Longitudinal Vibrations of Rods of Finite Length with Radial Deformation.
ARMY ARMAMENT RESEARCH AND DEVELOPMENT CENTER WATERVLIET NY LARGE CALIBER WEAPON SYSTEMS LAB
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An approximate theory of dynamics of rods was formulated by Mindlin and Herrmann in 1951. The unique feature of this theory is that it can model the longitudinal as well as the radial motions of a rod, and yet it can retain the simplicity of a one-dimensional problem in the axial direction. Solutions pertaining to rods of infinite length were also given by Mindlin and Herrmann. This report presents vibration solutions of this rod model with finite lengths. First, the set of two partial differential equations is recapitulated together with appropriate boundary conditions. For vibration problems, two sets of eigenvalue problems are formulated to satisfy the simultaneous partial parameters are defined to describe the dispersion relations. These dual eigenvalue matrix equation then solved numerically. For an infinite rod, a dispersion relation of frequency versus wave number which contains an imaginary branch has been obtained. The free vibration problem of a fixed-fixed Mindlin-Herrmann rod has been solved. The numerical values of six 6 lowest frequencies, the associated wave numbers and mode shapes are tabulated for three different slenderness ratios. Author
- Numerical Mathematics