Abstract
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.
Highlights
Bladed disk assemblies belong to a class of rotationally periodic systems in which cyclic symmetry is commonly exploited to predict vibrational response because it reduces the size of a problem to a single sector [2]
There are all indications that from the memory efficiency and accuracy point of view it is a good choice for the moderate order FEM models under low rank localized perturbation if narrow clustered areas of spectrum are analyzed
For very large scale industrial models as well as for the areas of spectrum where multiple mode families interact, a new method is proposed. It stems from the Jacobi-Davidson algorithm implementing a number of simple heuristic strategies based on the block-circulant structure of the nominal system and assumptions on perturbation
Summary
Bladed disk assemblies belong to a class of rotationally periodic systems in which cyclic symmetry is commonly exploited to predict vibrational response because it reduces the size of a problem to a single sector [2]. The SMC (Static Mode Compensation) method inspired by modal acceleration technique yields accurate approximates of perturbed eigenpairs under large geometric mistuning extracted from a low dimensional subspace. Should the linearization condition be satisfied, the precision of SMC is acceptable if we either select very narrow bands of nominal eigenpairs to correct or can provide a guess on the area of spectrum where the perturbed eigenvectors with larger angles to nominal eigenspace are most likely to occur, so that their corresponding eigenvalues would fall close enough to ω2c. Unlike Davidson method, its convergence is guaranteed whenever non-diagonal and non-positive definite preconditioners are used, which is often the case when we approximate the interior of the spectrum It avoids illconditioning of the linear correction equations when we cross the perturbed or nominal eigenvalues and we are not constrained with the choice of ωc.
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