Control of Period-Doubling and Chaos in Varying Compliance Resonances for a Ball Bearing

Abstract

Varying compliance (VC) is an inevitable parametrical excitation to rolling bearing systems due to time-varying stiffness from rolling element revolution. Period-doubling instability in the VC primary resonances of ball bearing is presented in many studies. Recently, this instability was demonstrated to be a probable indicator of occurrence of strong one to two internal resonances and chaotic motions, which has potential effects on the stability and safety of the bearing-rotor system. However, few studies have directly attempted to suppress this bifurcation instability. Here, a dynamic stiffness evaluating method is presented for assessing the threshold of the period-doubling and complex motions in VC primary resonances of ball bearings, where the elaborate evolution of the bifurcating process is obtained by harmonic balance and alternating frequency/time domain (HB-AFT) method and using Floquet theory. Our analysis indicates that by introducing certain additional stiffness, the period-doubling and corresponding subharmonic internal resonances can be suppressed. Besides, the evolution and mechanism of type I intermittency chaos in ball bearings will be clarified in depth. It is also shown that extensive chaotic motions for large bearing clearances (e.g., 40 μm) can vanish perfectly by action of additional stiffness.

Suppression of Complex Hysteretic Resonances in Varying Compliance Vibration of a Ball Bearing

“…e analysis of periodic motion types and their bifurcation mechanism of a nonlinear system is beneficial to abnormal vibration (e.g., super/subharmonic, combination, chaotic, bistable, and hystertic responses) control and even machine design. e HB-AFT method combined with Floquet stability analysis is an effective way to trace a VC periodic solution branch and its stability characteristics of ball bearing system [5,6].…”

“…e JIS6306 ball bearing is adopted in this paper, which has been studied commonly during recent years [5,6,[10][11][12], and the main parameters of the system studied are given in Table 1. Herein, the damping factor c is estimated by [31] c � (0.25 to 2.5)…”

“…e cyclic fold bifurcation and the subcritical bifurcation all can lead to instabilities and dynamic jump to the ball bearing systems. Herein, the period-doubling bifurcations come from one-to-two internal resonances [5,6], and it is clear that strong couplings between the two degrees of freedom of the system arise in the period-2 response range (see range B-B in Figure 4). At this time, the VC motions exhibit complex hysteresis characteristics of bistability and jumped phenomenon.…”

“…A major limitation of rolling bearings is that they are subjected to very high alternating and timevarying compliance (VC) oscillating Hertzian stresses at the rolling contacts. Due to the nonlinearities of Hertzian contact and bearing clearance, rolling bearing can bring complex dynamic hysteresis and jumping vibrations to its bearing-rotor systems, which is a basic scientific problem in the field of bearing-rotor dynamics [3][4][5][6].…”

“…e VC excitation is an inevitable parametrical excitation to generate vibrations to the rotor system, when the rolling elements carry external load moving along the bearing raceway [7]. Vibrations originating in other parts of the machine can also excite and influence the rolling bearing [2], and in recent years, indepth studies have been carried out gradually on the dynamic influences on some special operation conditions such as starved lubrication [8,9] and nonlinear instability [5,6]. As far as VC vibrations are concerned, Sunnersjö [7] studied a linear bearing model focusing on the severity of VC vibrations for a cylindrical roller bearing considering the rotor mass and bearing cage speed, which showed that large movements occur at low speed and heavy load.…”