COMPARISON OF TWO METHODS TO SOLVE PRESSURES IN SMALL VOLUMES IN REAL-TIME SIMULATION OF A MOBILE DIRECTIONAL CONTROL VALVE

Transcription

1 COMPARISON OF TWO METHODS TO SOLVE PRESSURES IN SMALL VOLUMES IN REAL-TIME SIMULATION OF A MOBILE DIRECTIONAL CONTROL VALVE Rafael ÅMAN*, Heikki HANDROOS*, Pasi KORKEALAAKSO** and Asko ROUVINEN** * Laboratory of Intelligent Machines Lappeenranta University of Technology PL 20, Lappeenranta, Finland ( rafael.aman@lut.fi) ** MeVEA Ltd. Laserkatu 6, Lappeenranta, Finland ( info@mevea.com ABSTRACT In fluid power systems, especially in many types of valves, there exists very small volumes which are particularly problematic in the case of dynamic analysis. Small volume with respect to the so-called normal sized orifice, formulate the system of equations which become mathematically stiff. This is a common source of numerical problems. To solve the stiffness problem, the present paper employs two alternative methods for conventional integration. Both implementations have in common that direct integration of pressures in small volumes is avoided and they are freely applicable regardless of used integration routine. Conventional numerical simulation with sufficiently small time increment is used as reference response. The valve studied is commonly available, proportional, load-sensing directional valve designed for mobile hydraulic systems, containing main spool and load-holding poppet with pilot spool. The present paper describes the methods in general level using a real-time simulation application of a relatively complex valve as a case study. Results are compared to those computed using conventional method. KEY WORDS Real-time, simulation, small volume, pseudo-dynamic, singular perturbation NOMENCLATURE Oil bulk modulus [Pa] Cv : Volume flow coefficient of the main spool [m 3 / s Pa] K: Volume flow coefficient of the load-holding poppet valve [m 3 / s Pa] k: Volume flow coefficient of the main spool depending on the input signal [m 3 / s Pa] L: Cylinder stroke [m] m: Mass connected to the cylinder [kg] p : Pressure [Pa] V: Volume [m 3 ] x : Cylinder position [m] x 0, x 1 : Threshold value in step function [Pa] y 0, y 1 : Threshold value in step function [m 3 / s Pa] : Variable in step function [-] p : Pressure drop [Pa] t : Time step length [s] Q : Compressional flow [m 3 / s] Q : Volume flow [m 3 / s] : Scaling factor : Typical relative volume or density change : Pressure in small volume [Pa] Subscripts A / B: Transmission lines A and B cyl : Cylinder inner : Inner loop of pseudo-dynamic model Leak : Leakage

2 Lock : Lock load-holding poppet min : Minimum value Ref : Reference model SPT/Pseu: SPT model and the outer loop of pseudo-dynamic model Pseudo : Variable in pseudo-dynamic solving method Tol : Pseudo-loop convergence criteria 1 : Refers to A1 2 : Refers to B1 A1 / B1: Variable related to the load-holding poppet A11 / B11: Variable related to the main spool : Variable related to the small volume between the load-holding poppet and the main spool INTRODUCTION In fluid power systems, especially in many types of valves, there exists very small volumes which are particularly problematic in the case of dynamic analysis. This is due the fact that from mathematical point of view small volumes in connection with larger, so-called normal volumes, formulate the system of equations which become mathematically stiff. Consequently, the system stiffness approaches infinity as the fluid volume approaches zero since it is related to fluid compressibility. For this reason, during dynamic analysis of the fluid power system the time integration of pressures in small fluid volumes is a common source of numerical problems. The presence of relatively small time constants makes numerical integration of the ordinary differential equation (ODE) system difficult. Conventional explicit integration methods become numerically unstable unless a very small time increment is used. This leads into excessively long computational times. For stiff systems implicit, especially L-stable, ODE algorithms are recommended. Their drawback is that they have to solve a set of nonlinear equations at each time step, which reduces the computational efficiency of the method. To solve the problem caused by small fluid volumes, the present paper employs two alternative methods for conventional integration. Both implementations have in common that direct integration of pressures in small volumes is avoided. Singular Perturbation Theory (SPT) is used in model reduction where the dynamics equations are pre-processed such that they can be integrated using routines for non-stiff systems. The Singular Perturbation Theory is originally introduced by Fenichel [2] and applied into fluid power simulation by Scheidl et al. [7] The other implementation is the pseudo-dynamic solving method that solves the pressure as a steady-state pressure at each time step. The solution is obtained by numerical integration and iterative solution of the steady-states of the pressures after transient state. To reach the steady-state, artificial volume for the stiff part of the system is used in a cascade integration loop. Pseudo-dynamic solving method is proposed by Åman [8]. The rule of thumb for using both above-mentioned solving methods is that the nominal frequency (time constant), created by the small volume, is not significant in comparison with the dynamics of the whole system. The hydraulic capacitance V/ of the parts of the circuit of which stiffness is reduced should be at least ten times smaller than that of those parts whose pressures are integrated conventionally. Both methods have the advantage of easy programming implementation and they are freely applicable regardless of used integration routine. Conventional numerical simulation with sufficiently small time increment is used as reference response when evaluating the accuracy of these two implementations. The valve studied is commonly available, proportional, load-sensing directional valve designed for mobile hydraulic systems, containing main spool and load-holding poppet with pilot spool. Between the load-holding poppet and the main spool there exists a very small volume compared to the other volumes in the valve structure. In order to simulate the dynamic behaviour of the valve in real-time, both the pseudo-dynamic solving method and singular perturbation technique are applied. The present paper describes the methods in general level using a real-time simulation application of a relatively complex valve as a case study. Results are compared to those computed using conventional method with a small time increment. MODELLING OF THE COMPLEX MOBILE DIRECTIONAL CONTROL VALVE The valve studied is commonly available, proportional, load-sensing directional valve designed for mobile hydraulic systems, containing main spool and load-holding poppet with pilot spool [5]. Between the load-holding poppet (Item no. 7 in Fig. 1) and the main spool (Item no. 2 in Fig. 1) there exists a very small volume (V A1 / V B1 ) compared to the other volumes in the valve structure. In order to simulate the dynamic behaviour of the valve in real-time, both the pseudo-dynamic solving method and singular perturbation technique are applied. Items no. 1 and 12 in Fig. 1 are not modelled. Modelling of the fluid power circuit shown in Fig. 1 is started by implementing the required differential-algebraic equations for all the volumes, respectively. As an example, only equations related to one fluid power transmission line (line A) are presented. The other is then derived similarly but naturally oppose direction.

3 , where = p Lock x 0 x 1 x 0 p Lock = A1 + p 1 - p A1 - p ref p A1 =0 ; U 1 < -1 x 10-4, p A1 = p 1 ; U 1-1 x 10-4 And the volume flow through the load-holding poppet is solved from Eq. (4) Q A1 =K A1 A1 p A1 (4) The opening of the main spool of the directional control valve is solved using Eq. (5). Figure 1 Modelled fluid power circuit [5] The volume through the main spool (Item no. 2) is described using Eq. (1a 1c) Q A11 =-U 1 Cv p 5 ; U 1 < - 1 x10-4 (1a) Q A11 = Cv Leak p 5 ; -1x10-4 < U 1 <1 x10-4 (1b) Q A11 =U 1 Cv p 0 ; U 1 > 1 x10-4 (1c) It is assumed that the steady-state opening of the lock valve orifice is a third order polynomial of the pressure drop and the valve dynamics can be described by a first order differential equation. Thus, the volume flow coefficient of the load-holding poppet is solved from the Eq. (2) K A1 = K A1, where y according to Eq. (3a 3c) y= y 0 ;p Lock x 1 (2) (3a) y= y 0 + (y 1 - y 0 ) 2 (3-2 ) ; 0 p Lock x 1 (3b) y= y 1 ; p Lock > x 1 (3c) U 1 = U 1ref U 1 (5) The pressure build up in each volume can be described by the continuity equation of Merritt, Eq. (6) [4]. p = V Q (6) The compressional flow is described by using Eq. (7). Q = Q in Q out + V (7), where is externally supplied volume flow into and out of the volume (e.g. pump or actuator flow). The flows in and out of the volume can be described by using Eq. (8). Q=fp) (8) SOLUTION OF PRESSURES IN SMALL VOLUMES IN DYNAMIC SIMULATION To solve the problem caused by small fluid volumes, the present paper employs two alternative methods for conventional integration. Both implementations have in common that direct integration of pressures in small volumes is avoided. Instead the degrees of simulation models are reduced using two different methods. These methods are the pseudo-dynamic solving method [8] and the Singular Perturbation Theory [7].

4 Degree Reduction by Pseudo-dynamic Solving Method The pseudo-dynamic solver is based on the basic assumption that if the volume in the system to be described is small enough, the pressure can be expressed by a steady-state pressure, as explained in [1]. The method has two key ideas. Firstly, the nominal frequency (time constant), which is created by the small volume, is not significant in comparison with the dynamics of the whole system. Secondly, instead of integrating the equations for pressure gradients in such volumes, their pressures are solved as steady-state pressures by using a pseudo-dynamic solver. The solver integrates the pressures in a separate integration loop while the volumes have pseudo-values providing a smooth and fast solution. The key idea in the proposed method is to find steady-state solutions for the pressures in small volumes at each integration step, while the pressures in larger volumes as well as the other differential equations are integrated normally [8]. In other words, the pseudo-dynamic solver consists of two cascade integration loops, the outer and the inner loop. The outer loop consists of the ODE solvers integrating all other variables except those which are related to small volumes. Inside the outer loop, there is a separate ODE solver (inner loop) encoded to produce steady-state solutions for pressures in small volumes. The inner loop is executed by iterative means, i.e. it is controlled using the criterion for convergence, it has its own time space, the outer loop is paused during the inner loop run and only the last value of the integrated variable is returned to the outer loop. As the convergence criterion, the first derivative of pressure is used. With this predetermined condition, can be ensured that the attained solution has reached the steady-state. The influence of convergence criterion into the simulation results is studied in reference [9]. Simulation is started by defining that the pressures A1 and B1 are solver in their own inner loops of the pseudo-dynamic solver. Initial parameters are substituted into differential and algebraic equations and pseudo-loop is started. Integration in inner loop is carried out until the defined stopping criterion is reached. Note that the outer loop is paused during the inner loop run and these loops have their own independent time spaces. After inner loops are executed the pressures A1 and B1 are directed to outer loop as initial parameters. Integration in outer loop at first time step is carried out according to initial parameters. Integrated values are updated into differential and algebraic equations as new initial parameters. Results are stored and handled in post-processing after outer loop integration time has run out. Pseudo-dynamic Solution of Steady-States of Fluid Power Circuits The idea behind this algorithm is to consider each pressure node as finite volume. By doing so, each node represents a volume in which pressure builds up or decreases dependent on the compressional flow of the node, i.e. the sum of total flow to and from the node. The three equations, Eq. (6), (7) and (8), make up the system formulation, which requires integration routine to update the pressures. For this a standard fixed step 4 th order Runge-Kutta implementation is used, where the time steps in the solver are set sufficiently low to the account for the pseudo-dynamics in the system. This, however, also means, as oppose to the static solver, that no update algorithm is used, as the pressures are directly updated by the integration routine. For the static solver the update law also had a filtering effect. For the pseudo-dynamic solver this effect is instead replaced with pressure build up in the nodes, but to make the routine numerically more robust it may be also beneficial to add some pseudo-dynamics to the components with discrete states [6]. Degree Reduction by Singular Perturbation Theory A system is described by a relation: F(u, ) = 0, where u is its state from a vector or function space, a small non-dimensional parameter (0 < 0 ; 0 1), and F some map. The system is called regularly perturbed in if [7]: lim u( ) =u 0, 0 F(u 0 ) = 0. Otherwise it is called singularly perturbed. u 0 is the solution of the so called reduced problem which is derived from the full or perturbed problem when the " is set to zero prior to solving the equation. u( ) is the solution of the full equation for different values of. In case of more than one solution regularity means that all solutions of the perturbed problem converge to a solution of the reduced problem [7] Singular Perturbation Theory in Modelling Complex Mobile Directional Valve The basic idea is to use the steady-state solution of two orifices in series connection. It is called as singularly perturbed i.e. its degree has been reduced so that the integration of the pressures A1 and B1 in small volumes can be avoided [3]. The drawback of this method is that

5 the pressures of the small volumes are needed in dynamic equation for the ambient volumes. It must then be reproduced by a steady-state equation from the surrounding pressures, orifice flow and cross-section area of orifice which leads into a term in which square of flow is divided by square of cross-section area of the orifice. This again leads into numerical problems. As an example, only equations related to one fluid power transmission line (line A) are presented. The other is then derived similarly but naturally oppose direction. Let us examine the transmission line A. The pressure build-up in small volume can be expressed as follows: A1 = V A1 ( k A11 p 0 A1 K A1 A1 -p 1 ) p 1 = V A1 ( K A1 A1 -p 1 Q cyla1 ) Let us use the following expressions: and A1 = p 1, = p 1 = A1, (9) (10) (11) (12) where is the scaling factor, the effective bulk modulus of the system and a typical relative volume or density change. For typical hydraulic fluids and pressures its magnitude is O(10-2 ) [7]. So, the set of equations can be written as follows: and V A1 A1 = k A11 p 0 K A1 A1 (13) V A1 = K A1 A1 Q cyla1 (14) V A1 = k A11 p 0 K A1 A1 V A1 = K A1 A1 Q cyla1 (16) When the relation between the pressure and modulus of compressibility approaches zero, the latter term in Equation (16) takes the form: lim 0 => k A11 p 0 K A1 A1 = 0 k A11 2 ( p 0 ) K A1 2 ( A1 ) = 0. Then by taking the square of the both sides and solving for A1, we finally bring Eq. (16) into the form: A1 = k A11 2 p 0 + K 2 A1 p 1 k 2 2 A11 + K A1 V A1 = K A1 A1 Q cyla1 (17) The volume flows Q A11 and Q A1 can be expressed in following form, Eq. (18) k A 11 K A1 Q A11 = Q A1 = k 2 2 A11 + K A1 p (18) The minimum value, k min, for volume flow coefficients is defined to avoid the situation that during calculation the denominator in Eq. (18) would become zero. This would lead into immediate crash of the simulation run. K A1 = max( k min,, K A1 ) Because the volume changes direction depending on the pressure drop, the following conditional statement of the directional spool position, Eq. (19) is needed. To avoid numerical problems caused by the pressure drop approaching zero, the absolute value of the pressure drop and the step function must be used. From Equations (2.3) and (2.4) we get A1 = (15) Now, by keeping Eq. (14) as it is and substituting Eq. (15) into Eq. (13), the model can be written as: if U 1 < -1 x 10-4 k A11 = max( k min, ( -U 1 Cv )); p = p 5 - p 1 elseif U 1 > 1 x 10-4 k A11 = max( k min, ( U 1 Cv )); p = p 0 - p 1 else k A11 = Cv Leak ; p = p 5 - p 1 (19)

6 Finally, the steady-state equation, Eq. (20) for the pressure in small volume can be written in simpler form: A1 = Q 2 A1 2 K +p 1 A1 (20) NUMERICAL EXAMPLE This study was started by modeling the fluid power circuit using three different approaches. The conventional 4 th order Runge-Kutta method was used as reference response and it was implemented in Simulink. The pseudo-dynamic and SPT methods were implemented in MATLAB M-Files. To simplify the implementations of the alternative solving methods, the Euler method is selected to be used for integration of the accessory calculations and the 4 th order Runge-Kutta method is only employed in the inner loop of the pseudo-dynamic solver. In all simulation runs step function is involved in calculations of the volume flows to ensure the smooth approaches and crossings of the zero pressure drop. Its influence on results has been minimized by setting threshold pressure as low as model still stand stable (threshold pressure 1 x 10 5 Pa). Without use of step-function simulation runs failed. To the hydraulic cylinder is connected the payload of kg. Due to the external dynamics of the system it is difficult to adjust the initial values such that the inner pressure in small volumes remains stable during simulation run. That is why in the pressure responses of A1 and B1 of the SPT model there appears vibrations while the directional valve is closed (U 1 =0). To ease this phenomena the boundary values of the step function has been increased to 5x10 5 Pa. This makes the calculation of volume flows smoother near zero pressure surroundings without any degradation of model accuracy or extension in calculation time. different models. Naturally, the goodness criterion for employing different solving methods was at least reasonable computational time. The simulated work cycle of the fluid power circuit is the following. First the cylinder is driven to (+) direction (out), then the movement is stopped and eventually the cylinder is driven to (-) direction. In Fig. 2 this is presented in the form of directional valve control reference signal. Also the realized valve spool opening is illustrated. Figure 2 Control reference signal and the feedback from valve spool Results The following results are achieved using three alternative solving methods for the pressures in small volumes. First, the response of the pressures in the small volumes A1 and B1 are studied. The responses are illustrated in Fig. 3 and 4. Reference response As a reference model the fluid power circuit is modelled as explained in Section 2. This carried out in Simulink which enables easy employment of different integrators. The 4 th order Runge-Kutta method is selected to be used for solving the equations. Time step length of t = 5 x 10-6 s was the longest possible for the use without notable changes in responses i.e. model stability. Used initial values are represented in Table 1. This method for finding the reference response is commonly acknowledged and can be stated as the most accurate one when time step length is set sufficiently short. The drawback for use of this conventional method is the computational speed. Computational times are not investigated within this study but the accuracy of Figure 3 Internal pressures A1 and B1 of directional control valve.

7 It can be stated that both proposed solving methods realize the piston position and piston velocity in acceptable accuracy. The responses of pseudo-dynamic solver show identical behaviour with the reference responses. In the response of piston position of SPT model there exist deviation within 1 mm tolerance. The volume flow coefficients of the load-holding poppet K A1 and K B1 are illustrated in Fig. 7 and 8. Figure 4 More focused view of Figure 3. From Fig. 3 and 4 can be seen that responses achieved using different solving methods correspond to the reference response mainly well. Only in switching points of the control reference there exist deviations. Pseudo-dynamic solving seems to be more accurate even there exist more oscillations in switching point. The cylinder piston position x and piston velocity is illustrated in Fig. 5 and 6. Figure 7 Volume flow coefficients of the load-holding poppet. Figure 5 Cylinder piston position and velocity. Figure 8 More focused view of Figure 7. Figure 6 More focused view of Figure 5. It can be seen from Fig. 7 and 8 that the differences between different solving methods come up in responses of the volume flow coefficients. This due to the fact that the load-holding poppet with the pilot-operated lock-up function represents the fastest dynamics in the system after the small volume in which the transients are very fast. The responses of pseudo-dynamic method follow the reference responses with small oscillations and deviation. But the SPT model suffers from numerical noise when the value of volume flow coefficient is

8 lower than 3 x 10-7 m 3 / s Pa. The steady-state deviation while the valve is closed is due the limitation of the minimum value of the volume flow coefficient to avoid numerical problems in SPT model. CONCLUSIONS Two alternative solving methods for pressures in small volumes were applied to fluid power circuit composing of mobile directional control valve and actuator. The pseudo-dynamic solving method was stated to meet the reference response more accurate. The reference response was achieved using explicit 4 th order Runge-Kutta integration routine and sufficiently short time increment. The reduced model by Singular Perturbation Theory provide less oscillation but more deviation from the reference responses than appear the pseudo-dynamic model. The pseudo-dynamic model provides better integrator stability since longer integration time steps compared to the conventional method can be used. It was then shown that using both of the proposed solving methods numerical problems apparent in calculations by conventional methods can be avoided. And both are suitable for the real-time simulation of complex mobile directional valve. Table 1 Initial values used in system simulation t Ref = 5 x 10-6 s t SPT/Pseudo = 1 x 10-5 t inner = 5 x 10-5 s s V A1 = 5 x 10-6 m 3 V B1 = 5 x 10-6 m 3 V pseudo = 1 x 10-3 m 3 V A1 = 4.4 x 10-3 m 3 V B1 = 16.2 x 10-3 m 3 L = 0.78 m D 1 = 0.2 m D 2 = 0.11 m x = 0.1 m p 0 = 290 x 10 5 Pa p 1 = 0 Pa p 2 = 0 Pa p ref = 5 x 10 5 Pa p Tol = 1 x 10 3 Pa p 5 = 0 Pa Cv = x 10-7 m 3 k min =1 x 10-7 m 3 Cv Leak = 1 x 10-7 m3 s Pa s Pa s Pa m = kg x 0 = 0 y 0 = 0 b=500 Ns x 1 = 1 x 10 5 Pa y 1 =10 x Cv m simtime = 0.75 s = 1.5 x 10 9 Pa REFERENCES 1. Ellman, A. Proposals for Utilizing Theoretical and Experimental Methods in Modelling Two-Way Cartridge Valve Circuits. PhD thesis, Tampere University of Technology, Fenichel, N. Geometric Singular Perturbation Theory for Ordinary Differential Equations. Journal of Differential Equations, Vol. 31, pp.53-98, Handroos, H. and Vilenius, M. Flexible Semi-Empirical Models for Hydraulic Flow Control Valves. Journal of Mechanical Design, Vol. 113, pp , Merritt, H. Hydraulic Control Systems. John Wiley & Sons, Parker Hannifin. Catalogueue HY /UK. PDF 07/05, Pedersen, H. Automated Hydraulic System Design and Power Management in Mobile Hydraulic Applications. PhD thesis, Aalborg University, Scheidl, R., Manhartsgruber, B., and Kogler, H. Model Reduction in Hydraulics by Singular Perturbation Techniques. In 2 nd Int. Conf. on Computational Methods in Fluid Power. Fluid Power Net Publication, pp.1-6, Åman, R. Methods and Models for Accelerating Dynamic Simulation Fluid Power Circuits. PhD thesis, Lappeenranta University of Technology, Åman, R. and Handroos, H. Optimization of Parameters of Pseudo-Dynamic Solver for the Real-Time Simulation of Fluid Power Circuits. In 7 th International Fluid Power Conference Aachen, (Aachen, Germany, March 2010), Vol. 1, pp , 2010.