# A Method to Minimize the Effort for Damper-Blade Matching Demonstrated on Two Blade Sizes
## 2 Method
### Proof of Accuracy  
It should be noted that, regardless of the chosen number of contact points (reduced to the strict minimum compatible with bending an torsion in the example of Figure 1 even if there is no theoretical limit to their number), the maximum size of the nonlinear problem to be solved never exceeds the value 6, i.e., the number of rigid body DoFs at point P. Nonetheless, this platform-centered reduction is performed without any significant sacrifice in the accuracy of blade/damper dynamics and stress at those critical points that have bearing on fatigue behavior because the platform effectively behaves, to a high degree, as a rigid body.  
The statement above has been proven by comparing the results of the PCR model to that of the full FE reference model shown in Figure 2. Further details on Blade A chosen as  
Figure 2: (Left) Assembling procedure to build the PCR model. (Right) Full FE (ANSYS) model: used for tuning the PCR model and for a final linearized response check. Fixed boundary conditions are applied at the connecting end of both models.  
a test case are given in Section 3. The full FE-PCR comparison has been performed both on linear (i.e., no damper) and nonlinear cases. Table 1 gives a summary of the assessment of equivalence.  
The full FE model allows displacements (static and modal) to be determined together with stresses at any location. The PCR model allows displacements to be calculated at any point which was predetermined on the airfoil (in our case, points E,A,B,C are the essential ones) plus displacements and rotations at beam-end points. While knowledge of cross-sectional moments at point R produces nominal stresses including \(\sigma_{max}\) at the airfoil root (see Figure 3a), which is the maximum stress location for the first mode of vibration with and without damper. For the sake of the comparison shown in Table 1, radial stresses are taken 1 mm above the airfoil-platform junction to avoid the influence of the notch effect dependent on the local fillet radius, a factor to be determined separately.  
Finally, the equivalence between PCR and full FE has been further confirmed by comparing the results of the PCR model to those of the full FE model, shown in Figure 3b for two representative nonlinear response cases, i.e., under mid-length periodic excitation force \(F_{E}\) having amplitude 4 and 5 N.  
\begin{table}
\begin{tabular}{c c c} \hline
**Equivalence Indicator** & **Symbol** & **Full FE** & **PCR** \\ \hline (dyn.) Resonance Frequency & \(f_{1}\) (Hz) & 381.2 & 381.3 \\ \hline \(u_{A}/u_{E}\) & 2.797 & 2.792 \\ \hline \(u_{B}/u_{E}\) & 3.065 & 3.061 \\ (dyn.) Mode shape & \(u_{C}/u_{E}\) & 2.948 & 2.943 \\ \hline \(\alpha_{P}/u_{E}\) (rad/m) & 8.98\(\cdot\)10\({}^{-3}\) & 8.67\(\cdot\)10\({}^{-3}\) \\ \hline \(\gamma_{P}/u_{E}\) (rad/m) & 2.407 & 2.402 \\ \hline (static) Platform rotation & \(\gamma_{P}\) (rad) & 4.148\(\cdot\)10\({}^{-6}\) & 4.146\(\cdot\)10\({}^{-6}\) \\ \hline (static) Airfoil tip displacement at point C & \(u_{C}\) (m) & 3.45\(\cdot\)10\({}^{-6}\) & 3.46\(\cdot\)10\({}^{-6}\) \\ \hline (static) Max. stress at airfoil “root” & \(\sigma_{max}\) (MPa) & 2.92\(\cdot\)10\({}^{-7}\) & 2.69\(\cdot\)10\({}^{-7}\) \\ \hline \end{tabular}
\end{table}
Table 1: Comparison between the Full FE and the PCR freestanding blade models for the smaller Blade A (see Section 3). Static analysis results refer to a unit force \(F_{E}\) applied at point E along the \(x\)-axis. Modal analysis results refer to the first bending mode.  
Figure 3: Full FE model vs. PCR model, Blade A. (**a**) Comparison of stresses at airfoil root; (**b**) comparison of nonlinear forced response results (tip trailing edge, point B) for increasing values of the excitation force \(|F_{E}|\). The response is here normalized by the excitation force magnitude \(|F_{E}|\).