## 3 Influence of Key Parameters on the Response
### Craig-Bampton Reduction: Influence of the Retained Contact Nodes
#### 3.3.1 Density of Nodes Incremented Along the Strip Long-Edge Direction X  
This section aims at investigating the two questions outlined below:  
1. Does the method (or pattern) the designer uses to refine its contact node selection affect results?
2. Why does the frequency decrease if the contact nodes density decreases?  
To answer question (1), the density of the retained contact nodes along the strip long-edge direction (i.e., node columns along the X axis in Figure 8a) is varied using different \"patterns\". Figure 8a shows patterns A and B. For a given pattern, nodes' density along the X direction is progressively doubled (for each increment the distance between neighboring nodes along the X direction is halved), while node rows along Y direction are kept fixed (in the present case to 3). Five different patterns have been investigated under low excitations (i.e., full stick case), and they are summarized in Table 2.  
Figure 8b plots the frequency variation as a function of the number of node columns for the patterns in Table 2. The results shown in Figure 8b confirm those in Figure 6b: by increasing the number of node columns the natural frequency of the system increases with a decreasing growth rate (i.e., moves closer to the asymptotic region). Conversely, for a low number of node columns the frequency plunges to low values (down to 92% of the asymptotic value).  
The position of the nodes in different patterns (A...E) has a negligible impact on the resulting full-stick frequency, especially if compared to other sources of uncertainty (e.g., contact parameter values, number of retained contact nodes etc.).  
Figure 7: Top blade-strip contact patch view: high mobility region (red) with the largest relative displacement.  
To answer question (2), the constraint level of the _high mobility region_ shown in Figure 7 is considered. It is observed that only the \"lower\" node rows intercept the high mobility region which propagates along the long-strip edge. If the number of constrained nodes belonging to the _high mobility region_ increases, the relative displacement of each node is reduced, as the mobility of each node is progressively impaired by the neighboring (constrained) points. This overall mobility reduction leads to a system stiffening. Such a concept is depicted in Figure 9.  
Lastly, it should be noted that with increasing node density the local contact stiffness is reduced (see Section 2.1), however the overall stiffness connected to the high mobility region (i.e., the sum of the local stiffnesses along the lower node rows) is not modified.  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Patterns** & \multicolumn{3}{c}{**Number of Node Columns (Corresponding Node Density [nodes/cm])**} \\ \hline
**A** & 3 (0.3) & 5 (0.5) & 9 (0.9) & 17 (1.7) & 33 (3.3) & 65 (6.5) \\
**B** & 4 (0.4) & 7 (0.7) & 13 (1.3) & 25 (2.5) & 49 (4.9) & - \\
**C** & 6 (0.6) & 11 (1.1) & 21 (2.1) & 41 (4.1) & - & - \\
**D** & 8 (0.8) & 15 (1.5) & 29 (2.9) & 57 (5.7) & - & - \\
**E** & 10 (1.0) & 19 (1.9) & 37 (3.7) & 73 (7.3) & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 2: Pattern of nodes retained during the reduction procedure.  
Figure 8: (**a**) Blade-strip contact patch discretization. (**b**) Frequency variation produced by variable number of master contact nodes.  
Figure 9: Effects of retained nodes on the model stiffness.