# A Comparison between Two Reduction Strategies for Shrouded Bladed Disks
## 1 Introduction  
Bladed disks (Figure 1) in gas turbine engines are interested by severe vibration problems, mainly caused by the non-stationary gas flow pressure [1], which can result in high cycle fatigue damage and failure. If resonances cannot be avoided, as in many practical cases due to the high modal density of bladed disks in certain frequency ranges, a forced response analysis is necessary to accurately assess the vibration levels [2]. Due to the presence of multiple contact surfaces in bladed disks (e.g., shrouds [3; 4], underplatform dampers [5; 6; 7], blade roots [8; 9], ring dampers [10; 11], inter-stage coupling [12]), nonlinear analyses must be performed in order to take into account the friction damping due to slip phenomena at the contact interfaces [13; 14] and predict accurately the response level. Under periodic excitation, which actually occurs in steady-state regimes, the state-of-the-art approach relies on the solution of the nonlinear governing equations in the frequency domain by means of the Harmonic Balance Method (HBM) [15], which allows huge time savings with respect to the Time Domain Analysis (TDA). Although bladed disks are supposed to be made of a set of identical sectors [16], the actual assemblies are never symmetric and small differences, called mistuning in the literature [17], always exist. Furthermore, even in case of a perfectly symmetric system, localization phenomena of the response might occur in the presence of either geometric nonlinearities [18] or scatter of the contact interface parameters [19]. Thus, on one side, for an exhaustive analysis of the dynamic response of bladed disks, it is necessary to study the full bladed disk, while, on the other side, the highly refined meshes typical of industrial bladed disks make the use of full Finite Element (FE) models unfeasible. The solution of this dilemma comes from the properties of cyclically symmetric structures, recently reviewed in [20]. Given the set of stiffness and mass matrices of the fundamental sector, obtained by applying all the allowable cyclic symmetry boundary conditions, it is possible to obtain the stiffness and mass matrices of the full system through simple algebraic operations. In the end, it is possible to obtain the model of the full bladed disk without actually modelling it, using as a fundamental brick the stiffness and mass matrices of the fundamental sector. Before continuing, it is worth mentioning that, although HBM allows for reducing the computational effort, the nonlinear forced response of a full bladed disk is still a formidable task. For this reason, reduced order models are commonly used. In case of localized nonlinearities, such as the contact interfaces, a typical reduction strategy is based on the Craig-Bampton (CB) method [21], originally developed for substructuring purposes. According to the CB method, a set of master degrees of freedom (dofs) is retained in the analysis, while all the other dofs are replaced by a set of normal modes with specific boundary conditions and, in the case of systems with contact interfaces, the set of master dofs must include those that lie on the contact surfaces, whose amplitude determines the contact forces that produce the nonlinear friction damping. In detail, in this paper, two different CB reduction strategies are described and compared. In both cases, the result of the process is a set of CB-reduced stiffness and mass matrices of the fundamental sector with cyclic symmetry boundary conditions. What is different in the two approaches is the order in which cyclic symmetry and CB reduction are applied to the fundamental sector of the bladed disk and it will be shown that the chosen sequence of operations can significantly affect the computation time necessary to obtain the ROMs.