Non-stationary processes of rotor/bearing system in bifurcations

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Abstract

Non-stationary processes of a rotor/bearing system were dealt with by taking the rotating angular speed, increases or decreases linearly at different levels of acceleration, as control parameter. The stationary bifurcation diagrams show that the period doubling bifurcation or quasi-periodic bifurcation, corresponding to the system with larger or smaller level of mass imbalance, respectively, occurs smoothly as the control parameter is increased or decreased in stationary manner. Then, the non-stationary processes of these two types of bifurcation were investigated by constructions of the non-stationary bifurcation diagrams using non-stationary bifurcation map technique. In the non-stationary bifurcation diagrams, penetrations can be easily observed during the forward and reverse transitions, and their absolute values increase with that of accelerations in all cases. Jumps exist only in forward period doubling transitions, which indicate the quick increases of the amplitudes of motion. Time flows and orbit trajectories are also presented to illustrate the non-stationary transition processes visually.

Introduction

Non-stationary processes occur naturally and widely in engineering field and physical world: in start-up and shut down of engines, in chemical and biochemical systems, in the movements of water, earth, air, etc. In stationary systems, all control parameters remain constant, whilst in non-stationary processes, one or more control parameters are time (process) dependent over extended ranges [1]. The early developments in non-stationary systems are attributed to two monographs by Mitropolskii [1] and Evan-Iwanowski [2]. The recent achievements can be found in extensive publications [3], [4], [5], [6], [7], [8], [9]. Besides the characteristics of non-stationary processes of Duffing's oscillators which have been investigated over many years, some significant phenomena have also been revealed in recent years from actual complex systems, e.q. the shear deformable orthotropic plates, the laminated angle-ply column, a string with time-varying length and a mass–spring system attached at the lower end (a basic model of an elevator system), etc. The most typical phenomena existing in non-stationary processes are known as elimination of the discontinuities or jumps, though the transitions in stationary processes are smooth and penetration (delay or memory) together with anticipation, i.e., the transitions appear later or before compared to that in the stationary case as the concerned control parameter is varied in same direction. Bifurcation diagram is a useful tool to reveal visually the panorama of the non-stationary processes, including jumps, penetrations and anticipations, over extended parameter ranges. To overcome the shortage of the commonly used phase portraits or Poincaré maps, which give inadequate information because of the overlapping dynamical responses within ranges of time, the non-stationary bifurcation map (NBM) was suggested in Ref. [4] which determines the non-stationary bifurcation diagrams by recording the responses at the consecutive peaks of the forcing.

Non-stationary processes occur in rotor system in the stages of start-up and shut-down. But compared with the extensive studies on stationary processes, studies on non-stationary processes are rather limited. Recently, the non-stationary motions of rotor systems in rub events are reported. Yanabe et al. treated a rubbing problem when the unbalanced rotor is supported by springs and dampers and accelerated at a constant angular acceleration. The non-stationary rotor vibration due to collision with annular guard during passage through the critical speed is calculated. The result shows that the rotor cannot pass through the critical speed due to occurrence of backward whirl [10]. Ding and Chen investigated the partial differential equations governing the shaft/casing system with rubs using an explicit stable finite difference scheme. The result suggests that during passage through the first several critical speeds accelerated at a constant angular acceleration, the instability of the system can lead to the shaft's structural failure ultimately due to unlimited increase of the bending moments and stresses [11].

This paper deals with non-stationary processes of a rotor/bearing system. The rotating angular speed is taken as control parameter and increased or decreased linearly at different levels of angular acceleration. First, the stationary bifurcation diagrams are determined using the customary Poincaré maps. Then the non-stationary bifurcation diagrams are determined using the NBM technique. Some phenomena occurring in the processes of the non-stationary period-doubling transitions and the quasi-periodic transitions are uncovered in cases of positive and negative accelerations. Processes of transitions are revealed by presenting time flows and orbit trajectories.

Section snippets

Equation of motion

An isotropic flexible shaft attached with two imbalanced disks and supported at its two ends on lubricated bearings is considered (Fig. 1). The masses of disks 1 and 2 are m1 and m2 with eccentricity e1 and e2; the lumped masses at two ends of the shaft are m3 and m4, respectively. The equivalent stiffnesses of the shaft in three spans arek1=k11+k12,k2=−k12,k3=k21+k22,where kij (i, j =1, 2) are deduced from the simple beam theory:k11=12L22L3EGL12Λ,k12=k21=−6(−L12+2L2L3−L22)L3EGL1(L3−L2,k22=12

Stationary bifurcations

Stability analysis shows that the perfectly balanced system (5) (i.e., ēi=0, i=1, 2), loses its stability through Hopf bifurcation. Analyzing the eigenvalues of the linearized perturbation system at static equilibrium position can result in threshold speed and dimensionless whirl frequency [13]. For the mass imbalanced system (5) (i.e., ē1≠0 and/or ē2≠0), behaviour of the periodically perturbed Hopf bifurcation depends on the level of mass imbalance. Generally, smaller level of mass

Non-stationary bifurcation map

On the basis of the NBM [4] and in consideration of the characteristic of the mass imbalanced excitation, we define the non-stationary maps by recording the responses X and Y of Eq. (5) at consecutive peaks of normal inertial imbalance force of disk 1 projected in positive X direction, i.e., at the time τN resulting from the maxima of the function cos(12ᾱτ2+ω̄0τ) :τN=ω̄0ᾱ(1+4πNᾱ/ω̄02)−1,N=0,1,2,…,where N denotes the number of evolution cycles. Note that the non-stationary maps can be

Conclusions

The non-stationary processes can be effectively revealed using non-stationary bifurcation diagrams, constructed on basis of the NBM, in terms of both the number of evolution cycle and rotating angular speed. The non-stationary period doubling transitions include forward transitions (from 2n−1P to 2nP), chaotic regime and reverse transitions (from 2n−1P to 2nP), whether the rotor speeds up or down. The limit periodic motions with periods 2 and 1 correspond to speeding up and down. Penetrations

Acknowledgements

This research was supported by the NNSF of China under Grant No. 10272078.

References (15)

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