Rail corrugation is a phenomenon that leads to a waving in the rails with wavelengths typically between 3 cm and 100 cm and amplitude levels of several microns. The genesis of this waving is complex. Rail corrugation is a recognized problem that leads to excess vibration on the rails and vehicles to a point of reducing their life span and compromising safety. In urban areas excess vibration noise is also a problem. A software tool was developed to analyze accelerometer signals acquired in the boggies of rail vehicles in order to quantify the rail corrugation according to their frequency and amplitude. A wavelet packet methodology was used in this work and compared with the One Third Octave Filter (OTOF) power representations, which is currently used in the industry. It is shown that the former produces better results.

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Fractional centred differences and derivatives definitions are proposed, generalizing to real orders the existing ones valid for even and odd positive integer orders. For each one, suitable integral formulations are obtained. The computations of the involved integrals lead to new generalizations of the Cauchy integral derivative. To compute this integral, a special two-straight-line path was used. With this the referred integrals lead to the well-known Riesz potential operators and their inverses that emerge as true fractional centred derivatives, but that can be computed through summations similar to the well-known Grünwald-Letnikov derivatives.

Generalised fractional centred differences and derivatives are studied in this chapter. These generalise to real orders the existing ones valid for even and odd positive integer orders. For each one, suitable integral formulations are presented. The limit computation inside the integrals leads to generalisations of the Cauchy derivative. Their computations using a special path lead to the well known Riesz potentials. A study for coherence is done by applying the definitions to functions with Fourier transform. The existence of inverse Riesz potentials is also studied.

Generalised fractional centred differences and derivatives are studied in this Chapter. These generalise to real orders the existing ones valid for even and odd positive integer orders. For each one, suitable integral formulations are presented. The limit computation inside the integrals leads to generalisations of the Cauchy derivative. Their computations using a special path lead to the well known Riesz potentials. A study for coherence is done by applying the definitions to functions with Fourier transform. The existence of inverse Riesz potentials is also studied.

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