Features of dynamic loading of transport vehicles in intensive operation

The paper proposes a method to construct mathematical models of technical objects exposed to intense vibrational interactions, which is typical for assemblies of transport vehicles. A technique has been developed to construct structural mathematical models in the form of structural diagrams, which are dynamically equivalent to automatic control systems. Analytical tools have been adopted from the theory of automatic control. The paper demonstrates the possibilities of changing dynamic properties of technical objects, design schemes of which are represented by mechanical oscillatory systems. Changes in the dynamic effects arising from the introduction of additional inertial couplings have been estimated. Effects of new dynamic properties have been considered. Analytic relations have been proposed for detailed estimates.


Introduction
Maintaining operational reliability and safety of railway rolling stock is one of the most important and relevant scientific and engineering problems, which have been reflected in studies of Russian scientists [1][2][3].The development prospects of transport systems require that a special attention is given to developing a methodological basis for dynamics and operation of vehicles under intense dynamic loading.It is important to assess dynamic states of transport vehicles because this assessment plays an important role in developing principles and specific techniques to be implemented in the systems that ensure the general operational safety of transport vehicles and reliability of their major units and assemblies [4][5][6].
An increase in speeds and axle loads has a significant effect on the dynamic state of a traction motor.This can be attributed to a change in dynamic stiffness and generation of dynamic bursts produced by a combination of disturbing factors.Some aspects of these issues are considered in [7][8][9][10].The available studies show that motions with increased velocities initiate the development of oscillatory processes with increasing oscillation amplitudes and resulting impact interactions.Passive means of adjusting dynamic states of motor mounting at high speeds of vehicles are not effective enough, especially in situations where natural frequencies and dynamic damping of oscillations are sufficiently close to each other.A promising solution in these situations is using motion conversion devices that are capable to widely vary the reduced mass of an object with a controlled dynamic state.
Traction electric motors of electric locomotives are among the technical devices that are subject to a lot of mechanical factors arising from both wheel-rail contacts and interactions with the entire complex mechanical system as a whole, which is determined by structural and engineering characteristics of a locomotive [11][12].
The novelty of this study is in examining scientific foundations underlying mechanisms of dynamic bursts or a certain bunch of oscillatory processes that arise in contacts between the traction motor case and elements connected to the truck frame.The authors put forward the idea of constructing frequency diagrams with families of frequency parameters (partial and natural frequencies, dynamic damping frequencies) constructed according to a parameter to be adjusted.The physical meaning of the burst generation mechanism implies uncontrolled convergence of frequency parameters due to simultaneous action of several external disturbances.
This study focuses on methodological aspects in the development of methods and tools aimed at maintaining the operational safety of locomotive traction motors.The key feature of the approach is that it takes into account possibilities of changing dynamic states of a motor mounting system in transport vehicles by introducing additional couplings.The paper gives a special focus to connectivity between parameters of mounting elements and simultaneous action of several external disturbances so that these aspects are taken into account in the approach.

Problem statement
Design diagrams representing mechanical oscillatory systems with lumped parameters and several degrees of freedom are used to assess dynamic properties of technical objects, including traction motors.Depending on tasks assigned, traction motors can be represented as mechanical chain systems with two or three degrees of freedom, as well as mechanical oscillatory systems with an object in the form of a solid body on elastic supports, which is in planar motion (Fig. 1).External influences on the traction motor are generated not only as external kinematic disturbances on the side of the bearing surface, but also can be created by irregular electrical loads on the traction motor armature and periodic force disturbances of a different nature [6].
Dynamics of traction motor units can be detailed according to the notions of vibrational effects from the traction motor case on the operation of the commutator-and-brush assembly.The commutator-and-brush assembly is subject to a large number of mechanical factors, among which contacts with surface roughnesses of the armature commutator are of particular importance.At the same time, vibrations from the case of the commutator-andbrush assembly significantly affect the operation of the traction electric motor [2,11].
Figure 1 shows the design diagram of a frame-and axle-mounted traction electric motor.A special feature of the design scheme is the introduction of additional couplings in the mechanical oscillatory system (Fig. 1) in the form of motion conversion devices with reduced masses L 1 and L 2 .Values of these parameters may change if the following condition is fulfilled: where α is the connectivity coefficient for parameters L 1 and L 2 .It is assumed that α is a positive quantity and can vary in a sufficiently wide range [13].The authors are developing an approach within the framework of structural mathematical modeling [14].The mathematical model of the initial system (Fig. 1) is presented in Figure 2. It is represented as a structural scheme that is dynamically equivalent to the automatic control system.Fig. 2. The structural mathematical model of the initial system taken from Figure 1 with kinematic disturbances.
Figure 2 assumes that p = jω as a complex variable; symbol "-" over the variable means its Laplace transform under zero initial conditions [15 ÷ 16].The system has two input actions: 1 z and 2 z .The dynamic state is estimated with variables 1 y , 2 y ; M, J -weight and moment of inertia of the solid body, respectively; a, b, cthe geometric parameters of the system.The purpose of the study is to assess the features of a dynamic system, which is simultaneously excited by several disturbances that create some specific dynamic effects.

Mathematical modeling
Taking into account the simultaneous action of external disturbances, transfer functions of the system can be represented as ) ( a , b , c are the geometric parameters. From ( 2) ÷ ( 4) it follows that the system has two frequencies at which resonant phenomena are possible.Frequencies of such processes can be determined from equation (4).In turn, the system for each coordinate 1 y , 2 y may be subject to the dynamic oscillation damping mode.The respective oscillation frequencies can be determined by setting the numerators of the transfer functions ( 2) and (3) to zero.
Using ( 1), ( 2) ÷ ( 4) can be transformed as follows where In this case, the partial frequencies of the system are given by expressions from which it follows that the partial frequency n 2 depends on the connectivity coefficient α.In our case, in contrast to conventional approaches [13,14], dynamic damping frequencies will no longer coincide with values of partial frequencies, but will be determined from the "zeroing" condition, i.e. setting the numerators of the rational expressions in the corresponding transfer functions to zero.This can be attributed, in particular, to features of solutions to two biquadratic equations., 0 )] ( Thus, the probability of dynamic damping conditions of oscillations will be assessed according to the corresponding roots of equations ( 10) and (11).In this respect, a graphoanalytical method has some capabilities and could be called a "frequency diagram construction method".This diagram is constructed as a set of dependency graphs show relationships between dynamic oscillation damping conditions.Conventional interaction schemes, in which L 1 = 0 , L 2 = 0 in the coordinate system y 1 and y 2 , with only a single external disturbance (e.g., z 1 (t) ≠0, z 2 (t) = 0), the dynamic oscillation damping mode can be realized only with respect to the coordinate 1 y .In this case, the dynamic damping frequency coincides with the partial frequency n 1 , i.e. ω n dyn  .In the case where z 1 (t) = 0, z 2 (t) ≠ 0, only one disturbing factor will be involved as well.In this connection, the dynamic damping frequency will also coincide with the corresponding partial frequency, i.e., the condition and L 1 = 0, L 2 = 0, dynamic damping frequencies can be realized with respect to both coordinates -1 y and 2 y , which is defined by expressions: Provided that k 1 bk 2 a = 0, two dynamic damping frequency modes can coincide with each other at a single frequency, which corresponds to certain symmetry relations of system properties (at а = b, the condition k 1 = k 2 is satisfied; this case needs to be addressed separately).

Estimation of dynamic properties based on the frequency diagram of the system
The following system parameters are selected to construct a frequency diagram for the model problem.M = 7,000 kg; J = 2,000 kg.m 2 ; a = 0.57; b = 0.43; c = 0.71; k 1 = 1,000 kN/m; k 2 = 2,000 kN/m; l 1 = 0.6 m; l 2 = 0.8 m; L 1 = 100 kg.
The dynamic damping frequency at which simultaneous dynamic damping of oscillations will occur is identified by intersection of the dependence graphs

Fig. 1 .
Fig. 1.The design diagram of a frame-and axle-mounted traction electric motor for motion conversion.