On some rotor-dynamical phenomena of high-speed trains

The paper is devoted to radial and out-of-plane vibration of railway wheels and to wheelset stability as key elements affecting high-speed vehicle dynamics, noise emission, and safety. In the present study, railway wheel tire is treated as a curved beam with various beam models, and the wheel plates are modeled as Winkler’s elastic foundation. New results are presented concerning the influence of the residual stresses on the corrugation and poligonalization of wheels as well as wave propagation in the wheel tire.


Effect of curvature of wheel rim modeled as a beam on elastic foundation
Consider a railway wheel with the wheel-tire modeled as a simple Bernoulli-Euler beam and the wheel plate treated as Winkler's elastic foundation as shown in Fig. 1. The wheel rim height is assumed to be small in comparison with the wheel radius, and the wheel rim cross section is symmetric.
The static normal (circumferential) force in the wheel rim as a curved beam can be described by the equation (see Mahrenholz [1] where N 1,2,3,4 are constants to be calculated from the boundary conditions for the radial displacement w: where F is the external contact force, as shown in Fig. 1. Under assumption that the normal force is constant (N 0 ), the curved beam equation has the following form: It is enough to neglect the first term in the brackets to obtain the equation for a straight beam subjected to longitudinal force N 0 .  One can see that lowering wheel plate stiffness, e.g., by placing rubber interconnection between rim and wheel plate to reduce vibration has to be undertaken together with the assumption that the rim must be modeled as a curved beam.

Phase velocity of wave propagated in the rim
After including inertia term into Eq. (2.1) with ρ denoting rim mass density and the following notations: we obtain the equation of the wheel rim radial motion in the form: where k and v f denote wave number and phase velocity, respectively. The phase velocity can be expressed as: Relation (3.4) between the phase velocity and the wave number for exemplary parameter values E = 2.1 × 10 11 N/m 2 , A = 9.26 × 10 −3 m 2 , ρ = 7,800 Ns 2 /m 4 are presented in Figs. 5 and 6, like in Ref. [2].
The above results allow one to evaluate the influence of the beam curvature and residual stress on the phase velocity and its critical value in the wheel-tire. These results comprise a first step to determine traveling waves generated by a moving and oscillating force and a next step for investigating stability of wheelset-track and train-track interaction. The oscillatory load results from periodical wheel structure, periodicity of sleeper spacing, and corrugations.
The equation of motion of the Bernoulli-Euler beam subjected to moving oscillating force can be described as follows: After transformation of Eq. (3.5) to a moving coordinate system connected with the contact force and introducing dimensionless variables, we obtain the equation of motion in the form: where R B and N B are dimensionless radius of the rim and dimensionless residual stress resultant, respectively, and To determine the proper solution of the problem formulated in Ref. [3], one has to find solution in the form of a traveling wave that satisfies the Sommerfeld condition of radiation [4], taking into account that energy is transported with the group velocity.
As shown in the graph on the left-hand side of Fig. 7, for a given value of speed V 1 and for positive values of wave number k 1 , the group velocity, V gr , is smaller than the phase velocity, V f -that is why this wave  should propagate from the source of excitation. The wave with wave number k 2 can propagate to the source of excitation because value of group velocity is greater than that of phase velocity.
In the case of straight Bernoulli-Euler beam, the dependence of displacement on velocity of force motion V oscillating with frequency Ω is shown in Fig. 8. The resonance curves divide the speed-frequency plane into the three regions of qualitatively different solutions.
It can be seen in Fig. 9 that large values of wheel-tire curvature (1/R B ) considerably influence the configuration of the three regions with qualitatively different solutions.
The influence of the residual stress is more important because compressive stresses reduce the critical wave velocity. This effect is shown in Fig. 10.
The visualization of the qualitative difference of the solutions in the first and second regions is shown in Fig. 11.
The tire of the wheel can be modeled in a more complicated way, i.e., as the Rayleigh beam or the Timoshenko beam, as the rail can be modeled. A very complicated wheel model was investigated in Ref. [5], where the curved beam could exhibit radial and circumferential displacements as well as lateral ones as a result of out-of-plane bending and torsion. The wheel plate was modeled by an elastic Winkler foundation. The crucial result of that investigation is connected with a very high resonance of the wheel rim at the vehicle speed of about 200 km/h in the case when the wheel is excited in the contact area by a force or a spin moment. An example is shown in Fig. 12. It can be seen that for the speed greater than zero, each eigenfrequency is divided into two values, corresponding to the waves traveling in the direction of the rolling motion and to the The description of such a complicated model is made by four partial differential equations, and the solution is obtained in the form of traveling wave. More details can be found in Ref. [5]. The solution of wheel dynamics makes possible an investigation of motion of a wheelset as a flexible system. For such an analysis, in case of rigid axle one will need 14 coupled differential equations for the wheels and the wheelset.
Using the fixture of symmetry (S) and anti-symmetry (A) and assuming constant speed along the track, it is possible to reduce the number of equations to nine. Solving the set of equations, we obtain trajectories of wheelset motion with positions of the wheelset axles as shown in Fig. 13. In Ref. [5], top and front views of a wheelset are presented.
The four situations shown in Fig. 13 correspond to the following motions: • S-S-wheels vibrate in-phase, symmetrically with respect to the wheelset center, the wheelset center moves vibrating sinusoidally in the vertical direction. The trajectory seen from the top is a straight line, and the axle is perpendicular to the track. • S-A-wheels vibrate in-phase, anti-symmetrically with respect to the wheelset center, the wheelset center moves vibrating sinusoidally in the horizontal direction and the axle is perpendicular to the track, as in the S-S case. • A-S-wheels vibrate out-of-phase, symmetrically with respect to the wheelset center, the wheelset center moves vibrating sinusoidally in the horizontal direction and the axle rotates with an oscillating inclination with respect to its initial position. • A-A-wheels vibrate out-of-phase, anti-symmetrically with respect to the wheelset center, the wheelset center moves vibrating sinusoidally in the horizontal direction and axle rotates horizontally and vertically with oscillating inclination with respect to the position perpendicular to the track. Details of various wheels eighenmodes are given in Ref. [6]. Two exemplary modes of wheelset according to Ref. [7] are shown in Fig. 14.
One can notice that in the case of anti-symmetric modes lateral forces are acting on the rail what needs further analysis (which does not belong to rotor-dynamical problems). In the case of modern light railway vehicles with independent drive systems the above mentioned instability phenomena do not appear, but the dynamic electromechanical coupling occurs [8].

Remark on wheel-rail contact with corrugation
Another very important dynamical effect is related to corrugated or poligonalized wheel motion on straight or wavy rail. Majority of investigations were conducted assuming that the wheel radius is small in comparison with the radius of the rail roughness. A study addressing this problem is devoted to the change of velocity of the load motion (horizontal) and the acceleration of the wheel center in the vertical direction. The position of the contact point and the wheel center and the trajectory of the wheel center in case of sinusoidal corrugation are shown in Fig. 15 left and right, respectively. It is interesting that in the ideal case of stiff contact and wheel curvature approaching curvature of corrugation, the acceleration tends to infinity. The elastic wheel-rail contact limited such picks of acceleration, but introduces resonance phenomena connected with the contact stiffness. A detailed discussion on this topic can be found in Ref. [9]. Some results of wavelets application one can find in Ref. [10].

Concluding remarks
The results presented in this paper demonstrate the importance of rotor-dynamical effects in high-speed train wheel-track interaction. Some of the considered effects are related to the fundamental phenomena of train-track dynamics in general, like traveling wave solution with regard to the group velocity of waves and particularly the wheel-rail and wheel-set-track interaction. The wheel-rail interaction is influenced by the load which causes deformations of the wheel plate and the wheel rim, what in turn changes the contact point position. This problem is discussed in Ref. [11].