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Abstract The ring-shaped periodic structures subjected to multiple moving loads are widely used in engineering practice, for example, the ring structure in a vibratory gyroscope, the ring gears in planetary gear trains, and the stators and rotors in electrical machines. Vibrations can be induced by the moving loads that significantly affect the working performance of the systems containing such structures. Since the moving loads can lead to time-variant effects, which can further cause parametric vibrations, the instability behaviors arise. Because such vibrations can damage the system on a permanent basis, the corresponding analyses and estimations become very important. The governing equation of motion for ring-shaped periodic structures is established by using Hamilton’s principle in the conventional inertial coordinate system, and thus a dynamic model with time-variant coefficients is obtained. In order to improve calculation efficiency, a mathematical transformation is introduced to remove the time-variant effect. Based on this model, the corresponding time-invariant version is obtained and discretized into ordinary differential equations using the Galerkin method. According to the time-invariant model, the eigenvalues are calculated on the basis of classical vibration theory to estimate the instability behaviors. For verification purpose, the unstable regions of the time-variant model are calculated using the Floquét theory. Then these regions are also calculated based on the time-invariant model. The consistent results verify the main idea of the transformation treatment, and at the same time, reveal the types of instability of the load-moving system, including the divergent and flutter instabilities. The results imply that the unstable regions are reduced with an increase in the vibration wavenumber. Meanwhile, this increase can also be obtained by selecting the combinations of the rotating supports’ rotation speeds and inclination angles. This study lays a theoretical foundation for the performance estimation and practical design of such ring-shaped periodic structures subjected to multiple moving loads.
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Received: 17 July 2018
Published: 11 April 2019
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2019, Vol. 40 
