On the chaotic rotation of a liquid-filled gyrostat via the Melnikov-Holmes-Marsden integral

The non-linear motions of a gyrostat with an axisymmetrical, fluid-filled cavity are investigated. The cavity is considered to be completely filled with an ideal incompressible liquid performing uniform rotational motion. Helmholtz theorem, Euler's angular momentum theorem and Poisson equations are used to develop the disturbed Hamiltonian equations of the motions of the liquid-filled gyrostat subjected to small perturbing moments. The equations are established in terms of a set of canonical variables comprised of Euler angles and the conjugate angular momenta in order to facilitate the application of the Melnikov-Holmes-Marsden (MHM) method to investigate homoclinic/heteroclinic transversal intersections. In such a way, a criterion for the onset of chaotic oscillations is formulated for liquid-filled gyrostats with ellipsoidal and torus-shaped cavities and the results are confirmed via numerical simulations. (c) 2006 Elsevier Ltd. All rights reserved.

Automatic measurement of vertebral rotation in idiopathic scoliosis

Study Design. Development of an automatic measurement algorithm and comparison with manual measurement methods. Objectives. To develop a new computer-based method for automatic measurement of vertebral rotation in idiopathic scoliosis from computed tomography images and to compare the automatic method with two manual measurement techniques. Summary of Background Data. Techniques have been developed for vertebral rotation measurement in idiopathic scoliosis using plain radiographs, computed tomography, or magnetic resonance images. All of these techniques require manual selection of landmark points and are therefore subject to interobserver and intraobserver error. Methods. We developed a new method for automatic measurement of vertebral rotation in idiopathic scoliosis using a symmetry ratio algorithm. The automatic method provided values comparable with Aaro and Ho's manual measurement methods for a set of 19 transverse computed tomography slices through apical vertebrae, and with Aaro's method for a set of 204 reformatted computed tomography images through vertebral endplates. Results. Confidence intervals (95%) for intraobserver and interobserver variability using manual methods were in the range 5.5 to 7.2. The mean (+/- SD) difference between automatic and manual rotation measurements for the 19 apical images was -0.5 degrees +/- 3.3 degrees for Aaro's method and 0.7 degrees +/- 3.4 degrees for Ho's method. The mean (+/- SD) difference between automatic and manual rotation measurements for the 204 endplate images was 0.25 degrees +/- 3.8 degrees. Conclusions. The symmetry ratio algorithm allows automatic measurement of vertebral rotation in idiopathic scoliosis without intraobserver or interobserver error due to landmark point selection.