^{1}

^{2}

^{3}

^{4}

^{2}

^{2}

^{3}

^{4}

^{1}

^{1}

^{1}

^{1}

^{2}

^{3}

^{4}

It is traditionally considered that, due to the Hertzian contact force-deformation relationship, the stiffness of rolling bearings has stiffening characteristics, and gradually researchers find that the supporting characteristics of the system may stiffen, soften, and even coexist from them. The resonant hysteresis affects the stability and safety of the system, and its jumping effect can make an impact on the system. However, the ball bearing contains many nonlinearities such as the Hertzian contact between the rolling elements and raceways, bearing clearance, and time-varying compliances (VC), leading great difficulties to clarify the dynamical mechanism of resonant hysteresis of the system. With the aid of the harmonic balance and alternating frequency/time domain (HB-AFT) method and Floquet theory, this paper will investigate the hysteretic characteristics of the Hertzian contact resonances of a ball bearing system under VC excitations. Moreover, the linearized dynamic bearing stiffness of the system will be presented for assessing the locations of VC resonances, and the nonlinear characteristics of bearing stiffness will also be discussed in depth. Our analysis indicates that the system possesses many types of VC resonances such as the primary, internal, superharmonic, and even combination resonances, and the evolutions of these resonances are presented. Finally, the suppression of resonances and hysteresis of the system will be proposed by adjusting the bearing clearance.

Rolling-element bearings are characterized by rolling motion of balls or rollers in bearing raceway, often referred to as antifriction bearings in comparison to the sliding motion of regular sleeve bearings [

There are two major effects of a health rolling bearing with respect to machine vibration [

Contact nonlinearity is a common nonlinear factor in scientific technology and engineering applications [

As a basic parameter of rolling bearings, bearing clearance has great significance for bearing life, installation, and thermal expansion capacity [

The motivation of the present paper is to investigate the mechanism of VC contact resonances and their hysteresis characteristics in a ball bearing system, considering control of the main structural parameters of the system such as bearing clearances and damping factors. For doing this, as our previous literature [

A classic 2-degree-of-freedom balanced ball bearing-rotor model (see Figure

There is an interference fit between the shaft and inner race of the radial ball bearings, and the outer race is fixed to a rigid support

The rotor shaft of the studied ball bearing-rotor system is assumed to be rigid

It is under pure rolling condition between the raceway and balls, and the whole system is fault-free

The inertia of balls introduces little dynamic effect on the system

Schematic diagram of the ball bearing and the relationship of the displacement about the

The equation of VC motion can be expressed as_{x} and _{y}_{b} are the equivalent mass, damping factor estimated from [_{i} and _{i} are radial deformation of _{b}) and instant angular location; 2_{0} is the radial internal clearance of the bearing; and Ω denotes the cage velocity, which is related to the shaft speed _{s}, ball diameter, and pitch diameter of the ball bearing system as

Due to the parametric VC excitation, primary parametric resonance occurs as Ω_{vc} = _{b}·Ω approaches one of the resonant frequencies of equation (_{avg_xx} and _{avg_yy}, discussed in the next section, are the averaged first resonant frequencies in the vertical and horizontal radical directions of the system.

Bearing stiffness as supporting spring is directly related to load and vibration characteristics of the bearing moving or rotating parts. Due to Hertzian point contact between each rolling element and its raceways of a rolling bearing, the total bearing load _{s} versus deformation _{Ws} along the load direction in the static condition exhibits approximately hard (i.e., stiffening) spring characteristics as [_{0} in equation (_{Ws} in the load direction can be expressed as_{s}) also can be given (see [

Compared with the static bearing stiffness derived in equation (

In order to descript the time-varying bearing support characteristics fully, from equation (

According to equation (_{int} is the number of numerical integration steps when solving equation (

The steady-state periodic response of a linear system is always stable, but it may get unstable in a nonlinear dynamic system. The 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. The 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 [

Introducing nondimensional time _{0} and _{0} to equation (

First, inserting equation (

Second, in order to solve

Herein, _{k} = _{k} + i_{k}; _{k} = _{k} + i_{k}, and ^{′} (

Third,

Finally, Hsu’s method is applied for Floquet stability analysis of obtained periodic solutions aided as shown in our previous work [

The JIS6306 ball bearing is adopted in this paper, which has been studied commonly during recent years [^{7}, so the damping factor is in range of 150–1500 Ns/m. If doing experimental research, the bearing damping can be measured by experimental modal analysis referring to literature [

Specifications and parameters of JIS6306 ball bearing.

Item | Value |
---|---|

Contact stiffness _{b} (N/m^{3/2}) | 1.334 × 10^{10} |

Ball diameter _{b} (mm) | 11.9062 |

Pitch diameter _{h} (mm) | 52.0 |

Number of balls _{b} | 8 |

Equivalent mass | 20 |

Damping factor | 150–1500 |

Radical load | 196 |

System dynamic stiffness of period-1 (black line) and period-2 (blue line) motions for _{0} = 6.0

In the following, periodic solutions are traced by HB-AFT method with the cage speed Ω as the controlled parameter, where a periodic VC motion is called period-n motion if the period of the response is _{vc} = 2_{vc}. Herein, the numerical verifications to the HB-AFT tracing results are simulated by the classical explicit Runge-Kutta numerical integration process of literature [

For _{0} = 6.0 _{1} and _{2}, cyclic fold bifurcations occur at turning points _{3} and _{4}, and secondary Hopf bifurcations include quasiperiodic motions at _{5} and _{6}, respectively. These bifurcation characteristics agree well with the numerical simulations as shown in Figure _{1} in Figure

Stable (solid) and unstable (dashed) period-1 frequency-response peak-to-peak curves in (a) the _{0} = 6.0

Numerical bifurcation diagram of (a) _{0} = 6.0

The 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 [

Time series and phase orbits of coexisting period-1 (black line) and period-2 (blue line) motions for _{0} = 6.0

Power spectra of _{0} = 6.0

Meanwhile, as shown in Figure

Influence of damping on (a) period-1 frequency-response peak-to-peak curves in the _{0} = 6.0

Unstable period 1 solution branch _{5}-_{6} in Figure _{vc} = _{b}·Ω. The values _{p} + _{q} = 2·Ω_{vc} occurs in the segment of

Time series _{0} = 6.0

Power spectra of _{0} = 6.0

In certain applications, particularly in high-speed machines or high precision machines, such as machine tools, it is possible to eliminate clearance and introduce small interference (negative clearance) in the bearings, because it increases the bearing stiffness and reduces the run-out and noise due to elastic deformation or clearance [_{0} reduces from 6 _{0} reduces to 0 _{0} = 2 _{0} = 0

Influence of bearing clearance _{0} on period-1 frequency-response peak-to-peak curves for

One can see in Figure _{0} reduces from 0.2 _{0} continues to reduce from −0.6 _{0} takes −1.2 _{1}-_{2} branch in Figure

Influence of bearing clearance _{0} on period-1 frequency-response peak-to-peak curves in the

Resonant response characteristics for _{0} = −1.2

In terms of hysteresis characteristics considered, due to the effect of clearance nonlinearity elimination as _{0} is negative, at this moment, the Hertzian contact nonlinearity of the system plays a dominant role. Therefore, as illustrated in Figure _{0} = −1.2

The system of (a) equivalent dynamic resonant frequencies and (b) waterfall plot of peak-to-peak

Figure _{peak-to-peak} + _{peak-to-peak}, where _{peak-to-peak} = _{max}-_{min}, and _{peak-to-peak} = _{max}-_{min}. In order to reflect the hysteresis characteristics in this figure, we get the DDP value when the controlled parameter Ω runs up and down by the way in literature [_{1} and _{2} in vertical and horizontal directions, respectively, and also superharmonic resonances _{3} and _{4} are aroused in these two directions. Moreover, the forms of _{p} + _{q} = 2·Ω_{vc} and 4·_{p} + _{q} = 8·Ω_{vc} combination resonances labeled as _{5} and _{6}, respectively, exist for this system. Figure _{p} = _{avg_yy} and _{q} = _{avg_xx}, so it is a valuable way for analysis of VC resonances for the ball bearing systems.

Power spectra of response amplitude for

Prediction of the locations of primary resonances (black lines), superharmonic resonances (green lines), and combination resonances (blue lines) from equation (

This paper focuses on the characteristics of Hertzian contact resonances and dynamic hysteresis of a ball bearing system. The HB-AFT method and Floquet theory are applied to trace the periodic response behaviors of the system. It is found that the system contains many types of resonances such as primary, combination, and superharmonic resonances, which makes the response characteristics of the system more complicated, and the linearized dynamic bearing stiffness can be used to assess the locations of these resonances. It should be emphasized that combination resonance can bring quasiperiodic motions to the system. Moreover, the coexistence of two types of time-varying stiffness characteristics is demonstrated. It is presented that bearing clearance has significant effects on the dynamic characteristics of ball bearing system, including the bearing stiffness, resonant amplitudes, and their hysteretic behaviors. Our analysis indicates that the hysteretic resonances can be suppressed effectively by adjusting the bearing clearances around the clearance-free operations, and this method may be beneficial for the engineering study and inspection of resonant vibration control and even bearing life optimization of rolling bearing systems. In the future, the intrinsic correlations between typical resonances, Hertzian contact nonlinearity, bearing clearance, and even asymmetric stiffness characteristics of the system should be further clarified.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work has been supported by the China Postdoctoral Science Foundation (Grant no. 2019M661849), Jiangsu Planned Projects for Postdoctoral Research Funds (Grant no. 2018K164C), Fundamental Research Funds for the Central Universities (Grant no. 30920021155), China Scholarship Council (ID: 201906845022), and Natural Science Foundation of Jiangsu Province (Grant no. BK20190434).