The Dynamics of Ball Bearings

1971 ◽  
Vol 93 (1) ◽  
pp. 1-10 ◽  
Author(s):  
C. T. Walters
Keyword(s):  
Fourth Order ◽  
Ball Bearing ◽  
Equations Of Motion ◽  
Numerical Results ◽  
Degree Of Freedom ◽  
Spin Axis ◽  
General Analysis ◽  
Ball Bearings ◽  
High Speeds ◽  
Six Degree Of Freedom

The details of the dynamics of the elements of a ball bearing become increasingly important at high speeds. A comprehensive general analysis of the motions of balls and a ball separator with realistic lubrication is summarized. The equations of motion consider four degree-of-freedom balls and a six degree-of-freedom separator and are integrated numerically with a fourth order Runge Kutta scheme. Numerical results are presented for a particular spin axis gyro bearing configuration.

Analysis of an Offset Unsymmetric Gyroscope With Oblique Rotor Using (3×3) Matrices With Dual-Number Elements

1969 ◽  
Vol 91 (3) ◽  
pp. 535-541 ◽  
Author(s):  
An Tzu Yang
Keyword(s):  
Equations Of Motion ◽  
Degree Of Freedom ◽  
Spin Axis ◽  
Dynamic Equations ◽  
Gyroscopic System ◽  
Dual Number ◽  
System A ◽  
Six Degree Of Freedom ◽  
Special Case

Using 3 × 3 matrices of dual-number elements, dynamic equations are obtained for an offset unsymmetric gyroscope with obliquely placed rotor, a generalized six-degree-of-freedom gyroscopic system (shown schematically in Fig. 3). Equations of motion for a special case of the system, a two-frame symmetric gyroscope, conventional in all aspects except the rotor is inclined relative to its spin axis, are deduced; these equations are applied to the study of the effects of a slightly inclined rotor on (a) a two-frame symmetric gyroscope in steady precession and (b) a Faucualt’s gyrocompass.

Dynamics of Rolling-Element Bearings—Part III: Ball Bearing Analysis

1979 ◽  
Vol 101 (3) ◽  
pp. 312-318 ◽  
Author(s):  
P. K. Gupta
Keyword(s):  
Ball Bearing ◽  
Dynamic Performance ◽  
Equations Of Motion ◽  
Ball Bearings ◽  
Rolling Element Bearings ◽  
Time Simulation ◽  
Rolling Element ◽  
Analytical Development ◽  
Critical Film Thickness ◽  
Bearing Analysis

An analytical formulation for the generalized ball, cage, and race motion in a ball bearing is presented in terms of the classical differential equations of motion. Ball-race interaction is analyzed in detail and the resulting force and moment vectors are determined. The ball-cage and race-cage interactions are considered to be either hydrodynamic or metallic and a critical film thickness defines the transition between the two regimes. Simplified treatments for the drag and churning losses are also included to complete a rigorous analytical development for the real-time simulation of the dynamic performance of ball bearings.

scholarly journals Dynamics and vibration analysis of the interface between a non-rigid sphere and omnidirectional wheel actuators

Robotica ◽  
2014 ◽  
Vol 33 (9) ◽  
pp. 1850-1868 ◽  
Author(s):  
A. Weiss ◽  
R. G. Langlois ◽  
M. J. D. Hayes
Keyword(s):  
Rotational Motion ◽  
Vibration Analysis ◽  
Equations Of Motion ◽  
Design Tool ◽  
Degree Of Freedom ◽  
Rigid Sphere ◽  
Motion Platform ◽  
Six Degree Of Freedom ◽  
Dynamics And Vibration ◽  
Omnidirectional Wheels

SUMMARYThis paper presents analysis of the dynamics and vibration of an orientation motion platform utilizing a sphere actuated by omnidirectional wheels. The purpose of the analysis is to serve as a design tool for the construction of a six-degree-of-freedom motion platform with unlimited rotational motion. The equations of motion are presented taking flexibility of the system into account. The behaviour of the system is illustrated by sample configurations with a range of omnidirectional wheel types and geometries. Vibration analysis follows, and sensitivity to various parameters is investigated. It is determined that the geometry of omnidirectional wheels has a significant effect on the behaviour of the system.

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.

Motion of Unbalanced Balls in High-Speed Angular Contact Ball Bearings

1990 ◽  
Vol 112 (1) ◽  
pp. 105-110 ◽  
Author(s):  
H. Kawamura ◽  
K. Touma
Keyword(s):  
High Speed ◽  
Ball Bearing ◽  
Three Dimensional ◽  
Ball Bearings ◽  
Axial Loads ◽  
Angular Contact Ball Bearing ◽  
High Speeds ◽  
Dimensional Ball ◽  
Ball Motion ◽  
Bearing Ball

This paper reports on an experimental study of the three-dimensional ball motion of an unbalanced ball in a 50-mm bore angular contact ball bearing operating at high speeds under axial loads. One bearing ball, which was unbalanced by making a small hole in it, was magnetized and the motion of the ball was determined using Hall-elements. The bearing was tested under various loads and speeds up to 12,000 rpm. The influence of unbalance eccentricity on the unbalanced ball’s motion was investigated.

Application of Dual Quaternions to the Study of Gyrodynamics

1967 ◽  
Vol 89 (1) ◽  
pp. 137-143 ◽  
Author(s):  
A. T. Yang
Keyword(s):  
Equations Of Motion ◽  
Degree Of Freedom ◽  
System Dynamic ◽  
Dynamic Equations ◽  
Dual Quaternions ◽  
Special Cases ◽  
Six Degree Of Freedom

Dual quaternions are used to derive equations of motion for an offset unsymmetrical gyroscope. From the equations, in view of the general characteristics of this six-degree-of-freedom system, dynamic equations for a variety of gyroscopic and pendulous systems may be deduced as special cases. Examples are given for illustration.

Ball Bearing Dynamic Analysis Using Computer Methods—Part I: Analysis

1996 ◽  
Vol 118 (1) ◽  
pp. 52-58 ◽  
Author(s):  
Crawford R. Meeks ◽  
Long Tran
Keyword(s):  
Ball Bearing ◽  
Integration Method ◽  
Equations Of Motion ◽  
Dynamics Model ◽  
Stiff System ◽  
Computer Methods ◽  
Wear Life ◽  
Rigorous Model ◽  
Six Degree Of Freedom ◽  
Bearing Dynamics

An analytical ball bearing dynamics model was developed that rigorously models all of the significant kinematic, structural, and dynamic effects. The model can analyze bearings of any material combination for the races, balls and ball cage. This model analyzes the stresses and deflections of the loaded elements due to (1) preload, (2) external axial, radial and moment loads, (3) centrifugal and gyroscopic ball loads. A rigorous six-degree-of-freedom model of ball cage motions was developed to analyze ball and cage dynamics. The ball cage equations of motion were written in a rotating coordinate system, which greatly simplifies the equations, resulting in a highly efficient, but rigorous, model of bearing dynamics. A computer program was developed, incorporating the algorithms, to solve the multiple simultaneous quasi-static ball-to-race load equations using modified Newton-Raphson methods. The Lawrence Livermore Ode package (LSODA) is employed for numerical integration of the dynamic equations of motion. This method assures convergence, while controlling the accuracy of the calculations as a function of computer run time and automatically selects the appropriate integration method for stiff and non-stiff system of ODE. The program analyzes ball and cage motions in time domain, wear life, fatigue life, lubricant film effects, ball-to-cage forces, torque noise and many other bearing parameters.

Calculation of Dynamic Tire Forces from Aircraft Instrumentation Data

2010 ◽  
Vol 38 (3) ◽  
pp. 182-193 ◽  
Author(s):  
Gary E. McKay
Keyword(s):  
System Performance ◽  
Equations Of Motion ◽  
Test Laboratory ◽  
Excellent Correlation ◽  
Friction Behavior ◽  
Aircraft Landing ◽  
Data Scatter ◽  
Tire Forces ◽  
Six Degree Of Freedom ◽  
Tire Friction

Abstract When evaluating aircraft brake control system performance, it is difficult to overstate the importance of understanding dynamic tire forces—especially those related to tire friction behavior. As important as they are, however, these dynamic tire forces cannot be easily or reliably measured. To fill this need, an analytical approach has been developed to determine instantaneous tire forces during aircraft landing, braking and taxi operations. The approach involves using aircraft instrumentation data to determine forces (other than tire forces), moments, and accelerations acting on the aircraft. Inserting these values into the aircraft’s six degree-of-freedom equations-of-motion allows solution for the tire forces. While there are significant challenges associated with this approach, results to date have exceeded expectations in terms of fidelity, consistency, and data scatter. The results show excellent correlation to tests conducted in a tire test laboratory. And, while the results generally follow accepted tire friction theories, there are noteworthy differences.

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