RTR 4/2010 8 n Frequencies of the ballasted track glecting the shear stiffness. The continuous mass is replaced by discrete masses. On the rail (UIC 60) 3 mass points are placed at equal distances between the sleepers. The sleepers get one mass for each support, and the mass of the wheel set is represented by 6 masses. The four brake discs on the ICE 1 axle are represented by 4 masses. The wheel discs at the ends of the axle are concentrated in 2 masses. Furthermore the moments of inertia of the 2 wheel discs about axles parallel to the rail are to be considered because of their mayor influence on the bending of the axle. So for the whole wheelset only 8 equations are needed, much less than in other papers. The wheel discs are supposed to be rigid and inflexible. Because of the flat conical shape of the wheel discs their large shape stiffness may be assumed. For the n mass points of our mathematical model we determine n displacement equations with n components in an (n∙n) matrix. For this matrix the characteristic polynomial of nth degree can be calculated, the zeros of which define the n eigenvalues λ1, λ2, λ3 … λn. The n eigenvalues λ are calculated from the matrix by known subroutines as roots of the characteristic polynomial. From the eigenvalues λ we get the eigenfrequencies in Hz by taking the reciprocal square root and divide by 2π. The n independent eigenfunctions (modes) belonging to the n eigenvalues are calculated by inserting the n eigenvalues into the original equation system and setting it equal to zero. No commercially available program system was applied. 3 Choice of the calculation parameters The fundamental postulate of this paper is to develop a mathematical model the results of which should not be contradictionary to the measured results. Therefore as few assumptions as possible had to be made. For example no multilayer ground was presumed, a possible damping was neglected, the influence of Hertzian stress between wheel and rail was neglected and the simpler Euler beam was chosen instead of the Timoshenko beam like in other papers. The reason for such simplifications is that we cannot guarantee the exactness and homogeneity of the ground parameters. Particularly for those a locally unavoidable error of 20% cannot be excluded, while the total error of the excluded assumptions will hardly be greater. Also the variation of vehicle properties and track due to wear is not understood partially. We are aware of the fact, that the interaction of the ballast layer with the other components has not yet been defined in this dynamic investigation. From the static point of view its distributing property is well known. Therefore this is considered in the calculation of the ground spring. Initially the dynamic interaction of the ballast mass was not defined. This had to be postponed to later stage at which a comparison between various calculated results and measured values were possible. For the wheel set a half worn wheel set was assumed. The parameters chosen as best fitting after multiple repeated calculations and comparisons with the measured results are shown in Tab. 1. The real ground stiffness proved to be about 20% higher than the minimum value required at the time of construction. For the three rail fastenings before and behind the load the parameters of a loaded rail pad (RP) were taken. 4 Calculated results of the spatial wheel-set-ballastedtrack-system In order to assess the calculated spatial results it was decisive to have one diagram for each frequency for to get the full survey at a glance. Additional the most important parameters and calculation results had to be readable numerically. As examples the displacement modes of the first two eigenvalues have been drawn in Figs. 5 and 6. The longitudinal curve of the mode is placed in the left part of the picture and the transverse mode in the right. In the right part we find the system above, below it is the displacement mode of the wheel set in a different scale and below we see the bisecting lines of the 3 sleeper modes before and behind the wheel set. The left side of wheel set and track in the transverse mode is shown in black and the right one in red. The longitudinal modes are seen from the right, so they are red for symmetric modes. On the left side important system values are printed. If the wheel set is not put in the middle between the sleepers but above the sleeper the eigenfrequency is raised by 2.6%. As for other modes a similar frequency deviation can be found, only this position is shown here. In Fig. 7 all eight modes of the wheel-setballasted-track-system are shown, in which elements of the wheel set have the greatest displacement. There are 4 symmetric and 4 anti symmetric modes. Their eigenfrequencies are situated between 47.5 and 754.3 Hz. If we are looking for modes in which there are strong interactions between wheel set Parameter Calculated value Bedding modulus of the ground 120 MN/m3 Stiffness of the unloaded RP 60 N/mm Stiffness of the loaded RP 160 N/mm Tab. 1: Calculation parameters Fig. 6: Spatial modes of wheel set and ballasted tracks at the second eigenfrequency of 58.7 Hz. Track section 975 cm long. Fig. 5: Spatial modes of wheel set and ballasted track at the smallest eigenfrequency of 47.5 Hz. Track section 975 cm long.
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