MATHEMATICAL MODEL OF WHEELSET OSCILLATIONS WITH INDEPENDENT WHEEL ROTATION IN THE HORIZONTAL PLANE

Dep. «Car and Car Facilities», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipro, Ukraine, 49010, tel./fax +38 (056) 776 84 98, e-mail sergeymyamlin@gmail.com, ORCID 0000-0002-7383-9304 Dep. «Car and Car Facilities», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipro, Ukraine, 49010, tel./fax +38 (056) 776 82 27, e-mail kirilchuk.o@mail.ru, ORCID 0000-0002-0565-1692 Dep. «Car and Car Facilities», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipro, Ukraine, 49010, tel./fax +38 (099) 709 85 34, e-mail VladR.K.I.S.M@yandex.ua, ORCID 0000-0001-5318-7559


Introduction
During movement along the track the car components perform complex oscillations.These oscillations are caused by dynamic forces and by the track irregularities, gaps on the rail junctions, wheel bearing surface conicity, as well as the irregularities on wheel bearing surface, variability of physical properties of track and wheelset materials, spring suspension type, changes of car speed, etc. [1,2,4,5].This paper examines in detail the oscillations caused by wheel bearing surface conicity.
Analyzing the movement of the wheelset on the rail track it is not difficult to see that it moves not only progressively along the track axis, but also makes lateral and rotational movements around its vertical axis.The conical wheel bearing surface causes the alternate lead of one wheel in relation to the other, herewith the geometric center of the wheelset axle deviates from the track central axis and at the same time the wheelset axle is rotated from the perpendicular to the track center position [6,9,10,13].So during the motion the wheelset traces out a complex wavy trajectory.This motion was first described in 1883 by Klingel [3].In this regard, at high speed the train becomes unstable, there are lateral oscillations and the motion quality becomes unsatisfactory.There is a danger of derailment.

Purpose
The purpose of the work is to study horizontal oscillation and to assess the motion stability of a single wheelset with independent wheel rotation, as well as to compare the stability indicators of the typical wheelset and the wheelset with independent wheel rotation.

Methodology
To achieve this purpose we developed the design model and composed the mathematical model describing the oscillations of a single wheelset with independent wheel rotation and in a horizontal plane.After that we analyzed the asymptotic solutions stability of linear homogeneous differential equations describing the oscillations of a single wheelset with independent wheel rotation in a horizontal plane on a straight track section.

Findings
The wheelset consists of two wheels firmly fixed on the relevant axle shafts.The axle shafts are connected with bearing units.Thus the wheelset may be represented as a single solid body, but the axle shafts can be rotated in a longitudinal plane on each other (have surplus degree of freedom).In the proposed mathematical model of the single wheelset motion, the friction in the bearing unit is ignored.To describe the motion of the wheelset with independent wheel rotation on the straight track section, we should calculate the railwheel interaction forces.In 1926, F. Carter found that the tangential force of contact wheel-rail interaction is proportional to the relative slip (creep) of the contacting bodies [3,7,8,11].The tangential force projections onto the longitudinal and transversal axis are, respectively: where ε xi , ε yi -relative ship towards x-axis and yaxis, respectively; k -creep coefficient (according to Carter's hypothesis we consider that the proportionality coefficients between longitudinal slip and longitudinal force and between lateral slip and lateral force are equal).Design model of wheelset with independent wheel rotation is shown in Fig. 1.The wheelset with independent wheel rotation moves at a speed V.According to the above design model (Fig. 1) the wheelset position is determined by the lateral swaying y, the hunting angle ψ, and the additional angle of rotation of each of wheelset axle shaft in the longitudinal plane φ 1 , φ 2 , so the angular velocity of each axle shaft is: Thus, we consider a system with four degrees of freedom.We accept that the hunting angle of the wheelset is small enough.The relative slip ε xi , ε yi can be determined by the formulas: where η xi , η yi -slip velocity at the wheel-rail contact points; The slip rate is determined as the difference between the rail and the wheel speeds: where µ -wheel tread grade; r -rim radius on neutral axis.
The relative slip is: Herewith we ignore the summand i y µ ϕ , given that it is by several orders of magnitude smaller than the others.Let us substitute the slip expressions into the formula 1 and we obtain the expression: Then the principal moment acting on the vertical axis is: The principal vector in the transversal direction is equal to: The moment acting transversely on the axle shafts respectively: The motion equations: (2) The resulting system of equations describes the motion of a single wheelset on a straight track section.
To assess the stability of the motion of the single wheelset with independent wheel rotation, we write the equation 2 in matrix form: where q -вектор обобщенных координат.
To determine the eigenvalues of the coefficient matrix the system of equations 3 can be written as:

= − +
After some transformations we obtain: Thus, calculation of the eigenvalues of the coefficient matrix for different speeds and different values of the wheel rate allows obtaining the diagrams of oscillation increment and frequency dependency on speed (Fig. 3, 4).
1 -Wheelset with independent wheel rotation; 2 -Standard wheelset.The obtained results indicate that for the wheelset with independent wheel rotation the form of hunting oscillations and swaying remains.Herewith the oscillation frequency at the same speed is half that of the standard wheelset.Oscillation increment is 5.5 times less than for the standard wheelset, although it remains positive over the entire range of the examined velocities.This indicates that the use of wheelsets with independent wheel rotation will increase the car movement stability at high speeds.However wheelset with independent wheel rotation is also characterized by non-oscillatory movement forms.In this regard, under certain conditions the wheelset loses its selfcentering in the track that is its significant disad-vantage.One of the variants to ensure wheelset self-centering in the track is the use of resistance element to the relative rotation of the wheelset axle-shafts [12].During the relative rotation of the wheelset axle-shafts such an element will create a moment of resistance in an effort to equal the angular velocities of the axle-shafts.This will lead to the wheelset rotation around the vertical axis in such a way that the wheelset will tend to take the central position in relation to the track axis [14,15].Of course, to confirm this assumption it is necessary to conduct experiments on a physical model or prototype of the wheelset as part of a bogie and a car.

Originality and practical value
The result of the work is the mathematical model of the sinuous movement of a single wheelset with independent wheel rotation and the estimate of the dynamic indices during its motion on a straight track section without any irregularities.The developed mathematical model of the motion of the single wheelset with independent wheel rotation can be used to create the advanced designs of railroad car undercarriage at the stage of selection of running gear parameters using mathematical modeling.

Conclusions
As a result of comparison of the increment and the oscillation frequency of a single wheelset, standard and with independent wheel rotation, it was found that the use of wheelsets with independent wheel rotation allows achieving higher dynamic qualities of movement.However, this requires experimental verification on the stand or dynamic tests on the line.

Fig. 1 .
Fig. 1.Design model of wheelset with independent wheel rotation