EP1523597B1 - Track with a levelling curve and force-minimal superelevation ramp - Google Patents

Abstract

A track comprising a track centre line of variable curvature on a horizontal section and having a variable superelevation angle. The curvature is determined on the basis of an assumed superelevation function such that the overall non-compensated lateral acceleration at a selected fixed aligning height, taking into account the amount of non-compensated lateral acceleration caused by the swaying movement, has a characteristic curve like that of said function and fulfils the following differential equation (formula I) wherein: KH(S) represents the curvature of the track centre line on a horizontal section, s represents the curve length along the track centre line, KC represents the constant reference curvature (in an arc), C represents the constant reference elevation angle (in an arc), (S) represents the elevation angle, h represents the aligning height, d represents the differential operator.

Description

The invention relates to a track with transitional bow and krafteminimal Überhöhungsrampe and on the track guidance of such a track.

The routing of the tracks of railways, subways and other track-guided vehicles is usually carried out as a succession of elements of constant curvature in plan, such as straight and circular arc, and elements with variable curvature. The transition from a straight track to a straight track deviating from it at an angle is carried out either in two parts, that is to say with two adjoining vertex transition arcs, or the transition is made in three parts, to which a straight track is followed by a conventional transitional bend Connects circular arc, which is followed by another conventional transition arc, which then joins the second straight track.

A track distortion, which is the excessive transition from one track to another, parallel track at a certain distance, is currently carried out either from two circular arcs or from a sequence of a transition arc, a circular arc, a turning transition arc, a circular arc and a transition arc.

In order to reduce the unbalanced lateral acceleration in the vehicle and the cornering forces in the curved track elements, the guide surface is rotated about the center line of the guide.

In railroads, that angular travel, measured at the top of the arched rail relative to the top of the arched rail, about which the transverse direction of the track is twisted about the track centerline, is referred to as elevation. To set this elevation is usually prescribed a proportional to the curvature of the track center line elevation. This results in no elevation on straight stretches and for curved stretches an increasing with the curvature of the track centerline cant on the outside of the track.

When traversing tracks, the most common element that has a variable curvature in plan is the clothoid. This is a curve in which the curvature changes linearly from one value to another. The associated, proportional to the curvature elevation is a linearly changing elevation, the straight Überhöhungsrampe. This results in kinks at the connection points to the neighboring elements with constant elevation. A unbalanced lateral acceleration is then obtained in the guide center line which changes linearly between the constant values in the neighboring elements.

It does not take into account that due to the rolling movement of the guided along the track vehicle outside the guide center line everywhere speed jumps occur and thus there the accelerations are infinite. That is, mathematically speaking, vehicle points on the guide center line are singular points where these infinite accelerations do not occur. The conventional kinematic description is therefore not sufficient.

The infinite accelerations are virtually avoided by planned or freely adjusting fillets of the rails at the kinks. However, the proportionality between curvature and superelevation is lost in the area of the kinks provided with fillets and thus also the desired course of the unbalanced lateral acceleration in the guide center line.

The disadvantages of this type of routing are known. To avoid this, instead of the linear gradient function, two squared quadratic parabolas or a cubic polynomial or a cosine half-wave are used to define the curvature in the plan and the elevation. This makes the speed curves in each vehicle point steady. The accelerations remain finite, but are unsteady, and their temporal changes - the jerks - still have infinity points. To remove these as well, a linear function is used together with a sine wave. Then the accelerations become steady and the jerks at the junctions have finite values.

Another aspect is that the routing of the tracks should be such that the desired kinematic properties not along the guide center line, but for a particular vehicle point, such as the center of gravity of the vehicle to be respected. In order to avoid geometric discontinuities in the guide elements, then the function outside the guide center line must be twice continuously differentiable.

From the AT 401 781 B is a track with a real surface strip with a steady curve of the curvature in the transition arc is known, in the routing of a fictitious surface strip is provided with a guideline along which a rated speed moving point has an unbalanced lateral acceleration of zero. From the guideline of the fictitious surface stripe, the guideline of the real surface stripe is created by shifting each individual guideline point along the stripe normal by a constant distance.

From the AT 402 211 B is a track with transitional arch known in the routing of each of the twice-continuous differentiable functions are used for the elevation angle and for the curvature of the guideline in plan. Furthermore, special functions based on the hyperbolic and sine square functions are given. The hyperbolic tangent function is arbitrarily often continuously differentiable and all its derivatives fit together at the edges of the transition arc with those of the subsequent tracing elements of constant curvature. In addition, the nonlinear relationship is given for large pitch and pitch angles for the balanced pitch at a given speed.

A track according to the preamble of claim 1 is known from DE-A-32 28553 known.

The practical application of this routing has several problems:

The actual course of the track is not known a priori. It arises only after application of the transformation from the fictional to the real track strip. In practice, however, is a direct indication of the function of the track centerline, as it is from is used to all conventional routes. Similar is the problem in assessing track position errors in terms of their effect on the kinematics of the vehicle. The track position errors must be transformed from the real track strip into the fictitious track strip. Only there, for example, the associated unbalanced lateral acceleration of the vehicle is determined.

When transitioning from one exaggeration level to another exaggeration level, a corresponding drastic change must occur somewhere in the affected derivatives. In the straight ramp, the first derivative (the angle) is minimal, and with it the track twist and the roll angular velocity, but all other derivatives are unlimited but at the edges. In the case of the ramp composed of two square parabolas, the second derivatives have minimum values corresponding to the ramp curvature. Thus, the roll angle accelerations are minimal in this embodiment, but the ramp is steeper in the middle and the other derivatives at the edges and in the middle do not exist and therefore not the roll angle. This is similar to the cubic polynomial and the cosine half wave. In the case of a linear progression with a superimposed sine wave, the jerks at the edges also exist, but the first and second derivative show higher values than the other superelevation ramps.

The well-known requirement that the second derivatives of the course functions still exist is also fulfilled by all known courses except the straight-sided clothoid. The infinitely variable differentiability of the course functions also has its disadvantages. Due to the very shallow transition at the edges, the higher discharges in between become unnecessarily large and thus also, for example, the roll angle accelerations and roll angle pressures.

Infinitely differentiable functions are transcendent, such as the hyperbolic tangent. They have a theoretically infinite mathematical smoothness at the junctions. Practically, however, their analytic differentiability is no longer present after a few derivations, since the expressions become unwieldy long. An analytical integrability, for example, the curvature to the position angle, which is also beneficial for practical work, is not given anyway. Thus, for the actual evaluation of transcendental functions, only the numerical differentiation and Integrate, where the continuity depends on the algorithm used, but is limited in each case.

What is desired is a function course which has just the necessary requirements for the differentiability at the transition points and, if possible, for all physical parameters, small values for a favorable design.

One aspect that has not yet been considered is the limitations due to the bendability of the continuously welded, originally straight rail. In the usual consideration of the rail as a continuous bedded carrier, that is, the effect of the rail fasteners is distributed, the elevation corresponds directly to the course function of the curved rail. Its second location derivative is according to elementary Bernoulli-Euler bending theory proportional to the bending moment in the rail, the third is proportional to the lateral force and the fourth derivative corresponds to the bedding distribution, with which the rail is brought into the desired ramp shape and must be maintained.

The object of the invention is to avoid the above disadvantages and to provide a track that can be produced in reality and achieves a smooth course of the unbalanced lateral acceleration.

wobei

• κ_H (s) .........Krümmung der Gleismittellinie im Grundriss
• s ..... Bogenlänge längs der Gleismittellinie
• κ_c ..... konstante Bezugskrümmung (in einem Kreisbogen)
• ψ _C ............. konstanter Bezugsüberhöhungswinkel (in einem Kreisbogen)
• ψ (s) ............Überhöhungswinkel
• h ..................Trassierungshöhe
• d ..................Differentialoperator

bedeuten.As a solution, the invention proposes a track having a track centerline with variable curvature in plan and a variable cant angle. This track is inventively characterized in that the curvature is determined from a function assumed for the elevation so that the total unbalanced lateral acceleration at a selected fixed Trassierungshöhe taking into account caused by the rolling motion portion of the unbalanced lateral acceleration has a course like this function, and following Differnzialgleichung met. $${\mathit{\kappa }}_{\mathit{H}}\left(s\right)=\frac{{\mathit{\kappa }}_{\mathit{C}}}{{\mathit{\psi }}_{\mathit{C}}}\mathit{\cdot }\mathit{\psi }\left(s\right)-\mathrm{<figure-callout\; id="2"\; label="H\; \cdot \; d"\; filenames="imgf0002.png"\; state="\{\{state\}\}">H\; </figure-callout>}\cdot \frac{{\mathrm{d}}^{}}{}\frac{{\mathrm{}}^2\mathit{\psi }}{{\mathrm{d}\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0002.png"\; state="\{\{state\}\}">s</figure-callout>}}^2}$$

in which

• κ _H ( s ) ......... Curvature of the track centerline in plan
• s ..... arc length along the track centerline
• κ _c ..... constant reference curvature (in a circular arc)
• ψ _C ............. Constant r reference elevation angle (in a circular arc)
• ψ ( s ) ............ elevation angle
• h .................. routing height
• d .................. differential operator

mean.

A further feature of the invention provides that the function can be differentiated at least four times in its entire course, including at the area borders, and also that the fourth derivatives of the function have finite values everywhere.

A further feature of the invention provides that in the determination of the curvature ( κ _H ) of the track center line in the floor plan, the route height zero is selected as the fixed route height.

mit $0\le \frac{s}{l}\le 1.$

A further feature of the invention provides that the normed function used is the seventh order polynomial given below in equation (2), this normalized function being used for the profile of the peaking angle and for the total, unbalanced lateral acceleration, and from this the curvature (κ _H ) the track centreline is determined in plan according to equation (1): $$f\left(\frac{s}{l}\right)=(\frac{s}{l}{)}^4\cdot \left(35-84\cdot \frac{s}{l}+70\cdot (\frac{s}{l}{)}^2-20\cdot (\frac{s}{l}{)}^3\right)$$

With $0\le \frac{s}{l}\le 1.$

• s ..................Bogenlänge längs der Gleismittellinie
• l ...................Länge des Übergangsbogens und der Überhöhungsrampe.

Where:

• s .................. arc length along the track centerline
• l ................... Length of the transitional arch and the elevation ramp.

mit $\tan \frac{Z}{2}=\frac{Z}{2}\approx 4,49340946,0\le \frac{s}{l}\le 1.$

A further feature of the invention provides that the function given below in equation (3) with a third-order polynomial in combination with the sine and the cosine and a constant value (Z) is used as the normalized function, this normalized function for the Course of the elevation angle and for the total, unbalanced lateral acceleration is used and from this the curvature (κ _H ) of the track center line is determined in the plan according to equation (1): $$f\left(\frac{s}{l}\right)=(\frac{s}{l}{)}^{2\cdot }\cdot \left(-2\cdot \frac{s}{l}+3\right)+\frac{6}{{\mathrm{<figure-callout\; id="2"\; label="Z"\; filenames="imgf0002.png"\; state="\{\{state\}\}">Z</figure-callout>}}^2}\cdot \left(+2\cdot \frac{s}{l}-1+\cos \left(Z\cdot \frac{s}{l}\right)-\frac{2}{Z}\cdot \sin \left(Z\cdot \frac{s}{l}\right)\right)$$

With $\mathrm{<figure-callout\; id="2"\; label="tan\; Z"\; filenames="imgf0002.png"\; state="\{\{state\}\}">tan\; </figure-callout>}\frac{Z}{}\frac{}{2}=\frac{\mathrm{<figure-callout\; id="2"\; label="Z"\; filenames="imgf0002.png"\; state="\{\{state\}\}">Z</figure-callout>}}{2}\approx 4.49340946.0\le \frac{s}{l}\le 1.$

• s ...................Bogenlänge längs der Gleismittellinie
• l ................... Länge des Übergangsbogens und der Überhöhungsrampe.

Where:

• s ................... arc length along the track centerline
• l ................... Length of the transitional arch and the elevation ramp.

mit $0\le \frac{s}{l}\le 1.$

A further feature of the invention provides that the function given below in equation (4) is used as a normalized function with a third-order polynomial in combination with the sine and the cosine, this normalized function for the course of the over-elevation angle and for the whole , unbalanced lateral acceleration is used and from this the curvature (κ _H ) of the track centerline is determined in the plan according to equation (1): $$f\left(\frac{s}{l}\right)=+\frac{1}{{\mathit{\pi }}^2-9}\cdot \left(+{\mathrm{<figure-callout\; id="2"\; label="\pi "\; filenames="imgf0002.png"\; state="\{\{state\}\}">\pi </figure-callout>}}^2\cdot (\frac{s}{l}{)}^2\cdot (-2\cdot \frac{s}{l}+3)-6\cdot \left(+1-\cos \frac{\pi \cdot s}{l}\right)\right)+\frac{3}{{\pi }^2-9}\cdot (+\frac{s}{l}-\frac{1}{2\cdot \pi }\cdot \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="imgf0002.png"\; state="\{\{state\}\}">sin</figure-callout>}\left(2\cdot \pi \cdot \frac{s}{l}\right))$$

With $0\le \frac{s}{l}\le 1.$

• s .............Bogenlänge längs der Gleismittellinie
• 1 ............ Länge des Übergangsbogens und der Überhöhungsrampe.

Where:

• s ............. arc length along the track centerline
• 1 ............ Length of transitional arch and elevation ramp.

mit $0\le \frac{s}{l}\le 1.$

Another feature of the invention provides that, as a normalized function, the function given below in equation (5) is used with a fifth-order polynomial in combination with only the sine, which normalizes Function is used for the course of the elevation angle and for the total, unbalanced lateral acceleration and from this the curvature (κ _H ) of the track center line is determined in plan according to equation (1): $$f\left(\frac{s}{l}\right)=+\frac{15}{15-{\mathit{<figure-callout\; id="2"\; label="\pi "\; filenames="imgf0002.png"\; state="\{\{state\}\}">\pi </figure-callout>}}^2}\cdot \left(+\frac{s}{l}-\frac{1}{2\cdot \pi }\cdot \sin (2\cdot \pi \cdot \frac{s}{l})\right)-\frac{{\mathrm{<figure-callout\; id="2"\; label="\pi "\; filenames="imgf0002.png"\; state="\{\{state\}\}">\pi </figure-callout>}}^2}{15-{\mathrm{<figure-callout\; id="2"\; label="\pi "\; filenames="imgf0002.png"\; state="\{\{state\}\}">\pi </figure-callout>}}^2}\cdot (\frac{s}{l}{)}^2\cdot (+6\cdot (\frac{s}{l}{)}^2-15\cdot \frac{s}{l}+10)$$

With $0\le \frac{s}{l}\le 1.$

• s ..................Bogenlänge längs der Gleismittellinie
• l ................ Länge des Übergangsbogens und der Überhöhungsrampe.

Where:

• s .................. arc length along the track centerline
• l ................ Length of the transitional arch and the elevation ramp.

mit $0\le \frac{s}{l}\le 1.$

A further feature of the invention provides that the function given below in equation (6) is used as a normalized function with a fifth-order polynomial in combination with only the cosine, this normalized function for the course of the over-elevation angle and for the entire, unbalanced one Side acceleration is used and from the curvature (κ _H ) of the track center line is determined in the plan according to equation (1): $$f\left(\frac{s}{l}\right)=+\frac{5}{10-{\mathit{<figure-callout\; id="2"\; label="\pi "\; filenames="imgf0002.png"\; state="\{\{state\}\}">\pi </figure-callout>}}^2}\cdot \left(+1-\cos \frac{\pi \cdot s}{l}\right)-\frac{{\mathrm{<figure-callout\; id="2"\; label="\pi "\; filenames="imgf0002.png"\; state="\{\{state\}\}">\pi </figure-callout>}}^2}{2\cdot (10-{\pi }^2)}\cdot (\frac{s}{l}{)}^2\cdot (+2\cdot (\frac{s}{l}{)}^3-5\cdot {\left(\frac{s}{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="imgf0002.png"\; state="\{\{state\}\}">l</figure-callout>}}\right)}^2+5)$$

With $0\le \frac{s}{l}\le 1.$

• s ..................Bogenlänge längs der Gleismittellinie
• l ...................Länge des Übergangsbogens und der Überhöhungsrampe.

Where:

• s .................. arc length along the track centerline
• l ................... Length of the transitional arch and the elevation ramp.

mit $0\le \frac{s}{l}\le 1.$

A further feature of the invention provides that the ninth-order polynomial given below in equation (7) is used as the normalized function, this normalized function for the course of the over-elevation angle and for the total, unbalanced lateral acceleration is used and from this the curvature (κ _H ) of the track center line is determined in the plan according to equation (1): $$f\left(\frac{s}{l}\right)={\left(\frac{s}{l}\right)}^{5\cdot }\cdot \left(126-420\cdot \frac{s}{l}+540\cdot {\left(\frac{s}{l}\right)}^2-315\cdot {\left(\frac{s}{\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="imgf0002.png"\; state="\{\{state\}\}">l</figure-callout>}}\right)}^3+70\cdot {\left(\frac{s}{\mathrm{<figure-callout\; id="4"\; label="l"\; filenames="imgf0002.png"\; state="\{\{state\}\}">l</figure-callout>}}\right)}^4\right)$$

With $0\le \frac{s}{l}\le 1.$

• s ..................Bogenlänge längs der Gleismittellinie
• l ................ Länge des Übergangsbogens und der Überhöhungsrampe.

Where:

• s .................. arc length along the track centerline
• l ................ Length of the transitional arch and the elevation ramp.

A further feature of the invention provides that a raised element which connects a straight track to a straight track diverging at an angle thereto is designed in one piece.

A further feature of the invention is that a four times finite value differentiable function is used for a one-piece and cambered track element connecting a straight track to a straight track diverging at an angle therefrom.

Another feature of the invention is that a canted track-tie connecting a straight track to a straight track parallel thereto is laid out in one piece.

A further feature of the invention is that a four times finite value differentiable function is used for a one-piece, over-the-top rail distortion connecting a straight track to a parallel straight track.

The invention of exemplary embodiments will be explained in more detail with reference to the drawings.

Fig.1

schematisch ein auf einem mit einer Überhöhung versehenen Gleis befindliches Fahrzeug,

Fig. 2

eine normierte Verlaufsfunktion einer erfindungsgemäßen kräfteminimalen Rampe mit ihren normierten Ableitungen und

Fig. 3

einen normierten Krümmungsverlauf für einen Übergangsbogen mit am Beginn und Ende verstärkter Krümmung.

In the drawings show:

Fig.1

schematically a vehicle located on a track provided with an elevated track,

Fig. 2

a normalized course function of a power minimal ramp according to the invention with their normalized derivatives and

Fig. 3

a normalized curvature curve for a transition arc with increased curvature at the beginning and end.

The vehicle on its guide, the track, is considered taking into account its height, Fig. 1 , The dimensioning height (h) is understood to be the height at which the unbalanced lateral acceleration is considered and evaluated.

In order to create a track with a continuous bedding, course functions are used for the course of the rails in plan and in the altitude, in which the fourth derivatives still exist. For an exact setting of a given geometry, its actual manufacturability and the associated requirement for limited fourth derivatives of the respective course function are of crucial importance. Thus, the jerk distribution in the entire vehicle cross-section is continuous and the kinematics of the vehicle meets all conditions.

The known requirement for the existence of the second derivatives is therefore not sufficient, the existence of infinitely many derivatives at the edges, however, brings the disadvantages described.

Fig. 2 shows by way of example the normalized course function of a force minimum ramp according to the invention with its likewise normalized derivatives, starting from the formula (2).

The function itself as the 0 derivative corresponds to the overshoot profile. Its 1st derivative is the ramp angle corresponding to the track distortion and the angular velocity of the vehicle about the longitudinal axis. The second derivative is still smooth and proportional to the rail curvature in the elevation, to the angular acceleration about the longitudinal axis of the vehicle and to the bending moment in the rail (s) forming the ramps. The 3rd derivative is still continuous and corresponds to the change of the rail curvature in the elevation, the angular pressure about the longitudinal axis of the vehicle and the lateral force in the rail (s) forming the ramps. The 4th derivative still exists. It has discontinuities at the edges and is proportional to the rail (s) forming on the ramps over the Rail fasteners acting force distribution per unit length, which is required to hold the ramp (s) in this form.

These course functions can be attached to a fictitious track strip, from which then the real track strip is obtained by projection. A special case would be conventional routing, where the two stripes become identical. The over-elevation ramp according to the invention with the curvature proportional thereto then produces a transitional arc with the corresponding smoothness at the connection points.

According to the invention, however, the procedure is different here: the elevation ramp itself is already known. We are looking for a direct stakeout of the track in such a way that the desired kinematic behavior of the vehicle running on the track is achieved.

As usual, the unbalanced lateral acceleration is considered. If one attaches these outside the track level, the known term, which consists of the product curvature times the squared driving speed, adds another term to the rolling motion, namely the roll angular acceleration about the vehicle longitudinal axis multiplied by the distance from the track centerline. If the curvature in the floor plan is chosen so that one part of it just compensates for the last-mentioned term and the other term is proportional to the superelevation, then the unbalanced lateral acceleration also becomes proportional to the superelevation. The curvature in the floor plan accordingly consists of two parts, a conventional proportion corresponding to the course of the elevation and a proportion proportional to the second derivative of the course of the elevation. It provides the well-known swinging of the transition arc, that is, at a transition from a straight line into a circular arc arise at the beginning first curvatures with opposite signs and a position on the other side of the circle to be reached. Fig. 3 shows a corresponding normalized curvature as it arises with the use of equations (1) and (12).

With this method, one obtains a completely independent of all kinematic variables description of the track center line, which, as in the conventional routing, are advantageously used purely geometrically can. Kinematic quantities are only needed for control in the sense of admissibility to certain rules.

The described method can generally be applied to paths in areas of variable curvature and elevation and not only transition curves.

There are three gradients which determine the properties of a path: the geometric functions of the curvature and the elevation and the kinematic function of the unbalanced lateral acceleration, preferably in the trassing height.

In the known paths, it is assumed that the geometric functions at the track centerline and also the calculation of the kinematic function always carried out only for the track centerline.

According to the invention, the procedure is different here: Accordingly, starting from a grading function that is at least three times differentiable or for fulfilling the requirements of bending theory starting from a four times differentiable grading function, the elevation and the unbalanced lateral acceleration in the trassing height, taking account of the component caused by the rolling motion, must compensate for the unbalanced lateral acceleration Follow function and from the curvature in the floor plan are determined.

With vanishingly selected track height (track height (h) equal to zero), a track course is then obtained in which the track center line likewise follows this function, as is currently the case.

At non-neglected trassing height occurs - due to the compensated proportion of unbalanced lateral acceleration due to the rolling motion - to a change in the curvature away from the course function, which then leads to a decay at the beginning in a transition arc from a straight line to a circle.

Written as a formula, the unbalanced lateral acceleration expediently expresses itself as an angle (Froude number) as follows: $${\beta }_Q=\frac{a_Q}{G}=\frac{i}{b}=\frac{{\mathrm{\kappa }}_H\cdot {\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="imgf0002.png"\; state="\{\{state\}\}">v</figure-callout>}}^2+H\cdot \alpha }{G}-\mathit{\psi }$$

• β_Q ............. Winkel der unausgeglichenen Seitenbeschleunigung
• a_Q .............. unausgeglichene Seitenbeschleunigung
• g ................ Fallbeschleunigung
• i ................. Überhöhungsfehlbetrag bei Zweischienenbahnen
• b ................ Lauflinienabstand (Spurweite) bei Zweischienenbahnen
• κ _H ............ Krümmung der Gleismittellinie
• ν .............. Fahrgeschwindigkeit
• h ..................Trassierungshöhe
• α ..................Rollwinkelbeschleunigung
• ψ ................Überhöhungswinkel.

Where:

• β _Q ............. Angle of unbalanced lateral acceleration
• a _Q .............. unbalanced lateral acceleration
• g ................ gravity acceleration
• i ................. Overstatement deficit in two-lane railways
• b ................ Track distance (gauge) for double-rail tracks
• κ _H ............ curvature of the track centerline
• ν .............. driving speed
• h .................. routing height
• α .................. Roll angular acceleration
• ψ ................ elevation angle.

The roll angular acceleration is calculated from the second time derivative of the roll-over angle, which is replaced by the second way-off by means of the driving speed: $$\alpha =\frac{{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="imgf0002.png"\; state="\{\{state\}\}">d</figure-callout>}}^2\mathit{\psi }}{{\mathrm{d}\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="imgf0002.png"\; state="\{\{state\}\}">t</figure-callout>}}^2}={\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="imgf0002.png"\; state="\{\{state\}\}">v</figure-callout>}}^2\cdot \frac{{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="imgf0002.png"\; state="\{\{state\}\}">d</figure-callout>}}^2\mathit{\psi }}{{\mathrm{d}\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0002.png"\; state="\{\{state\}\}">s</figure-callout>}}^2}$$

• α ..................Rollwinkelbeschleunigung
• d ..................Differenzialoperator
• ψ .................Überhöhungswinkel
• t ...................Zeit
• ν .................Fahrgeschwindigkeit
• s ................Bogenlänge längs der Gleismittellinie.

Where:

• α .................. Roll angular acceleration
• d .................. Differential operator
• ψ ................. elevation angle
• t ................... time
• ν ................. Driving speed
• s ................ arc length along the track centerline.

wobei

• κ_H (s) ..... Krümmung der Gleismittellinie im Grundriss
• s ..................Bogenlänge längs der Gleismittellinie
• κ _c ................konstante Bezugskrümmung (in einem Kreisbogen)
• ψ _c ...............konstanter Bezugsüberhöhungswinkel (in einem Kreisbogen)
• ψ(s)................Überhöhungswinkel
• h ..................Trassierungshöhe
• d ..................Differentialoperator

bedeuten.According to the invention, a track is provided so that the curvature is determined from a function assumed to be superelevated such that the total unbalanced lateral acceleration at a selected, fixed track height (h) taking into account the portion of the unbalanced lateral acceleration caused by the rolling motion is a course like this Function and thus fulfills the following differential equation: $${\mathit{\kappa }}_{\mathit{H}}\left(s\right)=\frac{{\mathit{\kappa }}_{\mathit{C}}}{{\mathit{\psi }}_{\mathit{C}}}\mathit{\cdot }\mathit{\psi }\left(s\right)-\mathrm{<figure-callout\; id="2"\; label="H\; \cdot \; d"\; filenames="imgf0002.png"\; state="\{\{state\}\}">H\; </figure-callout>}\cdot \frac{{\mathrm{d}}^{}}{}\frac{{\mathrm{}}^2\mathit{\psi }}{{\mathrm{d}\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0002.png"\; state="\{\{state\}\}">s</figure-callout>}}^2}$$

in which

• κ _H (s) ..... Curvature of the track center line in plan view
• s .................. arc length along the track centerline
• κ _c ................ constant reference curvature (in a circular arc)
• ψ _c ............... constant reference elevation angle (in a circular arc)
• ψ (s) ................ elevation angle
• h .................. routing height
• d .................. differential operator

mean.

This differential equation can be evaluated directly for a selected course function. For trassing height h = 0, a conventional routing is obtained. The reference curvature and the reference overshoot must be selected in the circular arc or at the same point on the track.

In order to achieve the assumed course in the reality of a continuously embedded track, an advantageous feature of the invention generally provides for a four times differentiability of the overshoot function. From equation (1) the corresponding curvature of the track center line is then calculated in plan.

mit $0\le \frac{s}{l}\le 1$

und Δ ψ = ψ ₂ - ψ ₁.For a transition arc from one banked arc to another banked arc, the increase by means of the normalized function ( f ( s / l ) is generally formed as follows: $$\mathit{\psi }\left(s\right)=\frac{u\left(s\right)}{b}={\mathit{<figure-callout\; id="1"\; label="\psi "\; filenames="imgf0002.png"\; state="\{\{state\}\}">\psi </figure-callout>}}_1+\mathrm{\Delta }\mathit{\psi }\cdot f\left(\frac{s}{l}\right)$$

With $0\le \frac{s}{l}\le 1$

and Δ ψ = ψ ₂ - ψ ₁ .

• ψ (s) .......... Überhöhungswinkel
• s ..................Bogenlänge längs der Gleismittellinie
• u(s) .............Überhöhung bei Zweischienenbahnen
• b ..................Lauflinienabstand (Spurweite) bei Zweischienenbahnen
• ψ _l .................konstanter Überhöhungswinkel am Beginn der Überhöhungsrampe
• Δψ ..............Überhöhungsdifferenz zwischen den Werten in den Kreisbögen
• f(s l) ...........zwischen 0 und 1 normierte Grundfunktion
• l ...................Überhöhungsrampenlänge
• ψ₂ ................konstanter Überhöhungswinkel am Ende der Überhöhungsrampe.

Where:

• ψ ( s ) .......... elevation angle
• s .................. arc length along the track centerline
• u ( s ) ............. Exaggeration in double-rail tracks
• b .................. Track distance (track width) in double-rail tracks
• ψ _l ................. Constant elevation angle at the beginning of the elevation ramp
• Δψ .............. Canting difference between the values in the circular arcs
• f ( sl ) ........... between 0 and 1 normalized basic function
• l ................... Cradle ramp length
• ... ₂ ................ constant elevation angle at the end of the elevation ramp.

The normalized function describes directly the course of the elevation ramp.

$$f\left(\frac{s}{l}\right)=(\frac{s}{l}{)}^{2\cdot }\cdot \left(-2\cdot \frac{s}{l}+3\right)+\frac{6}{Z^2}\cdot \left(+2\cdot \frac{s}{l}-1+\cos \left(Z\cdot \frac{s}{l}\right)-\frac{2}{Z}\cdot \sin \left(Z\cdot \frac{s}{l}\right)\right)$$

mit $\tan \frac{Z}{2}=\frac{Z}{2}\approx 4,49340946$

$$f\left(\frac{s}{l}\right)=+\frac{1}{{\mathit{\pi }}^2-9}\cdot \left(+{\pi }^2\cdot (\frac{s}{l}{)}^2\cdot (-2\cdot \frac{s}{l}+3)-6\cdot \left(+1-\cos \frac{\pi \cdot s}{l}\right)\right)+\frac{3}{{\pi }^2-9}\cdot (+\frac{s}{l}-\frac{1}{2\cdot \pi }\cdot \sin \left(2\cdot \pi \cdot \frac{s}{l}\right))$$

$$f\left(\frac{s}{l}\right)=+\frac{15}{15-{\mathit{\pi }}^2}\cdot \left(+\frac{s}{l}-\frac{1}{2\cdot \pi }\cdot \sin (2\cdot \pi \cdot \frac{s}{l})\right)-\frac{{\pi }^2}{15-{\pi }^2}\cdot (\frac{s}{l}{)}^2\cdot (+6\cdot (\frac{s}{l}{)}^2-15\cdot \frac{s}{l}+10)$$

$$f\left(\frac{s}{l}\right)=+\frac{15}{10-{\mathit{\pi }}^2}\cdot \left(+1-\cos \frac{\pi \cdot s}{l}\right)-\frac{{\pi }^2}{2\cdot (10-{\pi }^2)}\cdot (\frac{s}{l}{)}^2\cdot (+2\cdot (\frac{s}{l}{)}^3-5\cdot {\left(\frac{s}{l}\right)}^2+5)$$

$$f\left(\frac{s}{l}\right)={\left(\frac{s}{l}\right)}^{5\cdot }\cdot \left(126-420\cdot \frac{s}{l}+540\cdot {\left(\frac{s}{l}\right)}^2-315\cdot {\left(\frac{s}{l}\right)}^3+70\cdot {\left(\frac{s}{l}\right)}^4\right)$$

alle mit $0\le \frac{s}{l}\le 1.$

As at least four times differentiable and thus fulfilling the requirements of the Bernoulli-Euler bending theory normalized functions for the elevation ramp to the transition arc from a cambered circular arc to another, provided with camber arc according to the invention a polynomial of the seventh order, a third order polynomial in combination with the sine and cosine and a constant value (Z), a third-order polynomial in combination with the sine and cosine, a fifth-order polynomial in combination with only the sine, a fifth-order polynomial in combination with only the cosine, and a polynomial used ninth order: $$f\left(\frac{s}{l}\right)=(\frac{s}{l}{)}^4\cdot \left(35-84\cdot \frac{s}{l}+70\cdot (\frac{s}{l}{)}^2-20\cdot (\frac{s}{l}{)}^3\right)$$

$$f\left(\frac{s}{l}\right)=(\frac{s}{l}{)}^{2\cdot }\cdot \left(-2\cdot \frac{s}{l}+3\right)+\frac{6}{{\mathrm{<figure-callout\; id="2"\; label="Z"\; filenames="imgf0002.png"\; state="\{\{state\}\}">Z</figure-callout>}}^2}\cdot \left(+2\cdot \frac{s}{l}-1+\cos \left(Z\cdot \frac{s}{l}\right)-\frac{2}{Z}\cdot \sin \left(Z\cdot \frac{s}{l}\right)\right)$$

With $\mathrm{<figure-callout\; id="2"\; label="tan\; Z"\; filenames="imgf0002.png"\; state="\{\{state\}\}">tan\; </figure-callout>}\frac{Z}{}\frac{}{2}=\frac{\mathrm{<figure-callout\; id="2"\; label="Z"\; filenames="imgf0002.png"\; state="\{\{state\}\}">Z</figure-callout>}}{2}\approx 4.49340946$

$$f\left(\frac{s}{l}\right)=+\frac{1}{{\mathit{\pi }}^2-9}\cdot \left(+{\mathrm{<figure-callout\; id="2"\; label="\pi "\; filenames="imgf0002.png"\; state="\{\{state\}\}">\pi </figure-callout>}}^2\cdot (\frac{s}{l}{)}^2\cdot (-2\cdot \frac{s}{l}+3)-6\cdot \left(+1-\cos \frac{\pi \cdot s}{l}\right)\right)+\frac{3}{{\pi }^2-9}\cdot (+\frac{s}{l}-\frac{1}{2\cdot \pi }\cdot \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="imgf0002.png"\; state="\{\{state\}\}">sin</figure-callout>}\left(2\cdot \pi \cdot \frac{s}{l}\right))$$

$$f\left(\frac{s}{l}\right)=+\frac{15}{15-{\mathit{<figure-callout\; id="2"\; label="\pi "\; filenames="imgf0002.png"\; state="\{\{state\}\}">\pi </figure-callout>}}^2}\cdot \left(+\frac{s}{l}-\frac{1}{2\cdot \pi }\cdot \sin (2\cdot \pi \cdot \frac{s}{l})\right)-\frac{{\mathrm{<figure-callout\; id="2"\; label="\pi "\; filenames="imgf0002.png"\; state="\{\{state\}\}">\pi </figure-callout>}}^2}{15-{\mathrm{<figure-callout\; id="2"\; label="\pi "\; filenames="imgf0002.png"\; state="\{\{state\}\}">\pi </figure-callout>}}^2}\cdot (\frac{s}{l}{)}^2\cdot (+6\cdot (\frac{s}{l}{)}^2-15\cdot \frac{s}{l}+10)$$

$$f\left(\frac{s}{l}\right)=+\frac{15}{10-{\mathit{<figure-callout\; id="2"\; label="\pi "\; filenames="imgf0002.png"\; state="\{\{state\}\}">\pi </figure-callout>}}^2}\cdot \left(+1-\cos \frac{\pi \cdot s}{l}\right)-\frac{{\mathrm{<figure-callout\; id="2"\; label="\pi "\; filenames="imgf0002.png"\; state="\{\{state\}\}">\pi </figure-callout>}}^2}{2\cdot (10-{\pi }^2)}\cdot (\frac{s}{l}{)}^2\cdot (+2\cdot (\frac{s}{l}{)}^3-5\cdot {\left(\frac{s}{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="imgf0002.png"\; state="\{\{state\}\}">l</figure-callout>}}\right)}^2+5)$$

$$f\left(\frac{s}{l}\right)={\left(\frac{s}{l}\right)}^{5\cdot }\cdot \left(126-420\cdot \frac{s}{l}+540\cdot {\left(\frac{s}{l}\right)}^2-315\cdot {\left(\frac{s}{\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="imgf0002.png"\; state="\{\{state\}\}">l</figure-callout>}}\right)}^3+70\cdot {\left(\frac{s}{\mathrm{<figure-callout\; id="4"\; label="l"\; filenames="imgf0002.png"\; state="\{\{state\}\}">l</figure-callout>}}\right)}^4\right)$$

all with $0\le \frac{s}{l}\le 1.$

• s ..................Bogenlänge längs der Gleismittellinie
• l ...................Länge des Übergangsbogens und der Überhöhungsrampe
• $f\left(\frac{s}{l}\right)$
  
  .........zwischen 0 und 1 normierte Grundfunktion.

Where:

• s .................. arc length along the track centerline
• l ................... Length of the transitional arch and the elevation ramp
• $f\left(\frac{s}{l}\right)$
  
  ......... between 0 and 1 standardized basic function.

All of these normalized functions, which are used for a transitional arch with a bank ramp leading from one cambered arc to another cambered arc, are all either simple polynomials or simple combinations of trigonometric functions with short polynomials. They are not transcendental and can easily be evaluated in practice, for example analytically finely differentiated up to the physically still important order and also integrated.

The differentiability can also be increased slightly. By way of example, an increase by 1 in the normalized function Equation (7), a special polynomial of ninth order, is shown: With this profile as overshoot function, the bedding distribution of the rail is not only limited but also continuous and the jerk distribution not only continuous, but also smooth , For the amplitudes are again slightly larger than in the course of equation (2).

mit $0\le \frac{s}{l}\le 1$

und Δκ = κ₂ - κ₁.With these normalized functions, the overshoot ramps are formed from equation (10). Two differentiation according to the arc length along the track center line and insertion into the equation (1) adapted for transition curves in the following form provides the curvature (κ _H ) of the track center line in plan: $${\mathit{\kappa }}_{\mathit{H}}\left(s\right)={\mathit{<figure-callout\; id="1"\; label="\kappa "\; filenames="imgf0002.png"\; state="\{\{state\}\}">\kappa </figure-callout>}}_1+\mathrm{\Delta }\mathit{\kappa }\cdot f\left(\frac{s}{l}\right)-H\cdot \Delta \mathit{\psi }\cdot \frac{{\mathrm{d}}^2\mathit{\psi }}{{\mathrm{d}\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0002.png"\; state="\{\{state\}\}">s</figure-callout>}}^2}$$

With $0\le \frac{s}{l}\le 1$

and Δκ = κ ₂ - κ ₁ .

• κ_H (s) .........Krümmung der Gleismittellinie im Grundriss
• s ..................Bogenlänge längs der Gleismittellinie
• κ ₁ .............. konstante Krümmung im Kreisbogen am Beginn des ......................Übergangsbogens
• Δ κ .............Krümmungsdifferenz zwischen den Werten in den Kreisbögen
• $f\left(\frac{s}{l}\right)$
  
  .....zwischen 0 und 1 normierte Grundfunktion
• h ............Trassierungshöhe
• Δψ ...... Oberhöhungsdifferenz zwischen den Werten in den Kreisbögen
• d ...........Differentialoperator
• κ₂ .....konstante Krümmung im Kreisbogen am Ende des ..........Übergangsbogens.

Where:

• κ _H ( s ) ......... Curvature of the track centerline in plan
• s .................. arc length along the track centerline
• κ ₁ .............. constant curvature in the arc at the beginning of the ...................... transition arc
• Δ κ ............. Curvature difference between the values in the circular arcs
• $f\left(\frac{s}{l}\right)$
  
  ..... between 0 and 1 normalized basic function
• h ............ routing height
• Δ ψ ...... elevation difference between the values in the circular arcs
• d ........... Differential operator
• κ ₂ ..... constant curvature in the arc at the end of the .......... transitional arc.

The following table lists by way of example a numerical evaluation with the normalized function according to equation (2). This numerical evaluation applies to a transition arc with 200 [m] length of a circle (index 1) with -2000 [m] radius and -64 [mm] elevation to a circle (index 2) with +800 [m] radius and one Camber of 160 [mm] at standard gauge (1435 [mm] gauge, b = 1.5 [m]). Indicated are all sizes away from the beginning of the transition arc: the arc length, the arc length normalized with the transition arc length between 0 and 1, the cant angle, the elevation, the curvature in the ground plan, the local radius and the torsion important for the design. TABLE (Sheet A) Arc Length Arc Length Crescent Crescent Curvature radius twist normalized angle Layout local [M] [1] [wheel] [Mm] [1 / km] [M] [Mm / m] 0 0.00 -0.042667 -64,0-5,000000E-01 -2000 0,000 4 0.02 -0.042666 -64,0-5,010315E-01 -1,995.882 0.001 8th 0.04 -0.042655 -64,0-5,036866E-01 -1,985.361 0.009 12 0.06 -0.042608 -63,9-5,072155E-01 -1,971.549 0.028 16 0.08 -0.042491 -63,7-5,107841E-01 -1,957.774 0.063 20 0.10 -0.042259 -63,4-5,135152E-01 -1,947.362 0.114 24 0.12 -0.041865 -62,8-5,145237E-01 -1,943.545 0.185 28 0.14 -0.041258 -61,9-5,129488E-01 -1,949.512 0.274 32 0.16 -0.040389 -60,6-5,079798E-01 -1,968.582 0,381 36 0.18 -0.039213 -58,8-4,988790E-01 -2,004.494 0.504 40 0.20 -0.037687 -56,5-4,850001E-01 -2,061.855 0.642 44 0.22 -0.035777 -53,7-4,658022E-01 -2,146.834 0,792 48 0.24 -0.033453 -50,2-4,408612E-01 -2,268.288 0,952 52 0.26 -0.030697 -46,0-4,098773E-01 -2,439.755 1,117 56 0.28 -0.027495 -41,2-3,726788E-01 -2,683.276 1,285 60 0.30 -0.023845 -35,8-3,292241 E-01-3037.444 1,452 64 0.32 -0.019754 -29,6-2,796003E-01 -3,576.534 1,616 68 0.34 -0.015235 -22,9-2,240189E-01 -4,463.909 1,772 72 0.36 -0.010313 -15,5-1,628105E-01 -6,142.109 1,918 76 0.38 -0.005019 -7,5-9,641627E-02 -10,371.69 2,051 80 0.40 0.000609 0,9-2,537805E-02 -39,404.13 2,168 84 0.42 0.006525 9.8 4,967288E-02 20131.71 2,267 88 0.44 0.012680 19.0 1.280288E-01 7810.744 2,346 92 0.46 0.019017 28.5 2.089193E-01 4786.537 2,403 96 0.48 0.025477 38.2 2,915,264E-01 3430.221 2.438 100 0.50 0.032000 48.0 3.750000E-01 2666.667 2,450 104 0.52 0.038523 57.8 4,584736E-01 2181.151 2.438 108 0.54 0.044983 67.5 5,410807E-01 1848.153 2,403 112 0.56 0.051320 77.0 6.219712E-01 1607.791 2,346 116 0.58 0.057475 86.2 7,003271E-01 1427.904 2,267 120 0.60 0.063391 95.1 7.753780E-01 1289.693 2,168 124 0.62 0.069019 103.5 8.464163E-01 1181.452 2,051 128 0.64 0.074313 111.5 9,128105E-01 1095.518 1,918 132 0.66 0.079235 118.9 9.740189E-01 1026.674 1,772 136 0.68 0.083754 125,61,029600E + 00 971.2507 1,616 140 0.70 0.087845 131,81,079224E + 00 926.5916 1,452 144 0.72 0.091495 137,21,122679E + 00 890.7267 1,285 148 0.74 0.094697 142,01,159877E + 00 862.1602 1,117 152 0.76 0.097453 146,21,190861E + 00 839.7284 0,952 156 0.78 0.099777 149,71,215802E + 00 822.5022 0,792 160 0.80 0.101687 152,51,235000E + 00 809.7166 0.642 164 0.82 0.103213 154,81,248879E + 00 800.7181 0.504 168 0.84 0.104389 156,61,257980E + 00 794.9253 0,381 172 0.86 0.105258 157,91,262949E + 00 791.7977 0.274 176 0.88 0.105865 158,81,264524E + 00 790.8116 0.185 180 0.90 0.106259 159,41,263515E + 00 791.4428 0.114 184 0.92 0.106491 159,71,260784E + 00 793.1572 0.063 188 0.94 0.106608 159,91,257215E + 00 795.4086 0.028 192 0.96 0.106655 160,01,253687E + 00 797.6475 0.009 196 0.98 0.106666 160,01,251032E + 00 799.3404 0.001 200 1.00 0.106667 160,01,250000E + 00 800 0,000

Tables for transitional arches and overboost ramps, which are based on the other normalized functions of equations (3) through (7), can be easily obtained in an analogous manner by numerical evaluation of the formulas.

For the other specified functions can be proceeded analogously.

The method according to the invention can be analogously applied for a one-piece, excessive transition from a straight track to a straight track diverging at an angle therefrom. The choice of a corresponding function for the superelevation and the assumption of the total unbalanced lateral acceleration including the portion of the roll motion with the same function provides the curvature of the track in the floor plan. For the elevation, a function increasing from zero to a maximum value and then decreasing to zero is selected. In order to meet the requirements resulting from the bending of the rail, a function is selected which is also four times differentiable at the area edges.

Likewise, excessive track distortions that transition from a straight track to a parallel straight track can be performed in one piece. There, too, a suitable function, which is preferably four times differentiable everywhere, is assumed for the superelevation and the total unbalanced lateral acceleration, and from this the curvature profile of the track is calculated in the ground plan.

In an analogous way, the avoidance of an obstacle, that is, a route that emanates from a straight track, escapes an obstacle to one side, then runs back to the imaginary extension of the straight track and this crosses, then continues on the other side and in ending in a single piece on a straight track continuing to this side.

With the method according to the invention, in each case the route courses and the requirements of the bendability of the rails can be designed with ramp shapes with perfect dynamic properties for all conceivable applications.

Claims (13)

1. Track comprising a track centre line of variable curvature (κ_H) on a horizontal section and variable superelevation angle (ψ), characterised in that the curvature (κ_H) is determined from a function assumed for superelevation such that the overall non-compensated lateral acceleration at a selected fixed aligning height (h), taking into account the amount of non-compensated lateral acceleration caused by the swaying movement, has a characteristic curve like that of said function and fulfils the following differential equation: $${\mathit{\kappa }}_{\mathit{H}}\left(s\right)=\frac{{\mathit{\kappa }}_c}{{\mathit{\psi }}_c}\mathit{\cdot }\mathit{\psi }\left(s\right)-h\cdot \frac{{\mathrm{d}}^2\mathit{\psi }}{{\mathrm{d}s}^2}$$

   

   
   wherein:

   κ _H (s) represents the curvature of the track centre line on a horizontal section,

   s the curve length along the track centre line,

   κ _c the constant reference curvature (in an arc),

   ψ _c the constant reference superelevation angle (in an arc),

   ψ(s) the superelevation angle,

   h the aligning height,

   d the differential operator.
2. Track according to claim 1, characterised in that the function in its overall characteristic curve, even at the area edges, can be differentiated at least four times, for which reason even the fourth derivation of the function everywhere has finite values.
3. Track according to claim 1 or 2, characterised in that in determining the curvature (κ_H) of the track centre line on a horizontal section, the aligning height zero is selected as the fixed aligning height (h).
4. Track according to any one of claims 1 to 3, characterised in that as a standardised function, the following polynomial of the seventh order is used: $$f\left(\frac{s}{l}\right)=(\frac{s}{l}{)}^4\cdot \left(35-84\cdot \frac{s}{l}+70\cdot (\frac{s}{l}{)}^2-20\cdot (\frac{s}{l}{)}^3\right)$$

   

   
   with 0 ≤ s/l ≤ 1
   wherein

   s represents the curve length along the track centre line,

   l the length of the transition curve and the superelevation ramp,

   wherein this standardised function is used for the characteristic curve of the superelevation angle and for the entire non-compensated lateral acceleration and the curvature (κ_H) of the track centre line in a horizontal section is determined therefrom according to equation (1).
5. Track according to any one of claims 1 to 3, characterised in that as a standardised function, the following polynomial of the third order is used in combination with the sine and cosine and a constant value (Z): $$f\left(\frac{s}{l}\right)=(\frac{s}{l}{)}^{2\cdot }\cdot \left(-2\cdot \frac{s}{l}+3\right)+\frac{6}{Z^2}\cdot \left(+2\cdot \frac{s}{l}-1+\cos \left(Z\cdot \frac{s}{l}\right)-\frac{2}{Z}\cdot \sin \left(Z\cdot \frac{s}{l}\right)\right)$$

   

   
   with tan Z/2 = Z/2 ≈ 4.49340946 and 0 ≤ s/l ≤ 1
   wherein

   s represents the curve length along the track centre line,

   I the length of the transition curve and the superelevation ramp,

   wherein this standardised function is used for the characteristic curve of the superelevation angle and for the entire non-compensated lateral acceleration and the curvature (κ_H) of the track centre line in a horizontal section is determined therefrom according to equation (1).
6. Track according to any one of claims 1 to 3, characterised in that as a standardised function, the following polynomial of the third order is used in combination with the sine and cosine: $$f\left(\frac{s}{l}\right)=+\frac{1}{{\mathit{\pi }}^2-9}\cdot \left(+{\pi }^2\cdot (\frac{s}{l}{)}^2\cdot (-2\cdot \frac{s}{l}+3)-6\cdot \left(+1-\cos \frac{\pi \cdot s}{l}\right)\right)+\frac{3}{{\pi }^2-9}\cdot (+\frac{s}{l}-\frac{1}{2\cdot \pi }\cdot \sin \left(2\cdot \pi \cdot \frac{s}{l}\right))$$

   

   
   with 0 ≤ s/l ≤ 1
   wherein

   s represents the curve length along the track centre line,

   I the length of the transition curve and the superelevation ramp,

   wherein this standardised function is used for the characteristic curve of the superelevation angle and for the entire non-compensated lateral acceleration and the curvature (κ_H) of the track centre line in a horizontal section is determined therefrom according to equation (1).
7. Track according to any one of claims 1 to 3, characterised in that as a standardised function, the following polynomial of the fifth order is used in combination with the sine only: $$f\left(\frac{s}{l}\right)=+\frac{15}{15-{\mathit{\pi }}^2}\cdot \left(+\frac{s}{l}-\frac{1}{2\cdot \pi }\cdot \sin (2\cdot \pi \cdot \frac{s}{l})\right)-\frac{{\pi }^2}{15-{\pi }^2}\cdot (\frac{s}{l}{)}^2\cdot (+6\cdot (\frac{s}{l}{)}^2-15\cdot \frac{s}{l}+10)$$

   

   
   with 0 ≤ s/l ≤ 1
   wherein

   s represents the curve length along the track centre line,

   l the length of the transition curve and the superelevation ramp,

   wherein this standardised function is used for the characteristic curve of the superelevation angle and for the entire non-compensated lateral acceleration and the curvature (κ_H) of the track centre line in a horizontal section is determined therefrom according to equation (1).
8. Track according to any one of claims 1 to 3, characterised in that as a standardised function, the following polynomial of the fifth order is used in combination with the cosine only: $$f\left(\frac{s}{l}\right)=+\frac{15}{10-{\mathit{\pi }}^2}\cdot \left(+1-\cos \frac{\pi \cdot s}{l}\right)-\frac{{\pi }^2}{2\cdot (10-{\pi }^2)}\cdot (\frac{s}{l}{)}^2\cdot (+2\cdot (\frac{s}{l}{)}^3-5\cdot {\left(\frac{s}{l}\right)}^2+5)$$

   

   
   with 0 ≤ s/l ≤ 1
   wherein

   s represents the curve length along the track centre line,

   I the length of the transition curve and the superelevation ramp,

   wherein this standardised function is used for the characteristic curve of the superelevation angle and for the entire non-compensated lateral acceleration and the curvature (κ_H) of the track centre line in a horizontal section is determined therefrom according to equation (1).
9. Track according to any one of claims 1 to 3, characterised in that as a standardised function, the following polynomial of the ninth order is used: $$f\left(\frac{s}{l}\right)={\left(\frac{s}{l}\right)}^{5\cdot }\cdot \left(126-420\cdot \frac{s}{l}+540\cdot {\left(\frac{s}{l}\right)}^2-315\cdot {\left(\frac{s}{l}\right)}^3+70\cdot {\left(\frac{s}{l}\right)}^4\right)$$

   

   
   with 0 ≤ s/l ≤ 1
   wherein

   s represents the curve length along the track centre line,

   l the length of the transition curve and the superelevation ramp,

   wherein this standardised function is used for the characteristic curve of the superelevation angle and for the entire non-compensated lateral acceleration and the curvature (κ_H) of the track centre line in a horizontal section is determined therefrom according to equation (1).
10. Track according to any of claims 1 to 3, characterised in that an alignment element executed with or without superelevation, which combines a straight track with a straight track deviating therefrom at an angle, is designed with a one-part function.
11. Track according to claim 10, characterised in that a one-part function which forms the basis for an alignment element executed with or without superelevation, which combines a straight track with a straight track deviating therefrom at an angle, can be differentiated four times with finite values.
12. Track according to any of claims 1 to 3, characterised in that a track warping executed with or without superelevation, which combines a straight track with a parallel straight track, is designed with a one-part function.
13. Track according to claim 12, characterised in that a one-part function which forms the basis for a track warping executed with or without superelevation, which combines a straight track with a parallel straight track, can be differentiated four times with finite values.

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