Uncertainty quantification for an industrial mistuned bladed disk with geometrical nonlinearities

Recently, a methodology allowing a stochastic nonlinear reduced-order model to be constructed in the context of the nonlinear mistuning induced by geometric nonlinearities has been proposed. The present work is devoted to an industrial application for which the centrifugal stiffening due to rotational effects is also included. The nonlinear mistuned forced response is investigated in the time-domain by using a spatial cyclic load over a given excitation frequency range. The geometric nonlinear mistuned analysis is performed over the frequency range by using the Fast-Fourier Transform of the time response. A sensitivity analysis is conducted with respect to the load level, giving rise to secondary resonances, which appear outside the excitation frequency range and which can exhibit a particular sensitivity to uncertainties. Such new complex dynamical situation, induced by the coupling between the geometrical nonlinearities and the mistuning phenomenon, is analyzed in details. NOMENCLATURE bw,∞ Amplitude ratio bν,∞ Frequency ratio f0 Load level g(t) Time domain of the load ĝ(2πν) Frequency domain of the load j0,k0 Blade number t/t0 Dimensionless time v(t) Displacement vector in the time domain w(2πν) Frequency domain observation (mean NL-ROM) P Dimension of the NL-ROM W (2πν) Frequency domain observation (stochastic NL-ROM) Y (2πν) Amplification factor (stochastic NL-ROM) d Dispersion parameter ν/ν0 Dimensionless frequency Ω Angular speed (rotational motion) B Frequency band of analysis B 1 e , B 2 e Frequency band of excitation Bs,Bsub Sub-frequency range a Spatial discretization of the load f(t) External load vector u(t) Displacement vector in the time domain ̃ u(t) Displacement vector in the time domain ̂̃u(2πν) Fourier transform of the displacement vector in the frequency domain INTRODUCTION In general, the natural cyclic symmetry of turbomachinery bladed disks is broken because of manufacturing tolerances and material dispersions, which create small variations from one blade to another one. Such phenomena, referred to mistuning, can generate localization effects combined to a dynamic amplification of the forced response [1]. Many research efforts have been carried out on this subject, including reduced-order models with probabilistic approaches in the numerical modeling, for taking into account the random character of mistuning [2–4]