Nonintrusive turbomachine blade vibration measurement system

ARTICLE IN PRESS Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 21 (2007) 1717–1738

Nonintrusive turbomachine blade vibration measurement system Pierre Beauseroy, Re´gis Lengelle´ System Modelling and Dependability Group - FRE CNRS 2848, Charles Delaunay Institute, Universite´ de Technologie de Troyes BP2060 10010 TROYES Cedex, France Received 13 April 2006; received in revised form 13 July 2006; accepted 15 July 2006 Available online 29 September 2006

Abstract Conventional measurement systems for monitoring blade vibration generally use strain gauges attached to the surface of blades. Alternative noncontact measurement techniques are investigated in order to minimise the adverse effects of this method. In this paper, a new method based on groups of regularly spaced optical sensors and on apposite treatments is presented. An analytical model of the proposed monitoring system is introduced and some properties are studied. A general upper bound of the dynamic range is proposed. Some examples are presented and discussed. They assess the efficiency of this technique. Some specific situations are commented. The main advantage of the new system is its ability to analyse multicomponent blade vibrational signals on a large dynamic range with a reduced number of probes. Another interesting feature is that it enables the monitoring of all blades in a row. r 2006 Elsevier Ltd. All rights reserved. PACS: 89.20.Bb; 07.50.Qx; 07.07.Df Keywords: Turbomachine; Spectral analysis; Undersampled signal; Noncontacting vibration measurement; Multisampling; Dynamic range

1. Introduction Among all the components of a turbomachine, blades are especially important. While rotating, turbine blades experience vibrations which generate stress and can cause damage to the engine. Typically, blade vibration is dominated by the unsteady flow phenomena and the interaction effects set up by vibrations of blades within a high-velocity compressible fluid medium. The amplitude and frequency of these vibrations have a deep impact on fatigue life, performance and integrity of the engine. The accurate estimation of blade vibration characteristics according to rotational frequency is a crucial issue for the design of a new engine. Thus, turbine engine conception relies on instrumentation and test procedures to verify the design and to assess analytical models of engine behaviour.

Corresponding author. Tel.: +33 325 71 56 83; fax: +33 325 71 56 99.

E-mail addresses: [email protected] (P. Beauseroy), [email protected] (R. Lengelle´). 0888-3270/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2006.07.015

ARTICLE IN PRESS 1718

P. Beauseroy, R. Lengelle´ / Mechanical Systems and Signal Processing 21 (2007) 1717–1738

For a long time, monitoring systems were designed to measure blade vibration characteristics. Conventional measurement systems use strain gauges attached to the blade surface [1]. Each strain gauge gives out an analogue signal related to blade deformation. In order to be analysed, the signal of each gauge is transmitted radiotelemetrically to a recorder. This instrumentation is well established but it suffers from many drawbacks: it is costly, time consuming to set up and often interferes with the aerodynamic and mechanical properties of the engine. As a consequence, to limit these disadvantages, only a few blades are instrumented and monitored. So, the behaviour of noninstrumented blades has to be inferred from the information obtained. Alternative noncontacting measurement techniques have been developed such as magnetic, inductive, capacitive, optical or acoustic Doppler methods [2–10] and are still under investigation in order to minimise such adverse effects. The most promising and most commonly used system is based on stationary optical sensors placed in the engine’s casing. These sensors measure the arrival time of passing blades. It is known as the tip-timing system. As developed in part 2, blade deflection can be estimated using arrival time. For a given blade, the system gives a measure of deflection each time the blade passes in front of a sensor. The sequence of consecutive deflection measures of one blade generates a deflection signal which is sampled according to the engine speed and the optical probe’s positions. Contrary to strain gauges, with this system it is possible to analyse each blade deflection’s signal. One may expect to measure the vibration responses of all blades of a turbomachine. However, when using conventional spectral analysis techniques, the ability to estimate vibration response from deflection signal strongly depends on the number and the positions of optical probes. Due to engine design constraints, the number of probes has to be restricted. In such cases, the capability to identify vibration frequencies and amplitudes using a deflection signal is very limited if compared to strain gauge measure. To improve its performance, some specific methods have been developed [11,12,25–27]. These methods are based on the definition of a specific processing of the deflection signals and particular locations of a small number of probes. Their application is limited to cases where each blade responds at a single frequency. When no prior knowledge is introduced, this condition is not guaranteed. In this paper, we present a new method protected by patent number WOFR 0101128 [13]. This method is based on several groups of regularly spaced optical probes and on suitable processing. Its objective is to extend the ability of noninterfering optical monitoring system to perform spectral analysis in a wide frequency range when blades are vibrating at more than one frequency. A detailed review of tip-timing systems is presented and the problem’s formulation is introduced in Section 2. The proposed method is developed in Section 3 and applications are presented and discussed in Section 4. The concluding section synthesises the main results and possible extensions of this work. 2. Blade vibration measurement systems Measurement systems are used to obtain amplitudes and frequencies of blade vibration in relation to turbomachine speed and specific running situations. Among all blade vibration monitoring methods, strain gauge is the most widely used measurement system. It will be described first. Methods based on tip-timing are next discussed and an analytical formulation of the problem is introduced. 2.1. Strain gauge telemetry methods This method relies on measurements of the local deformation of the blade using strain gauges. The gauges are attached to the surface of critical points of the blades. The amplitude of their deformations is transmitted to monitoring and recording devices using a fairly complex and costly radio telemetry system. This monitoring system is a well established technique which provides extremely valuable information about blade behaviour while the turbomachine is in function. But it has to suffer from many problems. Significant difficulties are to be met in the installation process of strain gauges and telemetry devices. As a result, the number of monitored blades is limited. Thus, the system delivers incomplete information about blade behaviour. A second major consequence is the impact of the measurement system on the performance of the turbomachine. The installation of a telemetry system in the rotating part of the turbomachine is quite complicated and may influence the mechanical behaviour of the turbine. The presence of wire paths and gauges on the blade surface

ARTICLE IN PRESS P. Beauseroy, R. Lengelle´ / Mechanical Systems and Signal Processing 21 (2007) 1717–1738

1719

can also affect the reaction of blades and modify flow causing some bias to the measurement. A third consequence is the system reliability and availability. The life expectation of gauges and their temperature capability are limited, the wires attachment techniques also need to support high temperature. These different points may cause partial or global failures. Fixing system faults can cause serious prolongation of measurement campaign schedule resulting in design delay and additional costs. To overcome these problems, many experimental measurement systems have been designed and investigated. Inductive probes, laser doppler anemometry or acoustic doppler have been tested [14]. For example, Staeheli in [15] has proposed to attach a magnet at the tip of one blade and to install a meandershaped winding in the casing of the turbine. The vibrational response of a blade can be isolated by analysing the induced voltage signal in the winding. Finally among investigatory methods, much attention has focused on nonintrusive techniques and especially those based on tip-timing measurement. 2.2. Tip-timing methods This particular set of techniques relies on static probes located in the assembly casing of the turbine. Different kinds of probes may be used: optical, capacitive or inductive. These probes are mounted radially along rotor blades circular trajectory (Fig. 1). The experiments reported in literature up to now [1] indicate that only optical probes possess the required quality to design monitoring systems which meet industrial needs. The developments presented in this paper are illustrated and discussed on the basis of optical probes but are independent of the probing principle. In the absence of any structural vibration, the time at which the tip of a given blade arrives at a specific observation spot located on its trajectory (namely the arrival time) depends only on four variables:

   

the time, the location of the specified observation spot, the rotational speed of the rotor (which we first assume to be constant in time), a reference angular position (at time t ¼ 0 for example).

When a blade experiences vibrations, the arrival time also depends on vibration characteristics. Under some conditions, which will be discussed later, the blade tip motion and, by extension, its vibration can be characterised from such data. An analytical formulation of the problem is introduced in the next subsection to analyse the relation between signal data and vibration. Further developments are carried out in order to study the properties of tip-timing systems.

sensor

time reference recording, analysis, monitoring...

Fig. 1. Conceptual configuration of the nonintrusive vibration measurement system.

ARTICLE IN PRESS 1720

P. Beauseroy, R. Lengelle´ / Mechanical Systems and Signal Processing 21 (2007) 1717–1738

2.3. Measurement of blade deflection using tip-timing Consider a given rotor with L blades and K probes installed in the assembly casing of a turbine. The location of probe k can be referenced by yk its relative angular position compared to probe 0 in the casing. Without loss of generality the angular position of probe 0 (which is arbitrarily chosen) is set to 0. Assuming the turbine is rotating at a constant speed and without any vibration, the angular position j~ l ðtÞ of blade l is at any time given by the relation equation: j~ l ðtÞ ¼ fl þ 2pF r t,

(1)

where fl is the reference position of blade l at time t ¼ 0 and F r is the rotational frequency of the rotor. In this case, blade l is detected by probe k whose angular position is yk every time their angular position matches. In such a case, the following relation must be satisfied: ~ l ðtÞ ¼ yk þ 2pn, j

(2)

where n 2 N corresponds to the number of revolutions carried out by the rotor since an arbitrary time origin. Assume ft~n;k;l g is the set of time values solutions of Eq. (2). The time values t~n;k;l correspond to arrival times for a nonvibrating blade. In a real situation, while the turbine is running, blades experience vibrations. Let xl ðtÞ be the angular deflection of the tip of blade l at time t, due to vibrations. It follows from the composition of rotational motion and deflection that Eq. (1) is modified as follows: jl ðtÞ ¼ fl þ 2pF r t þ xl ðtÞ.

(3)

Let tn;k;l be the true arrival time of blade l in front of probe k at rotation index n. Since the probe is fixed, the angular position of blade l at time tn;k;l equals yk þ 2pn and consequently: jl ðtn;k;l Þ ¼ j~ l ðt~n;k;l Þ.

(4)

Substituting (1) and (3) in Eq. (4), we obtain: xl ðtn;k;l Þ ¼ 2pF r ðt~n;k;l  tn;k;l Þ ¼ 2pF r Dtn;k;l .

(5)

Fig. 2 illustrates a sensor’s signal caused by one blade. The pulses correspond to the detection of the presence of the blade tip in front of the sensor. The upper figure shows the signal when the blade is not vibrating; detection pulses are regularly spaced; their arrival time are ft~n;k;l g. The lower figure represents the signal when the blade is experiencing vibration; the arrival time of pulses are ftn;k;l g. The shifts of the passing time Dt are directly related to the vibratory state of the blade as proved by Eq. (5). The set ftn;k;l g is acquired by probes while values of set ft~n;k;l g have to be determined. The rotational frequency F r can be found with good accuracy by using a shaft-revolution sensor, probe locations yk are found out from the conception scheme of the monitoring system and relative blade position is also known (we assume that they are equally spaced around the rotor). Taking into account these

sensor's signal : non vibrating blade

Δt

sensor's signal : vibrating blade Fig. 2. Measured sensor’s signal.

(t)

(t)

ARTICLE IN PRESS P. Beauseroy, R. Lengelle´ / Mechanical Systems and Signal Processing 21 (2007) 1717–1738

1721

informations, Eq. (1) can be used to estimate t~n;k;l as follows: yk  fl þ 2pn 2pF r   fl yk ¼ Tr þ n þ Tr 2p 2p   f0 yk l Tr þ n þ  ¼ T r, 2p 2p L

t~n;k;l ¼

ð6Þ ð7Þ ð8Þ

where T r is the revolution period ðT r ¼ 1=F r Þ, L the number of blades, yk the relative position of probe k (which can be measured) and fl the angular position of blade l. fl ¼ f0 þ 2pl=L since blades are assumed to be regularly spaced. The constant f0 corresponds to the angular position of the first blade with respect to the position of the shaft-revolution sensor. At low speed (i.e. during ignition of the turbomachine) each blade experiences low level vibrations around an equilibrium position set arbitrarily, without loss of generality, to zero. Thus, the deflection signal mean equals zero and the sampled signal mean too: 1 X xl ðtn;k;l Þ ¼ 0. (9) NK n;k Relation (9) is true for all blades, thus: 1 X xl ðtn;k;l Þ ¼ 0. NKL n;k;l

(10)

The value of f0 has to be determined to calculate the passing times t~n;k;l . The analytical relation between the value f0 and the mean value of signals xl is obtained by substituting Eq. (5) in (10): X X tn;k;l . (11) t~n;k;l ¼ n;k

n;k

The value f0 can be estimated by introducing Eq. (8) in (11):  Xtn;k;l yk l c ¼ 2p þ  n  f . 0 NKL n;k;l T r 2p L

(12)

In experimental situations, observation can last as long as necessary to obtain a very precise estimate of f0 , therefore, the difference between the estimate and the real value is assumed to be negligible in the rest of the paper. Once this value is known, passing times t~n;k;l can be calculated for any blade, any probe at any revolution speed. As a consequence, Eq. (5) proves the equivalence between measuring angular deflection and measuring arrival time of the blade tip. Since this equivalence is true for any blade l, an arbitrary blade can be considered. Thus, to simplify notations, index l is dropped in the following sections. Let us now study the properties of time series xðtn;k Þ. 2.4. Spectral analysis of deflection signal 2.4.1. Sampling function Due to the nature of tip-timing measurements, the deflection signal xðtÞ is sampled at each arrival time tn;k . The ability of the monitoring system to perform spectral analysis is deeply related to sampling times. Considering a system composed of K probes and assuming the signal is measured during N rotations of the rotor, the sampled signal xe ðtÞ is defined by: xe ðtÞ ¼ xðtÞeðtÞ,

(13)

ARTICLE IN PRESS 1722

P. Beauseroy, R. Lengelle´ / Mechanical Systems and Signal Processing 21 (2007) 1717–1738

P PK1 where eðtÞ ¼ N1 n¼0 k¼0 dðt  tn;k Þ may be regarded as the sampling function. dðtÞ is Dirac’s delta function. In this case, the power spectrum of the signal is given by:  !2   N 1 K 1 X X  2 2  jX e ðf Þj ¼ jX ðf Þj  F dðt  tn;k Þ  , (14)   n¼0 k¼0 where FðÞ denotes the Fourier transform and ‘’ stands for convolution. The properties of the sampling function are very important. Our ability to perform spectral analysis for signals belonging to a certain class is highly dependent on its characteristics. 2.4.2. Regular sampling In the case of regular sampling the samples are taken at time tn;k ¼ t þ ðnK þ kÞT r =K (t is a time delay). The power spectrum jEðf Þj2 of the sampling function eðtÞ is given by:   1 X sinðpfNT r Þ 2 2 2 jEðf Þj ¼ ðKNÞ dðf  nKF r Þ  . (15) pfNT r n¼1 Considering Eqs. (14) and (15), the power spectrum of a finite time sampled signal appears to correspond to periodical duplications of the analogue power spectrum convolved by the power spectrum of the time window over which the signal is observed. The duplication period KF r is given by Eq. (15). It is well known that the Fourier transform allows us to reconstruct the analogue signal if the periodically duplicated patterns of the analogue spectrum are disjoint. Otherwise aliasing appears and the transformation is no longer invertible. In the case of a real signal, and without any prior information about the analogue spectrum jX ðf Þj, the invertibility of the sampling process is guaranteed when the Shannon condition is fulfilled. Low-pass filtering is usually carried out before sampling to force the signal to meet Shannon’s condition. In our case, the signal is sampled due to the nature of the monitoring system. No low-pass filtering can be introduced between the vibrational signal and its sampled measures to limit its spectral extent and avoid aliasing. 2.4.3. Irregular sampling In the case of nonregular sampling, the characteristics of the sampling function depend on each set ftn;k g. Thus it becomes very difficult to establish general rules to analyse and interpret the spectrum obtained [16–20]. However, some particular cases may be studied with valuable results such as random sampling (tn;k is drawn from a uniform distribution Uðn þ k=K; n þ ðk þ 1Þ=KÞT r , for example) or periodical sampling (tn;k ¼ tk þ nT r ) as shown by Fig. 3. These examples show that even if specific conditions are fulfilled, spectral analysis of periodical sampled signals becomes rapidly very complex (Fig. 4) while random sampling is theoretically very promising but simply impossible to implement using static sensors. 2.4.4. Tip-timing sampling In our case, the sampling times are arrival times ftn;k g. They are related to the angular position of the probe and to deflection signal by Eqs. (3) and (4). For a probe k whose angular position is yk arrival times ftn;k g are given by:   yk xðtn;k Þ f þ n Tr   ; n ¼ 0; . . . ; N  1, (16) tn;k ¼ 2pF r 2pF r 2p where f is the relative angle between probe indexed zero and the considered blade at revolution zero. As shown by Eq. (16), the sampling times depends on the signal itself. In order to establish precise properties based only on the physical implementation of the monitoring system a formulation has to be found to overcome the problem introduced by this nonconventional type of irregular sampling which cannot be modelled by random sampling. The signal samples xðtn;k Þ are not known and are surely not drawn from uniform distributions. To tackle this problem let us consider: xðtn;k Þ ¼ xðt~n;k Þ þ xn;k , where xn;k is a noise.

(17)

ARTICLE IN PRESS P. Beauseroy, R. Lengelle´ / Mechanical Systems and Signal Processing 21 (2007) 1717–1738

1723

⏐Xe (f)⏐

1.5 1 0.5 0 -5

-4

-3

-2

-1

0

1

2

3

4

5

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5

-4

-3

-2

-1

0

1

2

3

4

5

⏐Xe (f)⏐

1.5 1 0.5 0

⏐Xe (f)⏐

1.5 1 0.5 0 Normalized Frequency Fig. 3. Frequency representations of sampling functions: regular sampling (top), periodical sampling (middle), irregular random sampling (bottom).

⏐Xs (f)⏐

1.5 1 0.5 0 -5

-4

-3

-2

-1

0

1

2

3

4

5

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5

-4

-3

-2

-1 0 1 2 Normalized Frequency

3

4

5

⏐Xs (f)⏐

1.5 1 0.5 0

⏐Xs (f)⏐

1.5 1 0.5 0

Fig. 4. Spectral analysis of a real signal made of two sinusoids whose normalised frequencies are 2.7 and 3.1. Regular sampling (top), periodical sampling (middle), irregular random sampling (bottom).

Thus the studied signal xðt~n;k Þ is considered to be sampled at time t~n;k . Experimental results and analytical developments using realistic assumption about deflection amplitude show that the impact of this noise depends on frequency and that its amplitude is small in the monitored

ARTICLE IN PRESS 1724

P. Beauseroy, R. Lengelle´ / Mechanical Systems and Signal Processing 21 (2007) 1717–1738

frequency range. Therefore, sampling times can be regarded as independent of the signal. They are just considered as a function of probe position and, to lesser extent, of initial rotor position:   yk f þ n Tr þ ; n ¼ 0; . . . ; N  1. (18) t~n;k ¼ 2pF r 2p This simplification enables us to study the property of the sampling function independently of the deflection signal.

⏐Xsanalogique (f)⏐

2.4.5. Determination of the sampling function of the monitoring system Depending on probe positions and due to the spacial periodicity induced by the rotor’s rotations, two cases may be considered: periodical irregular sampling and regular sampling. As shown previously, the simplest situation is achieved with a regular sampling function. This solution is thus adopted for the on going developments. In this case, probes have to be located at regularly spaced locations ðyk ¼ 2pk=K þ y0 Þ. But, as mentioned before, the power spectrum of the sampling function is periodic, producing a periodic spectral representation of the sampled signal. This periodicity presents two adverse effects. The first one is that for a given analogue frequency it is impossible to distinguish the analogue component in the spectrum from its duplication (top of Fig. 4). The second one, called aliasing, occurs when two or more components are duplicated at the same locations (Fig. 5). In the first case the analysis of the spectrum is ambiguous and in the second some components are not measurable. These problems may jointly affect signal spectral representation and are only consequences of monitoring system design and constraints. Usually to limit these adverse effects, one can either use a low-pass filter to fulfil the Shannon condition or increase the sampling frequency. In our case, as mentioned before, the application of a filter is definitely impossible and increasing the sampling frequency implies an increase in the number of probes which is precisely what we want to limit.  But the problem of aliasing is rather marginal. Indeed, due to the nature of blade vibrations, the spectrum X ðf Þ is composed of several narrowband components where the occurrence probability of aliasing is low and the main problem is to recognise analogue components from their duplications.

2

1

⏐Xssampled (f)⏐

0 -5

-4

-3

-2

-5

-4

-3

-2

-1

0

1

2

3

4

5

-1

0

1

2

3

4

5

2

1

0 Normalized Frequency Fig. 5. Spectral analysis of a real signal made of three sinusoids whose normalised frequencies are 2.7, 3.1 and 3.7. Analogue spectrum (top), aliased spectrum of sampled signal (bottom): components at 2.7 and 3.7 are superimposed.

ARTICLE IN PRESS P. Beauseroy, R. Lengelle´ / Mechanical Systems and Signal Processing 21 (2007) 1717–1738

1725

In the next section, a new monitoring system based on multisampling rate is presented to solve this problem. This system has been developed to perform spectral analysis with signals composed of several narrowband components. An analytical development is proposed to establish its properties and limitations. 3. Multisampling analysis 3.1. Principle of the method In order to reach a wide analysis frequency range without placing many sensors in the casing of the turbines a possible solution is to simultaneously sample the signal at several different sampling frequencies. Using different sampling rates enables us to extend the frequency range in which the signal can be analysed without ambiguities, as mentioned before. Since the sampling rate is related to the number of probes, the proposed monitoring system is made of different sets of regularly spaced optical probes. Each set contains a small number of sensors and consequently undersamples the vibrational signal at a specific sampling rate. Each undersampled signal is a specific version of the vibrational signal whose sampling rate depends on the number of probes in the set. The next developments are dedicated to the study of the properties of a spectral analysis device based on several sampling rates. The following simplified analytical definition of vibrational signal xðtÞ has been adopted up to the end of this paper since the signals under considerations are assumed to contain only M þ 1 narrowband components: xðtÞ ¼

M X

am e2jpf m tþjjm .

(19)

m¼0

We assume now that P measurement devices, each composed of K p regularly spaced sensors (Fig. 6), are used to monitor xðtÞ during N rotations of the rotor. Each device supplies one undersampled measurement fxep ðtÞg of the vibrational signal with p ¼ 1; . . . ; P. The Fourier transform X ep ðf Þ of the signal fxep ðtÞg is: X ep ðf Þ ¼ NK p

1 X M X n¼1 m¼0

am ejjm

sinðpðf z ÞNT r Þ jpðf z ÞNT r e , pðf z ÞNT r

(20)

where f z ¼ f  f m  nK p F r . As already mentioned (Eq. (14)) the spectrum and, by extension, the Fourier transform of each measured signal xep ðtÞ repeats the spectrum (respectively, the Fourier transform) of xðtÞ convolved by the Fourier transform of the time window, with periodicity equal to K p F r . A first crucial observation is that analogue components are necessarily present in all spectrum at their true frequencies as shown in Fig. 7.

Commun probe to bothsets Blades

Probe of firstset (3 probes) Probe of second set (5 probes)

Casing of the turbine Fig. 6. Position of probes in the case of two measurement devices composed of K 1 ¼ 3 and K 2 ¼ 5 regularly spaced sensors (one is common to both sets).

ARTICLE IN PRESS ⏐Xsanalogique(f)⏐

1.5

⏐Xs3 sensors(f)⏐

P. Beauseroy, R. Lengelle´ / Mechanical Systems and Signal Processing 21 (2007) 1717–1738

1726

1.5

1 0.5 0 -5

-4

-3

-2

-1

0

1

2

3

4

5

2

3

4

5

-1 0 1 2 Normalized Frequency

3

4

5

Common components 1 0.5

⏐Xs4 sensors(f)⏐

0 -5

-4

-3

-2

-5

-4

-3

-2

-1

0

1

1.5 1 0.5 0

Fig. 7. Spectral analysis of a real signal made of two sinusoids whose normalised frequencies are 2.7 and 3.1. Analogue spectrum (top), spectrum of the signal sampled at F e ¼ 3 (middle), spectrum of the signal sampled at F e ¼ 4 (bottom).

The second observation derives from the fact that some duplications of analogue components in given Fourier transform X ek ðf Þ may not match any component in any other Fourier transform : fX ep ðf Þg with pak. In this case, as a consequence of the first observation, the considered component is a duplication and not an original one. Thus, we define a combined Fourier transform X c ðf Þ which is given by the following equation: X c ðf Þ ¼

1 1 1 X e ðf Þ if X e ðf Þ ¼ X e ðf Þ NK j j NK j j NK k k

X c ðf Þ ¼ 0

8ðj; kÞ 2 f1; . . . ; Pg2 ,

otherwise.

ð21Þ

By studying properties of X c ðf Þ one might expect to discard many duplicated components and to extend the spectral range at which analogue components can be identified. The purpose of the following section is to study the properties of a monitoring system composed of sets of sensors and whose measures are analysed by combining Fourier transforms of the signals acquired by each set of tip-timing detectors, according to Eq. (21). 3.2. Properties Different aspects of the combined spectrum have to be studied:

  

the spectral range to which analogue components can be identified in relation to the number of sets of sensors and the number of sensors in each set, the uniqueness of the solution in this range, the conditions to uniquely identify analogue components in the spectral range if the uniqueness condition is not met.

3.2.1. Spectral range The spectral representation X ep ðf Þ of the vibrational signal xðtÞ sampled by any set p of sensors is periodic. As introduced in a previous subsection (Eqs. (15) and (20)), its period is an integer, multiple of the rotor frequency F r .

ARTICLE IN PRESS P. Beauseroy, R. Lengelle´ / Mechanical Systems and Signal Processing 21 (2007) 1717–1738

1727

The spectral representation which is obtained by combining P periodic spectral representations of period K p F r is also periodic. Obviously its period is an integer multiple of F r . But it may be much larger than a single spectrum period. To determine the largest interval in which analogue frequency can be recognised without ambiguity, called dynamic range in [21], let us define the period f period as the smallest positive value such that X c ðf Þ ¼ X c ðf þ kf period Þ 8f 2 R and 8k 2 Z. As a consequence of Eq. (21): X ej ðf Þ ¼ X ej ðf þ kf period Þ ¼ X ej ðf þ kaj K j F r Þ

8j 2 f1; . . . ; Pg.

ð22Þ

Since Eq. (22) is satisfied for any set of sensors: aj K j ¼ lcmðK 1 ; K 2 ; . . . ; K P Þ 8j 2 f1; . . . ; Pg,

(23)

where lcmðÞ denotes the least common multiple. This result is basically the Chinese remainder theorem [22]. In the case of turbomachine monitoring, the normalised periodicity is given by lcmðK 1 ; K 2 ; . . . ; K P Þ, which is the dynamic range normalised by the rotor frequency F r (which does not depend on the monitoring device). A consequence of this result is that to achieve the largest periodicity, values K 1 ; K 2 ; . . . ; K P should be chosen pairwiseQrelatively prime moduli. In such a case the normalised periodicity of the mixed spectral representation equals Pp¼1 K p . The dynamic range equals the periodicity in the case of a signal containing only one complex component: xðtÞ ¼ ae2jpftþc . But analysed signals contain several real components. In such a case, without introducing any specific assumption about signal components, and despite this first promising result, the dynamic range depends both on the choice of sampling rates and on the number of components. The next sections discuss the relation between the number of components and the dynamic range. 3.2.2. Uniqueness of the solution 3.2.2.1. General case. The first important result has been demonstrated by Xia. In the case of complex signals (which corresponds to the chosen model formulated in Eq. (19)), Xia studied uniqueness conditions for multiple undersampled waveforms using different sampling rates in [23]. He proved the following theorem: Theorem 1 (frequency range ensuring uniqueness). Assume that a complex-valued signal xðtÞ contains M þ 1 different positive frequencies. Let F p ¼ K p F r , p 2 f1; . . . ; Pg be P different sampling rates. Let q be a nonnegative integer defined by: P ¼ ðM þ 1Þq þ r;

0proM þ 1.

(24)

Then the M þ 1 frequencies can be uniquely determined by using the P DFT of signals xep if maxðf 0 ; f 1 ; . . . ; f M Þo maxðK s ; K 1 ; K 2 ; . . . ; K P Þ,

(25)

where Ks ¼

min

0pr1 or2 oorq pM

Ks ¼ 0

lcmðK r1 ; K r2 ; . . . ; K rq Þ if q40,

otherwise.

ð26Þ

Readers interested in the proof of this theorem can find it in [23]. If the number of components is greater than or equal to the number of sets, this theorem implies that q ¼ 0 and K s ¼ 0. Thus it is equivalent to having many groups than to having only the most populated set with the Shannon condition ðmaxðf 0 ; f 1 ; . . . ; f M Þo maxð0; K 1 ; K 2 ; . . . ; K P ÞÞ. If q ¼ 1 then K s ¼ minðK 1 ; K 2 ; . . . ; K P Þ and once again the dynamic range is not improved compared to the dynamic range of the most populated set alone. This theorem shows that, to increase the analysis range it is necessary to have at least twice the number of complex components as the number of sets.

ARTICLE IN PRESS 1728

P. Beauseroy, R. Lengelle´ / Mechanical Systems and Signal Processing 21 (2007) 1717–1738

For example, assume that xðtÞ contains two complex components. Using four sets of respectively 5, 6, 7 and 11 sensors enables us to monitor a signal up to 30F r (the smallest lcm of two-sensor sets). Indeed q ¼ 42 thus: K s ¼ min lcmðK r1 ; K r2 Þ 0pr1 or2

(27)

which is equal to 5  6  F r . This is a very restricted range compared to the periodicity of the combined representation (2310F r in this case). This result shows that the search for a single solution implies the use of a large number of sensors. In particular, if the normalised dynamic range has to be large, due to Eq. (26), K s has to be large. To satisfy this condition with several real components it is necessary for q to be greater than one and thus the set number P has to be large. 3.2.2.2. Restriction to one real component. The special case of a real component having an integer normalised frequency has been studied by Pace et al. in [21]. The following developments slightly expand their results to any frequency. The reduced dynamic range can be defined by an interval ½0; f MIN ½ of the spectrum to which any single real component can be uniquely identified. Thus f MIN is the smallest frequency for which the spectral representations obtained using all the sets of probes can be exactly obtained from a pure sinusoid at an unknown frequency f 0 2 ½0; f MIN ½. Consider a single-frequency real signal xðtÞ ¼ aðe2jpf 0 tþc þ e2jpf 0 tþc Þ with frequency f 0 sampled at P different rates namely K 1 F r ; K 2 F r ; . . . ; K P F r . Power in each spectral representation is concentrated on certain frequencies given by: ( f 0 þ kK i F r ; (28) f 0 þ kK i F r ; where k 2 Z and i 2 f1; . . . ; Pg indicates the probe set number. Consider another single-frequency real signal xRS ðtÞ with frequency f RS 4f 0 sampled with the same monitoring system. The power of its spectral representation is concentrated on some frequencies given by: ( f RS þ kK i F r ; (29) f RS þ kK i F r : Representations of xðtÞ and xRS ðtÞ are identical only if two index sets R ¼ fr1 ; r2 ; . . . ; rR g and S ¼ fs1 ; s2 ; . . . ; sS g exist with R [ S ¼ f1; . . . ; Pg and R \ S ¼ ; so that: ( f 0 þ ki K ri F r ¼ f RS þ k0i K ri F r ; (30) f 0 þ kj K sj F r ¼ f RS þ k0j K sj F r for all ri 2 R and all sj 2 S. This equation expresses, on the one hand, that positive components of xðtÞ match positive components of xRS ðtÞ in spectral representations obtained from probe sets indexed by values of R, and, on the other hand, that positive components of xðtÞ match negative components of xRS ðtÞ in spectral representations obtained from probe sets indexed by values of S. Note that the matching of the negative component of xðtÞ is directly deduced from the positive matching by symmetry. Recombining equations related to both sets gives: ( f RS  f 0 ¼ ki K ri F r þ k0i K ri F r ; (31) f RS þ f 0 ¼ k00j K sj F r  k000 j K sj F r which is equivalent to: ( f RS  f 0 ¼ l lcmðK r1 ; K r2 ; . . . ; K rR ÞF r ; f RS þ f 0 ¼ l 0 lcmðK s1 ; K s2 ; . . . ; K sS ÞF r ; where ðl; l 0 Þ 2 N2 since f RS 4f 0 .

(32)

ARTICLE IN PRESS P. Beauseroy, R. Lengelle´ / Mechanical Systems and Signal Processing 21 (2007) 1717–1738

1729

Adding both equations gives the following result: Fr . (33) 2 The value of the upper bound of the dynamic range f MIN can be defined from this result since f MIN ¼ minR;S;l;l 0 f RS . For given sets R and S the minimum value of f RS is obtained for ðl; l 0 Þ ¼ ð1; 1Þ leading to the final result: f RS ¼ ðl lcmðK r1 ; K r2 ; . . . ; K rR Þ þ l 0 lcmðK s1 ; K s2 ; . . . ; K sS ÞÞ

f MIN ¼ minðlcmðK r1 ; K r2 ; . . . ; K rR Þ þ lcmðK s1 ; K s2 ; . . . ; K sS ÞÞ R;S

Fr . 2

(34)

The normalised upper bound of the dynamic range is defined by: 1 f norm MIN ¼ minðlcmðK r1 ; K r2 ; . . . ; K rR Þ þ lcmðK s1 ; K s2 ; . . . ; K sS ÞÞ2. R;S

(35)

According to the previous theorem, using four sets of, respectively, 5, 6, 7 and 11 sensors enable us to monitor signals containing two complex components up to 30F r . In the case of a real signal the bound is equal to 12ðlcmð6; 7Þ þ lcmð5; 11ÞÞF r ¼ 48:5F r . Preserving only the first three sets, the range is reduced to 7F r for two complex components and to 18:5F r in the case of a monochromatic real signal. Theorem 1 shows that, to insure uniqueness, the number of sets has to be very large. Since the number of components is not a priori known, this result cannot be used for the definition of a monitoring device in the case of a turbomachine. The second part of this section showed that taking into account the symmetry of the spectral representation of real signal allows us to increase the dynamic range. Both results give us bounds for the dynamic range that are indications for the choice of probes sets. However, obtaining a wide dynamic range generally implies a significant number of sensor sets and thus a significant number of sensors. Also, rather than seeking to guarantee the uniqueness of the solution on a large spectral range, we shall study in the next section the frequential properties that the signals must satisfy to preserve a satisfactory range. 3.2.3. Signal property ensuring uniqueness The reduction of the dynamic range for signal containing more than one complex component is basically due to incorrect matching between components in two or more spectral representations. Consider now two complex components whose true frequencies are f 1 and f 2 (f 1 af 2 ). Unfair matching occurs if, for at least one given couple of probes’ sets index ðj; kÞ 2 f1; . . . ; Pg2 : f 1 þ uK j F r ¼ f 2 þ vK k F r ,

(36)

where ðu; vÞ 2 Z2 . Such incorrect matching can occur if: f 1  f 2 ¼ l gcdðK j F r ; K k F r Þ,

(37)



where l 2 Z and gcd stands for greatest common divisor. The smallest difference Df between the frequencies of two components which enables incorrect matching is given by: Df ¼ min jf 1  f 2 j ðf 1 ;f 2 Þ

¼ min gcdðK j ; K k ÞF r ðj;kÞ

¼ lF r

ð38Þ

with l 2 N . This result demonstrates that a sufficient condition to avoid incorrect matching between two complex components is that the difference between their frequency is different from F r or a multiple of F r . If this constraint is satisfied for any two components of a signal xðtÞ, the extent of dynamic range for this signal is given by Eq. (23) and corresponds to the best possible case.

ARTICLE IN PRESS 1730

P. Beauseroy, R. Lengelle´ / Mechanical Systems and Signal Processing 21 (2007) 1717–1738

This is a very promising result but there is no guarantee that all measured signals will fulfil this condition. Previous developments may be summarised as follows:

  



In any given case, to uniquely identify M þ 1 complex components, at least 2ðM þ 1Þ sets of probes are required to extend the dynamic range further than Shannon’s limit. In the case of one complex component, the dynamic range equals the combined spectrum periodicity which is lcmðK 1 ; K 2 ; . . . ; K P ÞF r . In the case of one real component, the dynamic range is equal to ½0; f MIN  where f MIN is defined by Eq. (34). Compared to one complex component, the dynamic range reduction is due to possible incorrect matching between the two complex components of the real signal. But false matching can occur only if very restrictive conditions are fulfilled by the signal under consideration. First, the signal frequency must be multiple of F r =2. As a second constraint, complex signal components must verify ð1=NK j ÞX ej ðf þ uK j F r Þ ¼ ð1=NK k ÞX ek ðf þ vK k F r Þ which implies a special condition on signal phase. If these constraints are not met, the dynamic range extent remains equal to the spectrum periodicity. In general cases incorrect matching is possible only if the difference between frequencies is a multiple of F r and if ð1=NK j ÞX ej ðf 1 þ uK j F r Þ ¼ ð1=NK k ÞX ek ðf 2 þ vK k F r Þ. Thus, amplitude and phase of incorrectly matched components have to be identical. This situation is very unlikely and once again, if it does not occur the dynamic range extent remains equal to spectrum periodicity, which corresponds to the best possible case.

These results show that incorrect matching situations are very unlikely: this is most encouraging. But generally, the measured signal corresponds to the deflection signal plus additive noise. The relation ð1=NK j ÞX ej ðf 1 þ uK j F r Þ ¼ ð1=NK k ÞX ek ðf 2 þ vK k F r Þ which is used to discard incorrect matching, has to be considered with regard to signal-to-noise ratio. The number of accepted false matches will increase in relation to the noise level. Noise can be caused by imperfect sampling (previous development are based in the hypothesis of perfect sampling), sensor imperfection, casing vibrations or incorrect probe positions. The most critical factor according to the measurement system performance is probably the positions of sensors. As shown by Fig. 8 the sampling function is substantially disturbed by sensor shift (in the presented case one sensor has been shifted by 21). To limit the effect of noise and the occurrence of incorrect matching described above, the variation of rotor revolution speed may be used in addition to the described monitoring system. Obviously, the variation of rotor rotational speed induces a variation of the sampling frequency for each set of sensors. Blade vibrations vary continuously according to rotor speed and thus according to time. It follows from this remark that a time dependent combined Fourier transform X c ðt; f Þ of signals measured during rotor speed variation should exhibit continuous tracks. The problem of identifying analogue components of the vibrational signal consists of identifying the tracks which are not duplications due to sampling. If, by coincidence, two components match while they should not at a given rotating frequency F r ðtÞ, it is very unlikely that the condition of Eq. (21) holds over a wide range of rotation speed contrary to the analogue frequencies whose tracks are continuous. These last considerations, in addition to previous developments, are key elements that have enabled us to set up the principle of the multisampling monitoring system using variable rotation speed. 3.3. Multisampling monitoring system using variable revolution speed The proposed monitoring system is made of several sets of optical regularly spaced sensors attached to the casing of the turbomachine. The number of sensors of each set is chosen to ensure a large dynamic range according to previous analytical developments. To study the vibrational behaviour of blades, a sampled signal xl ðtÞ is acquired from tip-timing measurements of each set of probes according to Eq. (5), while rotational speed changes. The time frequency representation X c ðt; f Þ [24] is obtained by combining the time frequency representations related to each set of sensors. Then continuous tracks are selected in a given bandwidth as potentially analogue components of the blade vibrational signal and short tracks are discarded.

ARTICLE IN PRESS P. Beauseroy, R. Lengelle´ / Mechanical Systems and Signal Processing 21 (2007) 1717–1738

1731

5 regularly spaced sensors 1.4

⏐Xe (f)⏐

1.2 1 0.8 0.6 0.4 0.2 0

⏐Xe (f)⏐

0

5

10

15

20

25

30

35

40

45

50

5 regularly spaced sensors − one sensor shifted of 2°

1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

5

10

15

20 25 30 35 Normalized frequency

40

45

50

0

0

5

5 Normalized Frequency

Normalized Frequency

Fig. 8. Sampling functions with five sensors—equally spaced (upper curve)—one sensor shifted by 21 (lower curve).

10 15 20

10 15 20 25

25 30

30 1

1.2

1.4 1.6 1.8 2 2.2 2.4 2.6 Normalized Revolution Frequency

2.8

3

1

1.2

1.4 1.6 1.8 2 2.2 2.4 2.6 Normalized Revolution Frequency

2.8

3

Fig. 9. Time-frequency representations of a synthesised signal using two sets of three and four probes—revolution speed is varying.

Fig. 9 represents the time-frequency analysis of a synthesised signal. The time-frequency representations result from two sets of three and four probes. The synthesised signal is composed of two frequencies: 5.1 and 8.9 Hz. Revolution frequency varies linearly from 1 to 3 Hz. Fig. 10 provides the common components of the two previous representations. In this section, an analytical study of the proposed monitoring system composed of several sets of probes has been carried out. It has been shown that, in general cases, to insure uniqueness of solution, the number of sets has to be too large compared with our goal. But using good matching conditions (Eq. (21)) and time varying rotation speed enabling to overcome these limitations and to implement a monitoring system which fulfils our objective using a very small number of sensors. Note that situations where aliasing occur are much more complicated and are beyond the scope of this paper.

ARTICLE IN PRESS 1732

P. Beauseroy, R. Lengelle´ / Mechanical Systems and Signal Processing 21 (2007) 1717–1738

0

Normalized Frequency

5

10

15

20

25

30 1

1.2

1.4

1.6 1.8 2 2.2 2.4 Normalized Revolution Frequency

2.6

2.8

3

Fig. 10. Common tracks of both time-frequency representations.

In the next section the proposed monitoring system is simulated by carefully resampling strain gauge signals. Comparing results of the multisampling method with original strain gauge spectral analysis will give us a good overview of the new method’s ability and limitations in almost real cases. 4. Experimental results As shown in Eq. (5), the tip-timing measurement method is equivalent to sampling the vibratory amplitude of a considered blade at its passing time in front of optical probes. To undertake the experimental study of the suggested method without implementing the entire system we simulate the tip-timing measurement system. This approach was made possible because SNECMA has constituted a remarkable archive of strain gauge signals. Signals of interest were drawn from these records, resampled according to the suggested measurement system and analysed. The following section presents two examples of these simulations. 4.1. Simulation process Each strain gauge signal of the database is composed of two tracks, one for the blade deflection signal and the second for a revolution sensor signal. The amplitude of the blade vibration seen by the set of probes number p is deduced from the first track of strain gauge signal by sampling it at the theoretical passing time of the blade in front of the optical sensors. Fig. 11 shows a strain gauge signal and a simulated tip-timing signal. The theoretical passing times ftu;p g of the blade are deduced from the revolution’s sensor signal. The number and the position of these sensors are freely fixed. For each set of sensors, a tip-timing signal sp is interpolated using the strain gauge signal according to the equation:   1 X Fe N kF e 2jpðkF e =NÞt Sgauge (39) sp ðtÞ ¼ e N k¼0 N

ARTICLE IN PRESS P. Beauseroy, R. Lengelle´ / Mechanical Systems and Signal Processing 21 (2007) 1717–1738

1733

isplacement

strain optic

Time Fig. 11. Strain gauge signal and interpolated tip-timing signal.

at time t 2 ftu;p g where Sgauge ðf Þ is the Fourier transform of the strain gauge signal sgauge ðtÞ; F e is its sampling rate and N=F e is its duration. 4.2. Examples To illustrate the efficiency of the proposed method, two signals were chosen in the database. The first signal corresponds to a blade experiencing a single vibration at 3:2F r . The second signal is measured while a blade is vibrating at two different synchronous frequencies (frequencies are multiples of F r ). Both cases are complex since signal components are such that relation (38) holds, causing an incorrect match of components and reducing the dynamic range as discussed in Section 3.2. 4.2.1. Single vibration The monitored blade is vibrating at 3:2F r . Fig. 12 shows the spectral analysis of the sampled strain gauge signal. Some noise appears at low frequencies. Spectrogram of signals acquired (by simulation) with groups of three and five sensors are shown in Fig. 13. The duplication symmetry axes of the representations are indicated by dot lines. As expected, the combined spectrogram (Fig. 14) shows a continuous track at 3:2F r (dotted lines indicate multiples of F r ). An extra track remains at 11:8Fr which correspond to the duplication of the negative spectral component ð3:2F r Þ: ( 11:8F r ¼ 3:2F r þ 5  3F r ; 11:8F r ¼ 3:2F r þ 3  5F r : Note that with three and five sensors the maximum dynamic range is ðð3  5Þ=2ÞF r . For the simulated monitoring system this duplication is normal. To suppress this, another monitoring system has to be proposed by increasing either the number of probes in each set or the number of sets. 4.2.2. Several synchronous vibrations This strain gauge signal is made up of two major synchronous components at the frequencies 4F r and 13F r , respectively, indicated by dotted lines in Fig. 15. This figure shows that vibration amplitude of the second

ARTICLE IN PRESS P. Beauseroy, R. Lengelle´ / Mechanical Systems and Signal Processing 21 (2007) 1717–1738

1734

10 9

Frequency (arbitrary unit)

8 7 6 5 4 3 2 1 0 0

1

2

3

4 5 6 7 Time (arbitrary unit)

8

9

10

10

10

9

9

8

8

Frequency (arbitrary unit)

Frequency (arbitrary unit)

Fig. 12. Spectrogram of strain gauge sensor signal—first example—dot lines indicate multiples of F r .

7 6 5 4 3 2

7 6 5 4 3 2 1

1

0

0 0

1

2

3

4 5 6 7 Time (arbitrary unit)

8

9

10

0

1

2

3

4 5 6 7 Time (arbitrary unit)

8

9

10

Fig. 13. Spectrogram of simulated three and five sensors signals.

component is much more important than the first one. In this case the amplitude ratio is larger than 10. An attentive examination of the spectrogram reveals additional components at all multiples of F r . Their amplitudes are very low compared to the two main components and can be considered as negligible. In this case, two simulated sets of five and seven probes were used to monitor the blade. The dynamic range of such a monitoring system is at most equal to 17:5F r according to Eq. (23). Fig. 16 shows the spectrogram of the obtained signals. The duplication symmetry axes are represented by dot lines.

ARTICLE IN PRESS P. Beauseroy, R. Lengelle´ / Mechanical Systems and Signal Processing 21 (2007) 1717–1738

1735

10 9

Frequency (arbitrary unit)

8 7 6 5 4 3 2 1 0 0

1

2

3

4 5 6 7 Time (arbitrary unit)

8

9

10

Fig. 14. Combined spectrogram with two sets of three and five probes.

6

Frequency (arbitrary unit)

5

4

3

2

1

0 0

1

2 Time (arbitrary unit)

3

Fig. 15. Spectrogram of strain gauge sensor signal—second example.

Note that choosing a set of three probes would have introduced aliasing. Using such a group, the duplications of 4F r and 13F r undersampled components are located at the same frequencies f m : 8 > :

¼

3mF r þ F r

¼ ¼

4F r þ ðm  1Þ3F r 13F r þ ðm  4Þ3F r :

ARTICLE IN PRESS P. Beauseroy, R. Lengelle´ / Mechanical Systems and Signal Processing 21 (2007) 1717–1738

5

5

4

4 Frequency (arbitrary unit)

Frequency (arbitrary unit)

1736

3

2

1

3

2

1

0

0 0

1

2

3

0

Time (arbitrary unit)

1

2

3

Time (arbitrary unit)

Fig. 16. Signal spectrogram for two simulated sets of five and seven sensors.

5

Frequency (arbitrary unit)

4

3

2

1

0 0

1

2 Time (arbitrary unit)

3

Fig. 17. Combined spectrogram.

Fig. 17 represents the combined spectrogram. Components at 13F r and 4F r are present (indicated by dotted lines). A wideband component appearing at 10:3F r still remains. Components at 3F r and 6F r which are duplications of the 13F r positive and negative components are not completely removed. This is due to the small additional synchronous components which introduce noise and reduce the system’s sensitivity. The two short duplicated tracks may be correctly diagnosed as duplications by in depth examination of the spectrogram. This result illustrates the behaviour of the proposed system in a difficult situation. To obtain the same dynamic range using a standard monitoring system, 35 probes would need to be installed in the casing of the turbomachine. Since one probe may be common to both sets of sensors, only 11 probes are required in this case.

ARTICLE IN PRESS P. Beauseroy, R. Lengelle´ / Mechanical Systems and Signal Processing 21 (2007) 1717–1738

1737

5. Conclusion Nonintrusive blade monitoring system is of major concern for the development of new turbomachines. Measurement systems may be divided into two groups depending on the monitored variables used to obtain amplitudes and frequencies of blade vibration. The first group of systems is based on direct measurements of blade deflection. Strain gauges attached to the surfaces of blades is the most widely used method within this group and is considered as the reference for development of new methods. The second category of methods is based on indirect measurement systems. The monitored variables are not the deflection of blades but certain related measurement. The expected benefits of this type of methods are an easier implementation of the monitoring system and a more complete vibrational control of all blades. In this publication a nonintrusive indirect measurement system is studied. The specific monitoring system is based on the spectral analysis of tip-timing signals. The system consists of several sets of regularly spaced optical probes located in the casing of the turbomachine. The tip-timing signals acquired by each set are shown to be equivalent to sampled deflection signals of the blades. When the deflection signal contains a limited number of narrowband components, the use of several sets of probes enables us to enlarge the dynamic range of spectral analysis by combining all the individual spectra. Some upper bounds for the dynamic range are discussed depending on the number of complex components under consideration. It is shown that the upper bound can be reached if the difference between complex component frequencies are not a multiple of the revolution frequency. To reduce the probability of the occurrence of such a conjunction, a multisampling technique with variable rotor speed is introduced. To identify vibrational frequencies of a blade, the problem is to detect continuous tracks in a combination of the time-frequency representations obtained for each set of probes. The experimental study of the proposed method has been realised using a simulated monitoring system. The simulated tip-timing signals are interpolated true strain gauge signals at estimated passing times of a blade in front of virtual probes located in the turbomachine. The number of sets of probes and the number of probes in each set are freely fixed. This study assesses the quality of the proposed method. Its main characteristics are:

  

the ability to monitor each blade individually and thus to have a more complete knowledge of the turbomachine global behaviour; compared with a single group of regularly spaced probes, the capability to significantly enlarge the dynamic range with a restricted number of sensors; the capability of identifying more than one real vibrational component.

However, these advantages give rise to certain problems. The first problem is the difficulty in managing aliasing. This problem becomes particularly serious when the number of components is large and differences between components frequencies are multiples of the rotational frequency. When these conditions are encountered at a punctual rotational frequency, time varying analysis enable us to discard incorrect matching by in depth reading of the combined time-frequency representations. In the case of synchronous components the problem remains since frequential differences evolve according to rotational frequency. Two solutions may be considered to tackle this problem. Extra groups of probes, or a limited number of strain gauges may be used to eliminate some aliased components. But this solution is in contradiction with the initial aim of this study. The second solution is to design the monitoring system, which means choosing the number of groups and the number of probes in each group, by taking into account the predicted vibrational modes resulting from structural analysis. This strategy enables us to choose the duplication frequency of each group in order to reduce aliasing probability. Finally, in the case where aliasing does occur, complementary processing may be used to detect it without identifying precisely the true vibrational components. These latter parts of the study are still under investigations. The proposed method has been successfully applied by SNECMA in order to measure vibration frequencies and amplitudes. The noise level, which could result from errors in blades arrival time measurements, appears to be very low. While the extension of this method to vibration modes analysis has been studied jointly with

ARTICLE IN PRESS 1738

P. Beauseroy, R. Lengelle´ / Mechanical Systems and Signal Processing 21 (2007) 1717–1738

SNECMA, and considered to be potentially addressed by replacing single sensors by linear arrays of sensors aligned with the shaft, the corresponding developments do not appear in this paper. This extension will be addressed in further work. Acknowledgement The authors would like to thank SNECMA Moteurs for supporting this research and especially J.L. Eyraud who followed the development of this work for SNECMA Moteurs. References [1] R.C. Anderson, W. Atkinson, T. Bonsett, J. Osani, The Propulsion Instrumentation Working Group: Introduction and Discussion of Subteam Activities, American Institute for Aeronautics and Astronautics, 1999. [2] P.E. Mc Carty, J.W. Thomson, R.S. Ballard, Noninterference technique for measurement of turbine engine compressor blade stress, Journal of Aircraft 19 (1) (1982). [3] I. Davinson, J.E. Mutton, R. Parker, A. Parnell, J.P. Roberts, Optical fan blade profile measurement in rotating turbomachinery, International Gas Turbine and Aeroengine Congress and Exhibition, ASME, Brussel, June 11–14, 1990. [4] I. Davinson, The use of optical sensors and signal processing in gas turbine engines, Integrated Optics and Optoelectronics II, vol. 1374, SPIE, 1990, pp. 251–265. [5] H. Jones, A Nonintrusive rotor blade vibration monitoring system, International Gas Turbine and Aeroengine Congress and Exhibition, ASME, Birmingham, June 10–13, 1996. [6] A. Kurkov, J. Dicus, Synthesis of blade flutter vibratory patterns using stationary transducers, in: Gas Turbine Conference and Product Show, London, 1978. [7] H. Roth, Vibration measurements on turbomachine rotor blades with optical probes, in: Proceedings of the Joint Fluid Engineering Gas Turbine Conference and Product Show, New Orleans, 1980. [8] W.B. Watkins, W.W. Robinson, Noncontact engine blade vibration measurement and analysis, in: AIAA/SAE/ASMA/ASEE 21st Joint Propulsion Conference, Monterey, 1985. [9] I.Y. Zablotskiy, Y.A. Korostelev, L.B. Sviblov, Contacless measuring of vibrations in the rotor blades of turbines, Lopatochnyye Mashiny i Struynyye Apparaty, Sbornik Statey, vol. 6, 1972, pp. 106–121, English translation: Technical translation FTD-HT-23673-74, 1974, pp. 1–19. [10] M. Zielinski, G. Ziller, Optical blade vibration measurement at MTU, AGARD PEP 90th Symposium, 1997. [11] S. Heath, M. Imregun, An improved single parameter tip-timing method for turbomachinery blade vibration measurements using optical laser probes, International Journal of Mechanical Science 38 (10) (1996) 1047–1058. [12] S. Heath, A new technique for identifying synchronous resonances using tip-timing, International Gas Turbine and Aeroengine Congress and Exhibition, ASME, Indianapolis, IN, June 7–10, 1999. [13] P. Beauseroy, J.L. Eyraud, R. Lengelle´e, Proce´de´ de mesure sans contact de vibrations d’un corps tournant, Brevet Franc- ais No. FR0004768, Brevet Europe´en No. EP01400937.7, Brevet International No. WOFR0101128, April 2001. [14] J.R. Kadambi, R.D. Quinn, M.L. Adams, Turbomachinery blade vibration and dynamic stress measurements utilizing nonintrusive techniques, Transactions of ASME 111 (1989) 468–474. [15] W. Staeheli, Inductive method for measuring rotor blade vibrations on turbomachines, Sulzer Technical Review 57 (3) (1975). [16] J. Jimenez, J.C. Agui, Approximate reconstruction of randomly sampled signals, Signal Processing 12 (1987) 153–168. [17] E.L. Plotkin, L.M. Roytman, M.N.S. Swamy, Reconstruction of nonuniformly sampled band limited signals and jitter error reduction, Signal Processing 7 (1984) 151–160. [18] I.W. Sandberg, On the properties of some systems that distort signals, Bell System Technical Journal (1963) 2003–2046. [19] R.G. Wiley, Recovery of band limited signals from unequally spaced samples, IEEE Transactions on Communications 26 (1978) 135–137. [20] L.J. Yen, On nonuniform sampling of bandwidth limited signals, IRE Transactions on Circuit Theory CT-3 (1956) 251–257. [21] P.E. Pace, R.E. Leino, D. Styer, Use of the symmetrical number system in resolving single-frequency undersamples aliases, IEEE Transactions on Signal Processing 45 (1997) 1153–1160. [22] J.H. McClellan, C.M. Rader, Number Theory in Digital Signal Processing, Prentice-Hall, Englewood Cliffs, NJ, 1979. [23] X.-G. Xia, On estimation of multiple frequencies in undersampled complex valued waveforms, IEEE Transactions on Signal Processing 47 (1999) 3417–3419. [24] P. Flandrin, Temps-frquence, second ed., Herms, Paris, 1998. [25] U. Schaber, Non-contact vibration measurements of mistuned coupled blades, in: International Gas Turbine and Aeroengine Congress and Exhibition, ASME, Orlando, FL, June 2–5, 1997. [26] W.B. Watkins, R.M. Chi, A noninterference blade vibration system for gas turbine engines, in: AIAA/SAE/ASMA/ASEE 23rd Joint Propulsion Conference, San Diego, 1987. [27] R.M. Chi, H.T. Jones, Demonstration testing of a noninterference technique for measuring turbine engine rotor blade stresses, in: AIAA/SAE/ASMA/ASEE 24th Joint Propulsion Conference, Boston, 1988.