A constant phase approach for the frequency response of stochastic linear oscillators
Résumé
When studying a mechanical structure, evaluation of its frequency response function (FRF) over a given frequency range is one of the main interests. Computational cost aside, evaluating FRFs presents no methodological difficulty in the deterministic case. Doing this when the model includes some uncertain parameters may however be more difficult as multimodality can arise around resonances. Indeed, even for a single degree of freedom system, it can be shown that usual methods of the probabilistic frame such as generalized Polynomial Chaos may fail to properly describe the probability density function of the response amplitude. This study proposes another approach which involves a shift in the usual quantities used to draw FRFs. Instead of computing the stochastic response for a given excitation frequency, this work adopts a constant response phase point of view. For each phase value of the oscillator response, the uncertainty over some parameters is propagated to the corresponding uncertain amplitudes and excitation frequencies. This provides much smoother variations of the involved quantities which are much easier to describe using a simple Polynomial Chaos approach. This also provides a mean to conduct a straightforward stochastic study of special points such as the maximum of amplitude (which matches a given phase) or the point for which the response is in quadrature with the excitation. Both analytical and numerical results will be exposed for a single degree of freedom oscillator whose squared eigen frequency (or stiffness) follows a uniform law.
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