RTR 4/2010 12 n Frequencies of the ballasted track 80 kN/mm, we would not get these horizontal stages and a plateau, but oblique lines. With a value of 110 kN/mm we find horizontal stages and a plateau again, but they are displaced to different frequencies from those of Fig. 1 then. So the conformity of the shoulders in measured and calculated diagrams verify that in this case a stiffness of 60 kN/mm for the unloaded track (as listed in Tab. 1) is valid. At last we have found the explanation for the fact that in Fig. 1 the greatest amplitudes of the rail velocity – in the plateau between 550 and 850 Hz – are created by resonance on the excitations of short-pitch corrugation with about 1160 Hz. The length of the undulation is about 6 cm. This phenomenon is a good example of the fact that the excitation spectrum is widened by the impulse kind of corrugation. Remarkable in Fig. 11 is that there are 3 areas of accumulation at 81, 132 and 222 Hz. A fourth one comes together at 7596 Hz in Fig. 10P. This is the eigenfrequency of a rail beam with the length of a third of the sleeper distance, laying on its ends, whose mass is concentrated in the middle. With continuous rail mass the 4th accumulation point (AP) is shifted to infinity. Between 0 and 81 Hz as well as between 132 and 222 Hz we find two “MODE FREE” areas. The existence of two accumulation points (AP) in the eigenvalues of simply connected plane elastic areas is known from [1]. There Heise has shown the existence of 2 AP for the example of plane circular discs by 16 Integral Equations of geometric and static boundary value problems. Corresponding for tracks with the composite of the two elements rail and sleeper 4 AP have to exist. The AP at 81 and 132 Hz may be attached to the sleepers, while the AP at 222 Hz belongs to the rail as well as the AP in infinity. Considering the extreme amplitudes in the frequency spectrum of Fig. 1, we find very small values in the area between 125 and 220 Hz. This effect can be explained now by the fact that no resonance can happen in a mode free area. In Fig. 12 it is remarkable that the excitation frequencies have their direct response in the bands of eigenfrequencies of the systems of wheel-set-track and unloaded track. The self excitation of the wheel sets by the 2nd wheel harmonic at about 50 Hz is unfortunately close to the smallest eigenfrequency of the wheel sets on the track of about 48 Hz. Ballast is best consolidated by frequencies between 35 and 50 Hz. This is shown experimentally in [2]. The excitation by the 3rd wheel harmonic at 74 Hz is close to the smallest eigenfrequency of the unloaded ballasted track at Under “Rail” and “Sle” the biggest amplitudes of the rail and the sleepers are put together. The composition of these values in the range between 0 and 1000 Hz is shown in Fig. 11. In these graphs of the mode attributes we recognize in which part of the system the biggest dynamic displacements and the strongest interactions between rail and sleepers are happening at which frequency. With the graphs for rail and sleeper in Fig. 11 only we cannot yet find a connection to the measured graphs of Fig. 1. After several idle attempts with derived values a successful possibility was opened by using the biggest difference of displacements between rail and sleeper in the rail fastenings (named “DRS” in Figs. 8 and 9). In Fig. 11 these differences “DRS” are shown for all frequencies. In the graph of DRS for values higher than 222 Hz we find a parallel effect to Fig. 1 in the three “stages” or “shoulders” where the 3rd stage leads to a plateau between 650 and 850 Hz. If we would not choose a rail pad stiffness of 60 but and 800 Hz, 16 of the essential modes are presented in Fig. 10. Within the range of 80 to 100 Hz rails and sleepers are vibrating together in the same direction and with almost the same amplitude. Between 100 and 132 Hz the sleepers develop their own behaviour more and more, until in the mode 10K at 132.2 Hz nearly the sleepers only are vibrating. With these frequencies the rail is not yet vibrating as a beam, it is just bent by the opposed vibration of neighbouring sleepers (analogy to the pinned-pinned mode at about 1200 Hz). In the modes A to F the rail is quasi shifted, from G to K it is bent by the sleepers. Discontinuously the behaviour of the rail changes in the next mode L of Fig. 10 at 222 Hz. It vibrates in oppositely oriented phase to the sleepers and with increasing amplitudes compared with the sleepers. A full view over the properties of the modes in the range of frequencies seams to be desirable. Therefore in the lower left part of Figs. 8 and 9 mode attributes are shown. Fig. 12: Excitation and eigenfrequencies of the wheel-set-track-system at 250 km/h Fig. 11: Mode attributes of the unloaded track-system
RkJQdWJsaXNoZXIy MjY3NTk=