On general in-plane vibrations of rotating thick and thin rings

Abstract

Complete sets of equations of motion for general in-plane vibrations of rotating thick and thin rings on elastic foundations are derived. The thick ring theory takes account of the effects of extensibility, transverse shear, rotatory inertia and rotation, as well as the effects of initial stresses due to pressure loads and centrifugal force. The consistency of the formulation is demonstrated by reducing it to classical ring theories based on inextensional or negligible shear deformation assumptions. Predictions of frequencies for bending and extensional type vibrations are close to experimental data for non-rotating thick rings [1,2]. It is argued that inextensional assumptions in thick ring theories are improper because extensional coupling effects are as important as shearing effects, espically for the rotating ring. The correct quadratic displacement terms in the strain-displacement relations, necessary for the complete description of initial stress effects, are also discussed.