108 F. Schreyer et al. The assembly stiffness matrix Kcms is defined accordingly. The size of the system is still dominated by the large number of cyclic interface DOFs, which can be further reduced by an interface reduction. The modal reduction basis of the interface is obtained by solving the eigenvalue problem of the interface partitions from Eq. (9.12) QKred;ii !2 ii QMred;ii Q ii D0 (9.13) The eigenvectors of the interface Q ii span the same subspace as the constraint modes of the global interface between disk and blades, which can be truncated by keeping only the eigenvectors associated with the lowest k eigenfrequencies!k;ii. The transformation for the new set coordinates can be formulated as 2 66 4 Qpd f Qxi Npb f Nxb r 3 77 5 D 2 66 4 I 0 0 0 0 Q k;ii 0 0 0 0 I 0 0 0 0 I 3 77 5 „ ƒ‚ … i 2 66 4 Qpd f Qpi Npb f Nxb r 3 77 5 ; (9.14) and the assembled system matrices can be rewritten as Mcms;red D Ti Mcms i; Kcms;red D Ti Kcms i; (9.15) where i represents the interface reduction matrix. The assembled system can be further reduced by a final Craig-Bampton reduction of the system matrices, substituting the individual blade and disk modes by a set of global modes. The shroudcoupling DOFs are kept in the physical domain. 9.3 Mistuning The perfect cyclic symmetry of a bladed disk is destroyed, when mistuning is present. Especially blades are highly affected by variations of the manufacturing process as well as operational wear which leads to significant disturbance in the spread of vibrational energy in the system. Assuming only a variance of the Young’s modulus, a nondimensional mistuning parameter can be defined as ın D !2 n;mist !2 nom !2 nom : (9.16) The nominal eigenfrequency of the tuned blade is denoted by!2 nom, and!2 n;mist describes the mistuned eigenfrequency of the n-th blade. Furthermore, proportional mistuning is assumed, that means the percentage deviation of natural frequencies is the same for all modes, [4]. The mistuning deviations are small compared to nominal properties in the modal domain, i.e., jıj <<1, and it is hypothesized that the mode shapes are the same for a mistuned blade as for the tuned one. Note, that for a validation the mistuning parameters are preferably obtained from measured cantilevered modes. However, when sliding condition is applied, the shroud constraint modes also need to be mistuned to describe the motion of the blade more accurately, [4]. Therefore, the mistuning projection requires two sets of modes. On the one hand Ub shr, including the fixed interface modes b in combination with the shroud constraint modes, which were already obtained in Eq. (9.3). On the other hand the fixed-interface normal modes b cnt, which in contrast to b are obtained by neglecting the shroud coupling. The b cnt are associated with the mistuning parameters ın and therefore are used to transfer the mistuned stiffness matrix into physical domain. The modal stiffness deviation matrix Kb n;mist is obtained by adjacent projecting onU b shr, which leads to Kb mist;n DU b shr T Mb b cntın b cnt T Kb red;nU b shr; (9.17) with Ub shr D b Kb ffKb fishr 0 I ; b cnt D cnt 0 : (9.18)
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