Shooting With Deflation Algorithm-Based Nonlinear Response and Neimark-Sacker Bifurcation and Chaos in Floating Ring Bearing Systems

Abstract

The double-sided fluid film force on the inner and outer ring surfaces of a floating ring bearing (FRB) creates strong nonlinear response characteristics such as coexistence of multiple orbits, Hopf bifurcation, Neimark-Sacker (N-S) bifurcation, and chaos in operations. An improved autonomous shooting with deflation algorithm is applied to a rigid rotor supported by FRBs for numerically analyzing its nonlinear behavior. The method enhances computation efficiency by avoiding previously found solutions in the numerical-based search. The solution manifold for phase state and period is obtained using arc-length continuation. It was determined that the FRB-rotor system has multiple response states near Hopf and N-S bifurcation points, and the bifurcation scenario depends on the ratio of floating ring length and diameter (L/D). Since multiple responses coexist under the same operating conditions, simulation of jumps between two stable limit cycles from potential disturbance such as sudden base excitation is demonstrated. In addition, this paper investigates chaotic motions in the FRB-rotor system, utilizing four different approaches, strange attractor, Lyapunov exponent, frequency spectrum, and bifurcation diagram. A numerical case study for quenching the large amplitude motion by adding unbalance force is provided and the result shows synchronization, i.e., subsynchronous frequency components are suppressed. In this research, the fluid film forces on the FRB are determined by applying the finite element method while prior work has utilized a short bearing approximation. Simulation response comparisons between the short bearing and finite bearing models are discussed.

Nonlinear stability analysis and numerical continuation of bifurcations of a rotor supported by floating ring bearings

“…Rotors in floating ring bearings modeling presented in the literature consider either linear 2,20 or nonlinear models. 1,4,11,17,19 Since these latter are more suitable for the evaluation of the rotors supported by FRBs dynamics, a nonlinear model of a perfectly balanced symmetric rigid rotor supported by two identical floating ring bearings is considered in this study. Both bearings have the same motions, thus only one bearing is examined.…”

“…Kim et al. 17 used a numerical shooting/arc length continuation approach including a deflation algorithm to analyze the nonlinear behavior of a rigid rotor supported by floating ring bearings. Primary Hopf bifurcation and N-S bifurcation were identified depending on the L/D ratio of FRB.…”

A nonlinear optimization shooting method for bifurcation tracking of nonlinear systems

“…(2009) by using the shooting method. Kim and Palazzolo (2017) used the shooting method to analyze bifurcation behaviors of a rigid rotor supported by a floating ring bearing. Recently, Charroyer et al (2018) proposed a shooting method to calculate the self-excited vibration of mechanical systems subjected to friction-induced vibration.…”