# Key Action Mechanisms of Intentional Mistuning
## 3 Verification of AMM Results
### Alternate Mistuning Pattern  
We will use rotor A, which has an even number of sectors, as a very simple test for the results of the AMM about the action of intentional alternate mistuning pattern.  
Rotor A is shown in Figure 2, and its diagram for the tuned modes, frequency versus nodal diameter (ND), is presented in Figure 3. The forcing of the first modal family is analysed, with an engine order \(EO=8\). The forcing has been normalised to have maximum response amplitude equal to 1 for the tuned rotor.  
Figure 2: Rotor A with 24 blades and approximately 5800 DOF per sector.  
The relative variation of the first family modal frequencies with respect to the average frequency \(\omega_{0}\) what is actually plotted \(\Delta\omega_{j}=(\omega_{j}-\omega_{0})/\omega_{0}\) is represented in the top plot of Figure 4. The corresponding influence coefficients associated with the elastic coupling \(a_{1},\dots,a_{N}\) (the inverse Fourier transform of the \(\Delta\omega_{j}\) distribution) are shown in the lower plot. Both curves are symmetric so only one half is plotted.  
It can be clearly seen in Figure 4 that the main coupling comes from the coefficient \(a_{1}\), which corresponds to the coupling of one blade with the directly adjacent ones (see  
Figure 4: (**Top**): relative frequency deviations for the first modal family of Rotor A. (**Bottom**): corresponding influence coefficients.  
Figure 3: Rotor A natural vibration frequencies. Red thick line indicates the forcing case considered with engine order \(EO=8\).  
Figure 1). In this case, following the ideas extracted from the AMM analysis, an alternate mistuning pattern is expected to produce optimal attenuation of the vibration.  
In order to validate this result a significant number of simulations have been carried out. A total of 50 mistuning distributions have been tested, all of them composed of a random pattern with zero average and maximum amplitude 4%, and an alternate mistuning pattern \([1,-1,\ldots,1,-1]\) with amplitude \(Aim\) in the range 0-10%. The material damping is set to \(\xi=0.01\), which is much larger than the remaining coupling coefficients once \(a_{1}\) is cancelled by the alternate mistuning pattern (see Figure 4).  
The resulting vibration amplitude for the 50 different mistuning patterns is presented in Figure 5. It can be seen how the vibration amplitude is reduced to the tuned value as the amplitude of the intentional mistuning is increased from \(Aim=0\%\) to \(Aim=10\%\). The sensitivity to the random mistuning is also strongly attenuated, as predicted by the AMM theory in Section 2.1.  
The frequency sweeps and the maximum blade displacements for one random mistuned patterns (marked with a thick line in Figure 5) is represented in Figure 6. As the amplitude of the intentional mistuning grows there is a split in the resonance curve into two peaks, one for the odd and one for the even blades, and the amplification of the vibration is decreased. In the case of \(Aim=10\%\) (bottom plot), the even and odd blades are effectively decoupled and, at the maximum, only the even blades have a significative vibration amplitude.