# Determining Rotor Blade Multi-Mode Vibration Components +
## Appendix A The Least Square Tip-Timing Model [11]  
The i-th blade arrival times (t\({}_{1,n}\), t\({}_{2,n}\), \(\ldots\), t\({}_{1,n}\), t\({}_{1+1,n}\), \(\ldots\), t\({}_{5,n}\)) were derived from experiments for the number of sensors s and the number of consecutive rotations n = 1, 2, \(\ldots\) N, where N is the number of rotations. The actual velocity of the blade tips determines the angles between the probes (see Figure A1). Figure A1 schematically presents the disc, two blades, and two sensors, 1 and 1 + 1, above the blades. The angle between sensors 1 and 1 + 1 is \(\upalpha_{1+1}-\upalpha_{1}\). The specified blade tip linear velocity is:  
\[\mathrm{V}_{\mathrm{c}}(\mathrm{t})=\mathrm{V}_{\mathrm{\Omega}}(\mathrm{t})+ \mathrm{V}(\mathrm{t})\] (A1)  
where  
\[\mathrm{V}_{\mathrm{\Omega}}(\mathrm{t})=\mathrm{\Omega}(\mathrm{t})\mathrm{R}\] (A2)  
\[\mathrm{V}(\mathrm{t})=\sum_{\mathrm{i=1}}^{\mathrm{k}}\mathrm{A}_{\mathrm{i} }\mathrm{sin}(2\pi\omega_{\mathrm{i}}\mathrm{t}+\varphi_{\mathrm{i}})\] (A3)\(\Omega\)--the speed of rotation [rad/s], R--blade tip radius [m],  
\(\rm{A_{ie}}\)--the amplitude of velocity blade vibration [m/s],  
\(\omega_{\rm{i}}\)--i-th frequency [Hz],  
\(\rm{\varphi_{\rm{i}}}\)--i-th phase angle [deg], and  
\(\rm{k}\)--number of modes. \[\rm{t=t_{0}+\Delta t+\Delta t+\ldots}\]  
The blade angle of arrival (Figure A1) is:  
\[\rm{\varkappa(t)=\varkappa(t-\Delta t)+\Delta t\,(V_{\Omega}(t)+V(t))/R}\] (A4)  
\(\rm{\varkappa(t)\geq\varkappa_{\rm{i}}\,to\,t_{\rm{i}}=t}\),  
\(\rm{\varkappa(t)\geq\varkappa_{\rm{i+1}}\,to\,t_{\rm{i+1}}=t}\), and  
so for \(\rm{t=(t_{\rm{i+1}}+t_{\rm{i}})/2}\). \[\rm{V_{c}\,(I,1+1)=R\,(\varkappa_{\rm{i+1}}-\varkappa_{\rm{i}})/(t_{\rm{i+1}}- t_{\rm{i}})}\] (A5)  
From Equations (A1) and (A2)  
\[\rm{V(t)=V_{c}\,(I,1+1)-\Omega(t)\,R}\] (A6)  
The blade vibration velocity was:  
\[\rm{V(t_{v})=\sum_{i=1}^{k}\,(a_{i}sin(2\pi\omega_{\rm{i}}t_{v})+b_{i}cos(2\pi \omega_{\rm{i}}t_{v})\,)}\] (A7)  
where \(\rm{t_{v}=(t_{v1,n},\,t_{v2,n},\,\ldots,\,t_{v(s-1),n})}\), (n = 1, 2, \(\ldots,\,\) N, N number of rotations) (see Equation (A11)), M = N\(\times\) (s \(-\) 1)), and s \(-\) 1 is the number of calculated blade velocities in one rotor revolution  
\[\rm{A_{i}=(a_{i}^{2}+b_{i}^{2})^{0.5},\,\,\varphi_{i}=arctg(b_{i}/a_{i})}\]  
By integrating Equation (A7), the blade displacement is found  
\[\rm{X(t_{v})=C+\sum_{i=1}^{k}\,\Bigl{(}-a_{i}(2\pi\omega_{\rm{i}})^{-1}sin(2 \pi\omega_{\rm{i}}t)+b_{i}(2\pi\omega_{\rm{i}})^{-1}cos(2\pi\omega_{\rm{i}}t_ {v})\,\Bigr{)}}\] (A8)  
where C = 0. \[\mathrm{V}_{\mathrm{\Omega}}(\mathrm{t})=\mathrm{\Omega}(\mathrm{t})\mathrm{R}\]0  
Equation (A7) can be presented as:  
\[\mathrm{V}_{\mathrm{\Omega}}(\mathrm{t})=\mathrm{\Omega}(\mathrm{t})\mathrm{R}\]1 (A9)  
\(\rm{s=[sin(2\pi\,\,\omega_{\rm{1}}\,t_{v1,1}),\,\,cos(2\pi\,\,\omega_{\rm{1} }\,t_{v1,1}),\,\,sin(2\pi\,\,\omega_{\rm{2}}\,t_{v1,1}),\,cos(2\pi\,\,\omega_{ \rm{2}}\,t_{v1,1}),sin(2\pi\,\,\omega_{\rm{k}}\,t_{v1,1}),\,cos(2\pi\,\,\omega_{ \rm{k}}\,t_{v1,1}),\,cos(2\pi\,\,\omega_{\rm{k}}\,t_{v1,1}),\,cos(2\pi\,\, \omega_{\rm{k}}\,t_{v1,1}),\,cos(2\pi\,\,\omega_{\rm{k}}\,t_{v1,1})where rotation velocity \(\Omega_{\rm n}\) in successive n rotations (n = 1, 2, \(\ldots\), N, N being the number of rotations). Functional F in the Least Squares Technique for various t\({}_{\rm vm}\) ([PERSON] et al. [17], [PERSON], [PERSON] [18]) is:  
\[\mathrm{V}_{\mathrm{\Omega}}(\mathrm{t})=\mathrm{\Omega}(\mathrm{t})\mathrm{R}\]2 (A12)  
where V(t\({}_{\rm vm}\)) is the vibration velocity of the blade tip (Equation (A7)) and v(t\({}_{\rm vm}\)) is the measured velocity of the blade tip (Equation (A11)).