Time-Dependent Reliability Analysis of Rotor-Blade Systems with Floquet-Moment-EVD Interval Approximation

Document Type : Research Paper

Authors
INSA Rouen Normandie, Normandie Univ, LMN UR 3828, F-76000, Rouen, France
Abstract
We study the time-dependent reliability problem of the linear time-periodic rotor-blade model under stochastic forcing, encompassing a broad class of mechanical systems (wind and tidal turbines, helicopter rotors). The periodic modes from Floquet theory are used to implement efficient moment-propagation convolution expressions for the response of the system, thus characterizing the limit state function. We then propose an interval-based decomposition strategy to estimate the probability of failure using results from Extreme Value Theory. The approach is compared with raw Monte Carlo simulation for a simplified rotor-blade structure subjected to Gaussian and non-Gaussian non-stationary loads, showing substantial gains in computation time and excellent accuracy. The approach is original and stands in contrast to instant-based discretization approaches popular in the time-dependent reliability literature.
Keywords
Subjects

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[1] Carnegie, W., Vibrations of rotating cantilever blading: Theoretical approaches to the frequency problem based on energy methods, Journal of Mechanical Engineering Science, 1(3), 1959, 235–240.
[2] Khulief, Y.A., Laejoon, Y., Lead-lag vibrational frequencies of a rotating beam with end mass, Computers & Structures, 29(6), 1988, 1075–1085.
[3] Leung, T. C., Fung, A. Y. T., Spinning finite elements, Journal of Sound and Vibration, 125(3), 1988, 523–537.
[4] Ganguli, R., Finite Element Analysis of Rotating Beams: Physics Based Interpolation, Springer, Singapore, 2017.
[5] Christensen, R. H., Active Vibration Control of Rotor-blade Systems: Theory and Experiment, PhD Thesis, Department of Mechanical Engineering, Technical University of Denmark, Kgs Lyngby, 2004.
[6] Bottasso, C. L, Cacciola, S., Model-independent periodic stability analysis of wind turbines, Wind Energy, 18(5), 2015, 865–887.
[7] Hansen, M. H., Modal dynamics of structures with bladed isotropic rotors and its complexity for two-bladed rotors, Wind Energy Science, 1(2), 2016, 271–296.
[8] Firouzi, N., Lenci, S., Amabili, M., Rabczuk, T., Nonlinear free vibrations of Timoshenko–Ehrenfest beams using finite element analysis and direct scheme, Nonlinear Dynamics, 112(9), 2024, 7199-7213.
[9] Firouzi, N., Dohnal, F., Dynamic stability of the Mindlin-Reissner plate using a time-modulated axial force, Mechanics Based Design of Structures and Machines, 53(1), 2025, 446-463.
[10] Xu, J., Gasch, R., Modale Behandlung linearer periodisch zeitvarianter Bewegungsgleichungen, Archive of Applied Mechanics, 65, 1995,178-193.
[11] Floquet, G., Sur les équations différentielles linéaires à coefficients périodiques, Annales Scientifiques de l’École Normale Supérieure, 12, 1883, 47–88.
[12] Kelley, W. G., Peterson, A. C., The Theory of Differential Equations, Springer, New York, 2010.
[13] Ma, H., Lu, Y., Wu, Z., Tai, X., Li, H., Wen, B., A new dynamic model of rotor–blade systems, Journal of Sound and Vibration, 357, 2015, 168-194.
[14] Li, B., Ma, H., Yu, X., Zeng, J., Guo, X., Wen, B., Nonlinear vibration and dynamic stability analysis of rotor-blade system with nonlinear supports, Archive of Applied Mechanics, 89(7), 2019, 1375-1402.
[15] Wu, J., Rezgui, D., Titurus, B., Dynamics of nonlinear beam-propeller system with different numbers of blades, Nonlinear Dynamics, 112(2), 2024, 833-863.
[16] Forghani, S. M., Ritto, T. G., Stochastic analysis on the stability of the solutions of wind turbine blade vibrations applying the Floquet theory, 11th International Conference on Engineering Vibration, Ljubljana, Slovenia, 2015.
[17] Kumar, R., Ali, S. F., Gupta, S., Stochastic reduced order modelling and analysis of rotating bladed discs, Proceedings of the Royal Society A, 478(2260), 2022, 20210833.
[18] Bae, H., Boyd, I. M., Carper, E. B., Brown, J., Accelerated multifidelity emulator modeling for probabilistic rotor response study, Journal of Engineering for Gas Turbines and Power, 141(12), 2019, 121019.
[19] Sánchez Jiménez, O., On the stochastic response of rotor-blade models with Floquet modal theory: applications to time-dependent reliability of tidal turbine blades, PhD Thesis, Laboratoire de Mécanique de Normandie, Normandie Université, Saint-Étienne-du-Rouvray, 2023.
[20] Prelewicz, D. A., Response of linear periodically time varying systems to random excitation, AIAA Journal, 10(8), 1972, 1124–1125.
[21] Spires, J. M., Sinha, S. C., On the response of linear time-periodic systems subjected to deterministic and stochastic excitations, Journal of Vibration and Control, 2(2), 1996, 219–249.
[22] Han, X., Sánchez Jiménez, O., Pagnacco, E., Response and EPSD of rotor-blade nonlinear system with non-stationary non-gaussian stochastic excitation via PGHW method, Computers & Mathematics with Applications, 142, 2023, 140–156.
[23] Rackwitz, R., Reliability analysis-a review and some perspectives, Structural Safety, 23(4), 2001, 365–395.
[24] Zhang, B., Wang, W., Wang, Y., Li, Y., Li, C-Q., A critical review on methods for time dependent structural reliability, Sustainable and Resilient Infrastructure, 9(2), 2023, 91–106.
[25] Sudret, B., Uncertainty propagation and sensitivity analysis in mechanical models–contributions to structural reliability and stochastic spectral methods, H. D. R. Thesis, Université Blaise Pascal, Clermont-Ferrand, France, 2007.
[26] Andrieu-Renaud, C., Sudret, B., Lemaire, M., The PHI2 method: a way to compute time-variant reliability, Reliability Engineering & System Safety, 84(1), 2004, 75–86.
[27] Sudret, B., Analytical derivation of the outcrossing rate in time-variant reliability problems, Structure and Infrastructure Engineering, 4(5), 2008, 353–362.
[28] Hu, Z., Du, X., Time-dependent reliability analysis with joint upcrossing rates, Structural and Multidisciplinary Optimization, 48(5), 2013, 893–907.
[29] Hu, Z., Du, X., Mixed efficient global optimization for time-dependent reliability analysis, Journal of Mechanical Design, 137(5), 2015, 051401.
[30] Gong, C., Frangopol, D. M., An efficient time-dependent reliability method, Structural Safety, 81, 2019, 101864.
[31] Echard, B., Gayton, N., Lemaire, M., AK-MCS: an active learning reliability method combining kriging and Monte Carlo simulation, Structural Safety, 33(2), 2011, 145–154.
[32] Hu, Z., Mahadevan, S., A single-loop kriging surrogate modeling for time-dependent reliability analysis, Journal of Mechanical Design, 138(6), 2016, 061406.
[33] Attukur Nandagopal, R., Narasimalu, S., Chai, G. B., Probabilistic structural reliability analysis of a horizontal axis tidal turbine blade by considering the moisture effects on the blade material, Marine Systems & Ocean Technology, 15(4), 2020, 253-269.
[34] Val, D. V., Chernin, L., Yurchenko, D. V., Reliability analysis of rotor blades of tidal stream turbines, Reliability Engineering & System Safety, 121, 2014, 26-33.
[35] Friswell, M. I., Dynamics of rotating machines, Cambridge University Press, New York, 2010.
[36] Genta, G., Dynamics of rotating systems, Springer, New York, 2005.
[37] Géradin, M., Rixen, D. J., Mechanical vibrations theory and application to structural dynamics, John Wiley & Sons, New York, 2014.
[38] Rice, S. O., Mathematical analysis of random noise, The Bell System Technical Journal, 23(3), 1944, 282–332.
[39] Rice, S. O., Mathematical analysis of random noise, The Bell System Technical Journal, 24(1), 1945, 46–156.
[40] Lindgren, G., Gaussian Integrals and Rice Series in Crossing Distributions—to Compute the Distribution of Maxima and Other Features of Gaussian Processes, Statistical Science, 34(1), 2019, 100-128.
[41] Hasofer, A. M., Petocz, P., Extreme Response of the Linear Oscillator with Modulated Random Excitation, Statistical Extremes and Applications, Springer, Netherlands, 1984, 503–512.
[42] Lutes, L. D., Sarkani, S., Random vibrations: analysis of structural and mechanical systems, Elsevier, Massachusetts, 2004.
[43] Suptille, M., Caractérisation temporelle et spectrale de champs instationnaires non gaussiens: application aux hydroliennes en milieu marin, Ph.D. Thesis, INSA de Rouen Normandie, Saint-Étienne-du-Rouvray, 2015. 
[44] Ambetkar, V., Kuppa, R., Gupta, S., Multivariate Extreme Value Distributions for Vector of Non-stationary Gaussian Processes, Procedia Engineering, 144, 2016, 504–511.
[45] Konstant, D. G., Piterbarg, V. I., Extreme values of the cyclostationary Gaussian random process, Journal of Applied Probability, 30(1), 1993, 82–97.
[46] Clara, T., Falcão De Campos, J. A. C., Baltazar, J., Added mass effects on the natural frequencies of marine current turbine blades, Proceedings of the 6th International Conference on Mechanics and Materials in Design, Ponta Delgada, Azores, Portugal, 5524, 2014.