# Rotor Blades Diagnosis Method Based on Differences in Phase Shifts

Authors: Rafał Grądzki; Paweł Lindstedt; Zbigniew Kulesza; Błażej Bartoszewicz
Journal: Shock and Vibration
 (2018)

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## Abstract

In this work, an innovative diagnosis method of rotor blades is presented. A measurement signal of blade tip displacement is divided into two observation zones, in which two peaks of an additional template signal are introduced. Cross-correlation and power spectral density functions of the measurement and template signals in these two distinct zones are calculated. Next, the phase shift between the two cross-correlation functions is obtained by calculating the ratio of the two power spectral density functions. The changes in parameters of an analytical description of this phase shift are used to determine a distinct and easy-to-analyze image of the technical condition of the tested blade. A graphical portrait indicating the technical condition of all blades in the annulus is developed. Although not directly measured, environmental signals (e.g., external disturbances and sensor’s noise) affecting the diagnostic process are included in the method. It is demonstrated that, by using the proposed signal processing technique, a negative influence of these disturbances is eliminated and the reliability of the machine technical condition indications is improved. The approach is demonstrated using experimental results of turbine engine blade displacements. High effectiveness of the method and measurement equipment has been confirmed in a wide range of analyzed tip-timing results obtained in different periods of the measurement process.


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## Body

## 1. Introduction

One of the basic components of rotating machines on which reliable and safe operation depends is rotor blades. Practice proves that breaking off even a single blade out of several tens or even hundreds in a bladed rotor almost always leads to a serious machine failure and consequently to a tragic catastrophe. The knowledge about the technical state of the blade is also important to avoid unplanned machine downtimes and to plan the overhauls accurately. Therefore, the technical condition of the machine, and especially of its rotor blades, should be monitored continuously to detect damages as quickly as possible.Several methods of rotor blades diagnosis have been developed and tested, such as eddy current [1], ultrasonic [2, 3], thermographic [4, 5], radiographic [4, 6], and vibration methods [7–10]. A comprehensive literature review of various blade faults detection methods has been presented by Abdelrhman et al. [11].The most popular approach that can be applied for rotating blades is the frequency domain analysis of blade vibrations. The wide vibration spectrum of the tested machine can be compared with a vibration spectrum library to detect various blade faults such as foreign object damage, blade loss part, blade rub, or loose joints [7]. Different changes in blade pass frequency and its harmonics are reported as early indications of the occurring blade-casing rubbing in steady rotation [8, 9] or in the run-up or coast-down [10] of the machine. A modification of this approach based on cepstrum analysis has been applied to various blade faults by Satyam et al. [12] and Randall and Sawalhi [13]. Chang and Chen [14] used wavelet analysis methods to detect cracked blades. A similar approach has been applied by Aretakis and Mathioudakis [15], who demonstrated that each type of the analyzed blade faults (fouled, twisted, and mistuned blade) generates a unique wavelet signature. Lim and Leong [16] tested wavelet analysis methods for loose, rubbing, or creeping blades. Possible applications of ensemble empirical mode decomposition (EEMD), multiwavelet packet transform, Hilbert–Huang transform for multifault identification in a bladed rotor have been studied by Jiang et al. [17].Another variant of the blade vibration analysis, the so-called tip-timing method [18], involves the extraction of the vibrations of a single blade from a measurement signal obtained from displacement sensors. This signal contains the displacements of other blades. The extracted vibrations of this single blade can then be analyzed for possible occurrence of blade rubbing [19], blade cracks or loosing [20], or other blade faults [21, 22] by analyzing time differences of blade passages in different rotor revolutions. The tip-timing method requires special measurement equipment [23–26] and has its long history, well presented by Rzadkowski et al. [27].Displacement, velocity, and acceleration of blade tips are not the only quantities that can be used as diagnostic signals in the vibration-based diagnostic methods. Temperature [28, 29], pressure, and flow fluctuations [15, 30] inside machine chambers have also been used to detect various blade faults. A popular approach includes measurement of different parameters of acoustic emission [15, 31, 32] to detect blade faults.All these signals can be termed as “operational signals” because they are directly measured, and they carry on the information about the operational parameters of the machine, i.e., about its technical condition. However, there are many other signals such as measurement noises or external electromagnetic disturbances that affect the normal operation of the machine and of the measurement process but are difficult or even impossible to be measured directly. All these disturbing signals can be termed as “environmental signals,” as they carry on the information about the parameters of the technical and natural environment surrounding the machine. Usually, environmental signals are not included in traditional blade diagnostic methods, yet their influence on the obtained fault indications may be significant.The proposed method is based on a diagnostic model of the blade and includes the phase shift between cross-correlation functions of the displacement signal and an additional template signal calculated in two observation zones: the first zone when the blade tip approaches the sensor and the other zone when the blade tip recedes the sensor. In this model, environmental signals are included and it is demonstrated that, by using the proposed signal processing technique, a negative influence of these disturbances is eliminated. This way the reliability of the machine technical condition indications is improved. The method has been developed by Lindstedt and Gradzki [33–36], and the current paper improves and extends the approach by consolidating the form of the diagnostic model and the forms of damage maps. Furthermore, it introduces a new, compact form of a graphical damage representation to quickly assess the technical condition of all blades in the annulus. Provided experimental results confirm high effectiveness of the method in a wide range of analyzed blade tip vibrations obtained in different periods of the measurement process. An important part of the measurement equipment is a specially designed contactless inductive sensor, which is essential for a correct application of the method.It should be emphasized that the proposed method can be used to assess the technical condition of a blade in its normal operation. The analysis is conducted online. All external forces affecting this operation are included. No disassembly of an engine to perform static tests of the blades is required (which currently is a common practice). The method can be used to detect different types of blade malfunctions (e.g., mechanical wear or damage, fatigue crack, blade nicking or bending, and high-temperature damage) and gives early damage indications to plan overhauls accurately.


## 2. Diagnostic Model

### 2.1. Mathematical Description

Usually only one signal of blade tip displacementyt (apart from the rotating speed Ω) is directly measured. This signal is further termed as the “operational signal.” Operational signal is often disrupted with some sensor noise or other disturbances. The influence of these disturbing factors can be included by introducing an additional environmental signalxt. This abstract signal xt, gathering all possible environmental disturbances, is present in the operational signal yt, but it is difficult or even impossible to be measured. However, its influence on the diagnostic procedure results may be significant. Usually, environmental signal is represented as white noise.A typical operational signal of blade tip displacementsyt obtained using the tip-timing procedure is presented in Figure 1. The peaks in yt represent the so-termed cycles of blade tip operation.Figure 1
Typical blade tip displacement (operational) signalyt, environmental signal xt, and the proposed template signal zt.Using the proposed method, the signalyt is divided into two observation zones at each cycle. The first zone of time length Δt1 is from t0 to t1 and it is termed as the approaching zone. At this zone, the blade tip approaches the displacement sensor. The other zone is from t1 to t2, its time length is Δt2, and it is termed as the receding zone. At this zone, the blade tip recedes the sensor. The time t1 is related to the moment when the blade tip is just beneath the displacement sensor.Next, additionaltemplate signalzt is introduced. This signal is assumed as a series of approximations δ^t of Dirac’s delta distribution δt, where(1)δt=0,t≠tm,+∞,t=tm,∫−∞+∞δtdt=1.The peaks of these approximationsδ^t are located in the middle tm=t1−Δt1 of the approaching zone and in the middle tm=t1+Δt2 of the receding zone of the operational signal yt (Figure 1). As δ^t, any approximation of Dirac’s distribution δt may be used [37]. The only requirement is that there must be some correlation between δ^t and yt.The cross-correlation functionsRzy1τ and Rzy2τ between operational yt and template signal zt are then calculated in separate zones Δt1 and Δt2 as(2)Rzy1τ=∫t0t1ztyt−τdt,Rzy2τ=∫t1t2ztyt−τdt,where τ is the time shift variable.Next, using the Wiener–Khinchin theorem, power spectral density functionsSzy1ω and Szy2ω are calculated as Fourier transforms of the corresponding cross-correlation functions Rzy1τ and Rzy2τ [38]:(3)Szy1ω=FRzy1τ,Szy2ω=FRzy2τ,where Ffτ denotes the Fourier transform of a time function fτ.Using the corresponding form of Equation (3), the power spectral density functions Szx1ω and Szx2ω are calculated, where the autocorrelation functions Rzx1τ and Rzx2τ between the template zt and environmental xt signals in the two separate observation zones are defined as(4)Rzx1τ=∫t0t1ztxt−τdt,Rzx2τ=∫t1t2ztxt−τdt.The obtained power spectral density functionsSzy1ω, Szy2ω, Szx1ω, and Szx2ω are then used to present the relation between operational yt and environmental xt signals as(5)φ1ω=ArgSzy1ωSzx1ω,φ2ω=ArgSzy2ωSzx2ω,where φ1ω is a phase shift between signals Rzy1τ and Rzx1τ at the approaching zone, φ2ω is a phase shift between signals Rzy2τ and Rzx2τ at the receding zone, and ArgGω denotes the function which calculates the argument (the angle) of some complex function Gω.If time intervalsΔt1 and Δt2 are close to each other (i.e., when Δt2 immediately follows Δt1), then power spectral density functions Szx1ω and Szx2ω can be considered as equal, i.e.,(6)Szx1ω=Szx2ω=Szxω.This assumption can be explained by the observation that if the time distance between intervalsΔt1 and Δt2 is short, then environmental signal xt in those time intervals may be considered as unchanged (or only slightly changed in a small degree).Using Equations (5) and (6), the difference between phase shifts φ2ω and φ1ω can be calculated as(7)φ21ω=φ2ω−φ1ω=ArgSzy2ω/Szx2ωSzy1ω/Szx1ω=ArgSzy2ωSzxω×SzxωSzy1ω=ArgGω,where(8)Gω=Szy2ωSzy1ω.Note that according to Equation (3), the ratio of power spectral densities Szy2ω and Szy1ω is a transfer function denoted as Gω. The argument of this function is the phase shift between signal Rzy2τ at the receding zone and signal Rzy1τ at the approaching zone. This argument φ21ω can be considered as a unified representation of the technical condition of the tested blade, i.e., as a diagnostic model of the blade.Also note that by including Equation (6), the power spectral density functions Szx1ω and Szx2ω are eliminated from the model. That means that direct measurements of the environmental signal xt are not required. Of course, indirectly, this signal is included in the model, as it is a part of the operational signal. Thanks to a special organization of the diagnostic procedure, in which the analysis is conducted in two separate observation zones, and the ratio of two power spectral density functions Szy1ω and Szy2ω is calculated, and the influence of the environmental signal is eliminated.


### 2.2. Selection of Template Signal

The template signalzt can be chosen as any periodic function of time, i.e., the function which repeats the same pattern in time intervals Δt1 and Δt2 for subsequent blades. If this template signal repeats the same pattern in subsequent time intervals, then the changes in cross-correlation Rzy1τ and Rzy2τ functions (and consequently in the phase shift φ21ω between these cross-correlations) in the receding and approaching zones can appear only due to the changes in signal yt. Certainly, it is supposed that the changes in the operational signal yt reflect the changes in the technical condition of the blade.This is the main requirement for the template signal: periodically repeated patterns in subsequent time intervalsΔt1 and Δt2. The other one is there should be some correlation between zt and yt. Specifically, the template signal zt can be selected as a series of Dirac’s delta distributions (as proposed in the current paper) to better reconstruct the shape of the operational signal yt and consequently to obtain more robust values of correlation Rzy1τ, Rzy2τ and power density Szy1ω, Szy2ω functions.Another reason for introducing the template signalzt is to correlate it with the environmental signal xt (which is always present in the operational signal yt). If this template signal is identical in subsequent time intervals Δt1 and Δt2 and if the time distance between these intervals is small, then the assumption about identical cross-correlation functions Rzx1τ and Rzx2τ (and consequently about power spectral density Szx1ω, Szx2ω functions) can be taken. Finally, the influence of the environmental signal xt can be eliminated (as explained by Equation (7)). Of course, only the constant environment gives right to take such assumption, i.e., the environment in which the values of environmental signal xt do not change considerably between time intervals Δt1 and Δt2.


## 2.1. Mathematical Description

Usually only one signal of blade tip displacementyt (apart from the rotating speed Ω) is directly measured. This signal is further termed as the “operational signal.” Operational signal is often disrupted with some sensor noise or other disturbances. The influence of these disturbing factors can be included by introducing an additional environmental signalxt. This abstract signal xt, gathering all possible environmental disturbances, is present in the operational signal yt, but it is difficult or even impossible to be measured. However, its influence on the diagnostic procedure results may be significant. Usually, environmental signal is represented as white noise.A typical operational signal of blade tip displacementsyt obtained using the tip-timing procedure is presented in Figure 1. The peaks in yt represent the so-termed cycles of blade tip operation.Figure 1
Typical blade tip displacement (operational) signalyt, environmental signal xt, and the proposed template signal zt.Using the proposed method, the signalyt is divided into two observation zones at each cycle. The first zone of time length Δt1 is from t0 to t1 and it is termed as the approaching zone. At this zone, the blade tip approaches the displacement sensor. The other zone is from t1 to t2, its time length is Δt2, and it is termed as the receding zone. At this zone, the blade tip recedes the sensor. The time t1 is related to the moment when the blade tip is just beneath the displacement sensor.Next, additionaltemplate signalzt is introduced. This signal is assumed as a series of approximations δ^t of Dirac’s delta distribution δt, where(1)δt=0,t≠tm,+∞,t=tm,∫−∞+∞δtdt=1.The peaks of these approximationsδ^t are located in the middle tm=t1−Δt1 of the approaching zone and in the middle tm=t1+Δt2 of the receding zone of the operational signal yt (Figure 1). As δ^t, any approximation of Dirac’s distribution δt may be used [37]. The only requirement is that there must be some correlation between δ^t and yt.The cross-correlation functionsRzy1τ and Rzy2τ between operational yt and template signal zt are then calculated in separate zones Δt1 and Δt2 as(2)Rzy1τ=∫t0t1ztyt−τdt,Rzy2τ=∫t1t2ztyt−τdt,where τ is the time shift variable.Next, using the Wiener–Khinchin theorem, power spectral density functionsSzy1ω and Szy2ω are calculated as Fourier transforms of the corresponding cross-correlation functions Rzy1τ and Rzy2τ [38]:(3)Szy1ω=FRzy1τ,Szy2ω=FRzy2τ,where Ffτ denotes the Fourier transform of a time function fτ.Using the corresponding form of Equation (3), the power spectral density functions Szx1ω and Szx2ω are calculated, where the autocorrelation functions Rzx1τ and Rzx2τ between the template zt and environmental xt signals in the two separate observation zones are defined as(4)Rzx1τ=∫t0t1ztxt−τdt,Rzx2τ=∫t1t2ztxt−τdt.The obtained power spectral density functionsSzy1ω, Szy2ω, Szx1ω, and Szx2ω are then used to present the relation between operational yt and environmental xt signals as(5)φ1ω=ArgSzy1ωSzx1ω,φ2ω=ArgSzy2ωSzx2ω,where φ1ω is a phase shift between signals Rzy1τ and Rzx1τ at the approaching zone, φ2ω is a phase shift between signals Rzy2τ and Rzx2τ at the receding zone, and ArgGω denotes the function which calculates the argument (the angle) of some complex function Gω.If time intervalsΔt1 and Δt2 are close to each other (i.e., when Δt2 immediately follows Δt1), then power spectral density functions Szx1ω and Szx2ω can be considered as equal, i.e.,(6)Szx1ω=Szx2ω=Szxω.This assumption can be explained by the observation that if the time distance between intervalsΔt1 and Δt2 is short, then environmental signal xt in those time intervals may be considered as unchanged (or only slightly changed in a small degree).Using Equations (5) and (6), the difference between phase shifts φ2ω and φ1ω can be calculated as(7)φ21ω=φ2ω−φ1ω=ArgSzy2ω/Szx2ωSzy1ω/Szx1ω=ArgSzy2ωSzxω×SzxωSzy1ω=ArgGω,where(8)Gω=Szy2ωSzy1ω.Note that according to Equation (3), the ratio of power spectral densities Szy2ω and Szy1ω is a transfer function denoted as Gω. The argument of this function is the phase shift between signal Rzy2τ at the receding zone and signal Rzy1τ at the approaching zone. This argument φ21ω can be considered as a unified representation of the technical condition of the tested blade, i.e., as a diagnostic model of the blade.Also note that by including Equation (6), the power spectral density functions Szx1ω and Szx2ω are eliminated from the model. That means that direct measurements of the environmental signal xt are not required. Of course, indirectly, this signal is included in the model, as it is a part of the operational signal. Thanks to a special organization of the diagnostic procedure, in which the analysis is conducted in two separate observation zones, and the ratio of two power spectral density functions Szy1ω and Szy2ω is calculated, and the influence of the environmental signal is eliminated.


## 2.2. Selection of Template Signal

The template signalzt can be chosen as any periodic function of time, i.e., the function which repeats the same pattern in time intervals Δt1 and Δt2 for subsequent blades. If this template signal repeats the same pattern in subsequent time intervals, then the changes in cross-correlation Rzy1τ and Rzy2τ functions (and consequently in the phase shift φ21ω between these cross-correlations) in the receding and approaching zones can appear only due to the changes in signal yt. Certainly, it is supposed that the changes in the operational signal yt reflect the changes in the technical condition of the blade.This is the main requirement for the template signal: periodically repeated patterns in subsequent time intervalsΔt1 and Δt2. The other one is there should be some correlation between zt and yt. Specifically, the template signal zt can be selected as a series of Dirac’s delta distributions (as proposed in the current paper) to better reconstruct the shape of the operational signal yt and consequently to obtain more robust values of correlation Rzy1τ, Rzy2τ and power density Szy1ω, Szy2ω functions.Another reason for introducing the template signalzt is to correlate it with the environmental signal xt (which is always present in the operational signal yt). If this template signal is identical in subsequent time intervals Δt1 and Δt2 and if the time distance between these intervals is small, then the assumption about identical cross-correlation functions Rzx1τ and Rzx2τ (and consequently about power spectral density Szx1ω, Szx2ω functions) can be taken. Finally, the influence of the environmental signal xt can be eliminated (as explained by Equation (7)). Of course, only the constant environment gives right to take such assumption, i.e., the environment in which the values of environmental signal xt do not change considerably between time intervals Δt1 and Δt2.


## 3. Blade Condition Monitoring Using the New Diagnostic Model

To simplify further considerations, it is assumed thatt1=0 (Figure 1), i.e., the analyzed peak in signal yt is shifted to the beginning of the coordinate system. The operational signal yt of a given blade is measured and sampled with time period h>0. Then, prescriptive numbers N+1 of signal samples in two adjacent observation zones Δt1 and Δt2 are collected into two separable sets y1n1 and y2n2, where n1=−N,...,−1,0 and n2=0,1,...,N (Figure 1). The number N+1 of signal samples in each set must be chosen carefully to ensure proper statistical assessment of the measured operational signal y1n1 and y2n2.Next, to reduce frequency leakage, the signalsy1n1 and y2n2 are scaled by Hanning window Hwn:(9)Hwn=121−cos2πnN,to form new signals yH1n1 and yH2n2:(10)yH1n1=Hw1n1y1n1,yH2n2=Hw2n2y2n2.Discrete approximationsδ^n1 and δ^n2 of Dirac’s delta distribution δt are chosen as(11)δ^n1=1Te−π2n1+N2/T2,δ^n2=1Te−π2n2−N2/T2,to form template signals z1n1 and z2n2 in the approaching and receding zones, where z1n1=δ^n1, z2n2=δ^n2, and parameter T changes the amplitude of approximation.Then, discrete cross-correlation functionsRzy1n2 and Rzy2n2 between the signals yH1n1, yH2n2 and z1n1, z2n2 are calculated, as follows:(12)Rzy1n2=∑l=0N−n2−1z1l−n2−N+1yH1l−N+1,Rzy2n2=∑l=0N−n2−1z1l+n2yH1l.Cross-correlation functionsRzy1n2 and Rzy2n2 are then approximated with smooth approximating functions to obtain analytical representations Rzy1τ and Rzy2τ. Approximating functions polynomials may be used [35], resulting in the following forms of the cross-correlation functions:(13)Rzy1τ=arτr+ar−1τr−1+⋯+a1τ+a0,Rzy2τ=brτr+br−1τr−1+⋯+b1τ+b0,where ai and bi are coefficients of the polynomials, i=0,1,2,…,r.The orderr of the polynomials should be chosen carefully: too low order will result in inaccurate approximations, too high order will result in an excessive number of polynomial coefficients and longer calculation times.Based on analytical representations of the cross-correlation functionsRzy1τ, Rzy2τ, Laplace transforms FRzy1τ, FRzy2τ and power spectral density functions Szy1ω, Szy2ω are calculated according to Equation (3). Next, the power spectral density functions Szy1ω,Szy2ω are introduced into Equations (8) and (7). For polynomial approximation, the diagnostic model φ21ω is obtained in the following form:(14)φ21ω=ArgSzy2ωSzy1ω=ArgBrsr+Br−1sr−1+⋯+B1s+B0Arsr+Ar−1sr−1+⋯+A1s+A0,where Bi=br−ir−i!, Ai=ar−ir−i!, and i=0,1,2,…,r.The technical condition of the blade is then evaluated by calculating relative changesΔAi, ΔBi in model parameters, defined as follows:(15)ΔAi=Ai−A¯iA¯i,ΔBi=Bi−B¯iB¯i,where Ai, Bi are values of the φ21ω model parameters evaluated at the moment ϑ1 and A¯i, B¯i are mean values of these parameters evaluated during subsequent measurements within the period between ϑ0 (beginning of the monitoring process) and ϑ1 (the current moment at the monitoring process). Here, t is the time in terms of Newton’s definition (for diagnostic examinations) and ϑ is the time in terms of Bergson’s definition (for diagnostic inference).To simplify the evaluation of technical condition of a given blade, relative changesΔAi, ΔBi of the φ21ω diagnostic model are classified into three damage threshold ranges and presented in a convenient graphical way, termed as a “damage map” [33]. The damage map of the blade is created as described below.The three damage threshold ranges are defined with meanΔA¯i, ΔB¯i and standard deviation σAi, σBi values of relative changes ΔAi, ΔBi, where(16)ΔA¯i=1K∑kKΔAik,ΔB¯i=1K∑kKΔBik,and(17)σAi=1K∑kKΔAik−ΔA¯i2,σBi=1K∑kKΔBik−ΔB¯i2.Here,ΔAik and ΔBik are relative changes in model parameters evaluated at time moment ϑk, ϑ0≤ϑk≤ϑ1, and K is the number of those evaluations. For simplicity, ΔAi, ΔBi are further denoted as Δp, ΔA¯i, ΔB¯i as μ, and σAi, σBi as σ.Next, three damage threshold ranges are assumed and the damage map is created. The map has a form of a color table ofK rows and 2r columns, where each row contains color indications of whether at a given time moment ϑk the value of a given relative change Δp falls within one of the three damage ranges. The colors in subsequent cells in a given row are assigned as follows:(1)
If at a given time momentϑk, relative changes Δp are within the first damage threshold range, i.e., if(18)μ−σ≤Δp≤μ+σ,then the indication of Δp in the kth row, ith column becomes green.(2)
If relative changesΔp are within the second damage threshold range:(19)μ−2σ≤Δp≤μ−σorμ+σ≤Δp≤μ+2σ,then the indication in the kth row, ith column becomes blue.(3)
If relative changesΔp are within the third damage threshold range:(20)μ−3σ≤Δp≤μ−2σorμ+2σ≤Δp≤μ+2σ,then the indication in thekth row, ith column becomes red.(4)
If relative changesΔp are beyond any of the three damage threshold ranges, then the indication in the kth row, ith column becomes black.From the above procedure, it is clear that predominant black and red fields indicate a serious blade failure, predominant dark blue fields indicate excessive wear of the blade, and predominant green fields indicate slight or no wear of the blade.The algorithm of the proposed blade evaluation technique is presented in Figure2.Figure 2
Algorithm of the blade evaluation technique.


## 4. Experimental Test Stand

The proposed method has been verified experimentally at a turbine engine test stand at Air Force Institute of Technology, Warsaw (Figure3). The tests have been performed for first stage blades of an axial compressor of SO-3 turbine engine mounted in TS-11 “Iskra” aircrafts [39–41].Figure 3
Experimental test stand: (1) turbine engine, (2) signal amplifier, and (3) contactless inductive sensor [39].The contactless inductive sensor designed at Air Force Institute of Technology [26, 42] and mounted at the engine chassis has been used to measure blades displacements. Its principle of operation is based on electrodynamical interaction and therefore it may be referred as a passive eddy-current sensor. The sensor measures the signal of momentary location of the compressor blade tips during engine operation in time when blade approaches and recedes from the sensor (tip-timing method).


## 5. Results

An example blade displacement signal obtained with a sampling frequency of 500 kHz is presented in Figure4. Three ranges of engine rotating speed may be seen: 1st at 6900 rpm, 2nd at 12600 rpm, and 3rd at 15300 rpm.Figure 4
Blade tips displacement at three rotating speeds: (1) 6900 rpm, (2) 12600 rpm, and (3) 15300 rpm.The analysis has been conducted for the rotating speed of 12600 rpm, i.e., for the second range in Figure4. In that range, 364000 peaks of blade tips displacement have been recorded. Since the compressor blade annulus has 28 blades, each blade moves 13000 times beneath the sensor. The displacement of each of 28 blades beneath the sensor is termed as “one cycle.” The details of selected peaks in blade Nos. 3 and 8 displacement cycles measured at 12600 rpm (2nd range of the rotating speed) are shown in Figure 5. As can be seen, the quality of the displacement signals obtained with the special inductive sensor described above is very good.Figure 5
Selected peaks in blades displacement cycles at 12600 rpm.For blade analysis, 36 out of 13000 cycles are selected to form a statistical sample. To quickly form a sample, the selection is made in one of three ways:(i)
Logarithmic selection: 36 cycles are selected in such a way that each one from 1 to 9, each 10th from 19 to 90, each 100th from 100 to 900, and each 1000th from 1000 to 9000 form the sample(ii)
Selection of eachnth cycle: the first cycle, the cycle selection step, and the number of cycles are indicated(iii)
Random selectionIt is expected that damage maps obtained for each selection give the same image of blade technical condition (which will confirm the efficiency of the method). The damage maps are presented for two blades out of 28 in the annulus. The blades have been chosen in such a way that the first is in the best (blade No. 8), and the other is in the worst (blade No. 3) technical condition.After several initial calculations, the order of the approximating polynomials of cross-correlation functions has been selected asr=7, resulting in the coefficient of determination R2=99.95%. Lower polynomial orders lead to lower determination coefficient values (i.e., to less accurate approximations). Higher orders do not improve approximation’s accuracy noticeably (as R2 is already close to 1), yet considerably increase the calculation time. The amplitude parameter of environmental signal has been assumed as T=15.Example cross-correlation functionsRzy1m, Rzy2m and their 7th order polynomial approximations Rzy1τ, Rzy2τat the two observation zones of the selected peak of blade No. 3 displacement signal (shown in Figure 5) are presented in Figure 6.Figure 6
Cross-correlation functionsRzy1n2, Rzy2n2 and their polynomial approximations Rzy1τ, Rzy2τ for blade No. 3 at (a) the approaching zone Δt1 and (b) the receding zone Δt2.
(a)
(b)Thus, the technical condition of each blade is represented by 16 coefficients: 8 in the approaching and 8 in the receding zones. These zones are clearly visible as left and right parts of damage maps in Figures7–10. The samples have been selected using the nth selection (Figures 7(a) and 9(a) for each 7th cycle and Figures 7(b) and 9(b) for each 10th cycle), the logarithmic selection (Figures 8(a) and 10(a)), and the random selection (Figures 8(b) and 10(b)).Figure 7
Damage maps of blade No. 3 for (a) each 7th cycle and (b) each 10th cycle.
(a)
(b)Figure 8
Damage maps of blade No. 3 for (a) logarithmic and (b) random selection of cycles.
(a)
(b)Figure 9
Damage maps of blade No. 8 for (a) each 7th cycle and (b) each 10th cycle.
(a)
(b)Figure 10
Damage maps of blade No. 8 for (a) logarithmic and (b) random selection of cycles.
(a)
(b)As can be seen, the technical condition of each blade is clearly and explicitly indicated in the obtained damage maps. Predominant red and black fields in Figures7 and 8 are characteristics for the seriously damaged blade No. 3. For blade No. 8, the obtained damage maps (Figures 9 and 10) are almost green, confirming its relatively good technical condition. These results are independent of the selection type: the samples of nth, logarithmic, or random selection types give similar, clear indications of the technical condition of the blade. As damage thresholds are based on μ±σ, μ±2σ, and μ±3σ statistics, small deviations between the colors of a given damage map are possible. Therefore, it is important to analyze the map globally, for a reasonable number of rotation cycles (rows of the map).Using damage maps, the technical condition of all 28 blades can be presented in a compact, graphical form as shown in Figure11. This form is called the damage portrait of the blades in the annulus. At horizontal axis, all 28 blades are numbered in an ascending order. At vertical axis, the number of fields of a given color (green, blue, red, and black) at the related damage map is marked with a colored circle. As can be seen for blade No. 3, the number of red fields is high (above 100) while the number of green fields is low (below 350). This suggests a possible serious blade wear (or damage). For blade No. 8, the number of black, red, and blue fields is low (below 100), while the number of green fields is high (almost 500). This indicates that the technical condition of this blade is good.Figure 11
Damage portrait of all 28 blades in the annulus.The presented results reflect the state of each blade only in terms of its “better” or “worse” technical condition, as no real damage is introduced to the test stand. Experimental tests of the blade with a serious damage are difficult or even impossible to conduct as may quickly lead to a dangerous failure of the machine.


## 6. Summary

The proposed method of blades technical condition evaluation is based on a diagnostic model having the form of the phase shift between the cross-correlation functions of the measured vibration signal and an additional template signal calculated in two subsequent observation zones: when the blade approaches the sensor and when the blade recedes the sensor. If the time distance between these zones is short enough, then environmental signals affecting the operational signal in both zones can be assumed as identical. By taking this assumption, the direct measurement of the environmental signal is not required, yet its influence is indirectly eliminated by a special organization of the diagnostic process. This way the reliability of the machine technical condition indications is improved, which has been demonstrated by experimental results obtained at the turbine engine test stand. The technical condition of each blade has been projected into damage maps representing the relative changes in the parameters of the diagnostic model. Furthermore, the technical condition of all blades in the annulus has been presented in a compact, simple to interpret damage portrait. Such graphical representations are communicative and can be used to quickly interpret the technical condition of the machine.The proposed method can be used to assess the technical condition of advanced rotating machines, such as turbine engines. It can be applied online with no need to disassembly the engine and gives early damage indications to plan overhauls accurately.



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*Source: 9134607-2018-10-21.xml*