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

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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 de- termined 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 ⎤ K K Q = K (Y + Y )2 + J2 ⎢ T + p − p − T ⎥ 40 Dj S 04 ⎢ ()03 MB ⎥ (3) ()Y + Y 2 + J2 2 2 ⎣⎢ S 04 (YS + Y04 ) + J ⎦⎥ Qio non zero overlaps linear zone non linear zone (4) i=1, 2,3, YS <Y01 YS ≤ Y01 Y0i < YS < YSN 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 distri- bution 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 charac- teristic 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.
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