Calculation of probability density functions for nonlinear vibration systems

Technical systems are subjected to a variety of excitations that cannot generally be described in deterministic ways. Random excitations such as road roughness, wind gusts or loads on marine structures are commonly described by stochastic differential equations (SDEs). Given a set of SDEs, the main task is in finding probability density functions (PDFs), which yield statistical information about the system states. Monte-Carlo simulations represent a general way of generating PDFs, however, reliable integration methods can be time-consuming for complex systems. An alternative way of finding PDFs lies in the analysis of the Fokker-Planck equation, a partial differential equation of the PDF. Linear problems under Gaussian excitation can be solved analytically, which is the case only for a small class of nonlinear problems. As a result, there are a number of methods of approximating PDFs for general problems. Most of these are restricted to comparably low dimensions, such as d=4 ("curse of dimensionality"), limiting the relevance to technical applications. This paper presents solutions to problems of dimensions up to d=10, applying a Galerkin-method that expands approximate solutions into orthogonal polynomials. Problems include polynomial nonlinearities in damping and restoring terms, such as classical Duffing-elements, as well as other types of nonlinearities, demonstrated on a typical problem in vehicle dynamics. All results are compared with results from Monte-Carlo simulations or exact solutions, where available. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Ends of iterated function systems

Guided by classical concepts, we define the notion of ends of an iterated function system and prove that the number of ends is an upper bound for the number of nondegenerate components of its attractor. The remaining isolated points are then linked to idempotent maps. A commutative diagram illustrates the natural relationships between the infinite walks in a semigroup and components of an attractor in more detail. We show in particular that, if an iterated function system is one-ended, the associated attractor is connected, and ask whether every connected attractor (fractal) conversely admits a one-ended system.     

The hydrological response of heavy clay grassland soils to rainfall in south‐west England using δ2H

Stable isotopes of water have been previously used in catchment studies to separate rain‐event water from pre‐event groundwater. However, there are a lack of studies at the smaller scale looking at the separation of event water from pre‐event water. This is particularly relevant for heavy clay soil systems through which the movement of water is uncertain but is thought to be rainwater‐dominated. The data presented here were collected at a rural site in the south‐west of England. The historic rainfall at the site was isotopically varied but similar to the global meteoric water line, with annual weighted means of ?37‰ for δ²H and ?5.7‰ for δ¹⁸O and with no seasonal variation. Drainage was sampled from the inter‐flow (surface runoff + lateral through‐flow) and drain‐flow (55 cm deep mole drains) pathways of two 1 ha lysimeters during two rainfall events, which had δ²H values of ?68‰ and ?92‰, respectively. The δ²H values of the lysimeter drainage water suggest that there was no contribution of event water during the first, small discharge (Q) event; however, the second larger event did show isotopic variation in δ²H values negatively related to Q indicating that rainwater was contributing to Q. A hydrograph separation indicated that only 49–58% of the inter‐flow and 18–25% of the drain‐flow consisted of event water. This was surprising given that these soil types are considered retentive of soil water. More work is needed on heavy clay soils to understand better the nature of water movement from these systems. Copyright © 2010 John Wiley & Sons, Ltd.     

Certification of reference materials for ochratoxin A analysis in coffee and wine

Mycotoxins are important non-anthropogenic food and feed contaminants, which can be present on almost every agricultural commodity. Effective consumer protection therefore essentially depends on food surveillance by reliable quantitative analysis enabled by appropriate quality control. Certified (matrix) reference materials (CRMs) are versatile tools to support quality assurance. However, in the case of ochratoxin A (OTA), a hepato- and nephrotoxic mycotoxin, which is regulated in various foods, there is a lack of suitable CRMs. This lack has now been overcome by the development of two European Reference Materials (ERM^?) for the determination of OTA in roasted coffee (ERM^?-BD475) and red wine (ERM^?-BD476). This article discusses the material preparation process as well as the results of homogeneity and stability testing. Furthermore, the results of the in-house certification studies carried out at BAM Federal Institute for Materials Research and Testing are presented and discussed. Interlaboratory comparison studies involving selected expert laboratories with documented expertise in the field of mycotoxin analysis were conducted to confirm the certified values determined by BAM. The certified ochratoxin A values and their corresponding expanded uncertainties (k?=?2) were assigned in full compliance with the requirements of ISO Guide 35 and are as follows: (6.0?±?0.6)???g?kg^?1 for roasted coffee, ERM^?-BD475, and (0.52?±?0.11)???g?L^?1 for red wine, ERM^?-BD476.     

Verified convex hull and distance computation for octree-encoded objects

This paper discusses algorithms for computing verified convex hull and distance enclosure for objects represented by axis-aligned or unaligned octrees. To find a convex enclosure of an octree, the concept of extreme vertices of boxes on its boundary has been used. The convex hull of all extreme vertices yields an enclosure of the object. Thus, distance algorithms for convex polyhedra to obtain lower bounds for the distance between two octrees can be applied. Since using convex hulls makes it possible to avoid the unwanted wrapping effect that results from repeated decompositions, it also opens a way to dynamic distance algorithms for moving objects.     

One‐Step Synthesis of a [20]Silafullerane with an Endohedral Chloride Ion

Silicon analogues of the most prominent carbon nanostructures, namely, hollow spheroidals such as C₆₀ and the fullerene family, have been unknown to date. Herein we show that discrete Si₂₀ dodecahedra, stabilized by an endohedral guest and valence saturation, are accessible in preparative yields through a chloride‐induced disproportionation reaction of hexachlorodisilane in the presence of tri(n‐butyl)amine. X‐ray crystallography revealed that each silicon dodecahedron contains an endohedral chloride ion that imparts a net negative charge. Eight chloro substituents and twelve trichlorosilyl groups are attached to the surface of each cluster in a strictly regioregular arrangement, a thermodynamically preferred substitution pattern according to quantum‐chemical assessment. Our results demonstrate that the wet‐chemical self‐assembly of a complex, monodisperse Si nanostructure is possible under mild conditions starting from simple Si₂ building blocks.     

Boron‐Containing Polycyclic Aromatic Hydrocarbons: Facile Synthesis of Stable,Redox‐Active Luminophores

Herein we show that replacing the two meso carbon atoms of the polycyclic aromatic hydrocarbon (PAH) bisanthene by boron atoms transforms a near‐infrared dye into an efficient blue luminophore. This observation impressively illustrates the impact of boron doping on the frontier orbitals of PAHs. To take full advantage of this tool for the targeted design of organic electronic materials, the underlying structure–property relationships need to be further elucidated. We therefore developed a modular synthesis sequence based on a Peterson olefination, a stilbene‐type photocyclization, and an Si–B exchange reaction to substantially broaden the palette of accessible polycyclic aromatic organoboranes and to permit a direct comparison with their PAH congeners.     

Electrolyte-cathode interactions in 5-V lithium-ion cells

The electrolyte/electrode interactions on the anode side of a lithium-ion cell and the formation of the solid electrolyte interphase (SEI) have been investigated intensively in the past and are fairly well understood. Present knowledge about the reactions on the cathode side and the resulting cathode electrolyte interphase (CEI) is less detailed. In this study, the electrolyte/electrode interactions on the surface of the high-voltage cathode material LiNi_0.5Mn_1.5O₄ (LNMO), both bare and FePO₄-coated, were investigated. The gases evolving upon first time charging of the system were investigated using a GC/MS combination. The degradation products included THF, dimethyl peroxide, phosphor trifluoride, 1,3-dioxolane and dimethyl difluor silane, formed in the GC’s column as its coating reacts with HF from the experiments. Although these substances and their formation are in themselves interesting, the absence of many degradation products which have been mentioned in the existing literature is of equal interest. Our results clearly indicate that coating a cathode material can have a major influence on the amount and composition of the gaseous decomposition products in the formation phase.