1. Introduction

Modern rolling stock, including locomotives, passenger and freight cars is the most important constituent part of the technical facilities of railway transport characteristics, properties and parameters of which are determined by the growth of traffic, increase traffic speed, ensuring operational reliability, increase the comfort of passengers and the smooth progress of railway carriages, cargo safety, etc. Such requirements are directly connected with the improvement of their spring suspension, aimed at reduction of dynamic loads transmitted from the track of the body and crews and thus increasing the smoothness of the rail vehicles. The existing design elements spring suspension of rolling stock have a number of drawbacks, such as a high metal content, complexity of design, low reliability and, most importantly, the inability to change in the automatic mode of their damping characteristics.

2. Objectives

In view of the foregoing at the chair "Applied mechanics and engineering graphics" Yelets State University name I.A. Bunin for a number of years carried out the research works aimed at creation of perspective of a torsion spring suspension for railway vehicles, which includes adaptive torsion springs and adaptive hydromechanical dampers.

The subjects of such research is based on the plans of research and development JSC "Russian Railways" for the year 2009 and of its components include such sections as: "The Program of high-speed and high-speed movement", "The Program of the locomotive economy" and "The Program of carload economy", as well as on agreements on creative cooperation for the period 2007-2011, the prisoners YSU name I.A. Bunin with the Departament of the South-East railway (branch of JSC "Russian Railways") and Yelets Department of the South-East railway branch of JSC "Russian Railways" on a theme "Development of practical recommendations to improve reliability, technical, economic and operational characteristics of the rolling stock and other standard and non-standard equipment used at the enterprises of the Yelets Department of the South-East railway".

Today in the framework of the research developed a promising adaptive torsion suspension, designed for the main and industrial locomotives, including adaptive hydro-mechanical dampers, established on the level of inventions (RU2427737 and a positive decision of FIPS from 25.04.2012, according to the application №2011121597/11), schematic diagram of which is shown in figure 1. Therefore, the aim of the work is to conduct analytical studies on the kinematic and geometric parameters of adaptive torsion spring suspension, manufacturing model of and test it in poster conditions.

3. Methods

To solve this problem and provide the most detailed picture of the dynamic processes occurring in adaptive torsion suspension, formulated requirements to the developed calculation scheme and of the elaborated mathematical model, describing the damping characteristics of such a device.

At the first stage, to assess the efficiency of adaptive torsion spring suspension, with respect to the jaw cart trunk or industrial locomotive (Fig. 1) were carried out preliminary calculations of the basic parameters of the proposed technical solution on the basis of the adopted scheme (Fig. 2).

Figure 1. Adaptive torsion spring suspension locomotive

Figure 2. The calculation scheme of the torsion spring suspension

When forming of the mathematical model, according to a design diagram (Fig.2), used the equations of dynamics, theory of elasticity and hydromechanics. As the base of the calculation of the ratio was adopted by the equation of motion of a concentrated mass m₀, which for the cases of power, kinematic and mixed excitation is of the form:

In this case, the damping factor α_∑, depending on the hydraulic resistance in the channel, fluid viscosity, flow turbulence, as well as the relevant parameters, characterizing the working fluid in the throttle channels adaptive hydro-mechanical shock absorbers were defined in the following dependencies:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

As a result of calculations performed for the locomotive 2TE10L were identified piston diameter D_P = 80 mm, the diameter of the butterfly channel d_К = 1.5 mm, number of channels n = 4, stem diameter d_S = 20 mm, hydro-mechanical damping factor α_∑ = 374.8 N·s/cm, the force of resistance to the movement of the piston R_∑ = 206,96 N.

For pre-practical calculation of a torsion spring suspension used approximate method of determining the equivalent damping ratios proposed W. Kirpichew and further developed I. Chelnokow. In this case used the following differential equation and is included in the parameters describing the fluctuations of the concentrated masses on the springs in the form of:

(10)

(11)

(12)

As a result, were pre-installed with the most important geometric characteristics of the proposed adaptive torsion springs, designed just for jaw cart above mentioned locomotive 2TE10L. In this case it is considered, that the rod of the torsion springs made of steel 65С2ВА GOST 14959-79, and its diameter is equal to d_t = 60 mm. When you change the length of the working area, for example, from 600 mm in the statics of up to 550 mm under dynamic loading changing the torsional stiffness of the torsion increased in 1.6 times and amounts to, respectively, the c_s = 1127.6 N/mm and c_d = 1805 N/mm, which, in the end, allows you to reduce the movement of the crew part of the locomotive with Δ_s = 47 mm to Δ_d = 36.7 mm.

For the redetermined settlement, characterizing the smoothness of the locomotive 2TE10L, equipped with the technical solutions, and the study of oscillation and force loading its chassis is designed dynamic calculation scheme on the basis of elastic-plastic model Voigt-Kelvin (figure 3).

Under the С_φ provided torsional stiffness of the torsion springs, and under the α_∑ - total damping factor of hydromechanical shock absorbers.

At drawing up of the mathematical model uses the following assumptions:

1. Not taken into account bending oscillations of the rod, and other parts of hydromechanical damper.

2. Coefficient of friction of mating surfaces is taken constant.

3. The law of excitement due to the irregularities of the micro - and macro-profile conditional pavement changes in time according to the harmonic law.

4. Is not taken into account the gyroscopic effect of a rotating mass of wheels, as well as air resistance when its translational motion.

5. The processes occurring in the hydromechanical damper, believe isothermal to the specified coefficients of viscosity of the working body.

6. To reflect the movement of the piston and the rod relative to the frame with the longitudinal oscillations applied inverse form of the assignment of coordinates, and with torsional direct.

7. Vectors of dry friction forces in the forward kinematic pair the piston - working cylinder were heading in the opposite direction relative to the coordinate system.

8. The movement of the locomotive is on the straight stretch of road without bias.

9. Profile path under the two wheels of each wheelset is the same.

For a description of the rectilinear motion of the locomotive bogie choose the following generalized coordinates the movement's centre of gravity of the locomotive in the direction of motion; vertical movement of the centre of gravity Z locomotive; angular displacement φ crew of the locomotive in the vertical plane with respect to the transversal axis, passing through its center of gravity. The systems of generalized coordinates accept the Cartesian lefts.

Figure 3. The calculation scheme of the locomotive bogie 2TE10L

Using the technique of linearization, which gives opportunity to carry out, both quality and quantitative study of nonlinear systems of suspension with a high enough for practical tasks degree of accuracy, we write the equation of the oscillations of the crew part of the locomotive:

(13)

The force of P_i can be presented in the form of two forces:

(14)

wherePTi – force caused by the movement of the leaf suspension and characterized by the torsional rigidity of the shaft torsion;

PHMDi – force caused by the resistance of hydromechanical damper.

It is seen that the forces of P_i, attached to the wheels and motors unit of irregularities way, can be expressed in terms of generalized coordinates of φ, z, y, and the frequency of the external excitation p = 2πυ/α, where α is a dimensionless variable associated with the time t is a linear dependence of α=pt.

Then equation (1) can be written as follows:

(15)

(16)

Analyzing the formula (16) we see that the variable total hydro-mechanical damping factor α_Σ of the proposed hydromechanical damper will depend on the hydraulic resistance in the channel, the viscosity of the working fluid and flow turbulence α (t) arising in the channels of the piston and the rod, as well as from mechanical α_m (t), which are characterized by the torsional stiffness С_φS variable cross-section of the piston rod. Such an integrated its constructive characteristics allows to efficiently in the automatic mode and in a wide range of impact of the dynamic loadings arising in spring suspension rail crew, to produce vibration dampening the crew part of the locomotive.

The General solution of equations (15) looking for in the form of:

(17)

Solving equation (15) we can determine the amplitude of the steady-state oscillations of spring mass z₀ and the amplitude of φ₀, respectively, on dependencies:

(18)

For the resonance modes depending have the form:

(19)

For the decision of ordinary differential equations describing the fluctuations of the locomotive bogie, used widely professional applied mathematical package MATLAB 6.5., designed for the solution of practical engineering problems.

Construction of the model includes the solution of differential equations for the serial spring suspension of the above locomotive and the perspective of a torsion under the same initial conditions. For the series production of springs of the suspension of the locomotive equations describing the forced vibrations, have the form:

(20)

and for torsion, as it was mentioned above, according to equation (15).

The initial conditions:

For clarity performed analytical and numerical solution of the presented differential equations. The analytical solution obtained when using dsolve built-in package Tool Box Symbolic Math, the input arguments, which are the line to the equation, boundary conditions, if any, and the independent variable for which is the default t.

Numerical solution of a set of differential equations with initial conditions, i.e., the Cauchy problem, carried out with the help of built-in functions of MatLab, called solvers. For the above equations used solver ode45, which is based on the method of Runge-Kutta method of the fourth and fifth order of accuracy. The input arguments solvers are: the name of the file functions in apostrophes, vector with an initial and a final time value observations of the movements and the vector of initial conditions. Output argument two: the vector of values of time and a matrix of values of unknown functions. The values of the functions are located at the columns of the matrix in the first column is the value of the function, in the second - its first derivative.

Scheme of location of the numerical solution consists of the following stages:

1. Adjusting the differential equation to a system of differential equations of the first order. For the presented equations were obtained the following systems:

- for the series production of springs of the suspension:

- for the perspective torsion suspension:

2. Writing a special file of functions for the system of equations obtained. File-function must have two input arguments: the variable t, which is the differentiation, and the vector, the amount of which is equal to the number of unknown functions of the system. File-function for the initial equations (15) and (20) eq_serZ, eq_serFI and eq_modZ, eq_modFI are in the Listings.

3. Call solver ode45.

4. Visualization of the obtained results.

An analytical solution of the differential equations (20) after the start of the file-program solv_serial has the form presented in figure 4.

An analytical solution of the equations (15) split into 2 parts: figure 5, a - solution for Z, and figure 5, b - a solution for φ. From figure 6 we see that when using the serial spring suspension vertical movement lie in the range from 0.05 to 0.12 m, while the use of adaptive spring suspension they do not exceed 0.02 m. It is clear that the numerical value of the amplitude Z at the initial moment of time reaches of 0.055 m, but within 1.5 seconds it sharply decreases to the value of 0.02 m. A similar pattern is visible and the angular fluctuations of the body. When using the serial suspension amplitude of the angular oscillations of φ is constant on all the time interval in 10 seconds, and reaches a value of 0.15 rad. However, in the case of the use of adaptive spring suspension amplitude of such movements shall be no greater than 0.01 rad, starting from 4-th of a second movement of the locomotive, although at the initial moment of time they are large in size - 0.15 rad.

Figure 4. An analytical solution of the equations describing the forced vibrations of serial suspension

Figure 5. An analytical solution of the equations describing the forced oscillations of the torsion suspension

The result of the numerical solution of displayed in the graphs presented in figure 6.

Figure. 6. Amplitude-frequency characteristics of vertical and angular fluctuations

After the solution of the differential equations are constructed, amplitude-frequency characteristics for a torsion spring suspension. The formula for the amplitude of forced oscillations, has the form:

Taking into account the dynamic factor built a family of curves, the host various values of the ratios of n/p. The family of such curves presented in figure 7. This graph of the family of curves constructed for the perspective adaptive torsion suspension, shows that at the approach of the frequency perturbations ω to the frequency of oscillation of the p, dynamic factor increases, and the maximum amplitude of oscillation is achieved approximately in respect of ω/p=1. At the same time from 5 presented in figure 7 plots the most removed from the resonance zone chart with the ratio n/p=0.6 and maximum coefficient of dynamic equal to 1.7, which is characteristic of the proposed design of adaptive torsion spring suspension with adaptive damping elements at a total coefficient of damping α_Σ=420,0 kN∙s/m. While the decline in the last possible input suspension in resonance.

To determine the resonance mode of the amplitude of steady-state oscillations was used by the known method and then was used by the file-program rezonans, showing the dependence of the resonance amplitudes from the stiffness of the suspension in various types of dithering. The graphs obtained are shown in figure 8.

Figure 7. The family of curves for the dynamic factor of μ

Figure 8. Dependence of the amplitudes of resonance from the stiffness of the suspension

The analysis of the diagrams, characterizing the resonant modes of oscillation cart and depending on the torsional stiffness of the proposed torsion suspension, shows that the smallest fluctuations are obtained in the case when the torsional stiffness of the torsion spring suspension is in bounds of the order of 3,5∙107 up to 5∙107 N/rad.

For a more complete study of the oscillation process of the locomotive bogie with use of the developed models were calculated basic parameters characterizing the dynamics of motion of a locomotive at different values of F₀, α_Σ and с_φ, which, in particular reflect the change during the time interval in 10 seconds with the vertical movement of the body Z and the speed of this movement Z’. As a result of such calculations show that using the proposed design of adaptive torsion suspension of amplitude of oscillations of the body and the speed of its movement in the vertical plane much lower than the cart, equipped with leaf springs.

To confirm the correctness of the calculation was developed method benchmark test of experimental studies of the proposed design of perspective suspension.

The basic scheme of the test stand, shown in figure 9, represents the elastic rod, motionless fixed in support B and mobile, with the help of splines, in support A, which are established on the table hydraulic press, creating effort of compression of up to 160 kN.

were calculated rational parameters and characteristics of the perspective of adaptive torsion suspension, with respect to locomotives 2TE10L, which amounted to diameter equal to d_t = 69,5 mm, length of the working area of the rod of the torsion 637 mm with torsional stiffness C = 1213,0 N/mm. Such values of a good agreement with previously obtained estimates.

4. The Results

As a result of the bench experimental research, as well as using methods of mathematical modeling, received rational geometric and жесткостные characteristics of a number of torsions, which can be widely used in various models of the rolling stock.

According to the results of the research proposed a number of technical solutions, created at the level of 8 inventions of the Russian Federation, which are modifications of the above construction, and in each case, that permits to use them not only in the designs of locomotives, but also passenger and freight rolling stock. For such structures as the calculations on the substantiation of their rational kinematic and geometric parameters.

With account of the above-mentioned, the carried out researches have allowed to expand the well-known classification of the running parts of the rolling stock, as shown in figure 10.

Figure 9. The basic scheme of the test stand

Figure 10. The classification of the running gear of the rolling stock

5. Conclusions

Developed by promising adaptive torsion suspension, designed for the main and industrial locomotives, including adaptive hydro-mechanical dampers, established on the level of inventions (RU2427737 and a positive decision of FIPS from 25.04.2012, according to the application №2011121597/11), schematic diagram of which is shown in Fig. 1 allows you to: firstly, to reduce consumption of metal chassis of the rolling stock, secondly, to simplify the design of the spring suspension, third, to increase the maintainability of it and in the fourth increase the smoothness of the crew at the expense of its ability in the automatic mode effectively mitigate any dynamic components of the loads caused by the irregularities of the way when the motion of the rolling stock.

The results of the research are recommended both domestic and foreign scientific research institutes, design and production structures heavy machinery for further study and elaboration of proposed constructions of adaptive torsion spring suspension with a view to the possible introduction of it in practice.