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The article contains a model of wagon rolling down the hump profiles. It deals with motive and resistance forces. These forces made it possible to mathematically describe geometrical parameters of hump profile and kinematical characteristics of wagon movement depending on time. The conditions have been formulated, which makes the wagon movement down various marshalling hump profile sections possible. These conditions enable slowdown of the wagon as it moves from one hump profile element to the other until it stops in the hump-yard.

In [

In [^{st} BP) at speed which is lower than the entrance speed have been formulated. There have been derived equations of wagon rolling speed and obtained final analytical formulas for finding the distance travelled, the movement time on the given length of the braking section making it possible the movement of a wagon (cut) with slowdown before switching area. Analytical formulas for determining braking distance have been obtained in a case when wagon motive forces are lower than the braking ones which may occur in case of non-observance of certain conditions.

In [

On this basis it should be observed that the determination of forces perceived by the wagon rolling down the hump profile is part of a still unsolved problem.

It is required to find all motive forces and resistance forces acting upon the wagon and emerging during its rolling down the hump profile because of the friction of wheels against rail threads, friction in bearings in box units and due to some other accidental forces with allowance made for the action upon the wagon (or the cut) of transverse transferring inertia force and head or/and fair wind.

Just as in [

We assume that in case of absence of the action of aerodynamic resistance force from the wagon side face (i.e. under _{1}, being their speed instant center (SIC) has the speed equal to zero.

Let us suppose that under changing climatic conditions during winter causing the change of air density followed by snow and hoarfrost, within bundle switching area and on sorting tracks, during wagon entrance on a side track through point switch, under the action of force

We assume that during wagon passing along the length of hump and/or yard braking position (BP) truck wheel pairs move down the rail threads with sliding without rolling, but at the same time depending on the value of braking forces wheel sliding with rolling is not excluded.

Just as in [

We assume that in case of absence of the action of aerodynamic resistance force from the wagon side face (i.e. under _{1}, being their speed instant center (SIC) has the speed equal to zero.

Let us suppose that under changing climatic conditions during winter causing the change of air density followed by snow and hoarfrost, within bundle switching area and on sorting tracks, during wagon entrance on a side track through point switch, under the action of force

We assume that during wagon passing along the length of hump and/or yard braking position (BP) truck wheel pairs move down the rail threads with sliding without rolling, but at the same time depending on the value of braking forces wheel sliding with rolling is not excluded.

As in [

The forces of wagon (or a cut) wheel friction against the rail threads in the absence of defects in rolling bearings of box units come down to the forces of rolling friction the basic moment of which can be taken to be equal to M^{(e)} = f_{rf}F_{z}, where f_{rf} is a rolling friction coefficient, m; as this coefficient is equipotent to the arm of couple of rolling friction (wheel down the rail f_{rf} = 5 × 10^{−}^{6}, hardened steel on steel f_{rf} = 1 × 10^{−6}); F_{z} is the sum of projections of all active forces onto the vertical axis falling at each box unit according to the technology of cargo mass center location in relation to axes of wagon crossing [

For the wheel pair of the wagon rolling down the hump up to the first braking position (1^{st} BP) due to the fact that there is always observed the condition_{fr}/r_{w})F_{z} (where G_{x} = Gsinψ_{0,50} G_{x} = Gsinψ_{0,50} is projection of gravity force of a wagon with cargo upon axis Ox, kN; r_{w} is wheel radius equal for a freight car to 0.475 m). Here the relation f_{fr}/r_{w} for the majority of materials is smaller than friction coefficient of sliding f. The value f between contacting wheel surfaces of freight cars and rail threads is taken to be 0.25 [_{k}_{,50} = Gsinψ_{0,50} - F_{fr} causing wagon movement under its rolling down the hump with acceleration. An exception to this rule may be wet weather when coefficient f is reduced.

Hence, it is clear that rolling friction as the need arises can be replaced by symbolic sliding friction under pure rolling [

where n_{w} is the number of wheels in trucks, item, (n_{w} = 8); f_{rf} is coefficient of rolling friction, m, as this coefficient is equipotent to the arm of couple of rolling friction (wheel down the rail f_{fr} = 5 × 10^{−6}, hardened steel on steel f_{fr} = 1 × 10^{−6}), r_{w} is wheel radius equal for a freight car 0.475 m.

On the mechanical system there also act internal forces in the form of rolling friction moments _{трп} = M_{трпA} + M_{трпB.}

In points of contact with rolling elements of inner diameter of bearing inner ring there emerge internal forces N_{br}―normal component of bearing reaction and in the same point on the rolling element from the inner ring there acts reaction N_{b} which is the same according to the module but opposite in direction. Here, just as above,

(i is the number of wheels in one truck axis (i = 2));

The moment of rolling friction in bearings of box units of leading and rear wagon trucks

where n_{b} = 8 is the number of box units in trucks, item; f_{fr}_{0} is coefficient of friction of rolling elements down bearing rings (normally taken to be 0.001 × 10^{−3}), m; N_{b} is normal reaction falling on two cylindrical roller bearings, or the force acting upon the most loaded rolling element and determined according to the formula, kN [

taking into account the fact that n_{ffr} is the total number of rolling elements, perceiving the load in each bearing, units; k is constant coefficient which is taken according to the row lane and type of rolling bearings.

By analogy, (1) rolling friction can be replaced by symbolic sliding friction under pure rolling motion of rolling elements in bearings [

where n_{bb} is the number of bearings in truck box units, item. (n_{bb} = 16); r_{b} is an outer radius of inner ring of roller bearing, m.

We rewrite the above formula taking into account (1a) and (2a):

Joining (1) and (3) wheel rolling friction can be replaced by symbolic sliding friction under pure rolling and rolling elements in bearing box units:

where f_{0} is some symbolic (or modified) coefficient of sliding friction:

In agreement with the above, all given constraint reaction are broken into normals

In so doing, just as in [

It is also taken into account that the tangent constituents

Introducing the notion of “shearing”

The condition of wagon rolling down the first profile hump section with grades not steeper than 50% is

This implies that the excess of forces_{e}_{,50}(t) and acceleration a_{50} (t which depends mainly on the angle of rolling down the hump ψ_{0,50} and to some extent on the coefficient of sliding friction of wheel flanges down the rail and also on the state of roller bearings in truck box units. So, in order to ensure wagon movement at the end of the first profile hump section at speed v_{e}_{,50}(t) lower than the speed of entrance v_{ewx}(t) on the first braking position (1^{st} BP) i.e. v_{e}(t) < v_{ewx}(t), it is enough to select a rational value as a major geometrical hump parameter.

It is common knowledge that friction forces as resistance forces are always directed oppositely to the relative speed of wagon rolling down the hump. Friction force is a reactive (braking) force. In case of wagon (with cargo and/or without it) rolling down the hump friction force is a retaining force. That is why shunting masters consider this force to be “harmful” due to which the wagon (cut) may not reach the hump design point or, which is even worse, stop in the switching area impeding accelerated breaking up of the train. On the other hand, friction force as retaining force facilitates space-target regulation of cut speeds. This kind of speed regulation should be enough for switching operation and realization of design speeds of cuts at the exit from these positions. It is possible to provide cut approach to the yard braking position (3^{rd} BP) at speed not more than 23.4 km/h (6.5 m/s). In doing so the cuts should be able to roll up to the design point and the speed of cut collision in the yard shouldn’t exceed 5 km/h (1, 39 m/s).

It should be emphasized that while aiming at provision of space-target cut speed regulation it is necessary to slow down the wagon (or cut). At the same time it is necessary to slow down the rotation of wheels making them partially slide along the rail threads and at the expense of the emerging sliding friction between side surfaces of wheels and braking tires of wagon retarders due to the substantial value of compressed air maximum pressure in pneumo-system (0.75 MPa) it is possible to obtain considerable braking force. That is why the “desired” friction force in spite of generally accepted view shouldn’t be referred to incidental or “episodic” forces.

In a general case forces of sliding friction of wheels down the rails with rolling ^{st} BP, 2^{nd} BP) or yard braking position (3^{rd} BP) on mechanized humps:

where F_{τ} is tangent component of constraint reaction (rail threads), which is equal according to Coulomb law F_{τ} = f_{ск} N with allowance made for the fact that f_{sl} is friction coefficient of wheel sliding along rail threads (metal against metal) f_{sl} = 0.15/0.25), N is normal component of constraint reaction which is modulo equal to the sum of projections of all active forces on the vertical axis falling on each box unit, F_{z}. It should be noted the value of f_{sl} also depends on weather conditions;

F_{ffr} = f_{af}F_{af} is friction force emerging between lateral surfaces of wheels and retarder braking bars where f_{af} = 0, 14/0, 20 is friction coefficient of sliding of wheel lateral surfaces against retarder brake beams [_{af} also depends on the weather conditions; F_{af} - is application force of retarder brake blocks to the wheel lateral surfaces or average load on the wagon axis (normally it is 90, 100, 140, 150 kN depending on the type of a retarder and air pressure) [_{af} is equipotent to retarder braking force F_{br・f}, i.e. f_{af}F_{af} = F_{br・f}. While solving practical problems it is possible to take

In that way, when a single wagon (or a cut) is within BP, in a general case the speed of the wagon is reduced (i.e. the wagon slows down) at the expense of sliding of skating surface of wheels in combination with their rolling down rail threads and also on account of sliding between wheel lateral surfaces and retarder brake bars. Special reference should be made to the fact that in BP zone because of application of brake bars through the fault of controller of train sorting speed (human factor) there may occur pure wheel sliding against brake bars until complete stopping of the wagon. In this case the value of F_{тп} is large as compared to F_{τ} (i.e. F_{ffr} > F_{τ}) and it is possible that

We rewrite the above expression for the case of wheel sliding against brake blocks with rolling in the form

Here the normal component N of constraint reaction modulo is equal to the sum of projections of all active forces on the vertical axis F_{z}, falling on each box unit. In accordance with this statement we form the above ratio in the following way:

where _{0,b} is the angle of descend on the braking position of the hump section, rad.

Making allowance for the fact that each truck wheel enters hump the section of braking position consecutively, i.e. lagging in time τ, we rewrite [

where σ_{0}(t), σ_{0}(t - τ_{1}), σ_{0}(t - τ_{2}) and σ_{0}(t - τ_{3}) σ_{0}(t), σ_{0}(t - τ_{1}), σ_{0}(t - τ_{2}) and σ_{0}(t - τ_{3}) are dimensionless retarded Heaviside functions which make it possible to present time t_{i} as one analytical expressions appropriate under any value of t in the interval_{1}, τ_{2} and τ_{3} is the time lag of emerging of friction force while passing braking position by the second wheel of the leading truck (

bearing in mind that l_{tr} and l_{w} is the half of the truck and wagon base, m; v is speed of entrance of the wagon onto the 1^{st} BP before the dividing switch, m/s (lower than 30.6 km/h (8.5 m/s) or 23.4 km/h (6.5 m/s) according to the design of retarders).

Analyzing the results of the previous reasoning we can write that the condition of wagon movement at the hump section braking positions with gradients 7/15‰ is (q.v. (5))

Nonobservance of this condition (8) may lead to wagon stopping along the length of hump braking positions which may be the case in practice.

The analysis shows that excess of forces _{ebi}(t), and acceleration a_{bi}(t). These wagon kinematic parameters mainly depend on the gradient of hump braking positions ψ_{0,bi}, retarder braking force F_{br・f}_{.}, friction coefficient f_{sl} of sliding wheels against rail threads. It is necessary to ensure the movement of the wagon on hump sections with slowdown up to speed v_{b}(t) (considerably lower than entrance speed v_{ewx}(t), i.e._{0,b} as a geometrical hump parameter but also by close controlling of braking force of wagon retarder F_{br・f}_{. }In this case it is possible to ensure the movement of the wagon at speed v_{ebi}(t), which corresponds to acceleration with slowdown a_{50}(t), considerably lower in value than acceleration at the first profile hump section a_{50}(t), i.e.

Summing up the notion of friction force it is necessary to note that in addition to some other friction forces there are accidental or the so called “episodical” forces.

1) Friction forces of wheel flanges sliding along lateral sides of thrust rail_{ey} and projections of aerodynamic resistance force on the lateral surfaces of wagon with cargo_{ey} and

_{ey} and projection of force _{W} is moment from pair of forces _{W} will not be taken into account because it is not needed for solving the problem in question. It is should be remembered that moment M_{W} is necessary for solving problem on tilting of the wagon with cargo.

Force

where f_{rc・fr} is reduced coefficient of sliding friction for freight cars (i.e. a symbolic value) [

Here F_{z} is a projection of all active forces onto the vertical axis falling on each box unit, kN (_{wp} is gravity force of wagon wheel pair with account of rotation inertia, kN; f is friction coefficient of sliding between contacting surfaces of wheels and rail threads (for freight car it is taken to be 0.25); r_{bi} is rolling bearing inner radius in box units, m (0.079); r_{w} is wheel radius, for a freight car it is equal to 0.475.

2) Friction forces of sliding of wheels (with rolling) down the rail which emerge in winter conditions under the change of air density followed by snow and hoar frost within the limits of switching bundle zone and on sorting tracks. Here it is necessary to bear in mind that according to Coulomb law:

where f_{cr} is coefficient of resistance of the rail to the wheel movement (it is taken according to full-scale experiment data, or f_{cr} = f_{sl} (см. (6))),

The condition of wagon movement on this hump section with gradient 0.6/2.5‰ is also

It follows that excess of forces_{ebi}_{;0.6-2.5}(t) and acceleration a_{bi}_{;0.6-2.5}(t) (i just as previously are the numbers of braking positions). These wagon kinematic parameters mainly depend on the gradient of hump braking positions ψ_{0,bi;0.6-2.5} and coefficient of resistance of the rail to the movement of wheels f_{cr}. At that the movement of the wagon is with slowdown at speed v_{ebi}_{;0.6-2.5}(t) lower than the speed of exit from the braking positions v_{bi}(t), i.e._{cr} is an uncontrolled parameter, depending on weather conditions. In this case the wagon moves with slowdown a_{bi}_{;0.6-2.5}(t) at speed v_{ebi}_{;0.6-2.5}(t).

3) Friction force _{ey}) directed crosswise the wagon (axis Oy) and appearing under striking of wheel pairs at switch points (against switch blades, frogs, guard rails) [

where f_{fr・sl} is coefficient of sliding friction (assumed to be pure sliding) of wheel flanges being struck about against rail threads (for wagon wheels it is taken to be f_{fr・sl} = 0.25); σ_{0}(t), σ_{0}(t - τ_{1y}), σ_{0}(t - τ_{2y}) и σ_{0}(t - τ_{3y}) - f_{сп} = 0.25); σ_{0}(t), σ_{0}(t - τ_{1y}), σ_{0}(t - τ_{2y}) and σ_{0}(t - τ_{3y}) - dimensionless Heaviside retarded functions defined just as in (7a) and differing by the fact that τ_{1y}, τ_{2y} and τ_{3y} is retardation time of emerging friction force while passing point switch of the second wheel of the leading truck (

The condition of wagon movement down point switch is

Analyzing (13), special mention can be made of the fact that excess of forces_{e}_{・mp}(t) and acceleration a_{mp}(t). These kinematic parameters of the wagon depend mainly on the gradient, on which point switch ψ_{0・mp} = ψ_{0,bII} is located, and sliding friction coefficient of wheel flanges striking against rail threads f_{mp}. At that wagon movement down hump point switches is with slowdown speed v_{e}_{・mp}(t) lower than the speed of wagon entrance onto the zone of switch bundles v_{e}_{・bII}(t), i.e._{0,mp} as a hump geometrical parameter as sliding friction coefficient of wheel flanges striking against rail threads f_{mp} is an uncontrolled parameter depending on weather conditions. In this case there occurs wagon slowdown with acceleration a_{mp}(t), in value considerably lower than acceleration at section of braking positions with gradients 7/15‰ a_{bi}_{,7-15}(t), 7/15‰ a_{bi}_{,7-15}(t), i.e.

4) Sliding friction forces

Force

where

with allowance made for the fact that

Here n_{wb} is the number of wheels in trucks, item. (n_{wb} = 6); n_{bb} is the number of bearings in truck box units, units. (n_{bb} = 12);

where f_{slb} is coefficient of sliding friction of wheels against the rail (“metal against metal”)

_{1} = N/n_{wb} (here

Putting (15) and (16) in (14) we will get:

or, taking into account the fact that N_{1} = N/n_{wb},

Having regard to the fact that sorting yard tracks are designed with easy grades (1/1.5‰), for which cos(ψ_{0;1-1.5}) = 1 and sin(ψ_{0;1-1.5}) = 0,

where f_{frb} is coefficient of rolling friction of wheels with sliding in respect to the rail threads together with braking block under the wheels

The condition of wagon movement down sorting yard track with gradients is 1/1.5‰

Non-observance of this condition will result in wagon stopping at the prescribed length of the sorting yard track.

Hence it follows that excess of forces _{1-1.5}, which is, however, lower in value than acceleration at the sections of switching zone with gradients 1/2.5‰, i.e. a_{ss} < a_{1-2.5}. These kinematic parameters of the wagon mainly depend on the hump profile with gradients ψ_{0,1-1.5} and coefficient of rolling friction of the wheels with sliding against rail threads together with braking block under the wheel f_{frb}. In so doing it is necessary to ensure the movement of the wagon in the hump yard as far as the group of standing wagons with slowdown at speed v_{e}_{,1-1.5} (t lower than the speed of exit of the wagon onto the zone of point switches v_{e}_{・bss}(t), i.e._{0,ss}, as a hump geometrical parameter, but also by choosing material of contacting surface of the wheel with braking block f_{frb}, as the coefficient of sliding friction of the other wheel of the wheel pair of the leading track against the other rail thread is an uncontrolled parameter. In this case there occurs slowdown of the wagon with acceleration a_{1-1.5}(t), considerably lower in value than acceleration at the sections of point switching with gradients 10/12‰ a_{ss}(t), i.e.

Summing up the results of the conducted analysis concerning the character of resistance (braking) forces due to which there occurs wagon (or cut) movement with slowdown it can be noted that for ensuring wagon rolling down the hump up to the moment of its stopping together with the group of standing wagons it is necessary to fulfill the conditions:

Thus, there have been derived analytical formulas for determination of motive forces and resistance (braking) forces affecting the wheels of wheel pairs of the wagon during its rolling down hump longitudinal profile.

1) On the basis of classical statements of theoretical and applied mechanics, computable models of wagon rolling down the hump have been constructed and identified motive and resistance (braking) forces which will make it possible to model geometrical hump (parameters and kinematic characteristics of wagon movement).

2) Various conditions have been formulated, which makes it possible for a wagon to move down the hump profile elements allowing ensuring wagon (cut) slowdown in the process of movement from the hump crust down the design point of the sorting yard track.

The novelty of the derived analytical formulas of braking forces for each section of hump profile consists in correct account of all types of resistance affecting wagon movement. The obtained results of the research are a new stage in the development of this problem.

The significance of this methodology is presented by the opportunity of constructing a mathematical model of wagon rolling down the hump with allowance made for the influence of aerodynamic resistance due to the speed and direction.

In the long term the obtained results of the research can be used for solving technical problems of determination of hump rational geometrical parameters and wagon kinematic characteristics’ during its rolling.

Khabibulla Turanov,Andrey Gordiienko, (2015) Analytical Determination of Conditions of Wagon Rolling Down Marshalling Hump Profiles. Open Access Library Journal,02,1-11. doi: 10.4236/oalib.1101912