A Discrete Element Approach for Modeling Cage Flexibility in Ball Bearing Dynamics Simulations

2009 ◽  
Vol 131 (2) ◽  
Author(s):  
Nick Weinzapfel ◽  
Farshid Sadeghi
Keyword(s):  
Reference Frame ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Element Model ◽  
Discrete Element ◽  
Discrete Elements ◽  
Step Size ◽  
Deep Groove ◽  
Bearing Dynamics

A model for deep-groove and angular-contact ball bearings was developed to investigate the influence of a flexible cage on bearing dynamics. The cage model introduces flexibility by representing the cage as an ensemble of discrete elements that allow deformation of the fibers connecting the elements. A finite element model of the cage was developed to establish the relationships between the nominal cage properties and those used in the flexible discrete element model. In this investigation, the raceways and balls have six degrees of freedom. The discrete elements comprising the cage each have three degrees of freedom in a cage reference frame. The cage reference frame has five degrees of freedom, enabling three-dimensional motion of the cage ensemble. Newton’s laws are used to determine the accelerations of the bearing components, and a fourth-order Runge–Kutta algorithm with constant step size is used to integrate their equations of motion. Comparing results from the dynamic bearing model with flexible and rigid cages reveals the effects of cage flexibility on bearing performance. The cage experiences nearly the same motion and angular velocity in the loading conditions investigated regardless of the cage type. However, a significant reduction in ball-cage pocket forces occurs as a result of modeling the cage as a flexible body. Inclusion of cage flexibility in the model also reduces the time required for the bearing to reach steady-state operation.

A New Approach to Modeling Surface Defects in Bearing Dynamics Simulations

2008 ◽  
Vol 130 (4) ◽  
Author(s):  
Ankur Ashtekar ◽  
Farshid Sadeghi ◽  
Lars-Erik Stacke
Keyword(s):  
Dynamic Model ◽  
High Speed ◽  
Degrees Of Freedom ◽  
Surface Defects ◽  
Equations Of Motion ◽  
Infinite Domain ◽  
Step Size ◽  
Deep Groove ◽  
Dynamics Simulations ◽  
Bearing Dynamics

A dynamic model for deep groove and angular contact ball bearings was developed to investigate the influence of race defects on the motions of bearing components (i.e., inner and outer races, cage, and balls). In order to determine the effects of dents on the bearing dynamics, a model was developed to determine the force-deflection relationship between an ellipsoid and a dented semi-infinite domain. The force-deflection relationship for dented surfaces was then incorporated in the bearing dynamic model by replacing the well-known Hertzian force-deflection relationship whenever a ball/dent interaction occurs. In this investigation, all bearing components have six degrees-of-freedom. Newton’s laws are used to determine the motions of all bearing elements, and an explicit fourth-order Runge–Kutta algorithm with a variable or constant step size was used to integrate the equations of motion. A model was used to study the effect of dent size, dent location, and inner race speed on bearing components. The results indicate that surface defects and irregularities like dent have a severe effect on bearing motion and forces. Furthermore, these effects are even more severe for high-speed applications. The results also demonstrate that a single dent can affect the forces and motion throughout the entire bearing and on all bearing components. However, the location of the dent dictates the magnitude of its influence on each bearing component.

An Explicit Finite-Element Model to Investigate the Effects of Elastomeric Bushing on Bearing Dynamics

2016 ◽  
Vol 138 (3) ◽  
Author(s):  
Lijun Cao ◽  
Farshid Sadeghi ◽  
Lars-Erik Stacke
Keyword(s):  
Finite Element ◽  
Three Dimensional ◽  
Element Model ◽  
Constitutive Relationship ◽  
Explicit Finite Element ◽  
Deep Groove ◽  
Deep Groove Ball Bearing ◽  
Parametric Studies ◽  
Viscoelastic Behaviors ◽  
Bearing Dynamics

This work presents a numerical simulation which studies the effect of elastomeric bushing on the dynamics of a deep-groove ball bearing. To achieve the objective, a three-dimensional (3D) explicit finite element method (EFEM) was developed to model a cylindrical elastomeric bushing, which was then coupled with an existing dynamic bearing model (DBM). Constitutive relationship for the elastomer is based on the Arruda–Boyce model combined with a generalized Maxwell-element model to capture both hyperelastic and viscoelastic behaviors of the material. Comparisons between the bushing model developed for this investigation and the existing experimental elastomeric bushing study showed that the results are in good agreement. Parametric studies were conducted to show the effects of various elastomeric material properties on bushing behavior. It was also shown that a desired bushing support performance can be achieved by varying bushing geometry. Simulations using the combined EFEM bushing and DBM model demonstrated that the elastomeric bushing provides better compliance to bearing misalignment as compared to a commonly used rigid support model. As a result, less ball slip and spin are generated. Modeling with a bearing surface dent showed that vibrations due to surface abnormalities can be significantly reduced using elastomeric bushing support. It has also been shown that choosing a proper bushing is an efficient way to tuning bushing vibration frequencies.

Investigation of Train Dynamics in Passing Through Curves Using a Full Model

Joint Rail ◽  
2004 ◽  
Author(s):  
Mohammad Durali ◽  
Mohammad Mehdi Jalili Bahabadi
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Full Model ◽  
Bogie Frame ◽  
Nonlinear Characteristics ◽  
Train Derailment ◽  
Train Dynamics

In this article a train model is developed for studying train derailment in passing through bends. The model is three dimensional, nonlinear, and considers 43 degrees of freedom for each wagon. All nonlinear characteristics of suspension elements as well as flexibilities of wagon body and bogie frame, and the effect of coupler forces are included in the model. The equations of motion for the train are solved numerically for different train conditions. A neural network was constructed as an element in solution loop for determination of wheel-rail contact geometry. Derailment factor was calculated for each case. The results are presented and show the major role of coupler forces on possible train derailment.

Reduced Rough-Surface Parametrization for Use With the Discrete-Element Model

2009 ◽  
Vol 131 (2) ◽  
Author(s):  
Stephen T. McClain ◽  
Jason M. Brown
Keyword(s):  
Heat Transfer ◽  
Rough Surface ◽  
Surface Characterization ◽  
Rough Surfaces ◽  
Roughness Element ◽  
Three Dimensional ◽  
Element Model ◽  
Discrete Element ◽  
Diameter Distribution ◽  
Discrete Element Model

The discrete-element model for flows over rough surfaces was recently modified to predict drag and heat transfer for flow over randomly rough surfaces. However, the current form of the discrete-element model requires a blockage fraction and a roughness-element diameter distribution as a function of height to predict the drag and heat transfer of flow over a randomly rough surface. The requirement for a roughness-element diameter distribution at each height from the reference elevation has hindered the usefulness of the discrete-element model and inhibited its incorporation into a computational fluid dynamics (CFD) solver. To incorporate the discrete-element model into a CFD solver and to enable the discrete-element model to become a more useful engineering tool, the randomly rough surface characterization must be simplified. Methods for determining characteristic diameters for drag and heat transfer using complete three-dimensional surface measurements are presented. Drag and heat transfer predictions made using the model simplifications are compared to predictions made using the complete surface characterization and to experimental measurements for two randomly rough surfaces. Methods to use statistical surface information, as opposed to the complete three-dimensional surface measurements, to evaluate the characteristic dimensions of the roughness are also explored.

Trajectory Tracking of Underactuated Multibody Systems

2011 ◽  
Author(s):  
Stefan Reichl ◽  
Wolfgang Steiner
Keyword(s):  
Trajectory Tracking ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Inverse Dynamics ◽  
Feedforward Control ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Differential Algebraic Equations ◽  
State Variables ◽  
Algebraic Equations

This work presents three different approaches in inverse dynamics for the solution of trajectory tracking problems in underactuated multibody systems. Such systems are characterized by less control inputs than degrees of freedom. The first approach uses an extension of the equations of motion by geometric and control constraints. This results in index-five differential-algebraic equations. A projection method is used to reduce the systems index and the resulting equations are solved numerically. The second method is a flatness-based feedforward control design. Input and state variables can be parameterized by the flat outputs and their time derivatives up to a certain order. The third approach uses an optimal control algorithm which is based on the minimization of a cost functional including system outputs and desired trajectory. It has to be distinguished between direct and indirect methods. These specific methods are applied to an underactuated planar crane and a three-dimensional rotary crane.

A CUBIC ARRANGED SPHERICAL DISCRETE ELEMENT MODEL

2014 ◽  
Vol 11 (05) ◽  
pp. 1350102 ◽  
Author(s):  
WEI GAO ◽  
YUANQIANG TAN ◽  
MENGYAN ZANG
Keyword(s):  
Strain Energy ◽  
Element Model ◽  
Discrete Element ◽  
Discrete Element Model ◽  
Laminated Plate ◽  
Discrete Elements ◽  
Elastic Continuum ◽  
Vibration Process ◽  
Spring Constants ◽  
Dem Model

A 3D discrete element model (DEM model) named cubic arranged discrete element model is proposed. The model treats the interaction between two connective discrete elements as an equivalent "beam" element. The spring constants between two connective elements are obtained based on the equivalence of strain energy stored in a unit volume of elastic continuum. Following that, the discrete element model proposed and its algorithm are implemented into the in-house developed code. To test the accuracy of the DEM model and its algorithm, the vibration process of the block, a homogeneous plate and laminated plate under impact loading are simulated in elastic range. By comparing the results with that calculated by using LS-DYNA, it is found that they agree with each other very well. The accuracy of the DEM model and its algorithm proposed in this paper is proved.

Reduced Rough-Surface Parameterization for Use With the Discrete-Element Model

2007 ◽  
Author(s):  
Stephen T. McClain ◽  
Jason M. Brown
Keyword(s):  
Heat Transfer ◽  
Rough Surface ◽  
Surface Characterization ◽  
Rough Surfaces ◽  
Roughness Element ◽  
Three Dimensional ◽  
Element Model ◽  
Discrete Element ◽  
Diameter Distribution ◽  
Discrete Element Model

The discrete-element model for flows over rough surfaces was recently modified to predict drag and heat transfer for flow over randomly-rough surfaces. However, the current form of the discrete-element model requires a blockage fraction and a roughness-element diameter distribution as a function of height to predict the drag and heat transfer of flow over a randomly-rough surface. The requirement for a roughness element-diameter distribution at each height from the reference elevation has hindered the usefulness of the discrete-element model and inhibited its incorporation into a computational fluid dynamics (CFD) solver. To incorporate the discrete-element model into a CFD solver and to enable the discrete-element model to become a more useful engineering tool, the randomly-rough surface characterization must be simplified. Methods for determining characteristic diameters for drag and heat transfer using complete three-dimensional surface measurements are presented. Drag and heat transfer predictions made using the model simplifications are compared to predictions made using the complete surface characterization and to experimental measurements for two randomly-rough surfaces. Methods to use statistical surface information, as opposed to the complete three-dimensional surface measurements, to evaluate the characteristic dimensions of the roughness are also explored.

Performance Characteristics of a Smart Flexible Beam With Distributed ACLD Elements

2002 ◽  
Author(s):  
L. C. Hau ◽  
Eric H. K. Fung
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Viscoelastic Model ◽  
Element Model ◽  
Flexible Beam ◽  
Constrained Layer Damping ◽  
Optimal Output ◽  
Modes Of Vibration ◽  
Original Size

The finite element method, in conjunction with the Golla-Hughes-McTavish (GHM) viscoelastic model, is employed to model a clamped-free beam partially treated with active constrained layer damping (ACLD) elements. The governing equations of motion are converted to a state-space form for control system design. Prior to this, since the resultant finite element model has too many degrees of freedom due to the addition of dissipative coordinates, a model reduction is performed to revert the system back to its original size. Finally, optimal output feedback gains are designed based on the reduced models. Numerical simulations are performed to study the effect of different element configurations, with various spacing and locations, on the vibration control performance of a “smart” flexible ACLD treated beam. Results are presented for the damping ratios of the first two modes of vibration. It is found that improvement on the second mode damping can be achieved by splitting a single ACLD element into two and placing them at appropriate positions of the beam.

scholarly journals Effective 1D Time-Dependent Schrödinger Equations for 3D Geometrically Correlated Systems

Materials ◽  
2020 ◽  
Vol 13 (13) ◽  
pp. 3033
Author(s):  
Devashish Pandey ◽  
Xavier Oriols ◽  
Guillermo Albareda
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Longitudinal Direction ◽  
Time Dependent ◽  
Ab Initio Molecular Dynamics ◽  
Quantitative Accuracy ◽  
Dimensional Simulation ◽  
Transport Problems ◽  
Computational Resources

The so-called Born–Huang ansatz is a fundamental tool in the context of ab-initio molecular dynamics, viz., it allows effectively separating fast and slow degrees of freedom and thus treating electrons and nuclei with different mathematical footings. Here, we consider the use of a Born–Huang-like expansion of the three-dimensional time-dependent Schrödinger equation to separate transport and confinement degrees of freedom in electron transport problems that involve geometrical constrictions. The resulting scheme consists of an eigenstate problem for the confinement degrees of freedom (in the transverse direction) whose solution constitutes the input for the propagation of a set of coupled one-dimensional equations of motion for the transport degree of freedom (in the longitudinal direction). This technique achieves quantitative accuracy using an order less computational resources than the full dimensional simulation for a typical two-dimensional geometrical constriction and upto three orders for three-dimensional constriction.

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