Dynamic Analysis of Ball Bearings Due to Clearance Effect

2009 ◽  
Author(s):  
S. H. Upadhyay ◽  
S. C. Jain ◽  
S. P. Harsha
Keyword(s):  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Peak Frequency ◽  
System Response ◽  
Ball Bearings ◽  
Outer Race ◽  
Lagrange’S Equation ◽  
Rolling Elements ◽  
Machine Systems ◽  
The Individual

In this paper, the nonlinear dynamic behavior of ball bearings due to radial internal clearance and rotor speed has been analyzed. The approach presented in this paper accounts for the contact between rolling elements and inner/outer races. The equations of motion of a ball bearing are formulated in generalized coordinates, using Lagrange’s equation considering the vibration characteristics of the individual constitute such as inner race, outer race, rolling elements. The effects of speed of rotor in which rolling element bearings shows periodic, quasi-periodic and chaotic behavior are analyzed. The results also show the intermittent chaotic behavior in the dynamic response is seen to be strongly dependent on the speed of the rotor. The results are obtained in the form of frequency responses. The validity of the proposed model verified by comparison of frequency components of the system response with those obtained from experiments. The peak-to-peak frequency response of the system for each speed is obtained. The current study provides a powerful tool design and health monitoring of machine systems.

On the Dynamics of Elastohydrodynamic Mixed Lubricated Ball Bearings. Part II: Non-Linear Structural Vibration

2005 ◽  
Vol 219 (6) ◽  
pp. 423-433 ◽  
Author(s):  
M Sarangi ◽  
B. C. Majumdar ◽  
A. S. Sekhar
Keyword(s):  
Contact Stiffness ◽  
Equations Of Motion ◽  
Structural Vibration ◽  
Outer Race ◽  
Damping Coefficients ◽  
Lagrange’S Equation ◽  
Non Linear ◽  
Lubricated Contact ◽  
The Individual ◽  
Stiffness And Damping

Equations of motion of a ball bearing are formulated in generalized coordinates, using Lagrange's equation considering the vibrational characteristics of the individual constituents such as inner race, outer race, cage, and balls, in order to investigate the structural vibration of the bearing. This article is the second part of the present study dealing with structural vibration, whereas in the first part, elastohydrodynamic mixed lubricated contact stiffness and damping coefficients are determined. Utilizing these stiffness and damping coefficients, a non-linear load-deflection contact model is developed. This is then used in the equations of motion. The equations of motion are solved using Runge-Kutta integration technique. This work differs from the previous studies in the sense that the model simulates the vibration, considering that both the lubricated contact stiffness and damping correspond to the conservative and dissipative energies, respectively. It is observed that under undamped conditions, all the elements of the bearing actively participate in energy sharing and oscillate periodically, containing more than one frequency. The system vibration, however, died down rapidly in the presence of damping.

An improved model for investigating the vibration characteristics of ball bearing owing to outer race waviness under high speed

2021 ◽  
Vol 69 (2) ◽  
pp. 89-101
Author(s):  
Pingping Hou ◽  
Liqin Wang ◽  
Zhijie Xie ◽  
Qiuyang Peng
Keyword(s):  
High Speed ◽  
Ball Bearing ◽  
Vibration Measurement ◽  
Vibration Frequencies ◽  
Outer Race ◽  
Response Characteristics ◽  
Lagrange’S Equation ◽  
Rolling Elements ◽  
Initial Clearance ◽  
Improved Model

In this study, an improved model for a ball bearing is established to investigate the vibration response characteristics owing to outer race waviness under an axial load and high speed. The mathematical ball bearing model involves the motions of the inner ring, outer ring, and rolling elements in the radial XY plane and axial z direction. The 2Nb + 5 nonlinear differential governing equations of the ball bearing are derived from Lagrange's equation. The influence of rotational speed and outer race waviness is considered. The outer race waviness is modeled as a superposition of sinusoidal function and affects both the contact deformation between the outer raceway and rolling elements and initial clearance. The MATLAB stiff solver ODE is utilized to solve the differential equations. The simulated results show that the axial vibration frequency occurred at l fc and the radial vibration frequencies appeared at l fc fc when the outer race waviness of the order (l) was the multiple of the number of rolling elements (k Nb) and that the principal vibration frequencies were observed at l fc fc in the radial x direction when the outer race waviness of the order (l) was one higher or one lower than the multiple of the number of rolling elements (k Nb 1). At last, the validity of the proposed ball bearing model was verified by the high-speed vibration measurement tests of ball bearings.

Nonlinear Dynamic Analysis and Experimental Verification of an Unbalanced Rotor Supported by Ball Bearings

2011 ◽  
Author(s):  
S. H. Upadhyay ◽  
Satish C. Sharma ◽  
S. P. Harsha
Keyword(s):  
Dynamic Properties ◽  
Nonlinear Dynamic Analysis ◽  
Peak Frequency ◽  
Nonlinear Damping ◽  
Rotor System ◽  
System Stability ◽  
System Response ◽  
Ball Bearings ◽  
Unbalanced Rotor ◽  
Radial Forces

In this paper, a dynamic model is presented for studying the dynamic properties of unbalanced rotor system supported by ball bearings under the effects of radial internal clearance and unbalanced rotor effect. The Newmark-β method is used to solve the nonlinear equations. The dynamics behaviors of a rigid rotor system are studied through frequency responses of the system. Clearances, nonlinear stiffness & nonlinear damping, radial forces and unbalanced forces—all these bring a significant influence to bear on the system stability. The validity of the proposed model verified by comparison of frequency components of the system response with those obtained from experiments. The peak-to-peak frequency response of the system for each speed is obtained.

Dynamic Analysis of a High-Speed Rotor-Ball Bearing System Under Elastohydrodynamic Lubrication

2014 ◽  
Vol 136 (6) ◽  
Author(s):  
Yu-Yan Zhang ◽  
Xiao-Li Wang ◽  
Xiao-Qing Zhang ◽  
Xiao-Liang Yan
Keyword(s):  
High Speed ◽  
Ball Bearing ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Power Spectra ◽  
Elastohydrodynamic Lubrication ◽  
Lubricated Contacts ◽  
Rolling Elements ◽  
Stiffness And Damping ◽  
Bearing System

The nonlinear dynamic behaviors of a high-speed rotor-ball bearing system under elastohydrodynamic lubrication (EHL) are investigated. First, the numerical curve fittings for stiffness and damping coefficients of lubricated contacts between rolling elements and races are undertaken, and then the fitted formulae are introduced to the equations of motion of the rotor-ball bearing system to investigate its nonlinear characteristics. Furthermore, the time responses, power spectra, phase trajectories, orbit plots, and bifurcation diagrams for cases of ignoring and considering the lubrication condition in bearings are inspected and compared. The results indicate that, when lubrication is taken into account, the amplitudes of vibration displacements and velocities of the rotor system increase, and the appearance of different regions of periodic, quasi-periodic, and chaotic behavior is strongly dependent on the speed and load.

Dynamic Response of an Unbalanced Rotor Supported on Bearing With Outer Race Waviness

2013 ◽  
Author(s):  
P. K. Kankar ◽  
Satish C. Sharma ◽  
S. P. Harsha
Keyword(s):  
Dynamic Response ◽  
Chaotic Behavior ◽  
Mathematical Formulation ◽  
Rotational Frequency ◽  
Hertzian Contact ◽  
Radial Clearance ◽  
Outer Race ◽  
Unbalanced Rotor ◽  
Non Linear ◽  
Rolling Elements

The paper investigates the non-linear dynamic response of an unbalanced rotor supported on ball bearings with outer race waviness. The excitation is due to unbalanced force and waviness on outer race. The sources of non-linearities are both the radial clearance as well as the Hertzian contact between races and rolling elements. The nonlinear responses due to unbalanced rotor supported on bearings are investigated. The combined effects like non-linear stiffness and non-linear damping for unbalanced rotor with bearing waviness have been considered and analyzed in detail for a rotor bearing system. In the mathematical formulation, the contacts between the rolling elements and the races are considered as an oscillating spring-mass-damper system. The appearance of regions of periodic, sub-harmonic and chaotic behavior is seen to be strongly dependent on the number of waves in the outer race. The results show the appearance of instability and chaos in the dynamic response as the number of waves in the outer race is changed. The study indicates that the interaction of ball passage frequency (ωbp) due to outer race waviness and rotational frequency (X) due to the unbalanced rotor force. Poincaré maps and frequency responses are used to elucidate and to illustrate the diversity of the system behavior.

scholarly journals Using the Gottwald and Melbourne’s 0-1 test and the Hugichi fractal dimension to detect chaos in defective and healthy ball bearings

2018 ◽  
Vol 241 ◽  
pp. 01017 ◽  
Author(s):  
C. A. Kitio Kwuimy ◽  
T. Haj Mohamad ◽  
C. Nataraj
Keyword(s):  
Fractal Dimension ◽  
Experimental Model ◽  
Chaotic Behavior ◽  
Ball Bearings ◽  
Behavior Dynamics ◽  
Rolling Elements ◽  
Dynamic Map

The paper considers the identification of chaotic behavior dynamics using the data extracted from an experimental model of rotor supported on rolling elements. A description of the methodis provided as well as an illustration using a standard dynamic map. The 0-1 test for chaos and the Higuchi dimension are shown to be effective tool in the identification of chaotic behavior of the systembearing with and without faults.

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.

A DYNAMICAL SYSTEM WITH CHAOTIC BEHAVIOR

1989 ◽  
Vol 03 (15) ◽  
pp. 1185-1188 ◽  
Author(s):  
J. SEIMENIS
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Infinite Number ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Hamiltonian Dynamical Systems

We develop a method to find solutions of the equations of motion in Hamiltonian Dynamical Systems. We apply this method to the system [Formula: see text] We study the case a → 0 and we find that in this case the system has an infinite number of period dubling bifurcations.

scholarly journals Individual Alpha Peak Frequency, an Important Biomarker for Live Z-Score Training Neurofeedback in Adolescents with Learning Disabilities

2021 ◽  
Vol 11 (2) ◽  
pp. 167
Author(s):  
Rubén Pérez-Elvira ◽  
Javier Oltra-Cucarella ◽  
José Antonio Carrobles ◽  
Minodora Teodoru ◽  
Ciprian Bacila ◽  
...  
Keyword(s):  
Learning Disabilities ◽  
Pediatric Population ◽  
Peak Frequency ◽  
Normative Values ◽  
Z Score ◽  
Absolute Power ◽  
The Individual ◽  
And Behavior ◽  
The Waves ◽  
Alpha Peak

Learning disabilities (LDs) have an estimated prevalence between 5% and 9% in the pediatric population and are associated with difficulties in reading, arithmetic, and writing. Previous electroencephalography (EEG) research has reported a lag in alpha-band development in specific LD phenotypes, which seems to offer a possible explanation for differences in EEG maturation. In this study, 40 adolescents aged 10–15 years with LDs underwent 10 sessions of Live Z-Score Training Neurofeedback (LZT-NF) Training to improve their cognition and behavior. Based on the individual alpha peak frequency (i-APF) values from the spectrogram, a group with normal i-APF (ni-APF) and a group with low i-APF (li-APF) were compared in a pre-and-post-LZT-NF intervention. There were no statistical differences in age, gender, or the distribution of LDs between the groups. The li-APF group showed a higher theta absolute power in P4 (p = 0.016) at baseline and higher Hi-Beta absolute power in F3 (p = 0.007) post-treatment compared with the ni-APF group. In both groups, extreme waves (absolute Z-score of ≥1.5) were more likely to move toward the normative values, with better results in the ni-APF group. Conversely, the waves within the normal range at baseline were more likely to move out of the range after treatment in the li-APF group. Our results provide evidence of a viable biomarker for identifying optimal responders for the LZT-NF technique based on the i-APF metric reflecting the patient’s neurophysiological individuality.

Disk Position Nonlinearity Effects on the Chaotic Behavior of Rotating Flexible Shaft-Disk Systems

2012 ◽  
Vol 28 (3) ◽  
pp. 513-522 ◽  
Author(s):  
H. M. Khanlo ◽  
M. Ghayour ◽  
S. Ziaei-Rad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Disk System ◽  
Partial Differential ◽  
Rayleigh Beam ◽  
Flexible Shaft

AbstractThis study investigates the effects of disk position nonlinearities on the nonlinear dynamic behavior of a rotating flexible shaft-disk system. Displacement of the disk on the shaft causes certain nonlinear terms which appears in the equations of motion, which can in turn affect the dynamic behavior of the system. The system is modeled as a continuous shaft with a rigid disk in different locations. Also, the disk gyroscopic moment is considered. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed modes method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work are inclusive of time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The effect of disk nonlinearities is studied for some disk positions. The results confirm that when the disk is located at mid-span of the shaft, only the regular motion (period one) is observed. However, periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for situations in which the disk is located at places other than the middle of the shaft. The results show nonlinear effects are negligible in some cases.

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