Solution and asymptotic analysis of a boundary value problem in the spring–mass model of running

We consider the classic spring–mass model of running which is built upon an inverted elastic pendulum. In a natural way, there arises an interesting boundary value problem for the governing system of two nonlinear ordinary differential equations. It requires us to choose the stiffness to ascertain that after a complete step, the spring returns to its equilibrium position. Motivated by numerical calculations and real data, we conduct a rigorous asymptotic analysis in terms of the Poicaré–Lindstedt series. The perturbation expansion is furnished by an interplay of two time scales what has an significant impact on the order of convergence. Further, we use these asymptotic estimates to prove that there exists a unique solution to the aforementioned boundary value problem and provide an approximation to the sought stiffness. Our results rigorously explain several observations made by other researchers concerning the dependence of stiffness on the initial angle of the stride and its velocity. The theory is illustrated with a number of numerical calculations.

     

Betaine and Carnitine Derivatives as Herbicidal Ionic Liquids

This study focused on the synthesis and subsequent characterization of herbicidal ionic liquids based on betaine and carnitine, two derivatives of amino acids, which were used as cations. Four commonly used herbicides (2,4‐D, MCPA, MCPP and Dicamba) were used as anions in simple (single anion) and oligomeric (two anions) salts. The obtained salts were subjected to analyzes regarding physicochemical properties (density, viscosity, refractive index, thermal decomposition profiles and solubility) as well as evaluation of their herbicidal activity under greenhouse and field conditions, toxicity towards rats and biodegradability. The obtained results suggest that the synthesized herbicidal ionic liquids displayed low toxicity (classified as category 4 compounds) and showed similar or improved efficacy against weed compared to reference herbicides. The highest increase was observed during field trials for salts containing 2,4‐D as the anion, which also exhibited the highest biodegradability (>75 %).     

Ferromagnetic core coil hysteresis modeling using fractional derivatives

Nonlinear Dynamics - The modeling of a ferromagnetic core coil magnetic hysteresis has been considered. The measurement basis consisted of waveforms that have been recorded for various levels of...