User:UgoTORNAR/sandbox2/A Mathematical Model of the Road Surface

Abstract

The following model can be of interest for all automotive engineers who need to define mathematically the trepidations coming from the road and exciting a vehicle through its suspensions. By using the formulations provided by the present paper the mathematical modelizer can skip lengthy and costly measurements of road profiles that become soon outdated due to road weathering and aren’t anyway representative of all roads in the world.

The isotropic hypothesis was first proposed by Robson [4] and we make use of it to compute the Coherence Function of equation (37) by a new method which only requires the calculation of a FFT skipping the numerical calculations proposed by Robson.

Parts of the present theory were published in references [1] and [2] but only in the present paper the theory is provided in its entirety. The “Cost * f^(-2)” hypothesis for road spectra is well justified by measurements reported in reference [5] and is used in the present paper. Formula (37) was recently used in reference [3] for the calculation of bridges.

Keywords: Road, Computational model, Vibration, Power Spectral Density, Cross Spectral Density, Coherence, Fourier Analysis,
Mathematical model of a random surface.

As a first operation we have to devise a suitable sampling of the surface by dividing it into parts (surface samples). Let us divide the surface into squares of side length 2L. Each square is countersigned by an index n. In each square the surface is described by the function: