## 3 Influence of Key Parameters on the Response
### Craig-Bampton Reduction: Influence of the Retained Contact Nodes  
After meshing the FE model, a CB-CMS reduction is performed. This reduction technique requires the selection of master (i.e., physical) and slave (i.e., modal) degrees of freedom. Most of the selected physical DOFs are located on the blade-strip contact area: this selection is particularly relevant as only the retained physical DOFs allow for the application of the contact model. Ensuring an adequate representation of the contact of a flexible structure typically requires a high number of contact nodes. If the selected contact nodes are too few the local contact stiffnesses \(k_{n}\), \(k_{tx}\) and \(k_{ty}\) will be unnaturally high (see Section 2.1) and the dynamic behavior of the strip will be modified. Consider, as an example, a strip constrained at its four corners by very stiff contact springs: the inner unconstrained portion of the blade-strip contact will be free to move and deform. Conversely if a large number of uniformly distributed contact elements is applied, the additional stiffness at each contact node will be smaller, and the strip/platform contact will be continuous. For this reason, the present investigation takes as a benchmark choice a uniform fine discretization, under the hypothesis of non-deformed conforming contact surfaces. In detail each blade-strip contact patch (10 \(\times\) 100 mm) is represented using a density of 11 contact nodes/cm along the short edge dimension and 6.5 contact nodes/cm along the long-edge dimension (i.e., 11 rows and 65 columns of contact nodes respectively, for a total of 715 contact nodes, as summarized in Table 1). The concept of contact patch discretization into rows and columns of nodes is depicted in Figure 6: rows refer to contact nodes density along Y direction, while columns refer to nodes' density along X direction. In all cases, the distribution of contact nodes is uniform. As discussed in Section 2.1 efforts were made to maintain the global contact stiffness equal in all investigated cases even if the number of contact elements and the pattern varies. Although the global stiffness is kept constant, the natural frequency of the predicted forced response varies considerably, as shown in Figure 6. As discussed above, this influence is investigated in the full stick regime as contact springs are fully active during a complete period of vibration. A careful observation of Figure 6 allows to draw the following caveats:  
* If a sufficiently high contact nodes density is adopted (i.e., 4.5 node columns per cm and 7 node rows per cm respectively) the predicted frequency reaches an asymptotic value. * If the number of node rows decreases below its threshold value, the frequency starts increasing. * If the number of node columns decreases below its threshold value, the frequency starts decreasing. It can, therefore, be concluded that using a rough discretization of the contact patch may lead both to an underestimation or an overestimation of the frequency shift. Furthermore, the trend may change  
Figure 6: (**a**) Blade-strip contact patch discretization. (**b**) Frequency variation produced by variable number of master contact nodes. depending on the specific mode under investigation. It is, therefore, worth investigating the reason behind this trend in order to provide the designer with a general \"rule of conduct\", independent of the specific mode under investigation. The strong influence of the pattern and number of retained contact nodes is due mainly to the kinematics of the blade-strip interfaces. In detail, the numerical analysis performed in the \"asymptotic region\" (sufficient number of node rows and columns) has shown that, for a given mode, there are regions of the blade-strip contact patches where relative displacement is maximum. These regions are here termed _high mobility regions_. An example of _high mobility region_ is shown in Figure 7 for the reference bending mode (i.e., neck-bending).