Modeling the Dynamics of a Gyroscopic Rigid Rotor with Linear and Nonlinear Damping and Nonlinear Stiffness of the Elastic Support

Kuatbay Bissembayev,Azizbek Abduraimov,Nutpulla Jamalov,Zharilkassin Iskakov

doi:10.3390/machines9110276

Kuatbay Bissembayev, Azizbek Abduraimov + Show 2 more

Open Access

https://doi.org/10.3390/machines9110276

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Abstract

This study analytically and numerically modeled the dynamics of a gyroscopic rigid rotor with linear and nonlinear cubic damping and nonlinear cubic stiffness of an elastic support. It has been shown that (i) joint linear and nonlinear cubic damping significantly suppresses the vibration amplitude (including the maximum) in the resonant velocity region and beyond it, and (ii) joint linear and nonlinear cubic damping more effectively affects the boundaries of the bistability region by its narrowing than linear damping. A methodology is proposed for determining and identifying the coefficients of nonlinear stiffness, linear damping, and nonlinear cubic damping of the support material, where jump-like effects are eliminated. Damping also affects the stability of motion; if linear damping shifts the left boundary of the instability region towards large amplitudes and speeds of rotation of the shaft, then nonlinear cubic damping can completely eliminate it. The varying amplitude (VAM) method is used to determine the nature of the system response, supplemented with the concept of “slow” time, which allows us to investigate and analyze the effect of nonlinear cubic damping and nonlinear rigidity of cubic order on the frequency response at a nonstationary resonant transition.

Highlights

Rotary machines are widely used in many industries and have been studied for a long time
If we take into account that jumping effects pose threats to the safety of the system, including rotary, promising directions are the study of the effect of joint linear damping and nonlinear cubic damping, adopted on the basis of a phenomenological model, on the maximum amplitude and on the amplitude of the response of the system beyond the resonant region of the oscillation frequency, and on the boundaries of the bistability region to weaken the jumping effects, until they are completely eliminated, on the boundaries of the regions of stable and unstable modes of motion in order to completely narrow the instability region, on transients through the resonant region
Applications of nonlinearity in passive vibration control devices to provide an understanding of how nonlinearity is applied and useful in the implemented system are discussed in the review [18]

Summary

Introduction

Rotary machines are widely used in many industries and have been studied for a long time. If we take into account that jumping effects pose threats to the safety of the system, including rotary, promising directions are the study of the effect of joint linear damping and nonlinear cubic damping, adopted on the basis of a phenomenological model, on the maximum amplitude and on the amplitude of the response of the system beyond the resonant region of the oscillation frequency, and on the boundaries of the bistability region to weaken the jumping effects, until they are completely eliminated, on the boundaries of the regions of stable and unstable modes of motion in order to completely narrow the instability region, on transients through the resonant region. The method of varying amplitude used is supplemented by the concept of “slow time”

Related Work

Equations of Motion and Their Solutions

Nonlinear Frequency Characteristics

Non-Stationary Oscillations

Stability of Stationary Motion

Conclusions