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Sparse Representation Based Frequency Detection and Uncertainty Reduction in Blade Tip Timing Measurement for Multi-Mode Blade Vibration Monitoring Minghao Pan, Yongmin Yang *, Fengjiao Guan, Haifeng Hu and Hailong Xu Science and Technology on Integrated Logistics Support Laboratory, National University of Defense Technology, Changsha 410073, China; [email protected] (M.P.); [email protected] (F.G.); [email protected] (H.H.); [email protected] (H.X.) * Correspondence: [email protected]; Tel.: +86-731-8457-6109 Received: 21 June 2017; Accepted: 26 July 2017; Published: 30 July 2017

Abstract: The accurate monitoring of blade vibration under operating conditions is essential in turbo-machinery testing. Blade tip timing (BTT) is a promising non-contact technique for the measurement of blade vibrations. However, the BTT sampling data are inherently under-sampled and contaminated with several measurement uncertainties. How to recover frequency spectra of blade vibrations though processing these under-sampled biased signals is a bottleneck problem. A novel method of BTT signal processing for alleviating measurement uncertainties in recovery of multi-mode blade vibration frequency spectrum is proposed in this paper. The method can be divided into four phases. First, a single measurement vector model is built by exploiting that the blade vibration signals are sparse in frequency spectra. Secondly, the uniqueness of the nonnegative sparse solution is studied to achieve the vibration frequency spectrum. Thirdly, typical sources of BTT measurement uncertainties are quantitatively analyzed. Finally, an improved vibration frequency spectra recovery method is proposed to get a guaranteed level of sparse solution when measurement results are biased. Simulations and experiments are performed to prove the feasibility of the proposed method. The most outstanding advantage is that this method can prevent the recovered multi-mode vibration spectra from being affected by BTT measurement uncertainties without increasing the probe number. Keywords: blade tip timing; measurement uncertainty; vibration frequency spectrum recovery; sparse representation; multi-mode vibration

1. Introduction High-speed rotor blades in turbines and compressors are often subject to several sources of excitations that lead to various forced vibration responses. These vibrations are common causes of blade damages that may even result in catastrophic failures. As a non-contacting measurement technology, BTT [1] can be used to identify vibration problems in rotor blades through recording the arrival time of the blades at each probe installed in the casing. However, the sequence of blade tip deflections computed by BTT signals is inherently under-sampled [2], which makes the detection of blade vibration frequencies much harder. Many methods have been proposed for detecting vibration frequencies from the BTT data. Carrington [3] compared two global autoregressive-based vibration analysis methods. Salhi [4] used an interpolation technique to reconstruct under-sampled BTT signals. However, all of the above techniques only adapted to the case that each blade responds at a single harmonic frequency. Studies [5–7] have proved that multiple frequencies appear in the frequency response of blade Sensors 2017, 17, 1745; doi:10.3390/s17081745

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displacement when the blade is cracked or mistuned. Gallego-Garrido [8] proposed a class of auto-based methods to process BTT data from blades undergoing two simultaneous resonances. Kharyton [9] proposed a method for identifying multi-modes using FFT-like algorithms together with non-uniform Fourier transform. Lin [10] proposed a novel BTT analysis method based on compressive sensing theorem for reconstructing unknown multi-mode blade vibration signals. However, relatively more probes are required with the increase of responding modes in those methods. In addition, the influence of measurement uncertainties is not considered in the data processing. Thus, the proposed algorithms are not guaranteed to get the reliable vibration frequencies under the impact of measurement errors. Several factors that contribute to uncertainties in BTT measurement have been reported. Satish [11] provided an insight into many probable sources of BTT measurement errors in both engine systems and measurement systems using analytical and experimental techniques. Guo [12] performed a numerical analysis on the rising speed of tip-timing signals and indicated that a variable gain amplifier could minimize the tip-timing error. Berdanier [13] and Lawson [14] analyzed uncertainties on the capacitance probe measurement system and proposed two main sources due to analogue to digital (A/D) conversion processes and signal noise. However, those analyses lack quantitative expressions of the relationship between those mentioned uncertainties and blade vibration measurement results. Hu [15] estimated the reconstruction error affected by parameters in interpolating function respectively. However, approaches for removing uncertainties in those parameters was not proposed. To overcome the limitations stated above, a novel approach based on the sparse representation theorem for multi-mode vibration frequency detection is presented with the correspondent uncertainty reduction method. Sparse representation (SR) and compressed sensing (CS) have been widely applied to the recovery of signals with a sparse nature [16,17] from under-sampling data accurately. Compared with other predominant techniques, the proposed method has two main advantages: firstly, multiple harmonic frequencies can be extracted from under-sampled BTT data collected by no more than four probes. Experiments have confirmed that the upper boundary of the harmonic frequency number that can be detected simultaneously is no less than four. Secondly, the sensitivity of recovered frequency spectrum to BTT measurement uncertainty can be reduced. This paper is organized as follows: a BTT sampling mathematical model is built in Section 2. In Section 3, recovery of multi-mode vibration frequency spectra by the theory of sparse representation is introduced. Then the improved frequency spectra recovery method is adapted to the case of biased BTT data in Section 4. In Sections 5 and 6, simulation and experimental results are presented to evaluate the feasibility of proposed methods. Finally, conclusions are reported in Section 7. 2. Mathematical Model of the Blade Tip Timing Sampling 2.1. Formulations of Blade Tip Deflections Without attaching to the blades, the blade tip deflections will be determined from the time of arrival (TOA) of each tip relative to the arrival time of a rotating reference point in BTT measurement. As shown in Figure 1, I probes are mounted circumferentially around a rotor with K blades so that the times when the blades pass the probes can be measured. The locations of the ith probe and the kth blade can be described as angles αi and θk , respectively. The other probe r0 is mounted above the shaft to collect once per revolution signals as reference signals. Assuming that the blade disk rotates clockwise, when a non-vibrating blade k passes the ith probe at the nth rotation, the expected TOA ti,n expected can be calculated [18] as: ti,n expected =

Tn (2πn + αi − θk ), 2π

(1)

where Tn is the constant rotating period. Hence the ti,n expected can be used to calculate deflection of each blade.

collected by no more than four probes. Experiments have confirmed that the upper boundary of the harmonic frequency number that can be detected simultaneously is no less than four. Secondly, the sensitivity of recovered frequency spectrum to BTT measurement uncertainty can be reduced. This paper is organized as follows: a BTT sampling mathematical model is built in Section 2. In Sensors 17, 1745 of multi-mode vibration frequency spectra by the theory of sparse representation 3 of 19 Section2017, 3, recovery is introduced. Then the improved frequency spectra recovery method is adapted to the case of biased BTT data in Section 4. In Section 5 and Section 6, simulation and experimental results are presented By comparing the detected TOAs to the ideal ones of the blade, the blade tip displacement can be to evaluate the feasibility of proposed methods. Finally, conclusions are reported in Section 7. obtained as follows: ∆t = texpected − t actual , (2) 2. Mathematical Model of the Blade Tip Timing Sampling 2πRtip d(t actual ) = ∆t, (3) 2.1. Formulations of Blade Tip Deflections Tn where Rtip is attaching the radiustofrom the rotation axis of engine towill the be blade tip and t actual thetime actual Without the blades, the blade tipthe deflections determined fromisthe of TOA that the probe for each blade. arrival (TOA) of eachrecords tip relative to the arrival time of a rotating reference point in BTT measurement. Probe 1

Data Casing

Aquisition Probe 2 Probe r0

…… Probe I Blades

Figure Figure 1. 1. Blade Bladetip tiptiming timingsystem. system.

2.2. Function of BTT Signals Consider a continued blade vibration signal d(t) for an arbitrary blade k. The TOA of each blade trigger the probe once in a rotation based on the principle of BTT measurements. Assuming θk = 0 and the rotation period Tn is a constant, the BTT sampling function of an arbitrary blade k during N rotations can be formulated as follows: I

N

 αi  δ t − n − 1 T − Tn , ( ) n ∑ 2π i =1 n =1

ykI (t) = d(t) ∑

(4)

where δ( ) is the Dirac delta function, d(t) is the blade tip vibration signal. Therefore, I BTT data are αi sampled in each of N periodicities, where data sampled by the ith probe is shifted by 2π Tn from the origin one. As illustrated in Equation (4), the sampling frequency of the BTT measurement is determined by the number of probes and the rotating speed. Unfortunately, the rotating frequency is always much lower than the maximal blade vibration frequency. Therefore, BTT signals are under-sampled in most of the cases due to the bound of probe number. Generally, the response for a single mode blade vibration is modeled as a harmonic wave, which is classified into ‘Synchronous’ and ‘Asynchronous’ according to the vibration frequency [3]. However, actual blades may yield responses at coupled-modes vibrations. Therefore, the response of multi-mode blade vibration could be expressed as the following formula: M

d(t) =

∑ Ai sin(2π fi t + ϕi ),

(5)

i =1

where {Ai } are amplitudes of vibrations, {fi } are dominant frequencies, {φi } are phases of the response and M is the number of vibration modes. In general, harmonic features in d(t) are corresponding to different kinds of excitations and fatigues of blades.

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3. Vibration Frequency Detection Based on Sparse Representation Theory 3.1. Sparse Representation Mathematical Model of the BTT Measurement are L imaginary probes installed uniformly around the casing as shown in Figure 1. 4 of 19 For each rotation, L data are sampled from vibration signal. Thus, the sampling rate of this system equals to n ). According to Nyquist sampling theorem [19],[19], if the rate rate of this to Lf Lfrr( (frfr= =1/T 1/T to Nyquist sampling theorem if sampling the sampling of n ). According system satisfies Lf r /2 > f blade these calculated deflection data contain all information about the this system satisfies Lfr /2 > f blade these calculated deflection data contain all information about frequency of vibration signals. Considering Considering the the engine engine constraints, constraints, however, however, only a limited number of probes can be mounted. Therefore, as shown in Figure 1, only I probes are embedded in certain Therefore, as shown in Figure angular positions selected from these L ones. The sampling stream for an arbitrary blade vibration signal could be then viewed as discarding all but I actual samples in every every rotation rotation of of LL imaginary imaginary samples samples periodically. periodically. An An integer set  ={iλ|ii| 1, 2, the the position of Iof probes. TheThe ith ith probe placement is then written as Λ i= 1, 2,,·I· · ,describes I } describes position I probes. probe placement is then written as α = 2πλ /L. According to Equation (4), for an arbitrary blade, the vibration signals recorded by i  i2i / L . i According to Equation (4), for an arbitrary blade, the vibration signals recorded by I I probes can be represented as a vector of acquired samples y[ j] (1 ≤ j ≤ I N ), where the sampling probes can be represented as a vector of acquired samples y  j  1  j  IN  , where the sampling sequence is under-sampled. sequence is under-sampled. Suppose the vibration signal of an arbitrary blade can be recorded by L uniformly displaced Suppose theangular vibration signal of arbitrary can be recorded displaced probes, then the positions of an L probes can blade be formulated as 2πl/Lby(1L ≤uniformly l ≤ L) respectively. probes, then the angular positions of L probes can be formulated as respectively. 2 las: / L 1  l  L  Based on Equation (4), the imaginary measurement vector can be written Based on Equation (4), the imaginary measurement vector canbe written as:   hi L N  l 1 ≤ ej ≤ LN . (6) x ej = d(t) ∑ ∑L δN t −  (n − 1) Tn −l Tn x  j  ld=1tn=  t   n  1 Tn  LTn  1  j  LN  . (6)  1 there SensorsSuppose 2017, 17, 1745

l 1 n 1





L

hi Obviously, x xej j includes thethe frequency information of blade vibration signal,signal, whichwhich can be can solved Obviously, includes frequency information of blade vibration be   throughthrough FourierFourier transform according to the celebrated Shannon sampling theorem. solved transform according to the celebrated Shannon sampling theorem.

Displacement

signal signal signal signal

sampled sampled sampled sampled

from from from from

L probes probe 1 probe 2 probe 3

1

0.5

0

-0.5

-1

Tn 0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Time (s) Figure 2. 2. Blade Blade tip tip displacement displacement data data sampled sampled by by different different sensors sensors in in two two rotating rotating periods. periods. Figure

hi from a vibration waveform in vector y[ j] andy xj  ej and are presented Signals sampled sampledinintwo twoperiods periods from a vibration waveform in vector x  j  are hi presented Figure 2. the Therefore, the relationship and beaconverted into a in Figure 2.inTherefore, relationship between y[ j]between and x ej ycan converted into matrix equation. x  j  can  j  be

matrix equation.

1  0 where matrix Φ     0

hi y[ j] = Φx ej , y  j   Φx  j  ,

0 1

0 1

(7) (7)

0  0 is determined by the sampling time sequence of I probes   0  pq

in each rotation. The number of row p and column q of the matrix Φ, which determined by the scale

of y  j  and x  j  , satisfy p=IN and q=LN respectively. Matrix Ф has entries   , k   1 in each

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

 1 0 ··· ··· ··· 0   0 ··· 0   0 ··· 1  where matrix Φ =  . is determined by the sampling time sequence .    .. · · · · · · · · · · · · ..  0 · · · · · · 1 · · · 0 p×q of I probes in each rotation. h i The number of row p and column q of the matrix Φ, which determined by the scale of y[ j] and x ej , satisfy p = IN and q = LN respectively. Matrix Φ has entries ϕ(•, k) = 1 in each row. where Lk Tn equals to the arriving time of the blade in each rotation, i.e., k = λi + nL (i = 1, 2, ..., I, n = 1,2, ..., N). The frequency h i of the blade vibration can be derived in the frequency-domain. Hence the Fourier transform of x ej can be formulated as follows: hi ΨFFT x ej = θ( f ),

(8)

where ΨFFT ∈ Cq×q is the Fourier basis matrix. Apparently, θ( f ) ∈ Cq is the frequency spectrum of the vibration signal obtained by Fourier transform. As the blade vibration signal has a finite number of modes (harmonic features), θ( f ) has a sparse nature. So the non-zero coefficients of vector θ( f ) are constrained by number of modes. According to Equation (8), Equation (7) can be transformed into: y[ j] = ΦΨT FFT θ( f ).

(9)

Up to now, the sparse representation model of BTT measurement has been built. In addition, the recovery of vibration frequency spectrum θ( f ) from Equation (9) has been transformed into a classic Single-Measurement Vector problem [20]. As p < q, Equation (9) is referred to be an overcomplete transform. Thus, further information is needed to uniquely extract the solution of θ( f ). 3.2. Uniqueness of Solution to SR Model 3.2.1. Restricted Isometry Property and Orthogonalization Preprocessing The Restricted Isometry Property (RIP) is a useful notion for robust recovery of a sparse signal from an under-sampled measurement vector [21]. As for the single measurement vector model in Equation (9), RIP can be equivalently transformed into the goal that the columns of matrix ΦΨT FFT should be nearly orthogonal [17]. Referring to [22], the following orthogonalization preprocessing procedure can satisfy such property. Define a sensing matrix R as R = ΦΨT FFT . The orthogonalization process is done on both sides of Equation (9). QR† y[ j] = QR† Rθ( f ), (10) where Q = [orth(RT )]T , an orthogonal basis for the range of matrix RT , and R† is a pseudoinverse of matrix R. It has been proved [22] that QR† R = Q. Therefore, the vibration frequency problem formulated in Equation (7) can be reformulated as: z = Qθ( f ), (11) where z = QR† y[ j]. Hence the detection of vibration frequency spectrum can be transformed into estimating the solution of θ( f ) in Equation (11) with minimal sparsity. It is shown in [22] that the orthogonalized sensing matrix Q obeys the RIP. If the length of BTT sampling sequence also obeys the corresponding principles, then the vibration frequency spectrum θ( f ) has high probability to be solved from z by Basis Pursuit (BP) algorithm that solves the following l1 norm minimization problem [16]:

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θˆ  f   arg min θ  f  ;

1 (12) ˆθ( f )s.t= . arg z  Qmin θ  fkθ. ( f )k1 ; (12) s.t. z = Qθ( f ). Thus, the frequency of the blade vibration can be extracted from the solution of θˆ  f  . Thus, the frequency of the blade vibration can be extracted from the solution of θˆ ( f ). 3.2.2. Requirement of Probe Number 3.2.2. Requirement of Probe Number According to the theory of CS [17], the vibration frequency spectrum can be well recovered from to the theory of CS [17], the vibration frequency spectrum be well recovered rule: from z According with high probability when the number of validate sampling data pvcan obeys the following z with high probability when the number of validate sampling data pv obeys the following rule: pv  O Ks log  q  , (13) pv = O(Ks log(q)), (13) where Ks is the sparsity of θ  f  , i.e., θ  f  , q is the length of spectrum vector θ  f  . 0 where Ks is the sparsity of θ( f ), i.e., kθ( f )k0 , q is the length of spectrum vector θ( f ). However, the definition of pv various from synchronous modes to asynchronous ones. As for However, the definition of pv various from synchronous modes to asynchronous ones. As for synchronous vibration (engine-ordered) [3], whose response frequency is an integer multiple of the synchronous vibration (engine-ordered) [3], whose response frequency is an integer multiple of the rotational speed of the assembly, the same phase of the vibration cycle is detected in each rotation by rotational speed of the assembly, the same phase of the vibration cycle is detected in each rotation probes. Thus, the amount of data that is available for the estimation of the response frequency is by probes. Thus, the amount of data that is available for the estimation of the response frequency is limited. As shown in Figure 3a, no matter how many measurement results are sampled, the feasible limited. As shown in Figure 3a, no matter how many measurement results are sampled, the feasible results is equal to the number of probes, i.e., p = I. Asynchronous vibration, however, contains results is equal to the number of probes, i.e., pv = vI. Asynchronous vibration, however, contains different values of blade displacements in each rotation as shown in Figure 3b, Thus, pv is equal to different values of blade displacements in each rotation as shown in Figure 3b, Thus, pv is equal to the the length of sampling sequence, i.e., p = IN. length of sampling sequence, i.e., pv = I N.v

Time



Displacement

Displacement



Time

-1

-1

*

(a)

Probe 1

Probe 2

Probe 3

Probe 4

(b)

3. Vibration signal sampled by foursensors optical(a) sensors (a) synchronous (b) asynchronous Figure 3.Figure Vibration signal sampled by four optical synchronous vibrationvibration (b) asynchronous vibration. vibration.

Therefore, the requirement of probe number can be summarized from Equation (13) as follows: Therefore, the requirement of probe number can be summarized from Equation (13) as follows:

•  ForFor the the most idealideal case,case, which the blade responds at a synchronous mode,mode, at leastatfour are most which the blade responds at a synchronous leastprobes four probes 3 to 104 considering both needed to get the unique solution as the length of q is in the order of 10 3 4 are needed to get the unique solution as the length of q is in the order of 10 to 10 considering the both computational complexity and the time of engine rotate to in rotate a constant speed. speed. the computational complexity and the time oftoengine in a constant thethe multi-mode case, which usually mixed with asynchronous resonances caused by •  ForFor multi-mode case, which usually mixed with asynchronous resonances caused by flutter flutter three sensors abundant monitoring multiplemode modevibration vibrationasaslong longas asthe [23],[23], three sensors areare abundant forfor thethe monitoring ofof multiple the sampling sampling time time is is long long enough. enough. 3.2.3. TheThe Coherence of the Sensing Matrix andand Choice of Probe Placement Location 3.2.3. Coherence of the Sensing Matrix Choice of Probe Placement Location According to the theory of SR/CS, thethe coherence property of matrix Q also hashas effect on on thethe According to the theory of SR/CS, coherence property of matrix Q also effect accuracy of the recovery of vibration spectrum θ f . ( ) accuracy of the recovery of vibration spectrum θ  f  . The two-sided coherence [24] for matrix Q with columns qi is defined as: The two-sided coherence [24] for matrix Q with columns qi is defined as:

 qi , q j . (14) µ(Q) = max i,j;j6=i k qi k2 · k q j k 2

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For the linear system of equations z = Qθ(f ), if a nonnegative solution exists such that  1 1 kθ( f )k0 < 2 1 + µ(Q) , then several efficient algorithms are guaranteed to find it exactly [24]. As θ( f ) is the frequency spectrum, the sparsity of spectrum kθ( f )k0 represents the number of modes that contains in vibration signals. To get the high recovery probability of θ( f ), µ(Q) must be bounded with an certain value. According to Equation (7) and Equation (9), Q is defined by the number of sensors around the casing!and their positions. When the imaginary probe number L and probe number I are fixed, L−1 kinds of probe placements can be selected after locking the initial probe. In an attempt to I−1 reconstruct the frequency with high probability, minimal µ(Q) under proper probe placement need to be calculated. It can be concluded from the simulation results that several arrangements of probe placements, which satisfies the minimal two-sided coherence of matrix Q, can be chosen in BTT measurement. Assuming that L is set as a constant 25, the relationship between the two-sided coherence of Q and the number of probes is shown in Table 1. Table 1. Minimal two-sided coherence of matrix Q (L = 25). Number of Probes I

Minimal µ(Q)

2 3 4 5 6

1 0.893 0.691 0.500 0.478

Table 1 shows that increasing the number of probes can in some extent improve the incoherence of matrix Q. However, (1 + 1/µ(Q))/2 is not more than 12 × (1 + 1/0.4) = 1.75, which seems that spectra containing multi-mode frequencies can hard to be recovered. Fortunately, pioneers [25] have proved that the sparse solution still has high probability to be recovered with a µ(Q) exceeding the bounded value. In summary, optimized probe placements can be selected through finding the minimal µ(Q). It must be noted that the uniform placement of probes must be avoided because µ(Q) in that case is larger than any other kinds of placements regardless of the number of probes. 4. Quantitative Expression and Reduction of Measurement Uncertainties 4.1. Mathematical Model of Uncertainties in the BTT Measurement Measurement uncertainties are inherently present among the tip timing signals. Those uncertainty factors could cause aliasing in determining the TOA. Two main factors demonstrated in references are system noise [13] and probe position shift [11]. The deviations of blade tip displacement measurement results due to those factors will be represented in mathematic model respectively. 4.1.1. Uncertainty Due to BTT Signal Noise The best raw signal from a BTT probe is similar to a sharp square wave. During the measurement procedure, TOA must be selected from the square wave such as that shown in Figure 4b. TOA is defined as the time at which the signal exceeds a scaled threshold. However, the existence of BTT signal noise can actually cause difficulty for the timing system to choose consistent and accurate TOA points. As shown in Figure 4a, the system noise can cause rising edge start detection algorithms to be inconsistent. Moreover, noise added to the front face of the BTT signal could cause the optical fiber probe triggered at the incorrect time.

procedure, TOA must be selected from the square wave such as that shown in Figure 4b. TOA is defined as the time at which the signal exceeds a scaled threshold. However, the existence of BTT signal noise can actually cause difficulty for the timing system to choose consistent and accurate TOA points. As shown in Figure 4a, the system noise can cause rising edge algorithms to be inconsistent. Moreover, noise added to the front face of the8BTT Sensorsstart 2017, detection 17, 1745 of 19 signal could cause the optical fiber probe triggered at the incorrect time.

(a)

(b)

Figure case. Figure 4. 4. TOA TOA measurement measurement principle principle (a) (a) noise-corrupted noise-corrupted case case (b) (b) noise-free noise-free case.

In general, signal noise is always represented as zero mean white Gaussian noise. These In general, signal noise is always represented as zero mean white Gaussian noise. These assumptions assumptions would lead to the following actual blade arrival time: would lead to the following actual blade arrival time: 0 tactual  tactual  enoi , (15) 0 t actual = t actual + enoi , (15) 0 where tactual is the ideal TOA of the blade tip, enoi is zero mean white Gaussian noise with variance where t0actual is the ideal TOA of the blade tip, enoi is zero mean white Gaussian noise with variance  noi . According to Equation (2), the biased TOA can cause the fluctuation of finite time increments σnoi . According to Equation (2), the biased TOA can cause the fluctuation of finite time increments  t Thus, based based on byby the signal noise areare found as ∆t00.. Thus, on Equation Equation (3), (3),the thedeflections deflectionsofofblade bladetips tipsbiased biased the signal noise found as follows: follows: 2πRtip 2πRtip y = 2 R (∆t0 + enoi ) = y0 +2 R enoi , (16) T tip Ttipn y n  t  e  y  e , (16)  0 noi  2πR 0 noi Tn T where y0 is the true value of blade tip deflection. Since Tntip isn a constant, the blade tip measurement 2 Rtipwhite sequences be modeled the ideal results added noise sequence. where y0 is can the true value ofas blade tip deflection. Sincewith is a constant, the blade tip measurement T In practice, the noise level or the maximum measurement uncertainty of the sensor is generally n sequences can modeledas asan theimportant ideal results added with white noise sequence. provided in itsbe handbook parameter. So the uncertainty of tip deflection due to BTT In practice, the noise level or the maximum measurement uncertainty of the sensor is generally signal noise can be estimated in some extent. provided in its handbook as an important parameter. So the uncertainty of tip deflection due to BTT 4.1.2. Uncertainty to Probe Shift signal noise can beDue estimated in Position some extent. Reference [11] indicated that expansion of casing is unavoidable due to thermal and mechanical 4.1.2. Dueresonance to Probe Position Shift loads.Uncertainty Thus, engine and radial growth of casing can cause the shift of probe position. Sensors 2017,positions 17, 1745 9 of 19 The shifted of optical probes are displayed 5 from due different viewing points. Reference [11] indicated that expansion of casinginisFigure unavoidable to thermal and mechanical loads. Thus, engine resonance and radial growth of casing can cause the shift of probe position. The Initial Shifte shifted positions of optical probes are displayed in Figure 5 from different points. posit d positionviewing io n

Casing

Δα

Blades

Initial position of the probe

Shifted position of the probe

Blades

(a)

(b)

Figure5. 5.Shift Shiftininprobe probeposition. position.(a) (a)Top Topview viewofofblade bladeassembly assemblysystem system(b) (b)Front Frontview viewofofblade blade Figure assembly system. assembly system.

As can be seen from Figure 5b, the angular position of probes fluctuated under the shift of probe position. The error to original angular position is defined as  . When the position of the sensors are shifted, the time that sensors are expected to be triggered texpected must be changed simultaneously. texpected 

Tn T 0      2 n  texpected  n  , 2  2

(17)

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As can be seen from Figure 5b, the angular position of probes fluctuated under the shift of probe position. The error to original angular position is defined as ∆α. When the position of the sensors are shifted, the time that sensors are expected to be triggered texpected must be changed simultaneously. texpected =

Tn Tn ∆α, [(α + ∆α) + 2πn] = t0expected + 2π 2π

(17)

where t0expected is the expected arrival time without measurement error. As the shift of the probe position Tn has no effect on the accuracy of TOA, the finite time increment is shifted 2π ∆α from the ideal one. According to Equation (3), displacement of blade tip can be found as follows:

y=

  2π Tn Rtip ∆t0 + ∆α = y0 + ∆αRtip , Tn 2π

(18)

where ∆t0 is the ideal finite time increment and y0 is the ideal blade tip displacement. Under the irregularly interrupted load, the position shifts of probes ∆α in each rotation can be seen as independent random variables. As the long-time continuous testing will be done, measurement results are plentiful. The central limit theorem says that the computed values of the average of ∆α will be approximately normally distributed. Reference [11] indicate that true value is expected to lie within ±|∆α|max no less than 95 percent of time i.e., P{−|∆α|max ≤ ∆α ≤ |∆α|max } = 95%. In practice, the boundary of angular position can be determined by studying the Finite Element-based data of engine structural [26]. Thus, the inaccuracy level of probe position σpos can be estimated based on the theory of probability and statistics. 4.2. Uncertainties Reduction in Frequency Recovery Based on Equations (16) and (18), the blade tip displacement considering measurement uncertainties can be re-written as follows: y[ j] = y0 [ j] + enos + δpos ,

(19)

where y0 [ j] is the ideal BTT sampling It has been proved in the preceding part of the text  sequence, 
2 2 that enos ∼ N 0, σnos and δpos ∼ N 0, σpos . Variances can be selected from the parameters defining the level of corresponding normal distributions. According to Equation (9), the measurement sequence can be rewritten in the form: y[ j] = Rθ0 ( f ) + δmea ,

(20)

where δmea = enos + δpos is the additive error sequence. So the variance of δmea can be calculated through root-sum-squaring two uncertainty factors [27]. 2 2 2 σmea = σnoe + σpos .

(21)

Therefore, the uncertainty level of the total system can be estimated by combining the corrupting level of each uncertainty factor respectively. To make the sparse solution more closely to real frequency spectrum θ0 ( f ), Basis Pursuit De-noising (BPDN) method [28] is applied to this problem. First, the reconstruction problem can be transformed based on Equation (10) when measurement results are corrupted. Ty[ j] = Qθ0 ( f ) + Tδmea , (22) where T = QR† has been defined as the transformation matrix.

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BPDN method adapts to de-noised spectrum θˆ ( f ) through the solution of the convex unconstrained optimization problem: min θˆ ( f )

1 2 kz − Qθˆ ( f )k2 + λkθˆ ( f )k1 , 2

(23)

where z = Ty[j] is the transformed measurement sequence. The key step of BPDN involves finding a quadratically constrained linear program that is closely related to the following convex constrained optimization problems [29]: min kθˆ ( f )k1 s.t. kz − Qθˆ ( f )k2 ≤ ε, θˆ

(24)

where ε is nonnegative real parameters. If the error term of in Equation (24) satisfies kTδmea k2 ≤ ε, then the solution will be close to θ0 ( f ) with high probability. Therefore, it can be concluded that as far as the level of uncertainty factors is estimated accurately, efficient algorithms [29] have been proposed for obtaining θˆ ( f ) which close to the true value. In practice, a feasible way for estimation of ε is based on the estimation of δmea . Here the experimental biases are conducted to find out the uncertainty parameter σmea in the total measurement chain. Therefore the error sequence δmea can be regarded as a standard white Gaussian noise with a noise level σmea . Finally the boundary of ε can be determined by calculating kTδmea k2 . 5. Numerical Simulations 5.1. Recovery of Multi-Mode Signals This section a sequence of artificial multi-mode vibration signals is presented to validate the ability of BP and BPDN method in detecting the multiple harmonic frequencies. Based on Equation (5), the simulated signal is designed as follows: 3

d(t) =

∑ Ai sin(2π fi t + ϕi ),

(25)

i =1

where f i = {332, 482.2, 599.3}, which contains one synchronous mode and two asynchronous modes, amplitudes are set as Ai = {0.3, 0.2, 0.1}, ϕi ∈ [0, 2π ] are the randomly selected parameters. The BP method is firstly adopted in the simulated signals and the parameters of BP are set as: the parameter of rotating speed fr in simulation model is fixed as 5000 r/min (83.3 Hz); the parameters of probe number are selected to satisfy the uniqueness of solution and thus L = 25, I = 4; the parameter of probe positions Λ = {1, 3, 8, 19}; N is the number of rotation in testing which should be long enough to get the abundant information and thus N = 70. Therefore, the frequency 332 Hz is the synchronous mode whose engine order (EO) equals to four. All the simulated BTT signals are sampled using the SIMULINK function Hit Cross [30]. The reconstructed frequency spectrum of the vibration signal is displayed in Figure 6a. BPDN method is then introduced to detect the harmonic features of vibration signals. To compare with BP method, all the parameters in BPDN method are the same as those in BP. In addition, the parameter of noise level ε is set as zero because the simulated BTT signals are uncontaminated. The results of the extracted harmonic frequencies are shown in Figure 6b. Comparing the two proposed methods, it can be verified that both methods have well performance in detecting the harmonic frequencies of vibration signals without biased signals. Furthermore, the dominant mode of vibration can be clearly identified through the amplitudes of reconstructed spectrum.

of probe number are selected to satisfy the uniqueness of solution and thus L = 25, I = 4; the parameter of probe positions   1, 3,8,19 ; N is the number of rotation in testing which should be long enough

to get the abundant information and thus N = 70. Therefore, the frequency 332 Hz is the synchronous mode whose engine order (EO) equals to four. All the simulated BTT signals are sampled using the SIMULINK function Hit Cross [30]. The reconstructed frequency spectrum of the vibration signal is Sensors 2017, 17, 1745 11 of 19 displayed in Figure 6a. 6

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Figure 6. Comparison of frequency recovery of unbiased BTT data (a) Frequency spectrum recovered by BP (b) Frequency spectrum recovered by BPDN. by BP (b) Frequency spectrum recovered by BPDN. BPDN method is then introduced to detect the harmonic features of vibration signals. To compareofwith BP method, all the parameters in BPDN method are the same as those in BP. In 5.2. Reduction Measurement Uncertainties addition, the parameter of noise level ε is set as zero because the simulated BTT signals are

In uncontaminated. this section, simulation experiments are designed to illustrate the ability of the proposed method in reducing uncertainties infrequencies spectra recovery. The simulation signal can designed The resultseffects of the of extracted harmonic are shown in Figure 6b. Comparing thebe two proposed methods, it can be verified that both methods have well performance in detecting the as a double-mode signal corrupted by measurement uncertainties. harmonic frequencies of vibration signals without biased signals. Furthermore, the dominant mode of vibration can be clearly through amplitudes of freconstructed d(identified t) = A sin t) + A sin(2π t) + v(t). spectrum. (2π f the 1

2

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According to Equation (21), v(t) can be formulated as a zero-mean normally distributed signal In this simulation experiments are designed to illustrate theHence ability various of the proposed comprised of twosection, major uncertainty factors (noise and position shift). measurement method in reducing effects of uncertainties in spectra recovery. The simulation signal can be designed uncertainties has been converted into a single additive signal sequence. as a double-mode signal corrupted by measurement uncertainties. To describe the influence of measurement uncertainties and the accuracy of frequency recovery, A1 sin  2 f1t  of A2 sin t   probability  2 f2t   v t[32]  . are introduced respectively, (26) noise-to-signal ratio(NSR) [31] dand recovery(β) defined as: According to Equation (21), v t can be formulated  normally distributed signal   kd(t) as − avzero-mean (t)k2 , (27) NSR = 1/20 log comprised of two major uncertainty factors (noise andkposition v(t)k2 shift). Hence various measurement uncertainties Sensors 2017, 17, has 1745 been converted into a single additive signal sequence. 12 of 19 To describe the influence of measurement of frequency recovery, β= 1 − kθ( f )uncertainties − θˆ ( f )k2 /kand θ( fthe . (28) )k2accuracy noise-to-signal ratio(NSR) [31] and probability of [32] are introduced respectively, ˆ  frecovery(β) .  1  θ f   θ / θ f  2 BPDN   2 method respectively. The (28) Firstly, a specific simulated signal is applied to BP and parameters defined as:

of testing signal set as: simulated A1 = 0.4, signal A2 = 0.3, f1 = 600 Hz,and f2 =BPDN 280 Hz, NSRrespectively. = 0.3. Parameters in Firstly,are a specific is applied BP method The  d to t   v(t)  2 parameters of testing signal areasset as: A 1 =1Section 0.4, 2 =0.3, = 600waveform Hz, Hz, NSR = 0.3. data Parameters measurement system are the same those in 5.1.f1The sampled of simulated  , f2 = 280and NSR / 20 A log (27)  waveform and sampled data of v  t 5.1. The system are7.the same as those in Section signals in aremeasurement presented in Figure 2  simulated signals are presented in Figure 7. 1 noise-free data noise-corrupted data

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Figure 8a shows the recovered frequency spectrum by BP method. It can be seen that the noise has masked the dominant harmonic components. Considering the uncertainty level by importing the parameter Tv   , frequency spectrum recovered through BPDN method is shown in Figure 8b. 2

Obviously, the noise has been eliminated and the dominant frequencies can be observed more clearly. 5

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Figure 7. Ideal BTT signals and corrupted signals sampled by four probes in one rotation.

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Figure shows the the recovered recovered frequency frequency spectrum spectrum by BP method. method. It noise Figure 8a 8a shows by BP It can can be be seen seen that that the the noise has has masked masked the the dominant dominant harmonic harmonic components. components. Considering Considering the the uncertainty uncertainty level level by by importing importing the the parameter , frequency spectrum recovered through BPDN method is shown in Figure Tv   parameter kTvk ≤ ε, frequency spectrum recovered through BPDN method is shown in Figure 8b. 8b. 22

Obviously, the noise has been eliminated and the the dominant dominant frequencies frequencies can canbe beobserved observedmore moreclearly. clearly. Obviously, 5

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Figure Figure 8. 8. Comparison Comparison of of frequency frequency recovery recovery of of biased biased BTT BTT data data (a) (a) Frequency Frequency spectrum spectrum recovered recovered by by 𝑨𝟏 𝑨𝟏 A A 1 1 BP(β ≈ Frequency spectrum spectrum recovered recovered by by BPDN(β BPDN(β = = 56.03%, ≈ ≈ 2:1 2:1)) (b) (b) Frequency 56.03%, ≈ 4:3 4:3).). BP(β == 31.29%, 31.29%, 𝑨 A𝟐2

𝑨𝟐2 A

Then, of both both methods methods was was performed performed using using Monte Monte Carlo Carlo simulations. simulations. Then, comparison comparison of The parameters of simulated signals are set as follows: amplitudes A11 and A22 are selected The parameters of simulated signals are set as follows: amplitudes A and A are selected randomly in each each test test from from the the range range (0, (0, 1]. 1]. While While the the frequency frequency ff11 and randomly in and ff22 are are selected selected randomly randomly from from (100, 1000] Hz. (100, 1000] Hz. A number of ranging from to five five are are used used for for sampling sampling and and recovering recovering vibration vibration A number of probes probes ranging from three three to signal. angular position signal. For For each each probe probe number, number, aa set set of of angular position of of probes probes which which satisfies satisfies the the minimal minimal μ(Q) µ(Q) are selected and fixed during the simulation experiment. Six NSR values are selected from 0.05 to 0.3. are selected and fixed during the simulation experiment. Six NSR values are selected from 0.05 0.3. Sensors 2017, 17, 1745 13 to of 19 The noises noises in in different different level level will will be be added added to to original original signals signals in in test. test. 100 The 100 signals signals are are generated generated under under eachNSR NSRfor forthe thetesting. testing. The The average of ββ in each case of of testing and each in 100 100experiments experimentsare arecalculated calculatedinin each case testing displayed in Figure 9. 9. and displayed in Figure 100

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Figure BPand andBPDN BPDNmethods methods using 3 probes 4 probes (c) 5 Figure9.9.Reconstruction Reconstructionaccuracy accuracy of BP using (a) (a) 3 probes (b) 4(b) probes (c) 5 probes. probes.

Compared with BP, the BPDN method alleviated to a certain extent the effect of the noise and made an appealing improvement in identifying blade vibration frequencies without increasing the probe number.

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(c) BP method

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Figurewith 9. Reconstruction accuracy of BP and BPDN methods using (a) extent 3 probesthe (b) 4effect probesof(c)the 5 noise and Compared BP, the BPDN method alleviated to a certain probes. made an appealing improvement in identifying blade vibration frequencies without increasing the probe number. Compared with BP, the BPDN method alleviated to a certain extent the effect of the noise and

made an appealing improvement in identifying blade vibration frequencies without increasing the

6. Experiments probe number.

Two experiments are performed in this section to validate the ability of the proposed method in 6. Experiments multi-frequency detection and measurement uncertainty reduction using respective test rigs. Two experiments are performed in this section to validate the ability of the proposed method in multi-frequency detection and measurement uncertainty 6.1. Recovery of Multi-Mode Vibration Response in Frequencyreduction Domainusing respective test rigs.

6.1. Recovery of Multi-Mode Vibration Responsevibration in Frequency Domain in working conditions when suffered Rotor blades often generate multi-mode response from damage. Therefore, a vibration-based test is vibration performed to acquire the multi-mode deflection Rotor blades often generate multi-mode response in working conditions tip when signals. suffered The vibration-based test is runa vibration-based using the DV-100-1 electrodynamic shaker andtipthe rig is from damage. Therefore, test is (980N) performed to acquire the multi-mode deflection vibration-based is run using the DV-100-1 electrodynamic shown in Figure signals. 10. TheThe test specimen is test designed according to the (980N) size and material ofshaker a rotor blade and the rig is shown in Figure 10. The test specimen is designed according to the size and material for a third stage of a compressor. To match the working case more exactly, a cut is made on of the blade a rotor blade for a third stage of a compressor. To match the working case more exactly, a cut is made tip and the testing blade is attached to the fixture with dovetail joints, which is the same with the on the blade tip and the testing blade is attached to the fixture with dovetail joints, which is the same root attachment of compressor The tipdisk. displacement signals are measured by thebyacceleration with the root attachment ofdisk. compressor The tip displacement signals are measured the transducer on the transducer front of the under sampling 24,000rate Hz.24,000 Hz. acceleration onblade the front of thethe blade under therate sampling

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Figure 10. Vibration-based test rig (a) photo of testing rig (b) damaged testing blade of AL7075.

Figure 10. Vibration-based test rig (a) photo of testing rig (b) damaged testing blade of AL7075. Sensors 2017, 17, 1745

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The experiment is implemented at the excitation frequency of 136 Hz, which is close to the The experiment is implemented at the excitation frequency of 136 Hz, which is close to the resonantresonant frequency of the testing blade. Under the fixed excitation load, the vibration signal and its frequency of the testing blade. Under the fixed excitation load, the vibration signal and its frequency spectrum are shown in Figure 11.11.ItItcan thatthethe response of blade tip vibration frequency spectrum are shown in Figure canbe be found found that response of blade tip vibration under aunder single harmonic excitation multiple harmonic the recovery of a single harmonic excitationconsists consists ofof multiple harmonic signals.signals. Thus, the Thus, recovery of multimodesignals signals in domain has has beenbeen proved to be meaningful. multi-mode in frequency frequency domain proved to be meaningful. 200

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Figure 11. Tip vibration signals and its corresponding spectrums (a) tip deflection data sampled by acceleration transducer and BTT sampling system (b) frequency spectrum of original vibration signal acceleration transducer and BTT sampling and recovered spectrum by BP method. system (b) frequency spectrum of original vibration signal and recovered spectrum by BP method. Although the BTT signals of the compressor cannot be acquired directly, a sub-sampled sequence can be extracted from the continuous tip vibration signals to represent the BTT sampling sequence equivalently. Suppose the testing blade is attached to a disk and rotated under the speed fr = 200 Hz. Probe numbers are selected to satisfy the uniqueness of solution and thus L = 20, I = 4. The positions of probes   1, 3, 9,17 are selected by calculating minimal two-sided coherence. According to the sampling time of the imaginary BTT measurement above, equivalent BTT signals

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Although the BTT signals of the compressor cannot be acquired directly, a sub-sampled sequence can be extracted from the continuous tip vibration signals to represent the BTT sampling sequence equivalently. Suppose the testing blade is attached to a disk and rotated under the speed fr = 200 Hz. Probe numbers are selected to satisfy the uniqueness of solution and thus L = 20, I = 4. The positions of probes Λ = {1, 3, 9, 17} are selected by calculating minimal two-sided coherence. According to the sampling time of the imaginary BTT measurement above, equivalent BTT signals can be manually extracted from the continuous vibration signals sampled by the acceleration sensor as shown in Figure 10a. BP method is then introduced to recover the vibration frequency spectrum using the parameters mentioned above. The recovered frequency spectrum is shown in Figure 11b. Comparison of recovered frequency spectrum with the original one confirms that the proposed method is effective in detection of multi-mode frequency in the engineering applications. 6.2. Uncertainty Reduction in Frequency Detection Results The rotating blades often generate vibration at certain speed. Thus, the BTT measurement is performed under a single engine-ordered excitation. The experimental test rig is shown in Figure 12. Four optical-fiber probes are embedded in the circular case to sample arrival times. The key phase measurement probe is placed close to the rotating shaft for sampling reference time impulses. The system is driven by a motor whose speed could reach 8000 RPM. The experiment is performed through 32 blades made of corrosion-resistant chromium steel with overall sizes of 34 mm long, 11 mm Sensors 2017, 17, 1745 15 of 19 wide and 3 mm thick. Rotating shaft

Bladed-disk Platform

Motor Optical-fiber sensors Key phase sensor Figure 12. The test rig. Figure 12. The test rig.

The validation experiment is performed performed as as follows. follows. First, the Campbell diagram of rotating blades is completed based on the finite element (FE) finite element (FE) analysis analysis results results and and the the EO EO excitation excitation speed speed ffrr is selected Finally, selected according according to to the the diagram. diagram. Then the test is implemented at the certain excited speed frr.. Finally, the time impulse signals of the the blades blades are are sampled sampled to to calculate calculate the the tip tip deflection. deflection. Considering the fluctuation of the rotation system, a continuous testing is done to select data sampled undergoing a stable rotating speed. speed. The Campbell diagram of the rotating blade is shown in Figure 13 from which the EO excitation 14000 speed can be adjusted according to the point where the first bending frequency curve crosses one of the EO excitation lines.12000It has been proved that the blade willFirst beBending excited if the parameters of EO First Torsion excitation are set as EO = 18, fr = 126.3 Hz [33]. Here the parameter of BP method L is initialized as 41 Second Bending 10000 EO=72

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sensor Figure 12. The test rig.

The validation experiment is performed as follows. First, the Campbell diagram of rotating Sensors 2017, 17, 1745 blades is completed based on the finite element (FE) analysis results and the EO excitation speed fr15 isof 19

selected according to the diagram. Then the test is implemented at the certain excited speed fr. Finally, the time impulse signals of the blades are sampled to calculate the tip deflection. Considering the according to theofresults for the first bending modes in FEismodel. thesampled analysisundergoing in Section a3.2.2, fluctuation the rotation system, a continuous testing done toBased selecton data a probe placement method is decided as Λ = {1, 19, 28, 39} and shown in Figure 14. stable rotating speed. 14000

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Figure 13. Campbell diagram experimental setting setting blade with 18 18 EOEO crossing at at Figure 13. Campbell diagram ofofexperimental bladefirst firstthree threemodes modes with crossing 2278 Hz and 7580 rpm. 2278 and SensorsHz 2017, 17,7580 1745 rpm. 16 of 19

Campbell SensorsThe 2017, 17, 1745 diagram of the rotating blade is shown in Figure 13 from which the EO excitation 16 of 19 speed can be adjusted according to the point where the first bending frequency curve crosses one of the EO excitation lines. It has been proved that the blade will be excited if the parameters of EO excitation are set as EO = 18, fr = 126.3 Hz [33]. Here the parameter of BP method L is initialized as 41 Probe r modes in FE model. Based on the analysis in Section according to the results for the first bending Probe 1 (  =1) 3.2.2, a probe placement method is decided as Λ = {1, 19, 28, 39} and shown in Figure 14. Probe r 0

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Figure 14.14.Placement probes. Figure Placementof ofoptical optical probes. Figure 14. Placement of optical probes.

Vibration signals of blade 18 are shown in Figure 15. The tip deflections collected by four probes

Vibration signals of blade 18 are shown in Figure 15. The tip deflections collected by four probes are fluctuating around the same phrase respectively, which shows the blade resonate at a Vibration signals bladephrase 18 are shown in Figure 15. The tip deflections probes are fluctuating around theofsame respectively, which shows the blade collected resonateby at four a synchronous synchronous vibration and the sampling signals are corrupted. are fluctuating around signals the same respectively, which shows the blade resonate at a vibration and the sampling arephrase corrupted. Probe 1 Probe 2 Probe13 Probe Probe24 Probe

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Rotation number Figure 15. Signals of blade tip displacement. Figure15. 15.Signals Signals of of blade blade tip Figure tipdisplacement. displacement. The distribution of measurement results of four probes is shown in Figure 16. It can be seen that displacements that are detected by each probe approximately obey the normal distribution. Hence, it The distribution of measurement results of four probes is shown in Figure 16. It can be seen that can be concluded that the uncertainty measurement model built above is feasible. Furthermore, the displacements that are detected by each σprobe approximately obey the normal distribution. Hence, it uncertainty level of the measurement mea can be calculated through the distribution in Figure 16. can be concluded that the uncertainty measurement model built above is feasible. Furthermore, the As the optical probes are fixed to the platform in this experimental rig, the source of jitter for about uncertainty level of the measurement σmea can be calculated through the distribution in Figure 16. 20 μm is mainly caused by intrusion of optical noise and the limitation of sensor resolution rather As the optical probes are fixed to the platform in this experimental rig, the source of jitter for about than shift of probe position. 20 μm is mainly caused by intrusion of optical noise and the limitation of sensor resolution rather

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The distribution of measurement results of four probes is shown in Figure 16. It can be seen that displacements that are detected by each probe approximately obey the normal distribution. Hence, it can be concluded that the uncertainty measurement model built above is feasible. Furthermore, the uncertainty level of the measurement σmea can be calculated through the distribution in Figure 16. As the optical probes are fixed to the platform in this experimental rig, the source of jitter for about 20 µm is mainly caused by intrusion of optical noise and the limitation of sensor resolution rather than shift probe position. Sensorsof2017, 17, 1745 17 of 19 25

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With the sampled signals, a single-measurement vector model can be built and the frequency spectrum can be recovered by BP algorithm as Figure200017a. Then the parameter ε is calculated and 1000 2270 Hz the frequency spectrum is recovered based on BPDN method. The result of the advanced method is 800 shown in Figure 17b. 1500 By comparing Figure 17a with Figure 17b, it can be seen that the spectrum reconstructed by the 600 BP algorithm is mixed with some biased frequency characteristics while the spectrum reconstructed 1000 400 de-noise method is clean and the dominant frequency can be extracted easily. Furthermore, with the the frequency detected by the BPDN method was well 500 accorded with the Campbell diagram. 200 It must be noted that due to high stiffness of the material and the limited excitation method in our current equipment, only a single synchronous mode of the blades can be excited. Our further 0 0 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 work might focus on acquiring and analyzing BTT sampling data that contains multi-mode vibration Frequency (Hz) Frequency (Hz) components and measurement uncertainty simultaneously from compressors or jet engines to validate (a) (b) the proposed method. Figure 17. The recovered spectrum of sample streams based on (a) BP algorithm (b) BPDN algorithm.

It must be noted that due to high stiffness of the material and the limited excitation method in our current equipment, only a single synchronous mode of the blades can be excited. Our further work might focus on acquiring and analyzing BTT sampling data that contains multi-mode vibration components and measurement uncertainty simultaneously from compressors or jet engines to validate the proposed method.

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It must be noted that due to high stiffness of the material and the limited excitation method in 7. Conclusions our current equipment, only a single synchronous mode of the blades can be excited. Our further monitoring of the vibration characteristics of rotating blades is crucialvibration for fault workBTT-based might focus on acquiring and analyzing BTT sampling data thatengine contains multi-mode diagnoses. However, two of its main drawbacks are that BTT signals are under-sampled and tend to components and measurement uncertainty simultaneously from compressors or jet engines to be biased under the uncertainty factors in measurement. This paper explores a novel BTT analysis way validate the proposed method. for alleviating uncertainties in recovery of multi-mode blade vibration signals based on sparse theorem. With the probe number not more than those used in conventional methods, recovering four or even 7. Conclusions more frequency components from the blade vibration signals can be available. Moreover, without BTT-based monitoring of the vibration characteristics of rotating engine blades is crucial for fault increasing the probe number, the probability of frequency recovery can be maintained higher than 50% diagnoses. However, two of its main drawbacks are that BTT signals are under-sampled and tend to under a certain range of noise level. Simulation and experimental results confirm that the proposed be biased under the uncertainty factors in measurement. This paper explores a novel BTT analysis method dramatically improved the capacity of monitoring vibration characters of compressor blades, especially when the dynamic behavior of rotated blades is highly nonlinear. The main results are summarized as follows: 1.

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A single-measurement vector model for sparse BTT data is built and all the harmonic frequencies can be extracted from the solution of the sparse representation model if the measurement results are unbiased. The number and placement of optical-fiber probes play an important role in the effectiveness of frequency spectrum recovery. Considering the blade vibration conditions in practice, four probes located around the case, which satisfies the minimal two-sided coherence, is enough to recover the vibration frequency spectra well. The improved blade vibration recovery method based on BPDN obviously enhanced the robustness to the noise interferences. The dominant vibration frequencies can be identified from corrupted non-uniformly under-sampled BTT data through the proposed method without increasing the number of probes.

Acknowledgments: This work was supported by the National Basic Research Program of China (Grant no. 2015CB057400) and National Natural Science Foundation of China (Grant no. 11402296). The authors thank Weimin Wang and Sanqun Ren of Beijing University of Chemical Technology for assistance with the experiments. Author Contributions: This work conducted by Minghao Pan in collaboration with all authors. Minghao Pan designed the proposed method and wrote the paper. Yongmin Yang and Fengjiao Guan supervised the research. Haifeng Hu designed the experiments. Hailong Xu revised the final submission. Conflicts of Interest: The authors declare no conflict of interest.

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