3 VEHICLE OMA: THEORETICAL BACKGROUND 3.1 The road profile roughness excitation It is well known that the power spectral density (PSD) of the roughness of a road profile as a function of the spatial frequency, the wavenumber k , has the general decreasing behaviour reported in Fig.3a. In particular, in the figure one can actually see the trend of several possible analytical approximations of the PSD function, basically obtained by fitting the experimental data, that have been proposed and can be found in the technical literature. Those analytical expressions are reported in Fig.3b [5,6]. Since the wavenumber k is the inverse of the wavelength λ, the time frequency f uk = , where u is the car velocity. In Figures 4a and 4b the PSDs of the two road roughness profiles, Blue-kay and Florida, considered in the road tests are shown as functions of the time frequency f . Simple calculations lead to the conclusion that the wavelength range from 100 to 1 m, corresponds to a time frequency range from 0.2 to 20 Hz, which is that normally considered in vehicle dynamics for comfort, safety and road holding analyses and that is usually referred to as ride [7]. As it is possible to see, it is actually the range in which the PSD function of a road profile decreases. For time frequencies smaller than 0.2 Hz and, hence, for wavelengths higher than 100 m the PSD function is quite flat. From these considerations one can conclude that the road input excitation, on each wheel, is not actually a white noise sequence, as requested to be the basic OMA assumption strictly respected, but something more similar to a Brownian or red noise, whose PSD is inversely proportional to 2 f . This is still a suitable type of excitation for OMA, since it does not contain peaks in the spectrum that could lead to extra-peaks in the output responses, not related to structural modes. What needs to be more over said is that the input excitation on the front wheels is in a certain way correlated with that on the rear wheels, as a correlation could also exist between inputs belonging to the two different sides of the vehicle. This means that the input power spectral density matrix will have some of the off-diagonal terms not null. Even this second violation of the basic OMA assumption is considered weak in this paper. Aim of the analysis reported in the following sections is, indeed, to show that although the overall input excitation does not strictly respect the basic OMA hypotheses, it is still possible to obtain reliable modal models by processing the raw time output data recorded during a simple, cost-effective road test, with a classical operational modal parameter estimation algorithm. (a) (b) Fig.3 General behaviour of the road profile roughness PSD approximations (a) and their analytical expressions (b) (adapted from [5]) 316
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