Nonlinear Oscillations: Exact Solutions and their Approximations

The influence of blade vibration on the nonlinear characteristics of rotor–bearing system is non-ignorable in estimating system performance. The extensive studies simplify the rotor system as lumped mass points. The influence of shaft’s bending and shear and the flexibility are usually ignored. The present paper is aim to analyze the nonlinear dynamic behavior of a continuum model. The continuum model of flexible blade–rotor–bearing coupling system is established, simplifying the shaft as Timoshenko beam. The Lagrange method is utilized to derive the differential equation of motion of system. Then, the nonlinear equations of coupling system are numerically solved using the Newmark-
 $$\upbeta $$
 method. The results obtained through the proposed model are compared with the rotor–bearing system without the blades. The effect of several parameters such as rotational speed, the damping coefficient and the length of blade on the nonlinear dynamics of rotor system have been investigated. Inclusive of the analysis methods of bifurcation diagram, three-dimensional spectral plots, time-base analysis, Poincare maps and spectral plots are used to analyze the behavior of the coupling system under different operating conditions, which exhibits rich dynamic behavior of the system.

The effect of blade vibration on the nonlinear characteristics of rotor–bearing system supported by nonlinear suspension

Effects of location and aspect ratio of a flexible disk on natural frequencies and critical speeds of a rotating shaft-disk system

Effect of multi-frequency parametric excitations on the dynamics of on-board rotor-bearing systems

Dynamic analysis and experiment investigation of a cracked dual-disc bearing-rotor system based on orbit morphological characteristics

Large deflection model for nonlinear flexural vibration analysis of a highly flexible rotor-bearing system

The present article treats the nonlinear dynamic analysis of a lightweight flexible rotor–disk–bearing system with geometric eccentricity and mass unbalance. A large deflection model has been derived to represent a nonlinear flexible rotor–bearing system to study the bifurcation, stability, and route to chaos. This mathematical model includes a bidirectional flexible shaft characterized by nonlinear curvature and gyroscopic effect, geometric eccentricity, a rigid disk crooked with unbalance mass, and nonlinear flexible bearings. A perturbation technique has been used to obtain a set of nonlinear algebraic equations that govern the overall dynamics of the system. The system stability has been studied by investigating the bifurcation and route to chaos upon changing the design parameters such as geometric eccentricity, mass unbalance, and disk parameters under the resonance conditions. The present system exhibits a complex behavior traveling with periodic, quasi-periodic, period doubling and chaotic behavior on a gradual change in design variables. The system loses its stability due to S–N bifurcation, which leads to a sudden jump in the response amplitude. These complex behaviors have been studied in detail with the illustration of time history, phase trajectories, bifurcation diagrams, and Poincare’s map for each category. Qualitative assessment of bifurcation diagrams has been studied to explore the boundaries of the stable and unstable behaviors and essential dynamics of the systems. Special attention to predict its rich dynamics to highlight the route to chaos as a future diagnostic tool has been explored. The presented results offer significant understanding of the dynamic performances and its critical operating conditions of a rotor system subjected to geometric eccentricity and mass imbalance.

Nonlinear modeling, dynamics, and chaos in a large deflection model of a rotor–disk–bearing system under geometric eccentricity and mass unbalance

Nonlinear Frequencies and Unbalanced Response Analysis of High Speed Rotor-Bearing Systems

This research article investigated the rotor-casing contact phenomena of a high-speed rotor-bearing system under mass unbalance situation. Here, large deflection rotor-bearing model with a flexible shaft, a rigid disk loaded with an unbalance mass and contact phenomenon between a disk and casing of the rotating system has been used. A nonlinear mathematical model of a rotating system has been developed to study the steady-state stability by elucidating the state of bifurcation and route to chaos phenomena. The steady-state bifurcations and plausible route to chaos phenomena have been explored by numerically solving the governing motion upon gradually varying the design parameters i.e., flexible co-efficient, friction, shaft speed and unbalance mass. The rub-impact onto this large deflection model along with mass unbalance advocates the changes in dynamic state and ill-fated instability where the system losses its structural stability due to the presence of critical bifurcation and sensitivity toward chaotic responses. The present system enables a complex dynamic evolving with periodic, multi-period periodic solutions, sequences of period-doubling to chaotic motion on a gradual change in design variables. The results obtained will contribute to predict and identify the parametric conditions at which the system is safe from any contact between casing and rotor–disk.

Large deflection model for rub-impact analysis in high-speed rotor-bearing system with mass unbalance

The assessment of nonlinear phenomena with a focus on investigating the bifurcations and chaotic behaviour of an elastically induced flexible rotor-bearing system subjected to a harmonic ground motion has been studied. The higher-order Euler–Bernoulli deformation theorem has been adopted that governs the nonlinear dynamic characteristics of the rotating system. Nonlinear vibration analysis has been carried out to determine the critical speeds, i.e., Campbell diagram followed by demonstrating the inherent nonlinear signatures through the illustration of time history, Fourier spectrum and Poincare’s map upon varying the system design variables. The perturbation technique has been used to obtain the approximate nonlinear solutions from a set of polynomial algebraic equations under steady-state condition in various resonance cases. Further, stability analysis along with successive bifurcations of the solutions has been investigated. The present nonlinear model formulated based on Euler theory has been found to be capable enough to predict the correct value of critical speeds and dynamic responses in comparison with that of model developed based on Timoshenko theory. The present outcomes can offer immense practical importance for any application of the flexible rotor-bearing system when it performs high-speed operation.

Hanmant P. Phadatare

Papers

Large deflection model for nonlinear flexural vibration analysis of a highly flexible rotor-bearing system

Nonlinear modeling, dynamics, and chaos in a large deflection model of a rotor–disk–bearing system under geometric eccentricity and mass unbalance

Nonlinear Frequencies and Unbalanced Response Analysis of High Speed Rotor-Bearing Systems

Large deflection model for rub-impact analysis in high-speed rotor-bearing system with mass unbalance

Evaluation of nonlinear responses and bifurcation of a rotor-bearing system mounted on moving platform