Resonant vibrations of free cylinders and disks
A complete solution is obtained for nonaxisymmetric resonant vibrations of a free cylinder or disk involving infinite sums. For axisymmetric longitudinal vibrations an alternative to previous solutions is included. In principle, the solutions satisfy exactly the stress‐free boundary conditions , in contrast to the approximate bending‐mode solutions due to Pickett or approximate solutions based on a small diameter/length ratio and small shearing stresses at the ends.Subject Classification: 40.26, 40.24; 20.40.
SUMMARY ::: ::: In this paper six distribution-free (nonparametric) stereological methods for the solution of Wicksell's corpuscle problem—i.e. the determination of the distribution of diameters of spheres embedded in an opaque specimen from the diameters of their profiles on plane sections—are compared as regards their numerical stability, sensitivity to underlying distributions and certain error criteria. The study is based on the results of simulation studies for several types of distribution (one-point, normal, exponential, logarithmic normal) of sphere diameters. Recommendations are suggested for the choice of methods, sample size and the optimal number of classes for grouping sample data.