MS20.9 - UQ and Probabilistic Learning in Computational Dynamics

Forced vibration tests have always been a reliable method for dynamic characterization of concrete dams. In this field, LNEC has an extensive experience, having conducted a large number of tests on concrete dams. Test methodologies have continuously evolved with substantial improvements in the control of the dynamic actions applied to the dam, the reliability of the structural behavior records, and the processing techniques to identify the structural dynamic parameters. In this context, LNEC uses an eccentric masses vibration generator, which was designed and produced in-house and has been used for several decades in the concrete dam’s field observation. This equipment generates a controlled vibration using a set of masses eccentrically assembled in a rod connected to a vertical rotation shaft. This mechanical component is supported in two bearings inserted in a surrounding steel frame, which is fixed to the dam. The applied excitation force is controlled by the rotation frequency of the electrical engine and by the number and position the masses mounted in the rod. The main objective of this work was the determination of the measurement uncertainty of the excitation force applied by the mentioned dynamic testing equipment. Being a key issue for metrological quality evaluation, the obtained information is essential to assure confidence and rigorous knowledge about the applied excitation force, namely, in extreme situations near dynamical structural safety limits of the observed concrete dam and of the used equipment. In this case, since the excitation force is indirectly measured using a mathematical model, the measurement uncertainty evaluation consisted, in a first stage, in the probabilistic formulation of the input quantities (rotation frequency, masses, radial positions, dimension, diameter and density of the generator’s rod). Experimental work was performed to provide traceability to the International System of Units (SI) in the case of the generator’s masses and rotation frequencies, while information related to the remaining input quantities was obtained from design and production requirements. In a second stage, the dispersion of values related to the input quantities was propagated through the mathematical model, using the Monte Carlo method, allowing the quantification of the excitation force measurement uncertainty and the identification of the main uncertainty contributions by performing a sensitivity analysis. Two experimental cases were studied: (i) the use of five masses in the generator in the frequency range of 1 Hz up to 6 Hz; and (ii) the use of a single mass in the generator in the frequency interval comprised between 5 Hz and 15 Hz. In the first case, the excitation force estimates and expanded measurement uncertainties (considering a 95 % confidence interval) varied between 3,55 kN ± 0,14 kN and 127,68 kN ± 0,91 kN, being rotation frequency the major contribution for the obtained dispersion of force values. In the second case, the excitation force estimates and expanded measurement uncertainties varied between 16,71 kN ± 0,25 kN and 150,4 kN ± 1,8 kN, being the generator’s rod diameter the main contribution for the output measurement uncertainty.

Technologically, an interesting way for reducing the dynamical amplifications induced by the blade mistuning of turbine engines is to use a detuning strategy. This means that a few different blade designs (in general 2) are used in order to define a blades pattern that constitutes a detuned configuration of the full bladed-disk. The objective of this research is to validate a robust design methodology allowing for the detuning optimization in presence of random mistuning and in finite displacements. This latter consideration is justified by a green aviation context that involves lighter blades with thinner profiles. The main difficulty is related to the huge number of possible detuning configurations that exponentially increases with the cyclic order of the structure. In [1,2], many research efforts have been done to construct a high-fidelity nonlinear stochastic computational model (HFCM). In the mistuning context, the outputs characterize frequency peaks of the most unfavorable blade and the corresponding realizations usually present a large scatter. As a consequence, a careful attention has to be paid to the construction of the cost function in order to be representative of the vibrational behavior. Such cost function is parameterized with respect to a discrete parameter representing a given detuned configuration. It is then computed at a limited number of points during the training set, yielding an available small data training set. The main idea is then to construct a continuous approximation of this cost function, based on the use of the probabilistic learning PLoM tool [3], and that will be used in the learning step. Several difficulties inherent to the definition of the cost function (weak contrast and numerous local minima) require to reformulate the definition of the optimum. A numerical validation is proposed, using an available full data basis of a bladed-disk structure with 12 blades and for which a very few improving detuning configurations exist. It is shown that the proposed method is able to capture most of these optima using a small training set that only contains non optimal detuning configurations. [1] A. Picou, E. Capiez-Lernout, C. Soize, M. Mbaye, Robust dynamic analysis of detuned-mistuned rotating bladed disks with geometric nonlinearities, Computational Mechanics 65 (3) (2020) 711–730. [2] E. Capiez-Lernout, C. Soize, Nonlinear stochastic dynamics of detuned bladed-disks with uncertain mistuning and detuning optimization using a probabilistic machine learning tool, International Journal of Non-Linear Mechanics, Elsevier, 2022, 143, pp.104023. [3] C. Soize, R. Ghanem, Data-driven probability concentration and sampling on manifold, Journal of Computational Physics, 321 (2016) 242-258.

Within the structural health monitoring (SHM) field, consistent research efforts have been invested in developing automatic vibration-based indirect methodologies for inspecting existing heritage conditions. Current trends are mainly focused on output-only automatic operational modal analysis (AOMA), specifically throughout the stochastic subspace identification (SSI) technique among others. In the literature, a widespread workflow is implemented in a four-step solution: choice of the SSI control input parameters, computation of stabilization diagrams, stable poles' alignments detection, and their final clustering. However, the so far proposed solutions have not provided yet complete answers to some challenging and still open questions. For instance, an arbitrarily poor initial choice of the SSI control parameters may jeopardize the entire procedure. Therefore, in the current study, the authors present a novel intelligent-based AOMA framework in a machine learning perspective. Specifically, random-forest-driven Monte Carlo sampling of control parameters represents a promising intelligent way to automatically choose the proper SSI control parameters. Furthermore, the recurrent stable physical poles in the stabilization diagram among the Monte Carlo simulations deliver some special insights about mode shape confidence intervals. A numerical benchmark is herein analyzed illustrating some preliminary results and potentials of the proposed methodology.

Gantry robots, or cartesian manipulators, are popular choices for manufacturing due to their relatively simple control and movement precision. Currently, large gantry manipulators are being considered as part of the production of large structures, as in the case of wind turbine towers, offshore platforms, or vessels. Those gantry systems emerge as an excellent option to reduce manufacturing costs. However, their large size makes them susceptible to large vibrations which affect the precision of the end effector and overall fatigue life. The structural dynamics of flexible gantry manipulators comprise non-linear dynamics which are dependent on the position of the manipulator. This means that the structure will exhibit different modal characteristics depending on its configuration which makes this a challenging problem from a vibration control and fatigue life perspective. In this sense, a computationally simple modelling approach that can effectively represent the parameter-dependent dynamics could be of high value in the development of vibration control algorithms and as surrogate for fatigue-life estimation. In this work, we consider a simplified version of this problem featuring a beam with a moving mass, aiming at representing the constrained 1D horizontal motion of the manipulator and the resulting vertical vibration response. Our aim is to develop computationally efficient data-driven models for this simplified structure that could be easily generalized to the full range of motion of a gantry. To this end, we first consider a physics-based model of the structure, made up of an Euler-Bernoulli beam with a moving mass, where the inertial connection between the mass and the beam considers a random surface roughness. A finite element model of this system is built and is subsequently used to draw simulations of the vertical vibrations of the beam under different time-dependent mass trajectories. Next, we consider data-driven Linear Parameter Varying (LPV) Vector AutoRegressive (VAR) models to represent the vibration response of the beam. We consider various structures based on Bayesian non-linear regression and Gaussian Process regression of the LPV-VAR model coefficients. Due to the computational cost of model estimation, we introduce an input space sub-sampling method that reduces the size of the regression matrix, while preserving an even sample distribution over the input space. The proposed approach is evaluated on simulated data from the physics-based model. We compare the different modelling methodologies and assess their predictive accuracy, quality of the representation of the parameter-varying dynamics, and computational efficiency. While both modelling approaches lead to accurate representations of the beam’s vibration response the introduced input space sub-sampling method speeds up the estimation while preserving the modelling accuracy.