TRANSITION ANALYSIS ON RHYTHM MODES OF RAYLEIGH OSCILLATORS FAMILY COUPLED WITH MEMORY DAMPING DRIVEN BY JOINT NOISES

In this work, a design scheme on the parameters inducing transitions of the monorhythmicity, birhythmicity and trirhythmicity mode for Rayleigh oscillators family coupled with memory damping under joint noises is proposed. The stochastic averaging procedure and multi-scale expansion are used to analyze the oscillatory responses on the stochastic Rayleigh oscillators family (8-power, 6-power, and 4-power variants) with memory damping. The transition behavior of oscillators family is qualitatively and quantitatively evaluated from the most probable amplitude and the parameter relation determined by its inverse solution, and multiple probability densities of such as amplitude, joint phase state and its projection and section. The quantitative relationship of transitions on rhythmic modes caused by the coefficients of Rayleigh damping variant and the important parameters of memory damping is preliminarily identified. We notice that the trirhythmicity mode appears in response of 8-power Rayleigh damping oscillator, and a sample scheme for controlling the unstable vibration by combining the generalized elastic modulus and the feedback gains of improved memory damping model, the Rayleigh damping variant coefficients and the random loads parameters is obtained. Numerical techniques of Runge-Kutta are adopted synchronously to demonstrate evolutionary sequences for the phase state of the oscillators family, and intermittent phenomenon is observed, further revealing variation laws of the attractor. The distribution probabilities of vibrational states are simulated to describe reliability of the analytical method, and influence mechanism of joint noise on the oscillator is gained. Shannon entropy integrated based on the response probability, the entropy derivative and the combined indicator on entropy-entropy derivative varying with Rayleigh damping variant coefficients, important parameters of memory damping and random loads parameters are calculated separately. It is found that these indicators (especially the entropy derivative and its related indicator) changing significantly can indicate tipping points on transitions of the oscillators family. Instead of being limited to single-point estimation, these indicators can provide a reference range for the parameter design causing transitions on rhythm modes. The approaches and paths proposed in this paper develop a new improvement idea for the vibration control and the desired vibrational mode in potential engineering applications.