Vibrational resonances of nonrigid vehicles: Polygonization and ripple patterns

The well-known phenomenon of ripples on roads has its modern counterpart in ripple patterns on railroads and polygonization of wheels on state-of-the-art lightrail streetcars. Here we study an idealized mechanical suspension model for the vibrational frequency response of a buggy with a nonrigid body (typically, an aluminium chassis and coach). The finite flexural rigidity of the body is an important novel feature. Since the essential physics is described by only one extra material parameter (viz. the stiffness coefficient), the model retains its basic simplicity and can still be analysed exactly. The dynamics (i.e., the Lagrangian equations of motion) are solved in the frequency domain. The motion on a distorted surface is treated as a nonholonomic constraint. Thus we analytically calculate spectra, e.g., the wheel spectrum. This reveals a new, significant wheel resonance (typically near 30–35 Hz), which is confirmed by means of a novel analysis of the wheel’s lift force (taking care of traction forces). At moderate city speeds this resonance agrees with recently observed characteristic ripple patterns on lightrail tracks, with wavelengths of approximately 10–20 cm (amplitudes of the order of a millimeter), and correspondingly polygonized wheels.