Analysis On Parametric Vibration Stability Of Rotational Ring-shaped Periodic Structures

The ring-shaped periodic structure is widely used in engineering fields,such as the ring gear of the planetary gear transmissions,the stator and rotor of the rotating motors,the inner and outer ring of the rolling bearings,and micro-gyroscopes.In general,these mechanical constructions usually have ring-shaped symmetrical structures.Parametric vibration can be induced due to the contact/non-contact,distributed/discrete,or stationary/rotating loads.This thesis studies the dynamic characteristics of such structures deeply in order to improve their dynamics performance,and reveals the mapping relationship between the key parameters and vibration characteristics as well as instabilities.The study provides a theoretical guidance for parameters design and the reasonable choice of working conditions.The main contents include:(1)For the three-dimensional parametric elastic vibration of the rotational ring-shaped periodic structures,a dynamic model with time-variant coefficients is established by using Hamilton’s principle under the inertial frame.Convert it to the follow-up coordinate system by introducing coordinates transformation and an easily-solved time-invariant version can be obtained.A set of ordinary differential kinetic equations of the model are obtained by using Galerkin method.The modal properties and dynamic stability are estimated by means of the eigenvalue.Besides,the unstable regions are calculated based on Floquét theory with an aim to verify the analytical results.(2)Based on the three-dimensional complete model of the rotational ring-shaped periodic structures,the parametric vibration behavior of the simplified models is studied.The eigenvalues under different simplified models were calculated by using the general vibration theory respectively and are used for analytic study.Comparing with the parametric vibration law and stability of the complete model,the application scope of the extensional and inextensional modeling assumptions is focused on.The results imply that both simplified models can approximate the complete model well under the low rotational speed while weak parametric vibration system,but the simplified assumptions will not be applicable under the strong parametric vibration system.In addition,the analytical conclusions under the simplified models are verified by using the numerical computation method.(3)The out-of-plane parametric vibration of rotational ring-shaped periodic structures is examined.Considering the axial bending and tangential torsion pendulum vibration of the ring,an equivalent gyroscopic dynamic model is established under the coordinate system following the rotational load by using the Hamilton principle.Based on the eigenvalue of the system,the analytical results ofthe stationary ring with rotating supports and the inverse model are obtained.Comparing the dynamic characteristics of the two models,the results show that both models have parametric excitations,and the dynamic behaviors of the two models are similar under the low rotational speed system.However,the instability phenomenon of the stationary ring with rotating supports is more obvious and affected by the basic parameters more easily as the rotational speed increases.Furthermore,the time-domain dynamics responses of the models are solved by using the variable-step Runge-Kutta method so that the analytical conclusions are verified.