Hopf bifurcation analysis of asymmetrical rotating shafts

The Hopf and double Hopf bifurcations analysis of asymmetrical rotating shafts with stretching nonlinearity are investigated. The shaft is simply supported and is composed of viscoelastic material. The rotary inertia and gyroscopic effect are considered, but, shear deformation is neglected. To consider the viscoelastic behavior of the shaft, the Kelvin–Voigt model is used. Hopf bifurcations occur due to instability caused by internal damping. To analyze the dynamics of the system in the vicinity of Hopf bifurcations, the center manifold theory is utilized. The standard normal forms of Hopf bifurcations for symmetrical and asymmetrical shafts are obtained. It is shown that the symmetrical shafts have double zero eigenvalues in the absence of external damping, but asymmetrical shafts do not have. The asymmetrical shaft in the absence of external damping has a saddle point, therefore the system is unstable. Also, for symmetrical and asymmetrical shafts, in the presence of external damping at the critical speeds, supercritical Hopf bifurcations occur. The amplitude of periodic solution due to supercritical Hopf bifurcations for symmetrical and asymmetrical shafts for the higher modes would be different, due to shaft asymmetry. Consequently, the effect of shaft asymmetry in the higher modes is considerable. Also, the amplitude of periodic solutions for symmetrical shafts with rotary inertia effect is higher than those of without one. In addition, the dynamic behavior of the system in the vicinity of double Hopf bifurcation is investigated. It is seen that in this case depending on the damping and rotational speed, the sink, source, or saddle equilibrium points occur in the system.     

Carbon‐iodide bond activation by cyclometalated Pt (II) complexes bearing tricyclohexylphosphine ligand: A comparative kinetic study and theoretical elucidation

Cyclometalated Pt (II) complexes [PtMe(C^N)(L)], in which C^N = deprotonated 2,2′‐bipyridine N‐oxide (Obpy), 1 , deprotonated 2‐phenylpyridine (ppy), 2 , deprotonated benzo [h] quinolone (bzq), 3 , and L = tricyclohexylphosphine (PCy₃) were prepared and fully characterized. By treatment of 1–3 with excess MeI, the thermodynamically favored Pt (IV) complexes cis‐[PtMe₂I(C^N)(PCy₃)] (C^N = Obpy, 1a ; ppy, 2a ; and bzq, 3a ) were obtained as the major products in which the incoming methyl and iodine groups adopted cis positions relative to each other. All the complexes were characterized by means of NMR spectroscopy while the absolute configuration of 1a was further determined by X‐ray crystal structure analysis. The reaction of methyl iodide with 1–3 were kinetically explored using UV–vis spectroscopy. On the basis of the kinetic data together with the time‐resolved NMR investigation, it was established that the oxidative addition reaction occurred through the classical S_N2 attack of Pt (II) center on the MeI reagent. Moreover, comparative kinetic studies demonstrated that the electronic and steric nature of either the cyclometalating ligands or the phosphine ligand influence the rate of reaction. Surprisingly, by extending the oxidative addition reaction time, very stable iodine‐bridged Pt (IV)‐Pt (IV) complexes [Pt₂Me₄(C^N)₂(μ‐I)₂] (C^N = Obpy, 1b ; ppy, 2b ; and bzq, 3b ) were obtained and isolated. In order to find a reasonable explanation for the observation, a DFT (density functional theory) computational analysis was undertaken and it was found that the results were consistent with the experimental findings.     

Copper-doped silica cuprous sulfate (CDSCS) as a highly efficient and new heterogeneous nano catalyst for [3+2] Huisgen cycloaddition

The synthesis and characterization of copper-doped silica cuprous sulfate (CDSCS) as a new and efficient heterogeneous nano catalyst are described. CDSCS has been fully characterized by different microscopic, spectroscopic and physical techniques, including scanning electron microscopy (SEM), transmission electron microscopy (TEM), atomic forced microscopy (AFM), X-ray diffraction (XRD), inductively coupled plasma (ICP) analysis, thermogravimetric analysis (TGA) and FT-IR. CDSCS is proved to be a useful heterogeneous nano catalyst in Cu(I)-catalyzed ‘Click’ cycloaddition of organic azides with terminal alkynes. CDSCS catalyzes the 1,3-dipolar cycloaddition reactions of β-azido alcohols and alkynes at room temperature, in THF/H₂O (1:1, v/v) solution. Using CDSCS, 1,4-disubstituted 1,2,3-triazole adducts are mainly obtained, in good to excellent yields and in short reaction times. These compounds have featural resemblance to β-adrenoceptor blocking agents. CDSCS was approved as a chemically and thermally stable nano catalyst that can be reused for many consecutive trials without a significant decline in its reactivity.     

A Novel Polymeric Lead(II)‐Azido Compound: Synthesis,Structural Characterization,and DFT Calculations of [Pb(dmp)(N3)2]n

A novel 1D Pb^II coordination polymer containing Pb₂‐(μ‐N₃)₂ unit [Pb(dmp)(N₃)₂]_n (dmp =  2,9‐dimethyl‐1,10‐phenanthroline) has been prepared and characterized. Single‐crystal X‐ray diffraction analyses show that the coordination number for Pb^II ions is six, PbN₆, with “stereochemically active” electron lone pairs and the coordination sphere being hemidirected. The single‐crystal X‐ray data show the chains interact with each other through the π–π stacking interactions, which create a 3D framework. The structure of title complex has been optimized by density functional theory. Structural parameters and IR spectra for the complex are in agreement with the crystal structure.     

Forced oscillations and stability analysis of a nonlinear micro-rotating shaft incorporating a non-classical theory

In this paper, the stability and bifurcation analysis of symmetrical and asymmetrical micro-rotating shafts are investigated when the rotational speed is in the vicinity of the critical speed. With the help of Hamilton’s principle, nonlinear equations of motion are derived based on non-classical theories such as the strain gradient theory. In the dynamic modeling, the geometric nonlinearities due to strains, and strain gradients are considered. The bifurcations and steady state solution are compared between the classical theory and the non-classical theories. It is observed that using a non-classical theory has considerable effect in the steady-state response and bifurcations of the system. As a result, under the classical theory, the symmetrical shaft becomes completely stable in the least damping coefficient, while the asymmetrical shaft becomes completely stable in the highest damping coefficient. Under the modified strain gradient theory, the symmetrical shaft becomes completely stable in the least total eccentricity, and under the classical theory the asymmetrical shaft becomes completely stable in the highest total eccentricity. Also, it is shown that by increasing the ratio of the radius of gyration per length scale parameter, the results of the non-classical theory approach those of the classical theory.     

Engineering the nonlinear phase shift

We treat the nonlinear phase shift response in the weak perturbation limit as a linear digital filter that can be synthesized into the values of its poles and zeros and mapped onto an optical architecture. This procedure results in a significant enhancement in the nonlinear sensitivity with a response that is robust to frequency changes within the filter passband. A precompensation technique can be used to reduce distortions under strongly driven nonlinear operation to achieve a larger phase shift. We also show that nonlinear sensitivity improves with increasing filter group delay and can be increased within constant linear bandwidth by use of higher-order filters.     

Efficient access to novel tetra- and pentacyclic dihydroquinolin-2-ones by catalyst-free domino Knöevenagel hetero-Diels–Alder reactions from <Emphasis Type="Italic">N</Emphasis>-(2-formylphenyl)-<Emphasis Type="Italic">N</Emphasis>-methylcinnamamides and cyclic 1,3-dicarbonyls in water

An efficient catalyst-free synthesis of novel annulated hybrid derivatives of two known scaffolds, dihydroquinolinone and pyranopyranone, pyranopyrimidinedione, pyranocoumarin or chromenone is described. N-(2-Formylphenyl)-N-methylcinnamamides underwent a one-pot domino Knöevenagel hetero-Diels–Alder reaction with dimedone, N,N-dimethylbarbituric acid, 1,3-indanedione, 4-hydroxycoumarins and 4-hydroxy-6-methyl-2H-pyran-2-one in water, affording the desired tetra and pentacyclic pyranoquinolinones in excellent yields.     

One-pot synthesis of 3,4-dihydropyrimidin-2(1H)-ones,thiones and 2-selenoxo DHPMs using 1-butyl-3-methylimidazolium hydrogen sulfate as non-halogenated ionic liquid

Abstract

1-Butyl-3-methylimidazolium hydrogen sulfate ([BMIM][HSO₄]) as a non-halogenated ionic liquid (IL) was used for the synthesis of 3,4-dihydropyrimidin-2(1H)-ones and thiones or 2-selenoxo DHPMs in a Biginelli type multi-component reaction. By using this ionic liquid, the reaction time was significantly reduced and the products were obtained in good to high yields. Also, in this method, the synthesis of novel 2-selenoxo DHPMs is introduced in the presence of this ionic liquid and their structures were determined by ¹H and ¹³C NMR, FT-IR, and Elemental analysis.     

‘Click Synthesis’ of Some Novel Benzimidazol Oxime Ethers Containing Heterocycle Residues as Potential ß‐Adrenergic Blocking Agents

The ‘click synthesis’ of some novel O‐substituted oximes, 5a – 5j , which contain heterocycle residues, as new analogs of ß‐adrenoceptor antagonists is described (Scheme 1). The synthesis of these compounds was achieved in four steps. The formation of (E)‐2‐(1H‐benzo[d]imidazol‐1‐yl)‐1‐phenylethanone oxime, followed by their reaction with 2‐(chloromethyl)oxirane, afforded mixture of oil compounds 3 and 4 , which by a subsequent tetra‐n‐butylammonium bromide (TBAB)‐catalyzed reaction with N H heterocycle compounds (Scheme 1), led to the target compounds 5a – 5j in good yields.