Journal of applied science in the thermodynamics and fluid mechanics Vol. 1, No. 1/2012, ISSN 1802-9388
MATHEMATICAL MODEL OF PNEUMATIC PROPORTIONAL VALVE *Zdeněk VARGA, **Petri-Keski HONKOLA
*Technical *Technical University of Liberec, Faculty F aculty of Mechanical Engineering, Department of Applied Cybernetics Studentská Studentská 2, 461 17, Liberec, Czech Republic Email:
[email protected] **Aalto University, University, Engineering design and Production P.O.Box 14400, 00076 Espoo, Finland Email:
[email protected]
This paper presents a mathematical model of the proportional valve main stage flow characteristics. Model is verified with static measurements. Valve spool dynamics are modelled but not verified. Model uses variable sonic conductance and critical pressure ratio that are based on measurements. Valve control system and pilot stage are not included this model. This means that the model reacts faster than the real valve does. Model of the valve control electronics and pilot stage will be presented in our next paper. Keywords: proportional pneumatic valve; mathematic model; pressure ratio; sonic conductance
1 INTRODUCTION
The pneumatic proportional valve presented in this paper is used for fast and precise pressure control. Schema of the proportional pressure control valve is presented in Figure 1. In our case this proportional valve supplied pressurized air into an pneumatic muscle which was used in our earlier measuring arrangement [1]. The position of pneumatic muscle can be controlled with air pressure if the load is known [2]. For controlling a pneumatic muscle movement, it is necessary to know the mathematic model of both the muscle and the valve. This paper is about the main stage part of the Festo VPPM-6L-L-1-G18-0L6H valve model (part 5 in Figure 1). Previous research [3, 4] of the authors was used in building this model. In earlier research sonic conductance C and and the critical pressure ratio b were measured as the functions of the valve poppet position. Integral part of our previous research was to validate the model with static measurements. This was done by pressurizing a relatively large constant volume through the valve [3].For model presented here, the mechanism of the valve main stage is measured and modelled. However this part of the model can’t be verified by the earlier measurement because it is too slow to show any dynamic behaviour. No dynamic model verification has been done because control system and pilot stage of the valve haven’t been modelled yet and they have a major effect to valve response time.
Figure 1: Schema of the valve. 1 – primary control processor,
2 - pilot stage control processor, 3 -pilot valves, 4 - pilot stage surface, 5 – main stage spool, 6 – pilot pressure sensor, 7 – output pressure sensor.
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Journal of applied science in the thermodynamics and fluid mechanics Vol. 1, No. 1/2012, ISSN 1802-9388 2 MODELLING OF PROPORTIONAL PNEUMATIC VALVE
Creating mathematic model of pneumatic proportional split in two parts. First of them describes the air flow which goes through the valve. The second involves equations of motion for the poppet of the valve. Equations are different for filling and emptying the air through valve output port. Poppet mechanism is shown in Figure 2. 2.1 Description of air flow through the valve
Air flow was described by using ISO 6358 standard [5]. Sonic conductance C and the critical pressure ratio b were measured as functions of valve poppet position and then included in modified flow equations (1) and (2) q *m ( x ) = C ( x) p1 ρ 0
q m ( x) = C ( x) p1 ρ 0
where
ρ 0
p 1 p 2 T 0 T 1 C(x) b(x)
T 0
for
T 1
p2
≤
p1
p2 − b( x) T 0 p 1− 1 1 − b( x) T 1
checked flow
b
(1)
,
2
for
p2 p1
>
b
subsonic flow
,
(2)
- means density, - means upstream pressure, - means downstream pressure, - means upstream temperature, - means downstream temperature, - means sonic conductance in dependence on the stroke of poppet, - means critical pressure ratio in dependence on the stroke of poppet.
The sonic conductance C and critical pressure ratio b is determined as a function of poppet position in our previous research. Parameters C and b are determined by filling and emptying a constant volume tank with different openings of the poppet [6]. Obtained values of sonic conductance and critical pressure ratio were fitted by using polynomial functions of 7 th and 4th degree, equations (3) and (4). Note that the variable x is the stroke of poppet and has a range of [mm]. This was experimentally determined by dimension analysis on the actual valve. b( x) = p1 x 4 C ( x) = k 1 x 7
6
3
+ p 2 x + p3 x 5
4
2
+ p 4 x + 3
p5 2
(3)
,
+ k 2 x + k 3 x + k 4 x + k 5 x + k 6 x + k 7 x + k 8
,
where p 1,..,p n - means constant coefficients for critical pressure ratio polynomial approximation, k 1,..,k n - means constant coefficients for sonic conductance polynomial approximation, x - means stroke of poppet where [mm], - means sonic conductance dependence on poppet position C(x) b(x) - means critical pressure ratio dependence on poppet position
2
(4)
Journal of applied science in the thermodynamics and fluid mechanics Vol. 1, No. 1/2012, ISSN 1802-9388 2.2 Description of valve mechanics
This section describes the kinematic equations of the valve main stage mechanism. Valve construction determined that different equations of motion had to be used for pressurizing and depressurizing the output port. Equation (5) describes the parts of the valve that move during the filling, when the pressurized air goes from pressure port 1 to output port 2.Equation (7) describes the case of emptying, when the air goes from output port 2 to the exhauster port 3. Equations (6) and (8) describe the force generated by the pilot stage of the valve. Pilot stage is separated from the main stage with a metal disc and rubber membrane.
Output port 2 Pressure port 1
Exhauster port 3
Figure 2: Schema of the valve for the case of filling on the left and for the emptying of working space on the right
Integral part of creating mathematical model of proportional pneumatic valve was measuring the stiffness of the spring, properties of membrane and the weight of all parts. Equation of motion for filling the working space:
&&(t ) + b x& (t ) + k 1 (h1 + x(t ) ) + k 2 (h2 m p x F p (t ) = S ⋅ p p (t )
where m p b x k 1 k 2 h 1 h 2 p p S
+ x (t ) ) =
,
- means weight for filling working place - means damping constant, - means stroke of poppet where [mm], - means upper spring constant, - means bottom spring constant - means upper spring displacement in rest position, - means bottom spring displacement in rest position, - means controlling pressure for filling working place, - means area of metal plate.
3
F p (t )
,
(5) (6)
Journal of applied science in the thermodynamics and fluid mechanics Vol. 1, No. 1/2012, ISSN 1802-9388
The value of damping constant has been solved by measuring mechanical properties of membrane. The measured hysteresis data has been linearized and membrane stiffness is derived from this. Equation of motion for emptying the working space:
&&(t ) + b x& (t ) + k 1 (h1 mo x
+ x(t )) =
F o (t ) = S ⋅ p o (t )
where m p p o b x k 1 h 1 S
F o (t )
(7)
,
(8)
,
- means weight for emptying working place, - means controlling pressure for emptying working place, - means damping constant, - means stroke of poppet where [mm], - means upper spring constant, - means upper spring displacement in rest position, - means area of metal plate.
3 RESULTS
In the subsection 2.1 we discussed about the process of determining sonic conductance C(x) and critical pressure ratio b(x) as functions of poppet position. For the case where pressurized air is directed to output port, we were able to make an approximation from measured data by using polynomial functions. This was done in previous research done by the authors [3]. The graphs with measured data and our approximations are shown in figs.3 and 4. -8
2
x 10
1.8 )] a P * s( 3 / m [ C e c n at c u d n o c ci n o S
1.6 1.4 1.2 1 Measured and counted data Data approximation by polynomial 7th degree
0.8 0.6 0.4 0
0.1
0.2
0.3
0.4 0.5 0.6 Stroke of poppet x [x10 3 m]
0.7
0.8
0.9
1
Figure 3: Dependence of sonic conductance on the stroke of poppet 0.75 0.7 ] -[ b oi t ar er u s s er p l a ci t ri C
0.65
Measured and counted data Data approximation by polynomial 4th degree
0.6 0.55 0.5 0.45 0.4 0.35 0
0.1
0.2
0.3
0.4 0.5 0.6 Stroke of poppet x [x10 3 m]
0.7
0.8
0.9
1
Figure 4: Dependence of critical pressure ratio on the stroke of poppet
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Journal of applied science in the thermodynamics and fluid mechanics Vol. 1, No. 1/2012, ISSN 1802-9388
From the Figures 3 and 4 we can see that the sonic conductance C(x) and critical pressure ratio b(x) aren´t constant values but nonlinear functions of poppet position. Same conclusion can be found in literature with similar research [4] [8]. When the output port is opened to exhaust channel the poppet moves to same position every time and opens a constant size orifice. This means that the sonic conductance C and the critical pressure ratio b are constant. These constants were measured by emptying a known amount of pressurized air through the valve. The constant coefficient for polynomial function C(x) and b(x) for the case filling of working space are presented in Table 1. Table 1: Constant coefficients for polynomial functions C(x) and b(x) for the case of filling the working space
Critical pressure ratio b(x) Sonic conductance C(x) p 1 = 3,16180 k 1 = -1.1858e-6 p 2 = -8,31770 k 2 = 4.5064e-6 p 3 = 7,68470 k 3 = -6.9209e-6 p 4 = -2,73840 k 4 = 5.4385e-6 p 5 = 0,71748 k 5 = -2.2269e-6 k 6 = 3.7653e-7 k 7 = 2.1902e-8 k 8 = 5.0034e-9 Based on the knowledge gained from our previous research, it was possible to measure mechanical properties of the valve and to create the motion equations (5) to (8). Using these equations it was possible to construct a simulation model of the system in Simulink. Model for filling the working space is shown in Figure 5. It consists of valve main spool model (left side of the model) and model of the tank (right side of the model). Tank model is isentropic because during the filling gas temperature inside the 0.2 m3 tank rises approximately 18 °C. S -CT1 Step
-C-
kapa R0 sqrt
k -C-
k2 -Ch2 -C-
K-
T0
-C-
P(u) O(P) = 7 C(x)
q -Ck1 -Ch1 -C-
-C-
V2 -Kro
-Cmass
1/s
1/s
K-
x1
x2
ro 1
1/s
p1 -C-
x stroke of poppet P1 P(u) O(P) = 4 b(x)
P2 - pressure inside the tank
p20
MATLAB Function
measured_data
k2*(h2-x)
Figure 5: Mathematical model of the proportional valve joined with mathematical model of tank
5
Journal of applied science in the thermodynamics and fluid mechanics Vol. 1, No. 1/2012, ISSN 1802-9388 5
5.5
x 10
5 4.5 4 ] 3.5 a P[ 3 er us se 2.5 r P 2
Required value Mathematical model Measured data
1.5 1 0.5 0
20
40
60 Time [s]
80
100
120
Figure 6: Dependence internal pressure on time – combination mathematical model of valve join with mathematical
model tank
5 CONCLUSIONS
Comparison of simulation and measurement shows us that the model works well in slow processes. Measured data does not reach the same pressure as fast as model presumes. Main reason for this is probably the fact that the tank model was isentropic and in reality heat is transferred out of the gas to the steel tank. This also drops the pressure inside the tank. More precise model of the tank would be polytrophic. Using non constant sonic conductance and critical pressure ratio makes the model much more precise when the valve operates with different pilot pressures. This part of the model will be of its full use only after the rest of the valve model is implemented. In future, valve pilot stage will also be measured and modelled and control parameters evaluated. This full model of the valve will be used together with the model of the muscle to simulate muscle position system. Full valve model will also give us information about dynamic limitations of different components of the VMMP valve. This information is important if the valve or parts of the valve are used for very fast pressure control, for example in an active vibration damping systems. REFERENCES
[1] VARGA, Z. - MOU ČKA, M.: Mechanic of Pneumatic Artificial Muscle.Journal of Applied Science in the Thermodynamics and Fluid Mechanics.January 2009, vol. 3, No 2/2009, pp. 1–6, ISSN 1802-9388. [2] DAERDEN, F. - LEFEBER, D.: Pneumatic Artificial Muscles: actuators forrobotics and automation. European Journal of Mechanical and Environmental Engineering.Citeseer, 2002,issue 1,vol. 47,pp. 11– 21, ISSN 00353612. [3] VARGA, Z. - KESKI-HONKOLA, P.: Determination of Flow Rate Characteristic for Pneumatic Valves.Experimental fluid mechanics 2011.Jičín (Czech Republic), 22nd - 25th November 2011, vol. 1, Conference proceedings are listed in ISI-Web of Knowledge,ISBN 978-80-7372-784-0. [4] BEATER, P.: Pneumatic Drivers: System Design, Modelling and Control. Springer, Berlin, Germany, 2006.ISBN-10 3-540-69470-6. [5] ISO 6358: Pneumatic fluid power – Components using compressible fluids – Determination of flow-rate characteristics, 1989. [6] GIORGI, d. R. - KOBBI, N. - SESMAT, S., Bideauxthermal, E.: Model of a Tank for Simulation and Mass Flow Rate Characterization Purposes Toyama.Proceedings of the 7th JFPS International Symposium on Fluid Power.September 15-18, 2008, pp.225-230. [7] PRABEL, R. - SCHINDELE, D. - ASCHEMANN, H.: Nonlinear Control of an Electro-Pneumatic Clutch for Truck Applications using Extended Linearisation Techniques.The Twelfth Scandinavian International Conference on Fluid Power. Tampere, Finlad, May 18 – 20, 2011, vol. 1, pp. 125-136 , ISBN 978-952-152518-6. [8] NEVRLÝ, J.: Introduction to the modelling pneumatic systems (Original Czech title - Úvod do modelování pneumatických systémů). Česká strojnická společnost, Ústř ední odborná sekce Hydraulika a pneumatika. Praha, 2003,1. Vydání – náklad 40 výtisků, pp. 79, ISBN 80-02-01549-5.
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