# A Method to Minimize the Effort for Damper-Blade Matching Demonstrated on Two Blade Sizes
## 2 Method
### A Priori Estimate of the Excitation Limit Force  
The combined analysis of Figure 5a,b reveals the presence of an upper limit value for the excitation force, here called \(F_{lim}\), for which the damper ceases to be effective and the blade frequency (and mode shape) tends to its free-standing value. Being able to estimate a priori the limit force \(F_{lim}\), without resorting to nonlinear calculations, is undoubtedly a huge asset for blade-damper designers. The concept of limit excitation force is not new. In 1930, [PERSON] [27] remarked that, in the case of a single mass-spring system (Figure 6a), when the ratio between the energy dissipated by a constant friction force \(F_{F}\) over a sliding range of \(2u\) (\(E_{out}=4\cdot F_{F}\cdot u\)) and the energy injected in the system by the external sinusoidal excitation \(E_{in}=\pi\cdot F_{E}\cdot u\) falls below one, i.e., \(F_{F}/F_{E}\leq\frac{\pi}{4}\), the amplitude of motion of the system will grow infinitely large (limited only by hysteretic damping if present).  
This limit is valid for any friction damped system; however, its expression needs to be corrected if multiple DoFs are present. Indeed, when \(f\to f_{free}\), the mode shape tends to that of the free standing blade. It is therefore possible to express the displacement \(u_{E}\) of point E, as a function of the rotation of the platform \(\gamma_{P}\) using the quantity \(u_{E,free}/\gamma_{P,free}\), readily available through the computation of the eigenvector of the free-standing blade corresponding to the mode of interest (the first in the present case). Therefore, when the system tends to the free vibration condition:  
\[E_{in}=\pi\cdot F_{E}\cdot u_{E}=\pi\cdot F_{lim}\cdot\gamma_{P} \frac{u_{E,true}}{\gamma_{P,true}} \tag{1}\] \[E_{out}=\eta\cdot 4\cdot\Delta M_{P}\cdot\gamma_{P} \tag{2}\]  
where:  
* \(E_{out}\) is the area within the \(\gamma_{P}-M_{P}\) hysteresis cycle shown in Figure 6b;
* \(\eta\) is an efficiency coefficient \(<1\) due to the fact that, unlike the Den Hartog case, the cycle area is reduced due to the non-infinite stiffness of the tangential contact; however, at resonance, the gross-slip part prevails and \(\eta\to 1\).  
Figure 5: (**a**) Maximum bending stress on the blade (at airfoil root) as a function of \(F_{E}\), same case as Figure 4 for extended \(|F_{E}|\) range; (**b**) resonance frequency as a function of \(F_{E}\).  
The moments of the contact forces about point P are, see Figure 7, \(M_{p+}\) during clockwise rotation \(\gamma_{p+}\) and \(M_{p-}\) during counter-clockwise rotation \(\gamma_{p-}\). However, what matters here is the amplitude \(2\Delta M_{P}=M_{p+}-M_{p-}\).  
It should be noted that the gross slip limits \(M_{p+}\) and \(M_{p-}\) are simple functions of the gross-slip contact forces, then of platform angles \(\theta_{L}\) and \(\theta_{R}\), the friction coefficient \(\mu\), the centrifugal load on the damper [23]. As an example, with reference to the simplified 2D representation shown in Figure 8 valid for a pure bending mode, one can express \(M_{p+}\) as:  
\[\begin{split} M_{P+}&=(|F_{L1}^{+}\cdot(x_{L1}-x_{P })|+|F_{L2}^{+}\cdot(x_{L2}-x_{P})|)\cos(\theta_{L}-\arctan(\mu))\\ &-|F_{R}^{+}\cdot(x_{R}-x_{P})|\cos(\theta_{R}+\arctan(\mu))\\ &+|F_{L1}^{+}\cdot(y_{L1}-y_{P})|\sin(\theta_{L}-\arctan(\mu))\\ &-|F_{L2}^{+}\cdot(y_{L2}-y_{P})|\sin(\theta_{L}-\arctan(\mu)) \end{split} \tag{3}\]  
Finally, \(F_{lim}\) can be written as:  
\[F_{lim}=\frac{4\Delta M_{P}}{\pi\frac{u_{L,true}}{T_{P,free}}} \tag{4}\]  
Through a very straightforward linear procedure, Equation (4) can easily be extended to cases other than a single \"proof\" excitation force.  
Figure 6: (**a**) Den Hartog 1 DoF oscillator; (**b**) simplified representation of a blade. The hysteresis cycles produced by friction are shown below each system.  
Figure 7: Representation of contact forces and their moments about platform point P for an In-Phase mode of vibration.