Maksym Spiryagin - CQUniversity - Module 2: Contact mechanics

Contact Mechanics

Module 2

Dr Maksym Spiryagin, Centre for Railway Engineering, CQUniversity

Tuesday 20 May 2014

Brisbane

Module 2: Contact Mechanics

Contents

• Fundamentals

• Contact models and stresses

• Real contact processes – Creep forces at large creepages

• Contact models for rail vehicle multibody simulation

• Structure of a simplified contact model for a wheelset

Fundamentals

The wheel-rail contact problem can

be formulated as a rolling contact

problem between two nonlinear

profiles in the presence of friction.

This is a complex phenomenon,

which needs to be studied from the

point of view of mathematical-

analytical formulation and from the

numerical point of view.

Fundamentals The common way to solve a contact problem (adopted by

A.D. de Pater) is by subdivision into:

• Geometrical Problem: Wheel-Rail profiles coupling for the

identification of the location of contact points and of the geometrical

parameters of interest (local curvatures, etc.);

• Normal Problem: calculation of the constraint forces acting between

wheel and rail, evaluation of shape and dimensions of the contact

area and the corresponding pressure distribution;

• Kinematical Problem: determination of the condition of relative

motion between the wheel and rail, usually defined by the kinematic

creepages;

• Tangential Problem: calculation of the tangential forces generated by

friction and creepages in the contact area.

Fundamentals In the common case, the kinematical and

geometrical problems can be solved

analytically considering stylised profiles

(e.g., conical wheel on a cylindrical rail).

However, when considering real non-linear

profiles, the solution can be found only using

numerical methods.

The forces acting between wheel and rail,

generated by the contact constraints due to

the coupling of the two profiles, are strongly

influenced by the motion of the wheelset with

respect to the track.

z (

mm

)

y (mm)

Fundamentals

Constraint forces - Rigid contact

The wheel-rail interface is expressed by a set of algebraic equations that form a

bilateral constraint given by the equations (for a single wheelset):

wheelset roll

wheelset vertical

where the independent variables are the lateral displacement of the wheelset y,

the wheeset yaw angle ψ and the longitudinal coordinate along the track mean

line s.

where d is the distance between the profiles, yC is the lateral coordinate of the

contact point and q is the state vector of the system.

syf ,,1

syfz ,,1

0, CYydMin

C

q

Fundamentals

Constraint forces - Rigid contact

Advantages

• Easy to implement

• Good for generating bi-dimensional tables

Disadvantages

• the constraint is considered as bi-lateral, and therefore allows that traction

forces can act between wheel and rail preventing the lifting of the wheel even

in the case of track irregularities or other physical phenomena able to

realistically generate uplift.

• This formulation does not allow consideration of the case of double or multiple

point contact, and makes it difficult to apply it to the case or worn profiles.

Fundamentals

Constraint forces - Elastic contact

The wheel-rail constraint in the normal direction is presented though a single side

elastic contact element. The relative motion between the profiles originates areas

of possible intersection between the profiles, where a reaction force proportional

to the profiles intersection is applied. The normal force is given by:

Generally, the contact stiffness (KYc) is very high, is non-linear and depends on the

contact area, therefore it should be calculated at each time step.

0y,d if y,dK

0y,d if 0

CCy

C

,

Cqq

q

CyNF

Fundamentals

Constraint forces - Elastic contact

Advantages

• It allows working with worn profiles and the simulation of multiple contact

points (this obviously requires the definition of a normal force for each contact

point)

Disadvantages

• The contact stiffness (KYc) has a very high value and it is fully non-linear

because it is dependant on the contact area, so the calculation procedure is

time consuming.

• No table approximation can be adopted due to the independency of the state

variables ( 6 coordinates for wheelset position)

Fundamentals

Constraint forces – Quasi-static contact

This approach(proposed by G. Schupp, C. Weidemann and L. Mauer) allows the

simulation using simple algebraic equations of worn profiles and situations that

would produce multiple contact points.

where the distance function is weighted on the entire contact area [yc,max, yc,min]

and ɛ is a small positive parameter.

0 with 0

exp

lnmax,

min,

max,

min,

C

C

C

C

y

y

y

y

C

ds

dsyd q,

Fundamentals

Constraint forces - Quasi-static contact

Advantages

• Do not require a point by point function to define the profile distance

• Reduction of calculation time

Disadvantages

• It still has a problem with the prevention of wheel uplift

Fundamentals

Tangential Problem

Tangential forces arise due the relative motion between wheel and rail.

From early studies it was observed that the behaviour of a wheelset running on

the rail could not be considered as a “pure rolling” motion.

In fact, the evidence shows the motion is characterised by a “slow” sliding

phenomenon occurring at the contact. This phenomenon of small sliding has been

described as a pseudo sliding or micro-creepage or simply creepage.

The forces arising from this motion are therefore indicated as creep forces.

Fundamentals

Creepage

The term creepage is used to define the velocity differences in the longitudinal and

lateral directions as well as spin creepage due to yaw rotation, with the following

expressions for longitudinal, lateral and spin creepage:

(Longitudinal velocity of wheel – longitudinal velocity of rail) at the point of contactξ

Nominal velocity

(Lateral velocity of wheel – lateral velocity of rail) at the point of contact

Nominal veloci

ty

(Angular velocity of wheel – angular velocity of rail) at the point of contact

Nominal velocity

Fundamentals

Contact models – Carter’s theory

Coloumb Friction Force [1] Wheel-Rail Friction Force [1]

F = μ ⋅ N F = c⋅ ξ

where c is the coefficient of

proportionality in the elastic region

NrAc

Fundamentals

Contact models – Carter’s theory

The total tangential forces exchanged between wheel and

rail are calculated by integrating the tangential stress on

the entire contact area.

The shear stress τ can be calculated as the difference

between two circles, the first with a diameter equal to the

size of the contact area (Γ1) and the second with a radius

a2 (Γ2), which varies depending on the creepage.

The main limitations of the Carter’s theory:

• it is a mono-dimensional theory that is not suitable for

the study of lateral dynamics of vehicles

• the value of the coefficient of proportionality in the

elastic region has been incorrectly calculated

Tangential stress behaviour

according to Carter’s theory [1]

Fundamentals Contact models –

Vermeulen and Johnson, Haines and Ollerton

Vermeulen-Johnson’s model introduces the case of elliptical contact areas in the presence of

lateral and longitudinal creepages (neglecting the spin).

The pressure distribution in this model was such as to predict that the adhesion area was

also elliptical, as for the contact area, and tangential to the contact area at a single point

corresponding to the leading edge.

Haines and Ollerton’s method shows a more accurate distribution for tangential stresses for

the case of pure longitudinal creepage; this method, the strip theory, allowed them to obtain

the traction force by integrating the tangential stress using strips parallel to the creepage,

starting from the leading edge (where new particles of the wheel enter into the contact area)

and assuming slip at the trailing edge (τ=µp).

Fundamentals

Contact models – Kalker’s theories

Kalker investigated the problem using limitations

as low as possible, considering an elliptical

contact area with the simultaneous presence of ξ

(longitudinal), η (lateral) and ϕ (spin) creepages.

During his activity he developed several theories

and algorithms that can be considered as the

knowledge-base of the modern wheel-rail contact

theories.

First, he has described an exact analytical

method to calculate the contact forces in the

linear portion of the force-creepage curve. This is

known as Kalker’s linear theory.

Comparison between

the tangential stress distributions

obtained by different theories [1]

Formulation of the problem

With given wheel and rail profiles and vertical load, to determine traction:

and in particular

and it satisfies

where w is the local creepage and ẇ is the global slip.

),,(p zyx ppp

( , ) ( , )x y x y

contact

F F p p dxdy

0,||

)(

0,)(),(

ww

wpwp

wpwppp

z

zyx

Contact models and stresses

Tangential Forces

The total tangential forces, Fx and Fy, and the twisting moment, Mz, can be

obtained as follows [2]:


Kalker’s linear creep theory

This model introduces the case of elliptical contact areas in the presence of lateral and

longitudinal creepages.

where G is the modulus of rigidity and C11,C22, C23 and C33 are coefficients known as Kalker’s

coefficients and can be calculated as functions of the a/b ratio and of the Poisson’s module ν.

3323

2322

11

0

0

00

CbaCba

CbaC

C

baG

M

F

F

z

y

x


Kalker’s exact theory

This theory has no restrictions for creepage and spin. However, it is confined to Hertzian bodies

and an assumption of quasi-identity. For the exact theory [2]:

1 2 3

11 22 23

8 8 /, ,

3 3 4

a a a a bL L L

C G C G C G

2

11

1

2 31.5

22 23

2 3

8

3

8( )

3 4

x

y

a bF abGC

L

a b a bF abGC ab GC

L L


Kalker’s simplified theory

Using Hertz theory determines contact area in the simplified theory and the elastic body is

replaced by a set of springs.

Therefore, the simplified theory is based on a relaxation of the elastic relations between surface

deformations and tangential stresses, which is given by a single equivalent flexibility parameter

instead of the three parameters used in the exact theory. For the simplified theory [2]:

The value of flexibility parameter, L, is dependent on the the modulus of rigidity G, the semi-axes

of the contact ellipse a and b, and the creepages.

2

2 3

8 / (3 )

8 / (3 ) / (4 )

x

y

F a b L

F a b L a b L


Kalker’s simplified theory

Calculation algorithm:

Step 1

Determine tangential force with rigid slip

Step 2

Calculate slip

Step 3

Calculate tangential forces with slip

Step 4

Make a comparison

If we get a desired error, we have found a solution.

If we cannot get desired conditions then we need to go back to step 1.

The previous slip value can be used as the initial value to replace the rigid slip values in step 1.

( , ) ( ), ( , ) ( )xi z yi zp x x y p y p x x y p x

( , ) ( , )( , ) ( , ),

y yix xix y

p x y p x x yp x y p x x ys y L s x L

x x

( , ) ( ), ( , ) ( )xs z x ys z yp x x y p s p x x y p s

| | , | |xs xi ys yip p p p


Traction distribution for pure longitudinal creepage

Simplified theory [2] Exact theory [2]


Kalker’s programs:


FASTSIM

This algorithm is based on the simplified theory. The contact area is discretised into several

strips. The solution is obtained by numerical strip integration over the contact area which is

dimensionalised and discretised over a small number of elements (usually 20x20).

CONTACT

This is a program which refers to the exact theory. It is also based on discretisation of the

contact area but it covers a wide range of contact problems:

• Normal contact problem;

• Non-steady state rolling;

• Steady state rolling;

• Combination of normal and tangential problem;

• Contact bodies are elastic and visco-elastic materials.

Real contact processes –

Creep forces at large creepages

For the constant friction coefficient, the

friction force increases to the saturated

value when the slip occurs over the whole

contact area.

However, such an approach cannot be

used for the modelling of the real behaviour

because the static friction coefficient is

largely affected by material properties and

surface conditions.

Modelling of creep force characteristic

using falling friction coefficient and

different reduction factors k.

There is disagreement between the modelled

creep force characteristic shape and its typical

shape from measurements on wet rail. [3]

0

0.1

0.2

0.3

0.4

0 1 2 3 4 5 6 7 8 9 10

sx [%]

f x [

- ]

k = 1.0 k = 0.2 k = 0.05

Longitudinal creepage

Ad

he

sion

co

effi

cie

nt

Typical measurement

A

B

Contact models for rail vehicle multibody simulation

Algorithms

• Kalker’s linear theory

• FASTSIM

• Pre-calculated tables

• Shen-Hedrick-Elkins

• Polach

• Modified versions of FASTSIM


Shen-Hedrick-Elkins Heuristic method, which describes creep force law based on two theories[3].

From Kalker’s theory, it needs to evaluate

After that, using Vermulen and Johnson theory

Creep-force reduction coefficient

Creep force law

NFN

NFN

F

N

F

N

FN

F

r

rrrr

r

3,

3,27

1

3

132

r

r

F

F

22

yxr FFF

zz

yy

xx

MM

FF

FF


Polach (2005) Heuristic method, which applies the variable friction

coefficient and also represents the change of creep

force curves with increasing creep due to

nonlinearity of the tangential contact flexibility.

Polach algorithm [4] is based on his simplified

model developed for fast calculation purposes. The

model provides reliable results, but differences can

be observed compared with the exact theory and

the Fastsim code [5].

2

2

11

2 2

2arctan( ) , 1

1 ( )

2 1

3 4

2

, ,

, ,

AS S A

A

x

x y

A S

ii

ii

kQF k k k

k

G abkCC a bs s

Q Q

s s s

k kk

ws i x y

V

sF F i x y

s

Contact models for rail vehicle multibody simulation Modified versions of FASTSIM Generally, in order to find a solution in agreement with

measurements, it is necessary to introduce a variable

friction coefficient obtained from experimental

investigations.

Kalker and Piotrowski proposed two different coefficients

of friction, one in the area of adhesion and the other in

the area of slip, but this did not show a significant

difference in the results.

A slip velocity dependent friction coefficient, reducing

with increasing slip, has been implemented in Fastsim

by different researchers, however, the models

developed cannot represent the creep force

characteristics at large creep in agreement with the

typical measurements on locomotives and other traction

vehicles.

Comparison of calculated results

with measurements for locomotive

SBB 460 [6]


Modified versions of FASTSIM - Spiryagin et al. (2013)

This modelling methodology for computation of creep forces using the modified Fastsim

code, where the constant Kalker’s reduction factor by a variable reduction factor together

with a slip velocity dependent friction coefficient decreasing with increasing global

creepage [6].

The proposed variable stiffness reduction factor

where k0 initial value of Kalker’s reduction factor at creep values close to zero, 0< k0≤1;

αinf fraction of the initial value of the Kalker’s reduction factor at creep values approaching infinity, 0 ≤ αinf ≤ 1;

β non-dimensional parameter related to the decrease of the contact stiffness with the increase of

the slip area size, 0 ≤ β;

ε parameter describing the gradient of the tangential stress in the stress distribution transformed to a hemisphere

according to Polach.

Finally, the contact flexibility coefficient L defined by Kalker is increased and the new value

L* is calculated as:

inf0 inf

1

1k k

k

LL *



Example of the application for locomotive 12X [7]

Vehicle 12X

Wheel–rail conditions Wet Wet

Speed (km/h) 20 60

Figure Fig.4 Fig. 5

V (km/h) 20 60

k0 0.455

αinf 0.016 β 0.685

0.28

A 0.4

B (s/m) 0.4

S

Calculated dependence of adhesion

coefficient from longitudinal creepage and

vehicle speed [6]


Discussion

• How to model a friction coefficient ?

• How to verify it?

(no measurement technique for the determination of local slips currently exists)

• Variable contact flexibility has no explicit physical interpretation.

The flexibility depends on the asperities and the effects of contamination which convert it

to a non-steady condition, i.e., the process becomes a non-linear one.

The values of reduction for the contact flexibility coefficient can only be assumed from

experimental data because no value for the flexibility parameter is directly available, and

this is a reason why some parameters in the developed model are not related to a physical

meaning.


Structure of a simplified contact model for a wheelset

References [1] N. Bosso, M. Spiryagin, A. Gugliotta, A. Soma, Mechatronic modeling of real-time

wheel-rail contact. Springer, 2013. ISBN 978-364-236-245-3.

[2] J.J. Kalker, Rolling contact phenomena: linear elasticity. Internal report. Delft

University of Technology, Faculty of Electrical Engineering, Mathematics and Computer

Science, Delft Institute of Applied Mathematics, 2000.

[3] Z.Y. Shen, J.K. Hedrick, J.A. Elkins, A comparison of alternative creep-force models

for rail vehicle dynamic analysis. The Dynamic of Vehicles on Roads and Tracks,

Proceedings of 8th IAVSD Symposium, MIT, Cambridge, MA, 1983, pp.591-605.

[4] O. Polach, Creep forces in simulations of traction vehicles running on adhesion limit,

Wear, Vol. 258, 2005, pp. 992-1000.

[5] E. A.H. Vollebregt, S. D. Iwnicki, G. Xie, P. Shackleton, Assessing the accuracy of

different simplified frictional rolling contact algorithms, Vehicle System Dynamics, Vol. 50,

2011, pp. 1-17.

[6] M. Spiryagin, O. Polach, C. Cole, Creep force modelling for rail traction vehicles based

on the Fastsim algorithm, Vehicle System Dynamics, 51(11), 2013, pp. 1765-1783.

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Maksym Spiryagin - CQUniversity - Module 2: Contact mechanics