# Key Action Mechanisms of Intentional Mistuning
## 2 AMM Formulation
### Limit of Large Intentional Mistuning  
The mistuning coefficient of each blade is now assumed to have a random part plus an intentional part. An asymptotic expansion is performed in the AMM lumped system above to calculate the interesting limit of intentional mistuning amplitude much larger than that of the random mistuning. In order to make this derivation easier to follow, a simple alternate intentional mistuning pattern in a rotor with an even number of blades is considered (the results can be directly extrapolated to more complicated intentional mistuning patterns). The AMM lumped system with the alternate intentional mistuning included takes the form  
\[\left[ \begin{array}{cccc}\mathrm{i}_{\xi}^{x}-\Delta\omega+d_{1}+D&a_{1}& \ldots&a_{N-1}\\ a_{N-1}&\mathrm{i}\xi-\Delta\omega+d_{2}-D&\ldots&a_{N-2}\\ \vdots&\ddots&\ddots&\vdots\\ a_{1}&\ldots&a_{N-1}&\mathrm{i}\xi-\Delta\omega+d_{N}-D\\ &&&&\\ &&&&\\ &&&&\\ \end{array} \right]\left[ \begin{array}{c}x_{1}\\ \vdots\\ x_{N}\\ \end{array} \right]=\] \[=\mathrm{i}\xi\left[ \begin{array}{c}\mathrm{e}^{\mathrm{i}( \frac{2\pi}{N})1}\\ \vdots\\ \mathrm{e}^{\mathrm{i}(\frac{2\pi}{N})N}\\ \end{array} \right]\]  
where \(d_{j}\) corresponds to the amplitude of the random mistuning of blade \(j\), and \(D\) is the amplitude of the alternate intentional pattern. The intentional mistuning is now assumed to be much larger than the random mistuning and the elastic coupling and damping  
\[|D|\gg|d_{j}|,|a_{j}|,|\xi|\quad j=1,\ldots,N\]  
and the system above simplifies, in first approximation, to the homogeneous system:  
\[\left[ \begin{array}{cccc}-\Delta\omega+D&0&\ldots&0\\ 0&-\Delta\omega-D&\ldots&0\\ \vdots&\ddots&\ddots&\vdots\\ 0&\ldots&0&-\Delta\omega-D\\ \end{array} \right]\left[ \begin{array}{c}x_{1}\\ \vdots\\ x_{N}\\ \end{array} \right]=\left[ \begin{array}{c}0\\ \vdots\\ 0\\ \end{array} \right],\]  
Figure 1: Sketch of the lumped system (recall that \(a_{j}=a_{N-j}\)). This indicates that, when the intentional mistuning is dominating, the odd blades resonate at frequency \(\Delta\omega=+D\), and the even blades resonate at \(\Delta\omega=-D\), with arbitrary blade displacements:  
\[\left[ \begin{array}{c}x_{1}\\ 0\\ x_{3}\\ 0\\ \vdots\\ x_{N-1}\\ 0\end{array} \right],\left[ \begin{array}{c}0\\ x_{2}\\ 0\\ x_{4}\\ \vdots\\ 0\\ x_{N}\end{array} \right].\]  
In other words, we recover the well known result that the main effect of the alternate intentional mistuning pattern is to split the resonance frequency of even and odd blades. In order to see the detail of the resonance peaks, we have to go to next order in the perturbative procedure. Near the resonance of the odd blades the forcing frequency is written as \(D\) plus a small correction  
\[\Delta\omega=D+\Delta\widetilde{\omega}\]  
This expression is taken to the AMM lumped system and only the displacements of the odd blades are kept because the resonance peak of the even blades is at the distant frequency \(\Delta\omega=-D\). The resulting reduced system near the odd blades frequency peak is given by  
\[\left[ \begin{array}{cccc}\mathrm{i}\xi-\Delta\widetilde{\omega}+d_{1}&a_{N-2 }&\cdots&a_{2}\\ a_{2}&\mathrm{i}\xi-\Delta\widetilde{\omega}+d_{3}&\cdots&a_{4}\\ \vdots&\ddots&\ddots&\vdots\\ a_{N-2}&\cdots&a_{2}&\mathrm{i}\xi-\Delta\widetilde{\omega}+d_{N-1}\end{array}  \right]\left[ \begin{array}{c}x_{1}\\ x_{3}\\ \vdots\\ x_{N-1}\end{array} \right]=\mathrm{i}\xi\left[ \begin{array}{c}\mathrm{e}^{ \mathrm{i}(\frac{2\omega}{N})1}\\ \mathrm{e}^{\mathrm{i}(\frac{2\omega}{N})3}\\ \vdots\\ \mathrm{e}^{\mathrm{i}(\frac{2\omega}{N})(N-1)}\end{array} \right].\]  
Now the odd blades are only coupled among them through the elastic influence coefficients \(a_{2}\), \(a_{4}\),... This indicates that the split in frequency produced by the alternate mistuning pattern has cancelled the coupling between adjacent blades; that is, the coefficient \(a_{1}\) no longer appears in the blade motion equations.