Resolving Controversies in the Application of the Method of Multiple Scales and the Generalized Method of Averaging

I compare application of the method of multiple scales with reconstitution and the generalized method of averaging for determining higher-order approximations of three single-degree-of-freedom systems and a two-degree-of-freedom system. Three implementations of the method of multiple scales are considered, namely, application of the method to the system equations expressed as second-order equations, as first-order equations, and in complex-variable form. I show that all of these methods produce the same modulation equations.I address the problem of determining higher-order approximate solutions of the Duffing equation in the case of primary resonance. I show that the conclusions of Rahman and Burton that the method of multiple scales, the generalized method of averaging, and Lie series and transforms might lead to incorrect results, in that spurious solutions occur and the obtained frequency–response curves bear little resemblance to the actual response, is the result of their using parameter values for which the neglected terms are the same order as the retained terms. I show also that spurious solutions cannot be avoided, in general, in any consistent expansion and their presence does not constitute a limitation of the methods. In particular, I show that, for the Duffing equation, the second-order frequency–response equation does not possess spurious solutions for the case of hardening nonlinearity, but possesses spurious solutions for the case of softening nonlinearity. For sufficiently small nonlinearity, the spurious solutions are far removed from the actual response. But as the strength of the nonlinearity increases, these solutions move closer to the backbone and eventually distort it. This is not a drawback of the perturbation methods but an indication of an application of the analysis for parameter values outside the range of validity of the expansion.Also, I address the problem of obtaining non-Hamiltonian modulation equations in the application of the method of multiple scales to multi-degree-of-freedom Hamiltonian systems written as second-order equations in time and how this problem can be overcome by attacking the state-space form of the governing equations. Moreover, I show that application of a variation of the method of Rahman and Burton to multi-degree-of-freedom systems leads to results that do not agree with those obtained with the generalized method of averaging.Contributed by Prof. R.A. Ibrahim.     

A nonlinear composite plate theory

A general nonlinear theory for the dynamics of elastic anisotropic plates undergoing moderate-rotation vibrations is presented. The theory fully accounts for geometric nonlinearities (moderate rotations and displacements) by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvatures and strain-displacement expressions that contain the von Karman strains as a special case. The theory accounts for transverse shear deformations by using a third-order theory and for extensionality and changes in the configuration due to in-plane and transverse deformations. Five third-order nonlinear partial-differential equations of motion describing the extension-extension-bending-shear-shear vibrations of plates are obtained by an asymptotic analysis, which reveals that laminated plates display linear elastic and nonlinear geometric couplings among all motions.     

Monitoring of Anti Cancer Drug Letrozole by Fast Fourier Transform Continuous Cyclic Voltammetry at Gold Microelectrode

A continuous cyclic voltammetric study of letrozole at gold microelectrode was carried out. The drug in phosphate buffer (pH 2.0) is adsorbed at ?200 mV, giving rise to change in the current of well‐defined oxidation peak of gold in the flow injection system. The proposed detection method has some of advantages, the greatest of which are as follows: first, it is no more necessary to remove oxygen from the analyte solution and second, this is a very fast and appropriate technique for determination of the drug compound in a wide variety of chromatographic analysis methods. Signal‐to‐noise ratio has significantly increased by application of discrete Fast Fourier Transform (FFT) method, background subtraction and two‐dimensional integration of the electrode response over a selected potential range and time window. Also in this work some parameters such as sweep rate, eluent pH, and accumulation time and potential were optimized. The linear concentration range was of 1.0×10^?7?1.0×10^?10 mol/L (r=0.9975) with a limit of detection and quantitation 0.08 nmol/L and 0.15 nmol/L, respectively. The method has the requisite accuracy, sensitivity, precision and selectivity to assay letrozol in tablets. The influences of pH of eluent, accumulation potential, sweep rate, and accumulation time on the determination of the letrozol were considered.     

Keggin Heteropolyacid Catalyzed Synthesis of Isoxazolo[5,4-d]pyrimidine-4,6(5H,7H)-diones at Room Temperature

Isoxazolo[5,4‐d]pyrimidine‐4,6(5H,7H)diones 2a – 2f have been synthesized from the reaction of ethyl 5‐amino‐3‐methyl‐4‐isoxazole carboxylate ( 1 ) with aryl isocyanates in the presence of Keggin heteropolyacid H₃[PW₁₂O₄₀] as a green solid acid catalyst at room temperature in a one‐pot process in good yields.     

A convenient synthesis of 1,5-diarylpyrazoles from Baylis-Hillman adducts using HY-zeolite

<正>A facile and convenient protocol was developed for the synthesis of 1,5-diarylpyrazoles using Baylis-Hillman adducts in the presence of HY-zeolite as an efficient recyclable heterogeneous catalyst in reasonable reaction times(1.5-2.5 h) and high yields (78-90%).     

Selective and mild oxidation of sulfides to sulfoxides by H_2O_2 using DBUH-Br_3 as catalyst

<正>DBUH-Br_3 catalyzed selective conversion of sulfides to sulfoxides in the presence of H_2O_2 as oxidizing agent is described.The reaction was performed selectively at room temperature and relatively short reaction times.     

Comparative study of kinetic modeling for the oxidative coupling of methane by genetic and marquardt algorithms

Overall kinetic studies on the oxidative coupling of methane, OCM, have been conducted in a tubular fixed bed reactor, using perovskite titanate as the reaction catalyst. The appropriate operating conditions were found to be: temperature 750-775 ℃, total feed flow rate of 160 ml/min, CH4/O2 ratio of 2 and GHSV of 100 min-1. Under these conditions, C2 yield of 28% was achieved. Correlations of the kinetic data have been performed with lumped rate equations for C2 and COx formation as functions of temperature, O2 and CH4 partial pressures. Six models have been selected among the common lumped kinetic models. The selected models have been regressed with the experimental data which were obtained from the Catatest system by genetic algorithm in order to obtain optimized parameters. The kinetic coefficients in the overall reactions were optimized by different numerical optimization methods such as: the Levenberg-Marquardt and genetic algorithms and the results were compared with one another. It has been found that the Santamaria model is in good agreement with the experimental data. The Arrhenius parameters of this model have been obtained by linear regression. It should be noted that the Marquardt algorithm is sensitive to the first guesses and there is possibility to trap in the relative minimum.     

Deactivation studies of nano-structured iron catalyst in Fischer-Tropsch synthesis

A nano-structured iron catalyst for syngas conversion to hydrocarbons in Fischer-Tropsch synthesis (FTS) was prepared by micro-emulsion method. Compositions of bulk iron phase and phase transformations of carbonaceous species during catalyst deactivation in FTS reaction were characterized by temperature-programmed surface reaction with hydrogen (TPSR-H2), and XRD techniques. Many carbonaceous species on surface and bulk of the nano-structured iron catalysts were completely identified by combined TPSR-H2 and XRD spectra and which were compared with those recorded on conventional co-precipitated iron catalyst. The results reveal that the catalyst deactivation results from the formation of inactive carbide phases and surface carbonaceous species like graphite, and it will be increased when the particle size of iron oxides was reduced in FTS iron catalyst.     

Promoter effect on the CO2-H2O formation during Fischer-Tropsch synthesis on iron-based catalysts

The effects of Mg, La and Ca promoters on primary and secondary CO_2 and H_2O formation pathways during Fischer-Tropsch synthesis on precipitated Fe/Cu/SiO_2 catalysts are investigated. The chemisorbed oxygen atoms in the primary pathway formed in the CO dissociation steps reacted with co-adsorbed hydrogen or carbon monoxide to produce H_2O and CO_2 , respectively. The secondary pathway was the water-gas shift reaction. The results indicated that the CO_2 production led to an increase in both primary and secondary pathways, and H_2 O production decreased when surface basicity of the catalyst increased in the order Ca >Mg >La.