Moustapha Mbaye

A reduced-order model of detuned cyclic dynamical systems with geometric modifications using a basis of cyclic modes

A new reduction method for vibration analysis of intentionally mistuned bladed disks is presented. The method is built for solving the dynamic problem of cyclic structures with geometric modifications. It is based on the use of the cyclic modes of the different sectors, which can be obtained from a usual cyclic symmetry modal analysis. Hence the projection basis is constituted; as well as, on the whole bladed disk, each sector matrix is reduced by its own modes. The method is validated numerically on a real bladed disk model, by comparing free and forced responses of a full model finite element analysis to those of a reduced-order model using the new reduction method.

Robust Analysis of Design in Vibration of Turbomachines

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Robust analysis of design in vibration of turbomachines M. Mbaye, Christian Soize, J.-P. Ousty, Evangéline Capiez-Lernout

Nonlinear Modal Analysis of Mistuned Periodic Structures Subjected to Dry Friction

This paper deals with the dynamics of a cyclic system, representative of a bladed-disk subjected to dry friction forces, and exhibiting structural mistuning. The nonlinear complex modes are computed by solving the eigenproblem associated to the free response of the whole structure, and are then used to better understand the forced response to a traveling wave excitation.Similarly to the underlying linear system, the tuned model possesses pairs of modes that can be linearly combined to form traveling waves, unlike those of the mistuned structure. However, due to the nonlinearity, the modal properties are not constant but vary with the vibration amplitude in both cases.A qualitative analysis is also performed to assess the impact of the mistuning magnitude on the response, and suggests that further statistical investigations could be of great interest for the design of bladed disks, in terms of vibration mitigation and robustness.Copyright

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.Copyright

Uncertainty Quantification for Nonlinear Reduced-Order Elasto-Dynamics Computational Models

The present work presents an improvement of a computational methodology for the uncertainty quantification of structures in presence of geometric nonlinearities. The implementation of random uncertainties is carried out through the nonparametric probabilistic framework from a nonlinear reduced-order model. With such usual modeling, it is difficult to analyze the influence of uncertainties on the nonlinear part of the operators with respect to its linear counterpart. In order to address this problem, an approach is proposed to take into account uncertainties for both the linear and the nonlinear operators. The methodology is then validated in the context of the linear and nonlinear mistuning of an industrial integrated bladed-disk.

Uncertainty quantification in nonlinear structural dynamics for mistuned bladed disks

The recent improvements in turbomachinery design combined to the unavoidable requirements of kerosene savings require to analyze the exceptional operating regime of bladed disks, for which large deformations and large displacements can occur. It seems then quite appropriate to consider the geometrically nonlinear effects in the computational models dedicated to the analysis of mistuned turbomachinery bladed-disks. A special attention has to be first given to the case of geometrically nonlinear tuned bladed disks. The large set of nonlinear coupled differential equations issued from the finite element model of the tuned structure has to necessarily be solved in the time domain, leading us to establish a reduced-order strategy. The operators of the corresponding mean nonlinear reduced-order model are then deduced from its explicit construction as shown in the context of three-dimensional solid finite elements. One also has to focus on the modeling of the external load, corresponding to a frequency band chosen for the excitation, which has to be selected according to usual turbomachinery criterions. The external load has to be defined in the time domain but has to represent a uniform sweep in the frequency domain. We then propose to implement the mistuning uncertainties by using the nonparametric probabilistic framework. We then obtain a stochastic reduced-order model, which requires to solve a reasonable set of uncertain nonlinear coupled differential equations in the time domain, yielding appropriate efficient and dedicated algorithms to be constructed. Such computational strategy provides an efficient computational tool, which is applied on a finite element model of an industrial centrifugal compressor with a large number of degrees of freedom. This allows new high complex dynamical behaviors to be put in evidence.

A Contribution to the Reduced-Order Analysis of Bladed Disks Using Combined Approximations for Quick Campbell Diagrams Assessment

This paper proposes an original approach to the reduced-order modelling of integrally bladed disks. It is proposed to build a reduction basis which is independent from the rotational speed, from only one modal, cyclic-symmetry calculation performed at rest, and a few static computations. Based on previous works, a polynomial expansion is used which leads to a parametric approximation of the stiffness matrix for the entire operating range. Furthermore, the Kirsch Combined Approximation (KCA) method is used for building the final reduction basis. This method is based on successive approximations of the negative binomial expansion applied to the reanalysis eigenproblem. After giving a general overview of the main theoretical aspects, the paper focuses on the reanalysis problem based on the combined approximations method. Finally, the application of the extended reduction method to the case of a real bladed disk is presented. It is shown that the use of combined approximations provides a very accurate estimation of a Campbell diagram, and allows substantial computational time savings.© 2015 ASME

Computational Geometrically Nonlinear Vibration Analysis of Uncertain Mistuned Bladed Disks

The recent improvements in turbomachinery design requires the analysis of exceptional operating regime of bladed disks corresponding to geometrical nonlinear effects induced by the large displacements/deformations. In addition, the random nature of the mistuning has also to be modeled. First, a mean nonlinear reduced-order model of the tuned bladed disk is explicitly constructed in the context of the finite element method. The investigation is then devoted to the modeling of the mistuning through the nonparametric probabilistic approach extended to the nonlinear geometric context. The stochastic nonlinear equations are solved in the time domain using the Monte Carlo numerical simulation coupled with advanced arc-length methods adapted to high nonlinear response levels. Finally, the methodology is applied through a numerical example of a bladed disk and a nonlinear analysis is performed in both time and frequency domain.Copyright

Finite-Element Modelling of an Experimental Mistuned Bladed Disk and Experimental Validation

Integrally bladed rotors dynamic properties are known to be particularly sensitive to small geometric discrepancies due to the machining process or in-service wear. In this context, it is straightforward that setting up accurate numerical models which take into account real mistuning patterns is a key issue in the prediction of forced response amplitudes under operating conditions. The present study focuses on an experimental bladed disk. Due to strong inter-blade coupling, the geometric mistuning is supposed to result in severe mode localization for the studied bladed disk, thus emphasizing the need of a realistic, predictive finite-element model. This paper describes the procedure which leads to the development and validation of a high-fidelity FE model for a realistic bladed disk, based on coordinate measurements by means of fringe projection. After giving an overview of the coordinate measurement and model building for the studied bladed disk, the comparison of cantilevered-blade and full disk calculated eigenfrequencies to individual blade and full disk in quasi-vacuum measurements are presented.Copyright

Forced response prediction of a mistuned bladed disk in the presence of aeroelastic coupling and model uncertainties

The present study focuses on a sensitivity analysis of uncertain parameters for a mistuned bladed disk. After providing a general overview of the main theoretical aspects, this paper describes the way of computing the forced response with aeromechanical coupling. The bladed disk considered here is a representative naturally mistuned bladed disk whose geometry has been obtained by scan. A global sensitivity analysis is then performed using the Morris algorithm in order to rank a set of conservative and dissipative uncertain parameters in terms of their influence on the forced response level. Since a large number of response computations are needed for the sensitivity analysis, a reduced-order model that encompasses geometric mistuning is used.