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.

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.

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.

Motion of a Ball in a Ball Bearing

1973 ◽  
Vol 95 (1) ◽  
pp. 263-268
Author(s):  
H. Portig ◽  
H. G. Rylander
Keyword(s):  
Deformation Theory ◽  
Coulomb Friction ◽  
Ball Bearing ◽  
Digital Simulation ◽  
Equations Of Motion ◽  
Unsteady Motion ◽  
Hertzian Contact ◽  
Contact Deformation ◽  
Normal Forces ◽  
Simulation Language

A method is developed which allows the digital simulation of the unsteady motion of a single ball constrained only by two moving bearing races. Any desired motion of the races can be simulated. Normal forces acting on the ball are calculated by Hertzian contact deformation theory. If there is slippage between ball and races, Coulomb friction is assumed to occur. Solutions to the differential equations of motion were obtained on a computer with the digital simulation language MIMIC. The phenomenon of ball control as well as the behavior of the ball as it reached a controlled state from rest were observed. This analysis can produce more realistic results than methods that assume that the ball is controlled at all times, especially when the races are radially or angularly displaced with respect to each other.

An Efficient Transient Simulation Method for a Volume Dynamics Model

2018 ◽  
Author(s):  
Masahiro Kurosaki ◽  
Minoru Sasamoto ◽  
Kentaro Asaka ◽  
Keiko Nakamura ◽  
Daiki Kakiuchi
Keyword(s):  
Integration Method ◽  
Simulation Method ◽  
Euler Method ◽  
Test Model ◽  
Dynamics Model ◽  
Engine Operation ◽  
Turbine Engine ◽  
Engine Model ◽  
Influence Coefficients ◽  
Transient Simulations

This paper presents an efficient numerical integration method for a volume dynamics model in gas turbine engine transient simulations. It is a modified implicit Euler method that allows a time increment that would not be stable with the explicit Euler method. The Jacobian matrix of a nonlinear engine model is evaluated along the steady state engine operation line and scheduled as a function of the corrected shaft speed in advance, eliminating the necessity of computing during the simulation. The proposed method was applied to transient simulations of a compressor rig test model composed of a compressor, a nozzle with variable geometry and a volume placed between them. The eigenvalues of the simplified volume dynamics were analytically derived. It is shown that they are functions of the characteristic time of the volume defined by mass present in the volume divided by mass flow rate flowing into and out of the volume and dimensionless influence coefficients of nearby components.

scholarly journals Ball Bearing Dynamics at the Interference fit on Balls

2017 ◽  
Vol 176 ◽  
pp. 19-24 ◽  
Author(s):  
I.S. Barmanov ◽  
M.N. Ortikov
Keyword(s):  
Ball Bearing ◽  
Interference Fit ◽  
Bearing Dynamics

A Sequential Integration Method

1988 ◽  
Vol 110 (4) ◽  
pp. 382-388
Author(s):  
Liang-Wey Chang ◽  
James F. Hamilton
Keyword(s):  
Integration Method ◽  
Equations Of Motion ◽  
Slow Motion ◽  
Explicit Integration ◽  
Time Step ◽  
Lumped Model ◽  
Guaranteed Stability ◽  
Fast Motion ◽  
The Stability ◽  
Secular Terms

This paper presents a method for simulating systems with two inertially coupled motions, i.e., a slow motion and a fast motion. The equations of motion are separated into two sets of coupled nonlinear ordinary differential equations. For each time step, the two sets of equations are integrated sequentially rather than simultaneously. Explicit integration methods are used for integrating the slow motion since the stability of the integration is not a problem and the explicit methods are very convenient for nonlinear equations. For the fast motion, the equations are linear and the implicit integrations can be used with guaranteed stability. The size of time step only needs to be chosen to provide accuracy of the solution for the modes that are excited. The interaction between the two types of motion must be treated such that secular terms do not appear due to the sequential integration method. A lumped model of a flexible pendulum will be presented in this paper to illustrate the application of the method. Numerical results for both simultaneous and sequential integration are presented for comparison.

Dynamic Rocking Response and Optimization of the Nonlinear Suspension of a Railroad Freight Car

1980 ◽  
Vol 102 (1) ◽  
pp. 86-93 ◽  
Author(s):  
M. Samaha ◽  
T. S. Sankar
Keyword(s):  
Equations Of Motion ◽  
Sine Wave ◽  
Optimization Techniques ◽  
Model Accuracy ◽  
Frequency Responses ◽  
Optimum Parameters ◽  
Freight Car ◽  
Six Degree Of Freedom ◽  
Railroad Freight ◽  
Rocking Response

A modified mathematical model of a large capacity railroad freight vehicle is presented. The model for this investigation is constructed in such a way to describe the bounce, sway and rocking modes of the system and also to account for most of the vehicle non-linearity effects. Equations of motion of the six degree of freedom nonlinear model are derived assuming that the excitations from the track in vertical and lateral directions are purely periodic in the form of a rectified sine wave. The solution for the time and frequency responses on digital computer are compared with available measured data to investigate the model accuracy. Multivariable optimization techniques are employed to find the optimum suspension parameters that minimizes the maximum car rocking response over the frequency range of interest. The optimum parameters are presented in different forms either for the existing or for stabilized vehicle configuration.

Optimization of Vehicle Powertrain Mounting System and its Performance

2013 ◽  
Vol 441 ◽  
pp. 580-583
Author(s):  
Gong Yu Pan ◽  
Xin Yang ◽  
You Yan
Keyword(s):  
Vibration Isolation ◽  
Dynamics Model ◽  
Acceleration Response ◽  
Idle State ◽  
Decoupling Method ◽  
Before And After ◽  
Vibration Transmissibility ◽  
Six Degree Of Freedom ◽  
Vibration Isolation Performance ◽  
Isolation Performance

In order to solve the vibration problem of diesel engine powertrain assembly at its idle state, a six degree-of-freedom dynamics model of the powertrain mounting system is established and a optimization based on Adams/View is applied to simulation and analysis on the powertrain mounting system with energy decoupling method. The results show that the optimized repositioning mounts installation position can effectively improve decoupling rate in main vibration directions of mounting system. Based on this, the vibration transmissibility and acceleration response before and after optimization are simulated. The results show that the optimized engine mounting system makes a great improvement of vibration isolation performance.

Added Fluid Mass and the Equations of Motion of a Parachute

1983 ◽  
Vol 34 (3) ◽  
pp. 226-242 ◽  
Author(s):  
John A. Eaton
Keyword(s):  
Equations Of Motion ◽  
Added Mass ◽  
Unsteady Forces ◽  
Fluid Mass ◽  
Mass Tensor ◽  
Body Shapes ◽  
The Stability ◽  
External Forces ◽  
Six Degree Of Freedom ◽  
Nonlinear Solutions

SummaryWhile it has long been known that added fluid mass may be important in the dynamics of parachutes, due to inadequate or incorrect derivation and/or implementation of the added mass tensor its full significance in the stability of parachutes has yet to be appreciated. The concept of added mass is outlined and some general conditions for its significance are presented. Its implementation in the parachute equations of motion is reviewed, and the equations used in previous treatments are shown to be erroneous. A general method for finding the equivalent external forces and moments due to added mass is given, and the correct, anisotropic forms of the added mass tensor are derived for the six degree-of-freedom motion in an ideal fluid of rigid body shapes with planar-, twofold- and axisymmetry, These derivations may also be useful in dynamic stability studies of other low relative density bodies such as airships, balloons, submarines and torpedoes. Full nonlinear solutions of the equations of motion for the axisymmetric parachute have been obtained, and results indicate that added mass effects are more significant than previously predicted. In particular, the component of added mass along the axis of symmetry has a strong influence on stability. Better data on unsteady forces and moments on parachutes are needed.

Effect of Forcing Terms and General Characteristics of Torsional Vibrations of Marine Engine Systems with Variable Inertia

1974 ◽  
Vol 18 (02) ◽  
pp. 131-138
Author(s):  
W. D. Carnegie ◽  
M. S. Pasricha
Keyword(s):  
Equations Of Motion ◽  
Mean Value ◽  
Wave Form ◽  
Actual System ◽  
Computer Methods ◽  
Reciprocating Engine ◽  
Complex Wave ◽  
Wave Forms ◽  
Crankshaft Rotation ◽  
Engine Systems

The torsional vibration phenomenon in the running gear of reciprocating engine systems is usually dealt with by considering a series of constant inertias connected by sections of massless shafting. Such a simplified model does not reproduce the exact dynamic characteristics of the actual system. In recent years several cases of marine crankshaft failures have been attributed to the phenomenon of secondary resonance, which is explained by the fact that the effective inertia of each slider crankmechanism varies about a mean value in relation to the position of the crank. When the variableinertia effect is allowed for, the equations of motion taking into account the effect are nonlinear. Assuming small displacements, the equations can be linearized to predict important characteristics of the motion. The motions in the form of complex wave forms are studied at different speeds of engine rotation and some of the wave form solutions are analyzed in the range of present investigations. Computer methods making use of numerical analysis processes, namely, the modifiedEuler's equations and the Runge-Kutta constants, have been applied in the investigations. A study of the effect on the motion of the system due to variation of inertia ratio is carried out at a particular speed of the crankshaft rotation; also investigated are the variations in the motions due to the action of external excitations with respect to changes in phase angle and inertia ratio. General comments on Draminsky's work in the light of the present investigations are included.

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