Recursive Modeling of Dynamic Systems

POD-based filters for the representation of random loads on structures

“…At the moment, it is not necessary to specify the eigenvectors further, since any additional phase angle does not affect the expressions in Eq. (16), being elided in the product, and in Eq. (17), because of the uniform distribution of the phase angles of dZ v and dZ y .…”

“…(15) Thanks to these properties, S vv (u) can be factorized as (16) where QZ[q 1 /q N ] and G(u)Zdiag(g 1 /g N ). On the base of such representation of S vv (u), the orthogonal increments dZ v (u) of Eq.…”

“…(40) constitutes a recursive model that can be used to generate realizations of v (k) (t) starting from a sample of the initial state z k (0). The simulation technique here described has the typical computational advantages of the recursive models formulated directly in a discrete-time context such as AR, MA, ARMA [15,16], enabling the fast generation of long time series without the severe limitation concerning memory storage that affects the FFT-based techniques. For a PODbased recursive model, however, the selection of the order and the calibration of the coefficients are simplified since the optimization is performed referring to single-input systems corresponding to the POD-modes taken into account in the analysis.…”

“…(16) and (17) can be truncated retaining only a limited number N ! N of terms (in general different than the number of terms retained in CPT, even if identified for simplicity with the same symbol N).…”

“…If non-linearities are taken into account, Itô calculus has been used to develop time-domain solution [8][9][10], while the concepts of Volterra and Wiener series expansion have been widely applied in the frequency domain [11]. Monte Carlo simulation procedures can be formulated in the frequency domain, applying the usual superposition harmonic technique [12] or its more sophisticated derivations [13,14], as well as in the time domain by building suitable recursive models such as AR, MA and ARMA [15,16].…”