## 2 Methodology of the Analysis Method
### Two Degrees of Freedom Fitting Method  
One of the first methods proposed to fit tip-timing data is the single parameter method [9]. This method was developed based on the SDOF resonator model but it is only valid for a system with a single mode resonance. By fitting the SDOF model to the blade tip deflection at resonance, the vibration parameters can be determined. However, this method is not suitable in cases where two close resonance peaks are present. Two close peaks are quite common in the dynamic response of bladed disk. This happens when the frequency of one mode is split in two closely-spaced frequencies due to mistuning, or when two or more modes with close frequency are excited simultaneously by different EOs. In fact, in these conditions the blade response is dominated by only two modes' contribution around the resonance region. Therefore, the vibration response of the blade near the resonance can be modeled by only considering these two modes. Thus, by using the modal analysis technique [22; 23], the equations of motion of the blades can be represented by just two uncoupled equations in terms of generalized coordinates (modal coordinates). Each uncoupled equation in the modal coordinates is of the SDOF type. The solution in the original (physical) coordinate is obtained by the sum of the modal solutions of each SDOF system. Indeed, the concept is an extension of the SDOF method to 2 DOF system. The procedure is explained in detail below. Based on the modal analysis technique, the motion related to each mode \(\phi_{i}\) of a vibrating system can be represented by generalized coordinates, \(\eta_{i}\). Accordingly, the equations of motion of the system for the forced response oscillation can be written as  
\[\ddot{\eta}_{i}+2\zeta_{i}\omega_{n_{i}}\dot{\eta}_{i}+\omega_{n_{i}}^{2}\eta_{ i}=Q_{i}(t),\;i=1,2,\ldots,m \tag{1}\]  
where \(\zeta_{i}\), \(\omega_{n_{i}}\) and \(Q_{i}\left(t\right)\) respectively are the damping ratio, natural frequency and the generalized force (modal force) corresponding to the \(i\)th mode. Figure 1: Principle of the Blade Tip-Timing (BTT) method (solid line: non-vibrating blade, dashed line: vibrating blade). Equation (1) represents a set of \(m\) uncoupled equations and each equation can be treated as an SDOF system. The steady state response of an SDOF system under harmonic excitation \(Q_{i}\left(t\right)=\tilde{Q}_{i}\cos(\omega t+\tilde{\psi}_{i})\) is given by  
\[\eta_{i}\left(t\right)=\tilde{\eta}_{i}\cos\left(\omega t+\tilde{\psi}_{i}- \varphi_{i}\right) \tag{2}\]  
in which  
\[\tilde{\eta}_{i}=\frac{\delta_{i}}{\sqrt{\left(1-r_{i}^{2}\right)+\left(2\zeta _{i}r_{i}\right)^{2}}};\;\varphi_{i}=\frac{2\zeta_{i}r_{i}}{1-r_{i}^{2}};\;\;r_ {i}=\frac{\omega}{\omega_{n_{i}}};\;\;\;\delta_{i}=\frac{\tilde{Q}_{i}}{\omega _{n_{i}}^{2}} \tag{3}\]  
\(\omega\) is the excitation frequency and \(\tilde{\psi}_{i}\) is the initial phase angle in the forcing function. Finally, the response of the system can be obtained by the linear combination of the normal modes of the system, \(\phi_{i}\). When the contribution of two modes is present which corresponds to the particular case of \(m=2\), the response of the system in the physical coordinate, \(y(t)\), at an arbitrary point can be expressed as  
\[y\left(t\right)=\phi_{1}\,\eta_{1}\left(t\right)+\phi_{2}\,\eta_{2}\left(t\right) \tag{4}\]  
When the blade passes in front of a sensor located at an arbitrary angular position, \(\theta\), the phase angle \(\omega t\) at the \(k\)th rotation can be written as  
\[\left(\omega t\right)_{k}=EO\left(\varepsilon+\theta+2\pi k\right) \tag{5}\]  
where \(\varepsilon\) is the angular position of the blade at \(t=0\). By substituting Equation (5) into Equation (4), the blade tip response measured by the sensor is stated as  
\[