Maxbe D2.5 Axel Bearing Modelling and Analysis v 1maxbe/PU/wp2/MAXBE-D2.5 Axel... · 2014. 5....

Project number 314408 FP7‐SST‐2012‐RTD‐1

MAXBE

INTEROPERABLE MONITORING, DIAGNOSIS AND MAINTENANCE STRATEGIES

FOR AXLE BEARINGS

Deliverable D2.5 – Axle bearing modelling and analysis

Prepared by

IST

Document No.

MAXBE – WP – 2.5 – IST – 01 – 1.0

March 2014

Partners:

University of Porto (UPORTO) Rede Ferroviária Nacional (REFER) Ansaldo STS (ASTS) Institute of Transport, Railway Construction and Operation (IVE) COMSA Evoleo Technologies (EVOLEO) NEM Solutions (NEM) University College Cork (UCC) MERMEC SKF Industry

Instituto Superior Técnico (IST) Dynamics, Structures and Systems International (D2S) Vlaamse Vervoermaatschappij De Lijn (DL) Empresa de Manutenção de Equipamento Ferroviário (EMEF) I‐MOSS Krestos Limited (KRESTOS) University of Birmingham (UoB)

MAXBE‐WP‐2.5‐IST‐01 March 2014

Copyright © MAXBE Consortium 1 of 86

Revision history

Document No. MAXBE – WP – 2.5 – IST – 01 – 1.0

Title Project Documentation and Internal Distribution

Version Date Author Description/Remarks/Reasons for change

0.1 2014-03-17 IST Preliminary Report

Table 1.1 - Revision history

MAXBE‐WP‐2.5‐IST‐01‐1.0 March 2014

2 of 86 Copyright © MAXBE Consortium

CONTENTS

1. FORWARD............................................................................................... 4

2. INTRODUCTION ........................................................................................ 4

3. DYNAMIC ANALYSIS PROCEDURE .............................................................. 6

3.1. KINEMATICS OF A RIGID BODY ................................................................................ 6

3.2. DYNAMICS OF A MULTIBODY SYSTEM ...................................................................... 7

3.3. EXTERNAL AND INTERNAL FORCES ......................................................................... 9

3.4. SOLUTION OF THE EQUATIONS OF MOTION ............................................................ 10

3.5. STABILITY OF THE EQUATIONS OF MOTION ............................................................ 11

3.6. DIRECT INTEGRATION METHOD OF THE EQUATIONS OF MOTION ............................... 11

4. INITIALIZATION OF THE BEARING ROLLING ELEMENTS ............................... 14

4.1. SPHERICAL ROLLER BEARING ELEMENTS INITIALIZATION ........................................ 15

4.2. TAPERED ROLLER BEARING ELEMENTS INITIALIZATION ........................................... 20

5. GENERAL FORMULATION OF THE ROLLER CONTACT DETECTION ............... 27

5.1. CONTACT DETECTION BETWEEN A CIRCLE AND A GENERIC SURFACE ....................... 28

5.2. DESCRIPTION OF SPECIFIC SURFACE GEOMETRIES ................................................ 31

5.3. CYLINDRICAL BEARINGS...................................................................................... 33

5.4. TAPERED BEARINGS ........................................................................................... 34

5.5. SPHERICAL BEARINGS ........................................................................................ 35

5.6. RACE FLANGES .................................................................................................. 37

5.6.1. Cylindrical bearing flange contact .......................................................................................... 38

5.6.2. Tapered bearing flange contact ............................................................................................. 40

6. OTHER CONTACT DETECTION GEOMETRIES IN BEARINGS ......................... 42

6.1. CONTACT BETWEEN SPHERICAL CAP AND CONICAL SURFACE ................................. 42

6.2. CONTACT BETWEEN SPHERICAL CAP AND LINE ...................................................... 43

6.3. CONTACT BETWEEN CIRCLE AND LINE .................................................................. 45

6.4. CONTACT BETWEEN CIRCLE TOP AND LINE ............................................................ 46

6.5. CONTACT BETWEEN TWO CYLINDRICAL SEGMENTS ................................................ 48

7. FORMULATION OF THE ROLLER CONTACT FORCES ................................... 50



7.1. NORMAL CONTACT FORCES ................................................................................ 50

7.2. TANGENCIAL CONTACT FORCES ........................................................................... 56

7.2.1. Lubricant film thickness .......................................................................................................... 57

7.2.2. Equivalent friction coefficients ................................................................................................ 63

8. RAILWAY WHEEL-RAIL CONTACT FORCES ............................................... 65

8.1. WHEEL AND RAIL GEOMETRIC DESCRIPTION .......................................................... 65

8.1.1. Rail surface ............................................................................................................................ 66

8.1.2. Wheel surface ........................................................................................................................ 67

8.2. WHEEL–RAIL CONTACT FORMULATION ................................................................. 68

8.2.1. Wheel-rail contact algorithm ................................................................................................... 68

8.2.2. Contact point detection ........................................................................................................... 69

8.2.3. Two points of simultaneous contact ....................................................................................... 71

8.2.4. Normal contact forces in the wheel-rail interface ................................................................... 71

8.2.5. Tangential contact forces in the wheel-rail interface .............................................................. 73

9. REFERENCES ........................................................................................ 75

10. ANNEX 1 – INPUT DATA FOR BEARING MODEL ....................................... 81

10.1. GENERAL BEARING DATA ................................................................................. 81

10.2. SPHERICAL BEARING GEOMETRY, SURFACE AND MASS DATA ............................. 82

10.3. TAPERED BEARING GEOMETRY, SURFACE AND MASS DATA ................................ 83

10.4. RACE FLANGE GEOMETRY ................................................................................. 84

10.5. CAGE GEOMETRY, SURFACE AND MASS DATA ................................................... 85



1. FORWARD

The mechanical behavior of the axle bearing system is characterized by the rolling/sliding contact of the different mechanical components of the bearing. When subjected to different types of mechanical loading and friction modifiers the mechanical components deform, eventually leading to permanent defects, as those characterized in WP2. Associated to such failure modes and degradation processes, the relative kinematics of the system components is modified, leading to vibrations and thermal effects. The use of multibody dynamics tools have the advantage of allowing the characterization of the large rotations between the system components, include proper tribological constitutive behaviors and geometric deformations in the analysis, while accounting for the flexibility of the components via the incorporation of their finite element description in the models.

The models developed here, for the complete axle bearing system, provide a phenomelogical model that relates the loads, due to normal or abnormal conditions, including those resulting from the rail-wheel contact and transmitted to the bearing system, to the failure and degradation modes identified in task 2.1. The models for the axle bearing system developed are validated in the course of the tests identified in task 2.3 The work developed at task 2.4, on laboratory testing, will provide some of the data required to complete the models and improve their predictability. The data collected from the tests will be used to identify selected data in the models to improve their fidelity. An optimization approach based on genetic algorithms will be used for the purpose. The results of the deliverable will be further validated in WP6 and used in WP4 and WP5 to support the development of wayside monitoring systems and system integration.

2. INTRODUCTION

The dynamic performance of the roller bearings used in actual railway vehicles is the fundamental object of the MAXBE project. The monitoring of the bearings performance via wayside or on-board monitoring systems uses the vibration and/or thermal information to infer the health of the mechanical components and, consequently, to support the maintenance or operation actions to take. The noise and vibration output of the axleboxes of the rolling stock is in fact a measurable outcome of the bearing dynamic response, under the operating conditions, that is being characterized in this task. The approach used here requires the development of a dynamic analysis tool, referred to as BearDyn which serves as the acronym for Bearing Dynamic Analysis Program, able to handle bearing models representative of the actual bearings used in railway operations.

The code BearDyn uses a multibody formulation to describe the elements of the bearing and their interactions. This approach is used by most of the bearing dynamic analysis programs used today, being Adore [1], IBDAS [2] and Beast [3] just two representative codes that use this approach. The interactions between the rollers, cage, and raceways are described by continuous contact force models based on the Hertz elastic contact theory [4] and modified according to experimental evidence. The Hertzian contact describes well the normal contact force between elements and provides some of the information required to describe the friction forces. Tribological lubrication models are implemented to describe the tangential forces in the presence of lubricant [5,6].



One of the objectives of this work is to allow understanding the performance of the axle bearings in actual operation conditions, or at least in dedicated laboratory setups. For this purpose, the dynamic analysis tool must include the ability to handle the sub-systems of railway vehicles in which the bearings are mounted. In the case of this project, the basic railway vehicle subsystem is one boggie composed of two wheelsets, chassis frame, connecting rods and primary suspension spring and damper elements. For the purpose of this project only the bearings presented in Figure 1 are considered, i.e., spherical and tapered roller bearings with two rows. The bearings are included in the assembly by attaching their inner race to the wheelset and the outer race to the axelbox, which in turn is suspended from the chassis frame via the primary suspension systems. Note that by using this procedure the dynamic response of the roller bearings results not only from the internal contact between rolling elements but also from the external excitation, due to the wheel-rail contact in particular.

(a) (b)

Figure 1: Axelbox roller bearings of different vehicles used in the MAXBE Project. (a) Spherical bearing; (b) Tapered bearing.

By using appropriate wheel-rail contact models and suitable track geometries, or laboratory forcing procedures, the dynamics of the complete system is studied. Two scenarios are simulated with BearDyn: a laboratory testing in which the complete chassis, or only the axelboxes, are excited by known forces and displacements in the laboratorial environment; a field test in which the chassis, included in a railway vehicle, is simulated in a realistic operation scenario.

The outcome of the simulation of any scenario includes the time history of all kinematic variables of the rolling elements, i.e., positions, velocities and accelerations, and the forces generated between elements, i.e., contact, friction and elastohydrodynamic forces. These quantities are time histories that, in a raw form, provide little useful information. The dynamic response of the forces is post-processed to obtain the Frequency Response Functions (FRF), which serve as the basis for the evaluation of the bearing health. Dedicated graphical visualization is also used to understand the dynamics response of the rolling elements and their behavior in the face of abnormal situations.

This report constitutes the description of the dynamic analysis tool, BearDyn, and its fundamental formulations. It provides a description of the methods used to establish the dynamic analysis of the multibody system in the first place. Then, the initialization of the positioning of all the elements of the roller bearings and their velocities is described, this is, the position and velocities of the inner and outer raceways, rollers and cage for each type of bearing. Afterwards the methods selected to describe the contact mechanics between rolling elements is presented, with emphasis first on the contact detection and, after, on the normal contact force models used in the analysis tool. Finally the tribology of the bearing is discussed and the friction, hydrodynamic and elastohydrodynamic force models are presented. This report finalizes with the description of the data required to build the models used by the BearDyn.



3. DYNAMIC ANALYSIS PROCEDURE

This section presents the formulation of the general equations of motion to the spatial dynamic analysis of multibody systems, which may be simply the roller bearing or the complete railway vehicle boggie, including the roller bearings as sub-systems. A simple and brief description of the standard mechanical joints of spatial multibody mechanical systems is presented, namely of the ideal spherical and revolute joints. The methodology presented can be implemented in any general purpose multibody code, being tested in particular in the computer program BearDyn.

3.1. Kinematics of a rigid body

Due to its simplicity and computational easiness, Cartesian coordinates and Newton-Euler’s method are used to formulate the Eqs. of motion of the spatial multibody systems [7].

Figure 2 Definition of the Cartesian coordinates for a rigid body.

Let Figure 2 represent a rigid body i to which a body-fixed coordinate system ()i is attached at its center of mass. When Cartesian coordinates are used, the position and orientation of the rigid body must be defined by a set of translational and rotational coordinates. The position of the body with respect to global coordinate system XYZ is defined by the coordinate vector [ ]Ti ix y zr that represents the location of the local reference frame ()i. The orientation of the body is described by the rotational coordinate’s vector 0 1 2 3[ ]Ti ie e e ep , which is made with the Euler parameters for the rigid body [7]. Therefore, the vector of coordinates that describes completely the rigid body i is,

[ ]T T Ti i i iq r p (1)

A spatial multibody system with nb bodies is described by a set of coordinates q in the form,

1 2[ , ,..., ]T T T Tnbq q q q (2)

The location of point P on body i can be defined by position vector Pis with respect to the

body-fixed reference frame ()i and by the global position vector ri, that is,

P P Pi i i i i i r r s r A s (3)

where Ai is the transformation matrix for body i that defines the orientation of the referential ()i with respect to the referential frame XYZ. The transformation matrix is expressed as function of the four Euler parameters as [7],

Oi

i

i i

P

(i)X

Y

Zri

r Pi

sPi



2 2 10 1 1 2 0 3 1 3 0 22

2 2 11 2 0 3 0 2 2 3 0 12

2 2 11 3 0 2 2 3 0 1 0 3 2

i

i

e e e e e e e e e e

e e e e e e e e e e

e e e e e e e e e e

A (4)

Notice that the vector Pis is expressed in global coordinates whereas the vector P

is is defined in the body i fixed coordinate system. Throughout the formulation presented in this work, the quantities with (.)' mean that (.) is expressed in local system coordinates.

The velocities and accelerations of body i use the angular velocities iω and accelerations iωinstead of the time derivatives of the Euler parameters, which simplifies the mathematical formulation and do not require the use of mathematical constraint for Euler parameters. This is, the relation between the Euler parameters 0 1 2 3 0e e e e is implied in the angular velocity and, therefore, does not be used explicitly [8]. When Euler parameters are employed as rotational coordinates, the relation between their time derivatives and the angular velocities is expressed by [7],

T12i i

p L ω (5)

where the auxiliary 3×4 matrix L is function of Euler parameters,

1 0 3 2

2 3 0 1

3 2 1 0

i

i

e e e ee e e ee e e e

L (6)

The velocities and accelerations of body i are given by vectors,

[ ]T T Ti i i iq r ω (7)

[ ]T T Ti i i iq r ω (8)

The knowledge of the system response includes the evaluation of the position (and orientation), velocity and acceleration of all the components (bodies) of the system. For that purpose, the equations of motion of the system must be established and their solution evaluated and integrated in time.

3.2. Dynamics of a multibody system

In terms of the Cartesian coordinates, the equations of motion of an unconstrained multibody mechanical system are written as,

Mq g (9)

where M is the global mass matrix, containing the mass and moment of inertia of all bodies and g is a force vector that contains the external and Coriolis forces acting on the bodies of the system.

For a constrained multibody system, the kinematical joints are described by a set of holonomic algebraic constraints denoted as,

( , )t Φ q 0 (10)



Using the Lagrange multipliers technique the constraints are added to the equations of motion. These are written together with the second time derivative of the constraint equations. Thus, the set of equations that describe the motion of the multibody system is,

T

q

q

M Φ q gλ γΦ 0

(11)

where is the vector of Lagrange multipliers and is the vector that groups all the terms of the acceleration constraint equations that depend on the velocities only, that is,

( ) 2tt t q q qγ Φ q q Φ Φ q (12)

The Lagrange multipliers, associated with the kinematic constraints, are physically related to the reaction forces and moments generated between the bodies interconnected by kinematic joints. This system of equations is solved for q and . Then, in each integration time step, the accelerations vector, q , together with velocities vector, q , are integrated in order to obtain the system velocities and positions for the next time step.

As examples of the kinematic joints typically used in the modelling of vehicle systems, an ideal or perfect spherical joint, also known as by ball and socket joint, illustrated in Figure 3, constrains the relative translations between two adjacent bodies i and j, allowing only three relative rotations. Therefore, the center of the spherical joint, point P, has constant coordinates with respect to any of the local coordinates systems of the connected bodies, i.e., a spherical joint is defined by the condition that the point Pi on body i coincides with the point Pj on body j. This condition is simply the spherical constraint, which can be written in a scalar form as [7],

( ,3)s P Pi i i j j j

Φ r A s r A s 0 (13)

The three scalar constraint equations implied by Eq. (13) restrict the relative position of points Pi and Pj. Therefore, three relative degrees of freedom are maintained between two bodies that are connected by a perfect spherical joint.

Figure 3: Perfect spherical joint in a multibody system.

P

Pis

jr

ir

Pjs

X

Y

Z

(j)(i)

Oii

i

i

Oj

jj

j



Another kinematic joint often used in the modelling of railway vehicles is an ideal three-dimensional revolute or rotational joint between bodies i and j, as shown in Figure 4, built with a journal-bearing that allows a relative rotation about a common axis, but precludes relative translation along this axis. Eq. (13) is imposed on an arbitrary point P on the joint axis. Two other points Qi on body i and Qj on body j are also arbitrarily chosen on the joint axis. It is clear that vectors si and sj must remain parallel. Therefore, there are five constraint Eqs. for an ideal three-dimensional revolute joint [7],

( ,5)P P

i i i j j jr

i j

r A s r A s 0Φ

s s 0 (14)

Note that the cross product in Eq. (14) only has two independent constraints, being the third linearly dependent on the first two. The five scalar constraint equations yield only one relative degree of freedom for this joint, that is, rotation about the common axis of the revolute joint.

Figure 4: Ideal three-dimensional or spatial revolute joint in a multibody system.

Many other kinematic joints may be required to represent properly a vehicle, or any of its sub-systems. The aim of this work is not to provide a thorough description of kinematic joints, but simply to illustrate them. The interested reader is directed to reference [7] in which a detailed description of the formulation of kinematic joints in multibody systems is provided.

3.3. External and internal forces

Of particular interest in the application of the multibody dynamics formulation to the representation and solution of railway vehicle and bearing rolling element dynamics is the construction of the force vector g. This vector includes all the external and internal applied forces in the system, namely, the springs, dampers and actuator forces, the gravitational forces, the wheel-rail contact forces, the normal, hydrodynamic, elastohydrodynamic and friction forces between bearing rolling elements and the gyroscopic forces of the rigid bodies.

Pis

jr

ir

Pjs

X

Y

Z

(j)(i)

Oii

i

i

Oj

jj

j

js

isP

Qj

Qi



Among these the wheel-rail contact force and the internal roller bearing forces are of importance for this work. The evaluation of the wheel-rail contact forces, implemented in the BearDyn code are briefly described here, in section 4, with reference to the proper literature. The internal forces in the bearing rolling elements, in particular the normal forces, friction, hydrodynamic and elastohydrodynamic forces, are presented and discussed in detail in this report, in section 6 and those that follow. Notice that the fact that the health of the bearings is to be monitored through the models of the most common failure modes, the geometric description of the rolling elements and the description of common geometric and surface material irregularities is of fundamental interest. All remaining forces normally present in railway vehicles and roller bearings are modelled according to the best practice described in the literature, in particular in reference [7], and are not discussed here any further.

3.4. Solution of the equations of motion

According to the formulation outlined, the dynamic response of multibody systems involves the evaluation of the Jacobian matrix q and vectors g and γ , each time step. The solution of Eq. (11) is obtained for the system accelerations q . These accelerations, together with the velocities *q , are integrated to obtain the new velocities q and positions *q for the next time step. This process is repeated until the complete description of system motion is obtained for a selected time interval. Noted that, in vector *q , the angular velocities are substituted by the time derivatives of the Euler parameters using Eq. (5).

The system of Eqs. (11) can be solved by applying any method suitable for the solution of linear algebraic equations. The existence of null elements in the main diagonal of the matrix and the possibility of ill-conditioned matrices suggest that methods using partial or full pivoting are preferred. However, none of these formulations help in the presence of redundant constraints. The left-hand-side matrix of the system of Eqs. (11) can be inverted analytically, by rearranging and putting the acceleration vector in evidence in the left-hand-side. The result is substituted in the second row of Eq. (11), which is also rearranged as,

-1 1 -1 -1 1( ) ( )T T q q q q qM M g M (15)

In these equations it is assumed that the multibody model does not include any body with null mass or inertia so that the inverse of the mass matrix M exists. The substitution of Eq. (15) in the second row of Eq. (11) provides the expression for the system accelerations written as,

-1 -1 -1 1 -1 -1 -1 1( ) ( )T T T T q q q q q q qq M M M M g M M* - (16)

Eqs. (15) and (16) are now rearranged in a compact form as,

-1 -1 -1 1 -1 -1 -1 1

-1 1 -1 -1 1

( ) ( )

( ) ( )

T T T T

T T

q q q q q q q

q q q q q

* M -M M M M M gq=

M M M

(17)

The matrix in the right-hand side of Eq. (17) is the inverse of the system matrix that appears in Eq. (11).



3.5. Stability of the equations of motion

The system of the motion Eqs. (11) does not explicitly use the position and velocity equations associated with the kinematic constraints, that is, Eqs. (13) and (14), and their time derivatives. Consequently, for moderate or long simulations, the original constraint equations start to be violated due to the integration process and/or to inaccurate initial conditions. Therefore, special procedures must be followed to avoid or minimize this phenomenon. Several methods to solve this problem have been suggested and tested, being the most common among them the Baumgarte Stabilization Method [9], the Coordinate Partitioning Method [10] and the Augmented Lagrangian Formulation [11].

In addition to these three basic approaches, many research papers have been published on the stabilization methods for the numerical integration the Eqs. of motion of multibody systems, namely, Park and Chiou [12], Kim et al. [13], Yoon et al. [14], Rosen and Edelstein [15], Lin and Hong [16], Blajer [17], Lin and Huang [18]. Nikravesh [19] studied comparatively the standard or direct integration of the system’s Eq. of motion, the Baumgarte’s approach and the Coordinate Partitioning Method. More recently, Neto and Ambrósio [20] used different methodologies to handle the constraint violation correction or stabilization for the integration of DAE in the presence of redundant constraints discussing, in the process, the benefits and shortcoming of these methods.

Due to its simplicity and easiness for computational implementation, the Baumgarte Stabilization Method (BSM) is the most popular and attractive technique to control constraint violations. However, this method does not solve all possible numerical difficulties as, for instance, those that arise near kinematic singularities. Another drawback of Baumgarte’s method is the ambiguity in choosing feedback parameters. The choice of these coefficients usually involves a trial-and-error procedure [9]. The Augmented Lagrangian method, which also keeps the constraint violations under control, shares with BSM the problem of the parameter selection but is able to handle redundant constraints.

3.6. Direct integration method of the equations of motion

In this section, the main numerical aspects related to the standard integration of the equations of motion of a multibody system are reviewed. The standard integration of the equations of motion, here called Direct Integration Method (DIM), converts the nc second-order differential equations of motion into 2nc first-order differential equations. Then, a numerical scheme, such as the Runge-Kutta method, is employed to solve the initial-value problem [21].

The 2nc differential equations of motion are solved, without considering the integration numerical errors and, consequently, during the simulation the propagation of these kinds of errors results in constraint violations. The two error sources that lead to constraint violations for any numerical integration step are truncation and round-off errors. Truncation or discretization errors are caused by the nature of the techniques employed to approximate values of a function, y. Round-off errors are due to the limited numbers of significant digits that can be retained by a computer. The truncation errors are composed of two parts. The first one is a local truncation error that results from an application of the method in question over a single step. The second one is a propagated error that results from the approximation produced during the previous steps. The sum of the two is the total or global truncation errors.



The commonly used numerical integration algorithms are useful in solving first-order differential equations that take the form [21],

( , )f ty y (18)

Thus, if there are nc second-order differential equations, they are converted to 2nc first-order equations by defining the y and y vectors, which contains, respectively, the system positions and velocities and the system velocities and accelerations, as follows,

qy q and qy q (19)

The reason for introducing the new vectors y and y is that most numerical integration algorithms deal with first-order differential equations [22]. The following diagram can interpret the process of numerical integration at instant of time t,

( )ty Integration ( )t ty (20)

In other words, velocities and accelerations at instant t, after integration process, yield positions and velocities at next time step, t=t+t.

Figure 5 presents a flowchart that shows the algorithm of Direct Integration Method of the Eqs. of motion. At t=t0, the initial conditions on q0 and 0q are required to start the integration process. These values cannot be specified arbitrarily, but must satisfy the constraint equations defined by Eqs. (10) and their time derivative.

The direct integration algorithm shown in Figure 5 can be summarized by the following steps,

1. Start at time t0 with given initial conditions for positions q0 and velocities 0q .

2. Assemble the global mass matrix M, evaluate the Jacobian matrix q, construct the constraint equations , determine the right hand side of the accelerations , and calculate the force vector g.

3. Solve the linear set of the equations of motion (11) for a constrained multibody system in order to obtain the accelerations q at time t and the Lagrange multipliers .

4. Assemble the vector ty containing the generalized velocities q and accelerations q for instant of time t.

5. Integrate numerically the q and q for time step t+t and obtain the new positions and velocities.

6. Update the time variable, go to step ii) and proceed with the process for a new time step, until the final time of analysis is reached.

The Direct Integration Method of equations of motion is prone to integration errors, because the constraint Eqs. (10) and their time derivative are only satisfied at the initial instant of time. In the first few time steps, the constraint violations are usually small and negligible. However, as time progresses, the error in computed values for kinematic parameters is accumulated and constraint violations increase. Hence, the results produced can be unacceptable; therefore, the DIM requires the use of a constraint stabilization technique, especially for long simulations. It should be noted that the DIM is quite sensitive to initial conditions, which can be an important source of errors in the integration process.



Figure 5: Flowchart of computational procedure for dynamic analysis of multibody systems; Direct Integration Method (DIM).

Eq. (11) is a differential algebraic Eq. that has to be solved, being the resulting accelerations integrated in time. However, in order to avoid constraints violation during numerical integration, the [9] stabilization technique is used, being Eq. (11) modified to,

22

T

q

q

M Φ gqλΦ 0 γ Φ Φ

(21)

where and are positive constants that represent the feedback control parameters for the velocity and position constraint violations [8, 9].

STOP

Δt t t

Yes

No

Read input

0

0

0

t

t

t t

q q

q q

Evaluate

Mass matrix MJacobian matrix Фq

Position constraints ФVector Generalized forces g

Δ [ ]T T Tt t y q q

Integrate the auxiliary vector

is t>tend?

START

Solve for and qT

q

q

M Φ q g

Φ 0 λ

Form the auxiliary vector

[ ]T T Tt y q q

STOPSTOP

Δt t t

Yes

No

Read input

0

0

0

t

t

t t

q q

q q

Read input

0

0

0

t

t

t t

q q

q q

Evaluate



Evaluate







is t>tend?is t>tend?

STARTSTART

Solve for and qT

q

q

M Φ q g

Φ 0 λ

Solve for and qSolve for and qT

q

q

M Φ q g

Φ 0 λ


[ ]T T Tt y q q


[ ]T T Tt y q q



4. INITIALIZATION OF THE BEARING ROLLING ELEMENTS

The dynamic analysis of the roller bearings, or of the systems containing the roller bearings, requires that the positions and velocities of each component of the bearing is known and properly initialized. Typically each bearing includes an outer raceway, an inner raceway, a cage and a defined number of rollers, as illustrated in Figure 6. Due to the precision with which the bearings are mounted, with tight tolerances, and the complexity of the geometries it is necessary to build a pre-processor that, based on a limited set of data is able to evaluate the position, orientation and the velocities of all rolling elements of the bearing.

(a) (b)

Figure 6: Typical tapered rolling bearing elements. (a) Assembled roller bearing; (b) Individual elements of the rolling bearing including from left to right outer raceway, cage, rollers and inner raceway.

Common to all bearings possible to be modeled and analyzed in the Bearing Analysis program the data listed in Table 1 is required:

Table 1: Information required for the general dimensioning of a roller bearing

Bearing type reference 1 Ball 2 Cylindrical 3 Spherical 4 Tapered 5 Spherical tapered

Number of Rollers in the bearing nb Outer diameter of the bearing do Inner diameter of the bearing di Pitch diameter of the bearing dm Bearing width L

The dimensions listed in Table 1 are decribed in Figure 7. For more information on this data, refer to the BearDyn user manual presented in Annex A.



(a) (b) (c)

Figure 7: General information on the geometry of roller bearings. (a) Cylindrical roller bearing; (b) Spherical roller bearing; (c) Tapered roller bearing.

Because the roller bearings used in the axel boxes considered in the project are spherical and tapered double row bearings, only the initialization of these types of bearings is considered in this work. At this point it is assumed that the bearings are in full functional conditions and that they have no defects. At a later stage of the project a selected number and type of defects in the roller bearings is considered and models for the defected bearings developed. At that point, the program BearDyn is also upgraded to allow the simulation of bearings with defects.

4.1. Spherical roller bearing elements initialization

The geometric details required for the construction of the multibody model of the spherical roller bearing includes the information contained in Table 2. Notice that the data is structured equally in the case of single and double row bearings.

Table 2: Information required for the general dimensioning of a spherical roller bearing

Roller diameter Db Roller lenght Lb Roller crown radius Rcr Roller corner radius on the left side RcoL Roller corner radius on the right side RcoR Inner race radius Rir Outer race radius Ror Race clearance tc Inner race lenght Lil Tilt

di do

L

dm

j

j

dm

di do

Lj

j

do

Lo

di

Li

dm

j

j



The geometry of the spherical roller bearing is depicted in Figure 8 where the description of the data necessary to the geometric characterization of the bearing is depicted. Note that based on the information contained in Table 2 it is possible to define the complete initialization of the position of the rolling elements in the spherical roller bearing.

(a) (b)

Figure 8: Geometry of the spherical bearing elements. (a) Geometry of the cross-section of the roller bearing; (b) Geometry of the roller.

The body fixed coordinate systems of the outer and inner races and the cage of the spherical bearing are assumed to be coincident with the inertia frame being all the axis equally oriented, i.e., the frames (,,)i ≡ (X,Y,Z) for i=1,3. The initial positions and orientations of the elements of the spherical bearing, with the exception of the rollers are:

0 0 0000

inner outer cage

r r r (22)

0 0 0

1000

inner outer cage

p p p (23)

The positions and orientations of the rollers are dependent on the bearing having one or two rows. The tilt angle is used to define the existence of a second row of rollers, in case it is not null. In any case, the rollers are positioned on the pitch circle as implied in Figure 9. The rollers are numbered as b=1,2,…,nb, starting with the first roller positioned on the intersection between the pitch circle and Y.



(a) (b)

Figure 9: Position of the rollers in the pitch circle: (a) orientation of the rollers in the cross-section of the bearing; (b) position of each roller in the pitch circle.

The pitch angle m and the length of vector rk, shown in Figure 8, are obtained from the pitch diameter and tilt angle by

2

m nb

(24)

1

2 cosm

k

d

r (25)

For a single row spherical bearing =0 and the positions and orientations of the rollers are straight forward

1

01 1

1

0

cos 1 1,2,...,

sin 1b m m

m

r b b nb

b

r (26)

1

01

10

1,2,...,00

b b nb

p (27)

in which rm=½dm is the pitch radius. The orientation of the rollers is such that the body fixed referential (,,)b1 associated to each one of them remains parallel to the inertia frame (X,Y,Z).

For a roller bearing with two rows >0 and the positioning and orientation of the rollers becomes more intricate. First the position of the rollers in the pitch circle of each row is

Y

Z

nb

nb

2

2

3

3

1

1

1r

rm

m

k1

k1

k2

k2



shifted along X. Second each roller must be rotated about a vector ubk in order to obtain the orientation shown in Figure 9(a).

(a) (b)

Figure 10: Rotation of a roller about the tangent to the pitch circle (a) orientation of the rollers in the cross-section of the bearing; (b) rotation of each roller by bk about a vector ubk.

The position of the rollers in the pitch circle is similar to that of the single row bearing, only shifted along X. Defining k=1 for the row of rollers in the negative side of X and k=2 for the rollers in the positive, the rollers position are:

0

1

cos 1 1,2,..., ; 1,2

sin 1k

k

b m k m k

k m

tg

r b b nb k

b

r (28)

The orientation of the rollers in the bearings with two rows correspond to a rotation of each roller, equal to the tilt angle, about a vector tangent to the pitch circle, as depicted in Figure 10(b). Using the definition of the Euler parameters, which are used as rotation coordinates in this formulation, the initial orientation of the rollers in the two rows is

0

1

cos2

01,2,..., ; 1,2

1 sin sin 12

1 sin cos 12

k kb kk m

kk m

b nb kb

b

p (29)

Besides the initial position and orientations of the rolling elements it is also necessary to define the initial velocities of the elements in order to establish the initial conditions for the dynamic analysis. It is assumed that the roller bearing has all their element rotating about X,

k1

k1k2k2

Y

Z

rm

2

2

2

nb

nb

nb

1

1

1

3

3

3

3

u

1

u

nbu

b

b

b



i.e., the roller bearing is always in the plane YZ, as depicted in Figure 11. Note that at a later stage when assembling the bearing to the shaft of the wheelset and to the axel box the positioning and initial velocities will have to be transformed from the initial ones. Such transformation will be addressed in the framework of the complete multibody system.

Figure 11: Angular and translation velocity of the bearing rolling elements

Given the angular velocities of the inner and outer raceways about their local axis, ´inner and ´outer, the angular velocity of the cage is such that the velocity of its pockets equals, in average, the rollers velocity. Assuming that the points of contact between cage and rollers are close to the pitch radius, the angular velocity of the cage is the average of the inner and outer raceways angular velocities. Therefore, the initial velocities of these rolling elements are

0 0 0000

inner outer cage

r r r (30)

0 010 02

0 0 00 ; 0 ; 00 0 0

inner outerinner outer

inner outer cage

(31)

In the initial velocities for the rollers it is assumed that they start rotating without sliding in contact with the inner and outer raceways, as shown in Figure 11. Defining cos /b mD d with reference to Figure 11 the velocities of the contact points of the roller with the inner and outer raceways are

12 1inner inner mv d (32)

12 1outer outer mv d (33)

consequently, the speed of a roller is

dm

outer

bk

inner

vouter

vinner

vbk

Y≡ j

X≡jY≡ j

Z≡ j



12b inner outerv (34)

and its angular speed is

12 1 1b inner outer (35)

The roller translational velocity is tangent to the pitch circle, i.e., it is aligned with vector ubk of each roller, as shown in Figure 10, while its angular velocity is about its rolling axis, which is the roller local bz. Therefore, the initial velocities of the rollers in the spherical bearing are described by

0

0

sin 1 1,2,..., ; 1,2

cos 1kb b k m k

k m

v b b nb k

b

r (36)

0 0 1,2,..., ; 1,20

k

b

b kb nb k

(37)

Note that the initial conditions derived in this section ensure that the analysis starts with consistent velocities between all rolling elements and that there is no interference between the bodies at the start of the analysis. It is only during the evolution of the system that contact may develop and, eventually, sliding between elements.

4.2. Tapered roller bearing elements initialization

The geometric details required for the construction of the multibody model of the tapered roller bearing includes the information contained in Table 3. Notice that the data is structured equally in the case of single and double row bearings.

Table 3: Information required for the general dimensioning of a tapered roller bearing

Roller diameter in the large end Dl

Roller diameter in the small end Ds

Roller lenght LbRoller land lenght LcRoller crown radius Rcr

Roller end radius in the large end RcoR

Roller end radius in the small end RcoL

Outer race semi‐cone angle o

Inner race semi‐cone angle i

Outer race land lenght loInner race land lenght liOuter race width LoInner race width LiInner race land start eiEndplay epNumber of rows Nrow

Inter race spacing Si



The geometry of the tapered roller bearing is depicted in Figure 12 where the description of the data necessary to the geometric characterization of the bearing is depicted. Note that based on the information contained in Table 3 it is possible to define the complete initialization of the position of the rolling elements in the spherical roller bearing.

(a) (b)

Figure 12: Geometry of the tapered bearing elements. (a) Geometry of the cross-section of the roller bearing; (b) Geometry of the tapered roller.

The tapered roller bearing characterization uses a terminology different from that of other rollers. In particular, the outer raceway is designated by cup and the inner raceway by the cone. Also the pitch diameter, dm=2rm, locates the position of the rollers average radius, as seen in Figure 13. Another particular characteristic of the tapered bearing is that it can be mounted with endplay or with some preloading. Tapered bearings are more commonly assembled in pairs, i.e., as double row bearings. In this case they can be mounted back to back, also known as double cup assembly, or face to face, alternatively defined as double cone assemblies.

Figure 13: Basic dimensions defining the roller geometry and position.

Db

L

xin

b

b

inner

k

rm

X≡inner

io

Y



In order to position the rolling elements of the tapered roller bearing, with a single row of rollers, let the center of each roller be defined as the point of their axis that lays on its cross-section coincident with its average diameter Db, as depicted in Figure 12 and Figure 13. Furthermore, it is assumed that rollers central cross-section seats in the inner raceway halfway in the surface width li, seen in Figure 12. It is also assumed that the locations of the centers of mass of each raceway are coincident with their central cross-section, i.e., Li/2 for the inner raceway and Lo/2 for the outer raceway.

With reference to Figure 13 let the basic dimensions of the bearing required to position the raceways and rollers at the start of the analysis be defined as

12b s lD D D (38)

12k i o (39)

cos

sink

mk

L r

(40)

sinin b kx R (41)

in which Db is the roller average diameter, k is the roller axis semi-cone angle, L is the position of the apex of the roller cone in the X axis, xin is the position of the inner raceway inner axis with respect to Y and Rb=½Db. The position and orientation of the inner raceway are

0 00

in

inner

x

r (42)

0

1000

inner

p (43)

The initial position and orientation of the outer raceway is obtained as

0 0

0

p in

outer

e x

r (44)

0

1000

inner

p (45)

To establish the initial position and orientation of the cage it is assumed that, if guided, it is guided by the inner raceway and that its average diameter is similar, not necessarily equal, to the pitch diameter. Under these conditions the cage initial position and orientation is



0

000

cage

r (46)

0

1000

cage

p (47)

The position and orientation of the rollers in the tapered roller bearing follows the same procedure used for the spherical rollers, and depicted by Figure 9 and Figure 10, except for the X coordinate, which is null now. Therefore these initial positions and orientations are given by Eqs. (28) and (29) for k=1, rewritten here as

01 1

1

0

cos 1 1,2,...,

sin 1kb m m

m

r b b nb

b

r (48)

1

01

1

1

cos2

01,2,...,

sin sin 12

sin cos 12

k

b km

km

b nbb

b

p (49)

Tapered roller bearings are more common in double row assemblies, as shown in Figure 14, either as back to back mounting, also known as double cup assembly, or as front to front mounting, known as double cone assembly. For double row roller bearings the initial position and orientation of the raceways is coincident with the inertia frame, i.e.,

0 0

000

inner outer

r r (50)

0 0

1000

inner outer

p p (51)

The cages and rollers positions are shifted along X with respect to those used for the single row tapered roller bearing, depending on the size of the spacer and endplay. The orientation of these rolling elements in the double row assemblies are similar to that of the single row bearing, but adjusting the sign of the semi-cone angle.



(a) (b) (c)

Figure 14: Tapered roller bearing with two rows: (a) back to back (double cup assembly) mounting with play; (b) back to back (double cup assembly) mounting with preloading; (c) front to front (double cone assembly) mounting.

In the back to back mounting, illustrated in Figure 14(a) and (b), the position and orientation of the cages is

12

0

1 2

0 1,20

ki i p in

cage

L S e x

k

r (52)

0

cos2

01,2

sin2

0

cage

k

kk

p (53)

in which k=1 refers to the left side, or negative X, cage while k=2 is the right side, or positive X, cage. The position and orientation of the rollers follows the same logic of the cages, being

12

0

1 2

cos 1 1,2,..., ; 1,2

sin 1k

ki i p in

b m k m k

k m

L S e x

r b b nb k

b

r (54)

01

cos2

01,2,..., ; 1,2

1 sin sin 12

1 sin cos 12

k

k

kb kkk m

k kk m

b nb kb

b

p (55)

In the front to front mounting, illustrated in Figure 14(c), the position and orientation of the cages is

ep

LiLi Si LiLi Si Li LiSi



12

0

1 2

0 1,20

ki i p in

cage

L S e x

k

r (56)

0

1cos

20

1,21

sin2

0

cage

k

kk

p (57)

in which k=1 refers to the left side, or negative X, cage while k=2 is the right side, or positive X, cage, as before. The position and orientation of the rollers follows the same logic of the cages, being defined as

12

0

1 2

cos 1 1,2,..., ; 1,2

sin 1k

ki i p in

b m k m k

k m

L S e x

r b b nb k

b

r (58)

0

1

cos2

01,2,..., ; 1,2

1 sin sin 12

1 sin cos 12

k

k

kb kkk m

k kk m

b nb kb

b

p (59)

The initialization of the tapered roller bearings requires that the initial velocities of all rolling elements are still defined. The initial velocities for this type of bearing are similar to those of the spherical bearing, i.e., for the raceways and cages,

0 0 0

000

inner outer cage

r r r (60)

0 010 02

0 0 00 ; 0 ; 00 0 0

inner outerinner outer

inner outer cage

(61)

and for the rollers



0

0

sin 1 1,2,..., ; 1,2

cos 1kb b k m k

k m

v b b nb k

b

r (62)

0 0 1,2,..., ; 1,20

k

b

b kb nb k

(63)

in which the translational velocity vb and the angular velocity b for the tapered rollers are calculated using Eqs. (32) through (35) with cos /b k mD d ., i.e., the semi-cone angle of the tapered roller bearing plays the role of the tilt angle in the spherical roller bearing.



5. GENERAL FORMULATION OF THE ROLLER CONTACT DETECTION

The formulation of the contact between the bearing rollers and the raceway and cage is structured as a two stage problem. In the first stage the positions and geometries of the roller, raceways and cage are evaluated to identify the points of more proximity and the eventual existence of contact. The second stage consists in the evaluation of the contact forces that develop between the surfaces in actual contact, first, and in the calculation of the friction and other shear forces that develop due to lubrication. In the first evaluation stage only the kinematics of the bearing mechanical components and their geometry play a role while in the second stage the constitutive Eqs. that relate the interference between components with contact forces are used. The evaluation of the forces that act over each mechanical element of the bearing sets the ground for the formulation of the equilibrium equations of each body which have to be solved, being the resulting accelerations integrated afterwards. Here only the focus is on the calculation of the forces that act over each component of the bearing.

Let it be assumed that the roller approaches a generic surface, as represented in Figure 15(a), and that in the course of its motion it actually contacts the surface. The situation numerically perceived as contact is illustrated in Figure 15(b) being the shaded volume a representation of the penetration of the roller in the surface, i.e., the interference between the two bodies. Note that in penalty contact force models, foreseen for application in this work, the contact force is related with the quantification of the interference, designated hereafter as penetration.

(a) (b)

Figure 15: Contact between a roller and a surface. (a) Approach phase in which the minimum distance is calculated; (b) Contact phase in which the actual penetration, shaded volume, is evaluated.

The shape of the rollers in standard bearings is not perfectly cylindrical or conical, in the case of tapered rollers. They also include a crowned region and a corner radius in order to avoid edge loading, thus reducing fatigue or the contacting elements. Furthermore, the geometry of the surfaces of the raceways may not be cylindrical or conical and/or the axis of the rollers may be misaligned with the axis of the raceways resulting in skewing between the two rolling elements. Any of the conditions mentioned lead to a penetration depth between the roller and the contacting surface that varies along the roller axis. Consequently, instead of defining a common penetration depth for the complete roller the penetration can only be defined for each particular cross-section of the roller.

Let a roller with the shape represented in Figure 16(a) represent any generic roller, i.e., cylindrical, spherical, toroidal, tapered or spherical tapered roller. Consider now that the roller

i

ii

j

j

ji

ii

j

j

j



is divided in a user defined Nsl number of strips, i.e., cylindrical segments, which act as rigid bodies without any relative motion, as perceived in Figure 16(b). Now the contact problem of the complete roller can be described as Nsl independent contact problems of thin cylinders, being one of those highlighted in Figure 16(c), in which the contacting penetration depth is constant throughout the slice, or strip [23]. Therefore, each one of the contact problems, required to represent the roller to surface contact, is described by the contact of the central cross-section of the slice, as represented in Figure 17.

(a) (b) (c)

Figure 16: Geometry of a generic roller. (a) Three-dimensional geometry; (b) Division in a defined number of slices, or strips; (c) Individual strip with the representation of its central cross-section.

(a) (b)

Figure 17: Interaction between a circle and a surface. (a) Approach between the geometric figures; (b) Contact with the interference represented as a shaded area.

As a result of the approach followed here, the search for contact of the rollers with any potential contacting surface is reduced to the identification of the minimum distance between the central cross-section of each slice and the surface. By identifying the proper interaction conditions it is possible if such minimum distance corresponds to separation of effective contact. In what follows the central cross-section of the slice is designated by circular cross-section or simply by circle.

5.1. Contact detection between a circle and a generic surface

Let vector rPQ represent the distance between a point Q in a surface and another point P in the boundary of the circle that represents the central cross-section of a slice, as illustrated in

i

ii

j

jj

XY

Z

i

i

j

jj

XY

Z

i



Figure 18. Furthermore, let the circle be defined in the referential associated to the center of the roller, or body i, and defined as ()i and the surface in the referential ()j fixed to the center of mass of body j.

Figure 18: Geometric relations between a points P in a circumference and a point Q in a generic

surface.

With reference to Figure 18, the position of point P in body i is defined by the sum of the vectors ri, which is the position of the roller center of mass with respect to the inertial frame (XYZ), and vector sp representing the position of point P with respect to the body i fixed frame. Point P position is written as:

P i i P r r A s (64)

where Ai is the transformation matrix from the local referential attached to body i to the inertial frame and (•)′ means that the quantity (•) is expressed in body fixed coordinates. The coordinates of point P, belonging to the circumference, are expressed in the body i reference frame as

cossin

P

P s

s

RR

s (65)

In which angle is measured in a plane ()* parallel to plane ().

Similarly, the position of point Q is

Q j j Q r r A s (66)

in which the coordinates of point Q in the body j coordinate frame depend on the parametric description of the surface. For the surfaces of the raceways and cages considered in the bearing models developed in this work different parametric descriptions of the surfaces are considered, all discussed in forthcoming sections.

Still with reference to Figure 18, tP is a vector in body i that is tangent to the circle in point P, while in body j in Point Q the vector normal to the surface is nQ while tQ and bQ are tangent to the surface, forming an orthogonal basis. The tangent vector to the circumference, expressed in body fixed coordinates, is

i

P

j

jj

XY

Z

ri

rj

QnP

tQ

bQ

nQ

rPQ

sQ

sP

tP

ii



0sin

cosP

t (67)

The relation between the components of the different vectors defined in body j and the parameters used to define the surface depends on the specific geometry of the contacting surface. Such relations are developed in detail in the forthcoming sections. At this point it is assumed that expressions for nQ, tQ and bQ exist in close form.

The conditions for the minimal distance between the circumference and the surface is that points P, in body i, and Q, in body j, are described by

0

0

0

TPQ P

TPQ Q

TPQ Q

r t

r t

r b

(68)

in which rPQ=rP-rQ. Eq. (68) means that not only the normal to the surface must be collinear with the vector that connects the two points in closer proximity but also perpendicular to the tangent to the circumference taken at point P.

The circumference is in contact with the surface if besides fulfilling the conditions expressed by Eq. (68) there is effective penetration between the two entities, expressed by

0TPQ Q r n (69)

Once the contact is detected, the contact forces are applied in points P and Q that fulfil the conditions expressed by Eqs. (68) and (69). The contact force vectors, applied in bodies i and j, are related to the circumference and surface normal vectors by

P contact P

Q contact Q

f

f

f n

f n (70)

Besides the normal forces that develop during contact, described by Eq. (70), also tangential forces due to friction or to the lubrication fluid develop between the contacting bodies. For the evaluation of tangential, or creep, forces it is necessary to calculate the relative velocity between the bodies, namely its projection on the surface tangent to the contacting bodies on the points of contact.

The velocities of the contact points P and Q are obtained by taking the time derivative of the position vectors, given by Eqs. (64) and (66), and written as

P i i i P

Q j j j Q

r r A s

r r A s

(71)

where ir and jr are the velocities of the origin of the body fixed referential and i and j

the skew-symmetric matrices associated to the angular velocities of body i and j , respectively, expressed in the body fixed coordinate systems. The relative velocity between bodies i and j in the point of contact is obtained as

PQ P Q r r r (72)



The sliding velocity of the bodies in the contact point is just the projection of the relative velocity vector PQr in the tangent plane to the contacting surfaces in the contact point, as

TPQ PQ Q Q u r r n n (73)

Figure 19 illustrates the plane tangent to the contacting surface and circumference that passes by the contact point, as well as the sliding velocity calculated by Eq. (73).

Figure 19: Tangent plane, to both surface and circumference, passing by the contact point and sliding velocity vector.

In all normal contact force, friction force or lubrication force models used for the contact of the rolling elements in the bearings the kinematic quantities required, i.e., contact point positions, penetrations, normal vectors to the surfaces and relative sliding velocities are calculated using Eq. (64) through (73).

5.2. Description of specific surface geometries

The position the of surface points, rQ, and the normal, tangential and binormal vectors, nQ, tQ and bQ, to the contacting point of the surface, in local body fixed coordinates of body j, for the surfaces encountered in typical raceways and cages of roller bearings are identified here. In all that follows, the relation between the body fixed and inertial coordinates of the vectors that define the tangent to the contact surfaces is written as

Q j Q

Q j Q

Q j Q

n A n

t A t

b A b

(74)

Therefore, the identification of the body fixed coordinates of the surface defining vectors, n′Q, t′Q and b′Q, is required for the typical geometries encountered in the bearing components that contact with the rollers, i.e., raceways, flanges and cage. Being the roller bearings most commonly used in railway axelboxes tapered, cylindrical and spherical, this section focus on the derivation of the surface contact point and tangent surface defining vectors for these specific geometries.

All bearing raceways are surfaces of revolution, i.e., they are obtained by sweeping a plane line with a prescribed shape about an axis of revolution, as illustrated in Figure 20. Therefore, the coordinates of any point in the surface can be expressed in terms of the parameters that

j

jj

XY

ZQ

tQ

bQ

nQ

u



define the planar line and the sweep angle. For instance, in Figure 20 the line is described as a function of j, defined as R(j) and the sweep angle is .

(a) (b)

Figure 20: Axisymmetric surface obtained as the sweep of a line about an axis: (a) Parametric representation of the line and its tangential and normal vector in point Q; (b) Surface of revolution, with the sweep angle and the tangent surface defining vectors at point Q.

In the ()j plane the position of point Q is a function of a single parameter, which in the case illustrated in Figure 20 is the coordinate Q, and it is written as

*

0

Q

Q QR

s (75)

The normal and tangent vectors to point Q in the line, shown in Figure 20(a), have components that are also functions of the parameter defining the line. These vectors are expressed as

* *

* * * *;

0 0

Q Q

Q Q

Q Q

Q Q Q Q

n b

n b

n b (76)

The surface of revolution is obtained by sweeping the line around the axis of revolution, i.e., around j. This operation can be represented as a transformation of coordinates in which any point on the line, with coordinates given by Eq. (75), and any vector associated to such point, such as the normal and tangent vectors described in Eq. (76), are rotated as

*

*

*

Q Q

Q Q

Q Q

s A s

n A n

b A b

(77)

in which the rotation matrix A is obtained by

1 0 00 cos sin0 sin cos

A (78)

j

j Q *bQQ

*nQQ

Rr(j)

Q

*sQQ

j

j

j

sQ

tQ

Q bQ

nQ



By substituting Eqs. (75), (76) and (78) into Eq. (77), and rearranging, the components of the position, normal and tangent vector to the surface, in point Q, are

* *

* *

* *

0cos ; cos ; cos ; sinsin cossin sin

Q Q

Q Q

Q Q

Q

Q Q Q Q

n b

R n bR n b

s n b t (79)

In which R=R(Q) and the components of nQ*= nQ

*(Q) and bQ*= bQ

*(Q) continue to be function of the parameter defining the sweep line, generically defined as Q. The problem is now reduced to finding the parametric definitions of the lines that define the contour of the raceways in different types of bearings.

5.3. Cylindrical bearings

Raceways of cylindrical bearings are, typically, cylindrical geometries with which the rollers contact, as implied in Figure 21. In the inner raceway the contact takes place with an external cylinder while for the outer raceway the contact is with an internal cylinder.

Figure 21: Typical cylindrical roller bearing with an highlighted cross section (note that the details of the roller geometry, namely its crowning and the end radius, and the flanges of the raceways are not detailed).

With reference to Figure 22(a), the position of a point in the sweep line that defines the contact cross-section of the cylinder and the normal and tangent vectors, in the body fixed coordinate system, are

* * *

0 1; 1 ; 0

0 0 0

j

Q r Q QR

s n b (80)

in which Rr is the radius of the raceway. The definition of the normal vector is for the inner raceway, in which there is external contact with the cylinder. For the outer raceway the contact is interior to the cylinder and the negative of the normal vector must be used.

Note also that no limitation on the width of the raceway is used to limit the length of the cylinder. This is because in roller bearing applications the roller never travels beyond the raceway limits sue to the existence of flanges in the raceways that limit its travel.



(a) (b) (c)

Figure 22: Contact point Q and surface normal and tangent vectors in a cylindrical surface: (a) cross-section definition; (b) external contact, as in the inner raceway; (c) internal contact, as in the outer raceway.

5.4. Tapered bearings

Raceways of tapered bearings are, typically, conical geometries with which the rollers contact, as implied in Figure 23. In the inner raceway the contact takes place with an external conical surface while for the outer raceway the contact is with an internal conical surface.

Figure 23: Typical tapered roller bearing with an highlighted cross section (note that the details of

the roller geometry, namely its crowning and the end radius, and the flanges of the raceways are not detailed).

With reference to Figure 23(a), the coordinate of point Q in the sweep line is related with the apex position L and is the raceway angle. The coordinate of a point in the sweep line is a function of j given by

r jR L tg (81)

The normal and tangent vectors to the sweep line, at point Q, can be understood as a counter-clockwise rotation of the corresponding vectors defined for the cylindrical sweep line by an angle about the j axis. Using Eq. (81) and the rotation by in the vectors defined in Eq. (80) leads to the position of a point in the sweep line that defines the contact cross-section of the conical surface and the normal and tangent vectors, in the body fixed coordinate system, written as

j

j Q

*bQQ

*nQQ

Rr

Q

*sQQ

j

j

j

sQ

tQ

Q bQ

nQ

j

j

j

sQ

tQ

Q bQ

nQ



* * *

cos sin; sin ; cos

0 00

j

Q j Q QL tg

s n b (82)

The definition of the normal vector is for the inner raceway, named as cone in standard bearing terminology. For the outer raceway, defined as cup, the contact is interior to the conical surface and the negative of the normal vector must be used.

No limits are defined here for the axial coordinate of the bearing not only because tapered bearings are usually mounted in pairs but also because, depending in specific designs, one or more of the raceways have flanges that limit the relative travel of the rollers.

(a) (b) (c)

Figure 24: Contact point Q and surface normal and tangent vectors in a conical surface for: (a) contact point in the body fixed plane; (b) external contact, as in the inner raceway; (c) internal contact, as in the outer raceway.

5.5. Spherical bearings

Raceways of spherical bearings are, typically, toroidal geometries with which the rollers contact, as implied in Figure 25. In the inner raceway the contact takes place with an external inverted toroidal surface while for the outer one the contact is with an internal toroidal surface.

Most of the spherical bearings have two rows of rollers having the inner raceway geometry a more complex geometric description than the outer raceway geometry. As a result the description of the sweep line for the outer raceway is different from that used for the inner raceway. With reference to Figure 26(a), the coordinates of the center of the arc of circumference that defines the left side of the inner raceway is

sin

2 cos0

O ir or b cR R R t

s (83)

where the inner and outer raceway radius are depicted by Rir and Ror, respectively, Rb is the roller radius and tc the clearance between the roller and raceway. Angle is defined in Figure 25. The position of any point in the sweep line of the left inner raceway is obtained as

*

sincos0

Q O irR

s s (84)

j

j Q*bQQ

*nQQ

L

Rr(j)

Q

*sQQ

j

j

j

sQ

Q

tQ

bQ

nQ

j

j

j

sQ

Q

tQ

bQ

nQ



Figure 25: Typical Spherical roller bearing with an highlighted cross section (note that the details of the roller geometry, namely its end radius, and the flanges of the raceways are not detailed).

The normal and tangent vectors to the sweep line, at point Q, can be understood as a counter-clockwise rotation of the corresponding vectors defined for the cylindrical sweep line of the left inner raceway by an angle about the j axis. Using the rotation in the vectors defined in Eq. (80) leads to the normal and tangent vectors in the point Q, in the body fixed coordinate system, written as

* *

sin coscos ; sin

0 0Q Q

n b (85)

For the right side of the inner raceway, of a double row spherical bearing, according to Figure 26(b), the position of the center of the sweep line is.

sin

2 cos0

O ir or b cR R R t

s (86)

With reference to Figure 26(b), the position of any point in the sweep circle and the coordinates of the normal and tangent vectors are written as

* * *

sin sin coscos ; cos ; sin0 0 0

Q O ir Q QR

s s n b (87)

In which angle , for the right side of the inner raceway is taken as positive when measured as in Figure 26(b).

The outer raceway sweeping curve is an arc of circle with a radius Ror centered in the negative j axis, as depicted by Figure 27(a). The roller contact takes place in the interior of the surface of revolution resulting from the revolution of the arc circle about the j axis.



(a) (b) (c) Figure 26: Geometry of an inner raceway of a spherical roller bearing: (a) contact point in the body

fixed plane in the left side; (a) contact point in the body fixed plane in the right side; (c) Spatial representation.

(a) (b) Figure 27: Geometry of an outer raceway of a spherical roller bearing: (a) contact point in the body

fixed plane; (b) contact point and surface vectors in the inner raceway.

The arc of circle that constitutes the sweep line of the outer raceway is centered in the origin of the body fixed coordinate frame, which in turn seats in the axis or rotation of the raceway. Using the angle , positive in the counter-clockwise direction, the position of a point Q in the arc of circle and the coordinates of the normal and tangent vectors are

* * *

sin sin coscos ; cos ; sin0 0 0

or

Q or Q Q

RR

s n b (88)

The position of point Q with respect to the body fixed coordinate frame and the normal, tangent and binormal vectors associated to the surface tangent to the point are depicted in Figure 27(b).

5.6. Race flanges

In cylindrical and tapered bearings the normal dynamics of the rollers leads to occasional contacts between the roller tops and the flanges that limit the land length of the roller. Due to the roller end diameters, in the case of cylindrical rollers or small end of a tapered roller, the contact with the flange is achieved with the circular landmark that limits the roller, as seen in Figure 28(a). Due to its spherical shape, at least the large end but eventually the small end also, in the case of the tapered roller the contact with the flange is between a spherical surface and a conical solid, as depicted by Figure 28(b). The spherical roller bearing does not have flanges and, consequently, the eventuality of such contact does not need to be considered.

j

Q *bQQ

*nQQ

O

jRir

j

j

Q*bQQ

*nQQ

O

Rir

j

j

j

sQ

tQ

Q bQ

nQ

j

j Q

*bQQ

*nQQ

Ror ∙

j

j

j

sQ

tQ

QbQ

nQ



(a) (b) Figure 28: Contact geometries between rollers and flanges. Point Q refers to the contact point:

(a) Geometry of contact for a cylindrical roller; (b) contact geometry for a tapered roller.

5.6.1. Cylindrical bearing flange contact

The detailed geometry of the cylindrical roller bearing raceways, with flanges, are represented in Figure 29(a), for the inner raceway, and Figure 29(b), for the outer raceway. The raceway land geometry is given by the user, as well as the flange angles and heights, while all other features are obtained from the rollers and general bearing user data.

The flanges contact surface results from sweeping a line segment about the axis of the roller. In order to characterize such segment let the radius of the raceway flanges shown in Figure 29 be defined. In the inner raceway

1; cos ; cos

2i m c d oR i iR iR oL i iL iLR d t D R R h R R h (89)

while for the outer raceway

1; cos ; cos

2o m c d iR o oR oR iL o oL oLR d t D R R h R R h (90)

The position of a point of contact and the normal and tangent vectors are represented in Figure 30 for the flanges of the inner raceway, and in Figure 31 for the outer raceway. Note that the sweeping parameter is the same used for all other sweeping surfaces. However, j is used instead of j as the second parameter of the sweeping surface due to the relative orientation of the line segments that define the flanges.

(a) (b) Figure 29: Detailed geometry of flanges for cylindrical roller bearings: (a) Inner raceway; (b) Outer

raceway.

Q QQ

Q

iLiR

hiRhiL

RoRRiRoL di

Li

li j

j

oRoL

hoRhoL

j

j

RoRiRRiLdo

Lo

lo



(a) (b)

Figure 30: Contact point Q and surface normal and tangent vectors for the cylindrical roller bearing flanges of the inner raceway: (a) Left flange; (b) Right flange.

Using the same procedure adopted before for the different roller bearing raceways, the position of the points of contact and the respective normal and tangential vectors, as function of the parameters and j are obtained. For the right inner flange, the point of contact position and the vectors are written as

* * *

sin1

2 cos cos sin; sin ; cos

0 00

iRi j i

iR iR iR

Q j Q iR Q iR

l R

s n b (91)

Note that contact can only occur for a parametrical range i j oRR R . For the left inner flange the position of the contact point and the normal and tangent vectors are defined as

* * *

sin1


0 00

iLi j i

iL iL iL

Q j Q iL Q il

l R

s n b (92)

Just as for the right flange, also for the left inner raceway flange the contact can only occur for a parametrical range i j oLR R .

(a) (b)

Figure 31: Contact point Q and surface normal and tangent vectors for the cylindrical roller bearing flanges of the outer raceway: (a) Left flange; (b) Right flange.

j

jQ

*bQQ

*nQQ

R(Q)

Q*sQQ

iL

Ri

j

jQ

*bQQ

*nQQ

R(Q)

Q*sQQ

iR

Ri

oL

j

jQ

*bQQ

*nQQ

R(Q)

Q*sQQ

Ro

j

j Q

*bQQ

*nQQ

R(Q)

Q*sQQ

Ro

oR



The procedure adopted for the outer raceway follows the same steps taken before, but now with the definition of the raceway features provided in Figure 31. For the right outer flange, the point of contact position and the vectors are written as

* * *

sin1


0 00

oRo i j

oR oR oR

Q j Q oR Q oR

l R

s n b (93)

The contact can only occur for a parametrical range iR j oR R . For the left outer flange the position of the contact point and the normal and tangent vectors are defined as

* * *

sin1


0 00

oLo i j

oL oL oL

Q j Q oL Q oL

l R

s n b (94)

Just as for the right flange, also for the left outer raceway flange the contact can only occur for a parametrical range iL j oR R .

5.6.2. Tapered bearing flange contact

The detailed geometry of the tapered roller bearing inner raceways, with flanges, is represented in Figure 32. The raceway land geometry is given by the user, as well as the flange angles and heights, while all other features are obtained from the rollers and general bearing user data. Note that in the case of tapered roller bearings the outer raceways have no flanges, and, therefore, they are not considered here. However, in the case of any particular bearing design, for which flanges in the outer races would be used, their geometric definition would follow the basic steps given for the cylindrical roller bearings, with the corresponding adjustments due to the particulars of the tapered bearing geometry.

Figure 32: Detailed geometry of the flanges for tapered roller bearings inner raceway

The flanges contact surface results from sweeping a line segment about the axis of the roller, in the same way illustrated for the cylindrical roller bearing. In order to characterize such segment let the radius of the raceway flanges shown in Figure 32 be defined

iL

iR

hiR

hiL

L R

RoRRiRRidiRiLRoL

Li

li

j

j



1cos

2i m d kR d D (95)

Being the remaining geometrical features defined in Figure 13. The flange angles, in the definition of the sweeping surface obtained from the revolution of the line segments about the axis of the raceway, are referred to the orientation of j. The intermediate angles necessary to define such flanges are

;R iR i L iL i (96)

By the same token the intermediate radius necessary to define the tapered roller bearing inner flanges are

sin ; sin2 2i i

iR i i iL i i

l lR R R R (97)

cos ; cosoR iR iR R oL iL iL LR R h R R h (98)

The formulation of the geometry of the tapered roller bearing inner raceway is similar to that of the inner cylindrical bearing raceway, but with RiR and RiL, taking the role of Ri and R and L taking the role of iR and iL in Eqs.(91) and (92), and RoR and RoL taking the role of the equivalent quantities of the referred Eqs.. For the right inner flange, the point of contact position and the vectors are written as

* * *

sin1


0 00

Ri j iR

R R R

Q j Q R Q R

l R

s n b (99)

Just as before, contact can only occur for a parametrical range iR j oRR R . For the left inner flange the position of the contact point and the normal and tangent vectors are defined as

* * *

sin1


0 00

Li j iL

L L L

Q j Q L Q L

l R

s n b (100)

Just as for the right flange, also for the left inner raceway flange the contact can only occur for a parametrical range iL j oLR R .



6. OTHER CONTACT DETECTION GEOMETRIES IN BEARINGS

The most important roller contact, in any spherical, cylindrical or tapered roller bearing, takes place between its side and the raceway or flange surfaces, as described in section 0. However, other contact configurations play a role in the bearing dynamics, namely those of the roller spherical shaped ends with the flanges and cages and between the roller side and the cage. Also the cage guidance by either the outer or the inner raceway precludes contact that has a particular form of detection, as described in this section. In particular cases in which the breakage of the cage is considered, contact between rollers can also be considered. However, at this point such abnormal conditions are not considered here.

6.1. Contact between spherical cap and conical surface

The contact between the tapered roller large end, at least, and the inner raceway flange is represented as a contact between a spherical cap and a conical surface, as depicted by Figure 33. The parameterized conical geometry is described in section 5.6.2. A convenient parameterization of the spherical cap geometry, with reference to the data used to define the tapered roller, is pursued here.

Figure 33: Typical contact between spherical cap and conical surface, as in the contact between the tapered roller large end top and the flange of the inner raceway.

The geometry of the spherical cap is obtained as the surface resulting from sweeping the arc of circumference, depicted in Figure 34, about the roller axis. In this case the surface of the spherical cap is defined by two parameters, the angle , represented in Figure 34, and the angle k, which is equivalent of j in Figure 20(b), for instance.

(a) (b)

Figure 34: Contact point, surface normal and tangent vectors for the spherical cap of the tapered roller bearing end: (a) Contact point in the local plane: (b) spatial geometry of the spherical cap..

Q

Q

k

k Q

*bQQ

*nQQ

Rer

*sQQ

0

k

kk

k

Q

tQ

bQ

nQ

sQ



The position of any point in the arc of circle, and the corresponding normal and tangent vectors, are now written as

0

* * *

cos cos sinsin ; sin ; cos0 0 0

er

Q er Q Q

RR

s n b (101)

Contact can only occur for a parametrical range of max0 , where max is fixed by the definition of the roller geometry.

Note that the search for the contact between the two surfaces requires the identification of two parameters of each surface, associated to the location of the points that are either in contact or in closer proximity. This is illustrated in Figure 35, in which the tapered roller top spherical cap is referenced as k while the flange of the raceway conical surface is referenced as j.

Figure 35: Candidates to contact points between two parametric surfaces. Surface k is the spherical cap

of the roller and surface j is associated to the conical surface of the raceway flange.

The identification of the contact (or proximity) points corresponds to the fulfillment of the conditions

0 0

; 0 0

T Tk j k

TTkk j

n t d t

d bn b (102)

Effective contact occurs if, besides the fulfillment of Eq.(102) also Eq.(69) is fulfilled.

6.2. Contact between spherical cap and line

Contacts between spherical caps and lines, or very narrow rectangular patches, describe well the contact between the spherical top(s) of tapered rollers and the roller pocket of the cage, as depicted in Figure 36. In what follows it is assumed that the thickness of the cage is small enough to assume that the pocket shapes at the cage mid-thickness are lines that represent well the potential contact surfaces.

d

p , k

(j)

(k)

x

z

y

tj

bj

nj

q ,j j

Qk

Qj

nk

tk

bk



Figure 36: Contact between spherical cap and a line, as in the contact between the tapered roller large end top and the cage pocket.

The potential contact configuration between the tapered roller large end and the cage pocket top is shown in Figure 37. In order to develop the cage pocket configuration it is assumed that the center of mass of the cage, in which the body fixed coordinate frame ()c is attached, is located in its geometrical center. With reference to Figure 37, oc is half of the pocket long side dimension, Rc is the mid-thickness radius at the level of the pocket top, c is the angular position of the corner of the pocket with respect to the body fixed direction c and c is the angle from the vector defining the position of point Qo with respect to the body referential to the potential contact point Qc.

Figure 37: Contact between a spherical cap and a line, representing the contact between the tapered roller large end and the cage pocket top.

In order to define the contact conditions the position of the potential contact point in the line that defined the pocket top and the tangent vector at such point are written as

0* *

0cos ; sinsin cos

c c c c c c c

c c c c c

RR

s b (103)

k

kk

k

Qk

tQ

bQ

nQ

sQ

d

Qc

bc

sC

c

c

c

Qo

so

Rc

0c

c

c



Now, the conditions for points Qk, in the roller, and Qc, in the cage, to be in contact, or at least in the closest vicinity, are

0

0

0

TQ

TQ

TQ c

d t

d b

n b

(104)

Effective contact occurs if both Eq.(104) and Eq.(69) are fulfilled.

6.3. Contact between circle and line

The circle to line contact evaluation is required when checking on the collisions between the roller, represented by slices as described in Figure 16. In this case, the circle, representing the slice, contacts with the pocket long side, which is represented by the line of its mid-thickness. The situation is illustrated in Figure 38 for a tapered roller cage and roller, being the orientation and position of the pocket in the cage described as in section 6.2.

Figure 38: Contact between circle and a line, as in the contact between the roller and the side of the pocket of the cage.

The contact situation between the roller and the side of the pocket is depicted in Figure 39, where the circle corresponding to the central section of the roller slice approaches the line representing the mid-thickness of the side pocket.

With reference to the quantities depicted in Figure 39, the position of the contact point along the line that represents the left side of the cage pocket and its tangent vector are given as a function of the parameter c, which is also the coordinate of the point in the body fixed coordinate frame

* *1212

costg cos ; sin cos

sin sintg sin

c c

c c i c c c c c c

c cc i c c c

R WR W

s b (105)

where the quantities Rc, Wi, c and c are either given as features of the roller bearing cage or calculated during the initialization of the problem.



Figure 39: Contact detection between a circle and a line, representing the contact of the roller with the left side of the cage pocket.

The contact detection with the right side of the cage pocket is similar to that described by Eq. (105), being only the angle c substituted by c+p , which includes the angle of the pocket itself. Therefore, the position of a point in the line of the right side of the pocket is

* *12

12

costg cos ; sin cos

tg sin sin sin

c c

c c i c c c p c c c p

c i c c c p c c p

R W

R W

s b (106)

For the cylindrical roller c=0. Now, the conditions for points Qk, in the roller, and Qc, in the cage, to be in contact, or at least in the closest vicinity, are

0

0

Tc

TQ

d b

d b (107)

Effective contact occurs if both Eq.(107) and Eq.(69) are fulfilled. The contacts with the left and right side of the cage pocket must be checked independently. In any case, if a particular roller slice is in contact with one side of the cage it is unlikely that it also contacts with the other side, unless the width of the pocket is smaller than the roller diameter and the whole cage position is shifted such a way that it is ‘compressing’ the roller.

6.4. Contact between circle top and line

The terminal slice of the roller, i.e. the small end of the tapered roller or any of the ends of the cylindrical roller, can contact the top of the cage pocket either in one point, such as in the case of the circle to line contact of the roller side, or in a line along the end as shown in Figure 40. Therefore, the possibility of having two points of contact by the roller flat end(s) has to be accounted for.



Figure 40: Contact between circle top and a line, as in the contact between the roller flat top and the

top of the pocket of the cage.

The contact situation between the roller flat end the top of the pocket is depicted in Figure 41, where the circle corresponding to the end section of the roller approaches the line representing the mid-thickness of the bottom of the pocket. Two potential points of contact, Qc1 and Qc2, may develop.

Figure 41: Contact detection between a circle and a line, representing the contact of the roller fat

end with the bottom side of the cage pocket.

The coordinates and tangent vectors of the points of contact in the cage are obtained by expressions similar to those of the circle to line contact, described by Eq.(103), but with the modifications implied in Figure 41. The coordinates of the two contacting, or proximity, points are

0* *1 1 1 1

1 1

0cos ; sinsin cos

c cb c c c c c

cb c c c c

RR

s b (108)

0* *

2 2 2 2

2 2

0cos ; sinsin cos

c cb c c c c c

cb c c c c

RR

s b (109)

1d

Qc2

1bc

c

c

c

Qo

c1

c2

2d

Qc1

2bc

1sC

1bQ

2bQ

k

kk

k1Qk1

Qk2k2



where radius Rcb is equal to Rc in Figure 37 in case of cylindrical roller cages and the radius of the opposite side of the pocket cage, i.e., the radius of the pocket top mid-thickness that corresponds to the small end of the tapered roller.

Now, the conditions for points Qk1, in the roller, and Qc1, in the cage, to be in contact, or at least in the closest vicinity, and by the same token points Qk2 and Qc2, are

1 1

1 1

0

0

Tc

TQ

d b

d b (110)

2 2

2 2

0

0

Tc

TQ

d b

d b (111)

In the evaluation of Eqs.(108) through (111) the parameters that define the position of the points in the top circle of the roller must be kept inside the ranges +90º<k1≤+270º and -90º<k2≤+90º in order to guarantee their separation. Effective contact occurs if both Eq.(110) and/or (111) and Eq.(69) are fulfilled.

6.5. Contact between two cylindrical segments

The guidance of the cag by the inner or outer raceway is represented as the contact between two cylinders, one internal and the other external. Roller cages are not guided by both raceways, like illustrated in Figure 42(a), or by the same raceway in both ends as in Figure 42(b). The situations presented there only serve to illustrate the data supplied to define cage guidance in diverse situations.

(a) (b)

Figure 42: Data defining the geometry of roller cages including their guidance: (a) cylindrical roller cage; (b) tapered roller cage.

The cage guidance by one of the raceways is handled, in terms of contact, exactly in the same way as the contact between a roller slice and the raceway. Take the cage guided by the inner raceway, as illustrated in . Let the guide land be represented by two circles with nominal radius Rg and the raceway top flange have a radius Rr=Rg-Cg, for inner raceways, or Rr=Rg+Cg for outer raceways. Note that any of these quantities is used with another subscript L or R depending on if the guidance is done by the left or right side of the raceway, respectively, as implied in Figure 42.

Rou Rin

Wl

RgRRgL

CgR

CgL

WgRWgL

Cou

Cin

LgRLgL

Rou Rin

Wl

RgRRgL

CgRCgL

WgRWgL

LgRLgL

c



The search for the contact points is now pursuit for both circles independently. For each one of them a process described in section 5.3, dealing with the contact between a roller slice and the inner and outer raceways, is used. Note that the radius of the circles, representing the cage guide lands, and the raceway flange guide lands are easily obtained from Figure 42 and, therefore, it will not be pursued here anymore.

(a) (b)

Figure 43: Contact between two cylindrical sections, as in the contact between the cage and the raceway in the cage of cage guidance: (a) contact between raceway and cage guide land; (b) the guide land of the cage is represented as two circles.

c

c

c



7. FORMULATION OF THE ROLLER CONTACT FORCES

During the contact detection it is possible to identify the location of the points in contact in the surfaces of the rolling elements, the relative indentation and orientation between the contacting surfaces and the relative velocities. These are the kinematic variables required by the models of the normal contact forces and the friction and hydrodynamic forces. The loading of the rollers, raceways and cage is evaluated in two stages: first, the normal contact forces are evaluated using Hertzian contact models; second, the tangential contact forces are evaluated using friction and tribological models, recognizing the lubrication mode present in each contact.

7.1. Normal contact forces

The normal contact forces developed during the contact between the rolling elements, although large, are distributed over a small area when compared with the dimensions of the contacting surfaces. Typical examples of contact between rolling elements are described in Figure 44 for cases of wheel-rail contact and rolling bearings.

(a) (b)

(c) (d)

Figure 44: Contact patches for cases of Hertzian contact force models: (a) Elliptical contact; (b) Point contact; (c) Elliptical contact; (d) Line contact.

The stress distribution over the contact area, or contact patch, is described by Hertz elastic contact theory if some conditions are met [4]. In order for the Hertz elastic contact theory to be valid it is required that:

Rolling direction

Rolling direction

Rolling direction

Rolling direction



All deformations must be within the linear elastic limits, i.e., the strains are small and the body stresses can be described by a linear elastic constitutive relation.

All shear stresses are neglected, i.e., the loading is assumed normal to the contacting surfaces and, therefore, the contact is frictioneless.

The dimensions of the contact patch are small when compared to the dimensions of the surfaces in contact, i.e., the contact area is much smaller than the characteristic radii that define the surfaces curvatures, implying in turn that the surfaces are continuous and non-conforming.

Typically point contacts, described by elliptical contact patches as those described in Figure 45, and line contacts, described by the rectangular contact patch shown in Figure 46 are of importance in the contact of rolling elements, and are used extensively in the rolling bearing dynamics. It is not the objective of this work to provide thorough presentation of the Hertz elastic contact theory but simply to register its main findings that are of importance to the representation of the normal contact forces between rolling elements of bearings.

The point contact between two solids leads to elliptical contact patches with the geometry depicted in Figure 45, in which it is assumed that the rolling direction is along X. In roller bearings typically the contact between the roller end and the raceway flanges or the cage top are characterized as point contacts.

(a) (b)

Figure 45: Elliptical contact: (a) Geometric definitions; (b) Stress distribution and contact patch geometry

Let the interference, or compression, between two surfaces be described by , which can be found using Eq. (69). The normal contact force, in the case of point contact is

3

2n ptf K (112)

where proportionality factor Kpt, or contact stiffness, is given by

x

b

a

y

max

yb

xb

rbxrby

xa

ya

ray

rax

nf



13

34 2

*

10

2.79pt

p

K

(113)

with

1 1 1 1

pax ay bx byr r r r

(114)

1

3*2

2

2

(115)

being the characteristic radii of the contacting surfaces described in Figure 45, =a/b, with a and b being the semi-axis of the contact ellipse, and and are complete elliptic integral of the first and second kind [24]. Using a least-square relation, Brewe and Hamrock [25] find an approximate expression for the elliptic integrals and for are

0.636

1.0339

1.5277 0.6023 ln

1.0003 0.5968

y

x

y

x

x

y

R

RR

RR

R

(116)

where

111 1 1 1

;x yax bx ay by

R Rr r r r

(117)

It should be noted that in the contact of the large end of the tapered rollers with the cage or with the raceway flanges there is, in fact a contact between a sphere and a cone. In this case rax=ray=Rs, rbx=∞ and rby=Rf, being Rs the radius of the roller large end and Rf the flange approximate radius at the point of contact.

Of interest is also the evaluation of the maximum stress that develops during contact, as it is used in the evaluation of the tangential forces due to the lubricated rolling contact. For a point contact the maximum stress in the contact patch, shown in Figure 45 is written as [26]

max

3

2nf

ab

(118)

with the dimensions of the semi-axis of the elliptical contact patch found as

12 3

1

3

20.0236

20.0236

n

p

n

p

fa

fb

(119)



Note that the evaluation of a and b, and also *, by Eqs. (119) and (115) respectively, is approximate when the elliptical integrals and the ratio of the semi-ellipse radius are calculated by Eq. (116). The tables presented by Harris and Kotzalas [26] for a more precise evaluation of these quantities can be used instead.

The ideal line contact, illustrated in Figure 46, can be understood simply as a case in which the ratio →∞ due to the fact that a>>b. However, the relation between indentation and normal force becomes, in this case, a nonlinear relation that requires an iterative procedure to obtain the normal force when the indentation is known. Several elastic contact models for cylinders with parallel axis based on the Hertz elastic contact theory have been proposed. The Johnson model, used by the ESDU-78035 Tribology Series [27], is given by

*

*

4ln 1n

n

f L E R

f LE

(120)

where the composite modulus E*, for contact surfaces with identical material properties, and the characteristic radii R are given by

*

22 1

internal contactexternalcontact

ax bx

ax bx

EE

r rR

r r

(121)

where L represents the effective contact length, generally equal to the length of the shorter cylinder, E is the Young modulus and the Poisson coefficient.

(a) (b)

Figure 46: Line contact: (a) Geometric definitions; (b) Stress distribution and contact patch geometry

Different other solutions for the relation between the indentation and the normal force on the contact between cylinders with parallel axis have been proposed in the literature. In formulas

y

2b

2a

x

max

xa

nf

xb

ya

rax

rbx

yb

ray= rby= ∞



for stress and strain, by Roark [28], the expression first proposed by Radzimovsky [29] is presented as

*

4 42ln ln

3ax bxn r rf L

b bE

(122)

where the parameter b is

2.15 nf Rb

E (123)

with ( / )ax bxR r r R , being R given by Eq. (121). In Eq. (123) it is assumed that the materials used in the contacting surfaces have identical material properties. By substituting the relations of Eq. (123) into Eq. (122) leads to

* 2

2 8ln

3 2.15n

n

f L E R

E f L

(124)

Another cylindrical contact model has been proposed by Goldsmith and expressed as [30]

*

*1 lnn

n

f L E

f LRE

(125)

When used in the framework of computational dynamics, in which the normal contact force needs to be calculated based on a known indentation, resulting for the known positions of the contacting bodies, the line contact models considered so far are computationally inefficient. The iterative procedure required to calculate the contact force for a given indentation, required by the use of Eqs. (120), (124) or (125) not only is costly when a very large number of contacts have to be calculated, as in the case of roller bearing dynamics, but is also prone to convergence difficulties. Furthermore, the different line contact models discussed so far have limitations in their range of applications, as discussed by Pereira et al. [31].

To overcome the numerical difficulties and the physical limitations on the use of the Hertz based contact models by Johnson, Radzimovsky or Goldsmith, Pereira et al. proposed a cylindrical contact model that does not require an iterative procedure to calculate the normal contact force [32], written as

*Δ +

Δn

n

a R b L Ef

R (126)

in which

0.49 internal contact0.39 external contact

a (127)

0.10 internal contact0.85 external contact

b (128)

0.005 internal contact

1.094 external contactY Rn

(129)



In Eq. (129) the constant Y reflects the fact that for internal contact it is not possible to find a single expression to obtain a fit with good correlation between the Johnson and exponential fit function for the complete range of clearances R. The best fit is obtained with the constant Y given by

0.1921.56 ln 1000 if 0.005,0.7500.0028 1.083 if 0.750,10.00

R R mmY

R R mm

(130)

This cylindrical contact model matches in precision Johnson model in its range of validity and extends the range of application of current models due to the extensive use of finite element models for cylindrical contact, which are not prone to the Hertz elastic contact limitations.

For any of the normal contact force models presented, the maximum contact stress in the rectangular patch is given by

max

2 nf

L b

(131)

with the dimensions of the dimension b of the contact patch found as

1

2

0.00335 n

p

fb

L

(132)

All cylindrical contact models presented assume ideal line contact, i.e., the geometry of the cylinders is perfect and contact takes place in the full extension of the shorter cylinder. In roller bearings the rollers are crowned and the contact models presented do not represent correctly the relation between indentation and normal force. Based on laboratory testing, Palmgren [33] developed a relation, which is the basis of current contact models in roller bearing line contact, written as

9

10

810

92 102 1

3.81 nf

E L

(133)

which rearranged and having the normal force written as a function of the indentation, or relative elastic approach, is

8 10

9 9*0.71069nf E L (134)

In order to apply the normal contact force model for line contact to the roller contact with the inner and outer raceways and with the side of the cage pockets the roller is discretized in Nsl strips, as implied by Figure 16. The contact search for each strip is evaluated using the procedure described in section 5.1 and for the contact force is evaluated based on the strip indentation. The normal contact force of a strip of the roller is given by [34].

8 101

9 9 9*0.71069 1, ,ns sl s slf E N L s N (135)

where the counter s refers to the number of the slice in each of the rollers.



7.2. Tangencial contact forces

Surfaces in rolling contact, subjected to normal contact loads are also subject to tangential forces due to friction, in the case of dry contact such as for rail-wheel interaction, or due to the lubricant, in the case of lubricated contact as in normal roller bearing applications. Regardless of the type of contact, the relation between the tangential forces and the normal contact forces is given by

t nf f (136)

where is the equivalent friction coefficient. For the sake of brevity the subscripts t and n are used instead of ts and ns used in Eq. (135). Note also that the tangential force is applied in the opposite direction of the relative velocity between the contacting surfaces, obtained by Eq. (73).

The simple form of Eq. (136) hides the complexity of the calculation of for many important tangential forces, as in the case of lubricated contact. In the case of lubrication between the contacting surfaces, depending on the lubricant film thickness and on the roughness of the contacting surfaces the type of contact is different and the equivalent friction coefficient has to be evaluated differently. Fig. Figure 47 shows the different contact modes, from dry contact through full fluid film lubricated mode.

(a) (b)

(c) (d)

Figure 47: Types of contact ( Boundary lubricant layer, Lubricant): (a) Dry contact; (b) Boundary mode; (c) Mixed mode; (d) Full fluid mode.

For lubricated contact, the relation between the fluid film thickness and the roughness of the contacting surfaces defines the type of contact mode that is taking place. Let the lubricant film parameter be defined as

2 21 2

ch

(137)

where hc is the lubricant film central thickness and 1 and 2 the roughness of the contacting surfaces. Assuming for each lubrication mode a different equivalent friction coefficient, the relation between , in Eq. (136), such lubrication mode equivalent friction coefficients is written as [23]



6

6

bd bd

bd fmfm fm bd fm

bd fm

fm fm

(138)

The typical values for the lubricant film parameter used to define he transitions between the different lubrication modes are bd=0.01 and fm=1.5 [23]. The evaluation of the equivalent friction coefficient required the prior evaluation of the lubricant film parameter, which in trun imply the calculation of the lubricant film thickness, and the equivalent friction coefficients for each mode of lubrication, bd and fm.

7.2.1. Lubricant film thickness

Being the contacting surface roughness know, either from direct measurement of the rolling elements or by published data, the calculation of the lubricant fluid film thickness plays the central role in the decision on the lubrication mode experienced by the rolling elements. The lubricant film thickness varies along the contact region, being generally of importance its calculation in the center, designated by hc, or/and close to the exit of the contact region where it is minimum, hmin, according to the Elastohydrodynamic lubrication theory (EHL). Fig. Figure 48 shows the profile of the lubricant film thickness and contact pressure in a general contact.

Figure 48: Lubricant film thickness and pressure profiles for a typical rolling contact case.

Depending on the temperature and on the supply of lubricant fluid the lubricant film thickness may vary. Therefore, the computation of the lubricant film thickness is done by

c iso T Sh h (139)

where hiso is the isothermal central lubricant fluid thickness for fully flooded lubrication, T is the thermal reduction factor and S is the starvation factor, that need to be evaluated independently.

Before calculating any of the quantities appearing in Eq. (139) let the following dimensionless quantities be defined:

Film Thickness parameter chH

R (140)

X

Y

EHL pressure

hminhc

b

u1

u2



Speed parameter 0ˆ

o

uU

E R

(141)

Load parameter o

qW

E R (142)

Material parameter 1 oG E (143)

where the quantities used to define the dimensionless parameters are the effective radius in the rolling direction defined as

ax bx

ax bx

r rR

r r

(144)

the equivalent modulus of elasticity is *2oE E , with E* given by Eq. (121) for contact surfaces made of materials with equal material properties, while for materials with different properties is

12 21 2

1 2

1 12oE

E E

(145)

the average velocity of the contacting surfaces, depicted in Figure 48, is

11 22u u u (146)

the lubricant viscosity at reference temperature T0 and reference pressure is denoted by 0 and the pressure coefficient of viscosity 1 assume a rheological description of the lubricant fluid viscosity described by

1 0

0tp T Te (147)

being p the pressure difference for the reference pressure, T the temperature and t the viscosity temperature index. Finally, in Eq. (142), q is the normal compressive load, for point or elliptic contact, or the normal load per unit of length, for line contact.

Isothermal central lubricant fluid thickness

The estimation of the isothermal lubricant fluid thickness, assuming contact between rigid surfaces, isoviscous lubricant and isothermal lubrication was proposed by Martin as [35]. Under these conditions, the thickness film parameter for line contact is obtained as

14.9rmH U W (148)

where the subscript rm refers to the rigid mode assumption of the Martin Eq.. However, it was noticed that the Martin Eq., i.e., Eq.(148), underestimates the lubricant film thickness in most of the cases of interest.

Grubin relieved the assumption of rigidity between the contacting surfaces and assumed that the deformed shape of the contacting surfaces is the same that is observed for dry contact [36]. A new expression for the estimation of the film thickness parameter for line contact is obtained as



3 1

11 111.95H GU W

(149)

The estimation of the film thickness by Grubin expression, in Eq.(149) is very close to the actual experimental measurements [6]. Based on experimental work results, Pan and Hamrock [37] obtained a new relation by curve fitting the data to the film thickness measurements obtained from performing a large number of variations on the performance parameters. Pan and Hamrock estimation of the central film thickness for line contact is [37]

0.166 0.692 0.4702.922emH W U G (150)

Although Martin evaluation of the film thickness, given in Eq.(148), underestimates its value in most of the times, there are few cases in which it shows to be accurate. According to the suggestion by Harada and Sakaguchi for line contact [23] the central lubricant film thickness is calculated by the maximum of Martin and Pan estimations, i.e.

max ,c rm emH H H (151)

And consequently the isothermal central lubricant film thickness for line contact is given by

iso ch H R (152)

In the framework of rolling bearing dynamics, point contacts are observed, for instance, between in the presence of ball bearings or in the tapered rollers large ends with the raceway flanges or with the cage.For point, or elliptical, contact several expressions have been proposed to describe the lubricant film thickness. For instance, Archard and Cowking [38] suggest that the central lubricant film thickness for point contact is given by

0.74 0.0742.04H GU W (153)

where is the modified factor for side-leakage given as

12

13 r

(154)

in which r=Ry/Rx, being Rx simply the equivalent curvature radius calculated using Eq.(144) and Ry the equivalent curvature radius in the Y direction, i.e., /y ay by ay byR r r r r , while X the rolling direction and Y the perpendicular direction.

Table 4: Constants for the constants of the Cheng expression for the central film thickness parameter [39]

a/b C n1 n2

5 1.625 0.740 -0.220

2 1.560 0.736 -0.209

1 1.415 0.725 -0.174

0.5 1.145 0.688 -0.066

Also Cheng [39] propose a different expression for the central film thickness parameter as

2

1 max

0

nnH C GU

E

(155)



where the constants C, n1 and n2 are constants listed in [6] and a, b and max are the major and minor semi-axis and the maximum pressure for the case of ellipsoidal, or point, contact in the Hertz elastic contact theory.

Based on a numerical solution for the elastohydrodynamic lubrication for a point contact Hamrock and Dowson suggest that the central film thickness parameter for point contact is written as

0.67 0.53 0.067 0.732.69 1 0.61 kH U G W e (156)

in which k=a/b. For convenience, the relation the ratio between the major and minor semi-axis of the contact ellipse may be written as [6]

0.6361.0339 r

ak

b (157)

Under the assumption of the elastohydrodynamic lubrication theory, Chittenden et al. propose that the central lubricant film thickness parameter, for point contact, is evaluated as [40]

2/30.68 0.49 0.073 1.234.31 1 kemH U G W e (158)

For hydrodynamic lubrication, in which the contacting surfaces are assumed rigid, Brewe et al. suggest that the point contact film thickness parameter is written as [41]

2

1128 0.131 tan 1.6832

rrm r

UH

W

(159)

where the modified factor for side-leakage, , is given by Eq. (154).

Just as for line contact also for point contact although the hydrodynamic theory underestimates the thickness parameter value in most of the times, there are few cases in which it shows to be accurate. Also for point contact, Harada and Sakaguchi [23] suggest that the central lubricant film thickness is calculated by the maximum of Eqs.(158) and (159), i.e.

max ,c rm emH H H (160)

Certainly, depending on the operation mode of the roller bearings and on the materials involved other authors may suggest alternative expressions for the evaluation of the central lubricant film thickness. Regardless of their exact nature, the procedure described here can still be applied, provided that the parameter involved in the Eqs. are properly adjusted.

Thermal reduction factor

The thermal reduction of the film thickness is important at high rolling speeds and/or high slip velocities [42]. Based on the works by Cheng [39] and by Wilson and Sheu [43], Gupta et al. [42] propose a thermal reduction factor for line contact given by

0.42max

00.83 0.64

1 13.2

1 0.213 1 2.23

T

TT

LE

s L

(161)



where the thermal loading parameter LT and the slip ratio s are given by

2

0ˆ

tT

f

uL

K

(162)

1 2

ˆu u

su

(163)

being the average velocity of the contacting surfaces, u ,evaluated by Eq.(146), Kf the thermal conductivity and 0 the lubricant viscosity at reference temperature T0 and reference pressure and t the viscosity temperature index as used in the rheological Eq. of the lubricant given by Eq.(147).

The thermal reduction factor, given by Eq. (161), can be used for elliptical contact also if a/b>5 [5]. However, for non-elongated contact ellipses different expressions for this factor are available in the literature. For elongated contacts, i.e., for a/b>5, the elliptical contact and the line contact share the same thermal factors. For non-elongated contacts Eq.(161) can still be used. Although in some recent research work some alternative Eqs. are being proposed for point contact, the expression for line contact is still used for other types of contact as it does not lead to unreasonable errors.

Starvation factor

All the calculations of the lubricant film thickness done before assumed that the supply of fluid is enough to provide a fully flooded contact and the situation is defined as starvation. As indicated in Figure 49, the meniscus of the lubricant film approaches the contact area, ultimately leading to the reduction of the film thickness.

The influence of the starvation on the film thickness has been studied by a large number of researchers, among which Wolveridge et al. [44] propose a starvation factor for line contact as

0.8315 1.558 2.2961.68005 0.260137 0.016146

1S e (164)

in which

22 3

0

2

x b

b h R

(165)

being x0 the distance from the edge of the Hertzian contact zone to the film meniscus, depicted in Figure 49, or termination, h∞=hisoT the fully flooded film thickness, b the Hertzian contact half-width and R the equivalent radius of contact in the rolling direction, as defined in Eq.(144).

For elliptical contact Hamrock and Dowson [45] propose an expression for the starvation factor identified experimentally as

0.29

*

1

1S

m

m

(166)

in which



0.582* 1 3.06

Rm h

b

(167)

and m is the adimensionalized distance from the center of the contact zone to the film termination, as represented in Figure 50. Note that all quantities defined in Figure 50(a) are adimensionalized as x=x/b, ỹ=y/a and m=m/b, leading to the adimensionalized space depicted in Figure 50(b). Note also that m=x0+b, defined in Figure 49.

Figure 49: Starved lubrication with the identification of the dimensions required for the definition of the starvation factor.

Note that the starvation factor proposed by Hamrock and Dowson [45], for point contact, and by Wolveridge et al [44], for line contact, are being continuously the focus of new research leading to relations for the starvation factors that better fit specific lubricants. In any case, their computational implementation follow basically the procedures described here.

Figure 50: Representation of the contact area and distance to the lubricant fluid free boundary for starved lubrication of point contact.

X

Y

b

u1

u2x0

Hertziancontact area

Lubrication fluidMeniscus of lubrication fluid

X

Z

bx0

Hertziancontact area

Free boundary Lubrication fluid

Rolling direction

a

b

m

Y

X

Rolling direction

1

1

m

Ỹ

X



7.2.2. Equivalent friction coefficients

The evaluation of the tangential forces at contact, using Eq.(136), requires the evaluation of the equivalent friction coefficient using Eq.(138). In order to evaluate the equivalent friction coefficient for the specific type of contact it is necessary to evaluate the equivalent friction coefficient for boundary lubrication mode, bd, if the lubricant film parameter is <fm and the equivalent friction coefficient for full film mode,fm, if the lubricant film parameter is >bd.

Boundary mode equivalent friction coefficient

In the lubrication boundary mode the equivalent friction coefficient, bd, relation with the slip ratio is proposed by Kragelskii [46] as

181.460.1 0.1 22.28 sbm s e (168)

Note that for a slip ratio s=0 the equivalent friction coefficient is bd=0 while for very large slip ratios it tends to bd→0.1.

Full film mode equivalent friction coefficient

The evaluation of the full film equivalent friction mode the Muraki formula, reported by Harada and Sakaguchi [23] is used, i.e.,

0fmavg

S

(169)

where 0 is the lubricant characteristic stress, avg is the mean normal stress in the contact area for Hertzian elastic contact and S is the mean dimensionless shear stress that needs to be calculated. In order to calculate the mean dimensionless shear stress let the following parameters be defined: the dimensionless shear velocity of the lubricant is

1

0

0

avg

isoc

s e

h

(170)

the Debroah number is

1

0ˆ avg

iso

u eD

G b

(171)

and, the dimensionless length of the EHL contact area is

1sinh4

isoc iso

iso

Dx

(172)

Finally, the mean dimensionless shear parameter is written as a dependency of the dimensionless length of the EHL contact area as

1 1sinh 1 4 sinh 2

2iso iso iso iso c

iso iso c

D xS

D x

(173)



Alternatively, Smith et al. [47] based on measured tractions in an elastohydrodynamic rolling disk test rig proposed a relation for the traction coefficient, or full film equivalent friction coefficient. Also Gupta [42] based on experimental identification of different lubricant rheological behavior proposed a traction model that is adopted here.

Consider that the lubricant rheological model relating the viscosity to the pressure and temperature is given by

* *

0 1*0

p T Te

(174)

where is the viscosity and *0, * and * are constants to be identified experimentally for

the lubricant used. By calculating and integrating the shear stress on the contact area leads to the traction coefficient written as [5]

* *

max max

* 2/2 /210

* 2 * *max max

83 2sinhf

fm

Ke e

h

(175)

where the dimensionless parameter is given by

* *0

8 f

sK

(176)

and function () is defined as

21

0

sinhd

(177)

In the computer applications the integral in Eq.(177) can be tabulated and being that table interpolated suring the dynamic analysis.

Gupta found the values for the lubricant rheological constitutive Eq., given in Eq.(175) as [5]

2 3313 25*0 1

V T V uV e e (178)

2 3

*1 313 25

B BT uB

(179)

where T is the temperature, in Kelvin, and u is the rolling velocity, in m/s. The values for the constants involved in Eqs.(178) and (179) are depicted in Table 5

Table 5: Regression coefficients for Gupta Type I traction model [5]

Lubricant * V1 V2 V3 B1 B2 B3

1/Pa × 109 Pa s × 10 1/K × 102 s/m × 102 1/K × 10 × 10 MIL-L-23699 5.8015 2.6529 3.6358 1.7054 3.3398 1.3075 -3.9353 MIL-L-27502 5.8015 2.1655 3.9221 0.6441 3.2577 1.0723 -2.7538 MIL-L-7808 5.2214 1.1431 3.3723 2.4079 4.1745 0.60873 -4.7828 Santotrac 30 9.4275 4.0063 5.0459 1.5590 3.4104 1.0930 -2.5647 Mobil DTE 7.2519 2.8407 2.3743 4.5016 2.8939 2.1402 -5.2909



8. RAILWAY WHEEL-RAIL CONTACT FORCES

The objective of the MAXBE project is the identification of bearing failure, or damage, using acoustic monitoring systems. The acoustic emission of the axelbox system may be due to effective bearing damage or a result of the wheel-rail contact, or both. In order to allow for the identification of acoustic signatures associated to wheel-rail contact, and not to bearing damage, it is necessary to include in the BearDyn code the possibility to model and study the dynamic behavior of the complete boggie, including wheel-rail contact forces. Note that the inner raceways of the axel bearings are rigidly fixed to the shaft of the wheelset, represented in Figure 51. With this purpose in mind, the wheel-rail contact force model implemented is briefly described here being the interested reader forwarded to the references in the report.

Figure 51: Wheel set as a rigid body with the representation of its three-dimensional motion.

8.1. Wheel and rail geometric description

In this work the wheel and rail surfaces are considered as sweep surfaces, obtained by dragging plane curves on spatial curves. As a result, the problem of describing the surfaces reduces to the problem of defining plane curves, which represent the cross sections of the wheel and rail. Four independent surface parameters sr, ur, sw and uw are used to define the geometry of the wheel and rail surfaces, as shown in Figure 52. The parameter sr represents the arc length of the rail space curve, i.e., it positions the rail cross-section on which the contact point lies, while ur defines the lateral position of the contact point in the rail profile coordinate system (r,r,r). The parameter sw represents the rotation of the wheel profile coordinate system (w,w,w) with respect to the wheelset coordinate system (ws,ws,ws), i.e., it defines the rotation of the contact point, while uw defines the lateral position of the contact point in the wheel profile coordinate system [48,49].

Note that the origin of the wheel profile and wheel coordinate systems is the same and that w=ws are always coincident. Furthermore, w is always in the plane of the rail cross-section in which the contact point is located and pointing in the opposite direction of such contact point. In the text, subscripts (.)r and (.)w are referred to the rail and wheel, respectively, whereas the subscript (.)ws are referred to the wheelset.The problem of finding the position of each contact point of each wheel tread and flange consists in finding the four parameters sr, ur, sw and uw associated to each of these contact points. For the sake of computational efficiency,



as described in the next sections, for each wheel the position of two potential points of contact, one on the flange and other on the tread, are always monitored. By potential contact points it is meant that the pair of points, on the wheel and on the rail, are the ones that are closer to each other if no contact occurs. If contact effectively takes place then the appropriate set of forces is applied.

Figure 52: Wheel and rail surface parameters

8.1.1. Rail surface

The rail surface is generated by the two-dimensional curve that defines the rail profile, when it is moved along the rail space curve. The location of the origin and the orientation of the rail profile coordinate system, defined respectively by the vector rr and the transformation matrix Ar, are uniquely determined using the surface parameter sr [50]. Using this description, the global position vector of an arbitrary point Q on the rail surface is written as:

+ 'Q Qr r r rr r A s (180)

where 'Qrs is the local position vector that defines the location of the contact point Q on the rail surface with respect to the profile coordinate system. Note that due to the above-mentioned description of the rail geometry, the following relations hold:

; ; ' 0 TQr r r r r r r r r rs s u f u r r A A s (181)

where fr is the function that defines the rail profile. The transformation matrix Ar can be expressed in terms of a set of three orthogonal vectors, the unit tangent vector tr, the principal unit normal vector nr and the binormal vector br [51,52] that define the moving reference frame associated to the rail space curve. Hence, the transformation matrix is [7]:

r r r r r r r r rs s s s A A t n b (182)

The unit vectors are expressed uniquely in terms of the rail arc length, i.e., as function of the surface parameter sr. The Cartesian components of these vectors are obtained from the respective rail database previously described.

sr

Q

sww

w

ws

ws

Q

w

w

uw

fw(uw)

r

r

ur

fr(ur)



Figure 53: Rail profile parameterization using piecewise cubic interpolation schemes

Generally, the function fr, which defines the rail profile at each cross section, is not given by simple analytical functions. Here the rail profile is parameterized as function of the surface parameter ur using a piecewise cubic interpolation scheme. For this purpose shape preserving splines are used [53-56]. Hence, to obtain fr(ur) the user only has to define a set of control points that are representative of the rail profile geometry, as shown in Figure 53.

This methodology is general since it not only allows using any type of rail profile, obtained from direct measurements or by design requirements, but also allows changing the rail profile online during the analysis, if needed. For instance, using this formulation, the geometry of switches or crossings [57] can be easily modeled by a profile that changes as a function of the track length parameter. Also the change of the rail profile as a function of time or of the location along the track, due to a wear law for instance, can be easily performed without any particular modification to the formulation presented here.

8.1.2. Wheel surface

The wheel is a surface of revolution is obtained by a complete rotation, about the wheel axis, of the two-dimensional curve that defines the wheel profile [58]. The location of the origin and the orientation of the wheelset reference frame are defined, respectively, by the vector rws and the transformation matrix Aws. The global position vector of an arbitrary point Q on the wheel surface is written as:

+ + 'Q Qw ws ws w w wr r A h A s (183)

where 120 0 T

w Hh is the local position vector that defines the location of wheel profile coordinate systems with respect to the wheelset reference frame and H is the lateral distance between wheels profiles origin. The transformation matrix that defines the orientation of the wheel profile coordinate system with respect to the wheelset frame is [59]:

cos 0 sin 0 1 0

sin 0 cos

w w

w

w w

s s

s s

A (184)

The quantity 'Qws is the local position vector that defines the location of the contact point Q on the wheel profile coordinate system, written as:

' 0 TQw w w wu f us (185)

where fw is the function that defines the wheel profile. Since in general fw is not given by analytical functions, it is proposed to parameterize the wheel profile using a piecewise cubic

Nodal Points

r

r

ur

fr(ur)



interpolation scheme, as described for the rail surface. Hence, the user has to define a set of control points that are representative of the wheel profile geometry, as shown in Figure 54(a). Note that this procedure is general in the sense that it permits the dynamic analysis of railway vehicles using wheel profiles obtained from direct measurements or by design requirements, and just as for the rail profile, the wheel profile may change during the dynamic analysis, if needed.

In the multibody code used to solve the wheel-rail contact problem, it is necessary to devise a strategy to determine the location of the contact points between the parametric surfaces. The proposed formulation requires that the parametric surfaces are convex. Hence, when parameterizing the wheel profile, it is necessary to avoid the geometric description of the small concave region in the transition between the wheel tread and the wheel flange, depicted in Figure 54(b). To avoid this difficulty, the wheel profile is represented by two functions t

wf and f

wf that parameterize the wheel tread and flange, respectively, and the concave region is neglected, being the wheel surface made of two convex regions, as shown in Figure 54(a). Note that this limitation puts some restrictions on the use of this methodology for worn wheel profiles, if no modification on the procedure is implemented.

Figure 54: Wheel profile: a) Parameterization using cubic interpolation schemes; b) Concave region

Note that this methodology is general since it not only allows using any type of wheel profile, obtained from direct measurements or by design requirements, but also allows changing the wheel profile online during the analysis, if needed. For instance, using this formulation, the polygonization of the wheel [60] can be easily modeled by a profile that changes as a function of the wheel rotation. Also the change of the wheel profile as a function of time, due to some wear law for instance, can be easily performed without any particular modification to the formulation presented here.

8.2. Wheel–rail contact formulation

8.2.1. Wheel-rail contact algorithm

The first step of the wheel-rail contact formulation presented here is the accurate prediction of the location of the contact points between wheel and rail surfaces. The coordinates of the contact points have to be calculated online, for every time step, during the dynamic analysis. Once the coordinates of the contact points are determined, the normal contact forces that develop in the wheel-rail interface are computed. Then, the creepages, or normalized relative velocities between the bodies at the points of contact, are found. Once the creepages are obtained, different computer routines, using alternative approaches, can be applied to evaluate

Concave Region

w

w

Flange Nodal PointsTread Nodal

Points

uw

fwt (uw)

w

w

fwf (uw)



the creep forces. In this algorithm, three distinct methodologies are implemented to calculate the tangential contact forces that develop in the wheel-rail interface: Kalker linear theory[61-63]; Heuristic nonlinear force model [64]; and, Polach formulation [65].

A schematic representation of the wheel-rail contact algorithm [66-69] is presented in Figure 55. The procedure, used in the framework of a general multibody formulation, assumes that the position, orientation and velocities of each rigid body are known at every particular time step [7,70]. The remaining steps of the procedure used to calculate the contact forces are described hereafter.

Figure 55: Schematic representation of the algorithm proposed to study the wheel-rail contact problem

8.2.2. Contact point detection

The possible situations that may occur in the wheel-rail contact problem are sketched in Figure 56 where i and j represent the wheel and rail surfaces, respectively. When there is no penetration between the bodies, as for situations depicted by Figure 56(a) and (b), the minimum distance conditions are applied to find the surface parameters that define the coordinates of the contact candidate points as being the pair of points, each belonging to one surface, that are closer to each other.

If penetration occurs, the contact point on one body has to be located inside the volume of the other body, as shown in Figure 56(c). In this case, the contact points have to be selected from

t Δtq

t Δtq

endt Δt t

t t Δt

MULTIBODY FORMULATION

Begin

End

0t t

0tq

0tq

Initialize

0rs t

0ru t

0ws t

0wu t

Estimate

Normal Contact Forces

Creep Forces and Moments

rs t ru t

ws t wu t

Solve SystemNonlinear Equations

Coordinates of Contact Points

Polach Theory

Heuristic Model

Obtain

Update

No

Yes

KalkerTheory

Dimensions of Contact Area

Calculate Creepages



the points belonging to the volume that the two solids share. The contact points are defined as those that correspond to maximum indentation, i.e. the points of maximum elastic deformation, measured along the normal to the contact patch. In the situation described by Figure 56 (c), the minimum distance conditions are verified for the pairs of points I and III. However, only the points represented in case II should be considered to be candidates to contact points once they are the points of maximum penetration. In all situations described by Figure 56 the condition for contact, or contact potential, imply the alignment of the normal surface vectors ni and nj with the distance vector d.

Figure 56: Wheel-rail contact situations: a) No contact, i.e., wheel lift; b) Contact at a single point, without penetration; c) Contact with penetration

The coordinates of the contact points can be predicted online, during the dynamic analysis, by determining the four parameters that define the geometry of the contact surfaces. A two-step methodology is used here to determine the coordinates of the contact points between wheel and rail surfaces. First, four geometric Eqs. are defined and solved in order to find the surface parameters that define the coordinates of the candidates to be contact points between the surfaces. With reference to Figure 57, these Eqs. are written [51,66-69]:

1 1

2 2

0 0 ;

0 0

T Tr w wr wT Tr w wr w

n t d t

n t d t (186)

where nr is the vector normal to the rail surface, 1wt and 2

wt are two vectors tangent to the wheel surface and dwr = rw – rr is the distance between the potential points of contact.

The second step of the method consists in evaluating of the penetration condition in order to check if the points are, in fact, in contact or not. The penetration condition specifies that:

0Twr r d n (187)

By introducing four surface parameters, the coordinates of the contact points can be predicted during dynamic analysis even when the most general three-dimensional motion of the wheelset with respect to the rails is considered. The calculation of the surface parameters requires the solution of the preliminary system of nonlinear Eqs. (186). The computational implementation of this methodology leads to an efficient algorithm, outlined in Figure 55, since the information of the previous time step is used as initial guess to find the solution of the nonlinear Eqs. on the next step and, therefore, only few iterations are required to obtain the solution.

d = 0

(b)

(j)

(i)

ni

n j

d

(j)

I

(c)

II III

(i)

ni

ni

ni

n j

n j

n j

n j

ni

(a)

(j)

(i)

d



Figure 57: Candidates to contact points between two parametric surfaces

8.2.3. Two points of simultaneous contact

The methodology proposed here to determine the contact points location allows studying two points of simultaneous contact between one wheel and the rail by using an optimized search for possible contact points on the wheel tread and wheel flange. This strategy takes advantage of the fact that the wheel profile is parameterized by two functions, one for the tread, t

wf , and other for the flange, f

wf , as shown in Figure 54(a). The method used to look for the contact points is fully independent for the wheel tread and for the wheel flange. Due to the yaw angle of the wheelset with respect to the track, the second point of contact between the wheel flange and the rail can be located in a plane different from the plane that contains the wheel axis and the first point of contact. If the flange contact point is located ahead the tread contact point, as shown in Figure 58(a), the contact configuration is known as lead contact [48,49] and the wheelset is said to be in an under-radial position [71]. If the wheelset is in the so-called over-radial position, the flange contact point is located after the tread contact point, as shown in Figure 58(c), and the contact configuration is called lag contact. An intermediate situation occurs for radial wheelset position [71]. In this case, the flange and tread contact points are located in same plane, as represented in Figure 58(b).

In curve negotiation, when dealing with high angles of attack, or on switch transitions, it is very important to consider the lead and lag contact [48,49]. Such contact scenarios also have to be considered during the dynamic analysis of railway vehicles when investigating the hunting instability or the wheel climbing [71].

8.2.4. Normal contact forces in the wheel-rail interface

The Hertz contact force model with hysteresis damping [72,73] is used here to calculate the normal contact forces that develop at the wheel-rail interface. This force model accounts for the energy dissipation effect that occurs during contact. The normal contact force includes a component function of the indentation and a damping component, proportional to the velocity of indentation. The normal contact force is

t sj

t wi

d

p ,u w

q ,s t

tui

(j)

(i)

x

z

y

n j

t tj

ni



2

( )

3 1 = 1+

4n

n

ef K

(188)

where is the indentation, n=1.5 is the parameter used for Hertzian contact between metal surfaces, K is the Hertzian constant that depends on the surface curvatures and the elastic properties of contacting bodies, e is the coefficient of restitution, is the velocity of indentation and ( ) is the velocity of indentation as contact starts. The velocity of indentation is evaluated as the projection of the relative velocity vector of the bodies at the point of contact on the vector normal to the contact surfaces. Eq. (188) is valid for non-conformal contact, where the contact patch is described by an ellipse. For the case of conformal contact, as for worn wheel profiles, either another contact force model must be used or provisions have to be made to adjust the stiffness K to such conditions [74,75].

Figure 58: Possible positions of the wheelset relative to the track: a) Lead contact in the right wheel; b) Flange contact with radial wheelset position; c) Lag contact in the right wheel

The contact force model described by Eq. (188) requires the knowledge of the initial impact velocity ( ) . When the initial conditions of the contact problem are such that the simulation starts with wheel and rail already in contact the value for ( ) is not available. In this case, the indentation velocity of the contact ( ) = is stored for the contact pair when the simulation starts or when the compressive part of contact starts, if the simulation starts with the contact in the restitutive phase, and Eq. (188) is evaluated disregarding the term in parenthesis, while the compressive part of contact lasts, i.e., while the indentation velocity is positive. From the moment that contact reaches the maximum indentation onwards, i.e., from the moment =0 onwards, Eq. (188) is fully used with the dissipative term included.

Eq. (188) must be evaluated, at every time step and for each wheel-rail pair. The generalized stiffness coefficient K, for wheel-rail contact, is calculated as [76]:

4 =

3 h hw r

CK

A B

(189)

v

v

0

Flange contact

Tread contact

c)

v

0

v

Flange contact

Tread contact

b)

0

v

v

Flange contact

Tread contact

a)



The wheel and rail material parameters hw and hr, defined in reference [67], only depend on the elastic properties of the bodies and, therefore, are evaluated only once. The geometrical functions A and B are related with the principal curvatures of the wheel and rail surfaces at the contact point and are also defined in reference [67]. The parameter C is tabled in bibliography [76] being defined as a function of the ratio A/B. Therefore, at every time step during dynamic analysis, A and B are computed and the entries of this table are linearly interpolated in order to obtain the required value for C.

8.2.5. Tangential contact forces in the wheel-rail interface

Knowing the creepages and the normal contact forces, it is possible to compute the creep forces using one of the computer routines available in the literature. In this wheel-rail contact algorithm, the Kalker linear theory, Heuristic nonlinear model and Polach formulation are implemented as alternatives to each other. According with the Kalker linear theory [61,77] the longitudinal f and lateral f components of the creep force and the spin creep moment m that develop in the wheel-rail contact region are expressed as:

11

22 23

23 33

0 0

G 0

0

cf

f a b c ab c

m ab c ab c

(190)

where G is the combined shear modulus of rigidity of wheel and rail materials and a and b are the semi-axes of the contact ellipse that depend on the material properties and on the normal contact force. The parameters cij are the Kalker creepage and spin coefficients, obtained in references [61,77] and the quantities , and represent the longitudinal, lateral and spin creepages at the contact point, respectively. For sufficiently small values of creep and spin, the linear theory of Kalker is adequate to determine the creep forces. For larger values, this formulation is no more appropriate since it does not include the saturation effect of the friction forces, i.e., it does not assure that f N .

The Heuristic nonlinear model [64] involves the calculation of the creep force expected from the Kalker linear theory and its modification by a factor that takes into account the limiting creep force. First, the resultant creep force from Kalker linear theory is calculated as:

2 2 f f f (191)

where the notation (.) now means that the quantities are obtained with the Kalker linear theory. The limiting resultant creep force is determined by:

2 31 1

3 3 27

3

f f fN f N

f N N N

N f N

(192)

where is the friction coefficient. The new resultant creep force f is used to calculate the tangential forces as:

; f f

f f f ff f

(193)



In the Heuristic method the spin creep moment M is neglected. This theory gives more realistic values for creep forces outside the linear range than the Kalker linear theory. For high values of spin, the Heuristic theory can also lead to unsatisfactory results [71].

According to the Polach method [65], the longitudinal and lateral components of the creep force that develop in the wheel-rail contact region are expressed as:

; SC C C

f f f f f

(194)

where f is the tangential contact force caused by longitudinal and lateral creepages, C is the modified translational creepage, which accounts the effect of spin creepage, and fS is the lateral tangential force caused by spin creepage. The Polach algorithm requires as input the creepages , and , the normal contact force, the semi-axes of the contact ellipse, the combined modulus of rigidity of wheel and rail materials, the friction coefficient and the Kalker creepage and spin coefficients cij. The Polach algorithm allows the calculation of full nonlinear creep forces and takes spin into account.



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[42] P. Gupta, H.S. Cheng, D. Zhu, N.H. Forster, J.B. Schrand, Viscoelastic effects in MIL-L-7808-Type lubricant, Part I: analytical formulation, Tribology Transactions, 35(2), 269-274, 1992.

[43] W.R. Wilson, S. Sheu, Effect of inlet shear heating due to sliding on elastohydrodynami film thickness, Trans. ASME, J. of Lubrification Technology, 105, 187-188, 1983.

[44] P.E. Wolveridge, K.P. Baglin, J.F. Archard, The starved lubrication of cylinders in line contact, Proceedings of the Institution of Mechanical Engineers, 185, 1159-1169, 1970.

[45] B.J. Hamrock, D. Dowson, Ball Bearing Lubrication:The Elastohydrodynamics of Elliptical Contacts, John Wiley and Sons, New York, New York, 1981.

[46] I.V. Kragelskii, Friction and Wear, Butterworths, London, England, 1965.



[47] R.L. Smith, J.A. Walowit, J.M. McGrew, Elastohydrodynamic traction characteristics of 5P4E polyphenyl ether, Trans. ASME, J. of Lubrication Technology, 97, 353-360, 1973.

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[62] Kalker, J. J. "The Computation of Three-Dimensional Rolling Contact with Dry Friction", Numerical Methods in Engineering, 14, 9, pp. 1293-1307, 1979.

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10. ANNEX 1 – INPUT DATA FOR BEARING MODEL

10.1. General Bearing Data

Bearing.Type BType 1 Ball

2 Cylindrical

3 Spherical

4 Tapered

5 Spherical tapered

Bearing.NumberRollers BNumbRoller

Bearing.OuterDiameter do BOuterDiameter

Bearing.InnerDiameter di BInnerDiameter

Bearing.PitchDiameter dp BPitchDiameter

Bearing.Width L BWidth

(a) (b) (c) (d) (e)

Figure 1: Bearing types: (a) Ball; (b) Cylindrical; (c) Spherical; (d) Tapered; (e) Spherical tapered

(a) (b) (c)

Figure 2: Bearing general dimensions: (a) Cylindrical; (b) Spherical; (c) Tapered

Record 1 of the input deck:

Type; NumberRollers; OuterDiameter; InnerDiameter; PitchDiameter; Width

di do

L

dm

j

j

dm

di do

Lj

j

do

Lo

di

Li

dm

j

j



10.2. Spherical Bearing Geometry, Surface and Mass Data

Spherical.RollerDiameter Dd

Spherical.RollerLenght LbSpherical.RollerCrownRadius Rcr

Spherical.RollerCornerRadiusLeft RcoL

Spherical.RollerCornerRadiusRight RcoR

Spherical.InnerRaceRadius Rir

Spherical.OuterRaceRadius Ror

Spherical.RaceClearance tcSpherical.InnerRaceLenght LilSpherical.Tilt Spherical.RollerMass mbSpherical.RollerInertia Ib, Ib, Ib

Spherical.RollerSurfaceRoughness sbSpherical.InnerRaceMass miSpherical.InnerRaceInertia Ii, Ii, Ii

Spherical.InnerRaceSurfaceRoughness siSpherical.OuterRaceMass moSpherical.OuterRaceInertia Io, Io, Io

Spherical.OuterRaceSurfaceRoughness so

(a) (b) (c)

Figure 3: Spherical roller bearing: (a) Perspective view; (b) Bearing dimensions; (c) Roller dimensions

Record 2a of the input deck:

RollerDiameter; RollerLenght; RollerCrownRadius; RollerCornerRadiusLeft; RollerCornerRadiusRight


InnerRaceRadius; OuterRaceRadius; RaceClearance; InnerRaceLenght; Tilt


RollerMass; RollerInertia; RollerSurfaceRoughness; InnerRaceMass; InnerRaceInertia;

InnerRaceSurfaceRoughness; OuterRaceMass; OuterRaceInertia; OuterRaceSurfaceRoughness



10.3. Tapered Bearing Geometry, Surface and Mass Data

Tapered.RollerDiameterLargeEnd Dl

Tapered.RollerDiameterSmallEnd Ds

Tapered.RollerLenght LbTapered.RollerLandLenght LcTapered.RollerCrownRadius Rcr

Tapered.RollerEndRadiusLargeEnd RcoR

Tapered.RollerEndRadiusSmallEnd RcoL

Tapered.OuterRaceSemiConeAngle o

Tapered.InnerRaceSemiConeAngle i

Tapered.OuterRaceLandLenght loTapered.InnerRaceLandLenght liTapered.OuterRaceWidth LoTapered.InnerRaceWidth LiTapered.InnerRaceLandStart eiTapered.endplay ep If ep<0, the preload is reported here

Tapered.NumberRows Nrow If NRow<0, face‐to‐face mounting

Tapered.InterRaceSpacing Si If the bearing has two rows Si is used

Tapered.RollerMass mbTapered.RollerInertia Ib, Ib, Ib

Tapered.RollerSurfaceRoughness sbTapered.InnerRaceMass miTapered.InnerRaceInertia Ii, Ii, Ii

Tapered.InnerRaceSurfaceRoughness siTapered.OuterRaceMass moTapered.OuterRaceInertia Io, Io, Io

Tapered.OuterRaceSurfaceRoughness so

(a) (b) (c) Figure 4: Tapered roller bearing: (a) Perspective view; (b) Bearing dimensions; (c) Roller dimensions



Record 2b of the input deck:

RollerDiameterLargeEnd; RollerDiameterSmallEnd; RollerLenght; RollerLandLenght;

RollerCrownRadius; RollerEndRadiusLargeEnd; RollerEndRadiusSmallEnd


OuterRaceSemiConeAngle; InnerRaceSemiConeAngle; OuterRaceLandLenght; InnerRaceLandLenght;

OuterRaceWidth; InnerRaceWidth; InnerRaceLandStart; endplay; NumberRows; InterRaceSpacing


RollerMass; RollerInertia; RollerSurfaceRoughness; InnerRaceMass; InnerRaceInertia;

InnerRaceSurfaceRoughness; OuterRaceMass; OuterRaceInertia; OuterRaceSurfaceRoughness

(a) (b) (c)

Figure 5: Tapered roller bearing mounting: (a) Back to back with endplay, ep>0 and Nrow=+2; (b) Back

to back, ep≤0 and Nrow=+2; (c) Face to face, ep≤0 and Nrow=‐2.

10.4. Race flange geometry

Flange.OuterLeftAngle oL

Flange.OuterRightAngle oR

Flange.InnerLeftAngle iL

Flange.InnerRightAngle iR

Flange.OuterLeftHeight hoLFlange.OuterRightHeight hoRFlange.InnerLeftHeight hiLFlange.InnerRightHeight hiR

(a) (b) (c) Figure 6: Flanges in roller bearings: (a) Angle and height definitions; (b) Tapered roller bearing; (c)

Detail of the flanges in the tapered bearing

ep

LiLi Si LiLi Si Li LiSi

iL iR

oRoL

hoRhoL

hiRhiL

iL

iR

hoL=hoR=0

hiR

hiL

iR

hiRiL

hiL




OuterLeftAngle;OuterRightAngle; InnerLeftAngle;InnerRightAngle;

OuterLeftHeight;OuterRightHeight; InnerLeftHeight; InnerRightHeight

10.5. Cage Geometry, Surface and Mass Data

Cage.Guidance CGuide 0 No guidance

1 Outer race guidance

2 Inner race guidance

Cage.OuterDiameter Rou

Cage.InnerDiameter Rin

Cage.Width Wl

Cage.OuterRaceClearance CouCage.InnerRaceClearance CinCage.SemiConeAngle c

Cage.GuideLandRadiusRight RgR

Cage.GuideLandWidthRight WgR

Cage.GuideLandPositionRight LgRCage.GuideLandClearanceRight CgRCage.GuideLandRadiusLeft RgL

Cage.GuideLandWidthLeft WgL

Cage.GuideLandPositionLeft LgLCage.GuideLandClearanceLeft CgLCage.PocketShape PType 1 Cylindrical

2 Rectangular

3 Guided Surface

Cage.PocketLenght PlCage.PocketWidth PwCage.PocketDimension1 P1Cage.PocketDimension2 P2Cage.PocketDimension3 P3Cage.PocketDimension3 P4Cage.Mass mcCage.Inertia Ic, Ic, Ic

Cage.PocketSurfaceRoughness sc

(a) (b) Figure 7: Cage for roller bearings: (a) cylindrical, with semicone angle null; (b) Tapered. Note that a cage will not have both outer and inner guide lands.

Rou Rin

Wl

RgRRgL

CgR

CgL

WgRWgL

Cou

Cin

LgRLgL

Rou Rin

Wl

RgRRgL

CgRCgL

WgRWgL

LgRLgL

c



(a) (b) (c)

Figure 8: Cage pockets types: (a) Cylindrical; (b) Rectangular; (c) Guided surfaces


Guidance;OuterDiameter;InnerDiameter;Width;OuterRaceClearance; InnerRaceClearance;

SemiConeAngle


GuideLandRadiusRight; GuideLandWidthRight; GuideLandPositionRight; GuideLandClearanceRight;

GuideLandRadiusLeft; GuideLandWidthLeft; GuideLandPositionLeft; GuideLandClearanceLeft


PocketLenght; PocketWidth; PocketDimension1; PocketDimension2; PocketDimension3;

PocketDimension3


Mass; Inertia; PocketSurfaceRoughness

Pl

Pw

P3

P2P1 P3

P2

P4

P1

Maxbe D2.5 Axel Bearing Modelling and Analysis v 1maxbe/PU/wp2/MAXBE-D2.5 Axel... · 2014. 5....

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