## 2 Methodology  
Blade displacements are calculated based on the times of blade arrival (Figure 1) measured using blade tip sensors (\(\mathrm{S_{B}}\)) and a once-per-revolution sensor (\(\mathrm{S_{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{ \mathrm{ }}}}}}}}}}}}}}}\)). These displacements are then used to calculate the harmonics of multimode rotor blade vibrations (\(A(t)\) in Equations (1)-(4))  
To find the harmonics of multimode rotor blade vibrations using the tip-timing method, their amplitudes may be assumed as follows:  
\[A(t)=A_{0}\sin(2\pi f_{o}t+\varphi)+\widetilde{0}\text{ one harmonic} \tag{1}\]  
\[A(t)=A_{1}\sin(2\pi f_{1}t+\varphi_{1})+A_{2}\sin(2\pi f_{2}t+\varphi_{2})+ \widetilde{0}\text{ two harmonics} \tag{2}\]  
Figure 1: Tip-timing measurement diagram.  
\[A(t)=A_{1}\sin(2\pi f_{1}t+\varphi_{1})+A_{2}\sin(2\pi f_{2}t+\varphi_{2})+A_{3} \sin(2\pi f_{3}t+\varphi_{3})+\widetilde{0}\text{ three harmonics} \tag{3}\]  
where: \(A(t)\) refers to the known values of blade displacements in time, \(A_{i}\) is the amplitude for \(i\)-th harmonic, \(f_{i}\) is the frequency of the blade vibrations for \(i\)-th harmonics, and \(t\) is the known time for which the displacement \(A(t)\) was calculated, \(\varphi_{i}\) is the phase shift for \(i\)-th harmonic, \(\widetilde{0}\) is the \"0\"- noise level of the blade vibrations. In the case of more harmonics:  
\[A(t)=\sum_{i}A_{i}\sin(2\pi f_{i}t+\varphi_{i})+\widetilde{0},i=1,2,\ldots \tag{4}\]  
where: \(i\) is the number of harmonics.  
Equations (1)-(4) adequately fit the measured data in the nonlinear least-squares method to obtain the amplitudes \(A_{i}\), frequencies \(f_{i}\) and phases \(\varphi_{i}\) of \(i\)-th blade mode vibrations. The accuracy of each fitting can be calculated on the basis of the following average fitting error:  
\[\varepsilon=\frac{\sum_{i=0}^{n}\frac{|A_{im}-A(t_{i})|}{|A_{im}|}}{n} \tag{5}\]  
where: \(n\) is the number of points in the fitting sample, \(A_{im}\) is the \(i\)-th measured blade displacement in a sample, \(A(t_{i})\) is the amplitude for the fitting function in \(i\)-th time from a sample and \(\varepsilon\) varies from 5 to 15%.  
In the fitting for each blade, the smallest \(\varepsilon\) was chosen.  
In the algorithm, the initial values have to be assumed. Some problems of ill-conditioning and divergence can be corrected by estimating initial parameters. This can be done for amplitude in sinusoidal motion Equation (1) by using [29]:  
\[A_{0 start}=\sqrt{2}A_{RMS} \tag{6}\]  
where: \(A_{RMS}\) is the root mean square amplitude.  
The noise level does not have to be a fitting parameter and can be calculated beforehand:  
\[\widetilde{0}=\frac{\sum_{i=0}^{n}A_{im}}{n} \tag{7}\]  
where: \(A_{im}\) is the \(i\)-th measured blade displacement in a sample.  
The noise level is a sum of all measured displacements for a blade in a fitting sample, divided by the number of points in the fitting sample.  
It is impossible to estimate amplitudes in Equations (2)-(4), thus for each fitting attempt, an initial min-max value is randomly chosen:  
\[\begin{array}{l}A_{1 start}=random(A_{1 min},\ A_{1 max})\\ A_{2 start}=random(A_{2 min},\ A_{2 max})\\ A_{3 start}=random(A_{3 min},\ A_{3 max})\end{array} \tag{8}\]  
The starting value frequencies are:  
\[\begin{array}{l}f_{start}=random(f_{min},\ f_{max})\\ f_{1 start}=random(f_{1 min},\ f_{1 max})\\ f_{2 start}=random(f_{2 min},\ f_{2 max})\\ f_{3 start}=random(f_{3 min},\ f_{3 max})\end{array} \tag{9}\]  
Finally, the phase may be assumed as:  
\[\begin{array}{l}\varphi_{start}=random(0,\ \pi)\\ \varphi_{1 start}=random(0,\ \pi)\\ \varphi_{2 start}=random(0,\ \pi)\end{array} \tag{10}\]