PSI - Issue 24

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

847

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

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Fig. 7. (a) interface and active DoFs for a bladed disk; (b) inter-stage boundary of a cyclic symmetric stage. Sectors and radial line segments are denoted by i and k respectively.

The vector x i j can then be partitioned as follows:

x i j =    x R 1 j x R 2 j .. . x RNZ j

  

(20)

th radial line segment of the j th stage

where the subscript R stands for ”radial line segment”. The motion x Rk j of the k can be expressed by using the following relationship (Battiato et al. (2-2018)):

+

ϕ k ] +

˜ R − 1 h = 1

˜ R − 1 h = 1

2 NZ

2 NZ

1 √ NZ

1 √ NZ

( − 1) j − 1 a ˜ R

a 0 i j

a h i j , c cos[( j − 1)

a h i j , s sin[( j − 1)

x Rk j =

ϕ k ] +

(21)

where ϕ k = 2 π k / ( NZ ), a i j denote a vector of interface spatial Fourier coe ffi cients with subscripts c or s identifying the cosine and sine components respectively and ˜ R = NZ / 2 if NZ is even or ˜ R = ( NZ − 1) / 2 if NZ is odd. The last term Eqn. (21) does not exist if NZ is odd. If the spatial Fourier coe ffi cients are grouped as follows: a i j = ( a 0 i j ) T , ( a 1 i j , c ) T , ( a 2 i j , c ) T , . . . , ( a ˜ R − 1 i j , c ) T , ( a ˜ R i j , c ) T , ( a ˜ R − 1 i j , s ) T , . . . , ( a 2 i j , s ) T , ( a 1 i j , s ) T T (22) the interface DoFs can be written as:

x i j = ( F NZ , NZ ⊗ I n l ) a i j = F a i j

(23)