# Tip-timing data analysis for mistuned bladed discs assemblies

Cyrille Stephan, Marc Berthillier, Joseph Lardiès, Arnaud Talon

► To cite this version:

Cyrille Stephan, Marc Berthillier, Joseph Lardiès, Arnaud Talon. Tip-timing data analysis for mistuned bladed discs assemblies. ASME Turbo Conference on Controls, Diagnostics and Instrumentation, Jun 2008, Berlin, Germany. pp.447-455, 10.1115/GT2008-50825. http://00350174

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
TIP-TIMING DATA ANALYSIS FOR MISTUNED BLADED DISCS ASSEMBLIES

Cyrille Stéphan*

Marc Berthillier, Joseph Lardiès

FEMTO-ST LMARC

24, chemin de l'Epitaphe

25 000 BESANCON

FRANCE

Email: cyrille.stephan@univ-fcomte.fr

Arnaud Talon

Turboméca

64511 Bordes CEDEX

FRANCE

Email: arnaud.talon@turbomeca.fr

ABSTRACT

In turbomachine industry, bladed assembly vibration mea-
surements are very important for blades behaviour estimation.
These measurements are generally obtained from strain gauges.
However, one of the most promising methods for the analysis of
blade vibrations in rotating bladed assemblies is the Blade Tip
Timing or Optical Blade Vibration Measurement method.

A set of optical sensors is mounted on an engine casing, in
front of a disc, and measures the times of arrival of each blade.
These timings are then used to compute the vibrations of the
blades. However the fundamental problem for spectral analy-
sis of blade tip timing data is that the signals are undersampled
and aliased. We propose here a new method for spectral estima-
tion of blades responses from tip timing data that overcome these
difficulties.

The method proposed in this communication is based on the
use of a minimum variance filter associated with an iterative up-
dating of the autocorrelation matrix. That allows to process cor-
rectly a signal even if the number of known signal samples is less
than equivalent Nyquist criterion. This approach is suitable for
spectral analysis of undersampled and aliased signals.

Perfomances of the spectral estimator have been evaluated
for one simulated and one experimental test cases. The method
seems very promising for the monitoring of mistuned bladed
discs.

INTRODUCTION

The Tip-timing technology consists in the use of sensors
which are mounted on an engine casing around a rotating disc.
They measure the times of arrival of each blade. Then these tim-
ings are used to estimate the vibratory motions.

This measure is cheaper than strain gauges, but its analy-
sis is more difficult. Each sensor collects one time sample per
revolution and per blade. For a blade which can have several os-
cillations per revolution, this sampling is very weak. Traditional
Fourier methods are then useless to study this kind of signal. So
previous authors developed methods of analysis based on me-
chanical assumptions.

For each nature of vibratory response (asynchronous or syn-
chronous), a strategy was adopted. For example, the study of
synchronous vibration is based on the determination of the En-
gine Order which excites a mode during a acceleration (or decel-
eration) ( [1]). They give good results if bladed disc’s modes are
clearly distinct in frequency domain. Asynchronous vibration is
generally treated by an All-Blade-Spectrum method ( [2]). It is
designed for a perfectly tuned flutter mode : all the blades vi-
brate at the same frequency and with the same amplitude. It only
works well on real tip-timing data if the bladed disc respect this
assumption.

We prefer to choose a different way. Our main goal is to
study asynchronous vibrations in the case of non-stationary phe-
nomena, without making assumptions on the mechanical behav-
ior of the system. So the spectral estimation was retained as the
tool of identification for tip-timing data.

<footnote>*Address all correspondence to this author.</footnote>
In this paper we propose a method based on the minimization of the variance of output signal for spectral estimation of aliased, non-uniform and irregular data. With this method it is not necessary to make hypothesis about the nature of the observed phenomena by tip-timing sensors.

We will see in a first part the tip-timing measurement. Several remarks about the sampling's nature of the signals will be detailed in order to clearly understand the interest of a spectral estimator described in a second part. In a third part, a bladed disc simulator will be described. It aims to generate signals close to those obtained from experiments. Finally, the spectral estimator will be used on simulated signals for a particular case of mistuned bladed disc and on real test data.

## TIP-TIMING MEASUREMENTS
### Process

<figure><img src="image_2.png" /><figcaption>Figure 1. TIP-TIMING SYSTEM MEASURE</figcaption></figure>

Sensors are placed in an engine's casing in front of a disc (figure 1). They can be optical, capacitive or eddy-current. They measure the times of arrival of each blade (figure 2). Another

sensor is mounted in front of the shaft and takes only one information per revolution. The data provided by this sensor are then used to estimate the rotor speed.

## Vibratory displacement and sampling

<figure><img src="image_3.png" /><figcaption>Figure 2. SENSOR SIGNALS</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 3. SIGNAL FOR THE BLADE 1</figcaption></figure>

If there were no vibration, we could exactly determine the time of arrival of each blade in front of each sensor. But blades vibrate, and their vibratory displacements change these times (figure 3). The differences between the measured times and the estimated times of arrival without vibration are approximately proportional to the angular displacements of the blades, and so to the vibratory motions [1].

Let $x_k(t)$ be the angular deflection of the blade $k$ and $\alpha_k(t)$ its angular position. They are linked by the relation

$$ \alpha_k(t) = \int_{0}^{t} 2\pi F_r(u) du + x_k(t) \quad (1) $$

with $F_r$ the rotation's frequency. For simplicity, $F_r$ should be assumed constant in the rest of the paragraph. So equation (1) becomes

$$ \alpha_k(t) = \alpha_{k0} + 2\pi F_r t + x_k(t) \quad (2) $$
with $\alpha_{k0}$ the initial angular position of the blade $k$.

Let $t_{kns}$ be the time of arrival of the blade $k$ in front of the probe $s$ at the revolution index $n$ and $\theta_s$ the angular position of the probe $s$. The angular position $\alpha_k(t_{kns})$ is then given by

$$ \alpha_k(t_{kns}) = \alpha_{k0} + 2\pi F_r t_{kns} + x_k(t_{kns}) \quad (3) $$

$$ \alpha_k(t_{kns}) = (n - 1)2\pi + \theta_s \quad (4) $$

Equations (3) and (4) permit to calculate the fundamental relation of tip-timing sampling :

$$ x_{kns}(t_{kns}) = (n - 1)2\pi + \theta_s - \alpha_{k0} - 2\pi F_r t_{kns} \quad (5) $$

Several remarks should be made about equation (5). Firstly, the measured time $t_{kns}$ depends on the deflection $x_{kns}$, so the sampling's time is a function of the observed object. It is called an implicit sampling.

Furthermore, the blade's deflection is the sum of the static and the dynamical displacements. The static displacement is due to centrifugal and temperature effects. The dynamical displacement is the vibratory displacement of the observed blade.

If we suppose that the excitation's force contains a random part due to noise, then the blade's dynamical displacement (and the deflection $x_k$) contains a random part too. As seen previously, the sampling depends on the deflection, so if the deflection is random, then the sampling is random too.

Another point is that sampling is not uniform. Of course, sensors can be equally spaced, and if rotor speed is constant, sampling can be supposed uniform. But on a test bench, it is very lucky if all sensors work well, and we have to suppose that one sensor can have a failure; then a spectral estimator based on regular sampling is irrelevant.

For a blade, the entire signal is formed from samples of all sensors. For example, on the figure 2, a blade's signal is composed of three samples per revolution. Often, they are not equally spaced in the casing, so a true Nyquist frequency cannot be defined. But an equivalent "Nyquist frequency" $f_{Ny}$ in the case of irregular sampling (see [3]) can be defined by

$$ f_{Ny} = \frac{N_s F_r}{2} \quad (6) $$

with $N_s$ the number of sensors. Of course it is not the true mathematical Nyquist frequency, but it can give an idea of the undersampling of tip-timing data. Unfortunately, $f_{Ny}$ is generally very little compared to frequencies of interest. Data are then always undersampled. It is the main drawback of this technology.

## SPECTRAL ANALYSIS ESTIMATOR

Knowledge of times of arrival permits to estimate vibratory displacements. Unfortunately, signals are severely undersampled. In addition, the sampling is nonuniform. In this section, the feasibility to study these signals is discussed and the choice of the spectral estimator is explained.

### Sampling rate

For a real bandwidth signal $x(t)$ whose maximal frequency is $f_{max}$, let $X(f)$ be its Fourier transform. The minimal sampling rate is $2f_{max}$ (Nyquist theorem). A subset of this class is composed by multiband signals. They are distinguished by their spectral supports $F_s$, defined as the set of frequencies over which the spectrum $X(f)$ of the signal is nonzero. On the figure 4, $\chi(f)$ is the existence function of the Fourier transform $X(f)$ of a multiband signal. $\chi(f)$ is defined by:

$$ \chi(f) = \begin{cases} 0 & \text{if } X(f) = 0 \\ 1 & \text{if } X(f) \neq 0 \end{cases} $$

$F_s$ can contain several bands in the frequency domain. Landau showed ([4]) that the sampling rate of an irregular sampling for the class of multiband signals is lower-bounded by $F_{La}$, defined by:

$$ F_{La} = \sum_{p=1}^{P} F_p \quad (7) $$

with $P$ the number of frequency bands and $F_p$ the width of a frequency band. In practice, $F_{La}$ could be significantly smaller than the Nyquist rate. In our case, $F_s$ is unknown. But a bladed disc

<figure><img src="image_5.png" /><figcaption>Figure 4. FREQUENTIAL SUPPORT OF A REAL MULTIBAND SIGNAL</figcaption></figure>

behaves like an oscillator and filter frequency bands, generally
limited by the structural modes. So an estimator which can take advantage of irregular sampling should tend to the minimal rate sampling $F_{La}$ and should be relevant to analyse multiband signals obtained by the tip-timing technology.

## Choice and description of the spectral estimator

Experimental signals often contain a large part of randomness. Although they could not be periodic, spectral analysis can still be made by the use of the autocorrelation function. This probabilist tool measures the linear dependence of a process with itself and is directly linked to the Fourier transform of a random signal.

Several papers aim to estimate spectra for irregular and aliased data [5–7]. A few of them are based on the estimate of the autocorrelation function [8,9]. Among them, one was chosen because of its performance and its relative simplicity for non-specialists in spectral estimation.

This method is presented by Greitans [9–11]. It belongs to Capon’s family of spectral estimators and is based on properties of nonuniform sampling, filtering by minimization of the process output and an iterative algorithm on autocorrelation matrix. The main idea is to minimize the variance of the narrowband filter output signal. The frequency response of the filter adapts itself to the input signal spectral components on each frequency of interest. The variance of the output process is determined by $\rho = a^H Ra$ (see [12]), with $a$ the vector of coefficients filter, $H$ the conjugate transpose and $R$ the signal autocorrelation matrix. Coefficients have to verify that, on each frequency $f_0$, the gain of the filter response is one : $e^H(f_0)a = 1$, with $e_i(f_0) = e^{j2\pi f_0 t_i}$. It means that a sinusoid at frequency $f_0$ passes trough the filter designed for it without distortion.

Filter’s coefficients are given by :

$$a(f_0) = \frac{R^{-1}e(f_0)}{e^H(f_0)R^{-1}e(f_0)}. \qquad (8)$$

The spectral amplitude is obtained by $s(f_0) = xa(f_0)$, with $x$ the nonuniform undersampled signal.

According to equation (8), filter’s coefficients depend on the signal autocorrelation matrix. The usual way to estimate it is based on the average of the mutual products of signal samples. If the sampling is not uniform, this approach is irrelevant. Greitans proposed to use the Wiener-Khintchin theorem which links the autocorrelation function and the power spectral density (PSD) :

$$r(\tau) = \int_{-\infty}^{+\infty} P(f)e^{j2\pi f\tau} df \qquad (9)$$

with $P(f)$ the PSD of the signal. The easiest way to obtain a first approximation of the PSD is to operate an nonuniform discrete

Fourier transform :

$$P(f) = \frac{1}{N^2} \left| \sum_{k=1}^{N} x_k e^{-j2\pi f t_k} \right|^2. \qquad (10)$$

Values of the autocorrelation signal matrix $R$ computed by this first approximation of $P(f)$ are falsed by artifacts. An iterative updating algorithm described by Liepin’s [13] improves results. The $(i+1)$th order estimate of signal autocorrelation matrix is updated from the $(i)$th order estimate $\hat{P}^{(i)}$:

$$\hat{R}_{lk}^{(i+1)} = \sum_{m=1}^{M} \hat{P}_{m}^{(i)} \exp(j2\pi f_m (t_k - t_l)). \qquad (11)$$

Then, using $\hat{R}^{(i+1)}$, the estimates $\hat{S}^{(i+1)}$ and $\hat{P}^{(i+1)}$ are calculated by

$$\hat{P}^{(i+1)} = \left| \frac{E \hat{R}^{(i+1)} - 1}{diag(E \hat{R}^{(i+1)} - 1)} \right|^2 \qquad (12)$$

with $E_{mn} = \exp(-j2\pi f_m t_n)$.

This algorithm is very slow due to the nonuniform discrete Fourier transform used during the formation of the estimate autocorrelation matrix $\hat{R}$. As a Fast Fourier Transform can’t be used, the computational complexity is $O(N^2 M)$. Approximate Nonuniform Fast Fourier Transform are used to improve the speed [14] and the computational complexity reduces to $O(N^2 * \log(N) + \log(1/\epsilon)M)$, with $\epsilon$ the desired accuracy.

We choose a convergence’s criterion based on the PSD :

$$Error = \left| \frac{\hat{P}^{(i+1)} - \hat{P}^{(i)}}{\hat{P}^{(i)}} \right| * 100. \qquad (13)$$

The algorithm is stopped when the error is small enough. In practice, $Error = 5$ was chosen. If the algorithm does not converge, more sensors or data are required.

## AEROMECHANICAL MODEL

Simulator’s goal is to produce data close to those obtained on a test bench. It calculates the times of arrival of blades in front of sensors. Contrary to models previously made in the study of tip-timing [15, 16], this one aims to be able to simulate mistuned bladed discs and post-flutter behaviour.
<figure><img src="image_6.png" /><figcaption>Figure 5. CYCLIC MODEL</figcaption></figure>

## Cyclic model

A bladed disc is split into N sectors, with N the number of blades (figure 5). Each sector is constituted by one degree of freedom (dof) for the disc's part and two dofs for the blade. The disc's coupling is represented by the springs $k_i$. When a bladed disc is manufactured, some variations of dimensions appear. The symmetry can not be perfect. Random variations are introduced in the springs $k_b$ and $k_c$ to modelize it.

The displacement vector for the whole assembly is $x(t) = [x_1(t) \dots x_N(t)]^T$, with $x_i(t)$ the displacement vector for the sector i. The Lagrange's method gives the matrix equations :

$$M\ddot{x}(t) + (C + C_a(\Omega))\dot{x}(t) + K(\Omega)x(t) = F(t) \quad (14)$$

with M the mass matrix, C the damping matrix, $C_a$ the aerodynamic matrix, K the stiffness matrix, $F(t)$ the excitation force vector and $\Omega$ the rotor speed. Equations are integrated in the time domain in order to obtain the temporal responses.

The system can become unstable because of energy transfer between fluid and structure : it is the flutter phenomena. For real machines, the vibratory amplitude grows until collapse or until a limit cycle is reached due to non linear effects. For a detailed review of aeroelastic methods, see the paper of Marshall [17].

Here we use a simple model : fluid-structure coupling is represented by an aerodynamic matrix $C_a$ multiplied by the speed vector $\dot{x}(t)$. The N second-order differential equations are reduced to 2N first-order differential equations. A first-order state-

space model is then obtained :

$$\left( \begin{matrix} C + C_a M \\ M 0 \end{matrix} \right) \left\{ \begin{matrix} \dot{x} \\ \ddot{x} \end{matrix} \right\} = \left( \begin{matrix} -K 0 \\ 0 M \end{matrix} \right) \left\{ \begin{matrix} x \\ \dot{x} \end{matrix} \right\} + \left( \begin{matrix} F(t) \\ 0 \end{matrix} \right) \quad (15)$$

$$U\dot{y} = Ay + g(t) \quad (16)$$

The stability of the system is then studied by evaluating the generalized eigenvalues of the system. The imaginary part of each eigenvalue is related to the frequency of the aeroelastic mode and the real part to the damping. If a real part is strictly positive, the considered mode is unstable.

## Excitation model

Fluid goes upstream through a number of obstacles $N_{Ob}$ and takes a spatial shape of wakes. So each blade is harmonically excited by a traveling wave

$$F(t) = \sum_{k=1}^{n+1} A_k \cos(k N_{Ob} \Omega t) + B_k \sin(k N_{Ob} \Omega t) \quad (17)$$

with n the number of harmonics. In our case, we choose $B_k = 0$ for all k and n = 2 to simplify the model.

## TESTS

### Simulated case : mistuned flutter

Flutter is an asynchronous vibration and is generally treated by an all-blade spectrum method [2]. This method supposes that every blade vibrates at the same frequency and at the same amplitude. It works well for a quasi-perfectly tuned disc, but it is not suitable if mistuning causes vibratory energy localizations. Mistuning is sometimes used by industrials to suppress cases of flutter and unstable mistuned rotor analysis cannot be found in the litterature.

The simulated assembly has 25 blades. The mistuning's pattern is shown on the figure 6. We assume that its diameter is 440 mm and the rotor's speed is 18600 rpm. 5 sensors are disposed around the casing. Their positions are randomly chosen (0°, 38°, 47°, 88°, 103°). The equivalent Nyquist frequency is 775 Hz.

The frequency range of interest is between 0 and 20 000 Hz. A synchronous excitation is considered : a 7th Engine Order. It does not coincide to any eigenfrequencies. Furthermore the system is aerodynamically unstable (flutter). A part of the initial stability diagram is shown on the figure 7. One mode is unstable : it is the mode at 1370 Hz.

The mode shape of the flutter mode is shown on the figure 8. This mode is strongly localized on the blades 19-21.
<figure><img src="image_7.png" /><figcaption>Figure 6. MISTUNING PUT ON THE SPRINGS $k_b$ AND $k_c$</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 7. INITIAL STABILITY DIAGRAM</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 8. FLUTTER MODE SHAPE</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 9. BLADE 20 DISPLACEMENTS</figcaption></figure>

The displacement on the blade 20 seen by the sensor at 0° is shown on the figure 9. The flutter is weak, so the vibration slowly grows.

The proposed method was tested on 30 revolutions. The figure 10 shows the first obtained spectrum by a nonuniform Fourier transform. It seems to contain a lot of modes. The major part of them are replicas due to the severe undersampling. This spectrum cannot be studied directly without previous knowledge of true modes.

<figure><img src="image_11.png" /><figcaption>Figure 10. FIRST SPECTRUM</figcaption></figure>

The algorithm gives the spectrum shown on the figure 11. One mode can easily be identified at 1370 Hz. It's the flutter mode. The precision in frequency is 2 Hz. Amplitudes of artifacts due to undersampling disappear. Spectra are estimated on all blades and the flutter mode can be precisely studied (figure 12). As expected by the simulation, the mode is strongly influ-
<figure><img src="image_12.png" /><figcaption>Figure 11. FINAL SPECTRUM</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Figure 12. BLADES RESPONSES AMPLITUDES AT 1370 Hz</figcaption></figure>

enced by the mistuning. The major part of the vibratory energy is contained in the blades 19-21. There is a good correlation with the theoretical flutter mode (figure 8).

We can identify by this spectral estimator localizations of energy in mistuned bladed assemblies for a phenomena like mistuned flutter. The whole assembly can be studied, blade per blade : this is a great advantage compared to strain gauges which only monitor one or two blades.

## Experimental case

Tests were made on a bladed disc. 4 sensors were disposed around the disc at locations: 0°, 45°, 118° and 189°.

The response of the blade 1 is shown on the figure 13. An asynchronous vibration appears at $t = 137s$, grows until $t = 153s$ and suddenly stops.

A first spectrum computed by an nonuniform Fourier transform is shown on the figure 14. 50 revolutions of data were taken

during the growing part. The choice of the maximum frequency is arbitrary, and frequencies are normalized by this maximum. In practice, the use of the Finite Element Model can give an idea of the frequency range of interest.

The normalized equivalent Nyquist frequency is 0,19, so the frequency range of interest is approximately five times wider than the equivalent Nyquist frequency. As expected by the simulations, this spectrum is severely aliased. Several modes seem to vibrate at different frequencies from 0 to 1.

The described method gives the spectrum on the figure 15. Ten iterations are used to obtain it. The convergence is obtained for a criterion Error < 1%. We can observe that the Signal on Noise Ratio (SNR) grows. Artifacts due to undersampling disappear and permit to identify that the main part of the response has a frequential support above the equivalent Nyquist frequency,

<figure><img src="image_14.png" /><figcaption>Figure 13. DISPLACEMENT OF THE BLADE 1</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Figure 14. FIRST SPECTRUM ESTIMATION OF THE BLADE 1</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>Figure 15. FINAL SPECTRUM ESTIMATION OF THE BLADE 1</figcaption></figure>

between 0,2 and 0,3 normalized frequency.

By this experimental case, we can conclude that the algo-
rithm is suitable for real cases.

CONCLUSION

In this paper we propose a new method in order to analyse tip-timing data. A bladed disc simulator is described. It can simulate aeroelastic effects like flutter and provide tip-timing data close to those obtained on test rigs.

The main features of tip-timing measurements have been ex-
plained with an emphasis on the undersampled and irregular na-
ture of the data. A spectral estimator designed for this particular
sampling gives good results on simulated data of mistuned flut-
ter. The frequency range of interest of estimated spectra can be
several times the equivalent Nyquist frequency. We also showed
that the method is able to analyse experimental forced response
data.

By this method, the response level of each blade can be ob-
tained. Contrary to methods usually used for tip-timing data
analysis, no assumptions about mechanical behaviour of bladed
discs were made in order to obtain a parameter estimator. Hence
this method can analyse any phenomena, including forced re-
sponse and instability for tuned and mistuned rotors.

REFERENCES

[1] Carrington, I. B., 2002. “Development of blade tip timing data analysis techniques”. PhD thesis, University of Manchester. Dynamics and Aeroelasticity research Group, Mechanical Engineering Department, School of Engineering, U.K.

[2] Zielinski, M., and Ziller, G., 2000. "Noncontact vibration

measurements on compressor rotor blades”. *Measurement Science and Technology*, **11**, july, pp. 847–856. Issue 7.

[3] Stoica, P., and Sandgren, N., 2006. “Spectral analysis of irregularly-sampled data: Paralleling the regularly-sampled data approaches”. *Digital Signal Processing*(16), pp. 712–734.

[4] Landau, H. J., 1967. "Necessary density conditions for sampling and interpolation of certain entire functions". *Acta Mathematica*.

[5] Raman Venkataramani, Y. B., 2000. “Perfect reconstruction formulas and bounds on aliasing error in sub-nyquist nonuniform sampling of multiband signals”. *IEEE Transactions on information theory*, **46**, pp. 2173–2183.

[6] Patrick Vandewalle, Luciano Sbaiz, J. V. M. V., 2007. "Super-resolution from unregistered and totally aliased signals using subspace methods". *IEEE Transactions on signal processing*, **55**, pp. 3687–3703.

[7] Cormac Herley, P. W. W., 1999. "Minimum rate sampling and reconstruction of signals with arbitrary frequency support". *IEEE Transactions on information theory*, **45**, pp. 1555–1564.

[8] Lo, K. C., 1999. “A novel algorithm for computing autocorrelation of randomly sampled sequences”. *Circuits, Systems, and Signal Processing*, **18**(6), november, pp. 525–537.

[9] Greitans, M., 2001. “Multiband signal processing by using nonuniform sampling and iterative updating of autocorrelation matrix”. In Proceedings of the 2001 International Conference on Sampling Theory and Application, pp. 85–89.

[10] Greitans, M., 2003. “Enhanced signal processing in time and frequency domains in the case of non-uniform sampling”. In International Conference on Sampling Theory and Application SampTA’03, pp. 35–37.

[11] Greitans, M., 2005. “Advanced processing of nonuniformly sampled non-stationary signals”. *Elektronika ir elektrotechnika*, **3**(59).

[12] M.Kay, S., 1988. *Modern spectral estimation*. Prentice-Hall, p. 373.

[13] Liepin'sh, V. Y., 1996. "An algorithm for evaluating a discrete fourier transform for incomplete data". *Automatic Control an Computer Sciences*, **30**(3), pp. 27–40.

[14] Daniel Potts, Gabriele Steidl, M. T., 2001. *Modern Sampling Theory: Mathematics and Applications*. Birkhauser Boston, ch. Fast Fourier transforms for nonequispaced data: A tutorial, pp. 249–274.

[15] Gallego-Garrido, J., Dimitriadis, G., and Wright, J. R., 2002. “Development of a multiple modes simulator of rotating bladed assemblies for blade tip-timing data analysis”. In Proceedings of the 2002 International Conference on Noise and Vibration Engineering, ISMA, pp. 1437–1446.

[16] Salhi, B., 2007. "A subspace method for modal identifica-
tion of bladed assemblies using blade tip-timing data”. In Proceedings of GT2007 ASME Turbo Expo.

[17] Marshall, J. G., and Imregun, M., 1996. "A review of aeroelasticity methods with emphasis on turbomachinery applications". *Journal of Fluids and Structures*, **10**, pp. 237–267.