# A Method to Minimize the Effort for Damper-Blade Matching Demonstrated on Two Blade Sizes
## 4 Representation and Interpretation of Results  
This section applies the PCR method described in Section 2 to illustrate its results in the case of two damper-blade combinations characterized by very different dimensions, and how their suggested graphical representation effectively supports the analysis (and helps to draw conclusions) about blade-damper coupling. To this purpose:  
* the neck length \(L_{n}\) is assigned the values (20,30) mm to explore the influence of neck flexibility, which bears on the blade mode shape; it is worth noting that the stiffness and mass matrices of the PCR blade are parametrically linked to \(L_{n}\), and no re-meshing procedure is required;
* the centrifugal load on the damper is assigned its nominal value \(CF^{*}\) and \(0.5\cdot CF^{*}\); the 0.5 scaling factor is compatible with a realistic 50% reduction of the mass of the damper (e.g., by removing the lower portion not in contact with the blades) or with a 30% reduction of the disk rotational speed. The evaluation of the best match between the different possible combinations for the two blade sizes is based on the method that leads to Figure 5a and its interpretation. Preliminarily, it should be noticed that the diverging trend followed by the curves as they approach the respective \(F_{lim}\) asymptote is obtained here by calculating the nonlinear response for \(F_{E}=0.9F_{lim}\). However, the corresponding points are not visible in Figures 10 and 11 as the stress level is well above the 150 MPa value chosen for the representation. The cross-comparison of Figures 10 and 11 allows us to put into evidence what follows. * For both blades A and B, a longer, more flexible, neck is always beneficial: the limit \"proof\" excitation force \(F_{lim}\) is higher and the maximum fatigue bending stresses \(\sigma_{max}\) are lower than in the case of a shorter neck through the whole \(F_{E}\) span. This is mainly due to the fact that the ratio \(\gamma_{P,free}/u_{E,free}\) is larger (i.e., the platform rotation is larger for the same displacement at the excitation point) when the neck is more flexible. In the present case, the values of this ratio range from 2.5 to 3.5 for blade A, from 0.8 to 1.3 for blade B. It must be remembered that the damper's capability for In Phase blade vibration increases with the amount of rotation of the platform. * The consequence of reducing the \(CF\) value looks more complex, due to the curve crossing, but is readily justified. In effect, since in full sliding all the forces are proportional to the value of \(CF\), so are those of the frictional moment \(\Delta M_{P}\) and of \(F_{lim}\). Thus, with a reduced \(CF\), the asymptote is reached earlier. However, on the other hand, regardless of \(CF\), the initial full stick section on the curve is mostly determined by the stick constraint between damper and platform, which is independent on neck length. Since the first slip initiation in one of the contacts occurs for a lower contact \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Blade A** & **Blade B** \\ \hline \(L_{n}\) & 20 mm & 20 mm \\ \hline \(L_{p}\) & 10 mm & 15 mm \\ \hline \(L_{a}\) & 120 mm & 400 mm \\ \hline \(h_{n}-\) & - & 25 mm \\ \hline \(h_{n}+\) & - & 35 mm \\ \hline \(h_{n}\) & 8 mm & - \\ \hline \(b\) & 45 mm & 180 mm \\ \hline \(CF^{*}\) & 900 N & 4000 N \\ \hline \(f_{1,free}\) & 381 Hz & 211 Hz \\ \hline \(u_{E,free}/\gamma_{P,free}\) & 0.4 m/rad & 1.2 m/rad \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Relevant geometrical parameters and characteristics of Blades A and B. \(CF^{*}\) is the nominal centrifugal force on the damper (rounded to the next hundred) based on information on damper mass, disk radius, and rotational speed (not shown). force, the curve for smaller \(CF\) breaks off from full stick at a lower \(F_{E}\) value than that for higher \(CF\). This combination necessarily produces the intersection of curves.