The Analysis Of Synchronous Blade Vibration Using Linear Sine Fitting

Abstract In Blade Tip Timing several sensors installed circumferentially in the casing are used to record times of arrival (TOA) and observe deflections of blade tips. This paper aims to demonstrate methodology of model-based processing of aliased data. It focuses on the blade vibration excited by the forces synchronous with engine rotation, which are called integral responses. The driven harmonic oscillator with single degree of freedom (SDOF) is used to analyse blade vibration measured by tip-timing sensors during engine deceleration. When integral engine order EO is known, the linear sine fitting techniques can be used to process data from sensors to estimate amplitude, phase and frequency of blade vibration in each rotation. The oscillator model is implemented in MATLAB and used to generate resonance curves and simulate blade responses observed with tip sensors, installed in the axial compressor. Generated TOA data are fitted to the sine function to estimate vibration parameters. The validated procedure is then employed to analyze real test data.


Driven harmonic oscillator and resonance curves
A vibrating blade can be modelled as a single degree of freedom system. Forced vibration of blade tip is described by the equation of motion: with the following solution: Typical blades are characterized by low damping factors δ < 0.2. In this case amplitude reaches maximum values at a frequency of excitation close to natural [14]. It is assumed that the highest tip deflection is observed at the resonance speed.

Integral responses
In turbomachinery the driving force F(t) = F 0 cos (2 π f t) is often synchronous with the rotation:

EO -engine order, f -vibration frequency, f r -rotational frequency, equal rpm / 60
Tip deflection is measured synchronously with the excitation force, not only in the resonance: EO α i = 2 π f t i t i -time of blade arrival to the sensor i, installed at the angle α i This is why time can be eliminated from the oscillator solution: For integral responses tip deflection y i (ω) is time-independent speed function, related to angular sensor position ( fig. 3). The integral response is synchronized with the measurement and belongs to low-frequency (static) component of the signal. The vibratory data that remains after subtracting the static deflection is considered as a non-integral response or noise. During deceleration of the rotor, characteristic phase change (zigzag) is observed while passing a resonance. First generation single-sensor tip-timing systems located these zigzags to measure resonance speed and peak-peak amplitude. This technique, known as Zablotskiy method [14], works only for low-order, highamplitude resonances, lacks accuracy and is relatively difficult to be automated. Approximate resonant frequencies and engine orders exciting blade vibratory modes are usually known from previous measurements and FEM analysis and presented on Campbell diagram. Using this information engine orders can be assumed to determine the frequency, amplitude and phase of the observed responses.
Integral responses can be characterized by linear least-square fitting using the equation of vibration in the form [5], [12], [16]: In this case the fitting is linear and well-conditioned when sensors are properly distributed on the circumference [5]. The amplitude and phase can be calculated from the fitting result a and b.
The resonance frequency f 0 is proportional to the rotational speed, at which the highest amplitude is observed and the phase crosses zero: In practice, the problem of estimating the parameters of synchronous vibration is complicated, because higher-order responses have lower amplitudes and worse signal to noise ratio. Responses of different modes often occur at similar rotational speed ( fig. 1). At least 2 N + 2 sensors are required to measure N simultaneous resonances.

Least square fitting
The method of least squares is one of the most common numerical techniques. The linear sine fitting functions available in Matlab and LabView are used below.
Matlab even provides a special left-hand matrix division operator for this purpose: a = M \ y (mldivide). For a rectangular observation matrix M, it finds the approximate solution x of linear system of equations y = M a.

Linear regression
The following code performs the linear regression in MathScript, which is a clone of Matlab, available in LabView. Results of the function fit() and the left-hand division operator were compared ( fig. 4) (2); y x

Sine fitting
The following array of observation was created to perform sine fitting:

Numerical simulation
The harmonic oscillator was used to simulate tip vibration of the compressor blade, which was passing through the first-mode resonance during the deceleration of the turbojet. The speed profile from a real test was loaded from the text file ( fig. 6). We assumed that the natural frequency equals 500 Hz, the engine order EO = 2, and damping factor 0.05. The analysis of synchronous blade vibration using linear sine fitting Analiza drgań synchronicznych łopatek z wykorzystaniem liniowego dopasowania..
Amplitude and phase was calculated for each rotation, assuming the engine order EO = 2. Then the three virtual sensors were placed on the circumference of the compressor. The sensor placement and the equation of rotation were used to calculate times of arrival, while neglecting the error coming from blade vibration. Blade tip displacement y(tt) was computed for these times, using the equation of the oscillator. In this way continues vibration signal was sampled by the virtual sensors like in the real BTT system. The result shown in fig. 7 is incorrect, because the observations do not synchronize with the measurement. The arrival time tt depends on displacement of blade vibration, so it is difficult to calculate it directly. But for integral responses ωt = EO α and this is why the oscillator equation can be used in the alternative form: This time the sensors response y 2 ( fig. 8) is similar to the real test results and curves drown by Zablotskiy [14], [15].

Real test data
The displacement of a selected blade was recorded with 3 inductive sensors during the same deceleration ( fig. 6). The data included mainly M1EO2 response, which was coupled with some higher-order resonance at the rotation No. 950 (fig. 10). The static component of sensor's TOA signals should be adjusted to zero before fitting [13]. The amplitude shown by a single sensor is lower than the real resonance amplitude. This is why observation of several sensors should be combined and fitted into model to estimate the actual amplitude.

The identification of engine order
The engine order EO of the resonance can be identified on XY plot, showing the trajectory of the blade tip, measured by a pair of sensors [1][2][3][4], [8]. In practice the application of Heath method is limited as ellipses are incomplete and deformed for coupled responses. Two resonances are presented: the low order resonance in fig.11 (M1EO2, rotations No 1150-1300) and the higher order in fig. 12 (rotations No 850-1020 in fig. 10).

Data fitting
A sine fitting program was prepared in LabVIEW using the library function General LS Linear Fit ( fig. 13), which requires observation matrix, similar to the one used previously in Matlab. The program can perform one fit for each rotation or alternatively combine a few rotations (ileObr parameter) into one vector, so the solution should be more accurate.

Summary
The driven oscillator model was used to simulate blade vibration observed by BTT system installed in the axial compressor. The method of least squares and linear sine fitting functions available in Matlab and LabView were applied to estimate vibration parameters. This technique is widely used to analyze integral responses in gas-turbine industry.

The project was financed by the Polish National Science Centre (NCN) under the decision DEC-2011/01/D/ST8/07612
Radosław Przysowa PhD. Eng. has been working as a engineer and scientist supporting machine maintenance since 2002 (in the position of the Assistant Professor at the moment). He is also familiar with research management and commercialization and has remarkable programming skills, including object oriented programming, digital signal processing and data mining. He specializes in Blade Tip-timing, which is a non-contact vibration measurement method, involving processing of aliased displacement signals.