Chaotic vibrations of a beam with non-linear boundary conditions

Forced vibrations of an elastic beam with non-linear boundary conditions are shown to exhibit chaotic behavior of the strange attractor type for a sinusoidal input force. The beam is clamped at one end, and the other end is pinned for the tip displacement less than some fixed value and is free for displacements greater than this value. The stiffness of the beam has the properties of a bi-linear spring. The results may be typical of a class of mechanical oscillators with play or amplitude constraining stops. Subharmonic oscillations are found to be characteristic of these types of motions. For certain values of forcing frequency and amplitude the periodic motion becomes unstable and nonperiodic bounded vibrations result. These chaotic motions have a narrow band spectrum of frequency components near the subharmonic frequencies. Digital simulation of a single mode mathematical model of the beam using a Runge-Kutta algorithm is shown to give results qualitatively similar to experimental observations.     

Averaging methods for numerical simulations of flows through heterogeneous porous media

Many natural rock systems contain small patches of different permeability which affect the flow of fluids through them. As these heterogeneities become smaller and more numerous, they become harder to model numerically. We consider how to reduce the computational effort required in simulations by incorporating their effects in the boundary conditions at the edges of each grid block. This is in contrast with current methods which involve often arbitrary changes in the fluid properties. The method is restricted to the case of widely-spaced patches, which simplifies interaction effects. The system then reduces to an array of dipoles, and two averaging methods are proposed for finite grid blocks. Several infinite systems, including vertical and horizontal bands, are also considered as further approximations. There is a great wealth of existing results from different fields which lead to identical mathematical problems and which can be used in these cases. Finally, we consider how to use these techniques when the precise configuration of the grid block is not known, but only its statistical properties. This can lead to results which are very different from the deterministic case.     

Impulsive loading of ideal fibre-reinforced rigid-plastic beams—III Cantilever beam

The theory outlined in Part I is applied to the problem of a cantilever beam struck transversely at any point by a mass which subsequently adheres to the beam. In the subsequent motion, slope and velocity discontinuities propagate outwards from the point of impact. Solutions for the velocity and deflection of the various segments of the beam are obtained for the case of linear strain-hardening, and simpler approximate solutions are derived for the case of low impact velocity and/or slight strain-hardening. The discontinuity propagating towards the free end of the beam always comes to rest before it reaches this end, but for sufficiently high values of impact mass and velocity, and a strain-hardening parameter, one or more reflections of the discontinuity may occur at the fixed end of the beam and at the point of impact.     

Syntheses of (−)-gabosine A, (+)-4-epi-gabosine A, (−)-gabosine E, and (+)-4-epi-gabosine E

(+)-4-epi-Gabosine A 1 and (−)-gabosine A 2 have been synthesized starting from methyl α,d-glucopyranoside and methyl α,d-mannopyranoside, respectively, by utilizing Pd(0) catalyzed Stille coupling as the key step. On the other hand, syntheses of (+)-4-epi-gabosine E 3 and (−)-gabosine E 4 have been accomplished from methyl α,d-glucopyranoside and from methyl α,d-mannopyranoside, respectively, by utilizing DMAP catalyzed Morita-Baylis-Hillman reaction as the key step. Presence of acetyl group at C-6 position of sugar derived cyclic enone prevented the aromatization of MBH adduct. A plausible mechanism is also described.     

Redox non-innocence of thioether crowns: spectroelectrochemistry and electronic structure of formal nickel(III) complexes of aza-thioether macrocycles

The Ni^II complexes [Ni([9]aneNS₂‐CH₃)₂]²⁺ ([9]aneNS₂‐CH₃=N‐methyl‐1‐aza‐4,7‐dithiacyclononane), [Ni(bis[9]aneNS₂‐C₂H₄)]²⁺ (bis[9]aneNS₂‐C₂H₄=1,2‐bis‐(1‐aza‐4,7‐dithiacyclononylethane) and [Ni([9]aneS₃)₂]²⁺ ([9]aneS₃=1,4,7‐trithiacyclononane) have been prepared and can be electrochemically and chemically oxidized to give the formal Ni^III products, which have been characterized by X‐ray crystallography, UV/Vis and multi‐frequency EPR spectroscopy. The single‐crystal X‐ray structure of [Ni^III([9]aneNS₂‐CH₃)₂](ClO₄)₆?(H₅O₂)₃ reveals an octahedral co‐ordination at the Ni centre, while the crystal structure of [Ni^III(bis[9]aneNS₂‐C₂H₄)](ClO₄)₆?(H₃O)₃? 3H₂O exhibits a more distorted co‐ordination. In the homoleptic analogue, [Ni^III([9]aneS₃)₂](ClO₄)₃, structurally characterized at 30 K, the Ni? S distances [2.249(6), 2.251(5) and 2.437(2) Å] are consistent with a Jahn–Teller distorted octahedral stereochemistry. [Ni([9]aneNS₂‐CH₃)₂](PF₆)₂ shows a one‐electron oxidation process in MeCN (0.2 M NBu₄PF₆, 293 K) at E_1/2=+1.10 V versus Fc⁺/Fc assigned to a formal Ni^III/Ni^II couple. [Ni(bis[9]aneNS₂‐C₂H₄)](PF₆)₂ exhibits a one‐electron oxidation process at E_1/2=+0.98 V and a reduction process at E_1/2=?1.25 V assigned to Ni^II/Ni^III and Ni^II/Ni^I couples, respectively. The multi‐frequency X‐, L‐, S‐, K‐band EPR spectra of the 3+ cations and their 86.2 % ⁶¹Ni‐enriched analogues were simulated. Treatment of the spin Hamiltonian parameters by perturbation theory reveals that the SOMO has 50.6 %, 42.8 % and 37.2 % Ni character in [Ni([9]aneNS₂‐CH₃)₂]³⁺, [Ni(bis[9]aneNS₂‐C₂H₄)]³⁺ and [Ni([9]aneS₃)₂]³⁺, respectively, consistent with DFT calculations, and reflecting delocalisation of charge onto the S‐thioether centres. EPR spectra for [⁶¹Ni([9]aneS₃)₂]³⁺ are consistent with a dynamic Jahn–Teller distortion in this compound.     

Hydroxyl-directed cyclizations of 1,6-enynes

The palladium-catalyzed, hydroxyl-directed cyclization reactions of 1,6-enynes provide a highly diastereoselective process for the syntheses of stereochemically defined cyclopentanes. Consistently high levels of cis-selectivity are possible using homopropargyl alcohols in contrast to the corresponding propargyl alcohols. Hydroborylative enyne cyclizations coupled with this directing group effect provide a useful method for the syntheses of multifaceted compounds bearing all carbon quaternary centers.     

Chiral pyridin-3-ones and pyridines: syntheses of enantiopure 2,4-disubstituted 6-hydroxy-1,6-dihydro-2H-pyridin-3-ones, 2,3-disubstituted 4-iodopyridines, and enantiopure 2,3-disubstituted 4-pyridinemethanols

The development of an innovative method to access enantiopure 2,4-disubstituted 6-hydroxy-1,6-dihydro-2H-pyridin-3-ones starting from D-glucal via the aza-Achmatowicz transformation has been described. These highly functionalized pyridin-3-ones have been utilized for the synthesis of contiguously substituted pyridines through a rapid and efficient Et(3)N/Ac(2)O promoted cyclo-elimination, aromatization cascade, allowing the facile assembly of important pyridine-based building blocks like 2-substituted 3-acetoxy-4-iodopyridines and enantiopure 2-substituted 3-acetoxy-4-pyridinemethanols possessing benzylic stereogenic centers, whose synthesis otherwise would be tedious. The utilization of commercially available sugars as starting materials, mild reaction conditions, catalytic transfer hydrogen (CTH) of α-furfuryl azide derivatives, transfer of chiral aryl/alkyl methanols from enulosides to pyridin-3-ones and pyridines, high yields, and short reaction times are key features of this method. The utility of the method has been further exemplified by demonstrating the usage of the 2-substituted 3-acetoxy-4-iodopyridine for the construction of biologically significant molecules like 2,7-disubstituted furo[2,3-c]pyridines and 7,7'-disubstituted 2,2'-bifuro[2,3-c]pyridines.     

Non-hemolytic α-AApeptides as antimicrobial peptidomimetics

We report a new class of peptide mimetics, α-AApeptides, that display broad-spectrum activity against both Gram-negative and Gram-positive bacteria and fungi. With non-hemolytic activity, resistance to protease hydrolysis, and easy sequence programmability, α-AApeptides may emerge as a novel class of antibiotics.