This work aims to investigate the parametric vibrations of a ring subjected to external excitations. Since the external excitations change the initial stress, the geometric nonlinearity deformation needs to be considered. A model of a ring rested on the elastic foundation is established by using Hamilton’s principle and the nonlinear in-extensional assumption. Based on this assumption, the external excitations arise at the stiffness coefficients of vibration equations, which are different from the current researches. As the main portion for forced vibrations, the external excitations not only affect natural characteristics but also lead to parametric instability. According to the average of excitations, the natural frequencies are demonstrated. Motivated by the planetary gear system, this study only focuses on the ring gear, and the equally spaced external excitations are considered. Generally, the instabilities caused by the rotational excitations can be eliminated by certain simple relations in the system parameters such as the count of excitations N and wavenumber n satisfying non-integer 2n/N. However, thanks to the presence of multiple frequencies such as the rotation frequency, mesh frequency, and fluctuation frequency, which create multi-frequency parametric excitations, the eliminating instability conditions are changed. The rotational frequency can be eliminated as expected, but the modulation frequencies caused by rotational frequency and mesh frequency occur. By discussing the parameters, several configurations are examined and the parametric vibrations are classified into seven cases. The effects of different parameters on the instabilities of several cases are analyzed and compared by using Floquét theory, where the parameters include the count of excitations, the fluctuation amplitude of the excitations, orientation angle, tooth number, the order of harmonics, rotation speed, and wavenumber. The results show that the parametric instabilities induced by the multi-frequency excitations are not a simple superposition of instability zones regarding each single-frequency excitation. Furthermore, the intersection regions of instabilities caused by each frequency are larger than expected. Besides, the instabilities excited simultaneously by the internal and external excitations are illustrated. It is worth noting the effects of internal and external excitations on instabilities are mutually independent. However, the factor dominating the instabilities needs to be compared and discussed. In other words, if the change of the stiffness is more remarkable than the external force, the instabilities are dominated by stiffness; otherwise they are dominated by the external force. The demonstration and analysis of this model could be an innovative understanding for similar rotational symmetry structures.

Effect of the External Multi-Frequency Excitations on Dynamics of a Ring Structure Rested on Elastic Foundation

Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2021/volume/85543

ASME 2021 International Mechanical Engineering Congress and Exposition (IMECE2021)

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