56 Figure E.5b — Representation of the curves of kinematic rolling movement of the wheelset on track — Modified Wheel A / Rail A .... 60 Figure E.6b — Representation of the curves of kin
Integration of the equation of the wheelset movement of a conical profile
Figure 4 — Dimensions on the wheelset
The kinematic lateral movement of a free wheelset, with no inertia, running on a track, is defined by the following differential equation:
In accordance with this European Standard, the vehicle's forward movement speed, denoted as V, can be considered constant, leading to the conclusion that \$y = y_0\$ when \$x = 0\$.
This integration, based on the initial amplitude y 0 , leads to a periodic movement of the wheelset with a peak- to-peak amplitude of 2yˆ and a wavelength λ as presented in Figure 5.
Determining the wavelength of a conical profile
In the case of a wheelset whose wheels have a conical profile with a constant angle γ, the rolling radius difference is:
The differential Equation (4) then becomes: tan 0
2 + y = er dx y d γ (8) a second order differential equation with constant coefficients whose solution is a sinewave with a wavelength of λ λ = π γ tan
Definition of equivalent conicity for nonlinear profiles
When wheels lack a conical profile, it is necessary to employ linearization methods This allows for the application of linear differential equations by substituting tanγ with the "equivalent conicity," denoted as tanγ e.
The equivalent conicity is defined as the tangent of the cone angle of a wheelset with coned wheels, which experiences lateral movement with the same kinematic wavelength as the specified wheelset This condition applies solely to tangent tracks and very large-radius curves, where the equivalent conicity is represented as \( \tan \gamma_e = 0 \).
Annex B and Annex C give two possible methods for determining the equivalent conicity
5 Description of the reference procedure
General principles
For the described method of determination the following assumptions are used:
in the calculation both the wheel and the rail are considered rigid;
wheels are symmetrical in revolution and are represented by a single profile for each wheel;
rails are straight, parallel to each other and represented by a single profile for each rail;
wheel does not penetrate into the rail: only point contacts are considered;
no account is taken of an axle's rolling (rotation about an axis longitudinal to the track) as the wheelset moves laterally on the track;
It is important to consider this effect; however, it should be noted that in most applications to date, including all benchmark calculations in this standard, this effect has been neglected, assuming that the tangent planes of the rail and wheel are parallel at the point of contact.
NOTE 2 When using measured profiles the principles for averaging are given in Clause 6, point e) and point f)
The equivalent conicity, denoted as tanγ e, can be determined for any wheel and rail profile, whether theoretical or real, by analyzing the lateral movement yˆ of the wheelset on the track.
This is done by the following procedure:
The railway track consists of two rails, and the wheelset features two wheels, both defined within a track-centered coordinate system In this system, the x-axis is oriented longitudinally, the y-axis transversally, and the z-axis vertically Additionally, the profiles of the wheels and rails are characterized using a Cartesian coordinate system represented by (y_i, z_i).
for the rail, the profile is defined not only on top but also on the inner side;
for the wheel, the profile is defined not only on the classic wheel tread but also on the outer part and in the area of the wheel flange root
Specialized devices, including wheel and rail profile measuring apparatus, as well as automatic measuring systems mounted on dedicated railbound vehicles, are utilized to accurately determine profiles through measurement Additionally, track-based systems are employed for measuring wheel profiles.
It shall be stated in the report whether the profiles were measured in the loaded or unloaded condition
When theoretical profiles are used the inclination is to be considered
5.2.2 Accuracy of the measuring system
To ensure accuracy of the conicity calculation measurement accuracies shall lie within the following tolerances:
difference of wheel radii (measured at wheel tread plane): 0,2 mm;
wheel back-to-back distance: 0,5 mm;
The measured data for wheel and rail profiles must align with the evaluation processes, including smoothing and calculations It is important to note that tolerances for the diameter difference between wheels on the same axle are outlined in prEN 15313.
NOTE 2 For the assessment of equivalent conicity the sum of the tolerances of wheel back-to-back distance and track gauge is the relevant parameter
5.3 Determining the rolling radius difference function ∆∆∆∆ r
The determination of the Cartesian coordinates of the wheels, shall take into account the actual radius of each wheel
The Cartesian coordinates of the two wheels and two rails, based on a track-centered coordinate system, can be processed through computer techniques such as smoothing or interpolation to facilitate their use in subsequent steps of the procedure.
The characteristic ∆r = f(y) represents the difference in rolling radius between the right and left sides of a wheelset during lateral movement y on the track This characteristic is calculated relative to a centered position, with lateral movements reaching a maximum of ∆r = r1 – r2 = 5 mm, in increments of ∆y = 0.2 mm An example of this ∆r = f(y) characteristic can be found in Annex A It is important to note that the maximum lateral displacement for determining the geometric functions is influenced by the clearance between the wheel and rail.
The computation of the equivalent conicity shall be carried out for the amplitude yˆ from the actual movement of the wheelset
For practical purposes, the evaluation range for conicity changes in \$\hat{y}\$ is typically between 1 mm and 8 mm An example of this exercise's graphical representation can be found in Annex A It is important to note that the maximum value of 8 mm may not be achieved if the \$\Delta r = f(y)\$ characteristic is highly nonlinear or shows significant jumps.
To ensure the effectiveness of the evaluation system for equivalent conicity, it is essential to validate the chosen method through benchmark calculations using clearly defined test profiles, as the European Standard does not specify these parameters in practical applications.
sample rate of the profile measurement;
integration method for the evaluation of the wavelength
The validation of the equivalent conicity determination method and the profile data smoothing procedure will be conducted through a benchmark calculation using defined reference profiles along with random and grid errors This validation is accomplished through a three-step process.
step 1 checks the interpolation and calculation algorithm;
when Step 1 has been successfully passed, Step 2 applies defined errors to the data of Step 1;
when Step 2 has been successfully passed Step 3 then checks the whole candidate system
All calculations must be performed for every combination of the reference profiles outlined in Annex E, taking into account the effects of asymmetrical ∆r-functions caused by varying wheel diameters or different wheel profiles on both wheels, as specified in sections E.5 to E.9.
When applying EN 14363, it is essential to consider the equivalent conicity at a specific amplitude A rapid change in conicity at this amplitude necessitates additional measurements and calculations to accurately determine the results.
The results obtained will be evaluated against the reference results outlined in Annex E and the specified tolerance limits in Annex F For the calculation algorithm to successfully complete Step 1, the achieved results must fall within the tolerance range defined in Annex F.
Step 2 aims to evaluate the smoothing procedure by redoing the calculations from Step 1 This step involves applying the smoothing, interpolation, and calculation algorithm to the reference profiles outlined in Annex D, while incorporating all errors specified in Annex G.
The impact of random errors will be demonstrated through distinct calculations for scenarios where errors occur solely at the wheel, exclusively at the rail, and simultaneously at both the wheel and rail.
The effects shall be quantified in z-, y- and (z + y)-directions depending on the measuring system For non
Cartesian measuring systems appropriate coordinates should be used
For each case a sufficient number of calculations (≥ 20) shall be done for different randomized errors related to the grid origin position and random limit (see Annex H)
The assessment of the evaluation procedure (see Figure 2) shall be performed by a comparison of the achieved results with the reference results including the field of tolerances in Annex F
NOTE 2 The illustrations given in Annex G represent results of typical calculations These results are considered acceptable
Determining the wheel and rail profiles
Principles of measurement
The railway track consists of two rails, and the wheelset features two wheels, both defined within a track-centered coordinate system In this system, the x-axis is oriented longitudinally, the y-axis transversally, and the z-axis vertically Additionally, the profiles of the wheels and rails are characterized using a Cartesian coordinate system represented by (y_i, z_i).
for the rail, the profile is defined not only on top but also on the inner side;
for the wheel, the profile is defined not only on the classic wheel tread but also on the outer part and in the area of the wheel flange root
Specialized devices, including wheel and rail profile measuring apparatus, as well as automatic measuring systems mounted on dedicated railbound vehicles, are utilized to accurately determine profiles through measurement Additionally, track-based systems are employed for assessing wheel profiles.
It shall be stated in the report whether the profiles were measured in the loaded or unloaded condition
When theoretical profiles are used the inclination is to be considered.
Accuracy of the measuring system
To ensure accuracy of the conicity calculation measurement accuracies shall lie within the following tolerances:
difference of wheel radii (measured at wheel tread plane): 0,2 mm;
wheel back-to-back distance: 0,5 mm;
The measured data for wheel and rail profiles must align with the evaluation processes, including smoothing and calculations It is important to note that tolerances for the diameter difference between wheels on the same axle are outlined in prEN 15313.
NOTE 2 For the assessment of equivalent conicity the sum of the tolerances of wheel back-to-back distance and track gauge is the relevant parameter.
Determining the rolling radius difference function ∆∆∆∆r
The determination of the Cartesian coordinates of the wheels, shall take into account the actual radius of each wheel
The Cartesian coordinates of the two wheels and two rails, based on a track-centered coordinate system, can be processed using computer techniques such as smoothing or interpolation to facilitate their use in subsequent steps of the procedure.
The characteristic ∆r = f(y) represents the difference in rolling radius between the right and left sides of a wheelset during lateral movement y on the track This characteristic is calculated relative to a centered position, with lateral movements reaching a maximum of ∆r = r1 – r2 = 5 mm, in increments of ∆y = 0.2 mm An example of the ∆r = f(y) characteristic can be found in Annex A The maximum lateral displacement for determining the geometric functions is influenced by the clearance between the wheel and rail.
Determining the equivalent conicity
The computation of the equivalent conicity shall be carried out for the amplitude yˆ from the actual movement of the wheelset
For practical purposes, the conicity evaluation for changes in \$\hat{y}\$ typically ranges from 1 mm to 8 mm Annex A provides a graphical representation of this exercise However, the maximum value of 8 mm may not be achieved if the \$\Delta r = f(y)\$ characteristic is highly nonlinear or shows significant jumps.
Overview
Validating the method through benchmark calculations with defined test profiles is essential for evaluating the equivalent conicity, as the European Standard does not provide clear definitions for practical application.
sample rate of the profile measurement;
integration method for the evaluation of the wavelength
The determination of equivalent conicity and the smoothing procedure for profile data must be validated through a benchmark calculation using defined reference profiles along with random and grid errors This validation is accomplished through a three-step process.
step 1 checks the interpolation and calculation algorithm;
when Step 1 has been successfully passed, Step 2 applies defined errors to the data of Step 1;
when Step 2 has been successfully passed Step 3 then checks the whole candidate system.
Validation of evaluation method
All calculations must be performed for every combination of the reference profiles outlined in Annex E, taking into account the effects of asymmetrical ∆r-functions caused by varying wheel diameters or different wheel profiles, as specified in sections E.5 to E.9.
When applying EN 14363, it is essential to consider the equivalent conicity at a specific amplitude A rapid change in conicity at this amplitude necessitates additional measurements and calculations to accurately determine the outcomes.
The results obtained will be evaluated against the reference results outlined in Annex E and the specified tolerance limits in Annex F For the calculation algorithm to successfully complete Step 1, the achieved results must fall within the tolerance range defined in Annex F.
Step 2 aims to evaluate the smoothing procedure by repeating the calculations from Step 1 This step involves applying the smoothing, interpolation, and calculation algorithm to the reference profiles outlined in Annex D, while incorporating all errors specified in Annex G.
The impact of random errors and grid effects will be demonstrated through distinct calculations for scenarios where errors occur solely at the wheel, exclusively at the rail, and simultaneously at both the wheel and rail.
The effects shall be quantified in z-, y- and (z + y)-directions depending on the measuring system For non
Cartesian measuring systems appropriate coordinates should be used
For each case a sufficient number of calculations (≥ 20) shall be done for different randomized errors related to the grid origin position and random limit (see Annex H)
The assessment of the evaluation procedure (see Figure 2) shall be performed by a comparison of the achieved results with the reference results including the field of tolerances in Annex F
NOTE 2 The illustrations given in Annex G represent results of typical calculations These results are considered acceptable
Step 3 verifies that the candidate technique's entire system and process will produce results with adequate accuracy It involves recalculating the results from Step 2, utilizing the tolerances of the measuring and digitizing system under evaluation, rather than the specified errors from Step 2.
NOTE 3 The illustrations given in Annex G represent results of typical calculations These results are considered acceptable
Example of presentation of ∆∆∆∆ r function and conicity
Example of method for determining the equivalent conicity by integration of the nonlinear differential equation
Principle
The movement of the wheelset's centre of gravity on the track can be formalised on the basis of the angle: dx dy x y = &
Figure B.1 — Representation of dx , dy
Figure B.2 — Representation of ds , d ΨΨΨΨ ds = -R dΨ for a small angle Ψ, ds ≅ dx and, taking into account Equation (B.1), this yields: Ψ =
2 r r r + = is the nominal radius of each wheel r 1 - r 2 = ∆r e is the distance between the left and right contact points (approximately 1 500 mm for standard gauge)
This integration aims to determine the movement of the wheelset's center of gravity along the track, specifically focusing on the path that corresponds to half of the wavelength, starting from the minimum vertical displacement \(y_{\text{emin}}\) (where \(\Psi_{\text{emin}} = 0\)) and extending to the maximum vertical displacement \(y_{\text{emax}}\) (where \(\Psi_{\text{emax}} = 0\)).
The integral \(\int \Delta r \, dy\) must be computed only once for sufficiently large amplitudes to encompass the necessary domain for the subsequent calculations.
The wheelset's movement on the track is then obtained with the help of the following integration: dx = dy Ψ
NOTE Equation (B.5) is equivalent to the differential equation 0
∆ = Ψ + er r dx d where dx Ψ dy hence
∆ = + Ψ Ψ er d r dy such that Ψ d Ψ = - er 0
Steps of the procedure
This method is implemented by first determining the value of \( y_{em} \) that corresponds to \( \Delta r = 0 \) using the function \( \Delta r = r_1 - r_2 = f(y) \) Next, the function \( S(y) = -\int \Delta r \, dy \) is calculated, starting from \( y_{em} \) and progressing in increments of \( dy = +0.1 \, \text{mm} \) to \( +y \) and \( dy = -0.1 \, \text{mm} \) to \( -y \) The corresponding amplitudes \( y_{emin} \) and \( y_{emax} \) are then determined, allowing for the calculation of the mean lateral movement \( \hat{y} \) Subsequently, the functions \( y_{emin} = f(\hat{y}) \) and \( y_{emax} = f(\hat{y}) \) are found to ascertain the minimum and maximum amplitudes \( y_{emin} \) and \( y_{emax} \) for a given lateral movement of the wheelset \( 2\hat{y} \) Finally, the equivalent conicity \( \tan \gamma_e \) is computed for the specified movement \( \hat{y} \).
find the constant C of Equation (6) (Figure B.8) such that ψ emin = 0 for the corresponding y emin
calculate the angle ψ by integrating Equation (5) to give: Ψ = − er [ ∫ ∆ rdy + C ]
2 in steps of dy = 0,1 mm
calculate the abscissa of the wheelset movement: x = f(y) = ∫ dy Ψ between y emin and y emax which allows to find the wavelength λ of the wheelset's kinematic motion
In many instances, integration cannot be completed in a single step over the range from \( y_{\text{emin}} \) to \( y_{\text{emax}} \) Consequently, the wavelength \( x \) must be determined by summing \( dx \, \Psi \, dy \), with the step size for \( dy \) required to be less than or equal to 0.1 mm.
When the ∆r = f(y) characteristic has a negative slope, it is necessary to select for y em the most suitable point
Figure B.4 — ∆∆∆∆ r = f ( y ) characteristic with negative slope
In the example illustrated in the Figure, instead of choosing Point A for y em choose Point B or Point C according to the initial position of the wheelset
Figure B.5 — Calculation of ∫ ∆ rdy integral
Determination of: y emin , y emax corresponding and y ˆ = (y emax - y emin ) / 2
Figure B.6 — Determination of y em , calculation of ∫ ∆rdy and determination of yˆ
Figure B.7 — Determination of y emin = f ( y ˆ ) and y emax = f ( y ˆ ) functions
Example of method for determining the equivalent conicity by linear regression of the ∆∆∆∆ r function
For a linear ∆r function the slope of the ∆r function is equal to 2tanγ e
For a non linear ∆r function the slope of a linear regression gives an approximation of 2tanγ e
The regression of the ∆r function will be conducted over an interval of 2yˆ, specifically between the minimum amplitude (y emin) and maximum amplitude (y emax) of the wave's actual shape.
For symetrical ∆r functions y emin and y emax are equal to the mean amplitude yˆ , but for non-symetrical ∆r functions y emin and y emax normally are unequal to yˆ
For a mean amplitude yˆ the min and max amplitudes y emin and y emax can be calculated by the ∆r function as described in Annex B
This method is implemented by first determining the value of \( y_{em} \) where \( \Delta r = 0 \) based on the function \( \Delta r = r_1 - r_2 = f(y) \) Next, the function \( S(y) = -\int \Delta r \, dy \) is calculated, starting from \( y_{em} \) in increments of \( dy = +0.1 \, \text{mm} \) to \( +y \) and from \( y_{em} \) in decrements of \( dy = -0.1 \, \text{mm} \) to \( -y \) The corresponding amplitudes \( y_{emin} \) and \( y_{emax} \) are then determined, allowing for the calculation of the mean lateral movement \( \hat{y} \) Functions \( y_{emin} = f(\hat{y}) \) and \( y_{emax} = f(\hat{y}) \) are found to establish the minimum and maximum amplitudes for a given lateral movement of the wheelset \( 2\hat{y} \) Finally, the linear regression of the \( \Delta r \) function is calculated within the range of \( y_{emin} \) and \( y_{emax} \) to determine the equivalent conicity by \( \tan \gamma_e \).
B where B = slope of the regression f) Determine the function tanγ e = f( y ˆ) by applying Step C.2.e) For y ˆ amplitudes starting at 1 mm up to the maximum permitted by the ∆r = f(y) characteristic, with a maximum step ∆yˆ = 0,5 mm
When the ∆r = f(y) characteristic shows significant asymmetry around the origin within the 2 y ˆ interval, resembling two linear segments, it is advisable to perform separate regressions for the positive and negative sections This approach allows for the calculation of two conicities, tanγe,p and tanγe,n The overall equivalent conicity can subsequently be determined using the formula: tanγ e = 2.
Figure D.1 — Wheel A D.1.2 Analytic definition z coordinates of the profile are defined in the following ranges:
Table D.1 — Wheel profile: R-UIC 519-A — Right wheel
Figure D.2 —Wheel B D.2.2 Analytic definition z coordinates of the profile are defined in the following ranges:
Table D.2 — Wheel profile: R-UIC 519-B — Right wheel
Figure D.3 — Wheel H D.3.2 Analytic definition z coordinates of the profile are defined in the following ranges:
Table D.3 — Wheel profile: R-UIC 519-H — Right wheel
Figure D.4 — Wheel I D.4.2 Analytic definition z coordinates of the profile are defined in the following ranges:
Table D.4 — Wheel profile: R-UIC 519-I — Right wheel
Figure D.5 — Rail A D.5.2 Analytic definition z coordinates of the profile are defined in the following ranges:
Table D.5 — Rail profile: S-UIC 519-A — Right rail
Calculation results with reference profiles
The results of the combinations of reference profiles are presented in the following structure:
∆r = f(y) rolling-radius difference, necessary for calculating the equivalent conicity
The equation \$\tan\gamma_a = f(y)\$ represents the contact angle for both the right and left profiles, along with the total of these contact angles While this information is not essential for determining the equivalent conicity, it holds significance for independent wheels, serving as a relevant example.
Representation of the contact points
Representation of the curves of the kinematic rolling movement of the wheelset on track (normally not necessary)
The numerical values for the function ∆r = f(y)
The numerical values for the function tanγ e = f(y)
E.1.1 Diagram of ∆∆∆∆ r , tan γγγγ a , tan γγγγ e functions and representation of contact points
Figure E.1a — Diagram of ∆∆∆∆ r , tanγγγγ a , tanγγγγ e functions and representation of contact points —
E.1.2 Representation of the curves of kinematic rolling movement of the wheelset on track
Figure E.1b — Representation of the curves of kinematic rolling movement of the wheelset on track —
Wheel profile: R-UIC 519-A — Rail Profile: S-UIC 519-A
E.1.4 Numerical values for tan γγγγ e function
Table E.1b — Contact geometry wheel / rail: Conicity — Wheel profile: R-UIC 519-A — Rail profile: S-UIC 519-A y ˆ
E.2.1 Diagram ∆∆∆∆ r , tan γγγγ a , tan γγγγ e functions and representation of contact points
Figure E.2a — Diagram ∆∆∆∆ r , tanγγγγ a , tanγγγγ e functions and representation of contact points —
E.2.2 Representation of the curves of kinematic rolling movement of the wheelset on track
Figure E.2b — Representation of the curves of kinematic rolling movement of the wheelset on track —
Wheel profile: R-UIC 519-B — Rail profile: S-UIC 519-A
E.2.4 Numerical values for tan γγγγ e function
Table E.2b — Contact geometry wheel / rail: Conicity — Wheel profile: R-UIC 519-B — Rail profile: S-UIC 519-A y ˆ
E.3.1 Diagram of ∆∆∆∆ r , tan γγγγ a , tan γγγγ e functions and representation of contact points
Figure E.3a — Diagram of ∆∆∆∆ r , tanγγγγ a , tanγγγγ e functions and representation of contact points —
E.3.2 Representation of the curves of kinematic rolling movement of the wheelset on track
Figure E.3b — Representation of the curves of kinematic rolling movement of the wheelset on track —
Wheel profile: R-UIC 519-H — Rail profile: S-UIC 519-A
E.3.4 Numerical values for tan γγγγ e function
Table E.3b — Contact geometry wheel / rail: Conicity — Wheel profile: R-UIC 519-H — Rail profile: S-UIC 519-A y ˆ
E.4.1 Diagram of ∆∆∆∆ r , tan γγγγ a , tan γγγγ e functions and representation of contact points
Figure E.4a — Diagram of ∆∆∆∆ r , tanγγγγ a , tanγγγγ e functions and representation of contact points —
E.4.2 Representation of the curves of kinematic rolling movement of the wheelset on track
Figure E.4b — Representation of the curves of kinematic rolling movement of the wheelset on track —
Table E.4a — Contact geometry wheel / rail: ∆∆∆∆ r = f ( y ) — Wheel profile: R-UIC 519-I — Rail profile: S-UIC 519-A
E.4.4 Numerical values for tan γγγγ e function
Table E.4b — Contact geometry wheel / rail: Conicity — Wheel profile: R-UIC 519-I — Rail profile: S-UIC 519-A y ˆ
E.5.1 Diagram of ∆∆∆∆ r , tan γγγγ a , tan γγγγ e functions and representation of contact points
Figure E.5a — Diagram of ∆∆∆∆ r , tanγγγγ a , tanγγγγ e functions and representation of contact points —
E.5.2 Representation of the curves of kinematic rolling movement of the wheelset on track
Figure E.5b — Representation of the curves of kinematic rolling movement of the wheelset on track —
Diameter difference of 2 mm — Wheel profile: R-UIC 519-A — Rail profile: S-UIC 519-A
E.5.4 Numerical values for tan γγγγ e function
Table E.5b — Contact geometry wheel / rail: Conicity — Diameter difference of 2 mm — Wheel profile: R-UIC 519-A — Rail profile: S-UIC 519-A y ˆ
E.6.1 Diagram of ∆∆∆∆ r , tan γγγγ a , tan γγγγ e functions and representation of contact points
Figure E.6a — Diagram of ∆∆∆∆ r , tanγγγγ a , tanγγγγ e functions and representation of contact points —
E.6.2 Representation of the curves of kinematic rolling movement of the wheelset on track
Figure E.6b — Representation of the curves of kinematic rolling movement of the wheelset on track —
Diameter difference of 2 mm — Wheel profile: R-UIC 519-B — Rail profile: S-UIC 519-A
E.6.4 Numerical values for tan γγγγ e function
Table E.6b — Contact geometry wheel / rail: Conicity — Diameter difference of 2 mm — Wheel profile: R-UIC 519-B — Rail profile: S-UIC 519-A y ˆ
E.7.1 Diagram of ∆∆∆∆ r , tan γγγγ a , tan γγγγ e functions and representation of contact points
Figure E.7a — Diagram of ∆∆∆∆ r , tanγγγγ a , tan γγγγ e functions and representation of contact points —
E.7.2 Representation of the curves of kinematic rolling movement of the wheelset on track
Figure E.7b — Representation of the curves of kinematic rolling movement of the wheelset on track —
Diameter difference of 2 mm — Wheel profile: R-UIC 519-H — Rail profile: S-UIC 519-A
E.7.4 Numerical values for tan γγγγ e function
Table E.7b — Contact geometry wheel / rail: Conicity — Diameter difference of 2 mm — Wheel profile: R-UIC 519-H — Rail profile: S-UIC 519-A y ˆ
Figure E.8a — Diagram of ∆∆∆∆ r , tanγγγγ a , tanγγγγ e functions and representation of contact points —
E.8.2 Representation of the curves of kinematic rolling movement of the wheelset on track
Figure E.8b — Representation of the curves of kinematic rolling movement of the wheelset on track —
Diameter difference of 2 mm — Wheel profile: R-UIC 519-I — Rail profile: S-UIC 519-A
E.8.4 Numerical values for tan γγγγ e function
Table E.8b — Contact geometry wheel / rail: Conicity — Diameter difference of 2 mm — Wheel profile: R-UIC 519-I — Rail profile: S-UIC 519-A y ˆ
E.9.1 Diagram of ∆∆∆∆ r , tan γγγγ a , tan γγγγ e functions and representation of contact points
Figure E.9a — Diagram of ∆∆∆∆ r , tanγγγγ a , tanγγγγ e functions and representation of contact points —
(Right Wheel A – Left Wheel B) / Rail A
E.9.2 Representation of the curves of kinematic rolling movement of the wheelset on track
Figure E.9b — Representation of the curves of kinematic rolling movement of the wheelset on track —
(Right Wheel A – Left Wheel B) / Rail A
Wheel profile: right wheel R-UIC519-A / left wheel R-UIC 519-B — Rail profile: S-UIC 519-A
E.9.4 Numerical values for tan γγγγ e function
Table E.9b — Contact geometry wheel / rail: Conicity — Wheel profile: right wheel R-UIC 519-A / left wheel R-UIC 519-B — Rail profile: S-UIC 519-A y ˆ
The tolerances given in this annex are based on the following equation:
In scenarios where the slope of the tangent function is steep, meeting the required tolerances becomes challenging For benchmark calculations, it may be necessary to introduce additional tolerances, denoted as ∆y, to account for deviations in the y-axis The value of ∆y can be determined using a specific equation.
∆y = a 0 (1 – cos(arctan a 1 dy dtanγ e )) where a 0 = 0,2 mm; a 1 = 10 mm
Table F.1 — Benchmark calculations: Tolerances — Wheel profile: R-UIC 519-A — Rail profile: S-UIC 519-A y ˆ
[mm] tanγγγγ e tanγγγγ e max tanγγγγ e min tanγγγγ e max tanγγγγ e min including tolerance∆ (tanγe) including supplementary tolerance ∆y
[mm] tanγγγγ e tanγγγγ e max tanγγγγ e min tanγγγγ e max tanγγγγ e min including tolerance∆ (tanγe) including supplementary tolerance ∆y
Table F.2 — Benchmark calculations: Tolerances — Wheel profile: R-UIC 519-B — Rail profile: S-UIC 519-A y ˆ
[mm] tanγγγγ e tanγγγγ e max tanγγγγ e min tanγγγγ e max tanγγγγ e min including tolerance∆ (tanγe) including supplementary tolerance ∆y
[mm] tanγγγγ e tanγγγγ e max tanγγγγ e min tanγγγγ e max tanγγγγ e min including tolerance∆ (tanγe) including supplementary tolerance ∆y
Table F.3 — Benchmark calculations: Tolerances — Wheel profile: R-UIC 519-H — Rail profile: S-UIC 519-A y ˆ
[mm] tanγγγγ e tanγγγγ e max tanγγγγ e min tanγγγγ e max tanγγγγ e min including tolerance∆ (tanγe) including supplementary tolerance ∆y
[mm] tanγγγγ e tanγγγγ e max tanγγγγ e min tanγγγγ e max tanγγγγ e min including tolerance∆ (tanγe) including supplementary tolerance ∆ y
Table F.4 — Benchmark calculations: Tolerances — Wheel profile: R-UIC 519-I — Wheel profile: S-UIC 519-A y ˆ
[mm] tanγγγγ e tanγγγγ e max tanγγγγ e min tanγγγγ e max tanγγγγ e min including tolerance∆ (tanγe) including supplementary tolerance ∆y
[mm] tanγγγγ e tanγγγγ e max tanγγγγ e min tanγγγγ e max tanγγγγ e min including tolerance∆ (tanγe) including supplementary tolerance ∆y
F.5 Modified Wheel A (-2 mm on left wheel diameter) / Rail A
Table F.5 — Benchmark calculations: Tolerances — Wheel profile: R-UIC 519-A — Diameter difference of 2 mm — Rail profile: S-UIC 519-A y ˆ
[mm] tanγγγγ e tanγγγγ e max tanγγγγ e min tanγγγγ e max tanγγγγ e min including tolerance∆ (tanγe) including supplementary tolerance ∆y
[mm] tanγγγγ e tanγγγγ e max tanγγγγ e min tanγγγγ e max tanγγγγ e min including tolerance∆ (tanγe) including supplementary tolerance ∆y
Figure F.6 — Diagram modified Wheel B / Rail A
Table F.6 — Benchmark calculations: Tolerances — Wheel profile: R-UIC 519-B — Diameter difference of 2 mm — Rail profile: S-UIC 519-A y ˆ
[mm] tanγγγγ e tanγγγγ e max tanγγγγ e min tanγγγγ e max tanγγγγ e min including tolerance∆ (tanγe) including supplementary tolerance ∆y
[mm] tanγγγγ e tanγγγγ e max tanγγγγ e min tanγγγγ e max tanγγγγ e min including tolerance∆ (tanγe) including supplementary tolerance ∆y
F.7 Modified Wheel H (-2 mm on left wheel diameter) / Rail A
Table F.7 — Benchmark calculations: Tolerances — Wheel profile: R-UIC 519-H — Diameter difference of 2 mm — Rail profile: S-UIC 519-A y ˆ
[mm] tanγγγγ e tanγγγγ e max tanγγγγ e min tanγγγγ e max tanγγγγ e min including tolerance∆ (tanγe) including supplementary tolerance ∆y
[mm] tanγγγγ e tanγγγγ e max tanγγγγ e min tanγγγγ e max tanγγγγ e min including tolerance∆ (tanγe) including supplementary tolerance ∆y
Figure F.8 — Diagram modified Wheel I / Rail A
Table F.8 — Benchmark calculations: Tolerances — Wheel profile: R-UIC 519-I — Diameter difference of 2 mm — Rail profile: S-UIC 519-A y ˆ
[mm] tanγγγγ e tanγγγγ e max tanγγγγ e min tanγγγγ e max tanγγγγ e min including tolerance∆ (tanγe) including supplementary tolerance ∆y
[mm] tanγγγγ e tanγγγγ e max tanγγγγ e min tanγγγγ e max tanγγγγ e min including tolerance∆ (tanγe) including supplementary tolerance ∆y
F.9 (Right Wheel A – Left Wheel B) / Rail A
Table F.9 — Benchmark calculations: Tolerances — Wheel profile: right wheel R-UIC 519-A / left wheel R-UIC 519-B — Rail profile: S-UIC 519-A y ˆ
[mm] tanγγγγ e tanγγγγ e max tanγγγγ e min tanγγγγ e max tanγγγγ e min including tolerance∆ (tanγe) including supplementary tolerance ∆y
[mm] tanγγγγ e tanγγγγ e max tanγγγγ e min tanγγγγ e max tanγγγγ e min including tolerance∆ (tanγe) including supplementary tolerance ∆y
Examples of calculation results with introduced errors
G.1 Wheel A / Rail A – Random error in mm z y
Figure G.1 — Wheel A / Rail A — Random error in mm
G.2 Wheel A / Rail A — Random error in mm z y wheel ± 0,025 ± 0,025 rail ± 0,025 ± 0,025 z y
Figure G.3 — Wheel A / Rail A — Random error in mm
G.4 Wheel A / Rail A — Grid error in mm z y
Figure G.5 — Wheel A / Rail A — Grid error in mm
G.6 Wheel A / Rail A — Grid error in mm z y
Figure G.7 — Wheel H / Rail A — Random error in mm
Guideline for application of errors
Figure H.1 shows the result of a digitizing process of measured points with grid widths of ∆ y and ∆ z The real coordinates (y, z) of a point P are transformed to the nearest grid point (y', z')
In an "optimum" transformation to the nearest grid point (NINT), the area defined by the gridlines is divided into four equal quadrants, with all points within the same quadrant being assigned to the same grid point.
Figure H.1 — Transformation of the point P ( x , y ) to grid with grid widths ∆∆∆∆ y, ∆∆∆∆ z y i ' = ∆∆∆∆ y [NINT((y i +y 0 )/ ∆∆∆∆ y )] z i ' = ∆∆∆∆ z [NINT((z i +z 0 )/ ∆∆∆∆ z )]
NINT = Elemental Intrinsic Function (Generic): Returns the nearest integer to the argument i1 = NINT(2.783) ! returns 3
In Figure H.2 this method is applied to all points of a continuous, plane curve The result is a row of discrete points - here shown as linear interpolated function
Figure H.2 — Grid transformation with grid widths of 0,5 mm
To effectively evaluate the equivalent conicity, it is essential to select a sufficient number of different origin positions (y₀, z₀) for the grid O(y₀, z₀), with a recommended minimum of 50 positions This approach ensures a comprehensive investigation of the evaluation process, as illustrated in Figure H.3, which compares a data grid of 0.5 mm with an alternative grid of 0.5 mm featuring varied origin positions.
The arbitrary origin positions can be generated with random values (via the function RANDOM())
(Portability functions: Return real random numbers in the range 0,0 through 1,0) in the range ]0,1[ y 0 = [(upper limit - lower limit)RANDOM()+lower limit] z 0 = [(upper limit - lower limit)RANDOM()+lower limit]
In the example below the following limits are used for the variation of the grid origin:
The result is a evenly spread band of transformed points (see Figure H.4)
Figure H.4 — 50 variants of grid origins
To analyze the smoothing and calculation method, a random error is introduced to the reference profiles using the formula: \( y_i = y_i + \left[(\text{upper limit} - \text{lower limit}) \cdot \text{RANDOM()} + \text{lower limit}\right] \) and \( z = z + \left[(\text{upper limit} - \text{lower limit}) \cdot \text{RANDOM()} + \text{lower limit}\right] \).
In the example below the following limits are used for the generation of vertical random errors:
Figure H.5 shows a part of a profile with four different random errors
Figure H.5 — Random error of measuring points
Minimum requirements for the input parameter (real profiles) of equivalent conicity are defined
For effective applications, it is essential to utilize specialized devices for measuring profiles, such as wheel and rail profile measuring apparatus or automatic systems on rail-bound vehicles Reports must specify whether profiles were measured in loaded or unloaded conditions To analyze the dynamic running behavior of a railway vehicle, a track section should include at least 11 regularly spaced profiles over a 100 m line, with the average conicity and its standard deviation reported Additionally, rail profile measurements may be necessary to calculate mean conicities over specific lengths When measuring wheel profiles, it is crucial to ensure that the profile used for calculating equivalent conicity accurately represents the wheel's circumferential profile.
In specific scenarios, a real wheel can be defined by averaging measurements taken at regular intervals around the circumference The bending of the axle under load is significant, necessitating measurements and calculations under normal service conditions near the contact point For freight vehicles, load can greatly affect contact geometry, so both empty and laden conditions should be considered when using equivalent conicity to assess running behavior Vehicles typically encounter a wide variety of rail profiles, making it impractical to account for all types While equivalent conicity is crucial for dynamic behavior on straight tracks or large-radius curves, it becomes less relevant on sharp curves, where the ∆r function may impact wheelset or bogie performance In such cases, actual rail profiles should be utilized due to wear considerations.
Reference of the relevant equivalent conicity in the assessment process according to EN 14363:2005, 5.4.3.3 Measured a Theoretical b
Contribution to the assessment of vehicle’s behaviour Measured a Measured c
Study of severe instability, for example during acceptance tests
Assessment of rail profiles (incl gauge), check for need for track maintenance measures (reprofiling/grinding of the rails) Theoretical d Measured c
Assessment of wheelset conditions, check for need for wheel profile maintenance measures (reprofiling of the wheel set) Measured a Theoretical b
Theoretical evaluation during the design process
When measuring wheel profiles, it is essential to conduct these assessments under normal load conditions to accurately interpret results, as wheelset deformation can significantly impact findings Additionally, factors such as rail inclination and gauge must be considered For rail profile measurements, using an inspection car under normal load is recommended; otherwise, the effects of load, particularly on tracks with soft fastenings, should be factored into the analysis Furthermore, the distance between the active faces of the wheels is a critical consideration in these evaluations.
The European Standard is closely linked to the Essential Requirements outlined in EU Directive 2008/57/EC, which focuses on ensuring interoperability within the rail system across the Community This directive, established by the European Parliament and Council on June 17, 2008, aims to enhance the efficiency and safety of rail transport in Europe.
This European Standard was developed under a mandate from the European Commission and the European Free Trade Association to ensure compliance with the Essential Requirements outlined in Directive 2008/57/EC.