# Determining Rotor Blade Multi-Mode Vibration Components +
## 1 Introduction  
In this paper, blade tip-timing (BIT) is used for measuring multi-mode rotor blade vibration. The first tip-timing models work on the assumption that each blade vibrated with the same frequency [1, 2, 3, 4]. Two simultaneous resonances were analyzed by [PERSON] et al. [5] using the autoregressive methods BTT.  
[PERSON] and [PERSON] [6] and [PERSON] [7] carried out an analysis using a group of sensors to measure multi-mode blade vibration signals. Blade multi-modes using the non-uniform Fourier transform were identified by [PERSON] and [PERSON] [8]. A sparse reconstruction of the blade tip-timing signal for multi-mode blade vibration monitoring was proposed by [PERSON] et al. [9].  
The application of the sparse representation theorem to find multi-mode blade vibration frequencies with uncertainty reduction was presented by [PERSON] et al. [10].  
[PERSON] et al. [11] presented a method for finding the multi-modes of synchronous and non-synchronous blade vibrations from the tip-timing based on the blade vibration velocity using the Least Squares Technique. This method requires only two sensors in the casing, and a once-per-revolution (OPR) sensor for synchronous and asynchronousvibrations. The rotor blade velocities are obtained from the measured time of rotor blade arrival and used in the functional of the Least Square Technique.  
[PERSON] et el. [12] also used blade tip-timing vibration velocity obtained from the Taylor series expansion with compressive sensing-based methods in multi-mode blade synchronous vibrations. Five sensors in the casing and a once-per-revolution sensor were used. This method cannot be applied in engineering because of its low accuracy, inefficiency and difficult parameter selection.  
[PERSON] et el. [13] proposed an acceleration-based blade tip-timing method for synchronous vibration. This method captures rapid changes in vibration, with three sensors in the casing without an OPR sensor.  
In all the above papers, the multi-mode analysis assumed the coupling of successive n modes, and only then did a numerical analysis indicate which ones were important.  
This paper presents, for the first time, a technique to determine the frequencies and amplitudes of coupled modes in blade vibrations based on the algorithm presented in [11]. The essence of article [11] is to search for all harmonic components (amplitudes, frequencies and phases) of the rotor blades using the time of blade arrival measured by sensors in the casing. But the number of harmonics k must be assumed in advance and their number must be high enough to ensure a small calculation error.  
In this paper, a technique to determine the frequencies and amplitudes of coupled blade vibration modes based on the algorithm in [11] is shown. This allows us to determine the precise number of coupled blade vibration harmonics without having to make a prior assumption as to their number. Therefore, the calculation time is considerably shorter. This is the novelty.  
The technique is validated a numerical simulation. Here, the experimental and numerical analyses of the 1 st compressor rotor blade in an SO-3 engine during run-down are used to determine the multi-mode blade components for synchronous and asynchronous vibrations. The analyses are carried out using two sensors in the casing and an OPR sensor.