Radial Vibrations of Thin Rubber Rings

Abstract

The equations of motion of linear elasticity in cylindrical coordinates are reduced to an axisymmetric plane-stress situation valid for thin elastic rings. The eigenvalue problem for the natural frequencies of the system is solved in two cases: first, when both rims of the ring are subjected to stress-type (third type for the displacement) boundary conditions having real coefficients; second, when the inner rim is stress-free and the outer rim is fixed. The characteristic equations in both cases are determined and solved numerically for almost incompressible materials of Poisson's ratio v very close to O.S. The first few roots are plotted in a nondimensional form versus the outer-to-inner radii ratio. The eigenwavenumbers of the first case are shown to be substantially below those of the second. The transient vibration problem for the elastic displacement field in the ring is then formally solved by means of eigen-function expansions in the first case only. The analysis is extended to viscoelastic rings (Kelvin-Voigt model) and under the same conditions, the effect of viscosity is to shift the eigenvalues from the real line into the complex wavenumber plane. Parts of this analysis are applicable to the design of (underwater) acoustic resonant coatings.