# A Method to Minimize the Effort for Damper-Blade Matching Demonstrated on Two Blade Sizes
## 5 Platform-Centered Reduction into the Larger Picture  
The results presented in Sections 2 and 4 prove that concentrating all nonlinearities in the rigid body DOFs of the platform point P and of the damper center of mass D is an accurate and effective modeling choice. Namely, it draws attention to essential features of the blade-damper dynamic system such as the existence of a limit excitation force (Section 2.4) and the interaction between the most important design parameters and the resulting stress levels. The purpose of this section is to highlight the advantages of the PCR in terms of computational effort. As shown in Figure 12, this modeling choice produces a set of 12 nonlinear equations concentrated at points P and D, regardless of the chosen number of contact points, which cease to be nonlinear degrees of freedom. The PCR method has shown that, for applications where contact occurs between structures or portions of structures behaving as rigid bodies, it is possible to drop the proportionality between the number of contact points and the size of the system. This challenging task has been undertaken in the past with numerical reduction techniques [20; 29; 30] based on purposely developed projection bases. In this context, PCR configures as an alternative in which the limit on the number of contact points is inherently overcome. Another significant advantage of the PCR is that the <<time to result>> is reduced both by the size of the system and by the fact that a smaller system needs fewer iterations to reach convergence. Nevertheless, the PCR should not be intended as a substitute for a full nonlinear FE calculation, when it is a must to take into account the complete geometry of the bladder disk to offer a complete stress map at all points on the blade. A solution to the need \begin{table}
\begin{tabular}{l l l} \hline \hline  & **Blade A** & **Blade B** & **Comments** \\ \hline Material & René 41 & René 80 \\ \hline Airfoil mass & 0.12 kg & 4.17 kg \\ \hline Airfoil root area & 119 mm\({}^{2}\) & 2707 mm\({}^{2}\) \\ \hline Mean radius & 320 mm & 530 mm \\ \hline Mean blade speed & 198 m/s & 198 m/s \\ \hline Rotation speed & 6000 rpm & 3600 rpm \\ \hline Temperature & 650 \({}^{\circ}\)C & 450 \({}^{\circ}\)C \\ \hline Number of blades & 44 & 33 \\ \hline Root chord & 40 mm & 160 mm \\ \hline Mean pitch & 45 mm & 110 mm \\ \hline Blade circumf. force & 60 N & 490 N \\ \hline Blade bending force & 73 N & 590 N \\ \hline
1\% bend. force amplitude & 0.73 N & 5.9 N & Alternating comp. \\ \hline
5\% bend. force amplitude & 3.5 N & 30 N & Alternating comp. \\ \hline Mass radius & 296 mm & 454 mm \\ \hline Centrifugal force & 14 kN & 27 kN \\ \hline Nominal root stress \(\sigma_{m}\) & 120 MPa & 100 MPa & Mean on section \\ \hline Ultimate stress \(\sigma_{u}\) & 1060 MPa & 940 MPa & At temperature \\ \hline Yield stress \(\sigma_{y}\) & 810 MPa & 680 MPa & At temperature \\ \hline
10\({}^{6}\) cycl. fatigue limit \(\sigma_{a-1}\) & 205 MPa & 185 MPa & Alternating stress \\ \hline Stress inters. factor Kf \(\approx\) & 1.5 & 1.5 & At airfoil root fillet \\ \hline
10\({}^{6}\) cycl. fatigue limit \(\sigma_{a}\) & 185 MPa & 165 MPa & At mean root stress \\ \hline Allow bend stress \(\sigma_{a,bend}\) & 90 MPa & 80 MPa & Nominal at root \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Data relevant to the estimate of fatigue limits of blades A and B.  
of exploring different design configurations, while, at the same time, running a full FE nonlinear analysis, has been met by the use of a reduced surrogate model trained on data obtained through a Box-Behnken sampling of the design space [7]. As shown in Figure 13, we suggest here, instead, to use the PCR method to isolate a few optimized blade-damper configurations. These configurations will then be screened by a final full FE linear analysis. In fact, once contact forces have been calculated with the PCR, they can be fed ex-post into the full FE model excited with the corresponding force(s) and frequency.