液壓式組合建筑機械液壓部分設計【液壓式鋼筋彎曲切斷套絲多用機液壓系統(tǒng)設計】,液壓式鋼筋彎曲切斷套絲多用機液壓系統(tǒng)設計,液壓式組合建筑機械液壓部分設計【液壓式鋼筋彎曲切斷套絲多用機液壓系統(tǒng)設計】,液壓式,組合,建筑機械,液壓,部分,部份,設計,鋼筋,彎曲,曲折,切斷,割斷,多用 Nonlinear Dyn (2007) 47:219233 DOI 10.1007/s11071-006-9069-1 ORIGINAL ARTICLE Deregularization of a smooth system example hydraulics Friedrich Pfeiffer Received: 29 August 2005 / Accepted: 28 October 2005 / Published online: 1 November 2006 C Springer Science + Business Media B.V. 2006 Abstract Many technical systems include steep char- acteristics for force laws, which as a rule lead to stiff differential equations and large computing times. For the dynamical performance such steep characteristics are very near to laws with set-valued properties and might therefore be replaced by set-valued force laws. This is true for multibody dynamics including unilat- eral contacts, and it is in an approximate way true for fluid mechanical systems like hydraulics. In the follow- ing we present a new modeling scheme for hydraulic systems, which establishes the hydraulic equations of motion in the form of multibody system eqations with bilateral and unilateral constraints, and which is able to reduce computing times by three to four order of magnitudes. A large industrial example illustrates the excellent performance of the new theory. Keywords Deregularization of smooth systems Complementarities System models Hydraulics 1. Introduction Models approximate the physical or technical reality. They are more or less detailed, but all models include F. Pfeiffer Institute of Applied Mechanics, Technical University Munich, Boltzmannstrasse 15, D-85748 Garching, Germany e-mail: pfeifferamm.mw.tum.de some degree od approximation. The real world can- not be modeled in a perfect way. Therefore, in estab- lishing models we should keep in mind, that technical systems and especially technical mechanics are not de- ductive systems. This is true for classical analytical mechanics, which applies only partly to problems of some realistic significance. A further aspect concerns the goals connected with models. Do we want to map reality as perfect as possible, or do we want to consider certain parameter influences? Both objectives include difficult problems, because a model is by itself dumb. We have to make a model intelligent by introducing into it our knowledge of the problem under consider- ation, the physical and technical properties as good as we understand them, the parameter influences as good as we might expect them, the neglections as good as we can estimate them. This all is more an art than a science, but it is so essential, that no good model can be established without a preliminary phase of physical, of mechanical argueing leading to a sound imagination and a sound picture of the real world problem to be modeled. Simple models for the evaluation of tendencies with respect to the performance of a system include some re- ally difficult problems, because such simple models not only afford a perfect view of the overall system but also a perfect idea of what is important and of what is not important. In addition simple system modeling might be faulty, because the interference of many degrees of freedom might result in a completely different dynam- ical behavior as compared with simple considerations. Springer 220 Nonlinear Dyn (2007) 47:219233 A recently published finding with respect to the well known friction problem of self-excited oscillations con- nected with falling characteristics illustrates especially this danger of coming out with incorrect results. Includ- ing more than one degree of freedom it can be shown, that self-excited oscillations may happen also for an increasing and not for a falling characteristic 2. After having dealt with this model finding phase, which by the way is usually very much underestimated, we must find a decision on the mathematical and es- pecially on the numerical tools we want to apply. For one and the same problem we have a variety of possi- ble mathematical descriptions with again a certain va- riety of numerical algorithms. Some usual criteria for this choice are physical-mathematical correspondence, structural features of the resulting equations, transfor- mation capabilities with the goal of analytical or partly analytical solutions, convergence of the solutions, sta- bility of the numerical algorithms and finally the repre- sentation of the results allowing clear interpretations. Coming back to our problem of descibing systems with force laws including very steep characteristics we may apply two approaches, a smooth and a non- smooth one. In practical engineering characteristics with steep features occur in connection with contact problems, with fluidmechanical problems of hydraulic equipment, with cavitation or with electronic switch- ing problems, to name only a few examples. From the mathematical and from the resulting numerical stand- point of view such characteristics produce either stiff differential equations or they require a complementar- ity formulation. The decision which way to go depends mainly on the computation time. Considering contacts and related problems we might discretize a contact by evaluating the local stiffness properties of the contact, which allows the derivation of a force law. As contact stiffnesses are usually very large, we come out with stiff differential equations. The second way consists in assuming the local contact area as rigid, which does not imply that the whole body must be assumed rigid, and to formulate the contact proper- ties by complementarities 7, 8. In hydraulic networks we find such a complementarity behavior in connection with check valves, with servo valves and with cavitation in fluid-air-mixtures. For example a check valve might be open, then we have approximately no pressure drop, but a certain amount of the flow rate. Or a check valve might be closed, then we have a pressure drop, but no flow rate. A small amount of air in the fluid will be compressed by a large pressure to a neglectable small air volume, but for a very small pressure the air will expand in a nearly explosive way, a behavior, which can be approximated by a complementarity 1, 9. The area of non-smooth mechanics has been estab- lished during the last thirty years by Moreau in Mont- pellier and by Panagiotopoulos in Thessaloniki 6. During the nineties these theories have been transfered and applied to multibody system dynamics 7, 8 and since then furtheron developed in a very concise and rigorous way 4, 5. The research with respect to nu- merical methods is still on the way, because efficient numerical methods are the key for applications with respect to large technical systems 3. The paper will be mainly based on findings of the dissertation 1 and some publication 9. Therefore the description of the new hydraulic theory will be kept short. 2. Modeling hydraulics In order to set up a mathematical model we assume, that the hydraulic system can be considered as a network of basic components. These components are connected by nodes. In conventional simulation programmes these nodes are assumed to be elastic. In the case of rel- atively large volumes this assumption is reasonable whereas for very small volumes incompressible junc- tions, with unilateral or bilateral behavior, are a bet- ter approach. Complex components like control valves can be composed of elementary components like lines, check valves and so forth. In the following a selection of elementary components is considered. It is shown how the equations of motion are derived and how they are put together to form a network. 2.1. Junctions Junctions are hydraulic volumes filled with oil. The volumes may be considered as constant volumes or variable volumes as shown in Fig. 1. Junctions with variable volume are commonly used for hydraulic cylinders. 2.1.1. Compressible junctions Assuming compressible fluid in such a volume leads to a nonlinear differential equation for the pressure p. Springer Nonlinear Dyn (2007) 47:219233 221 Fig. 1 Hydraulic junctions with constant and variable volume Introducing the pressure-dependent bulk modulus E(p) =V dp dV (1) yields a differential equation for the pressure in a con- stant volume p = E V summationdisplay Q i (2) and p = E V (Q 1 A K x) (3) in a variable volume, respectively. A common assump- tion with respect to the fluid properties considers a mixture of linear elastic fluid with a low fraction of air. Fig. 2 shows the calculated specific volume of a mixture of oil and 1% air (at a reference value of 1 bar). For high pressure values the air is compressed to a neglectable small volume whereas the air expands abruptly for low pressure values, see Fig. 2 with the pressure p versus the specific volume v. This figure illustrates also that the curve for the pressure in dependency of the specific volume can be very well approximated by a unilateral characteristic. If we would choose a smooth model we would get stiff differential Equations (2) and (3) for very small volumes V 0. 2.1.2. Incompressible junctions To avoid stiff differential equations for small volumes it is obviously possible to substitute the differential equa- tions by algebraic equations. Assuming a constant spe- cific volume of the incompressible fluid yields for a constant and a variable volume, respectively, the fol- lowing algebraic equations: summationdisplay Q i = 0 respectively summationdisplay Q i A K x K = 0 (4) These equations consider neither the elasticity nor the unilaterality of the fluid properties. A fluid model cov- ering both elasticity and unilaterality is described in Section 2.1.1. In the case of neglectable small volumes the fluid properties can be approximated by a unilateral characteristic. As illustrated in Fig. 2 a unilateral law Fig. 2 Fluid expansion for low pressures Springer 222 Nonlinear Dyn (2007) 47:219233 can be established by introducing a state variable V = integraldisplay t 0 summationdisplay Qd (5) which represents the total void volume in a fluid vol- ume. Obviously this void volume is restricted to be positive, V 0. As long as the pressure value is higher than a certain minimum value p min , the void volume is zero. A void formation starts when the pressure p ap- proaches the minimum value p min . This idealized fluid behavior can be described by a so called corner law or Signorinis law 7. V 0; p 0; V p = 0 (6) The pressure reserve p is defined by p = p p min . By differentiation the complementarity can be put on a velocity level. V = summationdisplay Q i 0 , (7) The equality sign represents the Kirchhoff equation stating that the sum of all flow rates into a volume is equal to the sum of all flow rates out of the volume. If the outflow is higher than the inflow the void vol- ume increases, V 0. Substituting the flow rates Q into a fluid volume by the vector of the velocities in the connected lines and the corresponding areas, v = v 1 v 2 . . . v i W = A 1 A 2 . . . A i (8) yields the junction equation in the unilateral form W T v 0 . (9) It is evident that fluid volumes with non-constant vol- ume can be put also into this form by extending the velocity and area vectors by the velocity and the area of the piston, respectively. As long as the pressure is higher than the minimum value, p 0, the unilateral Equation (9) can be substituted by a bilateral equa- tion W T v = 0. In this case it is necessary to verify the validity of the assumption p 0 because the bilateral constraint does not prevent negative values of p. 2.2. Valves In the following we shall give some examples of mod- elling elementary valves and more complex valves as a network of basic components. Physically, any valve is a kind of controllable constraint, whether the working element be a flapper, ball, needle etc. 2.2.1. Orifices Orifices with variable areas are used to control the flow in hydraulic systems by changing the orifice area. As illustrated in Fig. 3 the pressure drop in an orifice shows a nonlinear behavior. The classical model to calculate the pressure drop Delta1p in dependency of the area A V and the flow rate Q is the Bernoulli equation. Delta1p = 2 parenleftbigg 1 A V parenrightbigg 2 Q|Q| (10) The factor is an empirical magnitude consider- ing geometry- and Reynoldsnumber-depending pres- sure losses. It must be determined experimentally. As long as the valve is open the pressure drop can be calculated as a function of the flow rate and the valve area, Equation (10). As shown in Fig. 3 the character- istic becomes infinitely steep when the valve closes. In most commercial simulation programmes this leads to numerical ill-posedness and stiff differential equations for very small areas. In order to avoid such numeri- cal problems the characteristic for the pressure drop of closed valves can be replaced by a simple constraint Fig. 3 Pressure drop in an orifice Springer Nonlinear Dyn (2007) 47:219233 223 equation. Q = Av = 0 respectively A v = 0 (11) This constraint has to be added to the system equations when the valve closes. In the case of valve opening it has to be removed again. This leads to a time-varying set of constraint equations. In order to solve the system equa- tions one has to distinguish between active constraints (closed valves) and passive constraints (opened valves). The last ones can be removed. The constraint equations avoid stiff differential equations. On the other hand they require to define active and passive sets 4, 7, 8. 2.2.2. Check valves Check valves are directional valves that allow flow in one direction only. It is not worth trying to describe all existing types, so only the basic principle and the mathematical formulation is presented. Figure 4 shows the principle of a check valve with a ball as working element. Assuming lossless flow in one direction and no flow in the other direction results in two possible states: a114 Valve open: pressure drop Delta1p = 0 for all flow rates Q 0 a114 Valve closed: flow rate Q = 0 for all pressure drops Delta1p 0 Again these two states can be described by a corner law Q 0; Delta1p 0; Q Delta1p = 0 . (12) Prestressed check valves with springs show a mod- ified unilateral behavior, see Fig. 5. The pressure drop curve of a prestressed check valve can be split into an ideal unilateral part Delta1p 1 and a Fig. 4 Check valve Fig. 5 Check valve characteristics smooth curve Delta1p 2 considering the spring tension and pressure losses, see Fig. 6 2.2.3. Combined components Many hydraulic standard components are combinations of basic elements. Since the combination of unilateral and smooth characteristics yields either non-smooth or smooth behavior it is worth to consider such compo- nents with a smooth characteristic separately. As an example we consider a typical combination of a throt- tle and a check valve. Figure 7 shows the symbol and the characteristics of both components. Since the flow rate of the combined component is the sum of the flows in the check valve and the throttle, the sum of the flow rates is a smooth curve. In such cases it is convenient to model the combined component as a smooth compo- nent (in the mechanical sense as a smooth force law). 2.2.4. Servovalves As an example for a servovalve we consider a one-stage 4-way-valve. It is a good example for the complexity of the networks representing such components like valves, pressure control valves, flow control valves and related valve systems. Multistage valves can be modelled in Fig. 6 Superposition of unilateral and smooth curves Springer 224 Nonlinear Dyn (2007) 47:219233 Fig. 7 Combination of smooth and non-smooth components a similar way as a network consisting of servovalves and pistons, which themselves are working elements of the higher stage valve. Figure 8 shows the working principle of a 4-way valve. Moving the control piston to the right connects the pressure inlet P with the output B and simultaneously the return T with the output A. If one connects the outputs A and B with a hydraulic cylinder, high forces can be produced with small forces Fig. 8 4-way valve acting on the control piston. The valve works like a hydraulic amplifier. Figure 9 shows a network model of the 4-way valve. The areas of the orifices A V 1 .A V 4 are controlled by the position x of the piston. The orifice areas are as- sumed to be known functions of the position x. The parameter covers a potential deadband. To derive the equations of motion the lines in the network are as- sumed to be flow channels with cross sectional areas A 1 .A 4 . The fluid is incompressible since the vol- umes are usually very small, and the bulk modulus of the oil is very high. The oil masses in the lines are m 1 .m 4 . Denoting the junction pressures with p i and the pressure drops in the orifices with Delta1p i , we get the equations of momentum as m 1 v 1 A 1 p 1 + A 1 p 2 + A 1 Delta1p 1 = 0 m 2 v 2 A 2 p 2 + A 2 p 3 + A 2 Delta1p 2 = 0 m 3 v 3 A 3 p 3 + A 3 p 4 + A 3 Delta1p 3 = 0 m 4 v 4 + A 4 p 1 A 4 p 4 + A 4 Delta1p 4 = 0 (13) Fig. 9 Network model of a 4-way valve Springer Nonlinear Dyn (2007) 47:219233 225 which can be expressed as M v + Wp+ W V Delta1p = W a Delta1p a . (14) where v is the vector of flow velocities, p the vector of junction pressures, Delta1p the vector of pressure drops in the closed orifices and Delta1p a the vector of pressure drops in the open orifices. The mass matrix M = diag(m i )is the diagonal matrix of the oil masses. The matrix W = A 1 A 1 00 0 A 2 A 2 0 00A 3 A 3 A 4 00A 4 is used to calculate the forces acting on the oil masses in the channels resulting from the junction pressures p. The junction equations are given by Q P Q A Q T Q B + A 1 00A 4 A 1 A 2 00 0 A 2 A 3 0 00A 3 A 4 v 1 v 2 v 3 v 4 = 0 (15) which can be written in the form Q in + W T v = 0 . (16) In order to determine the pressure drops Delta1p i one has to distinguish between open and closed orifices to avoid stiff equations, see Section 2.2.1. In case of open ori- fices the pressure drop can be calculated directly sub- ject to the given flow rates and the orifice area, whereas closed orifices are characterized by a constraint equa- tion. Delta1p ai = f (v i , A Vi (x) open orifices i A j v j = 0 closed orifices j (17) The constraint equations for the closed orifices are col- lected to give W T V v = 0 (18) Fig. 10 Coordinates for one-dimensional flow where the number of columns of W V is the number of closed orifices. Note that this matrix has to be updated every time an orifice opens or closes. 2.3. Hydraulic lines Hydraulic lines or hoses are used to connect compo- nents. For long lines the dynamics of the compress- ible fluid has to be taken into account. In order to get a precise system model, it is necessary to investigate pressure wave phenomena as well as the pipe friction. The pipe friction is rather complicated since the veloc- ity profile is not known a priory. In the case of laminar flow it is possible to derive analytical formulas for a uniform fluid transmission line in the Laplace domain. The so-called 4-pole-transfer-functions relate the pres- sure and the flow at the input and at the output of the line in dependency of Bessel functions. Many attempts have been made to approximate the transfer functions with rational polynomial functions which can be re- transformed into the time domain. Unfortunately the form of the equations of these models is not compati- ble with the equations in the framework of this paper, because the coupling with constraint equations might lead to numerical instability due to violation of the prin- ciple of virtual work. In the following a time domain modal approxima- tion is presented. This model can be extended to cover frequency dependent friction as well. The starting point are the linearized partial differential equations for one- dimensional flow. The coordinates are shown in Fig. 10. Partial derivatives of a arbitrary coordinate q are de- noted by q t = q and q x = q prime , respectively. The mass balance p + E A Q prime = 0 (19) Springer 226 Nonlinear Dyn (2007) 47:219233 with the flow rate Q = Au and the introduced state variable x = 1 A integraldisplay t 0 Qd ; x =