RTR 4/2010 7 Frequencies of the ballasted track n To produce the spatial mathematical model, we take two parallel rails of equal length, in the middle of which the wheel set is placed and rigidly connected with its wheel discs. For each of these two connections there is one equation of vertical displacement only for the wheel disc and the contact point of the rail. We are interested in the eigenvalues of the vertical vibration (natural frequencies) of the wheel set on the ballasted track as well as in the eigenfunction (mode) of the spatial structure. Therefore we build up the following mechanical system including its masses. The wheel set is connected to the vehicle body by the bogie and unsprung with the rails. The eigenfrequency of the vehicle body is tuned to about 1 Hz and that of the wheel set on the ballasted track to about 50 Hz and more. So the wheel set has a much softer connection to the vehicle body than to the permanent way. In order to simplify the vibration analysis of the wheel set we will neglect the connection to the vehicle body, allowing an error of about 1 Hz in the result. For the desired analysis of the wheel set vibrations an error of this magnitude is negligible, since the wear of the wheels causes much bigger deviations. In the mathematical model the axle of the wheelset and the rails are continuous beams with given bending stiffness but ne2 Mathematical models of the frequency analysis The mathematical model of the ballasted track is described by a continuous rail on discrete supports. The supports consist of the sleepers with elastic rail pads on top. The sleepers are supported by single elastic springs on top of an inflexible bottom plate (Fig. 3). The stiffness of these springs has to be determined from the stiffness of the ground. It must be taken into account that the 30 cm thick ballast bed distributes the sleeper’s load at an angle of 75°. Loads may be placed anywhere on the rail and the reaction in the rail can be calculated at arbitrary places. Since the mathematical model uses slab elements, the results are exact and no approximate solutions. The accuracy of the solutions can be proved by the equations of chapter 6. In order to simplify the plane model (in longitudinal direction) is calculated with half the ballasted track with one rail only (Fig. 4). So we use the wheel load only instead of the wheel set load. The ends are free to rotate and to deflect. The plane mathematical model may have arbitrary length. Here we use 15 to 20 supports, spaced at 65 cm each (9.75 to 13 m length). A change to the 60 cm spacing of the real track would raise the eigenfrequencies of the loaded track (chapter 4) by less than 2.3% only. For the unloaded track (chapter 5) they would be changed by less than 1‰ only. ated. Therefore it is important to find out the cause of each peak individually. The phenomena in the lower frequencies between 0 and 40 Hz – the so called quasistatic area – with the unavoidable, periodically repeated single loads are well known. p The first peak at 8 Hz is caused by wheelsets located about 9 m apart of each other. This are the wheelsets between the bogie pivots of each of two coupled coaches. p The second peak at 20 Hz is caused by wheelsets being about 4 m apart. This are the wheelsets beyond the bogie pivots of each of two coupled coaches, p whereas the distance of 2,50 m between the axles of each bogie is of minor importance. Above the 40 Hz we find true vibrations in the track, which partially change their amplitude with velocity. It can be seen later, that they are caused by resonances initiated by dynamic excitations of eigenfrequencies (natural frequencies) of the vehicle as well as ballasted track components. At 250 km/h dynamic excitation frequencies between 0 and 130 Hz arise from the out-of-roundness of the wheels. The 1st wheel harmonic initiates about 25 Hz, the 2nd 49 Hz, the 3rd 74 Hz, the 4th 98 Hz and the 5th 123 Hz (Fig. 2). When the train’s speed exceeds 200 km/h the out-of-roundness of the 3rd to 5th wheel harmonic are decisive. But all attempts to explain the high velocities between 400 and 1000 Hz in Fig. 1 by excitations of out of round wheels with frequencies between 50 and 123 Hz failed. A pure reflecting upon the out-of-roundness cannot explain the relation between speed and excitations that we have introduced up to now. Therefore the following calculations are applied to a wheel-set-track-system. A similar system was used in G.B. Morys’s dissertation which succeeded in simulating the formation of out-of-roundness of the wheels by calculation. Fig. 3: Longitudinal mathematical model of the ballasted track Fig. 4: Cross section of the mathematical model of the ballasted track Fig. 2: Frequencies of excitation at 250 km/h
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