Lyapunov definition and stability of regular or chaotic vibration modes in systems with several equilibrium positions

This paper deals with forced vibrations of two-DOF systems with more than one equilibrium positions. Such systems may be obtained by digitization of elastic post-buckling systems. A vibration mode, which is periodic at small force amplitudes and becomes chaotic as the force amplitudes are slowly increased, is selected. It is possible to formulate and solve the problem of stability of a periodic or chaotic vibration mode in a space with greater dimension using the classical Lyapunov stability definition and some calculating procedures. Instability of phase trajectories is used as a criterion of the chaotic behavior in dynamical systems. Trajectories with very close initial values are compared. Use of the Lyapunov stability definition shows mutual stability/instability of the trajectories. Calculations permit to observe an appearance and enlargement of the chaotic behavior regions. Specific results are obtained for the nonautonomous Duffing equation and pendulum system.     

Process of acceleration of filtration of viscose

Conclusions It is significant that the purification on a single passage of viscose through porous ceramic corresponds to the result of a two-stage filtration of it in industrial filter-presses with standard fillings.Kiev Combine. Kiev Technological Institute of Light Industry. Translated from Khimicheskie Volokna, No. 3, pp. 20–22, May–June, 1969.     

A study of Pt/WO<Subscript>4</Subscript><Superscript>2−</Superscript>/ZrO<Subscript>2</Subscript> catalyst deactivation in the hydroisomerization of heptane and a heptane–benzene mixture

The dynamics of the catalytic properties of a Pt/WO₄ ^2?/ZrO₂ catalyst (17 mol % WO₄ ^2?) in the hydroisomerization of heptane and a heptane–benzene mixture as a function of reaction time and temperature has been studied. It has been found that the heptane conversion decreases in the initial reaction period (within 2 h) and the isohexane selectivity symmetrically increases at 170–280°C, with the benzene hydrogenating activity decreasing at 250°C and higher temperatures. The observed changes in catalytic properties are attributed to the partial deactivation of both the acid and hydrogenation components of the catalyst. X-ray photoelectron spectroscopy has revealed the formation of carbon deposits on the catalyst surface; the structure of the deposits is determined by the reaction medium composition. The composition of the products formed on the catalyst after heptane hydroisomerization is mostly represented by aliphatic polymers of the “poly-С_хН_y” type, which are capable of blocking the active sites that catalyze the cracking reactions. The addition of benzene to heptane has led to the formation of graphite-like carbon deposits; this feature is apparently responsible for a decrease in both the acidity and hydrogenation activity of the catalyst. The phase state and textural characteristics of the catalysts are stable under the isomerization reaction conditions.     

Efficient Removal of Polyvalent Metal Impurities from KH<Subscript>2</Subscript>PO<Subscript>4</Subscript> by Crystallization

The separation factors of metal impurities for potassium dihydrogen phosphate crystallization from nonstoichiometric solutions have been experimentally determined. Based on the results obtained, a method has been proposed that makes it possible to obtain a material containing aluminum, iron, chromium, and titanium at a level of ~5 × 10^–2 ppm by weight from commercially available raw materials. In this method, a KH₂PO₄ solution is purified due to partial crystallization of the macrocomponent, so that the above impurities are concentrated in the solid phase. The purified solution is then boiled down with KOH additions.     

Nanocrystalline erbium oxide powders synthesized by the solid-state method from carbonates, oxalates, and hydroxides

Nanocrystalline erbium oxide powders are synthesized by the solid-state method from carbonates, oxalates, and hydroxides. Depending on the prehistory of their preparation, the average particle size decreases in the series: oxalates > carbonates > hydroxides; being 10 to 20 nm at a thermal treatment temperature of 680°C