# A Method to Minimize the Effort for Damper-Blade Matching Demonstrated on Two Blade Sizes
## 2 Method
### A Worked-Out Example of Application  
Through the example illustrated in this section, we can understand the reason why the proposed method is effective in offering the designer indications regarding the most important trends and relationships between the design parameters and the performance of the system.  
It should be noted that the validity of the proposed method is independent of the chosen blade-damper pair used as a test case. Further details on the test case used to produce the results shown in Figures 3b, 4 and 5 and Table 1 can be found in Section 3 (see Blade A).  
As shown in Figure 3b, the nonlinear frequency response function changes the location and value of its maximum depending on the dynamic excitation here represented by the \"proof\" force \(F_{E}\). Providing an estimate of \(F_{E}\) is not an easy task. If experimental data are available, methodologies typical of output-only methods [26], such as operational modal analysis may be used to estimate the excitation levels starting from the recorded mode shape. However, predicting the actual operating conditions and linking them to the excitation level during the early design stage may not be possible. It is therefore convenient to monitor, at the present stage, blade-damper configurations as a function of the \(F_{E}\) level and to rank them according to the maximum \(F_{E}\) they can withstand before reaching their HCF limit.  
The left-hand portion of Figure 4 shows the results of the forced response in terms of maximum bending stress of the blade. The forced response is run for different excitation levels \(F_{E}\) ranging from 0.1 N to 6 N. It is convenient to condense the information on the location of the peak of the different responses (see black points in Figure 4) in terms of frequency and stress amplitude. The right-hand portion of Figure 4 tracks the stress amplitude of the peak as a function of the excitation level \(F_{E}\).  
Figure 5a shows why it is convenient to perform the same \"tracking\" procedure for an extended range of \(F_{E}\) values. Figure 5b adds the frequency for the FRF maximum. The curve in Figure 5a can be split into three regimes:  
1. Stick/Microslip: the resonance frequency is close to \(f_{stick}\), the (low) hysteretic damping prevails and the stress levels are growing \(\approx\) linearly with \(F_{E}\).
2. Gross Slip: the resonance frequency is lower than \(f_{stick}\) but still higher than \(f_{free}\). This is the preferred working condition, dominated by friction and not by hysteretic damping. The stress levels still increase with a constant rate; however, the slope of the \(F_{E}-\sigma_{max}\) curve is significantly lower if compared with the one detected at regime 1.
3. Advanced gross slip: the resonance frequency is close to \(f_{free}\) as the damper slides freely between the platforms for the majority of the period of vibration. At the level of a single contact point, it can be observed that the distance traveled by the damper  
Figure 4: (**Left**) Maximum bending stress on the blade (at airfoil root) as a function of frequency. Different excitation levels are explored \(|F_{E}|=[0.1,6]\) N. (**Right**) Maximum bending stress vs. \(F_{E}\).  
with respect to the platform in stick condition is negligible (i.e., lower by more than one order of magnitude) with respect to the distance traveled in slip. In this condition, friction becomes incapable of dissipating the input energy, as already observed by [27]. This regime is once again dominated by hysteretic damping, which is typically low. As a result, the slope of the \(F_{E}-\sigma_{max}\) starts increasing exponentially. It can be argued that the higher the value of this \(F_{E}\) limit value, the better the matching between blade and damper.