Using Steady Flow Force for Unstable Valve Design - CiteSeerX

Using Steady Flow Force for Unstable Valve Design: Modeling and Experiments Qinghui Yuan Perry Y. Li1 e-mail: [email protected] Department of Mechanical Engineering, University of Minnesota, Minneapolis, Minnesota 55455

In single stage electrohydraulic valves, solenoid actuators are usually used to stroke the main spools directly. They are cheaper and more reliable than multistage valves. Their use, however, is restricted to low bandwidth and low flow rate applications due to the limitation of the solenoid actuators. Our research focuses on alleviating the need for large and expensive solenoids in single stage valves by advantageously using fluid flow forces. For example, in a previous paper, we proposed to improve spool agility by inducing unstable transient flow forces by the use of negative damping lengths. In the present paper, how steady flow forces can be manipulated to improve spool agility is examined through fundamental momentum analysis, CFD analysis, and experimental studies. Particularly, it is found that two often ignored components—viscosity effect and nonmetering momentum flux, have strong influence on steady flow forces. For positive damping lengths, viscosity increases the steady flow force, whereas for negative damping lengths, viscosity has the tendency to reduce steady flow forces. Also, by slightly modifying the non-metering port geometry, the non-metering flux can also be manipulated to reduce steady flow force. Therefore, both transient and steady flow forces can be used to improve the agility of single stage electrohydraulic valves. Experimental results confirm the contributions of both transient and steady flow force in improving spool agility. 关DOI: 10.1115/1.1997166兴 Keywords: Electrohydraulic Valves, Unstable Valve, Damping Length, Steady Flow Force, Viscosity, Non-Metering Flux, Computational Fluid Dynamics (CFD), Spool Agility

1

Introduction

In a single stage electrohydraulic flow control valve, the main spool is stroked directly by solenoid actuators. In high flow rate and high bandwidth applications, the solenoids must overcome significant forces. The force and power limitations of the solenoid actuators make single stage valves unsuitable for high performance applications. In these situations, multistage valves are generally used in which the spools are driven by one or more pilot stage hydraulic valves but they tend to be more costly to manufacture and maintain than single stage valves. Our research aims at improving the flow and bandwidth capabilities of single stage valves by alleviating the need for large and expensive solenoid actuators. Our approach is to utilize the inherent fluid flow forces to increase the agility of the spool, so that less power or less force is needed from the solenoids while achieving fast spool responses. Flow forces can be classified as either steady or transient. Steady flow forces are those exerted by the fluid on the spool during steady flow conditions, while transient forces are the additional forces due to the time variation of the flow condition. Our initial approach 关1兴 proposes to configure the valve to achieve transient flow force that has an unstable 共negative damping兲 effect on the spool, and then to stabilize the system by closed loop feedback. Our study in 关1兴 distinguishes from other studies of transient flow force induced instability in the 1960s such as summarized in 关2,3兴 in that our goal is to utilize the 1 Author to whom correspondence should be addressed. Contributed by the Dynamic Systems, Measurement, and Control Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received: July 25, 2003. Final revision: September 15, 2004. Associate Editor: Noah Manring. This research is supported by the National Science Foundation ENG/CMS-0088964. Submitted to the ASME Journal of Dynamic Systems, Measurement and Control, July 2003; Revised July 2004.

instability advantageously rather than to eliminate it as in the previous studies. Analysis and computer simulations in 关1兴 have shown that valves configured to have unstable transient flow forces can have faster step responses under solenoid saturation than their stable counterparts. They also take less positive power but more negative power 共braking兲 to track a sinusoidal flow rate. The saving in positive power becomes more significant at higher frequencies. A basic premise of this study is that the sign of the damping length determines whether the transient flow force is stabilizing or destabilizing 共see Fig. 1 for a definition兲. In the present paper, we investigate opportunities in which the steady state flow force can be manipulated to improve spool agility. Classical steady flow force model in 关1–3兴 is concerned mainly with the contribution by the axial fluid momentum at the metering orifices. In this model, steady flow force is monotonic with spool displacement and therefore acts like a spring with a positive spring constant. Since steady flow forces must be overcome by the solenoids, reducing steady flow force improves spool agility. Previously proposed steady flow force compensation techniques generally try to affect flow pattern at the metering orifice, and involve changing the chamber shape 关3兴. This approach becomes more attractive as commercial computational fluid dynamics 共CFD兲 software becomes readily available, making iterative design of the chamber geometry 关4兴 and of the metering notches 关5兴 feasible. However, because the chamber geometry that results tends to be complicated, manufacturing cost would still be high. Other ideas that have been proposed include the use of leakage in underlapped spool valves to reduce steady flow force 关6兴 at the expense of flow and energetic inefficiencies. Other recent studies of steady flow force include 关7兴 where the influence of asymmetry in the geometry of the four-way directional valve on steady flow forces is investigated. In this paper, we propose, highlight, and verify two often neglected components in steady flow force modeling: fluid viscosity

Journal of Dynamic Systems, Measurement, and Control Copyright © 2005 by ASME

SEPTEMBER 2005, Vol. 127 / 451

Fig. 1 Typical configuration of a four way direction flow control valve. The two “Q” ports are connected to the load „hydraulic actuator…, Ps is connected to the supply pressure, and Pt is connected to the return. In a single stage valve the spool is stroked directly by solenoid actuators. Damping length L is defined to be LªL2 − L1.

and non-metering momentum flux. Although both are alluded to in 关8, pp.85–87兴, detailed modeling, CFD, and experimental verification were not reported there, and nearly all other literature on steady flow force modeling focuses on the flow momentum at the metering orifices in 关2,3,1兴. The enhanced models are developed based on fundamental momentum theory and are verified using CFD analysis and experiments. It turns out that both viscosity and non-metering momentum flux have significant effects on steady flow forces, and hence spool agility. In particular, viscosity reduces the steady flow force for negative damping lengths, and increases it for positive damping lengths. For sufficiently negative damping lengths, the steady flow force can even become unstable. Similarly, the non-metering momentum flux can be manipulated by changing the angle of the non-metering inlet and outlet in order to reduce steady flow force. Thus, consideration of the viscosity effect and the non-metering momentum flux presents additional design opportunities to improve spool agility through relatively simple geometry modification. Nakada et al. 关9兴 presented experimental results for a four-way valve with a positive damping length that verify, to a large extent, the classical unsteady flow force models 关2,3,1兴, i.e., unsteady flow force acts like a damper. They found, as predicted by classical transient flow force model, that the measured flow force increases linearly with the flow frequency, and also that the rate of increase increases with damping length. Interestingly, they also noted that the steady flow forces 共i.e., flow force at low frequency flow兲 increases with 共positive兲 damping length. This result contradicts classical steady flow force model as it predicts that steady flow force is independent of damping lengths. This result is, however, consistent with the viscosity effect in our proposed enhanced model. The rest of paper is organized as follows. In Sec. 2, we present models for the fluid flow induced steady flow forces that take into account viscosity and non-metering momentum flux. Section 3 presents CFD analysis of the fluid forces and compare the accuracies of the various approximations. The experimental study on the viscosity effect and the non-metering flux effect is presented in Sec. 4. In Sec. 5, experiments and analysis of the combined effect of damping lengths on spool agility, as well as the relative contribution of the transient flow force effect versus the steady counterpart to improve spool agility are presented. Section 6 contains some concluding remarks.

2

Modeling of Steady Flow Forces

A typical four-way directional control valve such as in Fig. 1 consists of a meter-in chamber 共left兲 and a meter-out chamber 共right兲. The force acting on the spool is the sum of the flow force 452 / Vol. 127, SEPTEMBER 2005

Fig. 2 Meter-in valve chamber. The lower gray block is the control volume „C.V.… which includes all the fluid in the chamber, and is surrounded by the sleeve, rod, land ends and the inlet/outlet surfaces.

due to each chamber. The damping length is the distance between the inlet and outlet ports for the meter-out chamber subtracted by that of the meter-in chamber 共i.e., LªL2 − L1 in Fig. 1兲. We can consider the meter-out and meter-in chambers to have positive and negative damping lengths, respectively, and the total damping length to be their sum. The fluid flow induced force on the spool for the meter-in and the meter-out chambers are analyzed separately. In single chamber experiments, negative damping lengths correspond to meter-in chambers 共L = −L1兲 and positive damping lengths correspond to meter-out chambers 共L = L2兲. 2.1 Meter-in Chamber. Consider first the meter-in chamber 共Fig. 2兲. The force Fspool 共positive to the right兲 that the spool experiences from the fluid can be calculated in various ways. Most fundamentally, it is given by Fspool = Fland + Frod =

冕

Ar

PrdA −

冕

Al

PldA +

冕

␶rod dA 共1兲

rod

where Fland is the pressure force acting on the lands, Frod is the viscous force acting on the spool rod, Pr and Pl are the pressures on the right and left lands, Ar and Al are the annulus area of the right and left lands, and ␶rod is the shear stress exerted on the rod by the fluid. Fspool can also be computed from the reaction of the force that the spool acts on the fluid. The conservation of longitudinal 共left to right兲 momentum equation for the control volume 共C.V.兲 that encompasses the fluid in the meter-in chamber 共marked gray in Fig. 2兲 is given by

共2兲 where the left-hand side is the sum of the forces that the spool and the sleeve act on the fluid, and the right-hand side is the sum of the rate of change of the longitudinal momentum and the net momentum flux out of the control volume. Here, ␳ is the fluid density, v and vx are the fluid velocity and its longitudinal component, and n is the outward unit normal vector associated with the differential surface dA. The areas of and the shear stresses at the inlet and out-let surfaces are assumed to be negligible compared to the sleeve area and the sleeve shear force. From Eq. 共2兲, Fspool can be decomposed into a steady flow force component and a transient flow force component: Transactions of the ASME

共3兲 Previous models for steady flow force 关2,3兴 do not consider the viscosity effect, i.e., Frod = Fsleeve = 0. It is important to point out that because of the momentum analysis, the effect of viscosity on steady flow force model is due to the friction between the fluid and the sleeve but not between the fluid and the rod or the spool. Consider now Fefflux and Fsleeve separately. The net momentum efflux in Eq. 共2兲 is given by Fefflux,1 =

冕

␳vxv · ndA +

outlet

冕

␳vxv · ndA

共4兲

inlet

since, if there is no leak, the only surfaces that have non-zero flux through them are the inlet and outlet. The efflux at the nonmetering outlet is often neglected in previous studies 关1–3兴 on the assumption that the fluid velocity is nearly normal to the spool axis at the outlet surface 共vx = 0兲. However, CFD analysis in Sec. 3 reveals that the outlet efflux can be significant. Generally, knowledge of the flow profile at the inlet and outlet is needed to evaluate Eq. 共4兲. Often, the flow is assumed to be uniform in a Vena contracta at the metering orifice inlet 关3兴. If we further assume that the flow profile at the non-metering outlet is also proportional to the flow rate Q, then, Eq. 共4兲 can be approximated by

冉

冊

␳ cos ␪ Fefflux,2 = − cin + Q2 , CcAo共xv兲

共5兲

where Ao共xv兲 is the orifice area when the spool displacement is xv, ␪ is the Vena contracta jet angle, Cc ⬍ 1 is the contraction coefficient so that CcAo共xv兲 is the jet area, and cin is a coefficient that summarizes the normalized flow profile at the non-metering outlet. The two terms in Eq. 共5兲 correspond to the non-metering momentum flux, and the metering orifice flux, respectively. The sign convention in Eq. 共5兲 has been chosen so that cin ⬎ 0 in typical port geometries. Consider now the viscous sleeve force Fsleeve. If the fluid is Newtonian, then fundamentally, Fsleeve,1 = −

冕

sleeve

␶sleevedA =

冕

sleeve

␮

⳵ vx dA ⳵n

共6兲

where ␮ is the fluid dynamic viscosity and ⳵vx / ⳵n is the longitudinal velocity gradient with respect to the outward normal at the sleeve surface. To evaluate Eq. 共6兲, knowledge of the flow profile at the non-metering outlet is generally required. Using the assumptions that 共1兲 the friction force in the dead space between the outlet and the left land in Fig. 2 is negligible; 共2兲 the flow is laminar and fully developed along the full length L1 between the inlet and outlet; Eq. 共6兲 can be approximated by Fsleeve,2 = ␣␮L1Q

共7兲

for some coefficient ␣ ⬎ 0 that depends only on valve geometry. Generally, ␣ increases as the cross-sectional area of the flow passage decreases. For the cylindrical valve geometry in Fig. 2 共see the Appendix兲,

␣=

4共2R2o ln共Ro/Ri兲 + Ri2 − R2o兲 共R4o − Ri4兲ln共Ro/Ri兲 − 共Ri2 − R2o兲2

共8兲

By increasing the stem radius Ri, the viscosity effect can be emphasized since ␣ will be increased. The assumption that the flow in the chamber is laminar is valid when the Reynolds number Re ⬍ 2100 共e.g., when the flow rate is sufficiently low兲. For our experimental system in Fig. 19, Re ⬇ 1000 when Q = 38 lpm 共10 gpm兲. For higher flow rate, the flow may become turbulent and the shear stress must include also the Journal of Dynamic Systems, Measurement, and Control

Fig. 3 Meter-out valve chamber

Reynolds stresses caused by the random velocity components. The analysis can be extended to this case using friction/flow relation obtained from tabulated empirical friction factors. For example, in the fully turbulent regime, the friction factor is proportional to 1 / R0.25 关10兴 and Eq. 共7兲 can be replaced by e Fsleeve,3 = ␤␮0.25Q1.75L1

共9兲

where ␤ is a function of the geometry and the fluid density. In this paper, we focus on laminar flows. In summary, the steady flow force for the meter-in chamber is given by 共10兲

Fsteady = − Fefflux + Fsleeve

where Fefflux is given either fundamentally by Eq. 共4兲 or approximately by Eq. 共5兲; and Fsleeve is given either fundamentally by Eq. 共6兲 or approximately by Eq. 共7兲. Since the flow rate Q increases monotonically with the orifice area Ao共xv兲, and Ao共xv兲 is roughly linear with respect to xv, Eq. 共5兲 shows that the steady flow force component due to the metering orifice will be roughly linear in xv. This is the well-known stable linear spring effect on the spool 关2,3兴. On the other hand, both the non-metering flux component 共−cinQ2兲 in Eq. 共5兲 and Fsleeve in Eq. 共7兲 correspond to unstable quadratic and unstable linear spring forces with negative spring constants. Therefore, for the meter-in chamber, both the previously neglected viscosity effect and the non-metering flux effect tend to reduce the steady state flow force. 2.2 Meter-out Valve Chamber. A similar analysis can be performed for the meter-out valve chamber 共Fig. 3兲. The steady flow force is also given by Eq. 共10兲 with Fefflux given by one of the following: Fefflux,1 =

冕

␳vxv · ndA +

outlet

冕

␳vxv · ndA

共11兲

冊

共12兲

inlet

冉

Fefflux,2 = − cout +

␳ cos ␪ Q2 CcAo共xv兲

where cout is a coefficient that summarizes the flow pattern at the non-metering inlet for the meter-out chamber. The sleeve force is given by one of the following expressions: Fsleeve,1 = −

冕

␶sleevedA =

sleeve

冕

sleeve

Fsleeve,2 = − ␣␮L2Q

␮

⳵ vx dA ⳵n

共13兲 共14兲

where L2 is the distance between the entry port and the meter-out orifice in Fig. 3. For the valve geometry in Fig. 3, the same ␣ given in Eq. 共8兲 for the meter-in case can be used. To deemphasize SEPTEMBER 2005, Vol. 127 / 453

the viscosity effect in the meter-in chamber, ␣ can be made smaller by choosing a smaller stem radius Ri. By choosing different stem diameters in the meter-in and meter-out chambers, the meter-in and meter-out chambers can have different ␣’s. For the meter-out chamber, the steady flow force due to the momentum flux at the variable orifice can again be represented by a stable linear spring. The non-metering momentum flux 共−coutQ2兲 also acts like an unstable quadratic spring with a negative spring constant and serves to reduce the steady flow force. Unlike the meter-in chamber, the sleeve force component of the steady flow force in Eq. 共14兲 now acts like a stable spring with a positive spring coefficient. Therefore, the neglected viscosity effect tends to increase the steady flow force in the case of a meterout chamber. Moreover, since typically, 0 ⬍ cout ⬍ cin 共see CFD studies in Sec. 3兲, the reduction in steady flow force due to the non-metering flux in a meter-out chamber 共L = L2兲 will be less than that in a meter-in chamber 共L = −L1兲. Both of these effects can contribute to the observed improvement in spool agility in Sec. 5. 2.3 Valve With Both Meter-in and Meter-out Chambers. In a four way symmetric valve, the spool is acted on by the flow forces in both the meter-in and the meter-out chambers 共Fig. 1兲. Hence, the net force that acts on the spool is the sum of the forces and can be approximated by

共15兲 where the sign of A0 coincides with the spool displacement xv, and it is assumed that for a given spool displacement, the Vena contracta coefficient is the same for both meter-in and meterout chambers in a symmetric valve. In commercial valves, LªL2 − L1 is designed to be positive to ensure positive damping effect. In 关1兴 we proposed that by choosing L ⬍ 0 to induce a negative damping effect through the transient flow forces so as to improve the agility and responsiveness of the spool. The new results in this paper are that the steady flow forces can also be used to improve the agility of the single stage valve via the viscosity effect and the non-metering flux.

3

CFD Analysis of Flow Forces

In this section, we present CFD analysis to verify and evaluate the various flow force models presented in Sec. 2. The 3D computational models for a given xv are shown in Fig. 4. Notice that the valve is not axisymmetric because of the inlet and outlet ports. The mesh and the boundary condition for each geometry are generated by the GAMBIT pre-processor. Each computation volume uses about 1,000,000 nodes and 500,000 elements. The incompressible Navier-stokes equations without body forces are given by 关11兴. Continuity: ⵜ·v=0

Fig. 4 Two fluid volume models for a given xv used in the CFD study corresponding to two damping lengths. Each model corresponds to half of the fluid in the valve. The orifice is at the right-hand side port. To model the meter-in chamber „left chamber in Fig. 1…, the right-hand side port is the entry port, and the left-hand port is the outlet port. To model the meter-out chamber „right chamber in Fig. 1…, the left-hand side port is the entry port, and the right-hand side port is the outlet port.

= 6.35 mm. Sixteen models corresponding to combinations of four spool displacements xv = 0.635, 1.27, 1.905, and 2.54 mm, and four single chamber damping lengths L = −L1 = −0.216 m, ⫺0.118 m 共meter-in chamber兲 and L = L2 = 0.118 m, 0.216 m 共meter-out chamber兲 are investigated. Constant inlet pressure Ps = 689475.7 Pa. 共100 psi兲 and outlet pressure PT = 101300 Pa 共1 atm兲 are imposed. These parameters are chosen to be compatible with our experimental setup 共see Sec. 4兲. Solutions are obtained using FLUENT 6 共Fluent Inc., NH兲 on the IBM SP supercomputer at the University of Minnesota. The various components of the steady flow force and momentum fluxes can be directly measured and are used to evaluate the various models in Sec. 2. 3.1 Metering Orifice Flux. The metering orifice flux is first investigated. The meter-in and meter-out flow patterns are shown in Figs. 5 and 6. The key parameter group cos共␪兲 / Cc in Eq. 共5兲 is potentially a function of the spool displacement. The jet angle ␪ for both meter-in and meter-out chambers at different spool displacements may be obtained from the flow patterns shown in Figs. 5 and 6. For sufficiently large xv, ␪ ⬇ 69° as predicted in 关3兴. A

共16兲

Momentum:

␳

⳵v + ␳ v · ⵜ v = − ⵜ P + ␮ ⵜ 2v ⳵t

共17兲

where ␳ is the fluid density, v is the fluid velocity vector, P is the pressure, and ␮ is the dynamic viscosity. The SIMPLE pressure correction approach 关11兴 is applied to decouple the continuity and momentum equations. No slip conditions are imposed on all land faces, and rod and sleeve walls. Fluid density of ␳ = 871 kg/ m3, and dynamic viscosity ␮ = 0.0375 kg/ m / s are used. These correspond to a typical hydraulic fluid 共Mobil DTE 25兲 at 40°C. The rod and inner sleeve radii of the valve are Ri = 3.175 mm and Ro 454 / Vol. 127, SEPTEMBER 2005

Fig. 5 Flow patterns used to estimate the jet angles for the meter-in case: „from left to right… xv = −0.635, ⴚ1.27, ⴚ1.905, and ⴚ2.54 mm

Transactions of the ASME

Fig. 6 Flow patterns used to estimate the jet angles for meterout case „from left to right… xv = 0.635, 1.27, 1.905 and 2.54 mm

contraction coefficient of Cc = 0.6 has been cited in the literature 关3兴. This gives an estimate of cos共␪兲 / Cc ⬇ 0.60. Figure 7 shows the relationship between the measured metering orifice momentum flux and ␳Q2 / Ao共xv兲, where Ao共xv兲 is the orifice area of the circular ports used in this study. The linearity of the data indicates that a constant average of cos ␪ / Cc = 0.454, which is estimated using the least squares method suffices for all spool displacements and damping lengths in Eq. 共5兲. This value suggests an equivalent contraction coefficient of Cc ⬇ 0.79, which is larger than suggested in 关3兴. Figure 8 shows the relationship between the metering orifice fluxes and the damping length L. Note that the metering orifice fluxes estimated based on a constant cos ␪ / Cc = 0.454 closely approximate direct CFD measurements. The metering orifice flux obtained by direct CFD measurements and via estimation using Eqs. 共5兲 and 共12兲 with cos ␪ / Cc = 0.454 as functions of damping length are shown in Fig. 8. It demonstrates that the metering orifice flux is not a function of the damping length L. Since classical steady flow force model in 关1–3兴 considers only the metering orifice flux, it would therefore also predict steady flow force to be independent of damping length. 3.2 Non-metering Flux. Next, we consider the non-metering flux component of the steady flow force. To illustrate, the flow patterns at the non-metering inlet/outlet for a 2D 共i.e., 3D axial symmetric兲 valve model are shown in Fig. 9. Notice that despite the ports being normal to the spool axis, the flow patterns at the non-metering inlet or outlet are not normal to it, so that non-zero longitudinal momentum fluxes are expected. Figure 10 shows the relationship between the measured nonmetering flux and Q2 for various flow rates and damping lengths. The linearity of the meter-in and meter-out data confirms that the momentum fluxes in Eqs. 共5兲 and 共12兲 can indeed be

Fig. 7 CFD computed metering orifice momentum versus ␳Q2 / Ao„xv…. The four clusters of data, from the lower left corner to the upper right corner, correspond to xv = 0.635, 1.27, 1.905, and 2.54 mm. The dashed line is the estimate based on the least squares fit with cos„␪… / Cc = 0.454.

Journal of Dynamic Systems, Measurement, and Control

Fig. 8 Metering orifice fluxes as a function of damping length L, for various spool displacements xv. The dashed lines are directly obtained from CFD computation, while the solid lines are estimated using Eq. „5… based on cos„␪… / Cc = 0.454.

modeled using constants cin = 4.26⫻ 106 N s2 m−6 and cout = 1.79 ⫻ 106 N s2 m−6 for all flows, spool displacements, and damping lengths. Notice that the non-metering outlet flux coefficient cin in the meter-in chamber is more than twice that of the non-metering inlet flux coefficient cout in the meter-out chamber. This is consistent with the observation in Fig. 9 that the flow is more obtuse in the meter-in outlet than in the meter-out inlet. To further confirm that cin and cout are not dependent on damping length and spool displacement, extensive studies using 2D

Fig. 9 Non-metering flow pattern with the port designed to be normal to the spool axis. Note that the flow direction for „a… is downward at the port, while the flow direction for „b… is upward at the port.

Fig. 10 Non-metering flux as a function of Q2 for the sixteen 3D CFD models. The regression lines correspond to cin = 4.26 Ã 106 N s2 m−6 and cout = 1.79Ã 106 N s2 m−6.

SEPTEMBER 2005, Vol. 127 / 455

Fig. 12 CFD computed sleeve force Fsleeve as the function of ␮QL. The sixteen diamond points correspond to the sleeve forces computed from sixteen CFD models, and the dotted line is the least squares fit. The solid line is calculated from Eq. „7… or Eq. „14… with ␣ computed from Eq. „8….

Fig. 11 Estimated cin and cout for various spool displacements and damping lengths L in a 2D CFD model. Top: Various xv and L = 0.15 m. Bottom: Various L and xv = 0.645 mm.

共equivalent to 3D axis-symmetric兲 CFD models with multiple flow rates at each spool displacements and damping lengths were performed. Figure 11 shows that cin and cout can be treated as constants in these 2D cases as well. 3.3 Viscosity Effect. Figure 12 shows Fsleeve, computed from the fundamental equations 共6兲 and 共13兲 as a function of ␮QL for the 16 test geometries. The linear nature of the curve indicates that the laminar and fully developed flow approximations in Eqs. 共7兲 and 共14兲 are accurate. The least squares estimate for the coefficient ␣ is 1 ⫻ 106 m−2 and the one computed analytically from Eq. 共8兲 is 0.8⫻ 106 m−2. Figure 12 shows that these estimates are quite close. 3.4 Total Steady Flow Force. Consider the following expressions for the steady flow force that use different approximations: Fsteady,0 = Fland + Frod

共18兲

Fsteady,1 = − Fefflux,1 + Fsleeve,1

共19兲

Fsteady,2 = − Fefflux,2 + Fsleeve,2

共20兲

Fsteady,inviscid = − Fefflux,2

共21兲

where Fsteady,0 is the most fundamental, computed directly based on Eq. 共1兲; Fsteady,1 is the approach based on the fundamental momentum equation in Eqs. 共4兲, 共6兲, and 共13兲; Fsteady,2 is based on the approximate momentum equations—Eqs. 共5兲, 共7兲, 共12兲, and 共14兲—using the estimated parameters; Fsteady,inviscid is the estimate of the steady flow forces that neglects viscosity. Notice in Fig. 13 shows that Fsteady,0, Fsteady,1, and Fsteady,2 are almost exactly the same. Both the fundamental and approximate momentum methods to calculate the, steady flow forces are accurate. Furthermore, Fig. 456 / Vol. 127, SEPTEMBER 2005

13 shows that the viscosity effect plays a significant role in determining the steady flow forces. It reduces the steady flow force for negative damping lengths, while increasing it for positive damping lengths. The extent to which the steady flow force varies increases with the damping length. Figure 14 shows the dependence of the steady flow force for each meter-in/meter-out chamber on damping lengths according to Eq. 共20兲. Notice the discontinuity of the flow force at L = 0. Since the damping length L in Fig. 14 is produced using single chamber 共i.e., either L1 = 0 or L2 = 0 where only one of cin or cout is at play at a time兲, the discontinuity is related to the difference between the meter-in and meter-out non-metering flux, 共cin − cout兲Q2 so that it decreases for smaller xv 共hence Q兲. Figure 15 shows the total steady flow force for the two-chamber-four-way valve according to Eq. 共15兲. Compared to Fig. 14, no discontinuity is present because both the non-metering inlet and outlet fluxes are simultaneously present in all damping lengths. Figure 15 also shows that both the viscosity and non-metering flux are more significant at high flow rate. The non-metering effect is illustrated by the parallel shift and is given by 共cin + cout兲Q2. Additional CFD analysis 共not shown兲 reveals that the non-metering flux as a proportion of the metering orifice flux will be even larger for axisym-metric 共i.e., 2D兲 valves, than in the 3D axis-symmetric case in Fig. 4. 3.5 Modifying the Non-metering Flux. Since the nonmetering fluxes tend to reduce steady flow forces, it would be advantageous if it can be manipulated by design. One straightforward idea is to change the angle of the non-metering port to the spool axis 关8兴. Figures 9, 16, and 17 illustrate the differences in flow patterns for three port angles for a 2D 共or 3D axissymmetric兲 CFD model. Table 1 shows the non-metering flux values for various port angles, normalized by the value for the 0° port angle, meter-in chamber case. It can be seen that the non-metering flux is susceptible to changes in geometric configuration of the non-metering port. In particular, by rotating the inlet and outlet ports in the counterclockwise direction, the non-metering fluxes in both the meter-in and meter-out chambers increase. If the port is rotated 30° counterclockwise, the sum of the non-metering fluxes of the two chambers increase by 37%, thus having the effect of further reducing the steady flow force. Although the port angle affects the non-metering flux, flow patterns in Figs. 9, 16, and 17 show that the mean angles at which the Transactions of the ASME

Fig. 14 Steady flow forces as a function of the single chamber damping length

Fig. 15 Steady flow forces and their estimates in a two chamber four way valve as a function of the damping length for fours sets of spool displacements. “A” takes into account nonmetering flux, “B” does not take into account non-metering flux. The horizontal bars at L = 0 indicate the estimates if viscosity effect is ignored.

Fig. 13 Steady flow forces computed from various methods as a function of the orifice displacement. Four different damping lengths are considered.

Journal of Dynamic Systems, Measurement, and Control

Fig. 16 Non-metering flow pattern for a 2D CFD model with the port rotated ⴙ30° clockwise from the normal direction to the spool axis. Note that the flow direction for „a… is downward at the port, while the flow direction for „b… is upward at the port.

SEPTEMBER 2005, Vol. 127 / 457

Fig. 17 Non-metering port flow pattern for a 2D CFD model with the port rotated -30° clockwise from the normal direction to the spool axis. Note that the flow direction for „a… is downward at the port, while the flow direction for „b… is upward at the port.

flow enters or leaves the non-metering ports are typically not the angle of the ports, as assumed in 关8兴.

4

Experimental Study

The objective of the experiment is to verify the steady flow force models in Sec. 2 and the proposition that viscosity and nonmetering flux have significant effects on the steady flow forces, and hence on the spool agility. The experimental setup depicted in Fig. 18 consists of the custom-built valve in Fig. 19, an adjustable spool, and a force sensor. The geometry of the valve is similar to the one used in the CFD studies in Sec. 3. In order to measure the steady flow force, the shaft of the valve is connected in series with a stationary force sensor 共67M25A, JR3 Inc.兲 to measure the compression/tension forces along the spool axis. Various 共single chamber兲 damping lengths are achieved by connecting the inlet and outlet to different ports of the custom-built valve. The spool lands are adjusted on the threaded shaft to achieve the desired metering orifice opening, while the non-metering orifice is simply the size of the port. The inlet/outlet are connected to the pump in parallel with adjustable bleedoff needle valve. The flow rates are directly measured by a digital flow meter 共DFS-3, Digiflow Systems兲 connected downstream to the valve and controlled by the bleedoff valve. In the experiment, the orifice area is set to be 6 ⫻ 10−6 m2. The flow forces were measured in 24 conditions: combination of four positive 共L = L2兲, four negative 共L = −L1兲 single chamber damping

Table 1 Meter-in and meter-out non-metering flux for various non-metering port angles, normalized by that of the meter-in, 0° port angle case Port angle 30° 0° ⫺30°

Meter-in

Meter-out

Sum

1.17 1.00 1.20

0.09 0.45 0.80

1.26 1.45 2.00

Fig. 18 The diagram of experimental setup for measuring the steady flow forces

458 / Vol. 127, SEPTEMBER 2005

Fig. 19 The custom-built valve allows for different „both positive and negative… damping lengths L. The valve sleeve has eleven „11… ports, fitted with quick couplers for connection to the inlets and outlets. The ports that are not connected are blocked. By connecting the supply and return to different ports, the dimensions L1 and L2 in Fig. 1 and the damping length LªL2 − L1 can be changed. Although both meter-in and meter-out chambers „see Sec. 2… can be achieved simultaneously, our experiments investigate each single chamber separately so that either L1 = 0 or L2 = 0. The spool consists of a threaded hardened precision anodized aluminum shaft on which several bronze lands can be arbitrarily positioned to be consistent with the chosen inlet/outlet locations. By connecting the inlet/outlet and actuator to the various ports, different damping lengths can be achieved. The configuration shown in Fig. 18 has only one valve chamber.

lengths: L = ± 236, ⫾175, ⫾118, ⫾62 mm兲; and three flow rates Q = 3.7⫻ 10−5 m3 / s 共2.2LPM兲, 6.7⫻ 10−5 m3 / s 共4LPM兲, and 1 ⫻ 10−4 m3 / s 共6LPM兲. The procedure for measuring the steady flow forces is as follows. After adjusting the spool lands, a background force sensor reading is first taken when the pump is turned off. The pump is then turned on with the bleedoff valve adjusted to give the desired flow rates. After the flow condition and force sensor reading have settled down, the force sensor reading is taken again. The pump is then turned off and the second background force reading is taken to check if the spool positions have shifted. The difference between the force sensor reading when the pump is turned on and when it is turned off is the steady flow force for the particular condition. The readings are repeated and averaged over three times. Figure 20 shows the measured steady flow forces and the corresponding predictions based on Eq. 共10兲, and Eqs. 共5兲 and 共7兲 or Eqs. 共12兲 and 共14兲 for the negative and positive damping lengths cases, respectively. The parameters used in the predictions are estimated in the CFD study in Sec. 3 to be: ␣ = 1 ⫻ 106 m−2, cin = 4.26⫻ 106 N s2 m−6, cout = 1.79⫻ 106 N s2 m−6. The measured steady flow force values are qualitatively similar to the model in Fig. 14. First, the steady flow force has an affine dependence on the damping length. This is a direct verification of the effect of viscosity on steady flow force. In the absence of viscosity effect, the steady flow force will be independent of the damping length L. Second, the sensitivity of the steady flow force to damping length, kLª ⳵ Fsteady / ⳵L, increases with flow rate. This is predicted by Eqs. 共7兲 and 共14兲. Third, the steady flow force is discontinuous at Transactions of the ASME

Table 2 Estimates of ⌬1 and ⌬2 „N…

Fig. 20 Measurements „“squares” and dashed lines… and estimation „“triangles” and solid lines… of steady flow forces as a function of the damping length for various flow rates. The dashed lines are regression lines of the experimental data assuming a constant kLªFsteady / ⵲L for each flow rate. Top: Q = 2.2 LPM, middle: Q = 4.0 LPM, bottom: Q = 6.0 LPM.

L = 0 with the discontinuity increasing with flow rate. This is due to the non-metering orifice flux, as discussed in Sec. 3 共i.e., for single chamber case, either cin or cout but not both contribute to the non-metering flux, depending on the sign of L兲. Notice also that, as predicted, the steady flow force is reduced for negative damping lengths, and for sufficiently negative damping lengths, the steady flow force becomes marginally stable or even unstable 共reverses sign兲.

Fig. 21 Relationship between flow force sensitivity to damping length kL, and flow rate Q. The squares correspond to the slopes of the dashed lines in Fig. 20 that are the regression lines of the experimental data. The dashed line is the curve fit of the squares, whose slope should be ␣␮.

Journal of Dynamic Systems, Measurement, and Control

Q 共LPM兲

2.2

4.0

6.0

Meter-in 共L ⬍ 0兲

⌬1 共N兲 ⌬2 共N兲

⫺0.16 0.17

⫺0.20 0.27

⫺0.25 0.40

Meter-out 共L ⬎ 0兲

⌬1共N兲 ⌬2 共N兲

⫺0.14 ⫺0.058

⫺0.16 ⫺0.089

⫺0.20 ⫺0.141

In the study of unsteady flow force in a four-way valve in 关9兴, it was observed that the flow forces at low frequency 共i.e., steady flow forces兲 increase with damping lengths. This effect was previously unexplained but is exactly what the new flow force models that include viscosity effect would predict. Notice that for each Q, the steady flow force can be fitted by two straight 共dashed兲 lines 共L ⬎ 0 and L ⬍ 0兲 using the same slope kLª ⳵ Fsteady / ⳵L, in Fig. 20. The relationship between kL and Q is predicted by Eqs. 共7兲 and 共14兲 to be kL = ␣␮Q where ␮ is the dynamic viscosity. Experimental results in Fig. 21 confirm that kL indeed increases linearly with the flow rate. This implies that, in the high flow rate applications, negative damping lengths will be more effective in reducing steady flow forces. In addition, we can quantitatively calculate the parameter ␣ from Fig. 21. The straight dashed line has a slope which should be ␣␮. This gives a value of ␣ = 0.978⫻ 106 m−2 which is close to the values obtained by CFD in Sec. 3 共1.0⫻ 106 m−2兲 or via Eq. 共8兲 共0.8⫻ 106 m−2兲. Despite excellent match in the viscosity effect, there are some significant quantitative discrepancies between the experimental results in Fig. 20 and the model. First, the discontinuity at L = 0 between the meter-in and meter-out chamber is significantly larger than that predicted by the expected non-metering flux differences. Second, the experimental data tend to be more negative than the prediction, especially when L ⬎ 0. Extensive effort was spent to investigate the source of these disparities. Two limitations of our experimental setup have been identified as causes. First, since the valve lands are machined by hand, it turns out that the areas of the two spool lands are different. The area difference was estimated by measuring the spool force when the closed chamber formed using the two lands is pressurized. The land at the metering orifice was found to be smaller by ⌬A = 0.186 mm2. This leads to an extra orifice-closing force ⌬1 ⬇ ⌬A · P 共negative force in the figure兲 as the chamber is pressurized. Using the pressure measured by the digital pressure transducers during the experiments, the steady force offset ⌬1 are shown in Table 2. Second, the shape of the lands need to be considered. To avoid hydraulic locking and to reduce friction between the lands and the sleeve, the lands were intentionally machined to have a 2.5° taper to form a hydrostatic bearing. The effect of the taper can be seen in Fig. 22 where the control volume is different from that in Figs. 2 and 3. Modifying Eqs. 共1兲 and 共2兲 accordingly, the spool force of

Fig. 22 Modified control volume of the valve with tapered lands. The metering orifice land area is also smaller than the area of the left hand side land. The control volume is the gray colored block.

SEPTEMBER 2005, Vol. 127 / 459

Fig. 24 Schematic of the experimental setup used to investigate the effect of the damping length on spool agility

commercial valve, where the lands can be precisely machined, the land area difference is expected to be much smaller, and no taper or a much smaller taper will generally be used. Therefore, in practice, the ⌬1 + ⌬2 correction will not be needed.

5

Fig. 23 Measurements „“squares” and dashed lines… and estimation „“triangles” and solid lines… of steady flow forces as a function of the damping length for various flow rates. The estimates take into account limitation of the experimental setup via Eq. „22…. The dashed lines are regression lines of the experimental data assuming a constant kLª ⵲ Fsteady / ⵲L for each flow rate. Top: Q = 2.2 LPM, middle: Q = 4.0 LPM, bottom: Q = 6.0 LPM.

the tapered land valve is

Effect of Damping Lengths on Spool Agility

The central hypothesis in 关1兴 is that by configuring the valve to have a negative damping length, the transient flow force will have an unstable negative damping effect on the spool, thus increasing spool agility. On the other hand, Secs. 2–4 show that negative damping lengths also decrease the steady flow force via the viscosity effect, which would also increase spool agility. In this section, experimental results are presented to confirm the effect of damping length on spool agility as determined by stroking time, and the relative contributions of the transient and steady flow forces for a typical setup. The experimental setup in Fig. 24 consists of the single chamber 共meter-out or meter-in兲 spool valve with adjustable damping lengths. Unlike the stationary setup in Sec. 4, a solenoid is used to stroke the spool; and a potentiometer is used to measure the spool displacement. The solenoid is driven by the current driver circuit in Fig. 25, so that the control voltage Vc applied to the circuit would determine the solenoid current. A spring is used to push the spool to a fixed initial position where the orifice displacement xv = 0. The spool stops when the mobile armature of the solenoid touches the core. The control of the solenoid and the data acquisition system are coordinated by a PC within the Matlab 共Mathworks, MA兲 environment via the Matlab’s Realtime Toolbox 共Humusoft, CZ Rep.兲. Step voltage inputs of Vc = 1.6 V 共corresponding to solenoid current of 1.45 A兲 at t = 0.5 s are applied for the various damping

共22兲 where n is the outward normal vector of the surface, and nx represents the unit vector in the positive x direction. The offset force due to the taper, ⌬2, consists of the longitudinal pressure force applied on the orifice plane 共o-p兲 and on the side surface 共s-l兲, respectively. It can be seen that ⌬2 ⬎ 0 for the meter-in case because the first term of ⌬2 is greater than the second term, while ⌬2 ⬍ 0 for the meter-out case because of the opposite effect. The numerical values of ⌬2 were obtained via CFD analysis of a 2D 共i.e., axis-symmetric 3D兲 model at various pressure differences across the orifice. The ⌬2 values for the 3D axis-asymmetric case were then estimated from the 2D case with the same pressure difference, corrected for the port area difference 共3D case value is approximately 50% of the equivalent 2D case value兲. The specific values for the experimental conditions are listed in Table 2. The predicted flow forces after incorporating the offsets ⌬1 + ⌬2 match the experimental results very closely 共Fig. 23兲. Even better match can be obtained by slightly adjusting ⌬2. The fact that ⌬1 and ⌬2 are of different signs for L ⬍ 0 explains the observation that there are less discrepancies for L ⬍ 0 than for L ⬎ 0 in Fig. 20. In a 460 / Vol. 127, SEPTEMBER 2005

Fig. 25 Current driver circuit for solenoid. “Coil” refers to the coil of the solenoid. The current through the coil is Vc / R in which R = 1.1 ⍀ is the sensing resistor.

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Fig. 26 Spool displacement trajectories for various damping lengths „from left to right: L = 0.216, 0.118, ⴚ0.118, ⴚ0.216 m…, when a step input of Vc = 1.6 V is applied at the instant t = 0.5 s

lengths and the spool displacement trajectories are recorded. As shown in Fig. 26, as the damping length decreases, it takes less time for the spool to travel the full stroke of 4.5 mm 共transit time兲. In particular, the transit time for L = 0.216 m is over 60 ms, while the transit time for L = −0.216 m is about 37 ms 共Fig. 26兲 which is about 40% shorter. The experimental result shows clearly that damping length has a significant effect on spool agility. This result is qualitatively consistent with the effect of damping lengths on transient flow force 关1兴 as well as with their effects on steady flow forces due to viscosity. To determine the relative contributions due to transient flow force or viscosity induced steady flow force reduction, consider the dynamics of the spool in Fig. 24: mx¨v = Fsteady + Ftransient + Fsolenoid + Fspring

共23兲

where m is the lumped mass of the spool and the armature of the solenoid, Fsolenoid is the solenoid force which is a static nonlinear function of the current and the displacement 关12兴, Fspring is the spring force that a function xv. The total work done by the transient flow forces, the solenoid force, and the spring force are W=

冕

Fig. 27 Work done by the sum of the transient flow forces, solenoid force, and the spring force; and by the steady flow force, as functions of xv„tf… when step voltage input of Vc = 1.6 V is applied, for damping lengths L = ± 0.216 m.

sient flow forces assisting the spool to open. This is also the case for the work done by the steady flow force, indicating that for negative damping length, viscosity effect assists valve opening. In particular, for large valve openings, the work done contributed by the viscosity effect of damping lengths on steady flow force is of a similar magnitude as the work done by transient flow force effect. However, for small valve openings, the latter is more significant. This is consistent with the result in Fig. 21 that the effect of damping lengths on steady flow force is greater at higher flow rates. At least for the valve geometry we have investigated, both transient and steady flow force are important considerations for improving the spool agility.

xv共t f 兲

共Ftransient + Fsolenoid共xv兲 + Fspring共xv兲兲dxv

xv共t0兲

1 = mx˙v2共t f 兲 − 2

冕

6

xv共t f 兲

Fsteadydxv

共24兲

xv共t0兲

where xv共t0兲 is the initial spool displacement 共spool is at rest兲 and t f are the various sample times when the spool is in motion. x˙v is computed from the trajectory in Fig. 26. For each damping length, x 共t 兲 兰xv共t f 兲Fsteadydxv can be computed using the steady flow force modv 0 els 共22兲 and by assuming a quasi-static condition. The comparison of W and work done by Fsteady for two damping lengths, L = ± 0.216 m, and step voltage input Vc = 1.6 V, are shown in Fig. 27. Note that for each xv共t f 兲 and constant voltage Vc, the work done by the solenoid and by the spring are identical because the current through the solenoid is constant, and Fsolenoid共xv兲 + Fspring共xv兲 become a function of xv alone. Therefore, the observed difference in W in Fig. 27 between two damping lengths is the difference in the work done by transient flow forces alone. The differences between the Fsteady, on the other hand, is due to viscosity effect. Note that W is larger for negative damping lengths than for positive damping lengths indicating extra tranJournal of Dynamic Systems, Measurement, and Control

Conclusion

In this paper, we demonstrate that the fluid viscosity and the non-metering flux are very important in the estimation of the steady flow forces. On one hand, in the negative damping length region, the steady flow forces, conventionally regarded to be always stabilizing, can be reduced to be marginally stable or unstable, thus improving the agility of the spool. Paradoxically, using a valve with negative damping lengths, the spool’s agility will improve as the fluid becomes more viscous. On the other hand, the non-metering flux can be tuned by changing the angle of the non-metering port to the spool axis, so that the net momentum efflux, which is traditionally regarded to be always stabilizing, could possibly be reduced to be marginally stable or unstable. The viscosity and the non-metering effects predicted by the proposed models and CFD analysis are verified in experiment. The viscosity and non-metering flux effects generate other valve design parameters that can be used to improve spool agility in single stage electrohydraulic valves. Since the steady flow force is a significant force that the solenoid actuator has to overcome, the reduction of the steady flow force via the viscosity and nonmetering effect will be useful for developing high performance single stage valves. SEPTEMBER 2005, Vol. 127 / 461

Appendix: Detailed Derivation of Component of the Steady Flow Force

the

Viscosity

For the valve with geometry in Fig. 2, the sleeve force is fundamentally given by

冕 冏 冏 0

Fsleeve,1 = 2␲Ro␮

⳵ vx ⳵r

−共Ld+L1兲

dx

共A1兲

r=Ro

where 兩 ⳵ vx / ⳵r兩r=Ro is the longitudinal velocity gradient in the outward radial 共r兲 direction evaluated at the sleeve wall 共r = Ro兲. If the friction in the dead space is negligible,

冕冏 冏 0

Fsleeve,1 ⬇ 2␲Ro␮

−L1

⳵ vx ⳵r

共A2兲

dx. r=Ro

For laminar and fully developed flow, it can be shown that 关10兴

冉 冊

1 dP 1 d dvx r = r dr dr ␮ dx

共A3兲

where P共x兲 is the pressure. Applying the no-slip conditions that vx = 0 at the rod wall 共r = Ri兲 and the sleeve wall 共r = Ro兲, the velocity distribution can be determined to be vx =

冋

Ri2 − R2o 1 dP 2 r r − R2o + ln 4␮ dx ln共Ro/Ri兲 Ro

Since −

冕

册

共A4兲

Ro

2␲vxdr = Q,

References

Ri

⳵vx / ⳵r can be expressed as a function of Q so that ␣ in Eq. 共7兲 is given by

␣=

cout ⫽ meter-out chamber non-metering inlet flux coefficient Fefflux ⫽ net momentum efflux Fland ⫽ pressure force acting on the lands Frod ⫽ viscous force acting on the rod Fsleeve ⫽ viscous sleeve force acting on fluid Fsolenoid ⫽ solenoid force Fspring ⫽ spring force Fspool ⫽ fluid flow force acting on spool Fsteady ⫽ steady flow force L ⫽ damping length L1 ⫽ meter-in chamber damping length L2 ⫽ meter-out chamber damping length Ld ⫽ dead space distance n ⫽ unit vector outward normal to surface nx ⫽ positive x unit vector kL ⫽ steady force sensitivity to damping length Pl ⫽ pressure on the left land Pr ⫽ pressure on the right land Q ⫽ flow rate P ⫽ fluid pressure Ri ⫽ radius of the rod Ro ⫽ radius of the sleeve v ⫽ fluid velocity vector vx ⫽ longitudinal fluid velocity W ⫽ total work by solenoid, spring, and transient flow forces

4共2R2o ln共Ro/Ri兲 + Ri2 − R2o兲 共R4o − Ri4兲ln共Ro/Ri兲 − 共Ri2 − R2o兲2

.

Nomenclature ␣ ⫽ viscosity related geometry coefficient ⌬1 ⫽ land area difference compensation ⌬2 ⫽ land tapers compensation ␶sleeve ⫽ wall stresses on the sleeve ␶rod ⫽ wall stresses on the rod ␪ ⫽ Vena contracta jet angle ␮ ⫽ dynamic viscosity ␳ ⫽ density Al ⫽ annular area of the left land Ao ⫽ orifice area Ar ⫽ annular area of the right land Cc ⫽ Vena contracta contraction coefficient cin ⫽ meter-in chamber non-metering outlet flux coefficient

462 / Vol. 127, SEPTEMBER 2005

共8兲

关1兴 Krishnaswamy, K., and Li, P. Y., 2002, “On Using Unstable Electrolydraulic Valves for Control,” ASME J. Dyn. Syst., Meas., Control, 124共1兲, pp. 182– 190. 关2兴 Blackburn, J. F., Reethof, G., and Shearer, L. L., 1910, Fluid Power Control, MIT, Cambridge. 关3兴 Merritt, H. E., 1967, Hydraulic Control System, Wiley, New York. 关4兴 Borghi, M., Milani, M., and Poluzzi, R., 2000, “Stationary Axial Flow Force Analysis on Compensated Spool Valves,” Int. J. Fluid Power, 1共1兲, pp. 17–25. 关5兴 Yang, R., 2003, “Hydraulic Spool Valve Metering Notch Characterization Using CFD,” in ASME Fluid System Technology Division Publication - 2003 International Mechanical Engineering Congress, vol. 10, pp. 11–17. 关6兴 Mookherjee, S., Acharyya, S., Majumdar, K., and Sanyal, D., 2001, “StaticPerformance Based Comuter-aided Design of a DDV and its Sensitivity Analysis,” Int. J. Fluid Power, 2共2兲, pp. 47–63. 关7兴 Del Vescovo, G., and Lippolis, A., 2003, “Three-dimensional Analysis of Flow Forces on Directional Control Valves,” Int. J. Fluid Power, 4共2兲, pp. 15–24. 关8兴 McCloy, D., and Martin, H. R., 1980, Control of Fluid Power, Analysis and Design, 2nd ed., Ellis Horwood, West Sussex. 关9兴 Nakada, T., and Ikebe, Y., 1980, “Measurement of the Unsteady Axial Flow Force on a Spool Valve,” in Proceedings of the IFAC Symposium - Pneumatic and Hydraulic Components and Instruments in Automatic Control, Warsaw, Poland. IFAC, May 1980, pp. 193–198. 关10兴 Munson, B. R., Young, D. F., and Okiishi, T. H., 1998, Fundamentals of Fluid Mechanics, 3rd Edition, Wiley, New York. 关11兴 Tannehill, J. C., Anderson, D. A., and Pletcher, R. H., 1997, Computational Fluid Mechanics and Heat Transfer, Taylor and Francis, Philadelphia. 关12兴 Xu, Y., and Jones, B., 1997, “A Simple Means of Predicting the Dynamic Response of Electromagnetic Actuators,” Mechatronics 7共7兲, pp. 589–598.

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