Parametric Instability of a Rotating Circular Ring With Moving, Time-Varying Springs

Abstract

Parametric instabilities of in-plane bending vibrations of a rotating ring coupled to multiple, discrete, rotating, time-varying stiffness spring-sets of general geometric description are investigated in this work. Instability boundaries are identified analytically using perturbation analysis and given as closed-form expressions in the system parameters. Ring rotation and time-varying stiffness significantly affect instability regions. Different configurations with a rotating and nonrotating ring, and rotating spring-sets are examined. Simple relations governing the occurrence and suppression of instabilities are discussed for special cases with symmetric circumferential spacing of spring-sets. These results are applied to identify possible conditions of ring gear instability in example planetary gears.

Natural frequency splitting and principal instability of rotating cyclic ring structures

“…According to the Hamilton’s principle, in-extensional assumption v=-∂u/∂θ, 16,17,27 and omitting the nonlinear terms, the governing equation can be derived. Note that the geometric nonlinearity is included due to the ring rotation with an aim to develop a more accurate governing equation.…”

“…Thus, the governing equation in the support-fixed frame has time-invariant and gyroscopic terms as well. Such treatment has been used in Canchi and Parker, 17 which is employed in this section to verify the analytical predictions.…”

“…Canchi and Parker 16 formulated the parametric vibration of a stationary ring with moving springs, and then they studied the rotating ring with moving and time-varying springs. 17 Considering a typical application, they addressed the instability induced by the time-varying mesh stiffness in a planetary gear ring. 18 Lesaffre et al.…”

“…Relationships governing the splitting and instability are formulated. The interesting problem of the rotating ring with stationary supports and the inverse problem 17,26 are also discussed to further verify the analytical method. This method and analytical results have also been confirmed by the comparisons against the existing results and in particular the results comparison between the two types of frames.…”

Vibration analysis of rotating rings with complex support stiffnesses

“…There are fewer studies on the vibration of rotating rings with discrete supports. Canchi and Parker [11] studied the parametric instability of rotating rings connected to moving discrete springs with time-varying stiffness. Cooley and Parker [12] studied the vibration of high-speed rotating rings coupled to space-fixed stiffnesses.…”