ttgn5-1-bostraben-app1.pdf

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Appendix 1 Page 7Draft translation

Drafted March 2004-Issued by VDV May 2006Translated September 2008

Procedures

The following principles apply to 2-axle bogies (or trucks1) with parallel wheelsets

and/or with single independent wheels, in which the axes of the opposite wheels lie

on a common centre line as well as for simple switches, outside curved switches and

the branch tracks of interior curved switches.

In junction layouts with multiple crossings, such as is shown in Figure 1.8, it is

necessary to consider the relationship of the various crossings to each other in order

to identify the differences between common crossings, in which guidance is provided

by a check rail, and double crossings, where guidance in one crossing is provided by

the flangeway of the opposite crossing. Common crossings in multi-track layouts

must be considered in accordance with the differential calculation principles

(cf. Figure 8 of the Technical Rules (TR Sp)) applicable to main and branch tracks

In order to enable the recommended extensive use of deep groove crossings, the

calculation principles can if necessary be applied without difficulty to the straight main

track of switch and crossing work for both common crossings and also double

crossings. Single axle trucks must be checked to ensure that a possible flange

footprint position for the single wheels cannot arise as a result of widening of the

groove due to wear such that the flange can climb on to the top of the crossing nose.

Under this condition attention is drawn to the use of the calculation principle for

double crossings in the curved branch track of simple switches. However, in this

case, wear is then to be expected at the flanks of the crossing grooves and strict

inspection of any repair work on the crossing nose is required.

The determination of all measurement takes place in the GGE, the measurement

plane for switches and crossings in the gauge level. If the GGE and the gauging level

do not coincide, the measurements must be adjusted in accordance with the

inclination of the groove flanks!

1A truck is defined as a two-axle chassis supporting the car body and which has either no, or limited, axial rotationrelative to the car body. The calculations are equally valid for two-axle bogies and two-axle trucks, hence forconvenience, the remainder of the document will refer only to bogies.


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The most extreme outcomes are applicable to the determination of the track

curve radius at both the leading and trailing outer wheels (3 and 4) in the

crossing flangeway. and to explore for every marginal track curve radius,following 1.1.3, the largest track curve radius (using the new values for the

wheels) at the flange front face of the leading outer wheel (3) and the smallest

curve radius diameter (measured up to the minimum wear limits of the

wheelsets) at the flange back face of the trailing outer wheel. (4).

The theoretical calculation of these values requires consideration of the

following worst case combinations:-

The curve radius (R3) at the flange face of the leading outer wheel (3)

(potentially in contact with the crossing gauge face) must be determined for the

largest wheelset check gauge dimension from the following conditions:-

1.1.4.1 Inner wheels with smallest new flange thickness (nominal measurement minus

manufacturing tolerance) in the largest new check rail groove (nominalmeasurement plus positive manufacturing tolerance), outer wheels with largest

new flange thickness (nominal measurement plus positive manufacturing

tolerance)

1.1.4.2 Inner wheels with smallest new flange thickness (nominal measurement minus

manufacturing tolerance) in a fully worn down check rail groove (maximum wear

value), external wheels with largest new flange thickness (nominal

measurement plus positive manufacturing tolerance).

The curve radius (R4) touching the back of the flange of the trailing outer wheel

(4) (potential groove flank of the crossing groove) must be determined for the

smallest wheelset check gauge dimension from:-


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1.1.4.3 All wheels with minimum flange thickness (minimum worn value) in the largest

new check rail groove (nominal measurement plus manufacturing tolerance)

1.1.4.4 All wheels with minimum flange thickness (limit of permissible worn value) in afully worn check rail flangeway (maximum wear value)

The required gauge in the single track curve results from the difference of the

maximum value of the outer rail curve radius at the point of contact with the

front face of the leading outer wheel and the inner rail curve radius ( R3;max-

R2;min); the required crossing flangeways result from the difference between the

maximum value of the outer rail curve radius at the point of contact with the

front face of the leading outer wheel and the minimum value of the curve radius

in contact with the back face of the trailing outer wheel ( R3;max- R4;min). The

results are then rounded up to the nearest whole millimetre (converted first, if

necessary, into the gauging plane).

This calculation method, in considering all influential factors, definitively

prevents wear at the crossing gauge and/or wing rail faces / groove flanks. To

maintain compliance with this, control is required of the check gauge and of the

wear limits at the gauge face of the running rail and the guiding face of the

check rail in the check rail zone, or, in the case of grooved rail crossings, the

dimensions over the crossing and check rail grooves.

1.1.5 Crossings

The crossing groove widths derived from 1.1.4 represent worst case values

which are however not often realized and which impede the use of a deep

groove crossing (smallest wheel tread width!). From that it is a self-evident

conclusion that:-

1.1.5.1 Whilst it is advisable to avoid wear at the rail crossing gauge flank,it can be

permitted at thewing rail.


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The curve radius R3touching the front flange face of the outside wheel 3 is

investigated in accordance with 1.1.4.1 and 1.1.4.2; the track curve radius R4

touching the flange back face of the outside wheel 4 is, however, only

considered for the new state of the flangeway in the check rail zone (1.1.4.3).

Wear at the rail gauge face in the check rail area can therefore lead only to

wear at the wing rail.

The verification of the included wear at the check rail face has to be undertaken

via the inspection of the allowable check gauge.

1.1.5.2 Block crossings (also applicable to rail-built crossings)

With block crossings all four tips of the crossing are liable to wear and must be

inspected and protected if necessary. If the groove widths according to 1.1.4

are not viable, a calculation can be made from the unworn check rail groove.

The most unfavourable results are calculated from 1.1.4.1 and 1.1.4.3. For

wheels in the new state the check rail width initially results in no wear at the

crossing flanks; this starts gradually with wear of the gauge face in the check

rail zone as well as on the guiding face of the check rail.

The crossing noses are to be protected through tapering the gauge faces to

meet at a point behind the geometrical intersection (cf. VDV-permanent way

building guideline, chapter 14). The extent of this relief is dependent upon the

wear and has to be monitored and reworked as necessary.

Measurement of the gauge is not meaningful at this point. Controls must be

used to compare the permitted increase in the flangeway width of deep groovecrossings to the maintenance of the minimum wheel support width at the

crossing nose.

If the crossing groove width can be realized according to 1.1.4, (especially with

flat grooves), the compliance of the internally determined wear limit must be

controlled to determine the wear induced widening of a crossings flangeway,

and the simultaneous reduction in the wheel supporting surface through the

actual crossing nose.


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All values calculated for the gauge and flangeway width in the gauging plane

are rounded up to whole millimetres. By a control calculation it is to be proved

subsequently that with the most unfavourable allowable extreme value of the

table in the single limiting track curve radius there is no forcing (complying with

section 4.2.2 of the Technical Rules).


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1.2 Calculation principle for Double crossings

1.2.1 For the straight track [following the proven values under section 4.2.3 of the

technical rules] the required minimum gauge (nominal measurement minusbuilding tolerance) and the required minimum groove width is given by:-

WDH,min= 0.5 (SDH,min-KDH,max)

Thus the groove width, rounded up to full millimetres, is subsequently given by

KDH,min= SDH,max- 2 WDH,max

to check whether the proof can then still be provided against forcing with all

quantities in full millimetres.

1.2.2 For curved tracks, the additional width required for the flanges is determined

from 1.1.2 and appropriately selected from the measurement table the same

limiting track curve radius as in 1.1.3. The crossing noses are to be protected

through relief against the impact (cf. VDV -OR 14). This extent of this relief is

related to the wear on the approach side and has to be monitored and reworked

as necessary.

1.2.3 As wear occurs equally at the Gauge and Check rail faces, a direct inspection of

the check gauge is essential. With deep groove crossings, the growth of the

flangeways groove should be monitored with respect to the minimum support

width needed for the wheel at the crossing nose.

Just as under 1.1, the calculated values for gauge and flangeway width in the

gauging plane are rounded up to full millimetres. In a control calculation it is

subsequently to be proved that, for the most unfavourable allowable extreme

values of the table no forcing appears at the limiting track curve radius

(complying with section 4.2.3 of the technical rules).


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1.3 General references to the calculation

The basis for the investigation of the additional width required for the bogie in

curved track is the horizontal flange section in the GGE; slight differencesbetween the position of this level to the actual (continually changing) contact

level can be neglected. As a rule (for tapered flange profiles) this sectional

plane is defined on two sides by hyperbolae, with the tip rounding radii

represented by distorted ellipses. For ease of calculation, the hyperbolic curves

can be approximated with sufficient accuracy to circular curves; similarly, the

distorted ellipses can be approximated to true ellipses. For clarity, these will, for

the remainder of this document, normally be referred to as the circular andelliptic regions of the flange section.

All values that refer to the GGE and whose coordinates are not already specific

to this level, are marked by the asterisk symbol - *.

For the calculation, a horizontal coordinate system is used, whose x axis isparallel to the longitudinal axis of the bogie and whose y axis corresponds to the

transverse axis of the bogie.

The additional flangeway width required by the inner flange on curved track

differs negligibly from that by the outer flanges, for example:-

for a bogie with 1.90 m wheelbase and 700 mm diameter wheels on a curve

of 18m radius the difference is 0.5 mm, for 20 m radius it is 0.4 mm and for

25m radius, 0.3 mm.

From the decrease of the calculated values for the groove width in the check rail

is can be concluded that:-

Irrespective of whether the check rail is next to the outside rail of the curved

track (outside track of inside curved switches) or the inner rail (inside track of

inside curved switches) and/or the curve track of all other switches, it is


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2 Detailed Calculation Procedure(cf. section 1.3)

2.1 Common crossing including check rail zone

2.1.1 Flangeway width in the check rail zone

For the curved check rail, the additional flangeway required for the flanges of a

bogie with maximum flange thickness on the selected track curve radius is

investigated from the curve radii in contact with both the leading and trailing

flanges of the inner wheels of the bogie. The distance between the front and

back faces, covering the effective flange thickness, is radial to the track curve

and is thus the minimum required flangeway width in the check rail zone.

2.1.1.1 Geometry of the horizontal flange section

At the flange section, the hyperbolae defining the flange front and back faces

are replaced by circular curves of radius rA,Sand rA,Rto permit calculation, and

for the selection of the calculation procedure according to case 1a or 1b for:-

- the Circular / Elliptic transition point Sand R(at the flange front and back

faces respectively)

- the attack angles ,1and ,2and

- the track curve radii R,1and R,2at the wheels 1 and 2 and from that

- the related curve centreline radii R,1,mand R,2,m(Figures 1.1 and 1.2):

The radii RA,S/Rforming the lateral boundaries of the flange section formed by

the flange front / back faces are given by:

rA,S/R= (x,S/R2+ d,S/R*

2) / (2 d,S/R*) (1a/b)

x,S/R- distance in X axis in the flange section from the wheel centreline to the

Circular / elliptic transition points on the flange front / back faces:

x,S/R= [(0.5 dM+ a* + hU,S/R*)2 (0.5 dM+ a*)

2] (2 a/b)

dMWheel diameter at the tread contact point (MKFp)


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a* -vertical distance to the GGE from the line joining the wheel diameter points

(MKFp)

hS/R* - vertical distance from GGE to the flank/tip transition at the flange front/ back faces

hS/R* = h - a* - rK,S/R[1 cos (arc tan nS/R)] (3a/b)

h - flange height (maximum value in the new state)

rK,S/R- outer / inner flange tip rounding radius

nS/R- vertical inclination value of the flange front / back faces

d,S/R* - lateral offset of the flange front / back face between the GGE and

the junction between the flange flank and the tip radius (also the

hyperbola/ellipse transition)

d,S/R* = h,S/R* / NS/R (4a/b)

The curve radii R,1/2at the front / back face circular / elliptic transition point on

the flange section for wheels 1 and 2 are given by:

R,1= rA,R[(0.5 aF/ x,R) + 1] (5)

aF- bogie wheelbase

R,2= rA,S[(0.5 aF/ x,S) - 1] (6)

The track curve centreline radii R,1/2,mare given by:

R,1/2,mR,1/2+ 0.5 sN (7a/b)

sN- Nominal gauge in the GGE

The horizontal angles ,S/Rat the circular / elliptic transition points on the front /

back faces of the flange Section are given by:

,S/R = arc sin (x,S/R / rA,S/R) (8a/b)


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The following calculations for the contact points on both of the inner wheels of

the bogie differ according to their location of both points on the contour of the

flange section.

The criterion for determining this is:- If the mean track curve radius Rmthat is tobe explored is larger than R,1,mor R,2,m, the Wheel/Rail contact point lies on

the circular part of the flange front flank, otherwise it lies on the flange tip

rounding (elliptic part).


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Fig. 1.1: Sketch showing the geometry of the Horizontal Flange Section


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Fig. 1.2 Diagram for calculating the geometry of the Hyperbola/ellipse transition points on

the flange Section


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2.1.1.2 Contact point on the flange flank, case 1a (Figure 1.3)

The track curve radii R1/2at the back / front face contact points of wheels 1 / 2 lie on

the circular part of the horizontal flange section dependent upon the selected track

centreline curve radius Rm:R1= (R1-rA,R) + rA,R (9)

(R1- rA,R) = [(R1- rA,R)y2+ 0.25 aF

2] (10)

(R1- rA,R)y - y coordinate of (R1- rA,R)

(R1- rA,R)y= Rm,y 0.5 sN* + dmax* - rA,R (11)

Rm,y- y coordinate of Rm

Rm,y = [ Rm2- 0.25 aF2] (12)

Rm- selected centreline track curve radius

aF- bogie wheelbase

sN* - Nominal gauge in the GGE

sN*= SN-2 (a* - a) ns (13)

a* - vertical distance between the GGE and the line joining the

wheel:rail contact points

a - vertical distance between the gauge measurement line and the

line joining the wheel:rail contact points

nS/R - inclination of the flange front / back face

dN/max/min*- Nominal / maximum / minimum flange thickness in the GGE at

the horizontal attack angle of 0

dN/max/min* = dN/max/min [(a* - a)/nS] [a* - a)/nR] (14)

dN/max/min - Nominal / maximum / minimum flange thickness in the gauging

plane at the horizontal attack angle of 0

rA,R from (1 b)

R2= [(R2+ rA,S)y2+ 0.25 aF

2] rA,S (15)

(R2+ rA,S)y - y coordinate of (R2+ rA,S)


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Fig. 1.3 Diagram for calculating track curve radii for a contact point on the flange flanks,

chordal position


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Fig. 1.4 Diagram for calculating the track curve radius for contact point on the flange tip

rounding of the flange section


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The practical calculation by iteration by means of the objective value in Excel

2000 can be done by setting the value 0 as the Objective value", and in the

changeable cell" setting the value to that of xEII,2, and in the "objective cell"

entering the following expression (which will, when the correct value for xEll,2 has

been selected, become zero) :

[ (R2,y+ yEll,2 dK,S dS* + 0.5 sN*)2+ (0.5 aF)

2] Rm= 0

dK,S/R - distance between the intersection between the extended flange

front / back face with the y-axis of the flange tip

dK,S/R= rK,S/R tan (0.5 arc tan nS/R) (29a/b)

rK,S/R - outer / inner flange tip radius

nS/R - inclination value of the flange front / back face

dS/R* - horizontally flange front / back face reduction of d* between GGE

and not from round flange tip

dS/R* = (h - a*) / nS/R (30a/b)

h - Flange height (maximum value in the new state)

a* - vertical distance between the GGE and the line joining the rail head

contact points

sN* from (13)

aF - bogie wheelbase

over the confirmed value obtained from that for xEll,nat the contact point n. Rnis

the track curve radius calculated from (27).

The track curve radius R1 at the contact point of the inner flange tip rounding of

the flange section at the rail flank is determined in a similar manner. As before,

an arbitrary number is chosen for aEII,R, and from this the associated value for

yEll,1obtained from (23b) - as well as the value of the attack angle 1at the

contact point with the wheel 1 from (24b).

The horizontal distance R1,xfrom the transverse centreline of the bogie to the

start point Wheel/rail on the flange section is given by:

R1,x= xEII,R+ xEII,1

which generalises to:


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R2/3,x= xEII,S+ xEll,2/3 (31a)

and

R1/4,x= xEII,R+ xEll,1/4 (31b)

and the associated distance R1,yof the foot point from R1,xto midpoint of the track

curve radius R1from (26). The track curve radius R1at the contact point of the

flange back ellipse is then calculated from (27).

The practical calculation by iteration by means of the objective value in Excel 2000

can be done by setting the value 0 as the Objective value", and in the

changeable cell setting the value to that of xEII,1, and in the objective cell enteringthe following expression (which will, when the correct value for xEll,1 has been

selected, become zero) :

[ (R1,y- yEll,1+ dK,R dS* + 0.5 sN*)2+ (0.5 aF)

2] Rm= 0 (32)

yEll,1 - from (22b)

dK - Length of the flange tip straight lines until front and/or back face

extended to meet the flange tip rounding

dK,max/min= dmax/min* - dS* - dR* (33)

dmax/min* from (14)

With the value thus obtained for XEll,1 , the track curve radius R1at the contact point

is now calculated from (27).

2.1.1.4 Flangeway width in the check rail zone

For the calculation of the curved flangeway in the check rail zone of a common

crossing, an increase of the effective flange thickness must be allowed for

compared to that for the straight route.

For the back face the increase in the effective flange thickness amounts to:

dR* = (0.5 sN* - dN*) - (Rm- R1) (34)

sN* from (13)


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dN* from (14)

For the front face the increase in the effective flange thickness amounts to:

dS* = Rm- R20.5 sN* (35)

The overall increase of the operative flange thickness vis--vis the straight lines

amounts to:

d* = ds* + dR* (36)

the minimum required flangeway width in the GGE for the check rail zone WRthen

amounts to:

WR*= dN*+ d* (37)

If the GGE and the gauging plane do not coincide (A a*), another conversion must

be undertaken to convert the value that in GGE into the equivalent value in the

gauging plane, by allowing for the inclination of the rail flank.

The objective of the survey is, among other things, the compilation of a table for

gauge and flangeway width at the check rail, with the values calculated from (37) in

whole millimetres set against a graded series of curve radii, and to determine the

limiting track curve radius applying for that value

2.1.2 Check gauge over crossing flangeway and the check rail flangeway

This must be considered for both the largest new value and, where appropriate, at

the wear limit of the flangeway width in the check rail zone with the bogie in its

rotated position. The wheels must be considered in both the new state (for the

investigation of the largest check gauge, determined from the difference of the track

curve radii R1and Rg) and the fully worn state (for the investigation of the smallest

check gauge, determined from the difference of the track curve radii R2and R4).

The basis for the calculation is the limiting track curve radius that was determined

under 2.1.1.4 for the gauge and flangeway width table. The largest compliant

flangeway widths in the check rail zone are to be assigned these limiting track curveradii.


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The most unfavourable conditions in the branch track of a simple switch result from

the leading inner wheel (1) of the bogie being in contact with the guiding face of the

curved check rail, and the trailing inner wheel (2) of the bogie in contact with thegauge face of the inner rail of the curve (see Figure 1.5). For this position, the

flangeway dimensions are determined by the track curve radii R1and R2at the

contact points for wheels 1 and 2 for the given track curve radius, and R3and R4at

the contact points for wheels 3 and 4; R1results from R2plus WR,max. With the bogie

in this position, the curve radius R3touches the curve leading outer wheel (3) of the

bogie (and/or the first wheel pair) at the front face and the curve radius R4the

trailing outer wheel (4) of the bogie (and/or the second wheel pair) at the back face.

Depending upon the track curve radius and/or attack angle, different positions arise

for the location of these four contact points with the flange Section:

1 in large track curve radii and/or small attack angles all four contact points lie in

the range of the flange flanks (Hyperbolic) of the horizontal flange Section

2 in moderate track curve radii and/or attack angles some of the four contact points

lie in the flange flank range (Hyperbolic) and some in the range of the flange tip

rounding (ellipse range) of the horizontal flange section

3 in small track curve radii and/or large attack angles all four contact points lie in

the range of the flange tip rounding (ellipse range) of the horizontal flange section

As a rule, for all calculations that result in the contact point of the wheel 3, the value

of the new wheels are to be used; for all calculations that result in that contact point

of the wheel 4, the value of the worn down wheels should be used as this will then

give the extreme value of the check gauge. There that from.

Because the basis of the calculations for the check rail zone is the inner rail of the

curve, and therefore the contact points of wheels 1 and 2, there are therefore

required to be separate processes for the investigation of the contact point at wheel

3, with new wheels 1 and 2, and for the investigation of the contact point at wheel 4,


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with worn down wheels 1 and 2. This distribution is allowed for in the following

sections of the calculation of the contact point,.

The criterion for the location of the contact point is the attack angle of each wheel in

comparison with the angles

,Sand

,Rfrom (8a) and/or (8b) at theHyperbola/Ellipse transition point on the flange section front and/or back face.

Where the attack angles 2or 3are smaller than ,Sand/or 1or 4smaller than

,R, the contact point of the respective wheel lies in the range of the flange flank

(Hyperbola), otherwise they lie in the range of the flange tip rounding (ellipse).

The reference basis for the calculation is track curve radius at the gauge face of the

inner rail of the curve in the GGE, which is calculated approximately from:

R2= Rm 0.5 SN (38)

Rm - ca. track centreline curve radius

SN - Nominal gauge of the straight track in the gauge plane

2.1.2.1 The contact points for wheels 1 & 3 lie on the flange flanks (Hyperbolic

part)

The calculation results correspond to the calculation sketch, figure 1.5.

Connecting lines g1of the midpoints of the radii replacing the Hyperbolae in the

flange Section rA,Rin wheel 1 (back face) and rA,Sin wheel 2 (front face) :

g1= [aF2+ (rA,R+ rA,S- dmin*)

2] (39)

aF- Bogie wheelbase

rA,S/R from (1a / 1b)

dmin* from (14)


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Horizontal angle between g1and the wheelset:

= arc sin (aF/ g1) (40)

Horizontal attack angle 1at wheel 1:

1= (+ 1) (41)

(+ 1) - horizontal angle between g1and R1

(+ 1) = arc cos {[(R2+ rA,S)2 g1

2 (R1 rA,R)2] / [2 g1(R1 rA,R)]} (42)

R1= R2+ WR,max (43)

WR,max= WR+ W (44)

WR from 2.1.1.4

W - Manufacturing tolerance of the groove width at the check rail

Horizontal attack angle 2 at wheel 2:

2= (45)

- horizontal angle between g1and R2

= arc sin {[(R1 rA,R) sin (+ 1)] / (R2+ rA,S)} (46)

Horizontal attack angle 3 at wheel 3:

3= arc tan {(R1 rA,R)x/ [(R1 rA,R)y+ rA,R+ l max* -rA,S]} (47)

(R1 rA,R)x - x coordinate of (R1 rA,R)

(R1 rA,R)x = (R1 rA,R) sin 1 (48)

R2from (37)

(R1 rA,R)y - y coordinate of (R1 rA,R)

(R1 rA,R)y = (R1 rA,R) cos 1 (49)

l max* - maximum check gauge in the GGE at a horizontal angle of 0


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Track curve radius R3 at the contact point of wheel 3:

R3= [(R1 rA,R)x2+ [(R1 rA,R)y+ rA,R+ l max* - rA,S]

2] + rA,S (50)

2.1.2.2 The contact points of the wheels 1, 2 and 4 lie on the flange flanks (Circular part)

The calculation results correspond to the sketch, figure 1.5.

Values for the contact points of the wheels 1 and 2 are obtained from (39) to (46).

Horizontal attack angle 4at wheel 4

4= arc tan {[aF (R1rA,R)x] /[(R1-rA,R)y+ 2 rA,R- dmin* + l min*]) (51)

dmin* from (14)l min* - smallest check gauge measure in the GGE in the horizontal angle 0

Track curve radius R4at the contact point of wheel 4:

R4= {[ aF- (R1 rA,R)x]2+ [(R1 rA,R)y+ 2 rA,R dmin* + l min* - rA,S]

2} - rA,R (52)

2.1.2.3 The contact point of wheel 1 lies on the inner flange tip rounding (elliptic

part), the contact points of wheels 2 and 3 on the flange flanks (Circular part)

The calculation results correspond to the sketches figure 1.4 (in relation to wheel 1)

and figure 1.5 (in relation to wheels 2 and 3).

The correct value for 1will investigated through iteration over two variants to the

calculation of the y coordinate of (R2+ rA,S) :

1. (R2+ rA,S)y= R1 cos 1- yEll,1+ dK,R- dmin* + rA,S (53)

1 - selected horizontal attack angle at wheel 1

yEII,1 from (23b)

dK,R from (29b)

dR* from (30b)

dmin* from (14)


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Figure1.5: Calculation diagram for flange footprint, contact points on the flange flanks


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2. (R2+ rA,S)y= (R2+ rA,S) cos 2 (54)

2 - horizontal attack angle at wheel 2

2 = arc sin {[(aF (R1,x xEll,1 + aEll,R 0.5l A)] / (R2+ rA,S)} (55)

Rn,x - x coordinate of the flange section touching the curve radius Rn

Rn,x= Rn sin n (56)

xEll,1/4= aEll,R [ 1 (yEll,1/42/ bEll,R

2) from (23b) (57)

Putting the "correct" value of 1into both expressions (53) and (54) must yield the

same value for R2:

(R1 cos 1 yEll,1+ dK,R+ dR* - dmin* + rA,S) [(R2+ rA,S) cos 2] = 0 (58)

Horizontal attack angle 2at wheel 2 from (55).


3 = arc tan {(R1,x xEll,1 + aEll,R 0.5l A) / [(R2+ rA,S) cos 2 2 rA,S+ dmin* + l max*]}

(59)

l max* -maximum check gauge in the GGE at the horizontal attack angle 0

Curve radius R3at the contact point for wheel 3:

R3= rA,S+ [ (R1,x xEll,1+ aEll,R 0.5 l A)2+ [(R2+ rA,S) cos 2 2 rA,S+ dmin* +

l max*]2] (60)

2.1.2.4 The contact point of wheel 1 lies on the inner flange tip rounding, the

contact points of wheels 2 and 4 on the flange flank

Following the diagrams shown in figures1.4 (in relation to wheel 1) and 1.5 (in

relation to wheels 2 and 4).

Values for the contact point of wheels 1 and 2 after 2.1.2.3, (53) until (58).



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4= arc tan {[aF (R1,x xEll,1+ aEll,R 0.5 l A)] / [(R2+ rA,S) cos 2 rA,S+ l min* +

rA,R]} (61)

l min*-minimal check gauge in the GGE in the horizontal attack angle 0

Track curve radius R4at the contact point of wheel 4:

R4 = [[aF- (R1,x-xEII,1+ aEII,R 0.5 l A)]2+ [(R2+ rA,S) cos 2- rA,S+ l min*+ rA,R]

2

] -

rA,R (62)

2.1.2.5 The contact points for wheels 1 and 3 lie on the flange tip rounding (ellipse

zone), the contact point for wheel 2 on the flange flank (Hyperbolic zone)

Following the diagrams shown in figures 1.4 (in relation to wheels 1 and 3) and 1.5

(in relation to wheel 2).

Values for the contact points of wheels 1 and 2 after 2.1.2.3, (53) until (58).

The correct value for 3will be determined through iteration over two variants of the

expression for the curve radius R3:

1. R3= (R1 cos 1-yEII,1+ dK,R+ dR* +l

max* - dS* - dK,S+ yEll,3) / cos 3 (63)dS/R* from (30a/b)

dK,S/R from (28a/b)

l max* -maximum check gauge in the GGE at the horizontal attack angle 0

yEll,3 = bEll,S2 [ 1 / (aEll,S

2 tan23+ bEll,S2) ] from (23a/24a)

3-selected horizontal attack angle at wheel 3

2. R3= (R1,x-xEll,1+ aEII,R - aEII,S+ xEII,3) /sin 3 (64)

xEll,2/3= aEII,S [1 -(yEll,2/32/ bEII,S

2) ] from (23a) (57a)

Putting the correct value for 3into each of the expressions (63) and (64) should

generate the same result for R3, ie ( 63) minus (64) will give the result Zero.

[(R1 cos 1 yEll,1+ dK,R+ dR* + l max dS* - dK,S= yEll,3) cos 3]

[(R1,x xEll,1+ aEll,R aEll,S+ xEll,3) sin 3] = 0 (65)


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The curve radius R3at the contact point for wheel 3 is obtained from (63) or (64).

2.1.2.6 The contact points for wheels 1 and 4 lie on the flange tip rounding (ellipse

zone), the contact point for wheel 2 on the flange flank (Hyperbolic zone)

Following the diagrams shown in figures 1.4 (in relation to wheels 1 and 4) and 1.5

(in relation to wheel 2).

Values for the contact point of the wheels 1 and 2 after 2.1.2.3, (53) until (58).

The value for 4is determined by iteration using two variants of the equation for the

curve radius R4:

1. R4= (R1 cos 1-yEII,1+ 2 dK,R- dmin* + l min* - yEll,4) cos 4 (66)

dK,Rfrom (29b)

dR* from (30b)

dmin* from (14)

l min* - minimum check gauge in the GGE at the horizontal angle 0.

yEII,4= bEII,R2 [1 / (aEll,R

2 tan24+ bEII,R2) ] from (23b/24b)


2. R4= (aF R1,x+ xEII,1- xEII,4) sin 4 (67)

xEII,4 from (57b)

Putting the correct value for 4into each of the expressions (66) and (67) shouldgenerate the same value for R4.

[(R1 cos 1- yEII,1+ 2 dK,R+ 2 dR* - dmin* + lmin* - yEll,4) cos 4] -

(aF R1,x+ xEII,1- xEII,4) sin 4= 0 (68)

Curve radius R4in the contact point of the wheel 4 out of (66) or (67).


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2.1.2.7 The contact points for wheels 1 to 3 lie on the flange tip rounding

Following the diagram shown in figure 1.6.

Here a double iteration will be required, i.e. first the connection straight line g2of thecontact point 2 the intersection of the track curve radius R1at the contact point 1

with the track curve radius R2as well as the angle between these connecting

straight lines and the longitudinal axis of the bogie. In addition the attack angle 1at

the contact point of wheel 1 and the attack angle 2at the contact point of wheel 2

must be selected. For , 1and 2 :-

1 = 2+ (69)

The examination, as to whether the value was selected correctly for 1, is

undertaken over both calculation variants for the curve radius R2:

R2= 0.5 g2/ sin (1- ) (70)

R2 = 0.5 g2/ sin (2+ ) (71)

If 1has been selected correctly, the given value for R2will be equal to that obtained

from (38).

g2= 2[g2,x2+ g2,y

2] (72)

g2,x= aF+ x1- WR,X-X2 (73)

aF- bogie wheelbase

x1= 0.5 ( l A- aEII,R + xEII,1) (74b)

l A from (18)

aEII,R from (17b)

xEll,1 from (57b)

yEll,1 from (23b/24b)


WR,x= WR,max sin 1 (75)

WR,maxfrom (44)

x2= 0.5 l A- aEII,S+ xEII,2 (74a)

aEII,S from (17a)

xEII,2 from (57a)

yEll,2 from (23a/24a)

2- selected horizontal attack angle at wheel 2


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g2,y= WR,y- yEll,1- dK,G,min-yEll,2 (76)

WR,y= WR,max COS 1 (77)

dK,G,min - length of the straight flange tip relative to dmin

dK,G,min= dmin* - dS* - dK,S- dK,R- dR* (78)

dmin* from (14)

dS/R* from (30a/b)

dK,S/R from (29a/b)

= arc tan (g2,y/ g2,x) (79)

(1- ) - (2+ ) = 0 from (70/71 ) (80)

Moreover R2will be given by the "correct" selection of the angle of the track curve

radius complying with the given value:

R2- [0.5 g2/ sin (1- )] = 0 from (70)

The horizontal attack angle 3is now obtained through iteration over two variants for

the expression for R3,x:

1. R3,x= R1 sin 1-x1+ 0.5 l A-aEll,S+ xEll,3 (81)

xEII,3 - selected value

2. R3.x= [R3,x2+ R3,y

2] sin arc tan [(bEll,S2/ aEll,S

2) (xEll,3/yEII,3)] (82)

R3,y= yEII,3+ R1,y + dK,R+ dR* + l max* - dK,S- dS* (83)

yEII,3 from (23a)Rn,y= Rn cos n (84)

dK,S/R from (29a/b)

dS/R* from (30a/b)

If xEll,3has been selected correctly, the values for R2obtained from (81) and (82)

must be equal.

(R1 sin 1 x1+ 0,5 l A aEll,S+ xEll,3)


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{[R3,x2+ R3,y

2] sin arc tan [(bEll,S2/ aEll,S

2) (xEll,3/ yEll,3)]} = 0

(85)

Curve radius R3at the contact point for wheel 3 is obtained from (27).

Horizontal attack angle 3at the contact point of wheel 3:

n= arc tan (Rn,x/ Rn,y) (86)

2.1.2.8 The contact points of wheels 1, 2 and 4 lie on the flange tip rounding

(elliptic zone)

Following the diagram shown in figure 1.6:-

Values for the contact points at wheels 1 and 2 are obtained from section 2.1.2.7,

(69) to (80).

The horizontal attack angle 4can now be obtained through iteration over two

variants for the expression for R4,x:

1. R4,x = R2 sin 2+ x2 0.5 l A- aEII,R+ xEIl,4 (87)

xEII,4 - selected value

2. R4,x= [R4,x2+ R4,y

2] sin arc tan [(bEII,R2/ aEll,R

2) (xEII,4/ yEII,4)] (88)

R4,y= R1,y-yEII,1 + 2 dK,R+ 2 dR* -dmin* + l min* -yEll,4 (89)

dK,R from (29b)

dR* from (30b)

YEII,4 from (23b)

With the correct selection of the value for xEll,4in both formulas (87) and (88) the

values obtained for the curve radius R4from (87) and (88) should be equal.

(R2 sin 2+ x2 0,5 l A aEll,R+ xEll,4)

{[R4,x2+ R4,y

2] sin arc tan [(bEll,R2/ aEll,R

2) (xEll,4/ yEll,4)]} = 0 (90)

The curve radius R4at the contact point for wheel 4 is obtained from (27)

The attack angle 4at the contact point for wheel 4 is obtained from (86).


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2.1.2.9 Check gauge over crossing flangeway and/or over the flangeway at the

check rail

The provisional check gauges required (withoutconsideration of any lateral

elasticityof the wheel tire in the curved track!) are given by the extreme values of

the effective check gauge measure, i.e. from the flange positions in the rotated

bogie footprint as limited by contact with the rails (as per the calculations made

under sections 2.1.2.1 to 2.1.2.8).

The minimum required provisional check gauge over crossing flangeway (from

the difference between the track curve radii in the new state at the contact points of

wheels 3 and 1 for the respective centreline track curve radii)

LH,min* = R3 R1 (91)

The largest permissible provisional check gauge over the check railflangeway

(from the difference between of the touching track curve radii in the fully worn state

at the contact points of wheels 4 and 2 for the respective centreline track curve

radius)

LR,max* = R4 R2 (92)

With resilient wheels it is necessary to allow for the effects of the lateral elasticity of

the tyres in the check rail zone of the curved branch track, in that the gauge of the

leading wheelset is widened and that of the trailing axle is narrowed, and that under

the influence of the centrifugal force the gauge widening is larger. Further, the

effect of the lateral elasticity is proportional to the curvature. In the curved branch

track of inside curved switches, the opposite effect occurs: For the leading wheelset

a gauge reduction arises, for the trailing wheelset the gauge is increased. Here also

the centrifugal force effects are increased on the first axle and reduced on the

second axle.


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Minimum required check gauge over crossing flangeway

LH,min* = LH,min* + Qa

Qa - at the outside curve causing lateral elasticity of the wheel tire dependent

upon track curvature and centrifugal force

Largest allowable check gauge over the flangeway at the check rail

LR,max* = LR,max* Qi

Qi - at the inside curve causing lateral elasticity of the wheel tire dependent

upon track curvature and centrifugal force

Comment: Because only the maximum value for Q, being proportional to both the

curve radius and centrifugal force, is difficult to establish it is recommended that the

maximum value is assigned for small curve radii, an average value to mid-range

curve radii and a value of zero for very large curve radii and straight track.

2.1.3 Gauge width in the crossing zone

If it is supposed that there is to be complete clearance to the crossing groove flanks,

a relatively large crossing flangeway is required, which on the one hand is often not

achievable with a solid crossing in a moderate space whilst on the other hand the

required wheel support frequently cannot be guaranteed with deep groove

crossings.

For the calculation of the crossing flangeway, the user must decide whether to use

the unworn values for the check rail flangeway, or the maximum value within the

wear limits in determining the required value. Alternatively, the straight track of the

crossing can be dimensioned according to the principles of double crossings, with in

the curved track a mathematical overlap between the flange back and crossing

groove flank or, for the outside track of inside curved switches, overlap between the

flange front and the gauge face of the crossing.

The required minimum gauge in the crossing zone for the respective centreline

curve radius results from the total of the minimum required check gauge through the

crossing and the maximum flangeway through the check rail:


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Smin* = LH,min* + WR,max* (95)

At the same time, it is to be noted that the largestvalue of LH,min* must be obtained

for all bogies!

The nominal value of the respective gauge is then calculated with due regard for thenegative building tolerance:

SN*= Smin*+ |-S| (96)

|-S| - negative build tolerance for gauge

Comment: As a result of the coarse steps used for the value of Q in the

investigation of the required and/or allowable check gauge, the results that emerge

for the values for the gauge and crossing flangeway will appear discontinuous

instead of being a logical continuous variation. The user may correct this by the

addition or subtraction of correction values


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|-W| - Absolute value for the negative manufacturing tolerance for the

flangeway

With a rail-built crossing, a larger overlap (displacement) can be provided at thewing rail than at the crossing gauge face.

The comment under 2.1.3 is to be noted!

The clearance at the crossing gauge face is calculated from:

WH,Ff= Smin* -WR,max* - (l max* + Qa) (100)

The clearance at the crossing flangeway results from:

WH,Rf= l min* - Qi- (Smax* -WH,min*) (101)

If the clearance value is negative, the overlap lies before!


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Figure 1.6: Calculation diagram for flange footprint, contact points on the flange tip


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2.2 Double Crossings

Because of the equivalence of both grooves of the double crossings, the calculation

for these must ensure that the wheels are not forced between either the gaugefaces of the rails or the guiding faces of the check rails. Compared to the breadth

measure proof for the straight track the gauge in the curved track must be increased

by double the additional width requirement for the flange front and the check gauge

reduced by double the additional width requirement for the flange back.

The smallest allowable gauge Smin,B* in the curve results from:

Smin,B* = Smin,G* + 2 dS* (102)

Smin,G* - smallest allowable gauge on straight track

dS* from (35)

In (96) the nominal gauge is determined by taking into account the building

tolerance.

The largest permissible check gauge Kmax,B* in the curve is calculated from:

Kmax,B* = Kmax,G* - 2 dR* (103)

Kmax,G* - largest permissible check gauge in straight track

dR* from (34)

The smallest allowable flangeway WDH,min,B* in the curved track is then given by:

WDH,min,B* = WDH,min,G* + d* (104)WDH,min,G* - smallest permissible flangeway in the straight track

d* from (36)

By analogy to (99) the nominal flangeway is determined from consideration of the

fabrication tolerance.

Consideration of the lateral elasticity of the wheel tire is not meaningful in a double

crossing.


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An unwanted overlap value can emerge at the guiding or check flank and it must

therefore be checked with the rounded values, to determine how much free space

and/or overlap appears in the most unfavourable bogie/track configuration:

The clearance at the gauge face is calculated from:

WFf= Smin,B*- (smax,G*+ 2 dS*) (105)

smax,G* - maximum gauge at an attack angle of 0 in the GGE

The clearance at the groove flank results from:

WRf = kmin,G*-2 dR*- (Smax,B* - 2 Wmin,B*) (106)

kmin,G* - smallest check gauge at an attack angle of 0 in the GGE

General considerations for 2.1.3, 2.1.4 and 2.2:

In shallow depth groovesthe lifting of the wheel through flange tip / groove bottom

contact results in a reduction of the effective flange thickness, thus altering the

gauge, check gauge and groove width requirements (figure 1.7).

In raising the flanges in the flat groove, the GGE remains at the flange, in that way

the breadth mass of the bogie or wheelset (also on curved track) remains

unchanged compared with the deep groove. They must be converted however on

the gauging plane of the track.

The gauge is reduced in a one-sided flat groove relative to the deep groove by:

ST/F= (A + h -t - a*) / nS (107)

and in a two-sided flat groove by the double value:SFl/F= 2 (A + h -t - a*) / nS (108)

Compared to the deep groove, the groove width for a flat groove is reduced by:

WT/F = (A + h - t - a*) (1/nS+ 1/nR) (109)

Comment: This conversion assumes that the flank inclinations of the flat groove

match the respective inclinations of the flange flanks.

If the GGE and gauging plane for the deep groove are notcoincident (ie A a*),another conversion of all measurements must be undertaken taking into account the


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inclination at the rail flank to convert the GGE value into the gauging plane.

Fig. 1.7 Shallow Groove


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3 Simplified calculation procedure

(refer to Section 1.3)

3.1 Common crossing with Check Rail

3.1.1 Flangeway width in the check rail zone

For the curved check rail zone, the additional width required for the flanges of a

bogie with maximum flange thickness is investigated for the selected track curve

radius for the contact points of the front and back faces of both of the leading and

trailing inner wheels of the bogie respectively. The distance between the front and

back faces, determining the effective flange thickness is radial to the track curve and

is thus the minimum required flangeway in the check rail zone.

3.1.1.1 Geometry of the horizontal flange-section

At the flange section, the hyperbolae that are the flange front and back faces are

replaced by circles with radii rA,Sand/or rA,Rrespectively and for the selection of the

calculation procedure according to case 2a or 2b (cf. section 1.3) for the hyperbola /

flange tip transition points ESand ER(at flange front and back faces respectively),

the attack angles E,1and E,2and the track curve radii RE,1and/or RE,2at wheels 1

and 2 to calculate from that the track centreline radii RE,1,mand/or RE,2,m(figures 1.8

and 1.9) :

Radius rA,S/Ras lateral boundary of the flange section at the flange front / back flank

(replacement for Hyperbola) :

rA,S/R= (0,25 l A2 + dS/R*

2) / (2 dS/R*) (201a/b)

l Afrom (18)

dS/R* from (30a/b)

Curve radii RE,1/2are defined by the intersection point between of the flange flank /

tip section of wheels 1 and/or 2 and are tangential to the hyperbola:


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RE,1= rA,R[ (aF/ l A) + 1] (202)

aF- Bogie wheelbase

RE,2= rA,S[ (aF/ l A) - 1] (203)

The associated centreline curve radii RE,1/2,mare given by analogy from (7a/7b) :

RE,1/2,m~ RE,1/2+ 0.5 sN (204a/b)

sN - Nominal gauge in the Gauging plane

Horizontal attack angle Ell,S/Rat the front / back flank / tip transition point on the

flange section

The following calculations for the contact points on both the inner wheels of the

bogie differ according to the locations of both contact points on the outline of the

flange section.

The criterion for that is: When the centreline track curve radius Rmto be

investigated is larger than RE,1,mand/or RE,2,m, the back and/or front face wheel / rail

contact point lies on the range of the flange flank (Hyperbola), otherwise it is on the

flange flank (Hyperbola) / flange tip transition point.

3.1.1.2 Contact point on the flange flank, case 2a(figure 1.9)

The track curve radii R1/2at the back / front face contact point for wheels 1/2 in the

circular part of the horizontal flange section, dependent upon the selected centreline

curve radius Rm, is calculated from (9) to (16).

3.1.1.3 Contact point on the flange flank / tip transit ion point , case 2b (figure 1.10)

The track curve radii R1/2as back / front face covering of the wheel 1/2 at the

transition point of the horizontal flange section are dependent upon the centreline

curve radius Rm:

R1= [ R1,y2+ (0.5 aF+ 0.5 l A)

2] (206)


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R1,y - y coordinate of R1

R1,y= Rm,y 0.5 SN* + dS* + dK,max (207)

Rm,y from (12)

SN* from (13)

dK,maxfrom (33)

R2= [R2,y2+ (0.5 aF- 0.5 l A)

2] (208)

R2,y - y coordinate of R1

R2,y= Rm,y 0.5 SN* + dS* (209)

3.1.1.4 Flangeway width in the check rail zone.

For the calculation of the curved flangeway in the check rail zone of a common

crossing, an increase of the effective flange thickness must be allowed for relative to

that for the straight tracks. It is obtained from (34) to (37).

If the GGE and gauging plane level are not coincident, (a a*), another conversion

must beundertaken in accordance with the inclination at the rail flank to bring the

GGE value into the gauging plane.

As the objective of the investigation is, among other things, the setting up of a

gauge and flangeway width table, it is recommended, with the values derived from

(37) a graduated series of values for the flangeway in the check rail zone using

whole millimetres and to determine the limiting curve radius applying to that value.

[where appropriate: intermediate values should be used in the calculation.].

3.1.2 Check Gauge over crossing f langeway and/or over the flangeway in the check

rail zone

Basically the same rules apply here as also to under 2.1.2.

Depending upon the curve radius and/or attack angle, there arises for the situation

of the four contact points at the flange sections relative to 2.1.2 other definitions as a

result of eliminating the flange tip rounding, however:

1 with large track curve radii and small attack angles all four contact points lie on


51/72



the flange flanks (Hyperbolic zone of the horizontal flange section)

2 with medium-sized track curve radii and/or attack angles some of the four contact

points lie on the flange flanks (Hyperbolic zone) and some at the flank / flange tip

transition point on the horizontal flange section

3 for small track curve radii and/or large attack angles all four contact points lie on

the flank / flange tip transition points on the horizontal flange section.

The criterion for the location of the contact points is the attack angle of each wheel

in the comparison to the angles E,S/Rafter (205a/b) at the front face and/or back

face Hyperbola/flange tip transition point on the flange section. If the attack angles

2or 3are smaller than E,Sand/or 1or 4smaller than E,Rthe contact points of

the respective wheels lie on the flange flank, otherwise they lie on the flange flank /

flange tip rounding transition point.

3.1.2.1 The contact points for wheels 1 to 3 lie on the flange flanks

Following the diagram shown in figure 1.12 (also as shown in 2.1.2.1) the values

from (39) to (50) can be computed. It is to be noted that the value to be used for

rA,S/Ris obtained from (201a/b) and that for WRis obtained from 3.1.1.4.


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Figure 1.8: Calculation diagram for flange section without flange tip rounding


53/72



Fig. 1.9 Calculation diagram for curve radii with contact points at the flank / flange tip

transition point of the flange section tangential to the face (no flange tip rounding)


54/72



Figure 1.10: Calculation diagram for curve radii for contact points on the flange flanks

(Hyperbolic zone) of the flange section (no flange tip rounding), chordal

position


55/72



Fig. 1.11 Calculation sketch - curve radius for contact points on the flank / flange tip

transition point of the flange section (no flange tip rounding, chordal position)


56/72



3.1.2.2 The contact points of wheels 1, 2 and 4 lie on the flange flanks

Following the diagram shown in Figure 1.12, in the same way as in 2.1.2.2 the

values are obtained from (39) to (46), (51) and (52). It is to be noted that the valueto be used for rA,S/Ris obtained from (201a/b) and that for WRobtained from 3.1.1.4.

3.1.2.3 The contact point of wheel 1 lies at the flange flank: tip junction and those

for wheels 2 and 3 lie on the flange flanks

Following the diagram shown in figure 1.13:

The horizontal line g2tangential to the contact point in the transition point ERatwheel 1 to the midpoint of the radius rA,Sdefining the front face of wheel 2 (replacing

the front face hyperbola):

g2= [ (aF+ 0.5 l A)2+ (rA,S- dS* - dK,min)2] (210)

rA,S from (201a)

dS* from (30a)

dK,min from (33)

Total of the attack angles (1+ 2) between R1and (R2+ rA,S) :

(1+ 2) = arc cos [(R12+ (R2+ rA,S) - g2) / 2 R1(R2+ rA,R)] (211)

Horizontal angle between R1and g2:

= arc sin [sin (1+ 2) (R2+ rA,S) / g2] (212)

Horizontal angle 1between g2and the x axis:

1= arc cos [(aF+ 0.5 rA) / g2] (213)

Horizontal attack angle 1at the transition point of the flange section of wheel 1:

1= 90 - (- 1) (214)


2= ( 1+ 2) 1 (215)


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58/72



Figure 1.12: Calculation diagram for the flange footprint (no flange tip rounding), contact

points on the flange flanks


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3.1.2.5 The contact points for wheels 1 and 3 lie at the flank / flange tip transition

point; the contact point for wheel 2 lies on the flange front flank

In accordance with the diagram shown in Figure 1. 13 (relating to wheels 1 and 2)

and Figure 1. 14 (relating to wheel 3) the following can be calculated:

Values for the contact points at wheels 1 and 2 are obtained from (210) to (215), for

the curve radius R3at the contact point for wheel 3 from (27) and for the attack

angle 3at wheel 3 from (86). The value required for R3,xis calculated from (56); the

value for the y coordinate R3,yis given by:

R3,y= (R2+ rA,S)y- rA,S+ dmin* + l max* - dS* (223)

(R2+ rA,S)y from (218)

3.1.2.6 The contact points for wheels 1 and 4 lie at the flank / flange tip transition

point; the contact point for wheel 2 lies on the flange front flank

In accordance with the diagram shown in Figure 1.13 (relating to wheels 1 and 2)

and Figure 1.14 (relating to wheel 4) the following can be calculated: Values for the

contact point at wheels 1 and 2 from (210) to (215), for the curve radius R4at the

contact point for wheel 4 from (27) and for the attack angle 4at wheels 4 from (86).

The value required for R4,x,yis then given by:

R4,x= (R2+ rA,S)x 0.5 l A (224)

(R2+ rA,S)x from (221)

R4,y= (R2+ rA,S) - rA,S+ l min* + dR* (225)

(R2 + rA,S)y from (218)


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Figure 1.13: Calculation diagram for the flange footprint (no flange tip rounding), contact

point for wheel 1 at the flange flank / flange tip transition point, contact point

for wheels 2 to 4 on the flange flanks


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3.1.2.7 The contact points of wheels 1 to 3 all lie at the flank / flange tip

transition point

Following the diagram shown in Figure 1.14 will show:

Total of the horizontal attack angles + between R1and R2:

(1+ 2) = arc cos [(R12+ R2

2 g32) / (2 R1 R2)] (226)

g3- The connecting line from the back face contact transition point of the

flange section for wheel 1 to the front face contact transition point of the

flange section for wheel 2g3= [aF

2+ dK,min2] (227)

dK,minfrom (33)

Horizontal angle between R1and g3:

= arc sin [sin (1+ 2) R2 / g3] (228)


1= 90 - 1 (229)

1 - horizontal angle between g3and the x-axis

1= arc tan (dK,min/ aF) (230)

Horizontal attack angle 2at wheel 2 is obtained from (215), horizontal attack angle

at wheel 3 from (86) and curve radius R3from (27). The value required for R3,x/yis

then given by:

R3,x= R1 sin 1 from (56)

R3,y= R1 cos 1+ dR* + l max* - dS* (231)

dS/R* from (30a/b)

l max* - maximum check gauge in the GGE at the horizontal attack angle 0


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3.1.2.8 The contact points for wheels 1, 2 and 4 all lie on the flange flank / tip

transition point

The values at wheels 1 and 2, following figure 1.14, are obtained from (226) to (230)

and (215), and the value at wheel 4 from (27) and (86). The value required for R4,x/y

is obtained from:

R4,x= R2- sin 2 from (56)

R4,y= R2 cos 2-dS*+ l min*+ dR* (232)

dS/R* from (30a/b)

l min* - minimum check gauge in the GGE at the horizontal attack angle 0"

3.1.2.9 Check gauge over crossing flangeway and/or over the check rail flangeway

The investigation takes place by analogy to section 2.1.2.9, (91) to (94), using the

numerical values obtained from sections 3.1.2.1 to 3.1.2.8.

3.1 3 Gauge in the crossing zone

As in 2.1.3, the gauge in the crossing zone is derived from (95) to (96), using the

numerical value calculated from 3.1.2.9.

3.1.4 Crossing flangeway

As in 2.1.3, the crossing flangeway is obtained from (97) to (101) using the

numerical values calculated from 3.1.2.9 and 3.1.3.

3.2 Double Crossing

As in 2.2 the gauge, check gauge and flangeway width are obtained from (102) to

(106) using the numerical values calculated from 3.1.1.4.


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Figure 1.14 Diagram for calculating bogie footprint, all contact points at the flange flank /

tip transition (no flange tip rounding)


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Graphic procedure

4.1 Preliminary remarks

As an alternative to the analytical procedure, computer supported design technologyoffers a comparatively easy, but no less exact method for the investigation of the

flange clearance requirements and therefore the creation of a gauge table.

Alternatively, new wheel set geometries can also be constructed in an available

gauge system.

As CAD has become the rule, with its capability to generate 1:1 drawings, the field

of work for this requirement has grown. The automated production of the special

survey drawings in other formats is more difficult, especially using automatic

programs since for these applications intermediate steps are necessary and the

optical accuracy can be influenced. The quality of the width measurements thus

obtained are unaffected.

With the latter system, the gauge measurements can be investigated for all wheel

set and track geometries. The incremental advances in each individual case must

be aligned according to the capabilities of the CAD system.

The calculation of the gauge / check faces with the flange section(s) allowed in the

rule can be determined when a sufficiently enlarged cross section view is available.

Also by considering these optically, details can be established that are clearly less

than between 1/10 mm.

The procedure for the creation of the gauge widening table is identical with that for

the analytical procedure. The numerical values are then determined solely by

graphical methods instead of by mathematical calculation.

This procedure is based upon the graphic representation of the wheel sets and track

in any arbitrary geometry and the survey of the interface between wheel and rail in

plan elevation. The wheels become at the same time for the further observations of

the flange sections, the rails are reduced to the representation of the gauge- and

check faces in the collective geometry level (GGE).


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In the plan view, the horizontal flange sections of the wheels of the bogie at the

GGE are represented together with the gauge and check faces of the rails. In the

drawing, the projected contours of the flanges is represented with the wheel tirerotated to the appropriate attack angle together with the gauge- and check faces of

the rails.

4.2 Geometry of the horizontal flange section

4.2.1 Determination of the viewing level

Fundamentally all views in the graphic process are calculated in the collective

geometry level (GGE). The results are ultimately to be converted into the working

plane.

As a rule, a comparison of the transverse measurements in the GGE and the

working plane shows that for a normal tram wheel tire, working in the working plane

(GFT -14 mm) will yield sufficiently accurate results.

4.2.2 Construction of the flange section of the wheel set

The flange section in the two dimensional form is defined in section 2.1.1.1, figure1.1. With the analytical procedure it is, without loss of accuracy, permissible to

replace the hyperbolae that form the front and back faces of the sectioned flange by

circular arcs.

The construction of the flange section with defined geometric elements recommends

itself already therefore so that the figure remains amenable to later manipulation. In

its three dimensional form, slices or surface contours should serve only a plausibility

check on the correctness of the construction.


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Figure 1.18: View illustrating the contact point, GGE and gauging level

Figure 1.19 Classic construction of the flange section

4.3 Flange clearance requirement in straight line and curve

(Refer to Fig. 1.20)

The track curves to be observed are placed tangentially to the front and back flanks

of the flange section. The distance between them is equal to the flange width

requirement.

Concentric with the track axis, a circle with the radius (R s/2) and parallel (in

addition) with the flange width requirement is drawn a distance d from the straight

lines (shown dashed in the diagram), gives from its distance to the flange width

requirement limits in the curve the required additional width needed, separated into

dsand drfor the flange front and back faces (see Fig. 1.1). This view becomes

significant in the introduction of new bogie geometry to existing tracks.


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Fig. 1.20: Flange width requirement in the curve

4.4 Gauge and flangeway widthBy copying and mirror imaging of the flange section in accordance with the gauge

and wheelbase the sectional plan is produced through all four wheel sets. Where

necessary gauge variations (e.g. through bogie construction, wheel mounting,

loading state and/or elastically mounted wheel tires) must be allowed for.

By analogy to the flange room requirement survey, affecting curve radii in the

geometry to be observed are placed also in the groove expanse determination at

the flange sections

In the same way as for the examination of the flange width requirement, the track

curves in the geometry under investigation are placed against the flange sections,

the distance between them including allowance for the fabrication tolerances of the

wheel and rail, the total being rounded up to whole millimetres. This minimum

groove width corresponds to the flangeway width at the check rail.

The rounding up of results at the gauge survey in GGE is to be done after first re-

converting the measurements into the gauging level.

Particularly when determining the constraints for forcing, the most unfavourable

combinations of the tolerances have to be considered. This not only applies to

forcing between flange front faces and rail gauge faces at the smallest tolerable

gauge, but also to forcing between flange back flanks and check rails and/or check

gauge in combination with the largest tolerable gauge and the smallest tolerable

flangeway.


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4.4.1 Gauge and flangeway width in the track (cf. Figure 1.21)

Procedure:

1 Outer rail curve gauge face placed in contact with the flange sections

2 Inner rail gauge face placed at the nominal gauge, including any gauge widening,

with the opposite flange sections and checked for forcing - where appropriate.

3 Rotate the bogie flange footprint until the diagonally opposite flange sections

contact the rail flanks

4 Assign both the gauge and check groove head edges in conformance with the

selected (grooved) rail profile and check for forcing (where appropriate selectingthe profile with the largest flangeway width).


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4.4.2 Gauge and flangeway width in the crossing (cf. Figure 1.22)

Procedure:

1 Place the inner check rail face against the flange section.

2 Set the inner rail gauge face at the check rail flangeway width distance.

3 Rotate the bogie flange footprint until the flange face of the trailing wheel is in

contact with the rail.

4 Place outer rail gauge face so that it contacts the back of the flange section for

the trailing wheel, takes [that] but no contact takes place (where appropriate.

Observe Breadth measure premium note).

5 Place outer rail groove check face so at the flange section, so that no contacttakes place (ignoring the effects of any flare at the end of the wing rail)

6 Measure gauge and flangeway width.

Comment: In running through the main (outside-)track of an inside curved switch,

following the rules, it is the immersed flange footprint width of the trailing axle

that has to be determined.


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