Dynamics of a weakly nonlinear string on an elastic foundation with a partly prescribed discrete spectrum

In this paper the dynamics of a weakly nonlinear elastic string on a Winkler elastic foundation is studied. The foundation may be spatially heterogeneous. At one end of the string a mass-spring system is attached, and the other end of the string is fixed. The string is assumed to be long, and the lower part of the spectrum of the string is prescribed. It is shown that localized modes exist and that the dynamics of the string for large times is determined by these localized modes. The frequencies of these localized modes can be controlled by special choices for the spatial heterogeneities in the elastic foundation. Analytical and numerical results are presented to illustrate the findings.</p

A mathematical analysis of an extended model describing sea ice-induced frequency lock-in for vertically sided offshore structures

This paper presents a mathematical analysis of an extended model describing a sea ice-induced frequency lock-in for vertically sided offshore structures. A simple Euler–Bernoulli beam as model for the offshore structure is used, and a moving boundary between an ice floe and the structure itself is introduced. A nonlinear equation for the beam dynamics is found by using an asymptotical approach and a Galerkin procedure. It is shown analytically that a frequency lock-in regime occurs during ice-induced vibrations (IIV), when the dominant ice force frequency is closed to a natural frequency of the structure. For beams, perturbed by small nonlinearities and a small damping, the concept of quasi-modes is introduced. A quasi-mode is a linear combination of the usual eigenmodes. The large time behaviour of solutions at the instability onset is determined by a single quasi-mode, which is maximally linearly unstable.The beam model analysis leads to the conclusion that an interaction between a moving ice floe and a structure can lead to a “negative friction” for particular values of the ice floe parameters. From the analysis presented in the paper it follows that the lock-in regime occurs when simultaneously two phenomena are present: a forcing resonance and a “negative friction”.Mathematical Physic

Topological Shooting of Solutions for Fickian Diffusion into Core-Shell Geometry

King et al. (2019) introduced a model for Fickian diffusion into core-shell geometry. The purpose of this model is to study diffusion of oxygen through protective shells encapsulating pancreatic Langerhan islets. These core-shells are of interest for the preparation of artificial pancreas to treat diabetes. In this paper we prove the existence of viable core-shell solutions for King’s model using a topological shooting method. The governing equations of the diffusion model can be reduced to a 2-dimensional non-autonomous first order ordinary differential equation. Solutions which correspond to viable core-shell diffusion are required to satisfy global constraints and boundary conditions in both the core and the encapsulating shell. These boundary conditions each give rise to one free parameter. We call solutions satisfying the core boundary condition core solutions. We identify two parameter spaces corresponding to core solution families. The viable core-shell solutions are on the boundary of these two core solution families. Using analytically obtained bounds we apply the intermediate value theorem to prove the existence of core-shell solutions. In addition, we obtain rigorous approximations for the boundary conditions of the viable diffusion core-shell solution

Dynamic failure of dry and fully saturated limestone samples based on incubation time concept

This paper outlines the results of experimental study of the dynamic rock failure based on the comparison of dry and saturated limestone samples obtained during the dynamic compression and split tests. The tests were performed using the Kolsky method and its modifications for dynamic splitting. The mechanical data (e.g. strength, time and energy characteristics) of this material at high strain rates are obtained. It is shown that these characteristics are sensitive to the strain rate. A unified interpretation of these rate effects, based on the structural–temporal approach, is hereby presented. It is demonstrated that the temporal dependence of the dynamic compressive and split tensile strengths of dry and saturated limestone samples can be predicted by the incubation time criterion. Previously discovered possibilities to optimize (minimize) the energy input for the failure process is discussed in connection with industrial rock failure processes. It is shown that the optimal energy input value associated with critical load, which is required to initialize failure in the rock media, strongly depends on the incubation time and the impact duration. The optimal load shapes, which minimize the momentum for a single failure impact, are demonstrated. Through this investigation, a possible approach to reduce the specific energy required for rock cutting by means of high-frequency vibrations is also discussed