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Dynamic analysis of pilot operated pressure reduction valve Abstract In this article the dynamics of a pilot operated pressure reduction valve have been studied through Bond graph simulation technique. The governing equations of the system have been derived from the model. While solving the system equations numerically, the various pressure-flow characteristics of the valve ports are taken into consideration. The simulation study identifies some critical design parameters, which have significant effect on the transient response of the system. ©2002 Elsevier Science B.V. All rights reserved. Keywords: Bond graph; Piloted reduction valve; Dynamics; Simulation study; Transient response 34624
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1. Introduction The pressure reduction valve is used in almost every hydraulic system. The function of the reduction valve is to limit the maximum pressure that can exist in a system. Under ideal condition, the reduction valve should provide alternative flow path to tank for the system fluid while keeping the system pressure constant. There are two types of such valve that are commercially available: direct and pilot type. The direct operated reduction valve operates with a spring to pre-load the poppet of the valve. The use of such valve is gradually diminishing due to its poor pressure override characteristics. In order to improve such characteristics, a pilot stage is introduced. The pilot type can maintain a constant pressure in a hydraulic system; therefore such valves are frequently used in sophisticated hydraulic control systems. Since a small amount of flow rate is passed through the pilot spool, the set pressure can be maintained with nearly no effect on the pressure-flow characteristics of the valve. In the pilot type, the motions of the main and pilot spool are coupled through the hydraulic fluid. This makes the performance analysis more difficult. In the simulation studies on a single stage pressure reduction valve conducted by Ray and Watton several simplifications have been made to use the transfer function formulation technique. Borutzky analyzed the dynamics of the spool valve controlling the orifices using Bond graph simulation technique, where the various forces acting on the spool in the event of the opening and closing of the valve are considered. Chin conducted a study on a pilot operated pressure reduction valve using some form of linearization technique. However, in his study the detail dynamics of the pilot and the main valve were not considered (such as stopper reaction forces on the pilot spool which act when the valve ports are not being opened or fully opened). Those studies also neglected the oil compressibility effect on the main and pilot spool. The present study deals with the complete dynamic analysis of a pi-lot operated pressure reduction valve. The modeling of the system is performed through the Bond graph simulation technique. The effects of changes of various parameters on the dynamic performance of the system are investigated. The responsiveness and sensitivity of the system's performance with respect to the changes of some critical parameters have been investigated. These results show a good agreement with the earlier studies on a similar valve conducted by Chin. 2. Physical model of the system The simplified representation of a pilot valve is shown in Fig. 1. It is basically a two stage reduction valve which gives good regulation of pressure over a wide range of flow. It consists of a main spool controlled by a small direct acting pressure reduction valve. Pressure is sensed at the pilot reduction valve via the small hole. As long as the system pressure does not exceed the setting pressure of the valve, the control valve is closed. The main and the pilot spools are in hydraulic balance; however, they are held onto its seat by the preloaded springs against the seats. With the increase in system pressure in the main chamber L, the flow passes to the chamber A through a small orifice X. Then the flow from the chamber A passes to the chamber B through another orifice Y. The increase in the pressure in chamber B sufficient to open the pilot control valve (pre-loaded with the spring force) throws the main spool out of balance owing to the pressure drop across the pilot valve and the main spool is lifted, relieving the major flow from the chamber L to the tank port. The small amount of flow that passes through the passage Z is also returned to the tank port. The maximum displacement of the pilot and main spools are restricted by their respective stoppers. The pressure in the main chamber L is considered to be plenum pressure P,. The pressures in the other chambers and orifice Z are subsequently denoted by Pa, Pb, Pc and PZ. In analyzing the dynamic performance of the pilot operated pressure reduction valve, a simple hydraulic system shown in Fig. 2 is considered. In this system, a positive displacement pump is driven by a prime mover. In the event of opening of the direction control valve, the fluid supplied by the pump is directed to a linear actuator. As soon as the actuator hits the end stopper, the system pressure increases; exceeding the limit set at the reduction valve and the fluid is bypassed to the tank at a constant pressure. 3. Modeling of the system In the development of the dynamic model of the system: A constant stable source of supply to the valve inlet port is considered. Fluid inertia is neglected. Fluid considered for the analysis has Newtonian characteristics. Resistive and capacitive effects are lumped wherever appropriate. All springs are assumed to be linear. Outlet pressure is assumed to be zero. The masses of the main spring and the piston body are lumped into one inertial parameter, as are the damping forces. Similar considerations are also made for the pilot spool and its spring. The flow through the main and the pilot valve ports as well as through the orifices is assumed to be turbulent. With the opening of the valve ports, the dynamic flow forces acting on the valve spools are neglected, only the steady state flow forces are considered. The positional suction of the valve spools which may be obtained in real situation has not been accounted in the model. The dynamics of the reduction valve is only taken into consideration. The Bond graph model of the system is shown in Fig. 3, where the main valve and pilot valve models are separately marked for clarity. In this model the element SF1 represents the volume flow rate supplied to the valve from a constant stable source. The C3 and R2 elements representing the fluid bulk stiffness and the exit valve port resistance are connected with the 0 junction. This junction determines the plenum pressure P1 of the system. The R6 element connected with the 1 junction indicates the resistance of the orifice X. The C8 element on 0 junction (determines the chamber pressure Pa) indicates the bulk stiffness of the fluid of chamber A. R11 represents the resistance of the orifice Y. Similarly, the C13 element connected with 0 junctions indicates the bulk stiffness of the fluid in the chamber B. The R16 element connected with the 1 junction indicates the modulated resistance of the pilot valve port. The C18 and R19 elements with 0 junction (determines the pressure PZ) represent the bulk stiffness of the fluid and the resistance of the orifice Z. The I24, C25 and R23 elements connected with 1 junction indicate the pilot valve spool inertia, stiffness of the pilot spring and the damping of the pilot spool. The source element SE on the same junction takes into account the total force (Fp1) due to the stopper reaction force and the flow force acting on the pilot spool. Similar modeling has been made for the main valve, where the I30, C31，R29 and SE32 elements are connected with 1 junction. The movement of the pilot spool depends upon the difference in forces acting on it due to the pressures Pb and Pc at chambers B and C, respectively. Similarly the movement of the main spool depends upon the difference in forces acting on it due the load pressure P1 and chamber pressure P1. The transformer modulus ap1 and asp indicate the pilot valve spool area and the main valve spool area respectively. The system equations derived from the model are as follows: Referring to Fig. 4a, Forces acting on the pilot spool: The force equilibrium equation of the pilot spool is given by: In the above equation is the displacement of the pilot spool, Frfp stopper reaction force before opening of the pilot port and Frfp is the flow force acting on the pilot spool in the event of the opening of the pilot port. Before opening of the pilot port and after complete opening of the port, the reaction force Fp1, acts on the pilot spool due to the end stops. In the above equation bp, is used to modulate the force Fp, as follows: In the event of the opening of the valve port (i.e., 0< < ) Frpl= 0; and the above equation may be expressed as: (3.1a) Where Ffp1 is the steady state flow reaction force [5-7] expressed as: In this case the transient flow forces are neglected, since they are very small compared with the steady state flow forces in equilibrium condition [5]. When the line pressure is inadequate for overcoming the pre-compression of the pilot valve spring, i.e., (Pb一 Pc)ap1 < δ pplKp1 and δ pl=0, there is no pilot spool motion, ppl=0 and Ffpl=0 In such case the Eq. (3.1) may be expressed as, (3.1b) After complete opening of the valve port, i.e., bp,)bp,m, the valve spool is in contact with the stopper: where Khp, and Rhp, are the high stiffness and damping of the valve seat. In such case Eq. (3.1) is considered as: (3.1c) Velocity of the pilot spool is given by ( 3.2) The force acting on the main spool: The force equilibrium equation for the main spool is expressed (3.3) Frsp comes is the reaction force on the pre-compression (bpsp) main valve spool till the line pressure (P,) over-of the main spring and 凡 p is the flow reaction force acting on the valve spool with the opening of the main port: The above Eq. (3.3), bsP has been used to modulate FSP in the same way as for the pilot valve dynamics discussed above. The velocity of the main spool is expressed as (3.4) Depending upon the port opening areas, the resistances of the valve ports are modulated. As these resistances are in conductive causalities, they are considered as modulated flow sources, the value of which depend on the pressure differences across the port and the opening areas. The volume rate of change of the fluid in plenum (Ref: Fig. 4b) (3.5) where Vs is supply flow rate, YX is flow supplied through orifice X and Yo is exit flow rate. The volume rate of change of the fluid in chamber A is given by: (3.6) where VX and Vy are the flow passes through orifices X and Y respectively. Referring to Fig. 4a, The volume rate of change of fluid in chamber B (3.7) where Vk is the flow supplied through orifice K. The volume rate of change of fluid in the chamber C is given by: (3.8) where VZ is the flow passes through the long orifice Z. In the above equation, the flow supplied through orifice X is expressed as [6]: Similarly the flow through the orifice Y and Z are obtained. The general equation for the flow supplied through the spools is: （ In the above equation, C is the discharge coefficient, a(x) is the valve opening area and ΔP is the pressure difference across the port. Since the main and pilot spools have the conical shape as shown in Fig. 4a and b, the flow supplied through the orifice K is expressed as: The flow through the exit port is expressed as: The pressures at different chambers are expressed as: The Plenum pressure P1=K1V1 The pressure in chamber A Pa= KaVa The pressure in chamber B Pb= KbVb. The pressure in chamber C Pc= KcVc . The pressure in orifice Z Pz= KzVz . 4. Identification and estimation of the system parameters Some of the parameters associated with the system equations listed above are obtained from a commonly used pressure reduction valve made by Vickers India Ltd. And others are estimated suitably. The bulk stiffness of the fluid in the plenum and the different chambers are obtained theoretically based on their volume and pressure changes in the respective chambers. The stiffness and the pre-compression of the main spring are determined without dismantling it from the valve following the similar procedure described by Schoenau et al. [8]. These are measured from the static test only. However, the hysteresis effect on the springs during unloading at testing as observed has not been considered. This may be a source of error in transient response verification of the displacement of the main valve spool. The pilot valve spring has been taken out and its stiffness is measured with standard test. The values of Ca and Cd are considered from the earlier study of Chin [5] conducted on the similar valve.