Dynamics of Hydraulic Systems
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duction Hydraulic systems represent a steadily growing area of mechanical engineering with large applications in the automotive industry, in aerospace industry, in building and agricultural machinery and in machine tools, to name the most important fields. Large pressures, large forces generated by very compact equipment and for many cases acceptable control frequencies characterize hydraulics, which fundamentally comes from fluid mechanics. As a consequence we have various simulation tools on the market, which simulate hydraulics, though usually accompanied by large computing times due to a detailed consideration of all components, especially with respect to fluid compressibility. Compressibility generates stiff differential equations and steep characteristics. The basic idea of our treatment replaces steep characteristics by complementarities, where non-smooth set-valued force laws take the place of smooth, but steep single-valued forces [23]. In hydraulic networks we find such a complementarity behavior in connection with check valves, with servo valves and with cavitation in fluid-airmixtures. A check valve for example might be open, then we have approximately no pressure drop, but a certain amount of 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. The replacement of steep characteristics by complementarities in connection with the neglection of fluid compressibility for small volumes reduces computing time by 3-4 orders of magnitudes, which has been proven several times by simulation of very large hydraulic systems. We shall give examples ([195], [192]).
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4 Dynamics of Hydraulic Systems
With respect to compressibility we need to be cautious, nevertheless. Fluid nets can be approximately represented by two-terminal elements or by quadrupoles depending on the compressibility influence. The characteristic compressibility measure in form of the fluid capacitance is proportional to the fluid volume and inversely proportional to the fluid pressure of the fluid line under consideration, or the other way around, proportional to the fluid line volume and inversely proportional to the fluid density multiplied by the square of the fluid velocity of sound. The fluid volumes of hydraulic nets are usually very small leading also to very small capacitances, several orders of magnitude smaller than fluid resistance or fluid inductance. Therefore it makes not much sense to model such capacitances as representatives of compressibility, only to be on the safe side. We must check it and consider only really large fluid volumes with respect to compressibility (see also [174]). Borchsenius presents a nice example for illustrating the influence of compressibility [23]. According to Figure 4.1 we have a simple system with an oil storage, a fluid line and a hydraulic c
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