RTR 4/2010 14 n Frequencies of the ballasted track How close the results of this simplified calculation are to those of a track with 20 sleepers (less than 1‰) is shown in Tab. 2. In Fig. 14 we see the simplified calculated limits of a track with wooden sleepers and their low weight by the equations 1 to 5. In this calculation the weaker rail pad of the ZW 900 was replaced by the stiffer one of the ZW 687a. Additional the ground stiffness was weakened, as the springy bottom of the wooden sleeper with its supposed stiffness of 80 kN/mm had to be put in series with the ground stiffness. All the other parameters were kept. From the result in Fig. 14 we can see, that the lower weight of the wooden sleeper compared with the heavy weight of the B 70 sleeper has positive consequences in the range between 50 and 130 Hz, in spite of the fact that an additional spring under the sleeper had to be introduced. The limit 1 is about 12 Hz higher compared with Fig. 12. By this the distance to the frequency of the 3rd wheel harmonic is 18 Hz instead of 6 Hz with the B 70. So the amplitudes with the wooden sleeper caused by the 3rd wheel harmonic should be smaller than half the value of the concrete sleeper B 70. Besides this in Fig. 14 the limits 2 and 3 are placed much higher than for the B 70. That means, that the corrugation excitation in the lower frequencies is reduced by the 2nd mode free area between 400 and 700 Hz. Additional to the half space damping of the ground we get more damping by the wooden sleeper. This investigation allows some conclusions concerning elastically soled sleepers [3,4]. By the additional spring at the bottom of the B 70 the limit 1 is shifted to lower frequencies. By this the 1st eigenfrequencies of the wheel set (at about 50 and 60 Hz) could find resonance with the 1st eigenfrequency of the unloaded track, as the upper limit of the 1st mode free area is lowered. More promising it might be to choose damping soles [5] instead of elastic soles, if a sufficiently durable, non elastic material with pure damping could be found. In this case the limit 1 could not be shifted below the 80 Hz. References: [1] Heise U.: The spectra of some integral operators for plane elastostatical boundary value problems, Journal of Elasticity (8) 1978, H. 1, p. 47-79 [2] Fischer J.: Einfluss von Frequenz und Amplitude auf die Stabilisierung von Oberbauschotter, Dissertation Graz 1983 [3] Müller-Boruttau F.H., Kleinert U.: Besohlte Schwellen, ETR 50 (2001), H. 3, p. 90-98, published under D:\ Transferdateien\HWS\Büro\01-01-26ETR-Bericht. DOC, 23.01.01 14:12 (Vikipedia) [4] Trevin J.-M.: Gleis, Deutsche Patentanmeldung DE 691 13 884 T2 [5] Cronau H.: Querschwelle für Eisenbahngleise, Deutsche Patentanmeldung DE 297 21 118.8 including the mass of the rail we end up in differences of large numbers and consequently in large errors. But reliable values are available with the model B by neglecting the mass of the rail, but including the stiffness of the rail in the middle. So the model B contains one mass only, the mass of half the sleeper. In order to calculate the stiffness ctot we put the stiffness of the rail section in its middle in series with the stiffness of the rail pad. With “a” being the distance between two sleepers and with the bending stiffness of the rail we get c EI a 3 3 48 = (2) The total stiffness ctot of all springs acting at the sleeper is found as c c c c tot = + + 1 2 3 1 1 1 /( / / ) (3) With half the sleeper mass m1 we get the rotational frequency ω2 in eq. (4). ω2 1 = c m tot (4) By this the lower limit 2 of the second mode free area is known now. The conversion of the rotational frequencies ω1,2,3 into the frequencies f1,2,3 in Hz may be done by eq. (5). f1 2 3 1 2 3 2 , , , , / = ω π (5) With eq. 1 to 5 the exact calculation of the accumulation points (AP) of the infinite rail is possible now (Fig. 14), as the models in Fig. 13 belong to the infinite rail. Common in the three modes A, K and L is that with the mass of one sleeper and a rail section with the length of the sleeper distance a vibrator with two masses on a rigid base can be designed which has the same eigenfrequencies. In the models A and B of Fig. 13 there are the following parameters: c1 = stiffness of the ground spring c2 = s tiffness of the rail pad in the rail fastening m1 = mass of half the sleeper m2 = m ass of the rail between two sleepers, distance a With these symbols in the following eq. (1) we get the squared rotational frequency ω1 at the limit 1 (mode A) with the negative sign in front of the root and the squared rotational frequency ω3 at the limit 3 (mode L) with the positive sign before the root. Equation 1 is valid for the model A in Fig 13. ω2 1 3 1 2 2 2 1 1 2 1 2 2 2 1 1 2 2 2 , ( ) ( ) = + + ± + + c c m c m m m c c m c m m m − 2 1 2 1 2 c c m m (1) ω2 1 3 1 2 2 2 1 1 2 1 2 2 2 1 1 2 , ( ) ( ) = + + ± + + c c m c m c c c − 2 1 2 1 2 c c [ ] The eigenfrequency of Model B cannot be calculated from eq. (1). If we try to use the equations belonging to model B in Fig. 13 by calculating the rotational eigenfrequency at the limit 2 by Limit with 20 sleepers by eq. 1 to 5 1 81,469 81,464 2 132,249 132,313 3 222,270 222,657 Tab. 2: Frequencies [Hz] Fig. 14: Excitation frequencies of a wheel-set-track-system with wooden sleepers at 250 km/h and limits of the mode free areas.
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