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Recently, the Timoshenko beam, and generalizations thereof, have been considered as models for the vibration of single and multiple walled carbon nanotubes. Further, it has been shown that, in some instances, carbon nanotubes and nanotube sheets possess the common property that their Poisson’s ratio is negative. In these cases, the degenerate case of the Timoshenko beam problem, where E = k′G, becomes physically realizable. The asymptotic derivation of the vibration spectrum in this case is qualitatively different from that for the nondegenerate case. We performed such a derivation, and we compared the asymptotic results with numerical results. Further, we performed parameter studies based on the exact frequency equation for the undamped case for values of E/k′G on intervals including E/k′G = 1. The results show very interesting and complex behavior throughout a fairly large interval about E/k′G = 1.

A degenerate case of the Timoshenko beam problem with boundary damping: an asymptotic and numerical study