The rail can be considered as a discretely supported, quasi-infinite waveguide, i.e. a structure along which mechanical waves can travel without spreading in all directions. This work focuses on the forced vibration behavior of a rail which is excited at one of its cutoff frequencies. A cutoff frequency is the frequency limit below which a wave mode cannot propagate and above which it can propagate. The experimentally observed effects of can be understood by looking at a model of several elastically connected beams as illustrated in Figure 2.
Figure 2 shows a mechanical model consisting of n=3 beams interconnected by elastic layers.
This model is referred to as multiple mode model. A total number of n=3 modes can propagate along the z-direction of the model and each of these modes has a cutoff frequency. Analogous to the rail which is discretely supported on sleepers, the model is studied in a periodically supported configuration with supported regions (red) alternating with unsupported regions (green). Damping is introduced as modal damping to the system.
Figure 3 shows the magnitude of the transfer function of the multiple mode model between the excitation force at beam 3 (z=0m) and the displacement of beam 1 at z=0m.
If the multiple mode model is harmonically excited at a cutoff frequency of one of its wave modes, the amplitude of the system response reaches a maximum (green peaks in Figure 3). In the absence of damping the entire system would perform a synchronous motion and the amplitude would tend towards infinity. However, system damping prevents the amplitude from increasing to infinity and a local vibration zone close to the excitation exists. This type of system response with a local vibration zone is referred to as "type I vibration" (see e.g. Mode 2). Additional amplitude maxima of the system response are observed due to the discrete support (red peaks in Figure 3). At these amplitude maxima, however, no local vibration zone exists. This is referred to as "type II vibration" (see e.g. Mode 2,II). These additional amplitude maxima can occur close to the cutoff frequencies. In these cases the cross-sectional mode shape of the type I and the type II vibration are almost the same but the spatial behavior along the waveguide is completely different.
These statements are illustrated in the section below for all three modes of the multiple mode model. The results can be viewed as AVI or animated-GIF animations by pressing the corresponding buttons. The discussed characteristics of the forced vibration response of a discretely supported waveguide have also been experimentally observed at a test track as can be seen within the presentation of the experimental results.