The Method of the Pseudo Equivalent Deterministic Excitations (PEDE_M) to Bound the Random Response

The analysis of the response of a stochastic system, through a discrete coordinate set, can become computationally challenging, even when using a full modal representation. Nevertheless, many dynamic load cases, in engineering applications, have stochastic behaviour as the wall pressure fluctuations due to the turbulent boundary layer. In this work, a new method is proposed: it is named as frequency Modulated Pseudo Equivalent Deterministic Excitation, PEDEM, and it is based on the Pseudo Excitation Method, PEM. This latter can be considered as an exact representation since it uses a modal decomposition of the cross-spectral density matrix of the excitation; the extraction of the eigensolutions of the load matrix, at each frequency step, is a computational disadvantage. PEDEM overcomes this issue by introducing some approximations based on the analysis of the eigensolutions of the dynamic load matrix versus frequency. Mainly, two different approximations are proposed with reference to extreme frequency ranges (low and high) wherein the dynamic matrix of a random and convective load has different characteristics. A criterion to identify these frequency ranges is proposed versus a dimensionless representation of the frequency. Moreover, it is shown that the proposed approximations represent the bounding curves of the response for the whole frequency range. Fruitful comparisons with a full stochastic approach is discussed. The proposed approximations combine a good accuracy and representation of the stochastic system together with a significant reduction of the computational costs if compared to a full stochastic response or a PEM solution. The method is applied over two simple configurations (a chain of oscillators and two flexural plates) but the possible extensions to more complex cases are motivated by the quality of these preliminary results.