In this paper, a number of frequency-domain dynamic analysis procedures of randomly disordered structural systems in the medium frequency range are integrated into the stochastic finite element method. In all cases, frequencydomain model reduction strategies are used to minimize the computational effort in the mid-frequency range. Firstly, an energy operator approach (EOA) is investigated. In this procedure, an energy operator adapted to a fixed medium frequency band is defined whose dominant eigensubspace is used to construct a reduced model using a Ritz-Galerkin method. Secondly, the proper orthogonal decomposition method is used to extract the spatial dominant coherent structures in the vibration wave field in the mid-frequency band. The coherent structures are the eigenvectors corresponding to the dominant eigenvalues of the spatial autocorrelation function of the system response. Consequently, a close relationship between the energy operator approach and the proper orthogonal decomposition method is identified, although these two approaches are not mathematically identical. The proper orthogonal decomposition method appears to be more straightforward compared to the energy operator method from the viewpoint of numerical implementation. Thirdly, another approach, namely the dynamic element method based on frequency-dependent finite element shape functions, is considered. A stochastic reduction method is then utilized to represent the uncertain parameters, modelled as stochastic processes, in terms of their dominant scales of fluctuation. The Karhunen-Loeve and the polynomial chaos decompositions are used to that effect, in the context of a stochastic finite element formalism. The methodology adopted in the paper thus integrates efficient dynamical reduction techniques with a reduction scheme of stochastic processes for the mid-frequency vibration of linear random systems. The formulation is exemplified by its application to the analysis of the dynamics of a coupled uncertain rod assembly subjected to an external excitation. The example is also used to highlight some of the relative features of the three dynamic reduction.

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Journal Computer Methods in Applied Mechanics and Engineering
Sarkar, A, & Ghanem, R. (Roger). (2002). Mid-frequency structural dynamics with parameter uncertainty. Computer Methods in Applied Mechanics and Engineering, 191(47-48), 5499–5513. doi:10.1016/S0045-7825(02)00465-6