The non-linear flexural vibrations of thin circular rings are analyzed by means of the appropriate "shallow shell" equations. These partial differential equations are reduced to non-linear ordinary differential equations by assuming vibration modes and applying Galerkin's procedure. Vibrations involving primarily a single bending mode are investigated for three distinct cases, and the results indicate that the basic features of the problem are exhibited by an inextensional analysis.This information is then applied to simplify the analysis of vibrations in which several modes participate. A study of "self-coupled" bending modes shows that the single mode solution is valid only for certain combinations of amplitude and frequency: when the single mode exceeds a "critical amplitude", its companion mode is parametrically excited and participates in the motion.The general inextensional case (involving an infinite number of modes) is examined for two important sets of forces, and possible solutions are shown to be the excitation of primarily one or two bending modes. Stability analyses of these solutions indicate that when certain restrictions are met, all other bending modes play only a minor part in the vibration.An experimental study of the problem was also conducted. Theory and experiment both indicate a non-linearity of the softening type, the presence of ultraharmonic responses, and the appearance of the companion mode. Measurements of the steady-state response are in good agreement with the calculated values, and the experimentally determined mode shapes agree with the form of the assumed deflection.The analytical and experimental results exhibit several features that are common to the non-linear vibration of axisymmetric systems in general and to circular cylindrical shells in particular.
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