Influence of entrained air on frequency response of hydraulic actuator controlled by open center three way spool valve

Mechanism and Machine Theory. Vol. 20, No. 2, pp. 139-14.4, 1985 Printed in the U.S.A.

0094-114X/85 $3.00 + .00 © 1985 Pergamon Press Ltd.

INFLUENCE OF ENTRAINED AIR ON FREQUENCY RESPONSE OF HYDRAULIC ACTUATOR CONTROLLED BY OPEN CENTER THREE WAY SPOOL VALVE A K E L L A . S. R. M U R T Y t , G. L. S I N H A t and B. N. DATTA~t Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, India Abstract--A theoretical investigation on the effect of entrained air on frequency response of hydraulic actuator controlled by open center three way spool valve is made. It is shown that entrainment of air gives rise to additional non-linearity into the system equation. Both nonlinear and linearized system equations, derived in dimensionless form, are solved numerically and analytically respectively. Increase of air content or decrease of area ratio in presence of air increases the resonant peak amplitude and decreases the resonant frequency leading to a reduction in bandwidth. Similar effect is observed also when the process of change of state of entrained air is adiabatic rather than isothermal. Entrainment of air also reduces the allowable open loop gain making the stability in the closed loop inferior.

monly used definition is that of tangential bulk modulus. Goodwin[2] modified the expression develIt has now been established that the dynamic beoped by Rendel and Allen[l] to agree with the havior of hydraulic servosystem cannot be dedefinitions of isothermal and adiabatic tangent bulk scribed fully unless the influence of compressibility moduli. is taken into account, particularly for high pressure Meritt[3] developed an expression for equivalent applications. Compressibility effect appears, "in bulk modulus including the effect of mechanical general, in the form of bulk modulus of the working compliance. He showed that although the effect of fluid which, for pure oil, remains almost constant mechanical compliance is very small, the effect of over the range of pressures normally used in hysmall amount of entrained air reduces the equivadraulic systems and therefore, appears only as a lent bulk modulus considerably. He also discussed, constant parameter in the system equation. Howqualitatively, the adverse effects of the reduction ever, hydraulic systems can hardly be made abin bulk modulus on the dynamic behavior of hysolutely free from entrained air in spite of all predraulic servos, most of which conform to the findcautions being taken. A little entrained air can ing of the present investigation. cause a substantial change in the equivalent bulk Radhakrishnan and Sundaram[4] studied the efmodulus of the working fluid particularly at lower fect of entrained air on step response of hydraulic system pressures and produce severe effects on the servo using the simplified expression for bulk moddynamic behavior of the system. The influence of ulus developed by Rendel and Allen[l]. Urata[5] entrained air appears as additional nonlinearity in made similar study on the closed loop behavior of the system equation making it more complicated for hydraulic servo with four way critical centre valve analysis. Since hydraulic servos with lower system using the expression for equivalent bulk modulus pressures are quite often used in industry, e.g. hyoriginally developed by Goodwin[2]. He observed draulic tracer unit in machine tool, a study of the that the symmetricity of cylinder pressure change effect of entrained air on its performance seems imis remarkably disturbed due to entrained air despite portant. the geometric symmetricity of the system. Rendel and Allen[l] studied the effect of enIn the present work a theoretical investigation is trained air on the reversibility of hydraulic mechmade on the effect of entrained air on the open loop anism. They obtained an expression for equivalent frequency response behavior of a hydraulic actuabulk modulus of air-oil mixture neglecting the eftor controlled by an open center three way spool fect of change of pressure on bulk modulus of oil. valve. The expression for equivalent bulk modulus They, however, did not mention anything about the for air-oil mixture developed by Goodwin[2] has effect of the process of change of state of air enbeen used to define the compressibility parameter. trained into the system. Usually four bulk moduli, The system equation, developed in dimensionless viz., mean (secant) and tangent bulk moduli each form, is highly nonlinear in nature and is linearized associated with isothermal and adiabatic change of to yield an analytical solution by usual frequency state of air are defined. Of these, the most cornresponse technique. The linear solution shows directly the consequence of air entrainment on the t Professor. open loop gain, critical frequency and also on the * Research Scholar, on deputation from Applied Mestability of closed loop step response. The original chanics Department, B. E. College, Howrah-711103WB, nonlinear system equation is also solved numeriIndia. INTRODUCTION

139

140

A. S. R. MURTYet al.

cally by fourth-order Runge-Kutta Method to study the effect of air entrainment and area ratio on the open loop gain, critical frequency, resonant peak amplitude and cylinder peak pressure. Comparison of linear and nonlinear solutions shows very good agreement for small load parameter except in a narrow region around critical frequency, where the nonlinear solution shows a steep-rise in cylinder peak pressure breaking the assumption of small load pressure made for linear analysis.

where, Ce is the orifice discharge coefficient, d is the valve sleeve diameter, 13is the oil density, x is the input valve displacement, and u,, u, are supply and exhaust underlap, respectively. Combining eqns (1) and (2) we get: dy Vi + AiY de] A l - ~ - K L ( P , - P1) + 13mix dt = CdtrdX/~[(u~ + x ) X / - f f 7 - P1 -

(u,

-

(3)

x)V~l].

DERIVATION OF SYSTEM EQUATION The system under consideration is shown schematically in Fig. 1. The following assumptions are made in deriving the system equation: (i) turbulent port flow, (ii) rectangular valve ports, (iii) negligible fluid inertia, (iv) no reverse flow, (v) rigid conduit, (vi) constant supply pressure and (vii) pure inertia load. Under the above assumptions the flow rate Q to the piston end is given by: Q = flow for piston displacement - leakage flow + compressibility flow

For an external load XF, the condition of dynamic equilibrium gives: PIA1 - P~A2 = ~,F

where A2 is the rod end area of cylinder. Following Goodwin[2], the dimensionless compressibility parameter, h, the ratio of oil bulk modulus to the equivalent bulk modulus of air-oil mixture, for a polytropic change of state of air with an index n, can be written as:

(1) h=

dy Vi + A I y alP] = A1 ~ - Kz.(P~ - PI) + 13mix dt

13oil 13mix 1+-~

where, A1 is piston end area of cylinder, y is the output displacement, t, the time, KL is the laminar leakage flow coefficient, P, is the supply pressure, P1 is the piston end cylinder pressure, Vi is the initial enclosed volume of fluid in the piston end cylinder chamber and 13m,xis the bulk modulus of airoil mixture. From the orifice flow condition, the flow rate Q for operation within underlap region, can be written as:

(41

1+

rl Pat

/5) 1 _

P._! +

¢

1 +

P~

13o

where ~ is the percentage of air content by volume at atm. pressure, 13o(= 13on/PD is the dimensionless oil bulk modulus and pj( = P1/PD, Pat( = P,t,-,,oJPs) are the dimensionless piston end cylinder pressure and atmospheric pressure respectively. From eqn (4), the dimensionless cylinder pressure pl can be written as,

Q = Ca~rd'~/27"p[(u, + x ) ~ / - ~ - P, -

(u,

-

x)~P-~l]

p~ = k + p~

(2)

A2, Ps ]j AI~P1

I

.y J--.-.

V--q I

I-"

I

--1

Fig. 1. 3-way spool valve controlled asymmetric linear motor.

(6)

Frequency response of hydraulic actuator where k(= A2/AI) is the area ratio and pt(= ~F/ P,A1) is the dimensionless load pressure. F o r the present analysis it is assumed that the volume of oil in the conduit is negligible compared to that in the cylinder and initially, the piston is at m i d - p o s i t i o n o f t h e c y l i n d e r , s u c h that V~ = where Vt is the total volume of the cylinder.

V,/2,

141

In such cases the system equation, eqn (7) can be linearized, neglecting the effect of 0 and using eqns (8), (9) and (10), for small oscillation around the neutral equilibrium position (X = u* = pt = 0) to:

d 3y,

d 2y,

hotO*3--~-3 + 80tto.2 d,r2

Thus, for a sinusoidal valve input, x = r sin tot (r is the amplitude and ~ is the frequency), eqn (3) together with eqns (5) and (6), after nondimensionalization, can be written as:

+ to. dy* =

K~X

(11)

where, u* - *(1

2y*'~ dpt

- k - Pt) + hto*ct 1 + ---~-,]-~

= V ~ [ O ] + X ) V ' I - k - pt

8 =

rl

~(1 + k) 2k

+

(12)

(7) K~ = X/~(V'I - k + V~)

(13)

- (v'n - X ) V ' ~ + pt],

the dimensionless output velocity u* can be written

and compressibility parameter at null position,

as:

1 + -e l Pat 13-"9"° + n( u* = to* dy*

d'r '

(8)

k

X° =

Pa,/--k~-(l+l/n) (

1-~oo+~

Pt = '*to* dr '

(9)

where, -r(= tot) is the dimensionless time, to*(= to/ to,) is the dimensionless frequency, y*(= y/(Q/ A,to,)) is the dimensionless output displacement, @( = KLPfl(2) is the dimensionless leakage parameter, X ( = Z sin r) is the dimensionless input displacement, Z( = r/xv) is the dimensionless input amplitude, "q(= u,/xp) is the dimensionless supply underlap, v(-- ut/u,) is the lap ratio, Q(= C:r d x p . ~ ) is the reference flow, xp is reference port length, M is the mass of load, a( = Mto, Q/ PsA 2) is the dimensionless inertia load parameter, to,(= N/2~oivA2/MV,) is the system natural frequency (without air), 0(= LAjo~n/(2) is the dimensionless cylinder length and L is the length of the cylinder. Again, for equilibrium at neutral position, the lap ratio, v, is given by, v = ~/~_._k+~(1

- k) "qV~

For convenience we adopt a new time scale r' = r/to*, so that eqn (11) transforms to: d3y.___~ *

dy_..~* =

KvX

(15)

and X = Z sin(to%').

(16)

Taking Laplace transform of both sides of eqn (15) we get the system transfer function: G(s)

=

y*(s) X(s)

(17)

K~ S[hoS2 + 8as + I]' The frequency response characteristics can be obtained directly from eqn (17) by substituting s = jto*. Thus we get the sinusoidal transfer function: y*(jto*) G(jto*)

Equation (7), together with eqns (8), (9) and (10) represent the system against a sinusoidal valve input X = Z sin r.

It is clear that the system under consideration is highly nonlinear in nature. However, in many of the applications hydraulic servosystem works only with small change from its equilibrium condition.

d2y.___~ *

h0 dr,3 + Ba dr,2 + dr'

(10)

LINEARIZATION

(14)

l + - - Pat,/

and for pure inertia load: d//*

k ") -'/"

-

X(jto*)

(18)

Kv = jto*[(1 - hoto*2) + f6aco*] " Therefore the amplitude ratio and phase lag are given by: Amp. ratio, [ G(jto*) I =

Kv to*N/[(8otto*)2 + (1 - hoto*2)2]

(19)

A. S. R. MURTY eta[.

142

and phase lag, 6ato*

4zG(jto*)

= - ~r _ t a n - I 1 - hoto.2" 2

(20)

F r o m eqns (19) and (20), the critical frequency, at which resonance occurs for small load, is given by, ,

to cr

=

1

\ /-~oo "

With no air entrained into system Xo = 1, as can be seen from eqn (14) and as such the critical frequency is a constant. But in presence of air, Xo and hence the critical frequency, toc* changes remarkably with change in area ratio, k, and polytropic index, n. As the percentage air content increases, Xo increases thereby reducing the critical frequency. But as area ratio increases, ho decreases and therefore critical frequency increases.

solved simultaneously to give the values of y * , u ~ and p~ assuming their initial values to be zero for first cycle. The values of y*, u* and Pt at the end of each cycle were used as the initial values for the next cycle to repeat the process until steady-state was reached. Simultaneously, amplitude ratio, maximum and minimum load pressures were also calculated for each cycle. Amplitude ratios for five consecutive cycles were compared by five-point forward difference formula to note the change of amplitude ratio per cycle and also its rate. When the change of amplitude ratio per cycle and the rate at which it takes place becomes less than certain predetermined small amount, the results were printed out. assuming that the steady state has been reached and otherwise the process was made to repeat. Six hundred steps per cycle were used for the computation in HP-1000 system. The effect of the term containing 0 has been neglected. RESULTS AND DISCUSSION

STABILITY To state the stability condition of the corresponding closed loop, the output and input displacement should be written in the same scale. Thus changing the scale of output displacement y* to Y = y / x p , we can write:

Although eqn (5) gives the value of the dimensionless compressibility parameter, X, for an arbitrary process of change of state of air, only isothermal (n = 1) and adiabatic (n = 1.4) change has been considered for numerical computation. Figure 2 shows the effect of percentage air content on the

y*=o-Y-o. y

x p A 1to,,

Alton

35~- k =0.5,9=Oo,¢=0.0,Z=0.03,~I=0.06, I Pot,O.O2,e-o.),; !~.,

Thus the amplitude ratio can be written as:

30

Q

K,

l}'I

4L

t a ; [ = xpto*x/[(sato*) = + (1 - Xoto*~) =] A,to. "

Following Nyquist criteria the stability condition for the corresponding closed loop can now be written as: Q

K,.X0 --<1

Alto~ Xp&a

where

X001K~ - - < 213o8

or

0I

--

L Xp --

ii

I

i!Ji' -..oooo IV I IIl~J~ L.---°°°5

/ l ,II, AFL.--o.o,o

1,

•

Thus the increase of air content or decrease of area ratio in presence of air decreases the allowable open-loop gain making the stability inferior. NONLINEAR SOLUTION Although the linear solution shows the essential features of the effect of entrained air on the system behavior it fails to give good results in many practical situations due to its limitation in the assumptions made. The original nonlinear system equation has also been solved numerically by fourth order Runge-Kutta method. Equations (7)-(9) were

0.1

~

0.5

1.0

1.5

-90

w -180 o

e~

-270

I

0.1

i

I

I

I

I

I

0.5

I

[

1.0

I

l

I

I

1.5

CO u

Fig. 2. Effect of air content on frequency characteristics,

Frequency response of hydraulic actuator 35

k= 0.5, e=oo, I~ : 0.0, Z : 0.03, "q= 0.06 , Pot = 0.02 / /I, E::0.01 n : 1.4 -.-~ ~ . ~ - n = 1.0

30

.~ 20 G 15

n=!.0

,0

o, 1~ - - - - ~ . \ \ \

5 -

1

I

I

°o'.I

-

Non-lineor

I

I

i

I

I

o.5

I

I

"%~ I "~...~

i

1.o

ts

COf' Fig. 3. Effect of polytropic index on amplitude ratio.

1.0

g=°°, ~ = 0.0, Z= 0.03, "rl = 0.06, k = 0.5, 0.14 pat, 0.02 , n = 1

a=

~:= 0.000

o,

"t~

o:OOo __ii1

- 0.8

0"020~I

143

open loop gain and phase lag for small load parameter. For no air content (e = 0) resonance occurs as usual at ~o* = 1. But as e increases the resonant frequency decreases and the resonant peak amplitude increases. The effect is more prominent if the process of change of state of air is adiabatic rather than isothermal. The curve of phase lag also shifts accordingly. Thus, entrainment of air into the system causes reduction in effective " b a n d w i t h . " Figure 3 shows the comparison between linear and nonlinear solutions in the presence of entrained air for ~t = 0.14 and 3.5. It can be seen that for the small load parameter the agreement is extremely good except only in a very narrow region around the critical frequency. This can be explained through the observation made in Fig. 4, that the cylinder peak pressure response curve has a steep rise in this region making the assumption of small load pressure made for linear analysis invalid. For similar reason the difference increases as the load parameter is increased. Figure 5 shows the effect of area ratio, k, on the amplitude ratio for medium load. It is observed that as k decreases the critical frequency decreases and resonant peak amplitude increases. CONCLUSION

0.7

0.6

0.

0.s

,.0

,.s

al =

Fig. 4. Effect of air content on cylinder peak pressure.

A decrease in area ratio in presence of air, an increase in the value of the polytropic index, n or an increase in the amount of air entrained into the system all are responsible for increase of the resonant peak amplitude and decrease of the critical frequency at which resonance occurs. These effects also cause the reduction of allowable open-loop gain making the closed loop stability inferior.

REFERENCES e =o~ ~ .B--0.0, z= 0.03 ,~i= o.oE, ~t=o.02, 25-

,o\

00.1

E=0.01

~:o.7

n=1

FX'x ~ .

0.5

.-k=o.2,

1.0

1.5

W" Fig. 5. Effect of area ratio on amplitude ratio.

1. D. Rendel and G. R. Allen, The effect on the reversibility of a hydraulic mechanism of variations in the bulk modulus of the fluid. Aircraft Engng 23, 337-338 (Nov. 1951). 2. A. B. Goodwin, Power Hydraulics. B.I. Publication Bombay, India (1964). 3. H. E. Meritt, Hydraulic Control Systems. John Wiley, New York (1967). 4. P. Radhakrishnan and S. Sundaram, Effect of compressibility due to entrained air on the performance of a hydraulic system. Proc. 7th A1MTDR Conf. (June, 1976) pp. 285-288. 5. E. Urata, Influence of compressibility ofoil on the step response of a hydraulic servomechanism (3rd Report). Bull. JSME 25(203), 797-803 (1982).

144

A. S. R. MeaTY et al. A U S W l R K U N G E N VON MITGEFUHRTER LUFT AUF DEN FREQUENZGANG EINES HYDRAULISCHEN SYSTEMS MIT ASYMMETRISCHEM LINEAREM ANTRIEB Km-zfasstmg--Es ist bekannt, dab der fl/ichtige Anteil der Differentialgleichung zur Beschreibung eines hydraulischen Systems zum groflen Teil durch den Kompressionsmodul der Arbietsflfissigkeit bestimmt wird. Kleine Luftmengen k6nnen den Modul verringern, insbesondere, wenn das System bei niedrigen Drficken betrieben wird. Solche mitgefiJhrten Luftmengen, die unter ung/Jstigen Umst~inden bis 20% des gesamten Volumen bei Normaldruck betragen kOnnen, sind bei praktischen Anlagen nicht v611ig zu beseitigen. Die vorliegende Arbeit besch~tigt sich mit einer theoretischen Behandlung der durch mitgeflihrte Luft verursachten ~,nderungen der Verst/irkung ohne Gegenkopplung eines hydraulischen Systems mit asymmetrischem linearem Antrieb, das von einem Dreiwegesteurventil gelenkt wird. Bei der in dimensionsloser Form geschriebenen nichtlinearen Systemgleichung wirkt sich die mitgef/ihrte Luft als ein Kompressibilit~iterhaltendes zus~itzliches nichtlineares Glied aus. Die so erhaltene Systemgleichung ist nach zwei Methoden behandelt worden: (a) Durch Linearisierung der nichtlinearen Gleichung erhalten wir eine analytische L6sung, aus der unmittelbare Schl/isse auf die Verst~irkung ohne Gegenkopplung, die Resonanzfrequenz und auch auf die Stabilit~it des Systems bei Gegenkopplung gezogen werden kOnnen, (b) Die numerische Auswertung der ursprfinglichen Systemgleichung mit Hilfe des Viertenordnungs-Ringe-Kutta Verfahren gibt Auskiinfte fiber die Auswirkungen der mitgeftihrten Luft und des Querschnittsverhaltnisses auf die Verst~,rkung des Systems, die Resonanzfrequenz und den Spitzendruck am Zylinder. Bei niedrigeren Belastungen stimmen die zwei so abgeleiteten LOsungen sehr gut iiberein. Ausgenommen ist ein enger Bereich um die Resonanzfrequenz. In der Arbeit wird festgestellt, daft --mitgefiahrte Luft unerwiJnschte Auswirkungen auf die Stabilit~it bei Gegenkopplung hat - bei zu groSem Luftanteil kann die Anlage sogar unstabil werden - , --die Zunahme der eingeffihrten Luft einen Zuwachs der Resonanzspitzenamplitude, eine Herabsetzung der Resonanzfrequenz und eine Verschiebung der wirksamen Bandbreite gegen niedere Frequenzen verursacht, und --die Verringerung des Querschnittsverhaltnisses in Anwesenheit von Luft ebenfalls zu oben erwahnten Resultaten fflhrt.