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The longitudinal creep on a crane wheel has considerable effects, both for the design and the control of the drive technology of a modern crane. The creep values occurring during operation are, however, widely unknown. This article presents a calculation model for longitudinal creep of a driven crane wheel, which can be used for a fast analytical determination. Based on the contact area between the crane wheel and the rail, instead of the complex real point contact situation an equivalent line contact is calculated. The approach was validated at the Institute's wheel-rail test rig.

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Proceedings of the XXII International Conference MHCL' 17

Georg Havlicek

Project Assistant

Vienna University of Technology

Faculty of Mechanical and Indust rial

Engineering

Institute for Engineering Design and

Logistics Engineering

Stefan Krenn

Scientist

Excellence Centre of Tribology

AC²T research GmbH

Georg Kartnig

Professor

Vienna University of Technology

Faculty of Mechanical and Indust rial

Engineering

Institute for Engineering Design and

Logistics Engineering

Determination of the longitudinal

creep of a driven crane wheel on a

crowned rail

The longitudinal creep on a crane wheel has considerable effects, both for

the design and the control of the drive technology of a modern crane. The

creep occurring during operation is

, however, widely unknown. This

article presents a calculation model for

longitudinal creep of a driven

crane wheel, which can be used for a fast analytical determination. Based

on the contact area between the crane wheel and the rail an equivalent

line contact is calculated

instead of the complex real point contact

situation. The approach was validated at the institute's wheel-rail test rig .

Keywords: Longitudinal creep, crane wheel, rail-wheel-contact, analytic

method, running-in characteristic

1. INTRODUCTION

Currently applicable standards for crane design

(DIN EN 13001 and sub- standards [1]) consider only rail

geometries with flat heads at the contact between crane

wheel and rail. Nowadays, crowned rails are used almost

exclusively. As a result of the wear-specific run-in

behavior, a change in the rail head profile occurs with

increasing operating time (overrun cycles). This also

results in a change of the initially ideal elliptical contact

surface to an approximate line contact geometry, but not

over the entire width of the railhead.

For this reason, a research project was launched in

2014 at the Department of Transport, Handling and

Conveying Systems (KLFT) in cooperation with Hans

Künz GmbH, which aims at transferring existing

approaches of the static and structural design for the flat

rail head to the general case of a cambered rail. The

effects of the modified rail geometry on relevant system

parameters, e.g. contact pressure, rolling friction,

skewing forces, longitudinal creep, adhesion and wear of

wheel and rail are to be investigated. Furthermore, the

characteristics of the run-in behavior of the rail head due

to plastic deformation are considered in the context of the

project.

In 2016 the Excellence Center of Tribology (AC²T)

was included in the project to achieve a better

understanding of the tribological aspects with in the

contact surface.

In this publication, the longitudinal (tangential) slip

between a driven or braked crane wheel and a cambered

rail is to be described in more detail.

For various reasons it is necessary to know the

occurring slip:

• The maximum transferable braking and driving

force depends on the slip ratio (i.e. traction).

• The utilization of the coefficient of static friction

is associated with higher slip ratios and higher

wear.

• Different wheel loads on the individual wheels

result in different slip ratios. Production

tolerances on the wheel diameters also result in

different circumferential forces on the wheels and

thus deviating slip values. The crane clamps or

distorts itself according to different wheel speeds.

However, actual slip values at the crane wheel during

operation are widely unknown. Therefore an application-

oriented and quick-to- calculate analytical approach for

the longitudinal slip of current cranes is desirable.

When considering slippage a fundamental distinction

between micro- slip (creep) and macro-slip (sliding)

needs to be done. In the case of creep, the contact surface

is subdivided into a stick zone with the same speed and a

slip zone with a relative speed between the contact

partners. The coefficient of static friction is the limiting

factor for the tangential force that can be transmitted. If

the slippage becomes larger, the stick zone disappears

and the slip zone extends over the entire contact surface.

From this point on there is pure sliding (macro-slip), and

the coefficient of sliding friction is decisive (see Figure

1). On a driven or braked wheel, macro - slip corresponds

to wheelspin, which in principle is to be avoided in crane

construction. All slip curves considered in this work

concern the micro-slip region.

Figure 1. Traction-creep-relation (qualitative)

Dipl.-Ing. Georg Havlicek

Technische Universität Wien

9/301-7 , 1060 Vienna, Austria

-mail: georg. havlicek @ tuwien.ac.at

Proceedings of the XXII International Conference MHCL' 17

2. EXISTING CALCULATION APPROACHES

The following calculation approaches are only

described with respect to the tangential slip ratio. Axial

slip and spin (rotation around the ve rtical axis) are not

considered in this work, as far as the calculation methods

are concerned. Furthermore, a contact geometry between

a crane wheel and a crane rail according to DIN 536 has

been given [2].

The tangential slip is generally defined as a related

velocity difference between the circumferential speed of

the wheel and the absolute velocity.

(1)

2.1 Calculation approach for line contact according

to Carter

For the slip-bearing contact of a cylinder with a plane,

relationships for longitudinal creep were derived by F.W.

Carter [3] as early as 1926. T hese were again completely

elaborated by G. Heinrich and K. Desoyer [4] and

extended to incorporate lateral slip effects.

The following relationship for the longitudinal creep

is obtained as a function of the contact force FR , the

circumferential force at the wheel FT , the wheel radius R ,

the contact width in the contact bK , the coefficient of

static friction μ 0 and material constants G and ν . [4]











−−

−

F μ

(2)

Figure 2 shows the contact area as well as the shear

stress distribution in the contact surface between the

rolling cylinder and the plane under radial load and

transmission of a torque.

Figure 2. Contact between cylinder and plane

Figure 3. Distribution of shear stress and division in slip

and stick zone in contact

Along the contact length lK , the division into the slip

and stick zone as well as the shear stress distribution as

shown in Figure 3 are obtained.

The last term of Equation (2) corresponds to the

proportion of the stick zone over the entire contact length

and is a determining parameter for the creep.

(3)

The relationship between the contact length and the

contact width for a line contact according to Hertz is

included in Equation (2).

(4)

Thus, the formula can be rewritten to use the length

instead of the width of the contact surface. Along this

length the division into stick and slip zone is also

determined.











−−=

F μ

(5)

The correctness of the approach was confirmed by J.J.

Kalker amongst others using numerical methods [5].

For the contact between a crane wheel and a flat rail

head, Equations (2) or (5) can be used right away. Certain

deviations to the real contact situation between the wheel

and the rail are to be expected since the approach does

not take any edge effects at the boundary surfaces of the

cylinder into account .

2.2 Calculation approaches for point contact

Approaches for calculating the slip conditions at point

contact, as occurs with a cambered rail head, are not

trivially solved . The transmitted tangential force is not

constant over the width of the contact surface (here the

long semi-axis of the contact ellipse). The distribution of

the shear stress and the separation in the slip and stick

zone of the contact surface is shown in Figure 4.

Figure 4. Shear stress distribution at point contact

(qualitative)

Strip theory by Haines and Ollerton

The strip theory is a pure analytical calculation

method, in which the elliptical contact surface is divided

into thin strips parallel to the rolling direction and then

integrated. Each individual strip is treated as a line

contact, but any influence of the strips on each other is

Proceedings of the XXII International Conference MHCL' 17

neglected. The approach of B.J. Haines and E. Ollerton

for pure tangential creep was elaborated by J.J. Kalker

and later developed further in order to be able to take

transverse creep and a small proportion of spin into

account [6, 7, 8]. Figure 5 shows the discretization of the

contact surface and the shear stress distribution in the

strip.

Figure 5 . Strip theory according to Haines and Ollerton

The relationship between tangential force and creep

is defined by:

()

( )





















−+

−

−2

cos ζζ

μ F

(6)

with the factor ζ as a function of the slip ξ T and the

Hertzian pressure p 0 in the contact.

(7)

The results of this calculation method correspond

very well with experimental results and numerical

methods for slender contact areas (half-axis ratio

a/b ≈ 0.2). If, however, the contact surfaces deviate

severely from this shape, the errors become large due to

the lack of influence of the strips on each other [7].

The theory was not pursued further after the

development of the simplified theory of Kalker in favor

of the more exact numerical calculation.

Numerical methods according to Kalker

Calculation models of the exact and simplified theory

developed by Kalker can only be solved numerically.

They are implemented in the contact models of the

programs CONTACT (exact) and FASTSIM

(simplified). Both models divide the contact area into

rectangular parts, which must be balanced in relation to

the stress state over the entire contact surface. The exact

Kalker theory provides accurate results. At pure

tangential stress on the contact the simplified theory

deviates by up to 5% [9].

For more information on the numerical methods

according to Kalker, see [5] and [10].

Linear method according to Kalker

This theory uses the numerically determined Kalker

coefficients for the relation between slip and tangential

stress. These are defined in tabular form as a function of

the half- axis lengths of the contact ellipse and the Poisson

ratio. Interpolations are necessary for intermediate

values. The linear theory is applicable only for very small

slip values since the existence of a slip zone is neglected

in this approach. It represents the slope of the linear

branch of the creep curve from the origin. Larger slip

values caused by the influence of the increasing slip zone

are not reproduced correctly. The deviation of the linear

theory from the real creep curve is shown qualitatively in

Figure 6.

Figure 6. Discrepancy linear theory and real creep curve

According to Kalker's linear theory the dependence

of the tangential force on the longitudinal creep is defined

via the following relation:

(8)

For the pure ly tangential contact problems, only the

Kalker coefficient C11 is required. (For values see [5].)

3. CALCULATION APPROACH FOR THE ACTUAL

CONTACT GEOMETRY

The actual contact geometry between the crane wheel

and the rail does not correspond to an ideal point contact

after a short operating time of the crane system. Due to

plastic deformation, the curvature of the rail head

changes until the stresses inside the rail no longer exceed

the yield point.

Imprints recorded using pressure measuring films

already show run-in behavior on a new crane during the

commissioning phase, which leads to a leveling of the rail

head. Due to the limited measuring range of the Fujifilm

Prescale films, the occuring contact pressures cannot be

evaluated, but they provide very good information about

the shape of the contact surface. Figure 7 shows the

measuring film on the crane rail after loading by the crane

wheel. Figure 8 shows the resulting imprint on the film.

Proceedings of the XXII International Conference MHCL' 17

In addition, the calculated contact ellipse of an ideal point

contact is superimposed. Since the crane has was moved

across the film, the exact contour of the contact area is

not visible, but the contact width is significantly greater

than the result according to Hertz's theory.

Figure 7. Fujifilm pressure measurement film on crane rail

Figure 8. Imprint of the contact surface and theoretical

contact ellipse

The contact width stabilizes after a certain time, so

that even in the case of cranes that have been in operation

for years, the contact surface does not extend over the full

width of the rail. As an example of this a photograph of a

crane rail after approximately seven years of operation is

shown in Figure 9.

Figure 9. Run-in crane rail

Similar running-in behavior also takes place at the

test rig at the Institute for Engineering Design and

Logistics Engineering at the Vienna University of

Technology described in more detail in S ection 4. After

approximately twenty operating hours with maximum

wheel load the camber of the rail-wheel is flattened to a

permanent geometry. Figure 10, taken at a load of 50 kN,

compares a recorded imprint to the theoretical contact

ellipse according to Hertz.

Figure 10. Run- in contact area (dimensions in mm)

For the calculation of the creep ratios for such real

contact surfaces, the following approach uses the method

for line contact according to Carter as well as Heinrich

and Desoyer. This is adapted in such a way that a

fictitious linear contact is calculated in which the average

tangential stress in the contact surface coincides with the

actual contact situation.

For pure tangential slippage the length in the direction

of movement (lreal ) has been identified as a determinant

measure for the size of the contact surface. If the width

of the fictitious contact surface is calculated according to

the Hertzian theory for line contact as a function of the

contact force and this contact length, the same surface

areas as in the case of the real contact surfaces are

obtained. Figure 11 shows two imprints, on the right a

run-in rail, on the left in a new condition. The rectangle

drawn corresponds to the fictitious contact surface used

for slip calculation.

Figure 11. Dimensions of the contact areas (left: new, right:

run-in)

This correspondence of the surface areas was only

checked for the field of application and the geometries of

crane wheels and rails. In the case of strongly divergent

forms of the contact surfaces, the validity of this

relationship would have to first be verified.

The essential procedure for determining the slip of a

wheel according to this method is as follows:

When considering a new crane, or assuming that the

wheel loads are sufficiently low to prevent plastic

deformation, the required contact length lreal can be

determined by the Hertzian theory. The half-axis length

of the contact ellipse in the direction of motion can be

used as a good approximation for the contact length.

If the slip on a run-in crane system is to be calculated,

a determination of the real contact area is necessary. How

and to which geometry a crane rail runs in is part of the

investigations at the Vienna University of Technology,

but the contact surface cannot yet be estimated after

plastic deformation. A measurement by means of

pressure measurement films represents a simple and

favorable solution. The crane wheel to be evaluated is

lifted by means of hydraulics and put back on the rail

after the film has been placed. After two minutes of

exposure, the film is removed again and the impression

can be measured directly. Figure 12 shows an imprint on

a Fujifilm Prescale film.

Proceedings of the XXII International Conference MHCL' 17

Figure 12. Fujifilm Prescale pressure measurement film

with imprint

After determining the contact length lreal , the creep

curve can be computed without difficulty with Equation

(5) from Section 2.1 :











−−=

real

F μ

(9)

Additionally required factors for the calculation are

the radius of the wheel R , the coefficient of static friction

between wheel and rail μ 0 (according to DIN 13001-3- 3

in the range of 0.1 to 0.3), the wheel load FR , and the

tangential force FT to be transmitted. A requirement for

the validity of the approach are identical elastic

properties of the wheel and rail materials (modulus of

elasticity and Poisson's ratio).

The division of the contact surface into the slip and

stick zone, which is decisive for the slip, is determined

on the basis of the contact length used, and is assumed to

be constant for the entire contact width.

For a common configuration of a portal crane in the

new state, the creep curves shown in Figure 13 result

from this approach.

Contact details:

Radius of the crane wheel R = 315 mm

Radius of the railhead RS = 500 mm

Coefficient of static friction μ 0 = 0.3

Radial forces (wheel loads) FR = 25, 50, 75 and 100 kN

Tangential forces FT = 0 to 10,000 N

The dimensions of the real contact surface in this case

are calculated using the Hertzian theory for two generally

curved bodies (see [11]).

Figure 13. Creep over tangential force at various wheel

loads

The adhesion limit is reached with the assumed

coefficient of static friction of μ 0 = 0.3 and a wheel load

of 25 kN at FT = 7500 N. The creep values go to infinity

with a tangential force FT > μ 0 FR . The real elliptical and

the fictitious rectangular contact areas were calculated

for comparison and plotted in Table 1.

Table 1 . Dimensions of the contact areas

Radial

force

contact

Hertzian

area elliptic

fictitious

Deviation

25 kN 7 mm 52 mm² 52.1 mm² +0. 4%

50 kN 8.8 mm 8 2.5 mm² 82.8 mm² + 0.4%

75 kN 10.1 mm 108 mm² 108.5 mm² + 0.4%

100 kN 11.1 mm 130.9 mm² 131.4mm² +0.4%

The results of this simplified analytical calculation

approach are to be validated in the following section with

measurements on a wheel-rail test stand.

4. DETERMINATION OF TANGENTIAL CREEP AT

THE TEST STAND.

In order to examine the running behavior of crane

wheels on a cambered rail, a test stand was developed and

built in cooperation with Künz. Test stands of this type

were already used in the 1970s and 80s to research the

flat rail head. The wheel-rail test rig at the KLFT (Figure

14) consists of a rail bent to a circular ring (rail- wheel)

and a crane wheel. Both wheels can be independently

driven and braked. The contact force of the wheel can be

specified via hydraulic cylinders. On both drive units

there are incremental encoders to detect the exact

position of the wheels. The rail-wheel has a diameter of

2000 mm and a head shape corresponding to a rail o f the

form A55 according to DIN 536, while the crane wheel

has a diameter of 400 mm.

Figure 14. Wheel- rail test stand at KLFT, TU Wien

In the case of slip measurements, the crane wheel is

driven without power limitation, and the rail-wheel

brakes with a defined torque. After precise determination

of the diameter ratio, the rotational angle difference

between the wheel and the rail is used to calculate the

creep values after a defined number of revolutions.

Beforehand, the contact area is determined using Fujifilm

pressure measuring films for each load step.

0,1

0,2

0,3

0,4

0,5

02000 4000 6000 8000 10000

Creep (%)

Tangential force (N)

25 kN 50 kN 75 kN 100 kN

Proceedings of the XXII International Conference MHCL' 17

The measurements were carried out at various

conditions of the rail as well as at various friction values.

In order to influence the coefficient of friction between

the wheel and the rail, conditioning agents, also in

combination with water, were applied to the rail surface.

Since a flattening of the rail head radius occurs on the test

stand, in the same way as for a real crane rail,

measurement runs were carried out in the run-in state

first. After completion of the measurements in the run-in,

realistic state, the rail-wheel was re-profiled and the rail

geometry corresponding to the new state was restored. In

this state a real point contact with an elliptical contact

surface can be measured at low radial forces. These

measurements were used in order to be able to validate

the analytical approach also with this contact geometry.

2.3 Measurements in the run-in state

The measurements were carried out at wheel loads

(radial forces) of 25 to 100 kN. At each force, the braking

torque was varied between 250 and 1750 Nm, which

corresponds to a tangential force in the contact area of

1250 to 8750 N. For all combinations of tangential force

and wheel load, at least three valid measuring points were

recorded and averaged.

The evaluation of the contact areas shows that the

contact width increases only slightly in the run-in state.

The radius of curvature has flattened in the center of the

head of the rail, so the actual contact ellipse is no longer

recognisable. Figure 15 shows the contact surfaces for

the radial forces of 25 to 100 kN. The fictitious

rectangular contact surfaces of the analytical approach

are already drawn.

Figure 15. Contact areas on test stand at 25 to 100 kN

wheel load (dimensions in mm)

After analyzing the pressure measuring films in the

program GODAV the surface areas of the real and

fictitious contact surfaces can be compared.

Table 2. Dimensions of the contact areas on the test stand

force

lenght

area

contact area Deviation

25 kN 4.5 mm 52.5 mm² 51.3 mm² - 2.2%

50 kN 6 mm 86 mm² 77 mm² - 10.4%

75 kN 7 mm 108 mm² 99 mm² - 8.3%

100 kN 7.5 mm 127 mm² 123.2 mm² -3.0%

Figure 16 compares the results of the measurements

in the run-in, dr y, unconditioned state with the analytical

approach. The drawn creep values were averaged from

five measurement runs.

Figure 16. Creep over tangential force, measurement and

analytical approach (dry, run- in state)

The slip curves end at the maximum torque that can

be applied via the rail wheel drive. The coefficient of

static friction between wheel and rail was determined as

μ0 ≈ 0.4 in previous measurements. It exhibits very good

agreement between measured and analytically calculated

creep values.

The dependence of the creep curves on the coefficient

of friction between wheel and rail is shown in Figure 17

where the results in the conditioned state are plotted.

With the aid of solid lubricants, the coefficient of

static friction was reduced from μ 0 ≈ 0.4 to μ 0 ≈ 0.23 and

in further measurements to μ 0 ≈ 0.1. Again, one can see

the very good compliance between analytically

calculated and measured slippage.

For a contact force of 25 kN, the measured curves of

the creep for the different friction values are drawn. For

the two lower friction values, the maximum transmittable

force of FT = μ 0 ∙ FR is reached. When the maximum

tangential force is approached, the measured slip

becomes infinite and can no longer be determined with

the available measuring equipment. At this point, the

transition from creep to sliding occurs.

Figure 17. Creep over tangential force, measurement and

analytical approach at 25 kN (conditioned, run- in state)

0,1

0,2

0,3

0,4

0,5

02000 4000 6000 8000 10000

Creep (%)

Tangential force (N)

Measurement 25 kN Analytic 25 kN

Measurement 50 kN Analytic 50 kN

Measurement 75 kN Analytic 75 kN

Measurement 100 kN Analytic 100 kN

0,1

0,2

0,3

0,4

0,5

02000 4000 6000 8000 10000

Creep (%)

Tangential force (N)

Measurement µ≈0,4 Analytic µ≈0,4

Measurement µ≈0,23 Analytic µ≈0,23

Measurement µ≈0,1 Analytic µ≈0,1

Proceedings of the XXII International Conference MHCL' 17

2.4 Measurements in new condition

The measurements with actual point contact were

carried out only at 25 and 50 kN radial force, since at

higher loads the plastic deformation in the rail material

leads to the running-i n of the geometry. Figure 18

compares the measured curve of the creep with the

calculation approach for a dry rail, and in Figure 19 the

corresponding contact areas are shown.

Figure 18. Creep over tangential force, measurement and

analytical approach (dry, new condition)

Figure 19. Contact areas on test bench at 25 and 50 kN

wheel load in new condition (dimensions in mm)

These measurements prove that the chosen approach

also provides good results for point -shaped contact

geometries.

5. Summary and Outlook

The approach of Carter for line contact is successfully

applied to a point- shaped contact geometry. The

assumptions made, using an average tangential stress

over the surface, as well as a constant division in stick

and slip zone over the entire contact width, do not seem

to have any significant effect on the creep. Compared

with the measurements at the test rig, very good

agreement with the analytical calculation approach is

achieved both for various friction values as well as for

various contact geometries. In the case of a known

contact area, the creep curve can also be determined for

a run-in state in a fast manner without long computing

times.

In further ongoing studies, the consistency of the slip

ratios at the fictitious and the real contact surface is

examined more precisely by means of finite element

methods. Due to the wider possibilities of variation of the

contact geometry in the FEM calculations, the validity

limits of the analytical approach can also be determined.

ACKNOWLEDGMENT

Significant parts of this work were funded by the

"Austrian COMET-Programme" (Project XTribology,

no. 849109) and were developed in collaboration with the

"Excellence Centre of Tribology" (AC²T research

GmbH).

REFERENCES

[1] DIN EN 13001-3: Krane – Konstruktion allgemein

– Parts 3 -1 to 3-3.

[2] DIN 536-1: Kranschienen Form A.

[3] Carter, F.W. "On the action of a locomotive driving

wheel." Proceedings of the Royal Society of London

A: Mathematical, Physical and Engineering

Sciences. Vol. 112. No. 760. The Royal Society,

1926.

[4] Heinrich, G., and K. Desoyer. "Rollreibung mit

axialem Schub." Ingenieur-Archiv 36: 48-72, 1967.

[5] Kalker, J.J. Rolling contact phenomena. Springer,

Vienna, 2000.

[6] Haines, D.J., and E. Ollerton. "Contact stress

distributions on elliptical contact surfaces subjected

to radial and tangential forces." Proceedings of the

Institution of Mechanical Engineers 177: 95- 114,

1963.

[7] Kalker, J.J. "A Strip Theory for Rolling With Slip

and Spin. I.-IV." Koninklijke Nederlandse

Akademie van Wetenschappen-Proceedings Series

B-Physical Sciences 70: 10 -62, 1967.

[8] Johnson, K.L. Contact mechanics. Cambridge

University Press, 1987.

[9] Vollebregt, E.A.H., and P. Wilders. "FAST-SIM2:

a second-order accurate frictional rolling contact

algorithm." Computational Mechanics 47: 105- 116,

2011.

[10] Kalker, J.J. Three-dimensional elastic bodies in

rolling contact. Springer Science & Business

Media, 1990.

[11] Timoshenko, S., and J.N. Goodier. Theory of

Elasticity. McGraw-Hill book Company, Third

Edition, 1970.

NOMENCLATURE

Half- axis of the contact ellipse in the

direction of movement (half the contact

length)

Proportion of the stick zone of the

contact length

fict

Fictitious contact width for creep calculation

Contact width (width of the contact area)

real

Real (measured) contact width

Half length of the stick zone

0,1

0,2

0,3

0,4

0,5

02000 4000 6000 8000 10000

Creep (%)

Tangential force(N)

Measurement 25 kN Analytic 25 kN

Measurement 50 kN Analytic 50 kN

Proceedings of the XXII International Conference MHCL' 17

Radial force (wheel load)

Tangential force (traction force)

Contact length (length of the contact area)

real

Real (measured) contact length

Factor for slip calculation

Coefficient of static friction

ResearchGate has not been able to resolve any citations for this publication.

Edwin A. H. Vollebregt
P. Wilders

In this paper we consider the frictional (tangential) steady rolling contact problem. We confine ourselves to the simplified theory, instead of using full elastostatic theory, in order to be able to compute results fast, as needed for on-line application in vehicle system dynamics simulation packages. The FASTSIM algorithm is the leading technology in this field and is employed in all dominant railway vehicle system dynamics packages (VSD) in the world. The main contribution of this paper is a new version "FASTSIM2" of the FASTSIM algorithm, which is second-order accurate. This is relevant for VSD, because with the new algorithm 16 times less grid points are required for sufficiently accurate computations of the contact forces. The approach is based on new insights in the characteristics of the rolling contact problem when using the simplified theory, and on taking precise care of the contact conditions in the numerical integration scheme employed.

J. J. Kalker

In this paper, we treat the rolling contact phenomena of linear elasticity, with special emphasis on the elastic half-space. Section 1 treats the basics; rolling is defined, the distance between the deformable bodies is calculated, the slip velocity between the bodies is defined and calculated; a very brief recapitulation of the theory of elasticity follows, and the boundary conditions are formulated. Section 2 treats the half-space approximation. The formulae of Boussinesq-Cerruti are given, and the concept of quasiidentity is introduced. Then follows a brief description of the linear theory of rolling contact for Hertzian contacts, with numerical results, and of the theory of Vermeulen-Johnson for steady-state rolling. Finally, some examples are given. Section 3 is devoted to the simplified theory of rolling contact. In Section 4, the variational, or weak theory of contact is considered. First, we set up the virtual work inequality, and it is shown that it is implied by the boundary conditions of contact. Then the complementary virtual work inequality is postulated, and it is shown that it implies the boundary conditions of contact. Elasticity is introduced into both inequalities, and the potential energy and the complementary energy follow. Finally, surface mechanical principles are derived. In Section 5, we return to the exact half-space theory. The problem is discretized, and solved by means of the CONTACT algorithm. Finally, results are shown in Section 6.

K.L. Johnson

The thirteen chapters of this book are introduced by a preface and followed by five appendices. The main chapter headings are: motion and forces at a point of contact; line loading of an elastic half-space; point loading of an elastic half-space; normal contact of elastic solids - Hertz theory; non-Hertzian normal contact of elastic bodies; normal contact of inelastic solids; tangential loading and sliding contact; rolling contact of elastic bodies; rolling contact of inelastic bodies; calendering and lubrication; dynamic effects and impact; thermoelastic contact; and rough surfaces. (C.J.A.)

D J Haines
E Ollerton

The problem of Hertzian bodies in rolling contact and supporting radial and shearing forces in the rolling direction is considered. A modified form of the conventional photoelastic frozen stress technique has been used to study the particular case of flat elliptical contact surfaces. Existing theories are reviewed and new theories are presented which permit the analysis of the frozen stress results. The dependence of the measured stresses on the hysteresis of rolling is studied.

G. Heinrich
K. Desoyer

Das Problem des Schräglaufes eines elastischen Kreiszylinders auf der Oberfläche eines elastischen Halbraumes wird unter der Voraussetzung, daß das Coulomb'sche Reibungsgesetz für jedes Flächenelement des Berührgebietes gilt, im Rahmen der Elastizitätstheorie erster Ordnung untersucht. Nach Herleitung der dieses Problem beherrschenden nichtlinearen singulären Integralgleichungen werden für gleiche Materialkonstanten der beiden Körper Methoden entwickelt, die eine numerische Lösung ermöglichen. Die Ergebnisse sind in Schaubildern dargestellt, die die gesamte Lösungsmannigfaltigkeit für die spezielle Querzahl v = 0,3 (Stahl auf Stahl) umfassen.

J. J. Kalker

1 The Rolling Contact Problem.- 2 Review.- 3 The Simplified Theory of Contact.- 4 Variational and Numerical Theory of Contact.- 5 Results.- 6 Conclusion.- Appendix A The basic equations of the linear theory of elasticity.- Appendix B Some notions of mathematical programming.- Appendix C Numerical calculation of the elastic field in a half-space.- Appendix D Three-dimensional viscoelastic bodies in steady state frictional rolling contact with generalisation to contact perturbations.- Appendix E Tables.

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