114 A. Hatstatt and K. E. Tatsis This contribution is motivated by wind turbine structures, which are oftentimes difficult to instrument and the availability of Supervisory Control and Data Acquisition (SCADA) data has not been leveraged for vibration-based assessment. As such, the methodology proposed in this paper aims at bridging the gap between statistical features of vibration signals, such as the ones offered by SCADA systems, for the generation of vibration signals that can stochastically reproduce the response experienced by the system under various conditions of operation. To this end, a physics-informed Gaussian Process Latent Force (GPLF) [11, 12] is developed for the modeling of the modal response signals, which can be used to compose the entire system response. The model is constructed by leveraging the knowledge of the system structural properties and the features characterising the vibration response signals, such as the energy due to operational harmonics. The aim of the proposed surrogate approach is to served as a generative model that can produce time histories of the vibration response, such that their statistical and spectral information matches the one of the original signals. This paper is organized into three main sections. The first section outlines the methodology, focusing on the Gaussian Process Latent Force Model as the surrogate model, along with its formulation. The second section presents the application case, detailing the framework used to study the vibrational data of a wind turbine and the analytical approach taken. Finally, the third section showcases the results, offering a comparative analysis of the predictive capabilities of the surrogate model against the ground truth data obtained from simulations. Methodology In this section, the formulation of Gaussian Process Latent Force models is described for the representation of vibration response signals obtained from structural or mechanical systems. To this end, the response of the system is decomposed into the modal contributions, whose spectral content is easier to characterise and subsequently represented by a temporal Gaussian Process, whose dynamics are driven by a set of latent forces. Structural dynamics Consider ann-degree-of-freedom linear time-invariant dynamic system whose vibration response is governed by the following continuous-time differential equation M¨u(t)+C˙u(t)+Ku(t)=Spp(t) (1) where u ∈ Rn is the vector of displacements, M, Cand K ∈ Rn×n denote the mass, stiffness and damping matrices respectively, p(t) ∈Rnp is the vector of external forcing terms andSp ∈Rn×np is a selection matrix that assigns the forcing terms to their corresponding locations. In engineering systems, the dynamics are typically governed by a few vibration modes and as such, the response can be expressed as follows u(t)=Φq(t) (2) where Φ∈Rn×nm is a matrix whose columns contain the mode shapes and q(t) ∈Rnm denotes the vector of generalized modal coordinates. In view of this expression and under the condition of proportional damping, the initial equation of motion, described by Eq. (1), can be decomposed intomsecond order differential equations. Therefore, the dynamics of the i-th generalized coordinate qi(t) are governed by the following equation ¨qi(t)+2ξiωi ˙qi(t)+ω 2 i qi(t)=fi(t) (3) which applies for i = 1, 2, . . . ,nm. Moreover, ωi denotes the natural frequency of the i-th mode, ξi the corresponding damping ratio and fi(t) is the projection of the external loads to the modal space, which is obtained by the following expression fi(t)=ϕTi Spp(t) (4) inwhich ϕi denotes the i-th column of the mode shape matrix. Modal expansion Under the availability of vibration measurements from a number of sensing points, which are herein denoted byy(t) ∈Rns, the projection postulated by Eq. (2) can be written selectively for all sensing channels as follows y(t)= nmX i=1 ϕs,i qi(t) (5)
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