tempera tu re changes up to 25°C are quite normal and it will be c1ear that a reproducible behaviour of the motor cannot be obtained under such conditions. Hence turbulent port flow is to be preferred, if it were only for getting such a behaviour. With turbulent flow the coefficient a is independent of Reynolds number, viscosity

and temperature, so th at reproducibility is excellent. As we have seen before, turbulent flow leads to the lowest values of the velocity-time constant Tv. In this respect the reader is referred to the measuring results mentioned in Chapter 1, giving a value in the neighbourhood of 40 milliseconds for a number of hydraulic copying lathes. The servomotors developed at the Philips R~search Labs were found to have a velocity-time constant of about 4 milliseconds. This enormous difference is mainly caused by the properties of the port flow.

REFERENCES

1) W. Wuest, Strörnung durch Schlitz- und Lochblenden bei kleiner Reynolds-Zahlen,

Ingenieur Archiv 22, p. 357-367, 1954.

2) F. G. Johansen, Flow through pipe orifices at low Reynolds numbers, Proc. Roy. Soc. A. vol. 126, p. 231, 1930.

3) R. von Mises, Berechnung von Ausfiuss- und Überfallzahlen, Z. Ver. deutsch. Ing. 61,

p. 447, 1917.

4) H. E. Dali, Sorne aspects of fluid flow in orifices, nozzles and venturi tubes, Paper presented at a conference of the Institute of Physics, October 1950.

5) C. Zahor, Genauigkeit der Kopiersysteme, Schwerindustrie der Tschechoslowakei,

Nr. I, p. 46, 1955.

DIFFERENTIAL EQUATIONS FOR HYDRAULIC SERVOMOTORS 3.1. Genera}

In deriving the differential equations for hydraulic servomotors a number of fundamental equations are used, which can be divided into three categories : (a) equilibrium equation;

(b) flowequations;

(c) equations of continuity.

The th ree types of equations will be discussed fust, as they form the basis for the derivation ofthe differential equations for all kinds ofhydraulic servomotors described in Chapter 1. In the discussion a distinction wilt be made between turbulent and laminar port flow. The complete differential equations for servo-motors with 4,2 and 1 controlled port(s), successively, will be drawn up. It will be found that none of these equations is linear. However, it will become clear from graphical presentations that, in a wide region around the neutral position of the spool, th ere is an almost linear relationship between the piston speed

y,

the deflection x of the spool from the neutral position, and the total axial load

.EP on the ram. By series expansion of the complete differential equations, and by neglecting the higher derivatives, a set of differential equations is obtained which are linear in

y,

x and .EP. They can be written in a general form which is valid for all types of servomotors. Owing to the fact th at the dry friction Wo,

which is part ofthe totalload .EP, is a discontinuous function ofthe piston speed

y,

this' general differential equation is, however, basically non-linear.

Certain aspects of the non-linearity of the general differential equation are examined in more detail. At the end of the chapter a summary will be given enabling the various types of hydraulic servomotors to be compared.

3.2. Fundamental equations 3.2.1. Equilibrium equation

The pressures PI and P2 prevailing in the cylinder compartments exert a force on the piston, which balances the total axial ram load .EP (see Fig. 3.1). The latter can be considered as being composed of three components, to wit: (a) the external ram load P;

(b) the frictionalload W; (c) the inertia load

my.

In a provisional form, the equilibrium equation may be written as follows: P2F2 - PIFI

=

.EP

=

P - W -

my,

where P

=

external ram load; In

=

the mass ofthe ram; W

=

the frictional force.

When the ram rod is considered as a resilient link between the piston and the driven mass, then the externalload P can be thought of as a complex function of the "output signaI" of the servomotor. The effect of this complex nature is not considered in this thesis.

~ F,

l

- - j

~

• ~P:P-Wo-wj-mf 1 -P2 P,

I

4587 Q2- Q,

t

_{Q -Q 3 ,} Fig. 3.1. Fundamental equations.

The friction al force W is, of course, closely related to the forces acting late-rally on the ram. Blackburn 1) has caIculated these lateral forces under certain conditions, the piston not coming into "metallic" contact with the cylinder wal!. The oil flow between piston and cylinder wall, as caIculated by him, makes it possible to determine the frictional force.

Blackburn did not succeed in caIculating the forces when there was metallic contact between piston and cylinder wal!. These are governed by, among other factors, the clearance between piston and cylinder wall, the ratio between the pressures prevailing at either side of the piston and the speed at which the piston moves through the cylinder. Blackburn himself was aware of the fact that me-tallic contact wiJl as a rule take place and that the lateral forces wiII therefore be many times stronger than those calculated by him.

In practice, measurement of the frictional forces is also difficult. Me asure-ments carried out in the Philips Research Labs showed that the pressures at either side of the piston have astrong influence on friction. Without these pres-sures (cylinder empty), friction was found to be many times weaker than during normal operation of the servomotor, when the pressures were present. Also, the friction was found to depend strongly on the exactness of geometry (excentricity of the rod, out-of-roundness, etc.) and the surface quality of ram and cylinder. Even after much care had been devoted to obtain optimum conditions in these respects, friction was found to attain values of 100 to 500 N quite easily. There is some evidence that we may assume piston friction to be determinative oftotal friction. the contributions of friction occurring at other points in the mechanism (e.g. between a driven slide and its guides) being relatively smalI, provided suf-ficient care has been given to the design and construction of these parts.

Russian publications 2) give indications how to reduce piston friction by spe-cially shaped pistons.

Forced by the impossibility to calculate the friction forces or to determine them accurately by experiments, we assume a simple relationship to exist be-tween friction and piston speed, in analogy to what has been found in other techniques:

W= Wo

+

WIY.

(3,2) Fig. 3.2 shows this friction-versus-piston speed relationship. The friction can be regarded as the sum of the dry or Coulomb friction (the sign of which is reversed with the direction of movement of the piston) and the viscous friction

---+i'

Fig. 3.2. Friction vs ram velocity.

(which is proportional to the piston speed). As long as the piston is stationary, friction can have any value between

+

Wo and - Wo. We can now write for the friction when

Y

>

0

y

<

O

y=O

Wo

=

8 Wo; 8

=

+

1, 8=-1, -1 ~ 8~+ 1. (3,3)

We now use eqs (3,2) and (3,3) to bring the equilibrium equation (3,1) into its final shape

(3,4) 3.2.2. Flowequations

Before we can write down the differential equations we shall fust have to calculate the amounts of oil flowing through the ports. In doing this we shall confine ourselves to recapitulating the resuIts obtained in Chapter 2 for annular ports. In the case of turbulent flow we found:

Q

=

aOTrdh

h

11 pip, (3,5)

and for laminar flow:

In these equations ao represents the turbulent-flow coefficient, S the laminar-flow coefficient, 7] the dynamic viscosity, p the specific mass ofthe oil, dtheport diameter, h the port width and LIp the pressure drop across the port.

Wh en the pressure drop LIp across the ports is known we can determine port flow Q with eq. (3,5) or eq. (3,6). In the hydraulic servomotors under discussion the pressure drop is either equal to the difference between the constant pump pressure Ps and one of the pressures on the piston, PI or P2, or it is equal to the difference betweenpI or P2 and the pressure in the oil reservoir. Strictly speaking the pressure drops across the Iines ("line losses") should be accounted for. These line losses, however, can normaIly be ignored, as wiII be proved below. A loss occurring in a hydraulic line can easily be expressed in the line length 1,

the line cross-section j; and the line flow Q:

I lp .

LI PI

=

87T7] - Q

+

-

Q .

12

f

(3,7)

The fust term in the right-hand part of this equation represents the laminar-flow losses, the second term the pressure difference necessary for an acceleration of the oil flow. The foIlowing numerical example may iIIustrate that the line losses may be ignored. With Q = 60 cm3_/sec;

Q

_{= 6000 cm}3_/sec2 _{(i.e. Q} rises from zero to 60 cm3_{/sec in 0.01 sec); 1= 10 cm;f= 1 cm}2_{; 7] = 0.072} kg/m sec and p = 0.9 kg/dm3 _{we find LlPI = 1 kN/m}2

+

_{5 kN/m2 = 6 kN/m}2

(~ 0.06 kgf/cm2). Zeleny 3) took fuIl account of the line losses in an investi-gation of the stability and proved that only in very special cases (with special devices in the lines) does the pressure loss become important, having a favourable influence on stability.

The so-called leakage flow between the two cylinder compartments is neg-lected in comparison with the port flow. This is permissible provided that there is a continuo us connection between these compartments via the ports in the controI valve. As we have seen before this connection is necessary for obtaining a turbulent oil flow and so a small velocity-time constant Tv. This may again be ilIustrated by means of a numerical example. Let the piston diameter be 80 mm, the piston thickness 30 mm and the clearance between piston and cylin-der waIl 0.02 mm; Iet further LIp = 5 MN/m 2 and 7] = 0.072 kg/m sec. Then we find:

7TDh3 _LIp

QL = = 0.4 cm3_/sec.

12 7] b (3,8)

Wh en all ports are continuously open the leakage flow may be neglected. The infiuence of th is leakage flow on the properties of hydraulic servomotors, if such an influence is at all present, wiII be discussed further in Chapter 6.