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In this paper we investigate the non-linear Vlasov-Fokker-Planck (VFP) equation, a both physically and mathematically interesting modification of Vlasov's equation, which describes a plasma in a thermal bath. We prove existence, uniqueness and representation results for steady states of the VFP equation both in the case of a mollified interaction potential and for the VFP-Poisson system. The uniqueness and representation results are of special interest since they distinguish special solutions of the Vlasov equation.where E,, stands for some external electric field, Ein = -grad Vin, and p = p(t, x) is the spatial density of the plasma at time t. So iff=f(t, x, v) denotes the distribution of the plasma in phase space, p is given by p(t, x) = s f ( t , x, v)dv.

We study stationary solutions of the Vlasov‐Fokker‐Planck Equation, a modification of the Vlasov‐Poisson Equation, obtained by adding a diffusion term with respect to velocity. From a physical point of view it describes a plasma in thermal equilibrium. We prove existence of stationary solutions; for the mollified equation even an existence‐ and uniqueness theorem holds for sufficiently high temperature.

In recent years so-called 'virtual test rigs' have become more and more important in the development process of cars and trucks. Originally, the idea was to substitute expensive durability tests with computer simulation. Meanwhile, the focus has changed towards a more cooperative usage of numerical and laboratory rig simulation. For many safety critical issues laboratory tests remain indispensable. In early development stages, when no physical prototypes are available yet, numerical simulation is used to analyse and optimise the design. In this paper, we show how to build numerical simulation models of complex servo-hydraulic test systems and their test specimen using multi-body simulation for the mechanics in combination with simulation models for the hydraulics and controls. We illustrate this at two industrial application examples: a spindle-coupled passenger car suspension rig and a tyre-coupled full vehicle rig. We show how the simulation models are used to design and optimise better test rigs and to support the test rig operation by preparing the physical tests with new specimen, i.e. by performing numerical simulations including numerical drive file iteration before the physical tests start

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In diesem Artikel wird eine mathematische Beschreibung des sogenannten Rainflow- Rekonstmktionsproblems, d. h. des Problems, wie man eine Zeitreihe konstmied, deren Rainflow-Zahlung einer gegebenen Rainflow-Matrix entspricht, uorgestellt. Der vorgestellte Algorithmus ist mathematisch in dem Sinn korrekt, daj3 keine Naherungen oder Heuristiken uerwendet werden. Weiterhin erzeugt er eine Gleichverteilung unter allen miiglichen Rekonstruktionen. Das Verfahren arbeitet on-line, es ist leicht an die verschiedenen Varianten des Rainflow-Zahluerfahrens, wie symmetrische und unsymmetrische Versionen und uerschiedene Residuen, anpajlbar. This paper is devoted to the mathematical description of the solution of the so-called rainflow reconstruction problem, i.e. the problem of constructing a time series with an a priori given rainflow matrix. The algorithm we present is mathematically exact in the sense that no approximations or heuristics are involved. Furthermore, it generates a uniform distribution of all possible reconstructions and thus an optimal randomization of the reconstructed series. The algorithm i s a genuine on-line scheme. It is easy adjustable to all variants of rainflow such as symmetric and asymmetric versions and different residue techniques. MOS (1991): 62N05; 73M10; 47H30Fatigue 16 (1994), 287-293. Mech. A/Solids

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