On the averaging principle for one-frequency systems. Seminorm estimates for the error

We extend some previous results of our work [1] on the error of the averaging method, in the one-frequency case. The new error estimates apply to any separating family of seminorms on the space of the actions; they generalize our previous estimates in terms of the Euclidean norm. For example, one can use the new approach to get separate error estimates for each action coordinate. An application to rigid body under damping is presented. In a companion paper [2], the same method will be applied to the motion of a satellite around an oblate planet.Comment: LaTeX, 23 pages, 4 figures. The final version published in Nonlinear Dynamic

Estimation of cooling rates during close-coupled gas atomization using secondary dendrite arm spacing measurement

Al-4 wt pct Cu alloy has been gas atomized using a commercial close-coupled gas-atomization system. The resulting metal powders have been sieved into six size fractions, and the SDAS has been determined using electron microscopy. Cooling rates for the powders have been estimated using a range of published conversion factors for Al-Cu alloy, with reasonable agreement being found between sources. We find that cooling rates are very low relative to those often quoted for gas-atomized powders, of the order of 10 K s for sub-38 μm powders. We believe that a number of numerical studies of gas atomization have overestimated the cooling rate during solidification, probably as a consequence of overestimating the differential velocity between the gas and the particles. From the cooling rates measured in the current study, we estimate that such velocities are unlikely to exceed 20 m s

Understanding the rapid solidification of Al-4.3Cu and Al-17Cu using x-ray tomography

X-ray tomography using synchrotron radiation with 1-µm beam resolution at the European Synchrotron Radiation Facility was used to generate three-dimensional (3-D) images for rapidly solidified 660 µm A14.3 wt pct Cu and 500 µm Al17 wt pct Cu atomized droplets. Based on the observation of microstructural features in the 3-D images, physical phenomena of rapid solidification such as nucleation and initial growth, recalescence. and porosity were apparent and are reported. A single nucleation site was observed and the coordinates of this single nucleation site were evaluated. The volume fraction of the initial growth regime and recalescence during solidification was successfully quantified. It was also possible to semiquantitatively explain the porosity distribution within the droplets. These observations will be used to present a description of the sequence of the solidification events in these droplets