A simplified method for the analysis of the stochastic response in discrete coordinates

Nowadays, the problem of the computational cost vs. the accuracy of the results is an important field of research in engineering, particularly for the analysis of the response of a stochastic system. By using a set of discrete coordinates, this response can become computationally challenging, especially when using a modal representation. Many dynamic load cases, significant for engineering applications, have random and convective properties such as wall pressure fluctuations due to the turbulent boundary layer (TBL). In this work, a new method for the evaluation of the dynamic response of a linear system excited by a TBL like load is proposed: it is named as frequency modulated pseudo-equivalent deterministic excitation (PEDEM) and it is based on the pseudo-excitation method (PEM). The latter is an exact representation since it uses the modal decomposition of the cross-spectral density matrix of the load. Unfortunately, the extraction of the eigensolutions of the load matrix is required at each frequency step and thus can become computationally unacceptable when a large number of eigensolutions is required. PEDEM tries to overcome this computational bottleneck by introducing some approximations, which are based on the analysis of the eigensolutions of the dynamic load matrix vs. frequency. Three different approximations are proposed and thus modulated for the three frequency ranges wherein the dynamic matrix of the considered random and convective load has different characteristics. A criterion to identify the above-mentioned frequency ranges is also proposed by introducing a dimensionless representation of the frequency. The results show that the proposed approximations combine a good accuracy in representing the stochastic system response with a 40 per cent of reduction of the computational costs if compared to a full stochastic response. The method is successfully applied to a simple 1D configuration, a chain of oscillators, which still presents the main features of a generic system. The extension to 2D systems is also analysed considering a preliminary test case with a flexural plate.