PSI - Issue 24

G. Battiato et al. / Procedia Structural Integrity 24 (2019) 837–851 G. Battiato et al. / Structural Integrity Procedia 00 (2019) 000–000

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3.2. Gram-Schmidt interface reduction method

On the fulfillment of the partitioning of Eqn. 12 the undamped EQM of the j th component can be written as:

M mm j M ms j M sm j M ss j

¨ x m j ¨ η s j

+

K mm j K ms j K sm j K ss j

x m j η s j

=

f m j φ s j

(17)

By solving the eigenproblem defined by the partitions K mm and M mm the full set of characteristic constraint modes is obtained (Castanier et al. (2001)). These can be arranged for increasing eigenvalues as the columns of the following modal matrix: Φ mm j = ϕ 1 j . . . ϕ m j (18) The modal matrix of Eqn. 18 allows for a coordinate transformation involving all the master DoFs. If a coordinate transformation is desired just for a subset of n i j physical DoFs, Φ mm j has to be partitioned as in the following equation: x m j = x i j x a j = Φ mm j η i j η a j = Φ ii j Φ ia j Φ ai j Φ aa j η i j η a j (19) where η i j and η a j are two arbitrary set of modal coordinates with size n i j and n k j respectively. Previous studies on the GSI method proved that performing a Gram-Schmidt orthonormalization (Strang (1993)) on the columns of Φ ii j produces a good basis for the modal representation of the physical partitions x i j . The size of x i j can be actually reduced if a modal truncation on the GSI modes is performed as suggested by the Eqn. 14. 3.2.1. Application The GSI method is here used to compute the non-linear forced response of a structure consisting of a stator vane segment connected to the corresponding casing sector . The assembly is characterized by two friction joints that exhibit a non-linear behavior: the interlocking and the hooks (Fig. 4).

Fig. 4. (a) real vane segment of a low pressure turbine module for aeronautical applications; (b) schematic view of a stator vane segment connected at the casing by means of hook joints.