Investigation of the process of controlled starting of the open-pit locomotive for ensuring the maximum adhesion coefficient at the starting

In order to control the calculated coefficient of adhesion of the locomotive bandages with the rail, a controlled starting of the career locomotive is suggested, in which the relative traction coefficient varies over the time according to the exponential dependence. To realize a rational starting process the tractive effort during the 1⁄4 of the starting period should be 3⁄4 of the maximum which is recommended by the traction conditions for a given section of the track, and for the remaining time, i.e. in the interval of the subsequent three-fourths of the starting time, the tractive effort must be smoothly brought to the required level, acceleration during the same period must also be changed (decreased) and at the end of the starting process, when the train reaches the working speed at this track section, it must be made equal to zero. It is proposed a relationship for determining of the starting time of the locomotive, depending on the initial parameters and actual operating conditions, which makes it possible to develop technical requirements for the automatic launch system of a open pit locomotive, which ensure the implementation of the maximum possible values of adhesion coefficients closed to optimal after the starting process.


Introduction
For the locomotives which operate in the open-pit conditions with high track gradients and with the presence of movable rails, the choice of starting mode can be crucial for realizing the maximum coefficient of adhesion in the heaviest operating mode of the locomotivewhen train starts from the place, especially if this occurs when leaving the trench on a ruling grade.This circumstance is especially actual with the current trend of increasing depth of quarrying [1].
Professor I.P. Isaev [2,3] considered the effect of three different regimes of increasing the tractive effort (the running resistance forces were considered as quasi-constant during the starting period and were significantly lower than tractive effort) on the value of the realized adhesion coefficient.
The first starting mode is called the accelerated with monotonous (without inflection points) decreasing acceleration increment.The second mode is conditionally called the uniform, while the traction effort, minus the resisting forces, increases in proportion to the starting time and the third mode -accelerated with the monotonically increasing acceleration increment.
The researches [2,4,5,6,7] have found that the highest adhesion coefficients during the starting are provided in the regime when the increasing of the traction effort reaches 75% of the maximum value, approximately in the first quarter of the starting time segment.

Object methods of the research
In view of the above, let's present the scheme of the starting schedule in the following form (Figure 1).Let's introduce the concept of the relative traction coefficient .Here, F is the current value of the traction effort of the locomotive during the starting, Fmax is the limiting value of the tractive force of the locomotive according to the adhesion conditions (at the end of the start), t is the current time value during the starting, tn is the total start time.The coordinates of characteristic points: -A (0.00; 0.00) is the origin of coordinates (according to I.P. Isaev with reference to [4]); -B (0.23, 0.50) -the average point along the ordinate axis at F / Fmax = 0.5 (our proposal); -С (0.25; 0.75) -the first quarter of the starting time at F = 0.75 Fmax [2,4]; -D (1.00, 1.00) -completion of the starting (according to Isaev [2]).
Traction effort is proportional to the acceleration, if the moving bodies do not vary in mass.Since the loaded train has a practically unchanged mass, Fig. 1 in relative units also characterizes the change in the relative acceleration (а/аmax) in the function of the relative time t/tп.
To determine the mathematical dependence of the relative traction coefficient on the start time, polynomial and exponential approximations were considered, at that the assumed was divided into two intervals: 0 <  < 0.23 and 0.23 <  < 1.00.
The approximation of the dependence of the relative traction coefficient (i.e. the tractive effort, or the acceleration, which is the same) on the relative time during acceleration must pass through the characteristic points marked in Figure 3 and must take into account the physical properties of the acceleration process.In addition, from the physical considerations, the approximating function must satisfy the conditions 0 ≤ () ≤1 for any values of the time.
On the segment from 0 to 0.23 (Figure 1) the curve (τ) can be approximated by an exponential function The figure 2 shows the graphs of the polynomial and exponential dependences of the relative traction coefficient (tractive effort) in the process of starting the locomotive from the relative time.An analysis of the results of the theoretical studies has shown that the polynomial approximation of the powers 2, 3, 4 and 5 does not give satisfactory results, both on the whole time interval and on intermediate intervals of approximation.Therefore, for further studies the exponential approximation is taken as the basis.
As a result of integration of equation ( 1) in the range from 0 to 0.23 tп, the approximation of the relative velocity graph was obtained: ( ) By integrating (2) over the interval from 0.23tп to 1.00tп, the approximation of the relative velocity graph was obtained: In Eq. ( 3) and ( 4) 1 ( ) v τ  and 2 ( ) v τ  are the relative velocities during the starting at intervals from 0 to 0.23 tп and 0.23 tп to 1.00 tп respectively.
Figure 3 shows the combined graphs of the relative acceleration and the relative speed of the controlled starting process.

The results of the research and their discussion
Based on the conducted studies, it is possible to obtain a mathematical dependence of the change in the adhesion coefficient of the bandages of locomotives with the rails during the starting.For example, for a traction aggregate of the alternating current OPE 1 (for 0 <V <40 km / h) [8,9,10,11,12] 7 0.21 53 3 To determine the absolute value of the adhesion coefficient, the functions (3) and ( 4) are substituted in the Eq. ( 5) instead of the speed V.
Since the speed is calculated in relative units, we will assume that its largest value is 40 km / h, that is, Vmax = ) 00 . 1 ( Ṽ = 0.755 km / h.In order to convert from relative units to absolute ones, you can add a coefficient: Similar results can be obtained for direct current traction aggregate, for example PE 2M (for 0 <V <40 km / h) [5,6] 7.2 0.225 100 20 For the traction aggregates OPE 1 and PE 2M, the adhesion coefficients at the moment of starting (coefficient of friction of rest) are respectively 0.342 and 0.297.Calculations show that when using a rational mode of controlled start after the starting, these characteristics will be respectively equal to 0.26 and 0.235 at an operating speed of 40 km / h.It should be noted that the calculations were carried out for the conditions of dry rails without the supply of sand.
Figure 4 shows the dependence of the adhesion and speed coefficients for the controlled starting of the traction aggregates OPE 1 and PE 2M.The differential equation of motion of locomotive is where Fл(t) is the current value of tractive effort of the locomotive, N; Fсопр(t) is the resultant of forces of resistance to movement of locomotive, N; Мпр is the reduced mass of the locomotive, kg, defined as ( ) ( ) where Pсц is the coupling weight of the locomotive, N; Q is the weight of the trailing part of the train.N; γ is the inertia coefficient of the rotating masses of the locomotive, γ = 0.06; g = 9.81 m / s2 -acceleration of gravity.
Substituting the corresponding values in (7), we obtain the differential equation of motion of the locomotive ( ) ( ) ( ) ( ) To simplify the entries, it is assumed that F(t) = FЛ(t), Fсопр (t) = W =const, then After integrating (10) the equation for calculation of the working speed ν раб at the end of the descent is obtained .
from 0.23 to 1.00 we represent the approximation in the form of an exponential

Fig. 3 .
Fig. 3. Graphs of relative acceleration and relative speed of the starting process.

Fig. 4 .
Fig. 4. Dependences of the adhesion coefficients and the speed of the traction aggregates OPE 1 and PE 2 during the starting process.Application of the developed mathematical model of controlled starting of the open-pit locomotives allows to determine the optimum value of its starting time.The differential equation of motion of locomotive is