3. Some geometrical peculiarities of the wheel and rail interaction

The geometrical features of the wheel and rail interaction are stipulated by the designs of the rail track/bogie, wheel/rail, and their technical state. At lateral displacement of the wheelset relative to the rail, a contact point from the tread surfaces passes on the wheel flange root and rail corner, and the wheel and rail tread surfaces separate from each other. At further lateral displacement of the wheelset, the contact point passes on their steering surfaces, and the angle of inclination of the wheel flange increases up to 70° .

It is difficult to predict and control the friction forces, wear rate of various types, vibrations, and noise of the heavy loaded interacting surfaces of the railway transport running gear that decreases traffic safety, increases energy loses on friction, etc. Many works are devoted to the researches of dependences of the tribological

properties on various factors [1, 3–6], though their mechanisms of generation and variation are not always entirely clear that complicates the revelation of parameters influencing them [7, 8].

certain combination of such parameters, as radius of the rail track curvature, mass and speed of the rolling stock, friction coefficient between the wheel and rail, etc. Therefore, practically the outer wheel rolls on the less diameter than necessary, and in the case of the free wheelset (without bogie), it falls behind the inner wheel,

In the case of the non-free wheelset, the bogie makes the wheelset maintain a radial position, forcing the outer wheel to roll the greater distance not to fall behind the inner wheel. Thereat, the outer wheel rotates through the greater angle than the inner one, and the wheelset axle is twisted. The angle of twist of the wheelset can increase up to the value that is stipulated by the friction force between the wheel and rail. When this angle of twist reaches the limited value, the wheel slides on the rail due to action of the wheelset axle elastic moment tending to bring it back to the

Similarly, the wheel will slide on the rail at rolling in the straight rail track of the

At pure rolling of the free wheelset (without bogie) in the curved rail track with radius of curvature R of the internal rail, its axle will be inclined from radial position because both wheels will have passed equal distances l. However, in the wagon wheelset rolling with velocity V, the outer wheel is constraint to maintain the radial position and pass greater distance l + Δl, rotating relative to the inner wheel in the clockwise direction if it is seen from axial direction A (Figure 5). At that, the wheelset axle is twisted through angle φ equal to the ratio of the difference Δl of the outer and inner arcs to the radius D/2 of the wheel tread surface, supposing that

φ ¼ 2Δl=D (1)

Δl ¼ lΔR=R, (2)

φ ¼ 2 lΔR=DR: (3)

wheelset with the wheels of different diameters or with one wheel having an elliptical form. The mechanisms of the wheel sliding on the rail for the three noted

cases are considered and explained in the next paragraphs.

both wheels are rolling on the tread surfaces of equal diameters:

From the drawing α = l/R = (l + Δl)/(R + ΔR) = Δl/ΔR,

Movement of the wagon wheelset in the curve and wheelset shaft slope from the radial position.

3.1 Movement of the wagon wheelset in the curve

inclining the wheelset axle from the radial position.

DOI: http://dx.doi.org/10.5772/intechopen.89135

Influence of Tribological Parameters on the Railway Wheel Derailment

equilibrium position.

from where

and therefore

Figure 5.

211

There are many reasons of generation of vibrations and noise at movement of the train, part of which are well studied and predictable, and ways of their decrease are known. The interacting surfaces of the wheels and rails are characterized by the various types of irregularities, 5–20 mm gaps in the rail joints, where the rail tread surfaces are spaced by 0.5–2 mm in the vertical direction; the various wear traces (rail corrugation, fatigue, etc.) and deviations from the wheel roundness are the sources of vibrations and noise.

The wheel and rail interaction is accompanied by the forced and self-excited vibrations of various frequencies, as the main reason of the forced vibrations is considered macro- and micro-asperities of the rail (periodic and separate asperities) [9–13]. However, the main source of the self-vibrations is friction between the wheel and rail. It must be noted that to various working conditions of the heavy loaded contacting surfaces and wear types correspond typical micro-asperities, which can be different from the initial micro-asperities [27, 28]. The researches have shown an important role of the tread and steering surfaces in generation of the vibrations (self-vibrations) and noise, whose reasons are not studied sufficiently. There is quite vague information on the reasons of the self-vibrations generated at interaction of the wheel and rail [9].

Generation of vibrations of the heavy loaded interacting elements of the railway transport running gear is stipulated by the complex processes proceeding in the contact zone. As a result of interaction of the surfaces with the environment, they are coated by the layers of various physical and chemical origins that are the components of the third body in the contact zone and have a great influence on the tribological properties of the contacting surfaces. According to observations by Godet, dry friction is largely determined not by the properties of materials of the contacting pair but by the characteristics of the structure and composition of the thin film that is formed on the surfaces of both bodies because of compaction of the wear product and its chemical composition and oxidation. Destination of the third body in the tribological systems is separation of the contacting surfaces, providing with the stable friction forces of proper values and protection of the surfaces against damage of various types. Tribological properties of the third body greatly depend on the initial properties of its component elements and features of the contact zone. The sliding velocity, power and thermal loading, and the sliding distance have especially great influence on the destruction of the third body. For providing the stability of the third body in the contact zone of the wheels and rails and reduction of the derailment probability, energy consumed on traction, environment pollution, and maintenance expenses, the decrease of the sliding distance and relative sliding is especially important.

The wheel/rail squeal in curves is the most common type of vibrations and noise. It is especially typical for high-speed movements, when because of various reasons, the relative sliding and sliding distance increase. This contributes destruction of the third body, seizure of the surfaces at direct contact, subsequent destruction of the seized surfaces, and instability of the friction forces and relative movement of surfaces.

Many negative phenomena (wear, noise, vibrations) are generated because of the wheel sliding on the rail. For elimination of the wheel sliding in the curves, the wheel tread surface is given a conical form with the intention of making the outer wheel to roll on the greater diameter passing the greater distance than the inner wheel and rotate both wheels through the equal angles, maintaining this way radial position of the wheelset axle. However, this intention can be realized only for a

### Influence of Tribological Parameters on the Railway Wheel Derailment DOI: http://dx.doi.org/10.5772/intechopen.89135

certain combination of such parameters, as radius of the rail track curvature, mass and speed of the rolling stock, friction coefficient between the wheel and rail, etc. Therefore, practically the outer wheel rolls on the less diameter than necessary, and in the case of the free wheelset (without bogie), it falls behind the inner wheel, inclining the wheelset axle from the radial position.

In the case of the non-free wheelset, the bogie makes the wheelset maintain a radial position, forcing the outer wheel to roll the greater distance not to fall behind the inner wheel. Thereat, the outer wheel rotates through the greater angle than the inner one, and the wheelset axle is twisted. The angle of twist of the wheelset can increase up to the value that is stipulated by the friction force between the wheel and rail. When this angle of twist reaches the limited value, the wheel slides on the rail due to action of the wheelset axle elastic moment tending to bring it back to the equilibrium position.

Similarly, the wheel will slide on the rail at rolling in the straight rail track of the wheelset with the wheels of different diameters or with one wheel having an elliptical form. The mechanisms of the wheel sliding on the rail for the three noted cases are considered and explained in the next paragraphs.

#### 3.1 Movement of the wagon wheelset in the curve

At pure rolling of the free wheelset (without bogie) in the curved rail track with radius of curvature R of the internal rail, its axle will be inclined from radial position because both wheels will have passed equal distances l. However, in the wagon wheelset rolling with velocity V, the outer wheel is constraint to maintain the radial position and pass greater distance l + Δl, rotating relative to the inner wheel in the clockwise direction if it is seen from axial direction A (Figure 5). At that, the wheelset axle is twisted through angle φ equal to the ratio of the difference Δl of the outer and inner arcs to the radius D/2 of the wheel tread surface, supposing that both wheels are rolling on the tread surfaces of equal diameters:

$$
\boldsymbol{\upvarphi} = \boldsymbol{\upDelta l}/\boldsymbol{\upDelta l} \tag{1}
$$

From the drawing α = l/R = (l + Δl)/(R + ΔR) = Δl/ΔR, from where

$$
\Delta \mathbf{l} = \mathbf{l} \Delta \mathbf{R} / \mathbf{R},
\tag{2}
$$

and therefore

properties on various factors [1, 3–6], though their mechanisms of generation and variation are not always entirely clear that complicates the revelation of parameters

There are many reasons of generation of vibrations and noise at movement of the train, part of which are well studied and predictable, and ways of their decrease are known. The interacting surfaces of the wheels and rails are characterized by the various types of irregularities, 5–20 mm gaps in the rail joints, where the rail tread surfaces are spaced by 0.5–2 mm in the vertical direction; the various wear traces (rail corrugation, fatigue, etc.) and deviations from the wheel roundness are the

The wheel and rail interaction is accompanied by the forced and self-excited vibrations of various frequencies, as the main reason of the forced vibrations is considered macro- and micro-asperities of the rail (periodic and separate asperities) [9–13]. However, the main source of the self-vibrations is friction between the wheel and rail. It must be noted that to various working conditions of the heavy loaded contacting surfaces and wear types correspond typical micro-asperities, which can be different from the initial micro-asperities [27, 28]. The researches have shown an important role of the tread and steering surfaces in generation of the vibrations (self-vibrations) and noise, whose reasons are not studied sufficiently. There is quite vague information on the reasons of the self-vibrations generated at

Generation of vibrations of the heavy loaded interacting elements of the railway transport running gear is stipulated by the complex processes proceeding in the contact zone. As a result of interaction of the surfaces with the environment, they are coated by the layers of various physical and chemical origins that are the components of the third body in the contact zone and have a great influence on the tribological properties of the contacting surfaces. According to observations by Godet, dry friction is largely determined not by the properties of materials of the contacting pair but by the characteristics of the structure and composition of the thin film that is formed on the surfaces of both bodies because of compaction of the wear product and its chemical composition and oxidation. Destination of the third body in the tribological systems is separation of the contacting surfaces, providing with the stable friction forces of proper values and protection of the surfaces against damage of various types. Tribological properties of the third body greatly depend on the initial properties of its component elements and features of the contact zone. The sliding velocity, power and thermal loading, and the sliding distance have especially great influence on the destruction of the third body. For providing the stability of the third body in the contact zone of the wheels and rails and reduction of the derailment probability, energy consumed on traction, environment pollution, and maintenance expenses, the decrease of the sliding distance and relative sliding

The wheel/rail squeal in curves is the most common type of vibrations and noise. It is especially typical for high-speed movements, when because of various reasons, the relative sliding and sliding distance increase. This contributes destruction of the third body, seizure of the surfaces at direct contact, subsequent destruction of the seized surfaces, and instability of the friction forces and relative movement of

Many negative phenomena (wear, noise, vibrations) are generated because of the wheel sliding on the rail. For elimination of the wheel sliding in the curves, the wheel tread surface is given a conical form with the intention of making the outer wheel to roll on the greater diameter passing the greater distance than the inner wheel and rotate both wheels through the equal angles, maintaining this way radial position of the wheelset axle. However, this intention can be realized only for a

influencing them [7, 8].

Transportation Systems Analysis and Assessment

sources of vibrations and noise.

interaction of the wheel and rail [9].

is especially important.

surfaces.

210

$$
\mathfrak{q} = 2 \,\text{l}\Delta\text{R/DR}.\tag{3}
$$

Figure 5. Movement of the wagon wheelset in the curve and wheelset shaft slope from the radial position.

On the other hand, the maximum angle of twist of the wheelset axle φmax depends on the friction force

$$\mathbf{F} = \mathbf{f} \mathbf{Q} \tag{4}$$

and is calculated by the known, from the resistance of materials, formula

$$
\mathfrak{q}\_{\text{max}} = \mathbf{ML} / \mathbf{I}\_{\text{P}} \mathbf{G},\tag{5}
$$

where M is a torque caused by the friction force

$$\mathbf{M} = \mathbf{F} \mathbf{D} / 2 = \mathbf{f} \mathbf{Q} \mathbf{D} / 2; \tag{6}$$

f, friction coefficient; Q, vertical load (half of the load on the wheelset) of the wheel on the rail; L, length of the wheelset axle; Ip, polar moment of inertia of the wheelset axle cross section; and G, modulus of rigidity (share modulus) of the axle material.

We determine distance between the worn-out segments of the rail or path l (at traveling this path, the wheels are rolling on the rail without sliding), at rolling of which the axle is twisted on the maximum angle φmax, from (3) replacing φ by φmax

$$\text{I} = \text{DR } \mathfrak{q}\_{\text{max}} / 2\Delta\text{R} = \text{MLDR} / 2\text{I}\_{\text{P}} \text{G} \Delta\text{R} \tag{7}$$

and putting the found l into (2) we obtain difference of the paths passed by the outer and inner wheels at which the axle is twisted on the maximum angle φmax

$$
\Delta \mathbf{l} = \mathbf{M} \mathbf{L} \mathbf{D} / 2 \mathbf{I}\_{\mathbf{P}} \mathbf{G}.\tag{8}
$$

At one revolution, these wheels will pass the different distances, correspondingly l and l + Δl, deflecting the wheelset axle from its perpendicular position relative to the rail track (Figure 6a). However, in the wagon wheelset the axle being constraint to retain perpendicular position, the wheel with diameter D is forced to pass the same (greater) distance l + Δl and rotate relative to the elliptical wheel in the clockwise direction if it is seen from axial direction A. At that, the wheelset axle

Movement of the free wheelset in the straight rail track: (a) with the wheels of different diameters or with one

The difference of distances passed by the wheels at one revolution is Δl=L–πD,

The distance l at passing of which the wheelset axle will be twisted on the angle

I

In all the three cases considered above, at removing or decrease of the torque M acting on the wheel that takes place at its vertical vibrations when the friction force F decreases, the angle of twist of the axle will start to decrease. Suppose φmax falls down to zero during time t. This will take place at rotation of the inner wheel in the clockwise direction relative to the outer wheel on the angle φmax since the flange of the outer wheel is pressed on the rail and the friction force arisen between the flange and rail additionally restricts its movement. Obviously, during this time t the

l ¼ πDΔl

L ¼ π½ � 3 að Þ� þ b ð Þ 3a þ b ð Þ a þ 3b (10)

Δl ¼ π½3 að Þ� þ b ð Þ 3a þ b ð Þ a þ 3b � � πD (11)

<sup>I</sup> <sup>¼</sup> <sup>φ</sup>max <sup>D</sup>=<sup>2</sup> <sup>¼</sup> MLD=2IpG (12)

=Δ (13)

<sup>I</sup> corresponding to maximum angle of twist φmax is obtained con-

is twisted through angle φ that is determined by formula (1).

Influence of Tribological Parameters on the Railway Wheel Derailment

DOI: http://dx.doi.org/10.5772/intechopen.89135

Δl

where the length of the elliptical tread surface

elliptical wheel; (b) parameters of ellipticity.

or

Figure 6.

The value Δl

sidering formula (5)

φmax will be then

213

### 3.2. Movement of the wagon wheelset with the wheels of different diameters in the straight rail track

At rolling of the free wheelset (without bogie) with the wheels of different diameters D and D + ΔD in the straight rail track the distance l, the greater wheel passes a greater distance l + Δl, deflecting the wheelset axle from its perpendicular position relative to the rail track (Figure 2a). But in the wagon wheelset the axle being constraint to retain perpendicular position, the smaller wheel is forced to pass the same distance l + Δl and rotate relative to the greater wheel in the clockwise direction, if it is seen from axial direction A. At that, the wheelset axle is twisted through angle φ that is determined by formula (1), from where, considering (5), we obtain the value of Δl (see formula (8)) corresponding to the maximum angle of twist φmax.

The following proportion can be written from the drawing: (l + Δl)/l = (D + ΔD)/D or Δl/l = ΔD/D, from which we obtain distance l between the worn-out segments at passing of which the wheelset axle will be twisted through angle φmax:

$$\text{l} = \Delta \text{lD} / \Delta \text{D} = \text{MLD}^2 / 2 \text{I}\_\text{P} \text{G} \Delta \text{D} \tag{9}$$

### 3.3. Movement of the wagon wheelset with one elliptical wheel in the straight rail track

Consider a free wheelset with one wheel of diameter D and other elliptical wheel with the small D and bigger D + ΔD diameters moving in the straight rail track (Figure 6a and b).

Influence of Tribological Parameters on the Railway Wheel Derailment DOI: http://dx.doi.org/10.5772/intechopen.89135

#### Figure 6.

On the other hand, the maximum angle of twist of the wheelset axle φmax

and is calculated by the known, from the resistance of materials, formula

f, friction coefficient; Q, vertical load (half of the load on the wheelset) of the wheel on the rail; L, length of the wheelset axle; Ip, polar moment of inertia of the wheelset axle cross section; and G, modulus of rigidity (share modulus) of the axle

We determine distance between the worn-out segments of the rail or path l (at traveling this path, the wheels are rolling on the rail without sliding), at rolling of which the axle is twisted on the maximum angle φmax, from (3) replacing φ by

and putting the found l into (2) we obtain difference of the paths passed by the outer and inner wheels at which the axle is twisted on the maximum angle φmax

3.2. Movement of the wagon wheelset with the wheels of different diameters

At rolling of the free wheelset (without bogie) with the wheels of different diameters D and D + ΔD in the straight rail track the distance l, the greater wheel passes a greater distance l + Δl, deflecting the wheelset axle from its perpendicular position relative to the rail track (Figure 2a). But in the wagon wheelset the axle being constraint to retain perpendicular position, the smaller wheel is forced to pass the same distance l + Δl and rotate relative to the greater wheel in the clockwise direction, if it is seen from axial direction A. At that, the wheelset axle is twisted through angle φ that is determined by formula (1), from where, considering (5), we obtain the value of Δl (see formula (8)) corresponding to the maximum angle of

The following proportion can be written from the drawing: (l + Δl)/l = (D + ΔD)/D or Δl/l = ΔD/D, from which we obtain distance l between the worn-out segments at passing of which the wheelset axle will be twisted through angle φmax:

<sup>l</sup> <sup>¼</sup> <sup>Δ</sup>lD=Δ<sup>D</sup> <sup>¼</sup> MLD<sup>2</sup>

3.3. Movement of the wagon wheelset with one elliptical wheel in the straight

with the small D and bigger D + ΔD diameters moving in the straight rail track

Consider a free wheelset with one wheel of diameter D and other elliptical wheel

where M is a torque caused by the friction force

F ¼ fQ (4)

φmax ¼ ML=IpG, (5)

M ¼ FD=2 ¼ fQD=2; (6)

l ¼ DR φmax=2ΔR ¼ MLDR=2IpGΔR (7)

Δl ¼ MLD=2IpG: (8)

=2IpGΔD (9)

depends on the friction force

Transportation Systems Analysis and Assessment

in the straight rail track

material.

φmax

twist φmax.

rail track

(Figure 6a and b).

212

Movement of the free wheelset in the straight rail track: (a) with the wheels of different diameters or with one elliptical wheel; (b) parameters of ellipticity.

At one revolution, these wheels will pass the different distances, correspondingly l and l + Δl, deflecting the wheelset axle from its perpendicular position relative to the rail track (Figure 6a). However, in the wagon wheelset the axle being constraint to retain perpendicular position, the wheel with diameter D is forced to pass the same (greater) distance l + Δl and rotate relative to the elliptical wheel in the clockwise direction if it is seen from axial direction A. At that, the wheelset axle is twisted through angle φ that is determined by formula (1).

The difference of distances passed by the wheels at one revolution is Δl=L–πD, where the length of the elliptical tread surface

$$\mathbf{L} = \pi[\mathbf{3(a+b)} - (\mathbf{3a+b})(\mathbf{a+3b})] \tag{10}$$

or

$$
\Delta \mathbf{l} = \pi [\mathbf{\hat{3}}(\mathbf{a} + \mathbf{b}) - (\mathbf{\hat{3}a} + \mathbf{b})(\mathbf{a} + \mathbf{\hat{3}b})] - \pi \mathbf{D} \tag{11}
$$

The value Δl <sup>I</sup> corresponding to maximum angle of twist φmax is obtained considering formula (5)

$$
\Delta \mathbf{l}^1 = \varrho\_{\text{max}} \,\mathrm{D}/2 = \mathrm{MLD}/2\mathrm{I}\_\mathrm{P} \,\mathrm{G} \tag{12}
$$

The distance l at passing of which the wheelset axle will be twisted on the angle φmax will be then

$$\mathbf{l} = \pi \mathbf{D} \Delta \mathbf{l}^{\mathrm{l}} / \Delta \tag{13}$$

In all the three cases considered above, at removing or decrease of the torque M acting on the wheel that takes place at its vertical vibrations when the friction force F decreases, the angle of twist of the axle will start to decrease. Suppose φmax falls down to zero during time t. This will take place at rotation of the inner wheel in the clockwise direction relative to the outer wheel on the angle φmax since the flange of the outer wheel is pressed on the rail and the friction force arisen between the flange and rail additionally restricts its movement. Obviously, during this time t the inner wheel will roll and slide simultaneously on the rail and the rolling and sliding distance on the rail will be

$$\mathbf{S}\_{\mathbf{r}} = \mathbf{V}\mathbf{t} \tag{14}$$

We note that the rolling and sliding distance on the wheel tread surface is

$$\mathbf{S}\_{\mathbf{w}} = \Delta \mathbf{l} + \mathbf{S}\_{\mathbf{r}}.\tag{14'}$$

or for the variant of the elliptical wheel

$$\mathbf{S}\_{\mathbf{w}} = \Delta \mathbf{l}^{\mathrm{l}} + \mathbf{S}\_{\mathbf{r}} \tag{15}$$

here Δl or ΔlI is a sliding friction path and the wavelength of the worn-out rail (Figure 3)

$$\mathbf{W} = \mathbf{l} \mathbf{s}\_{\text{ers}} \tag{16}$$

We note that maximum velocity of the wheel contact point relative to the wheel

where A = φmax is an amplitude of the wheelset shaft torsion vibrations and ω =

� 100% and Kw <sup>¼</sup> Vsl

where i is the wear intensity and N, number of cycles which is determined as

where N1 is a number of the trains passing by a day; N2, number of wagons in the train; N3, number of wheels on one side of the wagon; and N4, number of days

Possibility of derailment or the wheel's rolling up on the rail is estimated by the criterion of the wheel flange contact point (point A, Figure 9) slipping down the rail lateral surface, based on the condition of equilibrium of forces acting on this point [24]. Lateral L and vertical V forces determined from the condition of equilibrium of these forces are.where N is a normal force; FI = fIN, friction force

The depth of the worn-out layer a year of the rail segment Sr.

ffiffiffi C I r

Vw

� D

Vsl ¼ Vw–Vr (21)

h ¼ iΔlN (23)

N ¼ N1N2N3N4 (24)

<sup>2</sup> <sup>þ</sup> Vr (20)

� 100% (22)

<sup>2</sup> <sup>þ</sup> Vr ¼�φmax

center.

Figure 8.

ffiffiffiffiffiffiffiffi

follows:

a year.

215

Sliding velocity

VI

The rolling and sliding distances on the rail and wheel.

DOI: http://dx.doi.org/10.5772/intechopen.89135

C=I p is cyclic frequency of vibrations.

Relative sliding velocities

4. Conditions of derailment

<sup>w</sup> ¼� <sup>A</sup>ω<sup>D</sup>

Influence of Tribological Parameters on the Railway Wheel Derailment

Kr <sup>¼</sup> Vsl Vr

This value of the wavelength assumes that at release of the inner wheel, the friction force acting on it from the rail is zero. When the friction force differs from zero, the wavelength will be less since its both components will decrease and its value depends on the friction force magnitude.

To determine time t, we present the wheelset as a one-mass torsional vibratory system (Figure 7a), where C is a torsional rigidity of the wheelset axle and I, total moment of inertia of the inner wheel. Then, angle of twist φmax will fall down to zero in conformity with a law of free vibrations of this vibratory system during the period P/4 (Figure 7b).

At that, period of free vibrations

$$\mathbf{P} = 2\pi\sqrt{I/\mathbf{C}} \tag{17}$$

and consequently, time t will be

$$\mathbf{t} = \mathbf{P}/\mathbf{4} = \frac{\pi}{2}\sqrt{I/\mathbf{C}}\tag{18}$$

The average velocity of the wheel contact point relative to the wheel center (Figure 8)

$$\mathbf{Vw} = -\frac{D\mathbf{q}\max}{\mathbf{2}t} + \mathbf{Vr} \tag{19}$$

where Vr = � V is a velocity of the rail contact point relative to the wheel center.

Figure 7. (a) One-mass torsional vibratory system; (b) graph of the system free vibrations.

Influence of Tribological Parameters on the Railway Wheel Derailment DOI: http://dx.doi.org/10.5772/intechopen.89135

Figure 8. The rolling and sliding distances on the rail and wheel.

We note that maximum velocity of the wheel contact point relative to the wheel center.

$$\mathbf{V}\_w^I = -\frac{A\alpha D}{2} + \mathbf{Vr} = -q\text{max}\sqrt{\frac{\mathbf{C}}{I}} \times \frac{\mathbf{D}}{2} + \mathbf{Vr} \tag{20}$$

where A = φmax is an amplitude of the wheelset shaft torsion vibrations and ω = ffiffiffiffiffiffiffiffi C=I p is cyclic frequency of vibrations.

Sliding velocity

inner wheel will roll and slide simultaneously on the rail and the rolling and sliding

We note that the rolling and sliding distance on the wheel tread surface is

Sw ¼ Δl

here Δl or ΔlI is a sliding friction path and the wavelength of the worn-out rail

This value of the wavelength assumes that at release of the inner wheel, the friction force acting on it from the rail is zero. When the friction force differs from zero, the wavelength will be less since its both components will decrease and its

To determine time t, we present the wheelset as a one-mass torsional vibratory system (Figure 7a), where C is a torsional rigidity of the wheelset axle and I, total moment of inertia of the inner wheel. Then, angle of twist φmax will fall down to zero in conformity with a law of free vibrations of this vibratory system during the

<sup>P</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup> ffiffiffiffiffiffiffiffi

2

where Vr = � V is a velocity of the rail contact point relative to the wheel center.

ffiffiffiffiffiffiffiffi

<sup>t</sup> <sup>¼</sup> <sup>P</sup>=<sup>4</sup> <sup>¼</sup> <sup>π</sup>

Vw ¼� <sup>D</sup>φmax 2t

(a) One-mass torsional vibratory system; (b) graph of the system free vibrations.

The average velocity of the wheel contact point relative to the wheel center

Sr ¼ Vt (14)

Sw ¼ Δl þ Sr: 14<sup>0</sup> ð Þ

W ¼ lasers (16)

I=C p (17)

I=C p (18)

þ Vr (19)

<sup>I</sup> <sup>þ</sup> Sr (15)

distance on the rail will be

(Figure 3)

period P/4 (Figure 7b).

(Figure 8)

Figure 7.

214

or for the variant of the elliptical wheel

Transportation Systems Analysis and Assessment

value depends on the friction force magnitude.

At that, period of free vibrations

and consequently, time t will be

$$\mathbf{V}\mathbf{sl} = \mathbf{V}\mathbf{w} - \mathbf{V}\mathbf{r} \tag{21}$$

Relative sliding velocities

$$\text{Kr} = \frac{V\_{sl}}{V\_r} \times 100\text{\% and Kw} = \frac{V\_{sl}}{V\_w} \times 100\text{\%} \tag{22}$$

The depth of the worn-out layer a year of the rail segment Sr.

$$\mathbf{h} = \mathbf{i}\Delta \mathbf{l} \mathbf{N} \tag{23}$$

where i is the wear intensity and N, number of cycles which is determined as follows:

$$\mathbf{N} = \mathbf{N}\mathbf{1}\mathbf{N}\mathbf{2}\mathbf{N}\mathbf{3}\mathbf{N}\mathbf{4}\tag{24}$$

where N1 is a number of the trains passing by a day; N2, number of wagons in the train; N3, number of wheels on one side of the wagon; and N4, number of days a year.

#### 4. Conditions of derailment

Possibility of derailment or the wheel's rolling up on the rail is estimated by the criterion of the wheel flange contact point (point A, Figure 9) slipping down the rail lateral surface, based on the condition of equilibrium of forces acting on this point [24]. Lateral L and vertical V forces determined from the condition of equilibrium of these forces are.where N is a normal force; FI = fIN, friction force

between the wheel flange and rail lateral surface; fI, friction coefficient between these surfaces; and β, angle of inclination of the wheel flange.

$$\mathbf{L} = \mathbf{N}\sin\beta - \mathbf{F}\mathbf{I}\cos\beta\tag{25}$$

Therefore, it is necessary to provide the criterion (27) with additional condition of impossibility of the wheel rolling on the contact point A, which, on the base of

where h is the value of the climbing advance; r is the radius of the wheel rolling

where f and fI are friction coefficients between the wheel and rail tread surfaces

Determining N and V correspondingly from (25) and (26), substituting them into (29) and then putting obtained P into (28), from the latter we obtain the

h sin β � f

If this criterion is not satisfied, the wheel starts to roll on the contact point A, and the contact between the wheel and rail tread surfaces is lost, or two-point contact at O and A passes into one-point contact at A. For obtaining a criterion of impossibility of the wheel rolling on the contact point A, it is necessary to put f = 0

h sin β � f

ð Þ <sup>r</sup> <sup>þ</sup> <sup>d</sup> <sup>f</sup> cos <sup>β</sup> <sup>þ</sup> f f <sup>I</sup>

I cos β

I cos β

ð Þ r þ d f

The criteria (30) and (31) provide both, the wheel flange contact point sliding down the rail lateral surface and impossibility of the wheel rolling on this point. Besides, the criterion (30) ensures less value (more conservative) of the allowable ratio of the lateral and vertical forces L/V than criterion (27), while criterion (31), depending on the value of the climbing advance h, gives the ratio L/V less or more than criterion (27). For illustration, consider two variants of numerical data of the

Allowable maximum ratios L/V for these variants calculated by the criteria (27),

Variant Criterion (27) Criterion (30) Criterion (31) a 1.39 0.31 1.04 b 1.39 0.44 1.47

Force P acting on the wheel axle cannot exceed the sum of the friction forces between the wheel and rail tread surfaces and between the wheel flange and rail

circle; d is the vertical coordinate of the contact point A.

Influence of Tribological Parameters on the Railway Wheel Derailment

and the wheel flange and rail lateral surfaces correspondingly.

L <sup>V</sup> <sup>≤</sup>

a. β = 60o; f = 0.4; fI =0.1; h = 62 mm and r + d = 485 mm;

b. β = 60o; f = 0.4; fI =0.1; h = 88 mm and r + d = 482 mm.

(30), and (31) are given in the following table:

following criterion of impossibility of the derailment:

L <sup>V</sup> <sup>≤</sup> Vh≥ P rð Þ þ d (28)

P≤F þ FI ¼ fV þ fIN (29)

sin β þ f <sup>I</sup> (30)

<sup>I</sup> (31)

Figure 10, can be written as

DOI: http://dx.doi.org/10.5772/intechopen.89135

lateral surface:

in (30), which gives.

parameters:

217

$$\mathbf{V} = \mathbf{N}\cos\beta + \mathbf{F}\mathbf{I}\sin\beta\tag{26}$$

It should be noted that the forces acting on point A are interdependent and equalities (25) and (26) are only valid for limited values of forces L and V, since the rise of the friction force FI is limited by the friction coefficient fI. Therefore, at a certain ratio of forces L and V, the friction force FI can no longer balance the contact point A, which will slip down on the rail lateral surface, and it is considered on this ground that the wheel cannot roll up on the rail. At that, equalities (25) and (26) become inequalities from where a criterion of impossibility of the wheel rolling up on the rail or derailment is obtained [24]:

$$\frac{L}{V} \le \frac{\tan \beta - f^I}{1 + f^I \tan \beta} \tag{27}$$

However, at sign of equality (=) in (27) and to a certain extent at sign of inequality (<) also, the wheel can rotate about contact point A and roll up on the rail if such possibility exists or if moment of the force P acting on the wheel axle exceeds the moment of the vertical force V about contact point A (Figure 10). In other words, under such condition, two-point (O, A) contact of the wheel passes into one-point contact at A. In the first case (at sign =), the wheel will roll on the immobile point A with pure rolling, and in the second case (at sign <), the wheel will roll on the mobile point A creeping slowly down the rail lateral surface with combined rolling and sliding. Both cases lead to the wheel climbing the rail and derailment.

Figure 10. Forces acting on the wheel axle.

Influence of Tribological Parameters on the Railway Wheel Derailment DOI: http://dx.doi.org/10.5772/intechopen.89135

Therefore, it is necessary to provide the criterion (27) with additional condition of impossibility of the wheel rolling on the contact point A, which, on the base of Figure 10, can be written as

$$\mathbf{Vh} \ge \mathbf{P}(\mathbf{r} + \mathbf{d}) \tag{28}$$

where h is the value of the climbing advance; r is the radius of the wheel rolling circle; d is the vertical coordinate of the contact point A.

Force P acting on the wheel axle cannot exceed the sum of the friction forces between the wheel and rail tread surfaces and between the wheel flange and rail lateral surface:

$$\mathbf{P} \le \mathbf{F} + \mathbf{F}\mathbf{I} = \mathbf{f}\mathbf{V} + \mathbf{f}\mathbf{I}\mathbf{N} \tag{29}$$

where f and fI are friction coefficients between the wheel and rail tread surfaces and the wheel flange and rail lateral surfaces correspondingly.

Determining N and V correspondingly from (25) and (26), substituting them into (29) and then putting obtained P into (28), from the latter we obtain the following criterion of impossibility of the derailment:

$$\frac{L}{V} \le \frac{h\left(\sin\beta - f^l \cos\beta\right)}{(r+d)\left(f\cos\beta + ff^l\sin\beta + f^l\right)}\tag{30}$$

If this criterion is not satisfied, the wheel starts to roll on the contact point A, and the contact between the wheel and rail tread surfaces is lost, or two-point contact at O and A passes into one-point contact at A. For obtaining a criterion of impossibility of the wheel rolling on the contact point A, it is necessary to put f = 0 in (30), which gives.

$$\frac{L}{V} \le \frac{h\left(\sin\beta - f^I \cos\beta\right)}{(r+d)f^I} \tag{31}$$

The criteria (30) and (31) provide both, the wheel flange contact point sliding down the rail lateral surface and impossibility of the wheel rolling on this point. Besides, the criterion (30) ensures less value (more conservative) of the allowable ratio of the lateral and vertical forces L/V than criterion (27), while criterion (31), depending on the value of the climbing advance h, gives the ratio L/V less or more than criterion (27). For illustration, consider two variants of numerical data of the parameters:

a. β = 60o; f = 0.4; fI =0.1; h = 62 mm and r + d = 485 mm;

b. β = 60o; f = 0.4; fI =0.1; h = 88 mm and r + d = 482 mm.

Allowable maximum ratios L/V for these variants calculated by the criteria (27), (30), and (31) are given in the following table:


between the wheel flange and rail lateral surface; fI, friction coefficient between

It should be noted that the forces acting on point A are interdependent and equalities (25) and (26) are only valid for limited values of forces L and V, since the rise of the friction force FI is limited by the friction coefficient fI. Therefore, at a certain ratio of forces L and V, the friction force FI can no longer balance the contact point A, which will slip down on the rail lateral surface, and it is considered on this ground that the wheel cannot roll up on the rail. At that, equalities (25) and (26) become inequalities from where a criterion of impossibility of the wheel rolling

tan β � f

1 þ f

However, at sign of equality (=) in (27) and to a certain extent at sign of inequality (<) also, the wheel can rotate about contact point A and roll up on the rail if such possibility exists or if moment of the force P acting on the wheel axle exceeds the moment of the vertical force V about contact point A (Figure 10). In other words, under such condition, two-point (O, A) contact of the wheel passes into one-point contact at A. In the first case (at sign =), the wheel will roll on the immobile point A with pure rolling, and in the second case (at sign <), the wheel will roll on the mobile point A creeping slowly down the rail lateral surface with combined rolling and sliding. Both cases lead to the wheel climbing the rail and

I

L <sup>V</sup> <sup>≤</sup>

L ¼ N sin β � FI cos β (25) V ¼ N cos β þ FI sin β (26)

<sup>I</sup> tan <sup>β</sup> (27)

these surfaces; and β, angle of inclination of the wheel flange.

up on the rail or derailment is obtained [24]:

Transportation Systems Analysis and Assessment

derailment.

Figure 9.

Figure 10.

216

Forces acting on the wheel axle.

Forces acting on the contact point a.

For analysis of the obtained results, suppose that ratio L/V = 1.3, i.e., criterion (27) is satisfied and derailment is not possible. However, it is seen from the table that for variant (a) neither criteria (30) nor (31) are satisfied and both predict derailment. For variant (b), criterion (30) is not satisfied, or it predicts derailment, and criterion (31) is satisfied, i.e., by this criterion, derailment is not possible. This means that the wheel starts to roll on the contact point A and two-point (O, A) contact passes into one-point contact at A. Then, this contact point slides down the rail lateral surface, the two-point contact restores, and so on, this process is repeated. However, at passing from two-point (O, A) contact into one-point contact at A, the lateral and vertical forces on the steering surfaces increase. Typical for these surfaces, increased relative sliding increases the power and thermal loads in the contact of these surfaces, generating the convenient conditions for destruction of the third body. This results in sharp increase of the cohesion forces, scuffing, and friction coefficient that promotes climbing of the wheel flange on the rail lateral surface. This is confirmed by the numerous laboratory researches carried out by us as well as the trace of the wheel climbing on the railhead lateral surface (Figure 1) that has a form of scuffing.

less value (more conservative) of the allowable ratio of lateral and vertical

For solution of the problem of derailment, an experimental–theoretical approach

Due to the existence of materials with quite different designations and properties in the contact zone, many new unanswered problems rise. They are related with the further increase of the derailment criterion informativity and precision, providing the contact zone with the third body having due properties, conditions of formation, and destruction of the third body. They also concern to the tribological properties of the interacting metal and nonmetal surfaces, direct interaction of their juvenile surfaces and generation of the strong adhesion bonds, cold welding, destruction and wear of the surfaces, variation of the value, and instability of the

On the base of experimental researches, we have ascertained dependence of the friction coefficient on the degree of destruction of the third body for the conditions of various relative sliding velocities, speeds, materials of interacting surfaces, roughness of the surfaces, friction modifiers, and loads at which the range of variation of the acting parameters is quite wide and therefore continuation of

George Tumanishvili\*, Tengiz Nadiradze and Giorgi Tumanishvili

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: ge.tumanishvili@gmail.com

Institute of Machine Mechanics, Tbilisi, Georgia

provided the original work is properly cited.

is needed because of the lack of comprehensive theoretical model of the wheel

Influence of Tribological Parameters on the Railway Wheel Derailment

forces than Nadal's formula.

DOI: http://dx.doi.org/10.5772/intechopen.89135

climbing on the rail.

friction coefficient.

researches is needed.

Author details

219

Thus, it is expedient to estimate possibility of derailment by criterion (30), since it provides both, the wheel flange contact point sliding down the rail lateral surface and impossibility of the wheel rolling on the same point, and ensures less value (more conservative) of the allowable ratio of the lateral and vertical forces L/V than criteria (31) and (27).
