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Dynamic analysis of pilot operated pressure relief valve
Simulation Modelling Practice and Theory 10 (2002) 35–49 www.elsevier.com/locate/simpat
Dynamic analysis of pilot operated pressure relief valve K. Dasgupta a
a,*
, R. Karmakar
b
Department of Engineering and Mining Machinery, Indian School of Mines, Dhanbad 826004, India Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur 721302, India
b
Received 23 March 2001; received in revised form 15 December 2001
Abstract In this article the dynamics of a pilot operated pressure relief valve have been studied through Bondgraph 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: Bondgraph; Piloted relief valve; Dynamics; Simulation study; Transient response
1. Introduction The pressure relief valve is used in almost every hydraulic system. The function of the relief valve is to limit the maximum pressure that can exist in a system. Under ideal condition, the relief 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 relief 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 *
Corresponding author. Tel.: +91-326-202487/202578; fax: +91-326-206372/203042.
1569-190X/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 1 5 6 9 - 1 9 0 X ( 0 2 ) 0 0 0 6 1 - 8
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Nomenclature asp apl Ca Cd dx dy dz dp ds Frpl Ffpl Frsp Ffsp lx ly Lz Ksp Kpl Khpl Khsp Msp Mpl j Pa Pb Pc Pz Pl ppl psp Vl Va Vb Vc Re Rpl Rsp Rhpl Rhsp V_x V_y
cross-sectional area of the main spool cross-sectional area of the pilot spool discharge coefficient of the flow rate discharge coefficient of the flow rate diameter of the orifice X diameter of the orifice Y diameter of the orifice Z diameter of the pilot port (orifice K) diameter of the main exit port reaction force on the pilot spool due to pre-compressed pilot spring flow force acting on the pilot spool reaction force on the main spool due to pre-compressed main spring flow force acting on the main spool length of the orifice X length of the orifice Y Length of the orifice Z stiffness of the main spring stiffness of the pilot spring high stiffness of the pilot valve seat high stiffness of the main spool mass of the main spool mass of the pilot spool it is a constant, the value of which is 0.5 pressure in chamber A pressure in chamber B pressure in chamber C pressure in line orifice Z plenum pressure momentum of the pilot spool momentum of the main spool charged volume of the fluid in plenum charged volume of the fluid in chamber A charged volume of the fluid in chamber B charged volume of the fluid in chamber C Resistance of the outlet port of the valve damping coefficient of the pilot spool damping coefficient of the main spool high damping coefficient of the pilot valve seat high damping coefficient of the main spool flow rate supplied through orifice X flow rate supplied through orifice Y
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V_k V_z V_o V_s a / dsp dpl dplm dppl dpsp q
37
flow rate supplied through orifice K flow rate supplied through orifice Z exit flow rate, m3 /s supply flow rate cone angle of the main spool cone angle of the pilot spool main spool displacement pilot spool displacement maximum displacement of the pilot spool pre-compression of the pilot valve spring pre-compression of the main spring fluid density it indicates the time derivative of the variable
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 relief valve conducted by Ray [1] and Watton [2] several simplifications have been made to use the transfer function formulation technique. Borutzky [3] analysed the dynamics of the spool valve controlling the orifices using Bondgraph simulation technique [4], where the various forces acting on the spool in the event of the opening and closing of the valve are considered. Chin [5] conducted a study on a pilot operated pressure relief valve using some form of linearisation 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 pilot operated pressure relief valve. The modelling of the system is performed through the Bondgraph simulation technique [4]. The effects of changes of various parameters on the dynamic performance of the system are investigated. The responsiveness and sensitivity of the systemÕs 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 [5]. 2. Physical model of the system The simplified representation of a pilot valve is shown in Fig. 1. It is basically a two stage relief 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 relief valve. Pressure is sensed at the pilot relief valve via the small hole. As long as the system
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Fig. 1. Simplified model of the pilot operated pressure relief valve.
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 pre-loaded springs against the seats. With the increase in system pressure Pl 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 Pl . The pressures in the other chambers and orifice Z are subsequently denoted by Pa , Pb , Pc and Pz . In analysing the dynamic performance of the pilot operated pressure relief 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 relief valve and the fluid is bypassed to the tank at a constant pressure.
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Fig. 2. The physical system.
3. Modelling 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 are 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 stiction of the valve spools which may be obtained in real situation has not been accounted in the model. The dynamics of the relief valve is only taken into consideration.
The Bondgraph model of the system is shown in Fig. 3, where the main valve and pilot valve models are separately marked for clarity.
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Fig. 3. Bondgraph model of pilot operated relief valve.
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 Pl 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 junction 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 ðFpl Þ 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 Pl and chamber pressure Pa . The transformer modulii apl 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:
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Fig. 4. (a) Pilot valve model and (b) main valve model.
p_ pl ¼ Fpl ppl
Rpl Mpl
ðdpl þ dppl ÞKps þ ðPb Pc Þapl ;
ð3:1Þ
Ffp ¼ Ffpl þ Frpl : In the above equation dpl is the displacement of the pilot spool, Frfp stopper reaction force before opening of the pilot port and Frpl 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
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after complete opening of the port, the reaction force Frpl acts on the pilot spool due to the end stops. In the above equation dpl is used to modulate the force Fpl as follows: In the event of the opening of the valve port (i.e., 0 < dpl < dplm ), Frpl ¼ 0; and the above equation may be expressed as: Rpl ðPb Pc Þapl ¼ Ffpl þ ppl ð3:1aÞ þ ðdpl þ dppl ÞKps þ p_ pl ; Mpl where Ffpl is the steady state flow reaction force [5–7] expressed as: Ffpl ¼ Ca2 pdp sin 2/ðPb Pc Þdpl : 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., ðPb Pc Þapl 6 dppl Kpl and dpl ¼ 0, there is no pilot spool motion, ppl ¼ 0
and
Ffpl ¼ 0:
In such case the Eq. (3.1) may be expressed as, ðPb Pc Þapl ¼ dppl Kpl þ Frpl :
ð3:1bÞ
After complete opening of the valve port, i.e., dpl P dplm , the valve spool is in contact with the stopper: Rhpl Frpl ¼ ppl þ dpl Khpl ; Mpl where Khpl and Rhpl are the high stiffness and damping of the valve seat. In such case, Eq. (3.1) is considered as: Rhpl ðPb Pc Þapl ¼ Ffpl þ Frpl þ ppl ð3:1cÞ þ ðdpl þ dppl ÞKhps þ p_ pl : Mpl Velocity of the pilot spool is given by ppl d_pl ¼ : Mpl
ð3:2Þ
Referring to Fig. 4b, The forces acting on the main spool: The force equilibrium equation for the main spool is expressed Rsp p_ sp ¼ ðPl Pa Þasp psp ðdsp þ dpsp ÞKsp þ Fsp ; Msp
ð3:3Þ
Fsp ¼ Frsp þ Ffsp Fsp : Frsp is the reaction force on the main valve spool till the line pressure ðPl Þ overcomes the pre-compression ðdpsp Þ of the main spring and Ffsp is the flow reaction force acting on the valve spool with the opening of the main port:
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Ffsp ¼ Cd pds sinð2aÞdsp Pl : The above Eq. (3.3), dsp 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 psp d_sp ¼ : ð3:4Þ Msp 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) V_l ¼ V_s V_o V_x d_sp asp ;
ð3:5Þ
where V_s is supply flow rate, V_x is flow supplied through orifice X and V_o is exit flow rate. The volume rate of change of the fluid in chamber A is given by: V_a ¼ V_x þ d_sp asp V_y ;
ð3:6Þ
where V_x and V_y are the flow passes through orifices X and Y respectively. Referring to Fig. 4a, The volume rate of change of fluid in chamber B V_b ¼ V_y V_k d_pl apl ;
ð3:7Þ
where V_k is the flow supplied through orifice K. The volume rate of change of fluid in the chamber C is given by: V_c ¼ V_k V_z þ d_pl apl ;
ð3:8Þ
where V_z is the flow passes through the long orifice Z. In the above equation, the flow supplied through orifice X is expressed as [6]: ð1=2jÞ V_x ¼ Ca ½Pl Pa :
Similarly the flow through the orifice Y and Z are obtained. The general equation for the flow supplied through the spools is: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s DP _V ¼ CaðxÞ : q In the above equation, C is the discharge coefficient, aðxÞ is the valve opening area and DP 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: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P P b c : V_k ¼ Cd pdp sin /dpl 2 q
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The flow through the exit port is expressed as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pl V_o ¼ Ca pds sin adsp 2 : q The pressures at different chambers are expressed as: The Plenum pressure
Pl ¼ K l Vl :
ð3:9Þ
The pressure in chamber A
Pa ¼ Ka Va :
ð3:10Þ
The pressure in chamber B
Pb ¼ K b Vb :
ð3:11Þ
The pressure in chamber C
Pc ¼ Kc Vc :
ð3:12Þ
The pressure in orifice Z
Pz ¼ K z Vz :
ð3:13Þ
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 relief 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. The geometrical parameters of the valve spools such as its cone angle (a and /), the length and diameter of the orifices (X, Y and Z) are all measured. The complete list of parameters used in the simulation studies are listed in the following Table 1.
5. Results of simulation The parametric studies on the dynamic performance of the pilot operated pressure relief valve have been carried out by solving the system equations obtained in Section 3 numerically with the help of software Symbols 2000 [9]. This particular package
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Table 1 Parameter asp apl Cd Ca ds dp dx dy dz Ka Kb Kc Kl Kpl Khpl Ksp
Value 2
80 mm 10 mm2 0.62 0.5 10 mm 3.5 mm 2.2 mm 3 mm 1.5 mm 30 106 MPa/m3 30 106 MPa/m3 100 106 MPa/m3 106 MPa/m3 40 kN/m 400 MN/m 20 kN/m
Parameter
Value
Khsp lx ly lz Msp Mpl Rpl Rplh Rsp Rsph V_s j a / q
200 MN/m 12 mm 12 mm 25 mm 100 gm 20 gm 1 kN s/m 5 MN s/m 0.5 kN s/m 0.5 MN s/m 7 103 m3 /s 0.5 1.2 rad 0.353 rad 850 kg/m3
readily lends itself to the Bondgraph representation and it can also take into account the various non-linearities of the model. The various system responses are obtained considering the step input V_s (supply flow rate). Fig. 5 shows the transient response of the system for different pre-compression of the pilot spring. It shows that with the increase in the pre-compression, the peak pressure as well as the steady state pressure increase. It increases the working range of the hydraulic system where such a valve is incorporated.
Fig. 5. Effect of the pre-compression on the overall response of the system.
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Fig. 6. Effect of the main spool cone angle (a) on the response of the system.
Fig. 6 shows the response of the system for the variation of the exit flow angle a. Figs. 7 and 8 show the settling and peak time of the main spool displacement for the different ratios of damping coefficients between main and pilot spools. Here the settling time means the time until the valve response reaches the steady state value when it is subjected to step input; whereas the peak rise time indicates the time taken to reach the peak of the first overshoot with the same input. It indicates
Fig. 7. Setting time versus damping change.
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Fig. 8. Responsiveness versus damping change.
Fig. 9. Effect of the orifice diameter (dx ) on the overall response of the system.
that both the settling and peak times are smaller when the ratio Rsp =Rpl varies from the value 0.01–0.1. However, it is found that when the damping coefficient of the main spool (Rsp ) is below 200 N s/m the valve response is unstable. The best performance of the system is attained when the value of Rsp is in the range of 700–900 N s/m. The results obtained by Chin [5] are also compared in the figure; where, this range varies from 500–700 N s/m. This difference may be due to the consideration of the oil compressibility effect and other minor differences in the geometrical parameters of the valve. Fig. 9 shows the system response for the variation of the orifice
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diameter dx . It indicates that with the increase in the orifice diameter, the settling time as well as the maximum overshoot decrease; whereas the peak time increases. From these simulation studies, it can be concluded that the damping coefficient of the main and pilot valve spools as well as their shape of the cone tip (a and /) are the important design variables for the improvement of the responsiveness of the system. Similarly, reducing the diameter of the orifice X may increase the system response, however, it would be at the cost of the increasing settling time.
6. Conclusion In this article a dynamic model of a two stage pilot relief valve is discussed. The effects of various design parameters on the overall response of the system are investigated through simulation. The verification of the simulation results with the earlier studies conducted by Chin [5], justifies the proposed model. The model takes into account the various non-linearities of the system. The effects of changes of the diameter of the orifice X and the cone angle (a) of the main spool on the peak rise time and settling time of the systemÕs performance have been presented. It implies that the shape of the tip of the main spool is an important design feature affecting the dynamic characteristics of the system. It is also shown that increasing the pre-compression of the pilot spring increases the operating range of the valve. The factors, having significant effect on the stability are the damping coefficients of the main and pilot spools. It may be difficult to control the damping coefficients of the valve spools in real situation; however, the experimental procedure described by Chin [5] may be a useful method for validating the same. The model can be further refined by incorporating the dynamic behavior of the pilot and main springs, dynamic flow forces on the valve spools. The model does lack the flexibility of the linearised model as proposed by Ray [1] and Chin [5]; however, its simplicity lends itself to be used for wider variation of system parameters. It is further concluded that this study provides a good base upon which an interaction study between other components and the pilot operated relief valve of a complete hydraulic system can be initiated. The model could be used in predicting trends that could occur under various loading conditions that are difficult to create experimentally. This consideration is important in future expansions of the study to include the dynamics of the pump and loads.
Acknowledgements The work is supported partially by INSA, Govt. of India, by sponsoring one of the authors for collaborative research work at IIT/Kharagpur, India and by a research grant from UGC, Govt. of India. All the above sources of support are gratefully acknowledged.
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References [1] A. Ray, Dynamic modelling and simulation of a relief valve, Simulation (1978) 167–172. [2] J. Watton, The design of a single-stage relief valve with directional damping, Journal of Fluid Control Including Fluidics Quarterly 18 (2) (1988) 22–35. [3] W. Borutzky, A dynamic bondgraph model of the fluid mechanical interaction in spool valve control orifice, in: Bondgraphs for Engineers, North Holland, 1992, pp. 229–236. [4] J.U. Thoma, Simulation by Bondgraph, Springer Verlag, Germany, 1990. [5] C.Y. Chin, Static and dynamic characteristics of a two stage pilot relief valve, ASME Dynamic Systems Measurements and Controls 113 (1991) 280–288. [6] J.F. Blackburn, G. Reethof, J.L. Shearer, Fluid power control, Technology Press of MIT and Wiley, USA, 1960. [7] T. Takaneka, Performance of a hydraulic pressure control valve, JSME, vol. 66, pp. 538–540. [8] J.G. Schoenau, R.T. Burton, G.P. Kavang, Dynamic analysis of a variable displacement pump, ASME Dynamic System Measurements and Controls 112 (1990) 122–132. [9] Symbols 2000, High Tech Consultants, STEP, IIT, Kharagpur, www.symbols2000.com.
Dynamic analysis of pilot operated pressure relief valve K. Dasgupta a
a,*
, R. Karmakar
b
Department of Engineering and Mining Machinery, Indian School of Mines, Dhanbad 826004, India Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur 721302, India
b
Received 23 March 2001; received in revised form 15 December 2001
Abstract In this article the dynamics of a pilot operated pressure relief valve have been studied through Bondgraph 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: Bondgraph; Piloted relief valve; Dynamics; Simulation study; Transient response
1. Introduction The pressure relief valve is used in almost every hydraulic system. The function of the relief valve is to limit the maximum pressure that can exist in a system. Under ideal condition, the relief 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 relief 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 *
Corresponding author. Tel.: +91-326-202487/202578; fax: +91-326-206372/203042.
1569-190X/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 1 5 6 9 - 1 9 0 X ( 0 2 ) 0 0 0 6 1 - 8
36
K. Dasgupta, R. Karmakar / Simulation Modelling Practice and Theory 10 (2002) 35–49
Nomenclature asp apl Ca Cd dx dy dz dp ds Frpl Ffpl Frsp Ffsp lx ly Lz Ksp Kpl Khpl Khsp Msp Mpl j Pa Pb Pc Pz Pl ppl psp Vl Va Vb Vc Re Rpl Rsp Rhpl Rhsp V_x V_y
cross-sectional area of the main spool cross-sectional area of the pilot spool discharge coefficient of the flow rate discharge coefficient of the flow rate diameter of the orifice X diameter of the orifice Y diameter of the orifice Z diameter of the pilot port (orifice K) diameter of the main exit port reaction force on the pilot spool due to pre-compressed pilot spring flow force acting on the pilot spool reaction force on the main spool due to pre-compressed main spring flow force acting on the main spool length of the orifice X length of the orifice Y Length of the orifice Z stiffness of the main spring stiffness of the pilot spring high stiffness of the pilot valve seat high stiffness of the main spool mass of the main spool mass of the pilot spool it is a constant, the value of which is 0.5 pressure in chamber A pressure in chamber B pressure in chamber C pressure in line orifice Z plenum pressure momentum of the pilot spool momentum of the main spool charged volume of the fluid in plenum charged volume of the fluid in chamber A charged volume of the fluid in chamber B charged volume of the fluid in chamber C Resistance of the outlet port of the valve damping coefficient of the pilot spool damping coefficient of the main spool high damping coefficient of the pilot valve seat high damping coefficient of the main spool flow rate supplied through orifice X flow rate supplied through orifice Y
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V_k V_z V_o V_s a / dsp dpl dplm dppl dpsp q
37
flow rate supplied through orifice K flow rate supplied through orifice Z exit flow rate, m3 /s supply flow rate cone angle of the main spool cone angle of the pilot spool main spool displacement pilot spool displacement maximum displacement of the pilot spool pre-compression of the pilot valve spring pre-compression of the main spring fluid density it indicates the time derivative of the variable
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 relief valve conducted by Ray [1] and Watton [2] several simplifications have been made to use the transfer function formulation technique. Borutzky [3] analysed the dynamics of the spool valve controlling the orifices using Bondgraph simulation technique [4], where the various forces acting on the spool in the event of the opening and closing of the valve are considered. Chin [5] conducted a study on a pilot operated pressure relief valve using some form of linearisation 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 pilot operated pressure relief valve. The modelling of the system is performed through the Bondgraph simulation technique [4]. The effects of changes of various parameters on the dynamic performance of the system are investigated. The responsiveness and sensitivity of the systemÕs 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 [5]. 2. Physical model of the system The simplified representation of a pilot valve is shown in Fig. 1. It is basically a two stage relief 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 relief valve. Pressure is sensed at the pilot relief valve via the small hole. As long as the system
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Fig. 1. Simplified model of the pilot operated pressure relief valve.
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 pre-loaded springs against the seats. With the increase in system pressure Pl 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 Pl . The pressures in the other chambers and orifice Z are subsequently denoted by Pa , Pb , Pc and Pz . In analysing the dynamic performance of the pilot operated pressure relief 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 relief valve and the fluid is bypassed to the tank at a constant pressure.
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Fig. 2. The physical system.
3. Modelling 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 are 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 stiction of the valve spools which may be obtained in real situation has not been accounted in the model. The dynamics of the relief valve is only taken into consideration.
The Bondgraph model of the system is shown in Fig. 3, where the main valve and pilot valve models are separately marked for clarity.
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Fig. 3. Bondgraph model of pilot operated relief valve.
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 Pl 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 junction 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 ðFpl Þ 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 Pl and chamber pressure Pa . The transformer modulii apl 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:
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Fig. 4. (a) Pilot valve model and (b) main valve model.
p_ pl ¼ Fpl ppl
Rpl Mpl
ðdpl þ dppl ÞKps þ ðPb Pc Þapl ;
ð3:1Þ
Ffp ¼ Ffpl þ Frpl : In the above equation dpl is the displacement of the pilot spool, Frfp stopper reaction force before opening of the pilot port and Frpl 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
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after complete opening of the port, the reaction force Frpl acts on the pilot spool due to the end stops. In the above equation dpl is used to modulate the force Fpl as follows: In the event of the opening of the valve port (i.e., 0 < dpl < dplm ), Frpl ¼ 0; and the above equation may be expressed as: Rpl ðPb Pc Þapl ¼ Ffpl þ ppl ð3:1aÞ þ ðdpl þ dppl ÞKps þ p_ pl ; Mpl where Ffpl is the steady state flow reaction force [5–7] expressed as: Ffpl ¼ Ca2 pdp sin 2/ðPb Pc Þdpl : 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., ðPb Pc Þapl 6 dppl Kpl and dpl ¼ 0, there is no pilot spool motion, ppl ¼ 0
and
Ffpl ¼ 0:
In such case the Eq. (3.1) may be expressed as, ðPb Pc Þapl ¼ dppl Kpl þ Frpl :
ð3:1bÞ
After complete opening of the valve port, i.e., dpl P dplm , the valve spool is in contact with the stopper: Rhpl Frpl ¼ ppl þ dpl Khpl ; Mpl where Khpl and Rhpl are the high stiffness and damping of the valve seat. In such case, Eq. (3.1) is considered as: Rhpl ðPb Pc Þapl ¼ Ffpl þ Frpl þ ppl ð3:1cÞ þ ðdpl þ dppl ÞKhps þ p_ pl : Mpl Velocity of the pilot spool is given by ppl d_pl ¼ : Mpl
ð3:2Þ
Referring to Fig. 4b, The forces acting on the main spool: The force equilibrium equation for the main spool is expressed Rsp p_ sp ¼ ðPl Pa Þasp psp ðdsp þ dpsp ÞKsp þ Fsp ; Msp
ð3:3Þ
Fsp ¼ Frsp þ Ffsp Fsp : Frsp is the reaction force on the main valve spool till the line pressure ðPl Þ overcomes the pre-compression ðdpsp Þ of the main spring and Ffsp is the flow reaction force acting on the valve spool with the opening of the main port:
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Ffsp ¼ Cd pds sinð2aÞdsp Pl : The above Eq. (3.3), dsp 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 psp d_sp ¼ : ð3:4Þ Msp 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) V_l ¼ V_s V_o V_x d_sp asp ;
ð3:5Þ
where V_s is supply flow rate, V_x is flow supplied through orifice X and V_o is exit flow rate. The volume rate of change of the fluid in chamber A is given by: V_a ¼ V_x þ d_sp asp V_y ;
ð3:6Þ
where V_x and V_y are the flow passes through orifices X and Y respectively. Referring to Fig. 4a, The volume rate of change of fluid in chamber B V_b ¼ V_y V_k d_pl apl ;
ð3:7Þ
where V_k is the flow supplied through orifice K. The volume rate of change of fluid in the chamber C is given by: V_c ¼ V_k V_z þ d_pl apl ;
ð3:8Þ
where V_z is the flow passes through the long orifice Z. In the above equation, the flow supplied through orifice X is expressed as [6]: ð1=2jÞ V_x ¼ Ca ½Pl Pa :
Similarly the flow through the orifice Y and Z are obtained. The general equation for the flow supplied through the spools is: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s DP _V ¼ CaðxÞ : q In the above equation, C is the discharge coefficient, aðxÞ is the valve opening area and DP 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: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P P b c : V_k ¼ Cd pdp sin /dpl 2 q
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The flow through the exit port is expressed as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pl V_o ¼ Ca pds sin adsp 2 : q The pressures at different chambers are expressed as: The Plenum pressure
Pl ¼ K l Vl :
ð3:9Þ
The pressure in chamber A
Pa ¼ Ka Va :
ð3:10Þ
The pressure in chamber B
Pb ¼ K b Vb :
ð3:11Þ
The pressure in chamber C
Pc ¼ Kc Vc :
ð3:12Þ
The pressure in orifice Z
Pz ¼ K z Vz :
ð3:13Þ
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 relief 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. The geometrical parameters of the valve spools such as its cone angle (a and /), the length and diameter of the orifices (X, Y and Z) are all measured. The complete list of parameters used in the simulation studies are listed in the following Table 1.
5. Results of simulation The parametric studies on the dynamic performance of the pilot operated pressure relief valve have been carried out by solving the system equations obtained in Section 3 numerically with the help of software Symbols 2000 [9]. This particular package
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Table 1 Parameter asp apl Cd Ca ds dp dx dy dz Ka Kb Kc Kl Kpl Khpl Ksp
Value 2
80 mm 10 mm2 0.62 0.5 10 mm 3.5 mm 2.2 mm 3 mm 1.5 mm 30 106 MPa/m3 30 106 MPa/m3 100 106 MPa/m3 106 MPa/m3 40 kN/m 400 MN/m 20 kN/m
Parameter
Value
Khsp lx ly lz Msp Mpl Rpl Rplh Rsp Rsph V_s j a / q
200 MN/m 12 mm 12 mm 25 mm 100 gm 20 gm 1 kN s/m 5 MN s/m 0.5 kN s/m 0.5 MN s/m 7 103 m3 /s 0.5 1.2 rad 0.353 rad 850 kg/m3
readily lends itself to the Bondgraph representation and it can also take into account the various non-linearities of the model. The various system responses are obtained considering the step input V_s (supply flow rate). Fig. 5 shows the transient response of the system for different pre-compression of the pilot spring. It shows that with the increase in the pre-compression, the peak pressure as well as the steady state pressure increase. It increases the working range of the hydraulic system where such a valve is incorporated.
Fig. 5. Effect of the pre-compression on the overall response of the system.
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Fig. 6. Effect of the main spool cone angle (a) on the response of the system.
Fig. 6 shows the response of the system for the variation of the exit flow angle a. Figs. 7 and 8 show the settling and peak time of the main spool displacement for the different ratios of damping coefficients between main and pilot spools. Here the settling time means the time until the valve response reaches the steady state value when it is subjected to step input; whereas the peak rise time indicates the time taken to reach the peak of the first overshoot with the same input. It indicates
Fig. 7. Setting time versus damping change.
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Fig. 8. Responsiveness versus damping change.
Fig. 9. Effect of the orifice diameter (dx ) on the overall response of the system.
that both the settling and peak times are smaller when the ratio Rsp =Rpl varies from the value 0.01–0.1. However, it is found that when the damping coefficient of the main spool (Rsp ) is below 200 N s/m the valve response is unstable. The best performance of the system is attained when the value of Rsp is in the range of 700–900 N s/m. The results obtained by Chin [5] are also compared in the figure; where, this range varies from 500–700 N s/m. This difference may be due to the consideration of the oil compressibility effect and other minor differences in the geometrical parameters of the valve. Fig. 9 shows the system response for the variation of the orifice
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diameter dx . It indicates that with the increase in the orifice diameter, the settling time as well as the maximum overshoot decrease; whereas the peak time increases. From these simulation studies, it can be concluded that the damping coefficient of the main and pilot valve spools as well as their shape of the cone tip (a and /) are the important design variables for the improvement of the responsiveness of the system. Similarly, reducing the diameter of the orifice X may increase the system response, however, it would be at the cost of the increasing settling time.
6. Conclusion In this article a dynamic model of a two stage pilot relief valve is discussed. The effects of various design parameters on the overall response of the system are investigated through simulation. The verification of the simulation results with the earlier studies conducted by Chin [5], justifies the proposed model. The model takes into account the various non-linearities of the system. The effects of changes of the diameter of the orifice X and the cone angle (a) of the main spool on the peak rise time and settling time of the systemÕs performance have been presented. It implies that the shape of the tip of the main spool is an important design feature affecting the dynamic characteristics of the system. It is also shown that increasing the pre-compression of the pilot spring increases the operating range of the valve. The factors, having significant effect on the stability are the damping coefficients of the main and pilot spools. It may be difficult to control the damping coefficients of the valve spools in real situation; however, the experimental procedure described by Chin [5] may be a useful method for validating the same. The model can be further refined by incorporating the dynamic behavior of the pilot and main springs, dynamic flow forces on the valve spools. The model does lack the flexibility of the linearised model as proposed by Ray [1] and Chin [5]; however, its simplicity lends itself to be used for wider variation of system parameters. It is further concluded that this study provides a good base upon which an interaction study between other components and the pilot operated relief valve of a complete hydraulic system can be initiated. The model could be used in predicting trends that could occur under various loading conditions that are difficult to create experimentally. This consideration is important in future expansions of the study to include the dynamics of the pump and loads.
Acknowledgements The work is supported partially by INSA, Govt. of India, by sponsoring one of the authors for collaborative research work at IIT/Kharagpur, India and by a research grant from UGC, Govt. of India. All the above sources of support are gratefully acknowledged.
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References [1] A. Ray, Dynamic modelling and simulation of a relief valve, Simulation (1978) 167–172. [2] J. Watton, The design of a single-stage relief valve with directional damping, Journal of Fluid Control Including Fluidics Quarterly 18 (2) (1988) 22–35. [3] W. Borutzky, A dynamic bondgraph model of the fluid mechanical interaction in spool valve control orifice, in: Bondgraphs for Engineers, North Holland, 1992, pp. 229–236. [4] J.U. Thoma, Simulation by Bondgraph, Springer Verlag, Germany, 1990. [5] C.Y. Chin, Static and dynamic characteristics of a two stage pilot relief valve, ASME Dynamic Systems Measurements and Controls 113 (1991) 280–288. [6] J.F. Blackburn, G. Reethof, J.L. Shearer, Fluid power control, Technology Press of MIT and Wiley, USA, 1960. [7] T. Takaneka, Performance of a hydraulic pressure control valve, JSME, vol. 66, pp. 538–540. [8] J.G. Schoenau, R.T. Burton, G.P. Kavang, Dynamic analysis of a variable displacement pump, ASME Dynamic System Measurements and Controls 112 (1990) 122–132. [9] Symbols 2000, High Tech Consultants, STEP, IIT, Kharagpur, www.symbols2000.com.