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1970 液压式 组合 建筑机械 液压 部分 设计

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黄河科技学院毕业设计(文献翻译） 第 8 页 以液压为例平稳系统的无规则性摘要：许多技术系统的力学规律包括陡峭的特点，它作为一个原则导致使用不同的方程和大量的计算时间。对动力性能如此陡峭的特点是非常接近有极值特性的规律，并且因此可能被极值力学所取代，对多体动力学（包括单方面）这是正确的，且对流体机械系统(如水力)是近似正确的。下面我们以液压系统为例，提出了一种新的建模方案，以有双边和单方面的限制多体系统方程确立了液压方程并能减少3到4个数量级的计算时间。一个大工业事例说明了新理论的优越性。关键词：光滑系统的无规则性，互补性，系统模型，液压力1 介绍模型符合物理或技术的现实。 他们是较为详细的，但所有的模型在一定程度上有近似性,但不能完全模拟现实世界，因此，在建立模型时，我们应该牢记，技术系统特别是技术力学不是演译系统。它对经典解析力学来讲是正确的，只适用于具有一些现实意义的部分问题。另一个方面关系到模型的目标。我们是要尽可能完美的模拟现实还是想考虑某些参数的影响呢？要实现这两个目标都会遇到复杂问题，因为一个模型本身不能叙述，我们必须做一个模型智能地引入我们的问题，并考虑：尽可能理解物理和技术特性、预测参数的影响、估计忽视条件、所有这些比科学更加艺术， 但它是如此重要,以至没有一个机械的、物理的模型,导致良好的想象和现实世界中被模拟的美好画面的问题,初步阶段就不能建立好模型。 以该系统的性能评价与倾向而建立的简单模型存在一些难点问题，因为如此简单的模式，不仅要负担对整个系统，而且还能准确地表达各种信息。因为除了简单的系统建模可能出错，很多自由度的干扰可能会导致系统完全不同的动态行为。最近发表的一份针对自激振荡与下降的特点的调查,说明了出现错误结果时的危险性。包括一个以上的自由度，它可以证明自激振荡都有可能表现为增加而不是下降的特点2 。 在处理这个模式过程中， 这一点往往被远远低估， 我们必须找到一种对数学特别是对我们想应用的数值工具。对同一个问题，我们有各种可能的数学描述和某种数值的算法。通常情况下采用物理-数学函数，利用该函数的结构特点来 分析和解决部分解析解的目标转化、收敛性，更好地表达数值算法的稳定性和结果的清晰度。我们在描述具有该特点的力学规律时，通常采用两种方式：光滑的和不光滑的。在实际的工程特性中，发生于结合接触问题、液压设备的流体力学问题、空穴或电子开关问题，上面只是列举的几个例子。从数学和最终的数值立场上看，这种特性产生刚性微分方程或需要一个有互补性的工作方式。最后采用那种方法主要取决于计算时间。 考虑接触及相关问题，我们可能通过局部刚度来离散处理，这需要力学规律的推导。由于接触刚度通常十分大，运用刚性微分方程。第二种方式假设以局部接触面积作为刚性，并由互补7，8制订了接触特性。在液压网络中，我们发现有这样一种与单向阀、伺服阀和在流体-空气混合物的汽蚀相关联的互补性行为。例如，一个单向阀可能打开，那我们就要假设没有任何压力降但有一定大小的流速。或者单向阀可能会被关闭，然后就有一个压力降，但没有流速。流体中少量的空气会被一个大的压力压缩成可以忽略不计，但对于一个非常小的压力，空气会以一种接近爆炸的方式膨胀，这种方式可近似为一个互补1，9。 非光滑力学的面积已经在过去三十年由蒙彼利埃的moreau和塞萨洛尼基 6 的 panagiotopoulos 确立。 在90年代，这些理论已被应用于多体系统动力学 7，8，从那时起furtheron以一个非常简洁和严格的方式4，5发展 。鉴于数值方法的研究仍然在继续，因为有效的数值方法是针对应用大型技术系统3的关键。该文章将主要基于对论文的研究1和一些出版物9发行 。因此对新型液压原理的描述将尽量简短。 2 液压学模型 为了建立一个假设的数学模型，即液压系统可被视为一个基本元件的网络。这些组件是由节点联系在一起的。在传统的模拟方案中，这些节点可假设为弹性的。在相对大容量的情况下，这个假设是合理的，而对于很小的容量，伴随单侧或双侧行为的不可压缩路口是一个更好的方法。复杂的部件如由线、单向阀基本组件组成的控制阀等等。基本组件的选择在下面被考虑。这表明运动方程如何推导，以及他们是如何组合在一起而形成一个网络。 2.1路口路口是填充油的液压卷。该卷可视为常数卷或可变卷如公式1所示。有可变卷的路口通常用于液压气瓶。 2.1.1 可压缩路口 假设一定体积的可压缩流体产生一个非线性微分方程的压力为p。 图1: 有常量和可变量的液压路口压力在不同体积弹性模量条件下的计算可根据下面的公式来计算: （ 1 ）在常数卷中流量-压力的差分方程 （ 2 ） 和 （ 3 ） 在一个可变容积下，鉴于流体性质的普遍性，假设考虑了具有少量空气的线性弹性流体的混合物。 例2显示含石油和百分之一空气的混合物计算的特定体积（参考值为1巴） 。在高压下空气被压缩到一个可忽视的体积，而在低压下空气突然扩大，见图2压力P-特别体积。这个图示也说明了取决于特别体积的压力曲线，可以有单侧行为很好地近似。如果我们将选择一个光滑模型，我们会得到刚性微分方程（2）及（3）对极少的体积 0 。 2.1.2 不可压缩路口 为了避免小体积的刚性微分方程，尽可能用代数方程代替微分方程。假设不可压缩流体流量的恒定体积为常量或变量，以下代数方程： 特别是 （4）这些方程即没有考虑弹性，也没有考虑单方面的流体性质。一个同时涵盖弹性和单方面性质的流体模型在第2.1.1部分已描述。在可忽略的小体积情况下，流体性质可近似由单方面特性。如图2说明，可以通过引入状态变量建立方程： （5）上式代表流体体积中总的无限体积。 由于，所以只要压力值高于某一个最低值，无限体积就为零。 当压力P趋近于最低值时，无效体积形成。这一理想的流体行为可以由一个所谓的角落法或signorini法7 来表示。图2 低压下的流体膨胀； （6）压力平均值被定义为。根据互补性的不同，压力平均值可以表述为与流速相关的方程。 （7） 等号标志代表了基尔霍夫方程即是流进容积的所有流速之和等于流出容积的所有流速之和。如果流出大于流入，则无效体积增加， v 0 。而由在连接线和相应的地方速度的方向流动的流率q 成液量， （8） 流量连接点方程以单方面形式. (9)很显然，有非恒卷的流体体积也能通过增大流速和面积而改变活塞的运动速度。只要压力高于最小值,单边方程（9） 可以由一个双边方程取代 。在这种情况下，由于双边约束并不能避免平均压力的负值，所以要核实假设的有效性 。2.2阀门下面我们将列举一些利用模拟简单的阀门和比较复杂的阀门作为网络的基本组成部分的例子。从物理角度讲，无论工作单元是一个插板、球，针头等， 任何阀都是一种可控的约束。 2.2管口 可变区的管口通过改变管口面积来控制液压系统中流体的流动。如图3说明 :在管口的压力降显示出非线性行为。伯努利方程是利用截面积和流率来计算压力降的传统模型。 （10） 该因子是一个经验值，决定于几何和依赖雷诺数的压力损失。它必须由实验确定。 只要阀门打开，压力降就可以用流速和阀门截面积来表示，如方程（10） 。如图3所示，当阀门关闭时，特点变得很明显。由于会导致数值不稳定性和刚性微分方程，该方程只应用于少数商业仿真软件中。为了避免这种数值问题，闭合阀门的压降的特点可由一个简单的约束方程取代。图3 孔口的压力损失 （11）当阀门关闭时，这个约束必须考虑到系统方程中。但在阀门开启时，又必须除去。这就引出一系列与时间变化相关的约束方程。为了解决系统方程，一要区分积极约束（关闭阀门）和被动约束（打开阀门） ，后者约束必须除去。 约束方程避免了刚性微分方程。另一方面，他们需要定义主动和被动套 4，7，8 。 2.3单向阀 单向阀是有方向性的阀门，只允许流体单方向流动。因为它不能描述所有现存的类型，所以只提出了基本原则及数学公式。 图4显示了有一个球的阀门作为工作单元的原则。假设只在一个方向流动的损失导致了下面两种可能情况。 图 4 单向阀阀门开启：对所有的流率q 0，压降为0阀门关闭：对所有的压降p 0，流动比率Q = 0 再次，这两种情况可以由一个边界规则描述 （12） 有弹性的预应力单向阀表现出一种单方面行为，见图5 。 图5 单向阀特性预应力单向阀的压力降曲线可以被划分为一个理想的单方面部分p1和一条考虑弹性收缩和压力损失的光滑曲线P2，见图六。图 6 叠加的单边性质和光滑曲线2.4结合组件 结合组件是许多液压标准组件组合的基本要素。由于组合单方面和平稳的特点，流量的通畅与不通畅则需要考虑这些部件与平稳的特点。我们举一个典型的组合：一个节流阀和一个单向阀。 毕业设计文献翻译 院（系）名称工学院机械系 专业名称机械设计制造及其自动化 学生姓名程昆鹏 指导教师邹景超 2012年 3 月 5 日Nonlinear Dyn (2007) 47:219233DOI 10.1007/s11071-006-9069-1ORIGINAL ARTICLEDeregularization of a smooth system example hydraulicsFriedrich PfeifferReceived: 29 August 2005 / Accepted: 28 October 2005 / Published online: 1 November 2006C?Springer Science+Business Media B.V. 2006Abstract Many technical systems include steep char-acteristics for force laws, which as a rule lead to stiffdifferential equations and large computing times. Forthe dynamical performance such steep characteristicsare very near to laws with set-valued properties andmight 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 forfluidmechanicalsystemslikehydraulics.Inthefollow-ing we present a new modeling scheme for hydraulicsystems, which establishes the hydraulic equations ofmotion in the form of multibody system eqations withbilateral and unilateral constraints, and which is ableto reduce computing times by three to four order ofmagnitudes. A large industrial example illustrates theexcellent performance of the new theory.Keywords Deregularization of smooth systems Complementarities System models Hydraulics1. IntroductionModels approximate the physical or technical reality.They are more or less detailed, but all models includeF. PfeifferInstitute of Applied Mechanics, Technical UniversityMunich, Boltzmannstrasse 15, D-85748 Garching,Germanye-mail: pfeifferamm.mw.tum.desome 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 technicalsystems and especially technical mechanics are not de-ductive systems. This is true for classical analyticalmechanics, which applies only partly to problems ofsome realistic significance. A further aspect concernsthe goals connected with models. Do we want to mapreality as perfect as possible, or do we want to considercertain parameter influences? Both objectives includedifficult problems, because a model is by itself dumb.We have to make a model intelligent by introducinginto it our knowledge of the problem under consider-ation, the physical and technical properties as good aswe understand them, the parameter influences as goodas we might expect them, the neglections as good aswe can estimate them. This all is more an art than ascience, but it is so essential, that no good model canbeestablishedwithoutapreliminaryphaseofphysical,of mechanical argueing leading to a sound imaginationand a sound picture of the real world problem to bemodeled.Simple models for the evaluation of tendencies withrespecttotheperformanceofasystemincludesomere-allydifficultproblems,becausesuchsimplemodelsnotonlyaffordaperfectviewoftheoverallsystembutalsoa perfect idea of what is important and of what is notimportant. In addition simple system modeling mightbe faulty, because the interference of many degrees offreedom might result in a completely different dynam-ical behavior as compared with simple considerations.Springer220Nonlinear Dyn (2007) 47:219233A recently published finding with respect to the wellknownfrictionproblemofself-excitedoscillationscon-nected with falling characteristics illustrates especiallythisdangerofcomingoutwithincorrectresults.Includ-ing more than one degree of freedom it can be shown,that self-excited oscillations may happen also for anincreasing and not for a falling characteristic 2.After having dealt with this model finding phase,whichbythewayisusuallyverymuchunderestimated,we must find a decision on the mathematical and es-pecially on the numerical tools we want to apply. Forone 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 forthischoicearephysical-mathematicalcorrespondence,structural features of the resulting equations, transfor-mation capabilities with the goal of analytical or partlyanalytical solutions, convergence of the solutions, sta-bilityofthenumericalalgorithmsandfinallytherepre-sentation of the results allowing clear interpretations.Coming back to our problem of descibing systemswith force laws including very steep characteristicswe may apply two approaches, a smooth and a non-smooth one. In practical engineering characteristicswith steep features occur in connection with contactproblems, with fluidmechanical problems of hydraulicequipment, with cavitation or with electronic switch-ing problems, to name only a few examples. From themathematical and from the resulting numerical stand-point of view such characteristics produce either stiffdifferential equations or they require a complementar-ityformulation.Thedecisionwhichwaytogodependsmainly on the computation time.Consideringcontactsandrelatedproblemswemightdiscretize a contact by evaluating the local stiffnessproperties of the contact, which allows the derivationof a force law. As contact stiffnesses are usually verylarge,wecomeoutwithstiffdifferentialequations.Thesecond way consists in assuming the local contact areaasrigid,whichdoesnotimplythatthewholebodymustbe assumed rigid, and to formulate the contact proper-ties by complementarities 7, 8. In hydraulic networkswefindsuchacomplementaritybehaviorinconnectionwithcheckvalves,withservovalvesandwithcavitationin fluid-air-mixtures. For example a check valve mightbeopen,thenwehaveapproximatelynopressuredrop,but a certain amount of the flow rate. Or a check valvemight be closed, then we have a pressure drop, but noflow rate. A small amount of air in the fluid will becompressed by a large pressure to a neglectable smallair volume, but for a very small pressure the air willexpand in a nearly explosive way, a behavior, whichcan 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 transferedand applied to multibody system dynamics 7, 8 andsince then furtheron developed in a very concise andrigorous way 4, 5. The research with respect to nu-merical methods is still on the way, because efficientnumerical methods are the key for applications withrespect to large technical systems 3. The paper willbe mainly based on findings of the dissertation 1 andsome publication 9. Therefore the description of thenew hydraulic theory will be kept short.2. Modeling hydraulicsInordertosetupamathematicalmodelweassume,thatthe hydraulic system can be considered as a network ofbasiccomponents.Thesecomponentsareconnectedbynodes. In conventional simulation programmes thesenodes are assumed to be elastic. In the case of rel-atively large volumes this assumption is reasonablewhereas for very small volumes incompressible junc-tions, with unilateral or bilateral behavior, are a bet-ter approach. Complex components like control valvescan be composed of elementary components like lines,check valves and so forth. In the following a selectionof elementary components is considered. It is shownhow the equations of motion are derived and how theyare put together to form a network.2.1. JunctionsJunctions are hydraulic volumes filled with oil. Thevolumes may be considered as constant volumes orvariable volumes as shown in Fig. 1. Junctions withvariable volume are commonly used for hydrauliccylinders.2.1.1. Compressible junctionsAssuming compressible fluid in such a volume leadsto a nonlinear differential equation for the pressure p.SpringerNonlinear Dyn (2007) 47:219233221Fig. 1 Hydraulic junctionswith constant and variablevolumeIntroducing the pressure-dependent bulk modulusE(p) = VdpdV(1)yields a differential equation for the pressure in a con-stant volume p =EV?Qi(2)and p =EV(Q1 AK x)(3)in a variable volume, respectively. A common assump-tion with respect to the fluid properties considers amixture of linear elastic fluid with a low fraction of air.Fig.2showsthecalculatedspecificvolumeofamixtureofoiland1%air(atareferencevalueof1bar).Forhighpressure values the air is compressed to a neglectablesmall volume whereas the air expands abruptly for lowpressure values, see Fig. 2 with the pressure p versusthe specific volume v. This figure illustrates also thatthe curve for the pressure in dependency of the specificvolume can be very well approximated by a unilateralcharacteristic. If we would choose a smooth model wewould get stiff differential Equations (2) and (3) forvery small volumes V . Incompressible junctionsToavoidstiffdifferentialequationsforsmallvolumesitisobviouslypossibletosubstitutethedifferentialequa-tions by algebraic equations. Assuming a constant spe-cific volume of the incompressible fluid yields for aconstant and a variable volume, respectively, the fol-lowing algebraic equations:?Qi= 0respectively?Qi AK xK= 0(4)These equations consider neither the elasticity nor theunilaterality of the fluid properties. A fluid model cov-ering both elasticity and unilaterality is described inSection 2.1.1. In the case of neglectable small volumesthefluidpropertiescanbeapproximatedbyaunilateralcharacteristic. As illustrated in Fig. 2 a unilateral lawFig. 2 Fluid expansion forlow pressuresSpringer222Nonlinear Dyn (2007) 47:219233can be established by introducing a state variableV = ?t0?Qd(5)which represents the total void volume in a fluid vol-ume. Obviously this void volume is restricted to bepositive,V 0.Aslongasthepressurevalueishigherthan a certain minimum value pmin, the void volume iszero.A void formation starts when the pressure p ap-proaches the minimum value pmin. This idealized fluidbehavior can be described by a so called corner law orSignorinis law 7.V 0; p 0;V p = 0(6)The pressure reserve p is defined by p = p pmin.By differentiation the complementarity can be put on avelocity level.V =?Qi 0 ,(7)The equality sign represents the Kirchhoff equationstating that the sum of all flow rates into a volumeis 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 Qinto a fluid volume by the vector of the velocities in theconnected lines and the corresponding areas,v =v1v2.viW =A1A2.Ai(8)yields the junction equation in the unilateral formWTv 0 .(9)It is evident that fluid volumes with non-constant vol-ume can be put also into this form by extending thevelocity and area vectors by the velocity and the areaof the piston, respectively. As long as the pressure ishigher than the minimum value, p 0, the unilateralEquation (9) can be substituted by a bilateral equa-tion WTv = 0. In this case it is necessary to verify thevalidity of the assumption p 0 because the bilateralconstraint does not prevent negative values of p.2.2. ValvesIn the following we shall give some examples of mod-elling elementary valves and more complex valves as anetwork of basic components. Physically, any valve isa kind of controllable constraint, whether the workingelement be a flapper, ball, needle etc.2.2.1. OrificesOrificeswithvariableareasareusedtocontroltheflowin hydraulic systems by changing the orifice area. AsillustratedinFig.3thepressuredropinanorificeshowsa nonlinear behavior. The classical model to calculatethepressuredrop?p independencyofthearea AVandthe flow rate Q is the Bernoulli equation.?p =2?1AV?2Q|Q|(10)The factor is an empirical magnitude consider-ing geometry- and Reynoldsnumber-depending pres-sure losses. It must be determined experimentally.Aslongasthevalveisopenthepressuredropcanbecalculated as a function of the flow rate and the valvearea, Equation (10). As shown in Fig. 3 the character-istic becomes infinitely steep when the valve closes. Inmost commercial simulation programmes this leads tonumerical ill-posedness and stiff differential equationsfor very small areas. In order to avoid such numeri-cal problems the characteristic for the pressure drop ofclosed valves can be replaced by a simple constraintFig. 3 Pressure drop in an orificeSpringerNonlinear Dyn (2007) 47:219233223equation.Q = Av = 0respectivelyA v = 0(11)This constraint has to be added to the system equationswhenthevalvecloses.Inthecaseofvalveopeningithastoberemovedagain.Thisleadstoatime-varyingsetofconstraintequations.Inordertosolvethesystemequa-tions one has to distinguish between active constraints(closedvalves)andpassiveconstraints(openedvalves).Thelastonescanberemoved.Theconstraintequationsavoidstiffdifferentialequations.Ontheotherhandtheyrequire to define active and passive sets 4, 7, . Check valvesCheck valves are directional valves that allow flow inone direction only. It is not worth trying to describeall existing types, so only the basic principle and themathematical formulation is presented.Figure 4 shows the principle of a check valve witha ball as working element. Assuming lossless flow inone direction and no flow in the other direction resultsin two possible states:rValve open: pressure drop ?p = 0 for all flow ratesQ 0rValve closed: flow rate Q = 0 for all pressure drops?p 0Again these two states can be described by a cornerlawQ 0;?p 0;Q ?p = 0 .(12)Prestressed check valves with springs show a mod-ified unilateral behavior, see Fig. 5.Thepressuredropcurveofaprestressedcheckvalvecan be split into an ideal unilateral part ?p1and aFig. 4 Check valveFig. 5 Check valve characteristicssmooth curve ?p2considering the spring tension andpressure losses, see Fig. 62.2.3. Combined componentsManyhydraulicstandardcomponentsarecombinationsof basic elements. Since the combination of unilateraland smooth characteristics yields either non-smooth orsmooth behavior it is worth to consider such compo-nents with a smooth characteristic separately. As anexample we consider a typical combination of a throt-tle and a check valve. Figure 7 shows the symbol andthe characteristics of both components. Since the flowrateofthecombinedcomponentisthesumoftheflowsin the check valve and the throttle, the sum of the flowratesisasmoothcurve.Insuchcasesitisconvenienttomodel the combined component as a smooth compo-nent (in the mechanical sense as a smooth force law).2.2.4. ServovalvesAsanexampleforaservovalveweconsideraone-stage4-way-valve.Itisagoodexampleforthecomplexityofthenetworksrepresentingsuchcomponentslikevalves,pressure control valves, flow control valves and relatedvalve systems. Multistage valves can be modelled inFig. 6 Superposition of unilateral and smooth curvesSpringer224Nonlinear Dyn (2007) 47:219233Fig. 7 Combination ofsmooth and non-smoothcomponentsa similar way as a network consisting of servovalvesand pistons, which themselves are working elementsof the higher stage valve. Figure 8 shows the workingprinciple of a 4-way valve. Moving the control pistontotherightconnectsthepressureinletPwiththeoutputB and simultaneously the return T with the output A.If one connects the outputs A and B with a hydrauliccylinder,highforcescanbeproducedwithsmallforcesFig. 8 4-way valveacting on the control piston. The valve works like ahydraulic amplifier.Figure9showsanetworkmodelofthe4-wayvalve.The areas of the orifices AV1. AV4are controlled bythe position x of the piston. The orifice areas are as-sumed to be known functions of the position x. Theparameter covers a potential deadband. To derive theequations of motion the lines in the network are as-sumed to be flow channels with cross sectional areasA1. A4. The fluid is incompressible since the vol-umes are usually very small, and the bulk modulus ofthe oil is very high. The oil masses in the lines arem1.m4. Denoting the junction pressures with piandthe pressure drops in the orifices with ?pi, we get theequations of momentum asm1 v1 A1p1+ A1p2+ A1?p1= 0m2 v2 A2p2+ A2p3+ A2?p2= 0m3 v3 A3p3+ A3p4+ A3?p3= 0m4 v4+ A4p1 A4p4+ A4?p4= 0(13)Fig. 9 Network model of a 4-way valveSpringerNonlinear Dyn (2007) 47:219233225which can be expressed asM v + Wp + WV?p = Wa?pa.(14)where v is the vector of flow velocities, p the vector ofjunction pressures, ?p the vector of pressure drops intheclosedorificesand?pathevectorofpressuredropsin the open orifices. The mass matrix M = diag(mi) isthe diagonal matrix of the oil masses. The matrixW =A1A1000A2A2000A3A3A400A4is used to calculate the forces acting on the oil massesinthechannelsresultingfromthejunctionpressures p.The junction equations are given byQPQAQTQB+A100A4A1A2000A2A3000A3A4v1v2v3v4= 0(15)which can be written in the formQin+ WTv = 0 .(16)Inordertodeterminethepressuredrops?pionehastodistinguish between open and closed orifices to avoidstiff equations, see Section 2.2.1. In case of open ori-fices the pressure drop can be calculated directly sub-jecttothegivenflowratesandtheorificearea,whereasclosed orifices are characterized by a constraint equa-tion.?pai= f (vi, AVi(x) open orifices iAjvj= 0closed orifices j(17)The constraint equations for the closed orifices are col-lected to giveWTVv = 0(18)Fig. 10 Coordinates for one-dimensional flowwhere the number of columns of WVis the number ofclosed orifices. Note that this matrix has to be updatedevery time an orifice opens or closes.2.3. Hydraulic linesHydraulic 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 geta precise system model, it is necessary to investigatepressure 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 laminarflow it is possible to derive analytical formulas for auniform 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 theline in dependency of Bessel functions. Many attemptshave been made to approximate the transfer functionswith rational polynomial functions which can be re-transformed into the time domain. Unfortunately theform 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 mightleadtonumericalinstabilityduetoviolationoftheprin-ciple of virtual work.In the following a time domain modal approxima-tion is presented. This model can be extended to coverfrequencydependentfrictionaswell.Thestartingpointare the linearized partial differential equations for one-dimensionalflow.ThecoordinatesareshowninFig.10.Partial derivatives of a arbitrary coordinate q are de-noted byqt= q andqx= q?, respectively.The mass balance p +EAQ?= 0(19)Springer226Nonlinear Dyn (2007) 47:219233with the flow rate Q = Au and the introduced statevariable x =1A?t0Qd ; x =QA; x =QA(20)can be solved analytically with respect to the pressurep:p(x,t) = EA?t0Q?d + p0(x)= E?t0 x?d + p0(x)(21)respectivelyp(x,t) = E x?+ p0(x)(22)The term p0(x) represents the initial pressure distribu-tion in the line. The equation of momentum u + p?+ fR+ fg= 0(23)with a friction force fRand a gravity force fgcan betransformed with Equation (20) to x E x?+ p?0+ fR+ fg= 0 .(24)Multiplying Equation (24) with arbitrary test func-tions w(x) and integrating over the length L of the lineyields the weak formulation?L0 xw(x)dx E?L0 x?w(x)dx+?L0p?0(x)w(x)dx +?L0fRw(x)dx+?L0fgw(x)dx = 0 w(x)(25)where the term w x?can be integrated by parts to fit theboundary conditions.?L0w x?dx = w x?|L0?L0w? x?dx= wL1E(p0(L) pE)w01E(p0(0) pA) ?L0w? x?dx(26)pA, pEare boundary pressures and w0, wLare thevalues of the test function w(x) at the boundaries (x =0,x = L).Forthesakeofsimplicitytheinitialpressuredistribution is assumed to be uniform, p0(x) = p0=const. In order to approximate the partial differentialequationsbyasetofordinarydifferentialequationsweintroducespatialshapefunctionsw(x)andaseparationof the variables x and t, x q(t)Tw(x)(27)According to Galerkins method the shape functionsw(x) are the same functions used in the weak formula-tion, Equation (25). The discretized equations of mo-tion are then?L0wwTdx q + E?L0w?w?Tdxq +?L0fRwdx+?L0fgwdx = w0pA wLpE (w0 wL)p0(28)which can be transformed with wA= Aw0, wE=AwLtoM q + Kq + Wp = A?L0fRwdx A?L0fgwdx (wA wE)p0(29)SpringerNonlinear Dyn (2007) 47:219233227with the abbreviationsM = A?L0wwTx(mass matrix)K = EA?L0w?w?T(stiffness matrix)W = (wAwE).Equation (29) is similar to the equations of motion ofa mechanical system. Suitable functions w(x) are har-monic functions and B-spline functions, as numericalexperiments confirmed.2.3.1. Frequency dependant frictionIncaseoflaminarflowthecross-sectionalvelocitypro-file of oscillatory flow can be calculated analytically asfunctions of Bessel functions. It turns out that the gra-dient of the velocity becomes higher with increasingfrequencies . Figure 11 shows calculated profiles fordimensionless frequenciesR where R is the radiusand the kinematic viscosity.For low frequencies the profile is the well-knownparabolic Hagen-Poiseuille profile for stationary flow.Since the friction force depends on the gradient at thepipe wall the friction force becomes higher with in-creasing frequency. Therefore the increasing frictionhas to be taken into account by a correction of thesteady-statefrictionfactor.Itcanbeshownthatthepipefriction for a parabolic velocity profile can be coveredby a damping matrix D0M q + D0 q + Kq = h(30)withD0= 8?L0wwTdx.(31)Sincetheparabolicvelocityprofileisvalidonlyforlowfrequencies this damping matrix has to be corrected. Inthe Laplace domain a friction correction factor can bederived asN =Ur|RU4R= R8?J1( R) J1( R)J2( R)?(32)where =?s.(33)With the dimensionless frequency kH=?R the fol-lowing approximation of the real part of Equation (32)can be given (see 1):N(kH)=?1+0.0024756 kH3.0253322kH 5(N(5) 0.175 5)+0.175 kHkH 5(34)In order to correct the damping matrix the eigenvaluesiand the eigenfrequenciesi=?i(35)of the matrix M1K are calculated. The eigenvectorsare collected in the orthogonal modal matrix = (1,2,.,n) .(36)Fig. 11 Normalized velocity profiles for oscillatory flowSpringer228Nonlinear Dyn (2007) 47:219233With a diagonal correction matrixN = diag(N(i) acorrected modal damping matrix is introduced in theformDm= TD0 N(37)which can be re-transformed to yieldD = (T)1Dm1(38)With this matrix the friction term in Equation (29)can be substituted:A?L0fRwdx = D q(39)2.4. Hydraulic networksHydraulic components are described by the state vari-ables velocity, piston position, pressure and the like.We collect these variables in two state vectors.rAllstatevariableswhicharegivenbyordinarydiffer-entialequationswithoutanyconstraintsarecollectedin a vector x RnxrThe vector v Rnvcontains velocities which are de-scribed by momentum equations with unilateral andbilateral constraints.Summarizing the equations of all componentsresults in the following set of equations.Momentum equations for vM v Wp WVpVW p = f(t,v,x) Rnv(40)Ordinary differential equations for x x = g(t,x,v) Rnx(41)Bilateral junction equationsWTv + w(t) = 0 RnI(42)Constraints for closed valves/orificesWVTv = 0 RnVa(43)Unilateral constraintsQ 0; p 0;QT p = 0mitQ =WTv R na(44)2.4.1. SolutionInafirststepwecombinetheactivebilateralconstraintEquations (42), (43) to giveWTGv + wG= 0 .(45)The number of independent constraints is the rank ofthe matrix WG,r = rank(WG)(46)Theconstraintequationscanbefulfilledbyintroducingminimal coordinatesvm Rnminwhere nmin= nv r(47)in the formv = v (vm,t) = Jvm+ b(t) .(48)The Jacobian J and the vector b can be calculatednumerically by a singular value decomposition of thematrix WGwith the benefit that also dependent con-straints can be handled. Since the column vectorsof the Jacobian are orthogonal to the column vec-tors of WGwe can write the momentum equationsasJTM(J vm+b) JTW p = JTf(t,x,v)(49)The square matrix JTMJ Rnminnminis the invertibleprojected mass matrix.As a next step we derive a linear complementaryproblem (LCP) for the unilateral constraint equations.For this purpose we put all active unilateral constraintson a velocity level and transform Equation (44) withEquation (48) toQ 0; p 0;QT p = 0whereQ =WT v =WTJ vm+WTb.(50)SpringerNonlinear Dyn (2007) 47:219233229Together with Equation (49) we obtainQ =WTJ(JTMJ)1JTW?ALCP p+WTWTJ(JTMJ)1JT( f Mb) +WTb?bLCP(51)This is a standard LCP in the formQ = ALCP p + bLCPQ 0; p 0;QT p = 0(52)whichcanbesolvedusingastandardLemkealgorithm.ExperienceshowsthattheLemkealgorithmsworksre-liable in many cases, also for large systems, but thecomputing times are long. In the meantime better al-gorithms are available, which are based on the ideas oftime-steppingandoftheAugmentedLagrangemethod.Numerical experiences in other areas indicate, that onthis new basis computing time can be shortened signif-icantly.With this solution of the complementarity problemthe time derivative of v can be calculated. The evolu-tion of x and v with respect to time is obtained by anumerical integration scheme, for example a Runge-Kutta-scheme.2.4.2. ImpactsSome examples for possible impacts in hydraulic sys-tems are:rValve closurerImpactsofmechanicalcomponents,forexamplepis-ton/housing contactrCondensation of vapor (waterhammer), cavitationIn connection with the use of constraint equations in-stead of stiff elasticities we must consider multiple im-pact situations. Due to the algebraic relationship be-tween some components sudden velocity changes inone component may cause also velocity jumps in othercomponents.Inordertocalculatethevelocitiesafteranimpact it is necessary to solve the impact equations forthecompletesystem.Thestartingpointarethemomen-tum Equations (49). We assume as usual an infinites-imal short impact time ?t 0 without any positionchanges. The integration of the momentum Equation(49) over the duration of the impact yields:?t+?tt(JTMJ vm)dt=?t+?tt(JTW p + JTf JTMb)dt(53)Denotingthebeginningoftheimpactwithandtheend with+, we can rewrite Equation (53) asJTMJ(v+m vm) =JTW p?t + JTf? JTMb?t .(54)Introducing unilateral pressure impulsesP =?t+?tt p dt = p?t(55)and letting ?t 0 we come out withJTMJ(v+m vm) = JTWP .(56)The unilateral constraint equations can be set up forthe end of the impact in the formQ+ 0;P 0;PTQ+= 0whereQ+=WTv+=WTJvm+WTb ,(57)which again can be transformed into an LCP.Q+=WTJ(JTMJ)1JTW?ALCPP +WT(Jvm+ b)?bLCP(58)With the solution vectorP of this LCP we can calcu-late v+mfrom Equation (56). Finally we get the velocityvector at the end of the impact as v+= Jv+m+ b.3. ExampleAsanexamplefromindustryweconsiderthehydraulicsafety brake system of a fun ride, the free fall tower1. It is manufactured by the company Maurer S ohneGmbH, Munich, Germany. Figure 12 shows the tower.Under normal operation conditions the cabin with thepassengersisliftedbyacablewinchtoaheightofaboutSpringer230Nonlinear Dyn (2007) 47:219233Fig. 12 Free fall tower60m.Subsequentlythecabinisreleasedandfallsdownnearly undamped. Before reaching the ground the nor-mal brake system stops the cabin softly via the cablewinch. For safety reasons a redundant brake system isnecessary. In the case of a failure of the normal op-erating brake or a cable rupture the safety brake sys-tem has to catch the cabin even under disadvantageousconditions.Thesafetybrakesystemisahydraulicsystemwhichmoves brake-blocks via hydraulic cylinders. For thispurpose steel blades are fixed at the cabin. The steelblades fall into a guide rail, where the brake-blocks arefixed.Thesafetybrakesystemconsistsofupto7identi-cal modules with 4 hydraulic cylinders each. The mod-ules are arranged upon each other. Figure 13 shows themodel of the system. The simulations were carried outwiththecomputerprogrammeHYSIM.AstheFig.indi-cates,theprogrammeHYSIMallowsmulti-hierarchicalmodelling of systems. It means that some componentscan be put together to form a group, which itself can beused and duplicated like any elementary component.Undernormaloperatingconditionsthebrake-blocksare moved outwards the guide rail quickly so that thesystem is decelerated softly by the winch. Only underbad conditions the safety brakes stay closed and thecabin will be stopped by the friction forces. Due tothe very short opening time of the brake the hydrauliccylindersmovequicklyandreachthestoppositionwithhigh velocities. The resulting impact forces caused insome cases damage of the cylinders. In order to reducetheimpactforcessimulationswerecarriedout.Theaimwas to find new values of adjustable parameters suchthat the impact velocities are reduced and the openingtime does not exceed a certain maximum value.Figure 14 shows some simulation results. The im-pactvelocityandthustheresultingforceissignificantlyhigh for the Cylinder 1. The reason for this behaviorliesinthedifferentlengthofthesupplylineandthedif-ferent movement direction (up/down) of the cylinders,i.e.hydraulicandmechanicalasymmetries.Theimpactvelocitiescanbereducedsignificantlybyincreasingre-striction parameters at some valves, as Fig. 14 shows.This simple measure has been adopted to the real sys-tem and no problems occurred since then.A comparison of measurement and simulation isgiveninFig.15.Asthecylindersstartmovin