Coupled twist-bending vibrations of incomplete elastic rings

Equations governing three-dimensional linear motions of elastic rings with generalized loadings and viscous damping are obtained making use of usual classical beam-theory assumptions. A list of admissible, or natural, boundary conditions is an important by-product of the minimal principle used. Particular attention is directed towards solving equations for the coupled out-of-plane bending and twisting dynamics. Results are given for incomplete rings with clamped ends. Two important parameters of the problem are the circular angle subtended, α, and the ratio of twisting to bending stiffness, k . The first four critical vibration frequencies for combinations of α = π , 3π 2 and 2π and k = 0·005, 0·2, 0·5, 1 and 1·625 are calculated and plotted. Several examples to demonstrate the generality of the results are offered and a practical range for the ratio of twisting to bending stiffness, 0 ⩽ k ⩽ 1·625, is derived.