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)     

The gonality sequence of a curve with an involution

If a curve X of genus g is a double covering of a curve C of genus h such that g ≥ 6h?3 ≥ 9, there is an explicit relation between the gonality sequences of X and C. In particular, it shows that X violates the slope inequalities if and only if C does. This provides new examples of curves X violating the slope inequalities.     

Comparison between 2-hydroxypropyl-��-cyclodextrin and 2-hydroxypropyl-��-cyclodextrin for inclusion complex formation with danazol

Complexation in solution between danazol and two different cyclodextrins [2-hydroxypropyl-??-cyclodextrin (HP-??-CD) and 2-hydroxypropyl-??-cyclodextrin (HP-??-CD)] was studied using phase solubility analysis, and one- and two-dimensional ¹H-NMR. The increase of danazol solubility in the aqueous cyclodextrin solutions showed a linear relationship (A_L profile). The apparent stability constant, K _1:1, of each complex was calculated and found to be 51.7 × 10³ and 7.3 × 10³ M^?1 for danazol?CHP-??-CD and danazol?CHP-??-CD, respectively. ¹H-NMR spectroscopic analysis of varying ratios of danazol and the different cyclodextrins in a mixture of EtOD?CD₂O confirmed the 1:1 stoichiometry. Cross-peaks, from 2D ROESY ¹H-NMR spectra, between protons of danazol and H3?? and H5??of cyclodextrins, which stay inside the cyclodextrin cavity, proved the formation of an inclusion complex between danazol and the cyclodextrins. For HP-??-CD, the inclusion complex is formed by entrance of the isooxazole and the A rings of danazol in the cyclodextrin cavity. For HP-??-CD, two different inclusion structures may exist simultaneously in solution: one with the isooxazole and A ring in the cavity and the other with the C and D ring inside the cavity. DLS showed that self-aggregation of the CD??s was absent in the danazol HP-??-CD system up to a CD concentration of 10% and in the danazol HP-??-CD system up to a CD concentration of 5%.     

Experimental and Theoretical Investigation of the Proton-Bound Dimer of Lysine

The structure of the proton-bound lysine dimer has been investigated by infrared multiple photon dissociation (IRMPD) spectroscopy and electronic structure calculations. The structures of different possible isomers of the proton-bound lysine dimer have been optimized at the B3LYP/6-31 + G(d) level of theory and IR spectra calculated using the same computational method. Based on relative Gibbs free energies (298 K) calculated at the MP2/aug-cc-pVTZ//B3LYP/6-31 + G(d) level of theory, LL-CS01, and followed closely (1.1 kJ mol^–1) by LL-CS02 are the most stable non-zwitterionic isomers. At the MP2/aug-cc-pVTZ//6-31 + G(d) and MP2/aug-cc-pVTZ//6-31 + (d,p) levels of theory, isomer LL-CS02 is favored by 3.0 and 2.3 kJ mol^–1, respectively. The relative Gibbs free energies calculated by the aforementioned levels of theory for LL-CS01 and LL-CS02 are very close and strongly suggest that diagnostic vibrational signatures found in the IRMPD spectrum of the proton-bound dimer of lysine can be attributed to the existence of both isomers. LL-ZW01 is the most stable zwitterionic isomer, in which the zwitterionic structure of the neutral lysine is well stabilized by the protonated lysine moiety via a very strong intermolecular hydrogen bond. At the MP2/aug-cc-pVTZ//B3LYP/6-31 + G(d), MP2/aug-cc-pVTZ//6-31 + G(d) and MP2/aug-cc-pVTZ//6-31 + G(d,p) levels of theory, the most stable zwitterionic isomer (LL-ZW01) is less favored than LL-CS01 by 7.3, 4.1 and 2.3 kJ mol^–1, respectively. The experimental IRMPD spectrum also confirms that the proton-bound dimer of lysine largely exists as charge-solvated isomers. Investigation of zwitterionic and charge-solvated species of amino acids in the gas phase will aid in a further understanding of structure, property, and function of biological molecules.     

Continuous synthesis process of hexagonal nanoplates of P6m ordered mesoporous silica

Hexagonally ordered mesoporous silica coined COK-12 was synthesized in a continuous process by combining streams of sodium silicate and citric acid/sodium citrate buffered solution of (ethylene oxide)(20)-(propylene oxide)(70)-(ethylene oxide)(20) triblock copolymer (Pluronic P123) from separate reservoirs. COK-12 precipitated spontaneously upon combining both streams at nearly neutral pH and ambient temperature. Stable intermediates of the COK-12 formation process could be prepared by limiting sodium silicate addition. Investigation of these intermediates using small-angle X-ray scattering revealed COK-12 formed via an assembly process departing from spherical uncharged core-shell P123-silica micelles. The sterical stabilization of these micelles decreased upon accumulation of silicate oligomers in their shell. Aggregation of the spherical micelles led to cylindrical micelles, which aligned and adopted the final hexagonal organization. This unprecedentedly fast formation of P6m ordered mesoporous silica was caused by two factors in the synthesis medium: the neutral pH favoring uncharged silicate oligomers and the high salt concentration promoting hydrophobic interactions with surfactant micelles leading to silica accumulation in the PEO shell. The easy continuous synthesis process is convenient for large-scale production. The platelet particle morphology with short and identical internal channels will be advantageous for many applications such as pore replication, nanotube or fiber growth, catalytic functionalization, drug delivery, film and sensor development, and in nano dyes as well as for investigation of pore diffusion phenomena.     

Surface modification of alumina nanofibres for the selective adsorption of alachlor and imazaquin herbicides

The effective removal of pollutants using a thermally and chemically stable substrate that has controllable absorption properties is a goal of water treatment. In this study, the surfaces of thin alumina (γ-Al(2)O(3)) nanofibres were modified by the grafting either of two organosilane agents, 3-chloro-propyl-triethoxysilane (CPTES) and octyl-triethoxysilane (OTES). These modified materials were then trialed as absorbents for the removal of two herbicides, alachlor and imazaquin from water. The formation of organic groups during the functionalisation process established super hydrophobic sites on the surfaces of the nanofibres. This super hydrophobic group is a kind of protruding adsorption site which facilitates the intimate contact with the pollutants. OTES grafted substrate were shown to be more selective for alachlor while imazaquin selectivity is shown by the CPTES grafted substrate. Kinetics studies revealed that imazaquin was rapidly adsorbed on CPTES-modified surfaces. However, the adsorption of alachlor by OTES grafted surface was achieved more slowly.     

Tridentate ionic hydrogen-bonding interactions of the 5-fluorocytosine cationic dimer and other 5-fluorocytosine analogues characterized by IRMPD spectroscopy and electronic structure calculations

Ionic hydrogen-bonding interactions have been found in several clusters formed by 5-fluorocytosine (5-FC). The chloride and trimethylammonium cluster ions, along with the cationic (proton-bound) dimer have been characterized by infrared multiple-photon dissociation (IRMPD) spectroscopy and electronic structure calculations performed at the B2PLYP/aug-cc-pVTZ//B3LYP/6-311+G(d,p) level of theory. IRMPD action spectra, in combination with calculated spectra and relative energetics, indicate that it is most probable that predominantly a single isomer exists in each experiment. For the 5-FC-trimethylammonium cluster specifically, the calculated spectrum of the lowest-energy isomer convincingly matches the experimental spectrum. Interestingly, the cationic dimer of 5-FC was found to have a single energetically relevant isomer (Cationic-IV) involving a tridentate ionic hydrogen-bonding interaction. The three sites of intermolecular ionic hydrogen bonds in this isomer interact very efficiently, leading to a significant calculated binding energy of 180 kJ/mol. The magnitude of the calculated binding energy for this species, in combination with the strong correlation between the simulated and IRMPD spectra, suggests that a tridentate-proton-bound dimer was observed predominantly in the experiments. Comparison of the calculated relative Gibbs free energies (298 K) for this species and several of the other isomers considered also supports the likelihood of the dominant protonated dimer existing as Cationic-IV.