# A Method to Minimize the Effort for Damper-Blade Matching Demonstrated on Two Blade Sizes
## 4 Representation and Interpretation of Results
A first criterion for judging the effectiveness of the damper in relation to the blade is certainly the \(F_{lim}\) limit value of the \"proof\" excitation force, which must be kept away from danger through a safety factor. In Figures 10 and 11, allowable limits \(F_{all}\) are tentatively set at 50% of \(F_{lim}\). They are based on the observation that in the curves for the lowest values of \(CF=0.5 CF^{*}\), and \(0.5F_{lim}\) corresponds to the beginning of the steeper asymptotic slope. A caveat to avoid misunderstandings: due to this choice, the \(F_{all}=0.5F_{lim}\) for \(CF=CF^{*}\) happens to coincide with the \(F_{lim}\) for the \(CF=0.5 CF^{*}\). However, second criterion, it is also necessary to judge the exciting forces by the alternating fatigue stresses they produce. Figures 10 and 11 show on the ordinates a maximum stress information (dotted line), which is a key parameter, being directly linked to the HCF safety of the blade. The procedure used to estimate the allowable nominal bending stress \(\sigma_{a,bend}\) is based on the airfoil data shown in Table 3 and is thoroughly described in the following paragraphs. Figure 11: Blade B. Maximum “nominal” bending stress vs. \(F_{E}\) for : **(a)** the shorter neck \(L_{n}\) = 20 mm, full and half \(CF\) values; **(b)** the longer neck \(L_{n}\) = 30 mm, full and half \(CF\) values. Shaded areas represent the maximum assumed span of the proof excitation force \(F_{E}\). The horizontal dashed line represents the allowable nominal bending stress \(\sigma_{a,bend}\). Figure 10: Blade A. Maximum “nominal” bending stress vs. \(F_{E}\) for: **(a)** the shorter neck \(L_{n}\) = 20 mm, full and half \(CF\) values, **(b)** the longer neck \(L_{n}\) = 30 mm, full and half \(CF\) values. Shaded areas represent the maximum assumed span of the proof excitation force \(F_{E}\). The horizontal dashed line represents the allowable nominal bending stress \(\sigma_{a,bend}\). Table 3 summarizes the relevant data for both blades. In the same line adopted in [24; 25], the variable part of the excitation force due to flow irregularities is a sinusoidal component whose amplitude is estimated within the limits of 1% to 5% of the nominal aerodynamic force on the blade. Blade A, already presented in [24; 25], has a circumferential force at 60 N and a bending component at 73 N; hence, the two limits for the time-variable bending force amplitude are 0.7 and 3.5 N, respectively. Blade B, scaled for the same velocity triangles, has a circumferential force at 490 N, a bending component at 590 N, hence 1% to 5% limits for the bending force amplitude at 5.9 and 30 N, respectively. From data on Table 3, for Blade A--at an airfoil temperature of 650 \({}^{\circ}\)C--the 10\({}^{6}\) cycles alternating fatigue limit of Rene 41 is set at about 205 MPa, the constant mean stress is at \(\sigma_{m}=120\) MPa, which times the stress intensity factor \(K_{f}\) gives a constant maximum stress at the junction with the platform (due to the centrifugal force on the airfoil cross-section) at 180 MPa. According to the Goodman line on the Haigh diagram, the 10\({}^{6}\) alternating fatigue limit at that constant maximum stress is about 170 MPa, which divided times \(K_{f}\) and a 1.3 safety factor gives an allowable nominal bending stress rounded at \(\sigma_{a,bend}=90\) MPa. Similarly for Blade B--at an airfoil temperature of 450 \({}^{\circ}\)C--the 10\({}^{6}\) cycles alternating fatigue limit of Rene 80 is set at about 940 MPa, the constant mean stress is at \(\sigma_{m}=100\) MPa, which multiplied by the stress intensity factor \(K_{f}\) is 150 MPa and gives the constant maximum stress at the junction with the platform. According to the Goodman line on the Haigh diagram, the 10\({}^{6}\) alternating fatigue limit at that constant maximum stress is about 155 MPa, which divided times \(K_{f}\) and a 1.3 safety factor gives an allowable nominal bending stress at \(\sigma_{a,bend}=80\) MPa.