To the question of action on the rods lying on elastic foundation, inertial load with a variable speed of its movement

. A method for calculating the rods on elastic foundation under the inertial load action when it moves at a variable speed is proposed. Test problems about a force or load movement with variable speeds along a hinged beam and about the movement with a high-speed railway car deceleration along a track section modeled by a hinged supported beam of great length on an elastic foundation are considered. The selection of the elastic foundation material of the rail track determines the dynamics of the high-speed railway car in different modes of its movement. To construct the methodology, the previously proposed by the author of the article solutions are used: a step-by-step procedure for solving the problems of unsteady dynamics of structures and the method of "nodal accelerations" to take into account the action on structures of a moving inertial load.


Introduction
The tasks of studying the interaction of high-speed rolling stock and railway tracks remain relevant [1][2][3][4][5][6][7][8][9]. The described method for solving the problems of a moving load takes into account any required number of vibration modes in the rod deflection function expansion and leads to a resolving system of equations when using an unconditionally stable integration scheme with a minimum number of the unknowns, as by the method of integral equations when solving problems at a constant movement speed. As the load on the rods, concentrated forces, loads and carriages moving at a variable speed are considered.

Problem statement, general formulas and test cases
At the beginning let us turn to the solution of the classical problem of a load movement along a beam on an elastic foundation with a variable speed, and then proceed to the case of a more complex load. Further an unconditionally stable step-by-step procedure in time and the method of taking into account the action of a massless moving load on rod systems, proposed earlier are used [4]. The cases of uniformly variable motion of the moving load on the rods will be considered.
The differential equation of vibrations of a beam on an elastic foundation when load P moves along it and mass M has the form     Here EJ defines bending stiffness of the beam, 1  ,  are the energy dissipation factors, k is the modulator of subgrade reaction,   can be written as [4,5]   From the conditions of the dynamic balance of the load we find: after substituting (5) into (4), at the step [tj,tj+1] the equation is taking into account (5), allows, taking into account the conditions of continuity at the load contact point, to calculate the initial conditions of the problem for the next integration step. Considering a beam on an elastic foundation, we use a series of test cases for a beam without an elastic foundation.
The step procedure (2) -(5) is implemented for the uniformly variable movement of the load along the beam with the parameters [1][2][3].
where 0 v and 1 v represent speed, respectively, of the load entry and exit from the beam.
It should be noted that the proposed method (2) -(5), for M =0, can be used to solve the classical problem, motion with variable speed along the concentrated force rod [1].  [1]. In all the cases considered in figure 2 (a, b, c, d,   The results obtained by different methods for the force movement case and presented in Fig. 2 a and in [1] (p. 313), practically coincide. Fig. 2 (a, b, c, d, e, f). 3 Method for solving the equations for the "car-track" system.
The algorithm (2) -(4) is easily implemented in the case of movement of a system of loads along a beam on an elastic foundation. The expressions of the form (4) when moving Ñ cargo form a system of Ñ linear, algebraic equations Here B A, are matrix and vector,  is an identity matrix, ko R is the vector of dynamic additions to the static pressure of loads on the beam, K q   is the vector of vertical acceleration of moving loads.
Let us consider a model for studying the vertical dynamics of the experimental system "track-train" for the possibility of studying their interaction. We will further denote this ( Fig. 3 a, 4), where h e is the rail-beam, on an elastic foundation,  o h e is railway car. Fig. 3 (a, b).
Let us construct a system of equations describing the vertical dynamics of the car In this case, we will assume that the initial conditions of the problem are zero, and the parameters determining the position  We will assume that the car braking when moving at a constant speed along the railbeam (Fig. 4) occurs at the moment * t , when the first wheelset of the carriage reaches the middle h e , while at the moment * t an inertial pair of forces is applied to the car body Fig. 4) Here А is a matrix characterizing stiffness, dissipative and inertial characteristics h e , Е is an identity matrix, B is the vector taking into account the initial conditions for h e in the moment tj and the action on the moving forces * P and o k R carriageway system.

Numerical simulation results
The step-by-step procedure (14) is implemented for a high-speed train from [6] when moving along the experimental track section. The system { v =0 and its support on a rigid base and the action of a moment suddenly applied to the body, equal to the inertial moment when the car is braking. As a result, Fig. 3b shows the changes over time t (c) dynamic reactions i R (kN) the first and fourth wheelsets. When the car is moving, its position on the track section is determined by the segment ) (t s (Fig. 4), and, accordingly, for the fourth wheel pair by the segment s4. The system of equations (14) changed its order from 1 to 4 in the process of numerical implementation, when the car was moving at a speed  0 v 250 km/h and from the beginning of the car braking at the moment when s =0.5  and 2   w m/s 2 [7]. The dynamic addition change R (kN) of the fourth wheelset of the car to the static pressure of this wheelset equal to P =170 kN depending on ) (t s is shown in Fig. 5a. Fig. 5 (b,c) shows, respectively, when the car is moving, depending on t and ) (t s (Fig. 4) (Fig. 5c)). Integration step j t  by п = 580 in (4) was chosen equal to j t  = 0.00072s, with the number of steps N = 2160 to implement the procedure (11) -(14). It should be noted that the expected coincidence of the graph ordinates in Fig. 5b, corresponding to the moment when the fourth wheel of the car reaches the middle of the experimental section of the rail at s = 75.4 m. The process emergence of unloading the fourth wheelset (in the form of a drop in the value of the dynamic additive R to the static reaction of the wheel (see Fig.  4a), which coincides in magnitude with the result shown in Fig. 3b, and caused by the action during braking of the inertial pair ) (ин M (Fig. 4), at the moment the car starts braking should also be noted.

Conclusion
The proposed method makes it possible to investigate the action of a movable inertial load with a variable movement speed on the rods on an elastic foundation, under various boundary conditions, while applying the corresponding fundamental functions using the step-by-step procedure proposed in [4]. Applied to the problems of railway transport, the method allows to investigate the interaction process (Fig. 5c) in the system "car -track" at different speeds of the train in the modes of the beginning of braking or its acceleration, at any position along the track section and the possible presence of various irregularities [5,8].