About the movement of the car on the high-speed sections of the sorting hill

. The content of the article is based on the classical provisions of mechanics (the Dalembert principle) for non-ideal communication. The analytical proof is given that the linear acceleration of the car with its equally accelerated movement along the descent part of the sorting slide depends on the force under the influence of which the car rolls down the slope of the slide, the strength of resistances of all kinds, on the reduced mass of the car with the load, taking into account the moment of inertia of the rotating parts. At the same time, the results of calculations proved that taking into account the mass of rotating parts practically does not affect the acceleration of the carriage along the slope of the slide. The acceleration formulas and the resulting forces acting on the car are presented in the generally accepted notation and in the usual sense, followed by the calculated data.

It is noted in [6] that the authors of the article [4], having mathematically unsubstantiated the correctness of the universal formula (4), stubbornly defend their opinion that it can be used for calculations on any sections [15] with a slope of i sorting slides. At the same time, in formula (4) [4], the acceleration of free fall ′ , taking into account the inertia of the rotating masses, is also taken into account in the sections of the braking positions (see subtracted 2 ′ ℎ т in formula (4)). In addition, the impact of accounting and/or not taking into account the inertia of rotating masses on the acceleration of the movement of the car along the profile of the sorting slide has not yet been assessed.
In the article, based on the classical provisions of mechanics (the Dalembert principle) for an imperfect connection [17,18], an analytical proof will be given that the linear acceleration of the car with its equidistant movement along the descent part of the sorting slide ai = aCi depends on the force Fxi, under the influence of which the car rolls down the slope of the slide, resistance forces of all kinds |Fсi|, as well as from the reduced mass of the carriage with the load Mred, taking into account the moment of inertia of the rotating parts JC. At the same time, it will be proved that taking into account the mass of rotating parts practically does not affect the acceleration of the carriage along the slope of the slide.

The purpose of this article
To prove and explain to the authors of the article [2,4] that in the design and technological calculations of the projected sections of the slide, the most important kinematic parameter of the movement of the car is the calculation of its acceleration, the value of which directly depends on the rest of the movement parameters (time, speed and path of passage of the studied sections).

Problem statement
It is required to give the results of calculations of acceleration with equidistant movement of the car on a specific section of the slide (for example, on the second high-speed section HS2), taking into account and without taking into account the mass of rotating parts, and to prove that in this case the relative error is THE PURPOSE OF THIS ARTICLE To prove and explain to the authors of the article [2,4] that in the design and technological calculations of the projected sections of the slide, the most important kinematic parameter of the movement of the car is the calculation of its acceleration, the value of which directly depends on the rest of the movement parameters (time, speed and path of passage of the studied sections).
Problem statement. It is required to give the results of calculations of acceleration with equidistant movement of the car on a specific section of the slide (for example, on the second high-speed section SK2), taking into account and without taking into account the mass of rotating parts, and to prove that in this case the relative error is δa2sw = 4,18 ≈ 4,2 %, which is less than the accuracy of engineering calculations (≈ 5%). which is less than the accuracy of engineering calculations (≈ 5%).

Research method
As in [6], the solution of the engineering problem of the carriage movement along the slope of the sorting slide is given on the basis of the application of the basic law of dynamics for non-ideal connections (the Dalembert principle) [17,18].

Mathematical description solution of the problem
Mathematically, the movement of the car on the high-speed sections of the sorting slide will be described according to the Dalembert principle [17,18] in projections on the descent part of the sorting slide.
Similarly [6], the analytical formula for finding the acceleration of the carriage aCi = ai along the descent part of the sorting slide has the form: Where ithe numbers of sections along the entire length of the sorting slide path profile (i = 1, … 9); |aCi| = |ai|acceleration of the center of mass of the Cc car to be found [19], mps 2 ; Mred is the reduced and/or imaginary mass of a car with a load, taking into account the moment of inertia of the rotating parts (wheel pairs) JD on all sections of the slide descent (see formula (5.39b) on page 411 in [20]), kg: Taking into account that in it M1 = 80,94•10 3the weight of the wagon with cargo, taking into account non-rotating parts (i.e. bogies, wagon body), kg; mwh = 1,937•10 3weight of one wheelset, kg; |∆Fi|the resultant force under the influence of which the car rolls down the descent part of the sorting slide, similar to the formula (7) in [6]), kN: Taking into account that in it, Fxithe projection of the gravity of the loaded car G on the direction of movement of the car, taking into account and/or without taking into account the projection of the force of the tailwind Fwx, under the influence of which the car moves along the slope of the descent part of the slide, kN: Note that the value Fwx, can be neglected due to its smallness: Fwx, << G (for example, 3.2 << 908 kN); ψithe slope angle of the descent part of the slide, rad.; |Fri|in general, the resistance force of any kind.
Here, the resistance force of any kind is |Fri| taking into account and/or without taking into account the projection of the headwind force of a small magnitude Fwx, which can be taken as a fraction of the gravity of the wagon with the load G, i.e. |Fri| = f(G), which does not contradict the force ratios of the hill calculations (see page 180 in [11], p. 141 in [12]), includes the following forces: sliding friction, taking into account the rolling friction forces in the bearings of axle assemblies, as a force from the main (running) resistance Ffri = Fmi; the resistances that appear during the transition of curves (and/or resistance from curves), which depend on the sum of the angles of rotation in the curves, including the switch angles in the section under consideration, and the speed of the car, Fcuri; resistances arising from switches (from wheel impacts on wits, crosses and counter rails) Fsw; resistance from air and wind Fra; resistance to overcome additional resistance from snow and frost within the switch zone of the bundles and on the sorting tracks of the Fsn).
Based on this, the resistance force of any kind |Fri|, taking into account the impact of the projection of the tailwind force Fwx, should be determined by the formula | | = ( + cur + sw + ra + sn ).
For the descent part of the slide, except for the sections of the braking positions, the following condition must always be met in formula (1): It follows from formula (1) that, if the condition |∆Fi| > 0 and /or, according to (6), Fxi > |Fri|, corresponding to the consideration of the effect of the projection of the force of a tailwind of a small magnitude Fwx, the movement of the car along the descent part of the slide at the speed of the car entrance to the investigated section of the slide vнi > 0 it will be uniformly accelerated, and if this condition is not met, it will be uniformly slowed down, which may correspond if the impact of the projection of the headwind force of a small magnitude Fwx, is taken into account.
In the latter formula, the force Fxi, under the influence of which the car rolls down the slope of the high-speed sections of the slide, is found by formula (4), and the resistance force of all kinds |Fсi|, according to formula (5), can be represented as: In formula (7) it is indicated: Kmi = 0,001the coefficient of rolling friction with sliding of hardened steel on hardened steel (see page 42 in [21]) and/or a coefficient that takes into account the strength of rolling resistances of sliding wheels Fmi in fractions of G; kcur.ia coefficient that takes into account the strength of the resistances during the transition of the curves Fcuri in fractions of G (the value is calculated); ksw ≈ 0.00025a coefficient that takes into account the strength of the resistances from the switches Fsw in fractions of G; kra ≈ 0.0005a coefficient that takes into account the strength of resistances from the air and wind Fra in fractions of G; ksn ≈ 0.00025 is a coefficient that takes into account the force of Fsn in fractions of G; kw.x ≈ 0.004 is a coefficient that takes into account the force of Fw.x in fractions of G; Thus, by the formula (1), it is mathematically proved that the linear acceleration of the car with its equidistant movement along the descent part of the sorting slide ai = aCi, similar to the expression (8) in [6], depends on the force Fxi, under the influence of which the car rolls down the slope of the slide, resistance forces of all kinds |Fri|, and also from the reduced mass of the carriage with the load Mred, taking into account the moment of inertia of the rotating parts JD, i.e. = ( , , red ).
As can be seen, the acceleration of the movement of the car along the descent part of the slide ai, determined according to the Dalembert principle, in particular, depends on the reduced mass of the Mred of the car with the load, taking into account the moment of inertia of the rotating parts JD.
Note that the speed (vspi = vi) and the path ((lspi = li = xi) of the carriage movement on the high-speed sections of the track profile at which vкi.sp ≠ 0, is determined by the formula of speed and path of elementary physics (see formulas (18) and (19) in [19]): Where viithe initial speed and/or the speed of the entrance of the car to the investigated section of the slide profile from the previous section, i.e. the value taken from the results of calculations of the previous sections of the slide; aithe acceleration of the carriage (the value calculated by the formula (1)). Jointly solving the last two formulas, it is possible to find the speed vi and the time ti of the carriage movement on the high-speed sections of the track profile (see formulas (20), (21) in [19): We will make a special reservation that the applicability of the formula of elementary physics for determining the speed vspi and the path lspi of the movement of the car on the highspeed sections of the track profile is justified on the basis of solving the differential equation of acceleration of the movement of the car, derived on the basis of the Dalembert principle (see formulas (16) and (19) in [19]).
Summarizing the results of studies of the dynamics of the car at equidistant acceleration when moving along the descent part of the sorting slide, it can be concluded that the derivation of formula (1) is correct, the theoretical basis of which is the classical D'Alembert principle of theoretical mechanics [17,18].
Thus, we emphasize that the D'Alembert principle, being an important tool of mechanics in the study of the dynamics of motion of solids and elastic systems, has the following advantages (see page 308 in [20]): firstly, it allows you to use the usual static equilibrium equations (see formula (4.45) in [20]); secondly, it makes it possible to immediately obtain equations resolved with respect to the higher derivatives (see formula (4.46) in [20]), and therefore does not require calculations for their allocation; thirdly, it allows the possibility to directly find the acceleration of a point in the absolute motion of the aаbs with known active and reactive forces, or coupling reactions (friction forces, or braking forces), if the acceleration of aаbs is known and causing its active forces (see condition (6)).

Results of the calculated data
Calculation example 1. For example, we study the second high-speed section (HS2), located after the dividing switch (S) of the sorting slide. The initial data are as follows: G = 908the gravity of the car with the load, kN; vi2sw = 7,285the initial speed and/or the speed of the entrance of the car to the section HS2 after the switch (S), mps; ψ2sw = 0.018the slope angle of the descent part of the slide, rad.; l2sw = 18.633the length of the descent parts of the slide, m; Fx2sw = 19.535projection of the gravity of the car G on the Cx axis, taking into account the projection of the tailwind force of a small magnitude Fwx (Fwx ≈ 3,2 kN) on the section of the slide HS2; Fо2sw = kо2swG = 0.0001×908 ≈ 0.908force from the main resistance to the movement of the car, kN; Fra = kraG = 0.0005×908 = 0.454resistance force from air and wind, kN; Fsn= ksnG = 2,5•10 -4 •908 = 0,227resistance force from snow and frost, kN; Fsw= kswG = 2,5•10 -4 •908 = 0,227resistance force when passing through the dividing switch, kN; Fcur2sw = kcur2swG = 5,317•10 -5 •908 = 0,048resistance force when passing curves (and/or resistance from curves), kN.
Calculation results [22]. 1) In the general case, the resistance force of any kind on the slope section HS2 of the slide, calculated by the formula (7), kN: |Fr2sw|•= Fо2sw + Frw + Fsn + Fsw + Fcur2sw = (0,908+ 0,454 + 0,227 + 0,227 + 0,048) = 1,864. This leads to an important conclusion for practice that in the future, the design and technological calculations of the projected sections of the slide should and / or should be performed without taking into account the moment of inertia of the rotating parts (wheel pairs) of the JC car with a relative error of 4.2%, which is within the accuracy of engineering calculations (≈ 5%). 5) We present the results of the calculation according to the formulas of elementary physics (9) -(12), the possibility of using which is analytically proved in [19] (see formulas (16), (18) -(20)).
Let's calculate the time of movement of the car according to the formula (12) at the initial speed and / or the speed of entry of the car to the section HS2 of the slide, vi2sw = 7.285 m / s, and the acceleration of movement a2sw = 0.191 mps 2 , calculated according to the formula (1): t2sw = 2.477 s.
Calculate the speed of the car using the formula (9,180) and/or, which is the same according to (11), at the initial speed and/or the speed of the entrance of the car to the section of the hill HS2 vi2sw = 7,285 m/s, a2sw = 0.191 mps 2 and t2sw = 2,477 s: v2sw = 7,758 mps with and/or v7 ≈ 27.9 kmph.
In accordance with the initial data of the calculation example 1, we note that the force Fxi, under the influence of which the car moves along the descent part of the slide, taking into account the projection of the tailwind force of a small magnitude Fwx, can be represented as a fraction of the gravity of the car with a load G in the form: Where E3S Web of Conferences 389, 05023 (2023) https://doi.org/10.1051/e3sconf/202338905023 UESF-2023 ix0i = ixi + kwxa dimensionless value that conditionally characterizes the designation of the slope of the descent part of the slide in fractions of G when taking into account the impact of the projection of the force of a tailwind of small magnitude Fwx, and when not taken into account: kwx = 0, i.е. kwx = f(G) Similarly to the force Fxi, the force of all resistances |Fri| and, therefore, in the general case, the resistance force of any kind (sliding friction force taking into account the rolling friction forces in the bearings of axle assemblies, as the main resistance Ffri = Fmi, from the curves Fcuri, from the switches Fsw, from the air and wind Fra, from snow and frost Fsn) taking into account and/or without taking into account the projection of the force of a passing and/or headwind of a small magnitude Fwx, which can be taken as a fraction of the gravity of the wagon with the load G, i.e. |Fri| = f(G), kN.
So, for example, in relation to the section of the first sorting path (SP1): Fx7 = kx07G, where kx07 = 0.0051 is a coefficient that takes into account the fraction of the driving force of Fx7, taking into account the projection of the tailwind force of a small magnitude (Fwx ≈ 3.2 kN) from G on the Cx axis, i.e. Fx7 = 0.0051×908 = 4.645 kN; Fо7 = kо7G, where kо7 ≈ 0.0001 is a coefficient that takes into account the fraction of the main resistance force of Fо7 from G; Fcur = kcurG, where kcur ≈ 0.00087 is a coefficient that takes into account the fraction of the resistance force during the transition of curves (and/or resistance from curves) Fcur from G; Fra = kraG, where kra ≈ 0.0005 is a coefficient that takes into account the proportion of the resistance force from the air and wind Fra from G.
As a result, the strength of any resistance |Fri| when taking into account the effects of a tailwind is determined by the formula (7), where kw.x = 0corresponds to the case of not taking into account the projection of the wind force Fwx, since Fвx << G and / or 3.2 << 908.
Finally, formula (1), according to [4,[11][12][13][14][15][16], will be presented in the generally accepted notation and in the usual sense in the following form (for comparison, see the second term of formula (4) in [4]): Where, aithe acceleration of the center of mass of the Cc car to be found (the figure is not given here), mps 2 ; gthe acceleration of gravity of the body, mps 2 : ix0i = ixi + kwxan abstract number and/or a dimensionless quantity represented in (13); |wi|an abstract number and/or a dimensionless quantity that conditionally characterizes the designation of the specific resistance to the movement of the car along the descent part of the slide in fractions of G (i.e. wi = f(G)), in contrast to [11, p. 180; 12, 14 -16], where |wi| has dimension in an off-system unit of measurement (kgfpt): | | = + cur + sw + ra + sn .
It follows from formula (14) that if the condition ix0i > |wi| is met, the movement of the car along the descent part of the slide will be uniformly accelerated (which corresponds to taking into account the effects of a tailwind), and at ix0i < |wi|uniformly slowed down, which may occur when taking into account the effects of a small headwind.
Note that formula (14) is apparently equivalent to dependence (1). As can be seen, the acceleration of the movement of the car along the descent part of the slide ai, determined according to the Dalembert principle, in the generally accepted notation and in the usual sense depends on the acceleration of the free fall of the body, the slope and the resistivity of the movement of the car along the descent part of the slide ixi and |wi|, i.e.
Which, apparently, is equivalent to the dependence (8). However, for the convenience of performing the calculation, it is more convenient to represent the resulting force |∆Fi| (see formula (7)), under the influence of which the car rolls down the slide, in the form of: Where kx0i = ixi + kw.xa conditional coefficient that takes into account the effect of the force contributing to the movement of the car along the slide profile.
As can be seen (see formula (15)), the acceleration of the movement of the car along the descent part of the slide ai, determined according to the Dalembert principle, in particular, depends on the reduced mass of the Mred of the car with the load, taking into account the moment of inertia of the rotating parts JC.
However, previously performed calculations proved that when taking into account the reduced mass of the Mred of a wagon with a load, taking into account the moment of inertia of the rotating parts JC, the relative error compared with the failure to take into account this moment of inertia (which is equivalent to calculating Mred0) is only 4.2%, which is less than 5%, which is negligible when performing engineering calculations.

Results of the calculated data
Calculation example 2. For example, we examine the intermediate section (IN) located after the dividing switch (S) of the sorting slide. The initial data are as follows: G = 908the gravity of the car with the load, kN; vi4s = 2.723the initial speed and/or the speed of the entrance of the car to the IN section after the switch (S), mps; ψ4d = 0.0011the slope angle of the descent part of the slide, rad.; l4d = 21.271the length of the descent parts of the slide, m; Fx4d = 13.18projection of the gravity of the car G on the Cx axis, taking into account the projection of the tailwind force of a small magnitude Fwx (Fwx ≈ 3.2 kN) on the section of the IN slide; Fо4dс = kо4dG = 0,0001•908 ≈ 0.908force from the main resistance to the movement of the car, kN; Fra = kraG = 0,0005•908 = 0.454resistance force from air and wind, kN Fsn = ksnG = 2,5•10 -4 •908 = 0,227resistance force from snow and frost, kN; ; Fsw = kswG = 2,5•10 -4 •908 = 0,227resistance force when passing through the dividing switch, kN; Fcur4d = kcur4dG = 6,052•10 -4 •908 ≈ 0,095resistance force when passing curves (and/or resistance from curves), kN; |Fr4d| = 1.911in general, the force resistance of all kinds on the site of the IN hill, book.
Calculation results [22]. 1) Dimensionless value ix04d, conditionally characterizing the designation of the slope of the descent part of the slide, taking into account the effect of the projection of the force of the tailwind Fwx, calculated on the basis of formula (13): ix04d = 0.015.
2) Dimensionless value |w4d|, conditionally characterizing the designation of the specific resistance of the movement of the car along the descent part of the slide, calculated in relation to the first sorting section slides based on the formula (16):