Beitrag zur Auswertung zwei- und dreidimensionaler spannungsoptischer Versuche

Übersicht Spannungsoptische Untersuchungen liefern als unmittelbares Ergebnis die Differenzen der Hauptnormalspannungen. In vielen Fällen genügen diese zur Beurteilung der Werkstoffanstrengung. Für die vollständige Bestimmung des Spannungszustandes sind unter anderem verschiedene zeichnerische und rechnerische Auswerteverfahren bekannt, die auf den Gleichgewichtsbedingungen und den Verträglichkeitsbedingungen beruhen. Es wird eine Gruppe von neuen Verfahren erläutert, die unter Verwendung der zweiten Ableitungen der Versuchsergebnisse die Auswertung auch in solchen Fällen erlaubt, für die die bisher bekannten Verfahren nicht anwendbar sind bzw. die experimentelle Durchführung vereinfacht.

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Summary Photoelastic experiments produce the differences of principal stresses. In some cases, these direct results give all information on failure. Separation of principal stresses by means of equations of equilibrium or compatibility supplies additional information in cases where it is needed. While the methods known are applicable only in certain simple cases, new methods using the second derivatives of the optical data allow to separate principal stresses in all cases or simplify the experimental procedure respectively.

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Herrn Professor Dr.-Ing. habil. Docteur ès Sciences F. Tölke zum 70. Geburtstag gewidmet.     

Separation of principal stresses in the freezing method

In order to separate principal stresses in the photoelastic freezing method, the stress sum is determined by measuring the dilatation or variation of specific weight of small particles of frozen-in models. A floating teat has proved to give very accurate results.     

The invariant density of a chaotic dynamical system with small noise

We consider a chaotic dynamical system perturbed by noise and calculate an approximate invariant density when the noise level is small. Because of the special structure of the dynamical system, the effective support of the invariant density is much smaller than the noiseless attractor. This behavior is captured by the asymptotic form of the invariant density, which is given explicitly.     

On Graph Products of Automatic and Biautomatic Monoids

It is shown that the graph product of automatic monoids is always automatic thereby improving on a result by Veloso da Costa [22] who showed this result provided the factors have finite geometric type. Secondly, we prove that, in general, the free product (and therefore the graph product) of biautomatic monoids need not be biautomatic. Imposing a restriction on the factors that is symmetric to Veloso da Costa's "finite geometric type", the biautomaticity of all graph products of biautomatic monoids is shown.     

Dynamics of Simple Balancing Models with Time-Delayed Switching Feedback Control

Time-delayed control in a balancing problem may be a nonsmooth function for a variety of reasons. In this paper we study a simple model of the control of an inverted pendulum by either a connected movable cart or an applied torque for which the control is turned off when the state of the pendulum is located within certain regions of phase space. Without applying a small angle approximation for deviations about the vertical position, we see structurally stable periodic orbits which may be attracting or repelling. Due to the nonsmooth nature of the control, these periodic orbits originate in various discontinuity-induced bifurcations. We also show that a coincidence of switching events can produce complicated periodic and aperiodic solutions.     

Symmetries of the Partial Order of Traces

This paper deals with the automorphism group of the partial order of finite traces. We show that any group can arise as such an automorphism group if we allow arbitrary large dependence alphabets. Restricting to finite dependence alphabets, the automorphism groups are profinite and possess only finitely many simple decomposition factors. Finally, we show that the partial order associated with the Rado graph as dependence alphabet does not give rise to a homogeneous domain thereby answering an open question from Boldi, P., Cardone, F. and Sabadini, N. (1993).     

Characterizing mixed mode oscillations shaped by noise and bifurcation structure

Many neuronal systems and models display a certain class of mixed mode oscillations (MMOs) consisting of periods of small amplitude oscillations interspersed with spikes. Various models with different underlying mechanisms have been proposed to generate this type of behavior. Stochastic versions of these models can produce similarly looking time series, often with noise-driven mechanisms different from those of the deterministic models. We present a suite of measures which, when applied to the time series, serves to distinguish models and classify routes to producing MMOs, such as noise-induced oscillations or delay bifurcation. By focusing on the subthreshold oscillations, we analyze the interspike interval density, trends in the amplitude, and a coherence measure. We develop these measures on a biophysical model for stellate cells and a phenomenological FitzHugh-Nagumo-type model and apply them on related models. The analysis highlights the influence of model parameters and resets and return mechanisms in the context of a novel approach using noise level to distinguish model types and MMO mechanisms. Ultimately, we indicate how the suite of measures can be applied to experimental time series to reveal the underlying dynamical structure, while exploiting either the intrinsic noise of the system or tunable extrinsic noise.