Non-uniformly parabolic equations and applications to the random conductance model

We study local regularity properties of linear, non-uniformly parabolic finite-difference operators in divergence form related to the random conductance model on $$\mathbb Z^d$$ Z d . In particular, we provide an oscillation decay assuming only certain summability properties of the conductances and their inverse, thus improving recent results in that direction. As an application, we provide a local limit theorem for the random walk in a random degenerate and unbounded environment.

Non-linear effects in the oscillating drop method for viscosity measurements

The contribution of non-linear fluid flow effects to the damping of surface oscillations in the oscillation drop method was investigated in a series of experiments in an electromagnetic levitation device installed on the International Space station, ISS-EML. In order to correctly evaluate the damping time constant from measured surface oscillation decays the effect of a modulated signal response on measured surface oscillation decay curves was investigated. It could be shown that various experimentally observed signal patterns could be well represented by a modulated response. The physical origin of such modulations is seen in rotation and precession. Over a temperature range of 220 K covered by different surface oscillation excitation pulses with an initial sample shape deformation of 5 – 10% the amplitude of surface oscillations as a function of time could be very well represented by a Lamb type damping with a temperature dependent viscosity. A direct comparison of surface oscillation decay times measured in the same temperature range but for different oscillation amplitudes showed no non-linear contribution to the damping time constant with a confidence level better 10%.

Reduction of Structural Dynamics Models Having Nonlinear Damping

The purpose of this work has been to explore the suitability of significantly reduced order structural dynamics models for simulation of nonlinear response of structures. In this paper oscillator chains having nonlinear damping and linear stiffness are considered. Both non-smooth and smooth nonlinear damping will be discussed for the case of isolated nonlinear damping and for the case of many equal nonlinear damping elements distributed throughout the structure. Most of the results to be shown will address the accuracy with which significantly reduced order models (e.g., 5, 10 and 20 DOF reduced models of a 1,000 DOF system) simulate the free oscillation decay caused by the nonlinear damping. The model reduction employed utilizes an “exact-for-the-linear-case” linear state transformation to eliminate slave coordinates, yielding a reduced model containing only specified master degrees of freedom. The ultimate objective of this work is to develop efficient, sufficiently accurate models that can be integrated numerically to determine response of very large nonlinear structures in situations for which numerical integration over reasonably long times is needed. The results obtained here are promising: even drastically reduced models provide accurate simulation of the oscillation decay for both smooth and non-smooth nonlinearity.

Bending of Plates on Thin Elastomeric Foundations

Closed-form and series solutions are presented for the bending of plates bonded to a thin elastomeric foundation which is in turn bonded to a rigid substrate. The standard fourth-order governing differential equation of a classical Winkler elastic foundation becomes a sixth-order equation for the case of an incompressible foundation. Oscillation decay rates are shown to be significantly different from those of the Winkler solution due to the incompressibility of the elastomer.