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IFPE 2008 Technical Conference

Paper number: 14.2

Estimated versus Calculated Viscous Friction Coefficient in Spool Valve Modeling

Medhat K. Bahr Khalil, Ph.D. Professional Education Instructor, Milwaukee School of Engineering

1025 North Broadway, Milwaukee, WI, 53202-3109 Tel: 414-940-2232 Email: khalil@msoe.edu

Abstract: In most hydraulic component mathematical models developed by design engineers, the term “viscous friction” is involved. Because of the lack of knowledge on how to calculate the viscous friction or directly measure it, the value of the viscous friction is usually estimated and adjusted to force the model to provide the desired performance. The model validity based on this estimation may be an issue. In addition, viscous friction is function of certain dimensional parameters and operating conditions. Assuming that it a constant is a harsh assumption that affects the model validity. In this paper a study has been made to investigate a proportional valve response where the viscous friction is calculated based on variable working conditions. Simulation runs were made to show the effect of variable viscous friction on the valve stability. Results are presented and discussed. While the study is applicable for many other hydraulic components, it is developed in this paper for a proportional directional valve dynamic response as a typical example. Introduction: Viscous friction between two surfaces that have relative motion between them depends on dimensional parameters such as contact area and clearance between the two surfaces, and also depends on fluid properties, e.g. fluid specific gravity and viscosity. Viscous friction, as will be shown later, was found linearly proportional to fluid viscosity, fluid specific gravity and the contact area between the two meeting surfaces. Viscous friction was found inversely proportional to the clearance between the two meeting surfaces. Fluid properties are significantly affected by the working temperature, which varies in the best case scenario within the recommended range of 25 to 65 oC. Clearances between spool and sleeve in the hydraulic valves are in the order of microns so that the wear due to abrasive contaminants in the hydraulic fluid has significant effect on this clearance.

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In modeling and simulation of hydraulic components, in general, product developers often assume constant viscous friction in order to simplify the model and keep their focus on the main criteria they are studying. In some cases, considering viscous friction to be a constant value may result in misleading analysis of the component characteristics and put the model validity in doubt. Since the damping ratio is directly related to the viscous friction, viscous friction variation with the operating conditions may affect the stability status of closed loop controlled hydraulic systems. Sample references who estimated viscous friction to be constant value are as follows. Khalil, Svoboda and Bhat [1] modeled and simulated closed loop dynamic characteristics of a proportional directional spool valve that is integrated with the variable displacement swash plate axial piston pump. The authors considered constant viscous friction of 90 N.s/m which resulted in a realistic damping ratio. Grabbel and Ivantysynova [2] have also considered viscous friction coefficient as constant within a Stribeck friction model in their study of control concepts for displacement pump actuators. Schoenau, Burton and Ansarian [3] discussed parameter estimation techniques to predict the spring constant and spring pre-compression in the main spool of a solenoid proportional hydraulic valve. They also assumed viscous friction coefficient to be constant, equal to 125 Ns/m. Some exhaustive researches were conducted to measure the friction between mating surfaces. Scharf and Murrenhoff [4] have developed a test stand to measure the friction between piston and bushings of an axial piston displacement unit. The authors concluded that the friction is influenced by the piston-bushing geometry. Our focus in this paper is to show the impact of calculating the viscous friction on a hydraulic proportional valve open loop and closed loop dynamic characteristics versus assuming viscous friction to be constant. Mathematical Model: Figure 1 shows schematically a proportional solenoid spool valve that is used typically as an electro-hydraulic pressure compensator. The prime objective of such a valve is to control its upstream pressure PU based on the change in its downstream pressure PD by controlling the spool displacement. The valve controller works to change the spool position in order to stabilize PU to be constant value if either the flow through the valve QS and/or the valve downstream pressure PD changes. So the simple goal of the controller here is to change the valve spool position based on certain inputs. As shown in Fig.1, the valve is normally closed and it opens gradually based on the strength of the input signal. The valve spool is assumed overlapped and the valve should move a displacement XO before it starts to open. The valve travels to the maximum position Xmax, equal 4 mm, when it receives the maximum input signal.

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Fig. 1- Schematic of the pressure compensator The valve can be represented by a single degree of freedom spring-mass system in which the spool mass mx is driven by a linear motor “proportional solenoid”. When the valve solenoid receives a control signal Ixi above zero, an electromagnetic force equals Ixiki proportional to the input signal acts on the valve spool and causes it to move until it is balanced at a displacement X against the return spring force. A simple second order equation of motion is used to calculate the spool acceleration as follows.

The spool velocity and displacement, respectively, can be found as follows

xi i v xx

1a = [I k - f v - k X] (1)

m

v = a dt (2)

X = v dt (3)

∫

∫

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Viscous Friction Calculation Figure 2 shows a spool moving in a sleeve. The resistive force (F) due to the viscous friction is directly proportional to the velocity (v) and the contact area (A), and is inversely proportional to the clearance (y). This can be written mathematically as follows

Fig. 2 - Viscous friction calculation According to the basic definition of a Newtonian fluid, the rate of fluid distortion (dv/dy) in the clearance (y) between two surfaces due to their relative motion is proportional to the shear stress acting on the fluid layers. Due to the very small clearance the rate of fluid distortion will be assumed linear. So, equation 4 can be rewritten to meet this definition as follows Where (µ) is the dynamic viscosity. Equation 5 can be rewritten as follows Where (ν) is the kinematic viscosity and (ρ) is the fluid density. As shown in equation 1, the resistive force due to viscous friction is equal the product of the viscous friction coefficient (fv) times the speed (v). Equation 6 can be used to deduce the viscous friction coefficient as follows

vAF α (4)

y

F v = µ (5)

A y

AvF = ν ρ (6)

y⋅ ⋅

-6wv

A ν ρ ×SGf = [ ] 10 (7)

y

× × ×

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Equation 7 shows, as it was indicated in the introduction, that the viscous friction is linearly proportional to fluid viscosity, fluid specific gravity and the contact area between the two meeting surfaces, and inversely proportional to the clearance between the two meeting surfaces. This demonstrates that it worthwhile to include the effect of these parameters on the viscous friction, rather than estimating a constant value. Simulation Results and Performance Analysis: A Matlab-Simulink model has been developed to simulate the valve spool position in an open loop. As shown in Fig.3, a potentiometer receives an input signal Xr and generates accordingly a control signal in voltage form Ixv. The valve amplifier card receives the control signal and generates the corresponding driving signal Ixi in current form with power enough to push the valve spool against the return spring. The proportional solenoid receives the driving signal and the valve responds dynamically based on the previously developed mathematical model. A normalized approach has been followed to compare the reference signal Xr(%) and the actual spool position X(%) as referred to a maximum spool displacement Xmax = 4 mm.

Fig. 3 - Valve open loop Matlab-Simulink model

An input ramp function was assumed to feed the program in order to simulate the open loop steady state characteristic of the valve. Simulation results, shown in Fig.4, confirmed that the valve design parameters were selected properly to obtain a linear input-output relationship. An input signal Xr of 50% value was used to investigate the step response of the valve under open loop conditions. Five combinations of parameters were used to simulate the valve step response. Simulation results are presented in Figs 5A through 5E. Case 1, Fig.5, considered the ideal values of fluid viscosity and spool-sleeve clearance that resulted in a viscous friction coefficient of 1.92 N.s/m and valve settling time of 0.7 s. In the following four cases the fluid viscosity was assumed to be decreased and the clearance increased. Results are concluded in Table2. Simulation results confirm that it is unrealistic to consider the viscous friction to be constant as these parameters change. If the viscous friction is considered constant, the simulation will show the valve to be a stable system while sometimes, in fact, it is not stable such as case 5 where the valve experiences unstable conditions.

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Fig. 4 - Valve open loop steady state characteristics

Fig.5A - Valve open loop step response, case 1

Fig.5B - Valve open loop step response, case 2

Fig.5C - Valve open loop step response, case 3

Fig.5D - Valve open loop step response, case 4

Fig.5E - Valve open loop step response, case 5

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Open Loop Case1 Case2 Case3 Case4 Case5 ν (cSt) 64 48 32 16 16 y (µm) 8 8 8 8 16 Fig. # 5A 5B 5C 5D 5E fv (N.s/m)

1.92 1.44 0.96 0.48 0.24

Settling Time (s)

0.7 1 >1 > 3 -

Stability Stable Stable Stable Stable Unstable

Table 1 - Open loop characteristics

Figure 6 shows a simple negative feedback control loop that is intended to accurately control the position of the valve spool. In the feedback path an LVDT displacement transducer is used to sense the instantaneous actual spool position and convert it to a calibrated voltage signal of voltage range (0 to 10) V. The actual spool position is subtracted at the comparator from the reference signal resulting in an error signal Ixve. Under steady state conditions, the error signal value is zero and the valve spool is held in its current position against the spring force. If the error signal is greater than zero, a PID controller drives a control signal in voltage form Ixv, which in sequence will be received by the amplifier, valve solenoid and then the position transducer to close the loop.

Fig. 6 - Feedback control loop for valve spool position

An input signal Xr of 50% value is used to investigate the step response of the valve in closed loop conditions. The same combinations of parameters have been used to simulate the valve step response. Simulation results are presented in Figs 7A through 7E. As shown in Fig.7A, the fixed parameter PID controller is tuned to stabilize the valve in 0.5 second. Simulation results of the following four cases reveal the same trend of results that show the viscous friction value and consequently the valve response is significantly affected by the working conditions. Results are concluded in Table 2. Therefore considering the viscous friction to be a constant value may be a misleading assumption.

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Closed Loop Case6 Case7 Case8 Case9 Case10 ν (cSt) 64 48 32 16 16 y (µm) 8 8 8 8 16 Fig. # 7A 7B 7C 7D 7E fv (N.s/m)

1.92 1.44 0.96 0.48 0.24

Settling Time (s)

0.5 0.7 >1 - -

Stability Stable Stable Stable Marginal Unstable

Table 2 - Closed loop characteristics Fig.7A -Valve closed loop step response, case 6

Fig.7B - Valve closed loop step response, case 7

Fig.7C - Valve closed loop step response, case 8

Fig.7D - Valve closed loop step response, case 9

Fig.7E - Valve closed loop step response, case 10

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Conclusion: In order to increase the accuracy of a hydraulic component mathematical model, viscous friction coefficients should be calculated interactively based on actual operating conditions, fluid properties and the dimensional parameters. Recommendations for future investigations are as follows. Investigate the effect of viscous friction on the valve frequency response, include pressure and temperature effects on hydraulic fluid specific gravity and confirm the results experimentally. Nomenclature:

Symbol Description Units Value A Spool-sleeve contact area m2 300 x 10-6 a Spool acceleration m/s2 F Force N fv Viscous friction constant N.s/m 1.92 Ixi Spool driving signal in current Amp Ixv Control signal in voltage Volt Ixve Error signal Volt ki Electromagnetic force constant N/Amp 4 kx Spring stiffness N/m 3000 mx Mass of the spool kg 0.1 SG Hydraulic fluid Specific gravity - 0.8 v Spool velocity m/s X Actual Spool displacement m Xmax Maximum spool displacement m 0.004 y Spool-sleeve clearance m 8 x 10-6 ν Hydraulic fluid kinematic viscosity cSt 64

µ Hydraulic fluid dynamic viscosity N.s/m2 ρ Hydraulic fluid density kg/m3 ρw Water density kg/m3 1000 References:

1. M. K. Bahr Khalil, J. Svoboda and R.B. Bhat, “Modeling of Swash Plate Axial Piston Pumps with Conical Cylinder Blocks”, Journal of Mechanical Design, ASME Transaction, Vol.126, pp 196-200, January 2004, USA.

2. Jeorg Grabbel and Monika Ivantysynova, “An Investigation of Swash Plate Control Concept for Displacement Controlled Actuators”, International Journal of Fluid Power, Vol. 6 Number 2, pp 19-36, March 2005. Germany.

3. Greg Schoenau, Rich Burton, Alireza Ansarian, “Parameter Estimation In A Solenoid Proportional Valve Using Ols And Mlh Techniques”, Mechanical Engineering Department of University of Saskatchewan, Canada.

4. Stephan Scharf and Hubertus Murrenhoff, “Measurement of Friction Forces between Piston and Bushings of an Axial Piston Displacement Unit”, International Journal of Fluid Power, Vol. 6 Number 1, pp 7-17, March 2005. Germany.

5. Franklin, Powell and Abbas, “Feedback Control of Dynamic Systems”, Fourth Edition, ISBN 0-13-032393-4, Prentice Hall, New Jersey, USA, 2002