A sparse preconditioned iterative method for vibration analysis of geometrically mistuned bladed disks

In this paper, the Static Mode Compensation method to predict geometrical mistuning effects on the response of bladed disks is reviewed and its limitations are analyzed. This method proved to be effective only for narrow clusters of modes under localized low rank perturbation. A new algorithm is introduced to address its deficiencies that draws on optimal preconditioned methods for generalized eigenvalue problem featuring sparse matrix vector multiplications, being more attractive under limited memory constraints of multi-millon DOF FEM models. The central idea of the SMC is to correct nominal eigenspace using modal acceleration method. It has been extended here by replacing the quasi-static set of modes with inexact solutions of the linear Jacobi-Davidson correction equations. Some heuristic strategies are discussed to lower the computational effort given the block-circulant structure of the nominal system. Numerical experiments on an industrial scale bladed disk model show that this leads to a very competitive tool. Computational performance and accuracy of both methods is compared in two areas of spectrum. The study demonstrates low accuracy of SMC method in the modal interaction zone, while validating efficiency and accuracy of the new algorithm in both areas.

[1]  Gerard L. G. Sleijpen,et al.  Jacobi-Davidson Style QR and QZ Algorithms for the Reduction of Matrix Pencils , 1998, SIAM J. Sci. Comput..

[2]  Alok Sinha.  Reduced-Order Model of a Bladed Rotor With Geometric Mistuning , 2009 .

[3]  Y. Saad,et al.  Experimental study of ILU preconditioners for indefinite matrices , 1997 .

[4]  Christophe Pierre,et al.  Compact, Generalized Component Mode Mistuning Representation for Modeling Bladed Disk Vibration , 2003 .

[5]  Jonathan A. Wickert,et al.  Spatial Modulation of Repeated Vibration Modes in Rotationally Periodic Structures , 2000 .

[6]  Christophe Pierre,et al.  A reduced-order modeling technique for mistuned bladed disks , 1994 .

[7]  G. Golub,et al.  Subspace iterative methods for eigenvalue problems , 1999 .

[8]  Marcus J. Grote,et al.  Parallel Preconditioning with Sparse Approximate Inverses , 1997, SIAM J. Sci. Comput..

[9]  Jerry H. Griffin,et al.  A Reduced-Order Model of Mistuning Using a Subset of Nominal System Modes , 2001 .

[10]  J. Griffin,et al.  A Normalized Modal Eigenvalue Approach for Resolving Modal Interaction , 1996 .

[11]  Christophe Pierre,et al.  A reduced-order modeling technique for mistuned bladed disks , 1994 .

[12]  Andreas Stathopoulos,et al.  Nearly Optimal Preconditioned Methods for Hermitian Eigenproblems under Limited Memory. Part I: Seeking One Eigenvalue , 2007, SIAM J. Sci. Comput..

[13]  H. V. D. Vorst,et al.  EFFICIENT EXPANSION OF SUBSPACES IN THE JACOBI-DAVIDSON METHOD FOR STANDARD AND GENERALIZED EIGENPROBLEMS , 1998 .

[14]  Christophe Pierre,et al.  Dynamic Response Predictions for a Mistuned Industrial Turbomachinery Rotor Using Reduced-Order Modeling , 2002 .

[15]  Mathias Legrand,et al.  A reduction technique for mistuned bladed disks with superposition of large geometric mistuning and small model uncertainties , 2009 .

[16]  A. Sameh,et al.  The trace minimization method for the symmetric generalized eigenvalue problem , 2000 .

[17]  H. V. D. Vorst,et al.  Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems , 1995 .

[18]  O. Bendiksen.  Localization phenomena in structural dynamics , 2000 .

[19]  D. J. Ewins,et al.  A new method for dynamic analysis of mistuned bladed disks based on the exact relationship between tuned and mistuned systems , 2002 .

[20]  Christophe Pierre,et al.  Vibration Modeling of Bladed Disks Subject to Geometric Mistuning and Design Changes , 2004 .

[21]  Alok Sinha.  Reduced-Order Model of a Bladed Rotor With Geometric Mistuning , 2007 .

[22]  Francois Maurice Garcin,et al.  Mistuning Phenomena on Bladed Disk: Industrial Methods and Applications , 2006 .