Theoretical Simulation of Static and Dynamic Behavior of Electro-Hydraulic Servo Valves

Scientific Bulletin of the The 6th International Conference on Politehnica University of Timisoara Hydraulic Machinery and Hydrodynamics Transactions on Mechanics Special issue Timisoara, Romania, October 21 - 22, 2004

THEORETICAL SIMULATION OF STATIC AND DYNAMIC BEHAVIOR OF ELECTRO-HYDRAULIC SERVO VALVES

Victor BALASOIU, Prof.* Mircea Octavian POPOVICIU, Prof. Department of Hydraulic Machinery Department of Hydraulic Machinery “Politehnica” University of Timisoara “Politehnica” University of Timisoara Ilare BORDEASU, Assoc. Prof. Department of Hydraulic Machinery “Politehnica” University of Timisoara *: Bv Mihai Viteazu 1, 300222, Timisoara, Romania, Tel.: (+40) 256 403681, Fax: (+40) 256 403700, (+) 256 4030682, Email: [email protected], [email protected]

ABSTRACT The electro-hydraulic servo-valves (EHSV) as in- range Yoi and the clearance J over the linearity degree terface in automatic hydraulic systems are in essence and the magnitude of the adjusted flow. hydroelectric directional control valves with cylindrical The model of the dynamic equilibrium of the spool spool, with integral reaction. The output quantity valve is defined starting with the computation of forces (flow rate, pressure) is modified proportional with acting on the spool. Introducing the concept of interac- the control signal (voltage, current) together link tion between the aggregate components nozzle-spool- (electrical, hydraulic or mechanical). Both the manner distributor it was defined the mathematical model for in which the reaction signal is produced and the the directional spool valve as a whole. position of the information circuit where it is applied, After defining the transfer function, both analyze characterises the type but also static and dynamic and syntheses of performances were possible for the behavior of the servo valve. The servo valves currently considered servo valve, in terms of time or frequency. used for automatic hydraulic systems are those with The frequency characteristics and the transfer function two stages able to develop great output hydraulic hodograph were plotted, putting into evidence the over- powers for small input electrical signals. lap degree Yoi, the input pressure at constant control In the paper, on the basis of electro-hydraulic electric current ∆ic or the control electric current for analogy it is developed the mathematical model able different constant input pressure. The analyze of the to give theoretical analyses of the static and dynamic dynamic behavior in time had in view the computation of the response at a unitary step input by determining behavior for the main stage, the valve with the cylin- the behavior of the output magnitude and the position drical spool. After the definition of the basic equations of the cylindrical spool Y (t). The variations of the of the aggregate spool-distributor for zero clearances S output magnitude represent the solution of the linear and overlaps it was analyzed the directional spool differential equation that is describing the running of valve with radial clearances and zero overlap. Fi- the servo valve by an III degree transfer function, nally the generalized equation of the adjustment char- which was obtained in the present work. acteristic for flow rate [QM = f(yS)∆pM=0], pressure In the framework of the paper was worked out a [∆pM = f(yS)QM = 0] and load [QM = f(∆pMyS)] for unitary mathematical model for the servo-valve with transition and laminar flow, with non-dimensional was two stage having pressure reaction permitting the study plotted both for linear and non linear zones, taking of the influence of geometrical and functional parameters into account the annular clearance J and the overlap upon the behavior of static and transient regimes. range Yoi ≠ 0. For the pressure valve taken into con- sideration there have been deter-mined and plotted the KEYWORDS adjustment characteristics [Q = f ∆p ,Y ], MA,B ()MA,B S Electro-hydraulic servo valves, spool-distributor, for the flowing directions A and B, for which there adjustment characteristics transfer function, mathe- have been emphasized the influence of the overlap matical model

303 1. INTRODUCTION motion (αD = f (∆p),(Re=100…2500), introduced Electro hydraulic servo valves (EHSV) in current by H. Weule and H. J. Feigel [1], the equation (1) use for the automatic hydraulic systems are those with written for the directional control valve became: two stages able to develop great hydraulic power, for small input electric signals. Running with constant pressure, regardless of the constructive solution, they have a functional dependence between the flow and the applied command current Q = f(∆iC), ensuring both a good stability and linearity. With the view to establish the dynamic and static characteristics, which are determined by a great number of physical, geometrical, electrical and mechanical parameters, there are taken into account the fundamental laws of mechanics and hydraulics, completed with the automatic system theory.

2. THEORETICAL ANALYSES OF THE DISTRIBUTION STAGE WITH LINEAR CYLINDRICAL SPOOL VALVE Appealing to the model introduced by W Backe [5] the stage spool valve-distributor body can be reduced to a scheme tip A + A, as is shown in Fig.1. For a permanent motion the continuity equation becomes:

2.∆pi,e Q = α .π.D . (1) Fig.1. D S ρ ⎡ K2 K ⎤ Q = K Y2 +1⎢ t − t ⎥ (2) Utilizing the modified flow equation both for the D j 2 2 ⎢ J j +1 Yj +1⎥ laminar (αD = f (∆p,ν, Re), and the transition ⎣ ⎦

⎡ 2 ⎤ K K Q = K (Y + Y )2 + J2 ⎢ T + p − p − T ⎥ 10 Dj S 01 ⎢ ()MA 04 ⎥ ()Y + Y 2 + J2 2 2 ⎣⎢ S 01 (YS + Y01) + J ⎦⎥ ⎡ 2 ⎤ K K Q = K (Y − Y )2 + J2 ⎢ T + p − p − T ⎥ 20 Dj S 02 ⎢ ()03 MA ⎥ Y Y 2 J2 2 2 ⎢ ()S − 01 + (YS − Y02 ) + J ⎥ ⎣ ⎦ ⎡ 2 ⎤ K K Q = K (Y − Y )2 + J2 ⎢ T + p − p − T ⎥ 30 Dj S 03 ⎢ ()MB 03 ⎥ ()Y − Y 2 + J2 2 2 ⎣⎢ S 03 (YS − Y03) + J ⎦⎥

⎡ 2 ⎤ 2 2 ⎢ KT KT ⎥ Q40 = KDj (YS + Y04 ) + J + ()p03 − pMB − (3) ⎢ 2 2 2 2 ⎥ ⎢ ()YS + Y04 + J (Y + Y ) + J ⎥ ⎣ S 04 ⎦

Qio non zero overlaps linear zone non linear zone (4) i=1,

2,3, YS YSN 4

− YSN < YS < −Y0i YS < −YSN

304 Q10 ⎡ 2 ⎤ ⎢ K K ⎥ 0 K (Y )2 + J2 T + p − p sign(p − p ) − T DJ S+Y01 ⎢ 2 2 MA 04 MA 04 2 2 ⎥ ⎢ ()YS + Y01 + J ()YS + Y01 + J ⎥ ⎣ ⎦ ⎡ 2 ⎤ ⎢ K K ⎥ − - K (Y )2 + J2 T + p − p sign(p − p ) − T DJ S+Y01 ⎢ 2 2 MA 04 MA 04 2 2 ⎥ ⎢ ()YS + Y01 + J ()YS + Y01 + J ⎥ ⎣ ⎦ Q 20 ⎡ K2 K ⎤ 2 2 ⎢ T T ⎥ KDJ (YS − Y02) + J + p03 − p sign(p − pMA ) − ⎢ 2 2 MA 03 2 2 ⎥ ()YS − Y02 + J ()Y − Y + J ⎣⎢ S 02 ⎦⎥

⎡ K2 K ⎤ ⎢ T T ⎥ − - KDJ.J + p03 − p sign(p − p ) - ⎢ J2 MA 03 MA J ⎥ ⎣⎢ ⎦⎥

Q30 ⎡ 2 ⎤ ⎢ K K ⎥ K ()Y − Y 2 + J2 T + p − p sign(p − p ) − T DJ S 03 ⎢ 2 2 03 MB 03 MB 2 2 ⎥ ⎢ ()YS + Y03 + J ()YS + Y03 + J ⎥ ⎣ ⎦ ⎡ 2 ⎤ ⎢ K T K T ⎥ – K DJ + p − p sign(p − p ) − ⎢ J 2 03 MB 03 MB J ⎥ ⎣ ⎦ Q40 ⎡ 2 ⎤ ⎢ K K ⎥ 0 K (Y + Y )2 + J 2 T + p − p sign(p − p ) − T DJ S 04 ⎢ 2 2 MB 04 MB 04 2 2 ⎥ ⎢ ()YS + Y04 + J ()YS + Y04 + J ⎥ ⎣ ⎦

⎡ 2 ⎤ ⎢ K K ⎥ − - K (Y + Y )2 + J 2 T + p − p sign(p − p ) − T DJ S 04 ⎢ 2 2 MB 04 MB 04 2 2 ⎥ ⎢ ()YS + Y04 + J ()YS + Y04 + J ⎥ ⎣ ⎦ (5) ⎡ ⎤ 2 ⎡ Q 1 K2 K ()Y + Y K2 MA = ⎢ T + p + p .sign(p − p ) − T ⎥ − J 01 ⎢ T + p + p . 2 03 MA 03 MA 2 2 03 MA QMN 2 ⎢ J J ⎥ p ⎢ (Y + Y ) + J p03(YSN +1) ⎣ ⎦ ⎣ S 01 ⎤ KT ⎥ .sign(p − p ) − pentru Y 0i < Y < YSN 03 MA 2 2 ⎥ (YS + Y01) + J ⎦⎥

Q 1 ⎡ K 2 K ⎤ MB = ⎢ T + p − p .sign(p − p ) − T ⎥ − 2 MB 04 MB 04 QMN 2 ⎢ J J ⎥ p03 ()YSN + 1 ⎣ ⎦ 2 ()Y + Y ⎡ K 2 K ⎤ J 04 ⎢ T + p − p .sign(p − p ) − T ⎥ 2 ⎢ (Y + Y )2 + J 2 03 MB 03 MB 2 2 ⎥ p03 (YSN +1) ⎣ S 04 (YS + Y04 ) + J ⎦

pentru Y0i < Y < YSN were: 2 C.ν 2 • K = C.π(D + J) -is a geometric constant, • KT = - is the coefficient for laminar flow. D S ρ 4.a2 ρ

305 The relation (2) valid for laminar and transition flow will be used for the theoretical analysis of the distribution stages. The directional control valve with negative overlap constructively differ from the ideal distributor by the fact that all four throttle openings are unsealed for YS = 0. For a spool valve stroke YS (depending on the value YS±Y0i) there is realized a throttle opening for which the flow capacities given by (1) become: Appealing to the computing model introduced by F. Klinger [11] and utilizing for the reference flow capacity the value 2 2 the QMN = K DJ YSN + J p03 Fig. 2.b. relation (4 ), valid in the sections Q10, Q20, Q30, Q40, is obtained. For the directional control valve with negative overlap and annular clearance using (3, 4) the relation (5) will be obtained. Similar relations can be obtained also for zones YS ≥ YSN , YS ≤ −YSN ,

− Y0i < YS < Y0i . The complex function (5) represent the generalized equation of the adjustment characteristic for the flow capacity QMA;QMB = f (∆pMAB, YS)∆p=ct with the pressure ∆pMAB = f(YS) and for the load QMA;QMB = f (∆pMAB, YS)∆p=0, in laminar and transition flow. In Fig. 2 are given the adjustment characteristics Fig. 2.c QMA; QMB = f (∆pMAB, YS)∆p=ct and QMA;QMB = = f (∆pMAB, YS)∆p=0 for the flow passing in directions The relations (3, 4, 5) put into evidence the influence A and B. The relations (5) emphasize the work of the different overlap degrees (Y0i ≠ 0) of the throttle ori- ensemble spool valve in three distinct domains: fices (Y01 ≠ Y02 ≠ Y03 ≠ Y04) which appear often negative or positive overlap zone YS < Y0i , linear in practice because technological it is impossible to obtain a perfect symmetry of the throttle edges. Simul- zone Y0i < YS < YSN and saturation zone YS > YSN . taneously these relations allow analyzing the influence The flow that passes through the distributor is upon the adjustment characteristics of variations affected by the overlap degree Y (Y = 1, 2, 3, 4) 0i 0i (Y ) of the four throttle edges. Fig. 2a, 2b, 2c. The selection of the overlap degree 0i Y and the size of the annular clearance J are of 0i 3.THE DYNAMIC ECHILIBRIUM MODEL OF great importance for numerous working characteristics of the system such as: consumption, precision, THE SPOOL VALVE stability and elasticity. During the work of distribution and adjustment elements, upon the spool valve operates a series of forces, their nature and magnitude determining the running performances. The resultant of the acting forces can be expressed as an algebraic sum:

Fpa + Ffrl + Ffrv + Fear +Fg +Fh ± Fin = 0 (6) which establishes the spool valve dynamics and finally the EHSV dynamics. These forces are: 2 d YS - Fisv = []mS + Kms(ma1 + ma2) – inertia (7) dt2 force; F = K (Y + Y ) − pressur forces ; (8) ear ear S oa DS dYS Ffrv = −π.ρ. .nm.Lm.Cfr.sign(Y&S) (9) J dt Fig .2.a. – viscous forces

306 In [1] there are given similar relations also for the 3.1. THE EHSV ANALYSIS THROUGH friction force, generated by non-balanced lateral forces. FREQUENCY The weight Fg is negligible in comparison with other The frequency analysis is characterized by plotting forces. The moving law of the spool valve is given by the transfer function in the imaginary plan and by the the dynamic equilibrium of the acting forces. For the functions that can be obtained at frequency variations distributor with non-null overlap and annular clearance (ω = 0 ⇒ ∞ ) for an input signal iC(t) = i0sinωt. the moving law of the spool valve is: Taking into account the transfer function equation for the ensemble directional control valve in complete d2Y M S = K + K p − p − p − p and normalized form (13,14,15) the sinusoidal re- S 2 {}frv HDY []03 MB MA 04 dt sponse is determined by substituting the complex op- .sign(Y&S).Y&S + []KHS(p03 − ∆pMAVB − p04 ) + Kear .YS − erator S = jω: 2 πDS K5 − ∆pcab. (10) H (Jω) = (16) 4 SV3 3 3 2 2 K1.j ω + K2.j ω + K3.jω + K4 The relation (10) was used for modeling mathe- Developing and separating the terms in (16) it result: matically the EHSV. Using together the equilibrium equation, [1; 5] between the stage nozzle-flap and the 2 K5(K4 −K3ω ) spool valve (Fig. 3). HSV3(jω) =Re(jω)+JIm(jω) = − (K −K ω2)2 +(K ω−K ω3)2 4 2 3 1 VC d(∆pcab ) 3 2 K ∆X − K .∆p = S .Y& + . K5(K3ω−K1ω ) QX Qp cab p S (11) −J (17) 2Eu dt 2 2 3 2 (K4 −K2ω ) +(K3ω−K1ω ) and the simplified dynamic equilibrium equation of with the terms : the spool valve: H (jω) = −20.lg Re2(jω) + Im2 (jω) Sp.∆pcab = MS.Y&&S + Kf .Y&S + KS.YS (12) sv3 dB

0 Im(jω) it was obtained the III order transfer function [1;5] for Φ (jω) = −arctg (18) SV3 the ensemble spool valve- distributor body, in complete Re(jω) form: The theoretical model was verified with the geo- KQX metric parameters of EHSV 2T-7.5 for the computation utilizing the program SIST-SERV. SP HSV3(S) = ⎡ ⎤ V .M KQP.MS VC.KQP C S S3 + ⎢ + ⎥S2 + 2E.S ⎢ 2 2 ⎥ P ⎣ SP 2E.SP ⎦ ⎡ ⎤ Kf .KQP V .K KS.KQP ⎢1 + + C S ⎥S + ⎢ 2 2 ⎥ 2 ⎣ SP 2.E.Sp ⎦ SP (13) Simplifying these equation it was obtained:

YS(S) K5 HSV3(S) = = = ∆X(S) K .S3 + K .S3 + K .S3 + K 1 2 3 4 (14) 1 = K YS 3 2 S + Q2.S + Q1.S + Q0 which can be written in normalized form, taking into account the experimental conditions: Y (S) 1 H (S) = Sn = (15) SV3n 3 2 Fig. 3 A0.∆Xn (S) A1.S + A2.S + A3.S +1 In fig. 4a, b there are represented the frequency The equation (15) allows determining the theoretical characteristics and the transfer plan, putting into evi- frequency characteristics in a plotting system compa- dence the influence of the overlap degree Y01, the input rable with the experimental results. pressure p03 = 5.5…10 MPa (Fig. 4a) and the influence

307 of control current ∆iC = 5…15 mA for input pressures p03 = 5.5…10 MPa (Fig. 4b). For all analyzed cases there were obtained similar frequency characteristics. On the whole, the studied cases attest the presence of a dominant proper frequency in the domain

ω −3dB ∈(10...30 )Hz , which correspond to a no periodic oscillation model and to a proper pulsation with great frequency ω ∈( 200...400 )Hz , r a.1. which correspond to a damped oscillation model. Taking into account the inertia of the dynamic system, EHSV can work in a frequency band of 10… 30 Hz 3.2. THE TIME ANALYSIS OF THE EHSV DYNAMIC BEHAVIOR This study has as objective to determine the variation in time of the system response YS(t) when it is excited with an input value i(t) of the type unitary step, unitary ramp or sinusoidal. This response is analyzed both for the adaptation period (transition stage) and for station- a.2. ary regime. The output value is obtained as the solution Fig.4.a of the linear differential equation, which describes the work of EHSV by III or V order transfer functions (14, 15). Applying the inverse Laplace transform to relation (14) the indicial response is obtained: −x t e 3 Y (t) = + (19) S 2 β − 2.ξgβ + 1

−ξg.ωgt β.e ⎡ 2 ⎤ + .sin ωgt 1− ξg − ψ 2 2 ⎣⎢ ⎦⎥ 1− ξg β − 2ξgβ + 1

maintaining the notation given in [1]. The determi- nation of the response YS(t) gives the time Tr that characterize the EHSV both in stationary and transient regimes. In Fig. 5a,b,c is plotted this response b.1. obtained for the EHSV 2T-7.5 with emphasize to both the influence of control current ∆iC for various input pressures (p03 = 5.5, 7.0, 10 MPa, Fig. 5a,b) and the input pressure for a constant control current ∆iC = 10 mA (fig.5.c). As become clear from Fig.5b the response time is diminished with the increase of the input pressure p03 for the constant control current ∆iC = 10 mA, that means it is diminished with the opening of the spool valve for the same input pressure (Fig. 5a, b). For all studied cases the weight of the oscillatory component is of little importance. In consequence it can be stated that the response of EHSV 2T-7.5 at a unitary step signal can be approximated with a transfer b.2 function of II order, which is specific for a rapid Fig. 4.b. dynamic process.

308 • with the view to put in evidence the influence of the overlap degree on the stationary behavior of EHSV the mathematical model was applied for eight values of overlap between Y01 = ±(0….15)YSN; • the equation of the dynamic equilibrium on the spool valve was established; • assuming as a basis the stability criteria enunciated in the techniques of automatic stability systems analysis it was effectuated the EHSV study both in the frequency domain (faze amplitude-frequency characteristics) and by the response characteristic

to a step signal; Fig.5.a Synthesizing the main elements of the dynamic analysis it results: • the dominant frequency ω-3dB = 10…40 Hz corresponding to the time constant values TA = = (0.0106… 0.0053 0 s, is significant to the behavior of a damp oscillatory hydraulic system (that means the system is stable in the analyzed variation range of control currents, processes, flow capacities and overlap degrees); • in the frequency domain ω-3dB = 10…30 Hz the EHSV behavior may be approximate with a linear system of first degree or at most of second degree; Fig.5.b • the response time tr = 10…30 ms put into evidence the feature of a rapid system with a high degree of stability.

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