Bah, M.T.
(2002)
Reduced order modelling for efficient prediction of the dynamics of mistuned bladed disks.
*University of Southampton, School of Engineering Sciences, Doctoral Thesis*.

## Abstract

The unavoidable existence of small differences between nominally identical sectors of bladed disks, called mistuning, can have an important impact on the dynamic behaviour of these structures. In particular, this has the potential to lead to responses qualitatively different from that of an ideal cyclic, tuned structure and, in turn, to significantly shorter life spans. Study of mistuning as a random phenomenon requires statistical analysis of a large number of mistuning patterns. This computational task is expensive especially when high-fidelity finite element models are used. This research is concerned with the development of reduced order computational modelling techniques for the dynamic analysis of mistuned bladed disks. These techniques combine accuracy and computational efficiency for a reliable statistical assessment of the effects of mistuning on the dynamics of such systems. For free vibration, the nominal assessment of the effects of mistuning on the dynamics of such systems. For free vibration, the nominal periodicity is exploited, leading to an approximation that greatly reduces the order of the original model. The natural frequencies and mode shapes for a passband are found by treating the unknown complex amplitudes between the nominally identical sectors as the generalized co-ordinates of the problem. In spite of a very large reduction in the computational effort, the results obtained are very accurate both for frequencies and mode shapes even when strong mode localization is observed. To test the perfor4mance of the proposed approximation further, a situation where two passbands are brought close to each other is also considered. This method is general in its formulation and has the potential of being used for complex geometries. It is also extended to the frequency response problem. The great advantage is that the statistics of ‘blades’ forced responses can be numerically generated at a cost of a single degree-of-freedom per blade/disk sector model. Furthermore, a stochastic reduced basis approach is developed for the approximation of these statistics. This approach allows for a complete stochastic analysis of the effects of mistuning. The system response is represented using a linear combination of complex stochastic basis vectors with undetermined coefficients. The terms of the preconditioned stochastic Krylov subspace are used as basis vectors. Two variants of the stochastic Bubnov-Galerkin scheme are employed for computing the undetermined terms in the reduced basis representation. Explicit expressions for the response quantities are then derived in terms of the system random parameters. This allows for the possibility of efficiently computing the response statistics in the post-processing stage. This novel approach can be applied either on the original model in the physical domain or on a reduced model in the modal domain as a secondary reduction technique. The accuracy of the response statistical moments computed using this approach can be orders of magnitude better than classical perturbation methods. Finally, component mode synthesis or substructuring and probabilistic methods are combined to generate reduced order models. The Craig-Bampton reduction procedure is applied while using stochastic component modes instead of deterministic modes. An additional calculation of sensitivities of fixed interface modes, constraint modes and substructure-matrices is required with respect to the physical random variables. In the case of turbomachinery mistuned bladed disks composed of nearly identical substructures, the sensitivity analysis can be targeted to only one substructure. One great advantage is that the physical variations can be used as input in the reduced order model. This novel approach allows for an efficient computation of the statistical characteristics of responses and a complete stochastic analysis of the effects of mistuning.

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