scholarly journals Mechanism and Characteristics of Global Varying Compliance Parametric Resonances in a Ball Bearing

2020 ◽  
Vol 10 (21) ◽  
pp. 7849
Author(s):  
Zhiyong Zhang ◽  
Thomas Sattel ◽  
Yujie Zhu ◽  
Xuan Li ◽  
Yawei Dong ◽  
...  
Keyword(s):  
Ball Bearing ◽  
Primary Resonance ◽  
Period Doubling ◽  
Hertzian Contact ◽  
Rolling Bearings ◽  
Bearing Clearance ◽  
Parametric Resonances ◽  
Rolling Elements ◽  
The Stability ◽  
Varying Compliance

Varying compliance (VC) is an unavoidable form of parametric excitation in rolling bearings and can affect the stability and safety of the bearing and its supporting rotor system. To date, we have investigated VC primary resonance in ball bearings, and in this paper other parametric VC resonance types are addressed. For a classical ball bearing model with Hertzian contact and clearance nonlinearities between the rolling elements and raceway, the harmonic balance and alternating frequency/time domain (HB–AFT) method and Floquet theory are adopted to analyze the VC parametric resonances and their stabilities. It is found that the 1/2-order subharmonic resonances, 2-order superharmonic resonances, and various VC combination resonances, such as the 1-order and 2-order summed types, can be excited, thus resulting in period-1, period-2, period-4, period-8, period-35, quasi-period, and even chaotic VC motions in the system. Furthermore, the bifurcation and hysteresis characteristics of complex VC resonant responses are discussed, in which cyclic fold, period doubling, and the second Hopf bifurcation can occur. Finally, the global involution of VC resonances around bearing clearance-free operations (i.e., adjusting the bearing clearance to zero or one with low interference) are provided. The overall results extend the investigation of VC parametric resonance cases in rolling bearings.

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

2020 ◽  
Vol 2020 ◽  
pp. 1-11 ◽  
Author(s):  
Zhiyong Zhang ◽  
Thomas Sattel ◽  
Aditya Suryadi Tan ◽  
Xiaoting Rui ◽  
Shaopu Yang ◽  
...  
Keyword(s):  
Ball Bearing ◽  
Hertzian Contact ◽  
Rolling Bearings ◽  
Dynamical Mechanism ◽  
Nonlinear Characteristics ◽  
Bearing Clearance ◽  
Rolling Elements ◽  
Bearing Stiffness ◽  
Hysteretic Characteristics ◽  
The Stability

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.

Hysteretic Resonances and Intermittency Chaos in Ball Bearings

2018 ◽  
Author(s):  
Zhiyong Zhang ◽  
Xiaoting Rui ◽  
Yushu Chen ◽  
Wenkai Dong ◽  
Lei Li
Keyword(s):  
Ball Bearing ◽  
Harmonic Balance Method ◽  
Rolling Bearing ◽  
Hertzian Contact ◽  
Ball Bearings ◽  
Rolling Bearings ◽  
Rotor Systems ◽  
Revolute Joints ◽  
Rolling Elements ◽  
Varying Compliance

Ball bearings are essential parts of mechanical systems to support the rotors or constitute the revolute joints. The time-varying compliance (VC), bearing clearance and the Hertzian contact between the rolling elements and raceways are three fundamental nonlinear factors in a ball bearing, hence the ball bearing can be considered as a nonlinear system. The hysteresis and jumps induced by the nonlinearities of rolling bearings are typical phenomena of nonlinear vibrations in the rolling bearing-rotor systems. And the corresponding hysteretic impacts have direct effects on the cleavage derivative and fatigue life of the system components. Therefore, the behaviors of hysteresis and jumps are given full attentions and continued studies in the theoretical and engineering fields. Besides, many researchers have done a lot of calculations to depict the various characteristics of bifurcations and chaos in the rolling bearings and their rotor systems, but few researches have been addressed on the inherent mechanism of the typical intermittency vibrations in rolling bearings. With the aid of the HB-AFT (the harmonic balance method and the alternating frequency/time domain technique) method and Floquet theory, this paper will investigate deeply the resonant hysteresis and intermittency chaos in ball bearings.

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

2019 ◽  
Vol 87 (2) ◽  
Author(s):  
Zhiyong Zhang ◽  
Xiaoting Rui ◽  
Rui Yang ◽  
Yushu Chen
Keyword(s):  
Ball Bearing ◽  
Dynamic Stiffness ◽  
Period Doubling ◽  
Rolling Bearing ◽  
Type I ◽  
Ball Bearings ◽  
Bifurcation Instability ◽  
Internal Resonances ◽  
Chaotic Motions ◽  
Varying Compliance

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.

scholarly journals Nonlinear Vibration of a Continuum Rotor with Transverse Electromagnetic and Bearing Excitations

2012 ◽  
Vol 19 (6) ◽  
pp. 1297-1314 ◽  
Author(s):  
Haiyang Luo ◽  
Yuefang Wang
Keyword(s):  
Nonlinear Vibration ◽  
Periodic Motion ◽  
Primary Resonance ◽  
Period Doubling ◽  
Time History ◽  
Oil Film ◽  
Rotor Bearing ◽  
Transverse Electromagnetic ◽  
The Stability ◽  
Bearing System

The nonlinear vibration of a rotor excited by transverse electromagnetic and oil-film forces is presented in this paper. The rotor-bearing system is modeled as a continuum beam which is loaded by a distributed electromagnetic load and is supported by two oil-film bearings. The governing equation of motion is derived and discretized as a group of ordinary differential equations using the Galerkin's method. The stability of the equilibrium of the rotor is analyzed with the Routh-Hurwitz criterion and the occurrence of the Andronov-Hopf bifurcation is pointed out. The approximate solution of periodic motion is obtained using the averaging method. The stability of steady response is analyzed and the amplitude-frequency curve of primary resonance is illustrated. The Runge-Kutta method is adopted to numerically solve transient response of the rotor-bearing system. Comparisons are made to present influences of electromagnetic load, oil-film force and both of them on the nonlinear vibration response. Bifurcation diagrams of the transverse motion versus rotation speed, electromagnetic parameter and bearing parameters are provided to show periodic motion, quasi-periodic motion and period-doubling bifurcations. Diagrams of time history, shaft orbit, the Poincaré section and fast Fourier transformation of the transverse vibration are presented for further understanding of the rotor response.

scholarly journals Experimental Rig for Measuring Lubricant Film Thickness in Rolling Bearings

2014 ◽  
Vol 658 ◽  
pp. 381-386
Author(s):  
Xing Nan Zhang ◽  
Karolina Jablonka ◽  
Romeo Glovnea
Keyword(s):  
Film Thickness ◽  
Lubricant Film ◽  
Hertzian Contact ◽  
Lubricant Film Thickness ◽  
Rolling Bearings ◽  
The Past ◽  
Rolling Element ◽  
Rolling Elements ◽  
Experimental Rig ◽  
Made In

Electrical capacitance has been applied in the past for measuring the lubricant film thickness in rolling element bearings. The main difficulty arises from the fact that the measured capacitance is a combination of the capacitances of many rolling elements, which come in contact with both the inner and outer rings. Besides, the capacitance of the Hertzian contact itself and the surrounding area must also be separated. It results in a complex system which, in order to be solved for the film thickness at a particular location on the bearing many approximations have to be made. In the present study the authors use an experimental rig in which the capacitance of a single ball can be isolated. Moreover the capacitance of the ball – inner ring and ball – outer ring contacts can be measured separately.

Bifurcations of Periodic Orbits of a One-Dimensional Pre-Compressed Granular Array

2018 ◽  
Author(s):  
Gizem Dilber Acar ◽  
Balakumar Balachandran
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Bifurcation Point ◽  
Granular Media ◽  
Period Doubling ◽  
Initial Step ◽  
Hertzian Contact ◽  
One Dimensional ◽  
The Stability ◽  
External Forces

Bifurcations of periodic orbits of a one-dimensional granular array are numerically investigated in this study. A conservative two-bead system is considered without any damping or external forces. By using the Hertzian contact model, and confining the system’s total energy to a certain level, changes in in-phase periodic orbit are studied for various pre-compression levels. At a certain pre-compression level, symmetry breaking and period doubling occur, and an asymmetric period-two orbit emerges from the in-phase periodic orbit. Floquet analysis is conducted to study the stability of the in-phase periodic solution, and to detect the bifurcation location. Although the trajectory of period-two orbit is close to the in-phase orbit at the bifurcation point, the asymmetry of the period-two orbit becomes more pronounced as one moves away from the bifurcation point. This work is meant to serve as an initial step towards understanding how pre-compression may introduce qualitative changes in system dynamics of granular media.

Nonlinear Vibration Analysis of an Unbalanced Flexible Rotor Supported by Ball Bearings With Radial Internal Clearance

2008 ◽  
Author(s):  
T. C. Gupta ◽  
K. Gupta ◽  
D. K. Sehgal
Keyword(s):  
Ball Bearing ◽  
Natural Frequencies ◽  
Power Spectra ◽  
Hertzian Contact ◽  
Phase Portraits ◽  
Integration Scheme ◽  
Flexible Rotor ◽  
Deep Groove ◽  
Varying Compliance ◽  
Bearing System

In the present work, the nonlinear dynamic response of an unbalanced horizontal flexible rotor supported by deep groove ball bearing is studied. Nonlinearity effects in rolling element bearings arise from Hertzian contact force deformation relationship and clearance between rolling elements and races. The system is bi-periodically excited due to varying compliance of ball bearing and rotating unbalance. The flexible rotor bearing system is modeled by finite element method, taking into account the gyroscopic moments, rotary inertia, shear deformation, proportional damping, nonlinear stiffness and radial internal clearance of ball bearing. The implicit type numerical time integration scheme Newmark-β and Newton-Raphson methods are used to numerically solve the nonlinear equations of motion. The mathematical model is validated for the natural frequencies of the flexible shaft and whirl frequencies. On account of variation in the ball bearing stiffness, the variation in natural frequencies of the rotor ball bearing system is estimated. The influence of ball bearing nonlinearity on dynamic behavior is analyzed by time histories of steady state response, phase portraits and power spectra. Effect of radial internal clearance and varying compliance on the unbalance response of flexible rotor is studied in detail.

scholarly journals A theoretical model for predicting the vibration response of outer race defective ball bearing

2018 ◽  
Vol 7 (2) ◽  
pp. 289
Author(s):  
Samir Shaikh ◽  
Sham Kulkarni
Keyword(s):  
Theoretical Model ◽  
Ball Bearing ◽  
Mathematical Formulation ◽  
Vibration Response ◽  
Contact Forces ◽  
Hertzian Contact ◽  
Outer Race ◽  
Rolling Element ◽  
Rolling Elements ◽  
Extended Type

The theoretical model with 2 degree-of-freedom system is developed for predicting the vibration response and analyze frequency properties in an extended type defective ball bearing. In the mathematical formulation, the contact between the races and rolling element considered as non-linear springs. The contact forces produced during the collaboration of rolling elements are obtained by utilizing Hertzian contact deformation hypothesis. The second order nonlinear differential equation of motion is solved using a state space variable method with the help of MATLAB software and the vibration acceleration response of the defective ball bearing presented in the frequency spectrum. The effects of variation in speed and size of the defect on characteristic frequency of extended fault on the outer raceway of the ball bearing have been investigated. The theoretical results of the healthy (non defective) and defective bearing are compared with each other.

Study on Nonlinear Dynamic Response of an Unbalanced Rotor Supported on Ball Bearing

2009 ◽  
Vol 131 (6) ◽  
Author(s):  
G. Chen
Keyword(s):  
Nonlinear Dynamic ◽  
Ball Bearing ◽  
Period Doubling ◽  
Dynamic Responses ◽  
Hertzian Contact ◽  
Frequency Spectra ◽  
Rotating Speed ◽  
Unbalanced Rotor ◽  
Motion Period ◽  
Contact Position

An unbalanced rotor dynamic model supported on ball bearings is established. In the model, three nonlinear factors of ball bearing are considered, namely, the clearance of bearing, nonlinear Hertzian contact force between balls and races, and the varying compliance vibrations because of periodical change in contact position between balls and races. The numerical integration method is used to obtain the nonlinear dynamic responses; the effects of the rotating speed and the bearing clearance on dynamic responses are analyzed; and the bifurcation plots, the phase plane plots, the frequency spectra, and the Poincaré maps are used to carry out the analyses of bifurcation and chaotic motion. Period doubling, quasiperiod loop breaking, and mechanism of intermittency are observed as the routes to chaos.

Nonlinear Excitation Model of Ball Bearing Waviness in a Rigid Rotor Supported by Two or More Ball Bearings Considering Five Degrees of Freedom

2001 ◽  
Vol 124 (1) ◽  
pp. 82-90 ◽  
Author(s):  
G. H. Jang ◽  
S. W. Jeong
Keyword(s):  
Degrees Of Freedom ◽  
Ball Bearing ◽  
Angular Displacement ◽  
Equations Of Motion ◽  
Hertzian Contact ◽  
Ball Bearings ◽  
Rigid Rotor ◽  
Nonlinear Load ◽  
Rolling Elements ◽  
Nonlinear Equations Of Motion

This research presents a nonlinear model to analyze the ball bearing vibration due to the waviness in a rigid rotor supported by two or more ball bearings. The waviness of a ball and each races is modeled by the superposition of sinusoidal function, and the position vectors of inner and outer groove radius center are defined with respect to the mass center of the rotor in order to consider five degrees of freedom of a general rotor-bearing system. The waviness of a ball bearing is introduced to these position vectors to use the Hertzian contact theory in order to calculate the elastic deflection and nonlinear contact force resulting from the waviness while the rotor has translational and angular motion. They can be determined by solving the nonlinear equations of motion with five degrees of freedom by using the Runge-Kutta-Fehlberg algorithm. Numerical results of this research are validated with those of prior researchers. The proposed model can calculate the translational displacement as well as the angular displacement of the rotor supported by two or more ball bearings with waviness. It also characterizes the vibration frequencies resulting from the various kinds of waviness in rolling elements, the harmonic frequencies resulting from the nonlinear load-deflection characteristics of ball bearing, and the sideband frequencies resulting from nonlinearity of the waviness interaction.

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