New step to improve the accuracy of blade tip timing method without once per revolution

Mechanical Systems and Signal Processing 134 (2019) 106321

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

New step to improve the accuracy of blade tip timing method without once per revolution Kang Chen a, Weimin Wang a,b,⇑, Xulong Zhang b, Ya Zhang b a

Chemical Safety Engineering Research Center of the Ministry of Education, Beijing University of Chemical Technology, Beijing 100029, PR China Beijing Key Laboratory of Health Monitoring and Self-recovery for High-end Mechanical Equipment, Beijing University of Chemical Technology, Beijing 100029, PR China b

a r t i c l e

i n f o

Article history: Received 24 January 2019 Received in revised form 10 June 2019 Accepted 20 August 2019 Available online 29 August 2019 Keywords: Blade vibration None OPR BTT Compound reference Uncertainty Straight line fitting

a b s t r a c t The mechanical vibration has always been one of the main factors that restrict the improvement of turbomachinery performance such as aero-engines. Blade vibration is one of the most important obstacle. As a non-intrusive vibration measurement method, blade tip timing (BTT) technology has the potential to replace strain gauges for blade dynamic stress monitoring. However, it still faces enormous challenges, one of which is the accuracy and reliability of the once per revolution (OPR) probe. Here, an efficient and high-precision compound reference (CR) based BTT method without OPR is proposed. The method is based on straight line fitting (SLF), and taking the blade at the midpoint of the SLF interval as the reference blade. The vibration displacement of the reference blade is obtained by the SLF of the time of arrival (ToA) of probe No. 1. Further, taking the blade as the reference, the vibration displacement of other blades is obtained. With this method, the actual blade installation angle and the probe installation angle can be measured simultaneously based on the none OPR method so as to further improving the measurement accuracy. By the numerical simulation, high speed test-rig and large scale industry turbo fan verification, the results indicate that the method has the merit of stronger anti-noise ability, less uncertainty, and much higher computational efficiency. Whereby, high-efficiency blade vibration measurement can be achieved in the whole operating speed range. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Blade condition monitoring is considered an important aspect of turbomachinery health management, especially with the current trend of operating conditions at higher rotating speeds, aerodynamic loads and temperatures. Condition monitoring is one of the way to improve their reliability and reduce risk especially for aircraft engine blade. Monitoring the vibration parameters of the blade is necessary for their crack detecting prior to blade failure. In the scope of blade vibration monitoring method, it is currently mainly divided into intrusive and non-intrusive measurement methods. The intrusive method is mainly depend on strain gauge, and the representative of the non-intrusive method is BTT. After years of development, strain gauge technology has been used as a large-scale test for turbomachinery such as aero-engines and steam turbine, and is considered to be the most reliable test method at present [1]. However, due to the complicated installation, interfering with

⇑ Corresponding author at: DSE research center, School of mechanical and electrical engineering, Beijing University of Chemical Technology (BUCT), No.15.North San Huan East Road, POBox#130, Beijing, 100029, China. E-mail addresses: [email protected] (K. Chen), [email protected] (W. Wang). https://doi.org/10.1016/j.ymssp.2019.106321 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

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K. Chen et al. / Mechanical Systems and Signal Processing 134 (2019) 106321

Nomenclature fs fn R nb np nEo

m X tCR-BTT tSLF-BTT

sampling frequency of probe natural frequency radius of bladed disc number of blades number of probes engine order rotating speed rotor frequency time comparison of CR-BTT time comparison of SLF-BTT

Subscripts b blade index p probe index n revolution index fit obtained by straight line fitting t theoretical

the local fluid flow field and the inability to work in extreme environments for a long time [2]. It is not fully applicable to the testing of modern advanced turbomachines which need to work under more extreme conditions and require better performance. The Blade Tip-Timing is a well-known non-intrusive measurement technique currently employed for the identification of the dynamic behaviors of rotating bladed disks [3]. The measurement system has become a typical industry equipment for bladed disks vibration surveys. While, the type of sensors, the positioning of the sensors around the bladed disk and the used algorithm for data post-processing are still not standard techniques, and their reliability has to be proved for different operation conditions [4,5]. So the methods is need to be developed further. Many BTT data analysis methods have been developed and published in recent years. Gallego-Garrido et al. [6] presented BTT method to multimodal vibration. Zhong-Sheng Chen et al. [7,8] and Pan et al. [9] contributed to the signal reconstruction of the tip timing method. Battiato et al. [5] proposed a new type of probe arrangement and two kinds of experimental bench designed for blade vibration mode and nodal diameter (ND) identification using BTT. At the same time, the uncertainty of the BTT method is receiving more and more attention [10]. Mohamed et al. [11] proposed a method for determining the movements of the sensing position of probes. Rossi et al. [12] analyzed the sources of uncertainty in various BTT methods and tried to promote a new calibration technique. Hajnayeb et al. [13] designed and implemented a new reference test bench for researching the uncertainty of BTT. Although, good result have been achieved, an OPR probe is needed in these methods to calculate the blade vibration. The OPR probe is used as the stationary reference for the position on the disk or the shaft. Obviously, the reliability and precision of the whole BTT system was largely depended on the OPR probe. Russhard [10] revealed the factors affecting the uncertainty and points that the monitored blade vibration is derived from the arrival time at the probe and expected time, which in turn is derived from the OPR probe. Jousselin et al. [14] pointed out that the signal to noise ratio (SNR) will have an additional impact on the accuracy of the triggered ToA data points for each blade tip passing each optical probe. These additional effects result in an increased level of uncertainty on the extracted blade tip amplitudes. Furthermore, in many industrial case, it is very difficult to install an OPR probe and maintain its reliability. So, it is urgent need to develop one none OPR BTT method with high accuracy and efficient to monitor blades vibration. A novel none OPR BTT algorithm was reported by Russhard [15]. It showed a significant effect in reducing measurement uncertainty. Later, Duan et al. [16] proposed a blade resonance parameter identification method based on BTT method without OPR probe. The principle of the none OPR BTT is to find other objects that can be used as stationary references to replace the OPR probe, no matter the references is assumed to be static or actual static. Comparing with the published none OPR BTT methods, it can be divided into the following three forms in principle: (1) reference to theoretical reference based on straight line fitting (SLF-BTT) [15]; (2) reference to one probe (RTP) [16]; (3) reference to one blade (RTB) [17]. Through the theoretical analysis of the above three kinds of methods, we can find: (1) The SLF-BTT method can directly obtain the vibration displacement of the blade, and have chance to realize online real-time monitoring. However, due to the disadvantage of the least squares linear fitting method itself [18,19], that is, the fitting reliability is different at each fitting point. For blades far from the midpoint of the blade number, the reliability of the result is not as high as the midpoint. It is easy to imagine that a slight error in slope will have a significantly greater effect on the points at both ends than on the midpoint of the line segment. Therefore, in the low operation speed or synchronous vibration occurs, the difference in the TOA of the blades within one rev is large, which will result in a large slope error of the fitted straight line, and the error of the fitting result increases. (2) The RTB or RTP method cannot obtain the blade vibration reference to static state directly, and a series of separation is required. It can only be used as a means to identify the parameters of the synchronous vibration of the blade especially for mistuned blade. (3) Least squares linear fitting requires more computation time (or cpu-time) than the simple

K. Chen et al. / Mechanical Systems and Signal Processing 134 (2019) 106321

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four arithmetic operations, and the calculation process is complicated and time consuming. In practice, multiple probes are required to perform real-time monitoring at the same time. This has great requirements on the computing performance of the device, and requires more equipment investment. At the same time, a large number of fitting calculations will have an impact on the immediacy of real-time monitoring when the operating speed is high, and it is difficult to achieve a fast transition from the ToA to the vibration displacement of the blades. Therefore, there is an urgent demand to develop a novel none OPR BTT method to achieve more accurate and real-time on line blade vibration monitoring. The key to the solution is to find a more reliable static reference, and only the SLF-BTT method can provide a theoretical static reference in the current feasible method. Based on this, further research is carried out, and a novel method called compound reference based blade tip timing (CR-BTT) is developed. Furthermore, the blade installation angle error, probe installation angle error, bladed disk mistuning and other factors are considered. Simulation and experiments show that the none OPR BTT method can eliminate most of the noise caused by the rotational speed fluctuations and other factors compared with the BTT method with OPR. Moreover, CR-BTT can better preserve the true vibration amplitude of the blade compared to the SLF-BTT method. What’s more, the calculation efficiency can be much higher. 2. Theory of compound reference based blade tip timing method 2.1. BTT method without OPR based on straight line fitting (SLF-BTT) The motion of high-speed rotating blade can be separated as two independent processes: rotation and vibration. The principal accumulation degree equation represents the rotating process of the blade (rotating equation). The accumulation degree U1ðtÞ of the blade tip at time t after sampling is


 1 2 at þ f 0 t : 2

U1ðtÞ ¼ 2p

ð1Þ

where a represents the angular accelerated in Hz/s, f 0 represents the angular frequency at the beginning of sampling in Hz. Sampling time t is

t ¼ N=f s :

ð2Þ

where, N represents the number of sampling points, and f s represents the sampling frequency. The vibration of the blade during the rotation is

U2ðtÞ ¼

qb ðtÞ : R

ð3Þ

where qb ðtÞ is the vibration displacement of blades at time t. R is the distance from the tip of the blade to the axis of rotation which can also called the radius of bladed disc. By analyzing the blade rotation and blade vibration independently, the accumulation degree of the blade in sampling number N (or time t ¼ N=f s ) which is obtained from the rotational Eq. (1) and the vibration Eq. (3) is


 1 2 q ðtÞ at þ f 0 t þ b : 2 R

WðtÞ ¼ U1ðtÞ þ U2ðtÞ ¼ 2p

ð4Þ

When the blade passing the probe, the time of arrival (ToA) is recorded by the probe, and at the same time the accumulation degree is

Wðn;bÞ ¼ n  2p þ Uðp;bÞ ðn ¼ 1; 2; 3 . . .Þ:

ð5Þ

where n represents the nth revolution of operation, Uðp;bÞ represents the initial arrival distance of the probe NO.p from the blade NO.b, the corresponding general formula is

Uðp;bÞ ¼ Uðp;1Þ þ ðb  1Þ

2p : nb

ð6Þ

nb is the number of blades. For none OPR BTT method, when the first probe record the first ToA of blade NO.1, the sampling is officially begin. Therefore, there is Uðp;1Þ ¼ 0. Combined Eqs. (4)–(6), if the blade is not vibrating, then the time of each blade passing through the probe t tðn;bÞ is the theoretical time of arrival. The time and the cumulative angle of the blade passing through the probe accords with a fixed functional relationship. That is

pat2tðn;bÞ þ 2pf 0ðnÞ ttðn;bÞ  Wðn;bÞ ¼ 0:

ð7Þ

During acceleration or deceleration (a–0 Hz=s), according to the root solving formula of the quadratic equation with one unknown, there is

ttðn;bÞ ¼

2pf 0ðnÞ 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4p2 f 0ðnÞ þ 4paðnÞ Wðn;bÞ 2paðnÞ

:

ð8Þ

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K. Chen et al. / Mechanical Systems and Signal Processing 134 (2019) 106321

That is to say, the accumulative angle Wðn;bÞ of the blade b in the nth revolution accords with the relationship shown in Eq. (8). In the actual case t  0, the corrected formula of Eq. (8) is

ttðn;bÞ

f 0ðnÞ ¼ þ aðnÞ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 f 0ðnÞ Wðn;bÞ : þ aðnÞ paðnÞ

ð9Þ

When running at constant speed (a ¼ 0 Hz=s), the Eq. (7) becomes

2pf 0ðnÞ ttðn;bÞ  Wðn;bÞ ¼ 0:

ð10Þ

The corresponding ttðn;bÞ is

ttðn;bÞ ¼

Wðn;bÞ 2pf 0ðnÞ

:

ð11Þ

However, taking the possible effects of speed fluctuations and other factors into account, the modified formula (11) is

ttðn;bÞ ¼

Wðn;bÞ 2pf 0ðnÞ

þ BfitðnÞ :

ð12Þ

where BfitðnÞ is the correction coefficient. When the data acquisition begins, the accumulative angle Wðn;bÞ and the actual arrival time t ðn;bÞ of each blade. Since the dimensionless vibration value qb ðtÞ=R of the blade is very small compared with the accumulative angle Wðn;bÞ , the point ðWðn;bÞ ; tðn;bÞ Þ must fluctuate around the Eqs. (9) or (12). The theoretical arrival time t tðn;bÞ can be obtained by fitting the col  lected data Wðn;bÞ ; tðn;bÞ with the least square method. Usually there is a certain speed fluctuation in the rotor operation, and the blade vibration value is different under different rotational speeds. So only the single revolution data are fitted separately in application. Considering that Eq. (9) is a complex nonlinear fit, it is more complicated than a straight line fit of Eq. (12). The fitting time required for nonlinear fitting is often several times or even tens of times that of a straight line fit. What’s more, the rotating speed in one revolution can be considered as nearly unchanged. So Eq. (12) is used as the fitting objective function in practice no matter variable speed or constant speed. It can be simplified as

tðn;bÞ ¼ kfitðnÞ  Wðn;bÞ þ bfitðnÞ :

ð13Þ

where kfitðnÞ is the slope of straight line, bfitðnÞ is the intercept of straight line. So t fitðn;bÞ can be gotten by fitting this straight line by least squares, and it can be considered as the theoretical arrival time. The blade vibration value obtained by tip timing method is

    xðn;bÞ ¼ t ðn;bÞ  t tðn;bÞ  v ðnÞ  t ðn;bÞ  t fitðn;bÞ  v n :

The line speed

vn ¼

ð14Þ

v ðnÞ is

2pR  Tn t

R

 ¼

tðn;nb þ1Þ t tðn;1Þ

R : kfitðnÞ

ð15Þ

2p

Where


 2p þ bfitðnÞ ttðn;nb þ1Þ ¼ kfitðnÞ  Wðn;nb Þ þ nb

ð16Þ

Therefore, the vibration signal of the SLF-BTT method can be finally obtained. 2.2. Compound reference based blade tip timing method (CR-BTT) For the least squares straight line fitting, the fitting result at the midpoint of the fitting interval is often the most reliable [18,19]. This property is extremely important for the accuracy of the BTT method without OPR. The impact will be explained in the simulation below. Taking the fitting result at the midpoint of the fitting region obtained by SLF-BTT as the reliable static reference, this paper proposed an efficient and high-precision blade tip timing method without once per revolution for blade vibration real-time monitor. The specific steps are as follows: selecting the blade at the midpoint of the straight line fitting region as the reference blade, and obtaining the vibration displacement of the reference blade by fitting the arrival time of probe No. 1 with SLF-BTT method.

  x1ðn;KÞ ¼ t1ðn;KÞ  tfitðn;KÞ  v n :

where

ð17Þ

K. Chen et al. / Mechanical Systems and Signal Processing 134 (2019) 106321

K ¼ ceil

n  b ; 2

5

ð18Þ

tfitðn;KÞ ¼ kfit ðnÞ  Wðn;KÞ þ bfit ðnÞ;

ð19Þ

ceil is a round up function. Based on the vibration displacement of the reference blade, the vibration displacement of other blades can be derived. The specific process is as follows: As shown in Fig. 1, according to the principle of BTT with OPR probe, when the blade NO.b rotating through probe NO.p in the nth revolution, the vibration displacement xpðn;bÞ can be expressed as follow:

  xpðn;bÞ ¼ t pðn;bÞ  t ðn;OPRÞ  v ðn;OPRÞ  Uðp;bÞ ;

v ðn;OPRÞ ¼

ð20Þ

2pR 2p R ¼ : Tn tðnþ1;OPRÞ  t ðn;OPRÞ

ð21Þ

where Uðp;bÞ is the arc length between the tip probe NO.p and the blade NO.b when the OPR signal arrives. tðn;OPRÞ is arrival time of OPR signal in the nth revolution. And v ðn;OPRÞ is the rotor speed calculated by the OPR probe. Similarly, in the nth revolution, by taking blade NO.K as the reference, the vibration displacement xKpðn;bÞ of blade NO.b relative to blade NO.K can be expressed as

  xKpðn;bÞ ¼ t pðn;bÞ  t pðn;KÞ  v ðn;KÞ  UðK;bÞ :

v ðn;KÞ ¼

ð22Þ

2p R 2p R :  Tn t pðnþ1;KÞ  t pðn;KÞ

ð23Þ

Combined Eqs. (20) with (22), there is

xKpðn;bÞ ¼

 

 

t pðn;bÞ  t pðn;OPRÞ  v n  Uðp;bÞ  tpðn;KÞ  t pðn;OPRÞ  v n  Uðp;KÞ :

ð24Þ

And simplified Eq. (24) as follows

  xKpðn;bÞ ¼ xpðn;bÞ  xpðn;KÞ ¼ t pðn;bÞ  tpðn;KÞ  v n  UðK;bÞ :

ð25Þ

From Eq. (25), there is

  xpðn;bÞ ¼ xpðn;KÞ þ xKpðn;bÞ ¼ xpðn;KÞ þ t pðn;bÞ  t pðn;KÞ  v n  UðK;bÞ :

ð26Þ

where

UðK;bÞ ¼

2pRðK  bÞ : nb

ð27Þ

From this, the vibration displacement x1ðn;bÞ of all blades under probe No. 1 are obtained. In general, the monitoring system monitor blades by the BTT method with multi-probes. In real-time on-line monitoring, if the data of all probes are straight line fitted, the real-time performance of the monitoring system will be greatly affected, and the processing speed cannot keep up with the sampling speed.

x p n ,b

t n ,OPR

( p ,b )

(t p n ,b

t n,OPR ) v n,OPR

( p ,b )

(20)

tp(n,b)

xp(n,b)

Fig. 1. Schematic of blade tip timing.

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K. Chen et al. / Mechanical Systems and Signal Processing 134 (2019) 106321

Based on the vibration of each blade under probe No. 1, the vibration of each blade under other probes can be calculated combining with the installation angle of the probe. The specific process is as follows. As Fig. 2 showed, the mounting angle between the two probes which called probe NO. 1 and probe NO. 2 is Da. The vibration displacement of the blade NO.b when it passing the probe NO. 1 and the probe NO. 2 on the nth revolution is named as x1ðn;bÞ , x2ðn;bÞ . For the BTT system with OPR probe, from Eq. (20), the vibration displacement x1ðn;bÞ of the blade rotating through the probe No. 1 in the nth revolution is as follows:

x1ðn;bÞ ¼ ðt1ðn;bÞ  tðn;OPRÞ Þ  v ðn;OPRÞ  Uð1;bÞ :

ð28Þ

Similarly, the vibration displacement x2ðn;bÞ through probe No. 2 is:

x2ðn;bÞ ¼ ðt2ðn;bÞ  tðn;OPRÞ Þ  v ðn;OPRÞ  Uð2;bÞ

ð29Þ

Therefore,

xSðn;bÞ ¼ x2ðn;bÞ  x1ðn;bÞ ¼ v ðn;OPRÞ Dt ðn;bÞ  RDa:

ð30Þ

where Dtðn;bÞ is the difference of actual arriving time between the blade No.b reaches the probe No. 1 and the probe No. 2.

Dtðn;bÞ ¼ t 2ðn;bÞ  t 1ðn;bÞ :

ð31Þ

In summary, xSðn;bÞ is called the blade vibration of probe 2 referenced to probe 1. From Eq. (30), there is

  xpðn;bÞ ¼ xSðn;bÞ þ x1ðn;bÞ ¼ ðtpðn;bÞ þ t 1ðn;bÞ Þv n  RDa þ x1ðn;bÞ p ¼ 2; 3; . . . ; np :

ð32Þ

where np is the number of probe used for blade tip timing. The vibration displacement of the blade under other probes are obtained. However, as a method to be applied to engineering applications, the above solve process fails to take the actual possible blade and probe mounting angle error into account and needs to be corrected reasonably. 2.2.1. Acquisition of the blade installation angle error For the ideal bladed disc shown in Eq. (27), in the case without blade vibration, the arrival time and the angle should be the perfect linear relationship. However, the installation angle of the blade on the actual bladed disc will be different due to the machining error and installation error, and will not be perfectly uniform. Eq. (26) shows that accurate blade installation angle is also very important for the accuracy of CR-BTT method. For blade NO.b, the relationship between the error Dab of the actual installation angle and the installation angle under ideal uniform distribution and the error DUb resulted from this is as follows.

DUb ¼ Dab R:

ð33Þ

Assuming that the installation angle error of the blade Dab ¼ 0:001 rad  0:057 and the tip radius of the blade R ¼ 0:1 m, if the angle error is not considered, the blade vibration error DUb ¼ 0:0001 m ¼ 100 lm will be caused, which is a very large amount for the measurement of blade vibration. Therefore, in the case of existing the installation angle error of the blade, the Eq. (14) is rewritten as follows:

t1(n,b)

Processor t2(n,b) xp(n,b)

Fig. 2. Schematic of blade tip timing without OPR.

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K. Chen et al. / Mechanical Systems and Signal Processing 134 (2019) 106321

One-to-one correspondence between ToA and blades Selection of static reference n K = ceil( b ) 2

(18)

Calculate the actual blades mounting angles N

Δ

n 1

b

t n ,b

tfit n ,b

vn

(37)

N

Calculate the actual probe mounting angle vn Δt( n ,K )

N

Δ

( i ,i 1)

n 1

xi 1 n ,K R N

xi n ,K (39)

Calculation of vibration displacement of static reference blade under probe 1 x1 n,K t1 n,K tfit n ,K vn Δ K (35) Calculation of vibration displacement of other blades under the probe1 x p n,b = x Kp n ,b x p ( n,K) x Kp n ,b t p ( n,b ) t p ( n,K) vn (K ,b ) (26) Calculation of vibration displacement under other probes (t p ( n,b ) +t1( n,b ) )vn RΔ +x1 n,b p 2,3,..., np (22)

x p n,b =xSn,b +x1 n,b

Fig. 3. CR-BTT flowchart.

  xðn;bÞ þ DUb ¼ t ðn;bÞ  t fitðn;bÞ  v n :

ð34Þ

And the actual vibration displacement of blade is

  xðn;bÞ ¼ t ðn;bÞ  t fitðn;bÞ  v n  DUb :

ð35Þ

Because of the dynamic characteristics of the blade, the vibration of the blade is usually very small at low speed or at a speed far away from the resonance region, which is much smaller than DUb , or even can be considered xðn;bÞ ¼ 0 lm. In this case, there is

  DUðn;bÞ ¼ t ðn;bÞ  t fitðn;bÞ  v n :

ð36Þ

Therefore, it is feasible to obtain blade installation angle error by tip timing method without OPR. In order to avoid accidental error, the average can be obtained by fitting different revolution data with several times.

PN 

DUb ¼

n¼1



t ðn;bÞ  t fitðn;bÞ  v n N

ð37Þ

Normally, N = 20 is enough to meet the accuracy requirement. Usually a larger N is also allowed. But considering the calculation efficiency, 20 is enough. So that the blade installation angle can be obtained for subsequent CR-BTT method, thus ensuring the accuracy of CR-BTT method. So, the Eq. (27) is corrected as

UðK;bÞ ¼

2pRðK  bÞ þ DUK  DUb nb

ð38Þ

The other effect of blade installation angle calculation error is reduce blade vibration data offset. If the error is not controlled, it will result in a significant data offset. And the blade installation angle error can be also a way to plot the stack plot.

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K. Chen et al. / Mechanical Systems and Signal Processing 134 (2019) 106321

2.2.2. Acquiring the accurate angle of probe installation In the actual blade vibration monitoring, the installation angle of the probes are usually not exactly the same as the design due to machining errors of mounting hole and structural limitation of the case. At the same time, more accurate mounting angle of the probes is needed when the multi-probe signal conversion carried out by CR-BTT method. And the accuracy of the mounting angle of the probe also have a significant impact on the subsequent blade vibration parameters analysis and identification and vibration signal reconstruction. Therefore, it is necessary to obtain the accurate mounting angle of probes. This paper also presents a method to determine the mounting angle of probe based on tip timing method without OPR. The specific steps are as follows: The vibration displacement xiðn;KÞ and xiþ1ðn;KÞ of the blade corresponding to the midpoint position of the fitting region under the probe are obtained by processing the arrival time of two probes in N revolution with SLF-BTT method. From Eq. (30), there is

PN

Daði;iþ1Þ ¼

n¼1

½v n Dtðn;KÞ ðxiþ1ðn;KÞ xiðn;KÞ Þ R

N

:

ð39Þ

Usually, N = 20 is enough to meet the requirements of the probe installation angle accuracy. In summary, the flow chart of the complete process of CR-BTT method is as Fig. 3. 3. Simulation verification of CR-BTT method 3.1. Modeling of simulation The structure of the blades used in industry is complex with infinite degrees of freedom. The complex structure and motion is difficult to be expressed by mathematical expressions. Finite element analysis can effectively solve this problem, which is helpful for further application of blade tip timing analysis. A mathematical model of blade vibration will be established by finite element method. The differential equation of blade forced vibration with multi-degrees of freedom can be

€ þ Cq_ þ Kq ¼ fðtÞ: Mq

ð40Þ

In the formula, M, C, K are the mass matrix, damping matrix and stiffness matrix of the whole structure, respectively. f(t) is the excitation vector, and q is the vibration displacement of blades with multi-degrees of freedom. The excitation force function of each blade can be

f b ðtÞ ¼ F b sinðnEO Xt þ ub Þ:

ð41Þ

where F b represents the amplitude of exciting force. nEO is the engine order of blade vibration. X is the blade rotating speed. Subscript b is the blade number, ub is the delay phase of blade NO.b. In the simulation, the delay phase of the excitation force of blade NO.b is

ub ¼ ðb  1Þ

2p : nb

ð42Þ

In conclusion, the vibration displacement of each blade in the simulation region can be obtained by solving Eq. (40) with Runge-Kutta method. In the analysis of blade synchronous vibration, the relationship of the engine order of blade vibration nEO , the center frequency of blade vibration x and the rotor operating frequency X is

nEO ¼

xn X

ðnEO ¼ 1; 2; 3; 4 . . .Þ:

ð43Þ

In order to verify the validity and feasibility of the above method without the OPR probe, a simulator is developed by Fortran, and blade vibration signals are generated by a single-degree-of-freedom blade model [20,21] shown as Fig. 4. It is assumed that each blade in the assembly has only one significant mode: the first bending mode. The parameters of pffiffiffiffiffiffiffiffiffiffi the tuned blade are blade mass m ¼ 0:05 kg, natural frequency f n ¼ k=m=ð2pÞ ¼ 2200 Hz, damping ratio  pffiffiffiffiffiffiffiffiffiffiffi n ¼ c= 2  m  k ¼ 0:0005 and amplitude of exciting force F b ¼ 10 N. The number of blades is nb ¼ 32. The sampling frequency is f s ¼ 20 106 Hz ð20 MHzÞ. The radius of bladed disc is R ¼ 0:1 m. However, due to the presence of small blade-to-blade variations of structural and (or) geometrical properties, mistuning bladed disk is very common and almost all the bladed disk are mistuning. In order to simulate mistuning on the bladed disk, mistuning is modelled by adding mass variations to nominal blade mass. The ith-mistuned mass is expressed as

mi ¼ m þ di ði ¼ 1; 2; 3; . . . ; nb Þ:

ð44Þ

where di is the ith mistuning value which is given as

di ¼ dmax randðxÞ

ð45Þ

K. Chen et al. / Mechanical Systems and Signal Processing 134 (2019) 106321

9

mi ci

ki

Disk

Fig. 4. Assembly model of blades vibration (Forces not shown).

Here the max value of di is set as dmax ¼ 0:05 m in the simulation. In addition, randðxÞ is the function to get a random number between 0 and 1. In order to simulate the variable speed and constant speed, the two different case is considered in this part as (a) Case 1: nEO ¼ 10, the rotation frequency X rises from 219 Hz to 221 Hz (13,140–13,260 in rpm) and the speed-up frequencies is 2 Hz/s; (b) Case 2: nEO ¼ 10, the rotation frequency X is 218.5 Hz (13,110 in rpm). The ToA of blade tip was sampled by two probes, and the angle between them was 6°. At the same time, the actual vibration displacement of each blade at the corresponding time can be extracted in the simulation, which can be used as the accuracy evaluation criterion of the results obtained by BTT method.

3.2. Result of SLF-BTT

Average / m

30 20 10 0

0

10 20 Blade number (a) Average of ME

30

Standard deviation/ m

Figs. 5 and 6 show the difference between the actual vibration displacement of each blade and the results calculated by SLF-BTT, namely the average and standard deviation of the measurement error (ME) at each sampling point of SLF-BTT method under the above two cases. Combining Figs. 5 and 6, it can be clearly seen that the average of measurement errors of blade vibration displacement obtained by SLF-BTT method are different under both constant speed and variable speed cases. At the same time, the fluctuation of measurement errors of each blade is also different according to the standard deviation. From the overall trend, the average and standard deviation of measurement errors decrease with the increase of blade number. Meanwhile, when the

20

10

0

0

10 20 30 Blade number (b) Standard deviation of ME

Fig. 5. Average and standard deviation of ME of each blade obtained by SLF-BTT method under case (a).

K. Chen et al. / Mechanical Systems and Signal Processing 134 (2019) 106321

Average/ m

30 20 10 0

0

10 20 Blade number (a) Average of ME

30

Standard deviation/ m

10

6 4 2 0

0

10 20 30 Blade number (b) Standard deviation of ME

Fig. 6. Average and standard deviation of ME of each blade obtained by SLF-BTT method under case (b).

blade number increases to the midpoint of the total number of blades, the average and standard deviation of measurement errors will increase with the increase of blade number, which shows the obvious convergence. The error is minimum at the midpoint of the straight fitting interval (corresponding blade number interval) and the fluctuation as well. Therefore, the blades corresponding to or near the midpoint of the linear fitting interval will be an ideal and reliable ‘‘static” reference. 3.3. Result of CR-BTT

Average / m

8 6 4 2 0

0

10 20 Blade number (a) Average of ME

30

Standard deviation/ m

In order to verify the validity of the CR-BTT method and its high accuracy and efficiency, the above two kinds of simulation data are processed by the CR-BTT method. The result of data of blade No. 16 sampled by probe NO. 1 dealing with SLFBTT method is used as static reference. Figs. 7 and 8 show the average and standard deviation of the measurement errors of probe NO. 1 with the CR-BTT method. As shown in Figs. 7 and 8, for the two cases considered, with CR-BTT method to process the tip timing signal, the average and standard deviation of vibration displacement ME of different blades is basically close. Meanwhile, compared with the processing results by SLF-BTT method (in Figs. 5 and 6), the average value and standard deviation of the errors are greatly decreased, either reaching or very closing to the minimum mean error and standard deviation of SLF-BTT method. More intuitively, the vibration displacements of blade No. 1 and blade No. 32 processed by SLF-BTT method and CR-BTT method are compared with the real values in Figs. 9 and 10, respectively. The advantages of CR-BTT method over SLF-BTT method in vibration accuracy can be clearly seen from Figs. 9 and 10. The CR-BTT method basically coincides with the real vibration value in the small error range, while the SLF-BTT method does not. For SLF-BTT method, there are obvious differences. And the greater the vibration value, the more obvious the difference is. Especially near the resonance center, the difference between the amplitude and the actual value is the most obvious. And normally the calculated result is less than the actual value. For multi-probe, with the CR-BTT method, only one probe’s data needs to be processed. The data of other probes are calculated by Eq. (32). The average and standard deviation of the measurement errors of the results obtained by CR-BTT method under probe No. 2 are shown in Figs. 11 and 12. Compared with Figs. 7 and 8, the error level of probe No. 2 is still close to that of probe No. 1, without phenomenon of error amplification. It can be seen that CR-BTT method is effective and stable for multi-probe tip timing system.

4

2

0

0

10 20 30 Blade number (b) Standard deviation of ME

Fig. 7. Average and standard deviation of ME of each blade obtained by CR-BTT method under case (a).

11

K. Chen et al. / Mechanical Systems and Signal Processing 134 (2019) 106321

Standard deviation/ m

Average/ m

4

2

0

0

10 20 30 Blade number (a) Average of ME

3 2 1 0

0

10 20 30 Blade number (b) Standard deviation of ME

Fig. 8. Average and standard deviation of ME of each blade obtained by CR-BTT method under case (b).

0

SLF-BTT CR-BTT Real

400

Displacement/ m

Displacement/ m

800

0

-100

-200 -300

-400

0

SLF-BTT CR-BTT Real

-400 100

50

100 150 200 250 Rotation (a)Vibration displacement of blade1 of case (a)

150 200 Rotation (b)Vibration displacement of blade1 of case(b)

Fig. 9. Vibration displacements of blade No. 1.

Displacement/ m

SLF-BTT CR-BTT Real

400 0

-400

0

50

100 150 200 250 Rotation (a)Vibration displacement of blade32 of case (a)

Displacement/ m

300

800

200

SLF-BTT CR-BTT Real

100 0 100

150 200 Rotation (b)Vibration displacement of blade32 of case (b)

Fig. 10. Vibration displacements of blade No. 32.

Because the CR-BTT method only fits the data on one probe with the least square straight-line fitting method, the calculation speed and efficiency of the subsequent data processing of other probes have been greatly improved compared with SLF-BTT method. The measurement accuracy and stability have not reduced but been greatly improved. At the same time, with the increase of the number of probes and continuous sampling, the advantages of CR-BTT method in calculating efficiency for on-line monitoring system will be enormous, which provides a reliable guarantee for long-term blade vibration monitoring. In terms of algorithm, whether CR-BTT or SLF-BTT method, the main time-consuming is fitting the ToA series with least squares straight-line fitting method. The CR-BTT method only needs one time, but the SLF-BTT method needs to fit each probe, the times required of which is np . It can be seen from the analysis that, besides a common probe for straight-line fitting, the ratio of the subsequent calculation of np  1 probes in time can be regarded as a constant 1/b, which is determined

12

K. Chen et al. / Mechanical Systems and Signal Processing 134 (2019) 106321

Standard deviation/ m

Average/ m

8 6 4

2 0

0

10 20 Blade number (a)Average of ME

30

4

2

0

0

10 20 30 Blade number (b)Standard deviation of ME

Fig. 11. Average and standard deviation of ME of each blade obtained by CR-BTT method under case (a).

Standard deviation/ m

Average/ m

4

2

0 0

10 20 Blade number (a)Average of ME

3 2 1 0

30

0

10 20 30 Blade number (b)Standard deviation of ME

Fig. 12. Average and standard deviation of ME of each blade obtained by CR-BTT method under case (b).

by the performance of the computer, usually far less than 1. Therefore, when the number np of probes to be calculated approaches infinity, comparing the calculation time of CR-BTT method with that of SLF-BTT method, there is

lim T cpu ¼ lim

np !1

np !1

1 þ 1b ðnp  1Þ 1 tCR - BTT ¼ lim ¼ : b tSLF - BTT np !1 np

ð46Þ

In a small range,

T cpu ¼

tCR - BTT 1  : tSLF - BTT np

ð47Þ

The data processing of CR-BTT and SLF-BTT method is performed on 1–11 probes respectively, and the calculation time is compared. The results are shown in the following table (Table 1).

Table 1 Time comparison of CR-BTT and SLF-BTT. np

tCR-BTT (unit: s)

tSLF-BTT (unit: s)

tCR-BTT/tSLF-BTT

1 2 3 4 5 6 7 8 9 10 11

10.1210 10.5446 10.9157 11.5061 12.2249 12.5794 13.1122 13.8577 14.0311 14.6097 15.2896

10.1158 20.2655 30.92 40.0404 48.9579 58.3757 68.2916 77.9198 86.7968 96.7177 106.9476

1.0005 0.5203 0.3530 0.2873 0.2497 0.2154 0.1920 0.1778 0.1616 0.1511 0.1430

K. Chen et al. / Mechanical Systems and Signal Processing 134 (2019) 106321

13

Considering that the number of probes is x , and the time consumed by CR-BTT and the time consumed by SLF-BTT are y. The straight lines obtained are tCR - BTT ¼ 0:5188np þ 9:5043 and tSLF - BTT ¼ 9:5699np þ 1:2486 respectively, there is

1 t CR - BTT 0:5188np þ 9:5043 0:5188 ¼ lim  0:0542: ¼ lim ¼ np !1 t SLF - BTT np !1 9:5699np þ 1:2486 b 9:5699 In the simulation, the angle between probes is set to 6°. Probe mounting angular interval measured by the CR-BTT method is 6.000004°. Which shows that the accuracy is very high. In a number of other simulations, by changing the mounting angle of the probe and comparing it with the CR-BTT measurement result, the error is less than 0.1% degree. Table 2 shows partial results as an example. Based on the above simulation, the accuracy and stability of CR-BTT method are verified in many aspects in simulation and comparison. 4. Test rig and experimental steps In order to validate the method described in this paper, a high-speed straight blade test rig is developed. The test rig and acquisition system are shown in Figs. 13 and 14, the detail introduction of which is in Ref. [22]. The number of blades is nb ¼ 32, the diameter is D ¼ 2R ¼ 138 mm. There are 12 mounting holes for excitation on the top of the shroud, and each excitation component is embedded in a cobalt-based permanent magnet to provide excitation force. We can carry out the experiment with different combinations by adjusting the number and distribution of magnet excitation. The shroud is circumferentially bored a number of holes to install probes at a specific angle (6° minimum). In the experiment, six magnets are uniformly installed in the circumferential of the test rig. The bladed disk is a mistuning bladed disk. Three tip fiber optic probes are used, and the probe mounting angles between two adjacent probes are all designed as 6°. The sampling frequency of the probe is 5 MHz. The rotational frequency range is 40–132 Hz, and the speed-up rate is 2 Hz/s during frequency sweeping, which has a certain speed fluctuation. 5. Experiments results and analysis The errors of blade installation angle measured by three probes are shown in Fig. 15. Fig. 15 shows that the installation angle errors of each blade measured by different probes are different, which is caused by the axial error of the installation angle of the probe and the uncertainty of the tip timing method. Most likely, it is the uncertainty about the measurement point of the tip timing method. The effect is very hard to eliminate by only one probe on axial. The angle between the positions of probes measured by CR-BTT is shown in Table 3. By obtaining the interval angle of the probes, they are different from the 6 degree of the design. Obtaining the actual probes’ position will greatly reduce the uncertainty of tip timing signal processing and subsequent analysis. The angles obtained by CR-BTT are more realistic than the design drawings. It can be considered as the actual installation angle. In the processing of experimental results, the vibration value of blade No. 16 under probe No. 2 processed by SLF-BTT is used as static reference, and CR-BTT method is used to process data on subsequent blades and other probes. Fig. 16 shows a comparison of the results of static reference processing using BTT with OPR method and SLF-BTT method, respectively. Taking blade No. 32 under probe No. 3 as an example for comparison, Figs. 17–19 are processed by CR-BTT method, SLFBTT method and BTT with OPR method respectively. Because of the motor drive in the experiment, there is obvious speed fluctuation. Compared with the above results, it is obvious that the OPR-BTT is greatly affected by the speed fluctuation. None OPR BTT methods can eliminate the errors caused by the speed fluctuation, and the errors are relatively small. Comparing the two none OPR methods with the OPR-BTT, it is obvious that the vibration amplitudes of the two none OPR methods are very close in the non-resonance region, and the vibration amplitudes of CR-BTT method in the resonance region are closer to OPR-BTT than those of SLF-BTT method, especially near 4000 and 14,200 rotation. The accuracy of vibration displacement is very important for subsequent signal reconstruction and stress inversion. At the same time, for the case shown, it takes 11.1985 s CPU-time to calculate the vibration displacement of all blades on all probes by CR-BTT method, and 31.1547 s CPU-time by SLF-BTT method. The calculation efficiency of CR-BTT method is 2.782 times than that of SLF-BTT method.

Table 2 Mounting angle between probes measured by CR-BTT method. Set/°

Measured/°

Error/°

20 45 90 180

19.9996 45.0002 90.0001 180.0000

0.0004 0.0002 0.0001 0.0000

14

K. Chen et al. / Mechanical Systems and Signal Processing 134 (2019) 106321

Fig. 13. High-speed straight blade test rig [22].

Fig. 14. Probe installation and testing system [22].

0.06 Error Of Blade Angle/(°)

probe1

0.04

probe2 probe3

0.02 0

-0.02 -0.04

-0.06 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

Blade number Fig. 15. Errors of blade installation angle.

15

K. Chen et al. / Mechanical Systems and Signal Processing 134 (2019) 106321

Name

Probe1 to Probe2

Prebe2 to Probe3

Interval angle/°

6.1668

6.0314

200

140

100

120 100

0

80 -100

60

-200 -300

40 0

2000

4000

6000

8000

10000

12000

14000

16000

20 18000

Rotation frequency/Hz

Vibration / m

Table 3 Interval angle of probes.

Rotation 200

140

100

120 100

0

80 -100

60

-200

40

-300 0

2000

4000

6000

8000

10000

12000

14000

16000

Rotation frequency/Hz

Vibration/ m

(a) BTT with OPR

20 18000

Rotation (b) SLF-BTT Fig. 16. Vibration displacement of the static reference blade.

140 120

100

100

0

80 60

-100

40

-200 0

2000

4000

6000

8000 10000 Rotation

12000

14000

16000

20 18000

Rotation frequency/Hz

Vibration/ m

200

Fig. 17. Processed by CR-BTT.

Furthermore, intercepting the resonance region near 124 Hz, the vibration amplitude of each blade in the resonance region is analyzed by using Single Parameter Technology [23]. The corresponding blade vibration amplitudes of probe No. 1 and probe No. 3 are shown in Fig. 20. As shown in Fig. 20, near blade No. 16, the vibration amplitudes obtained by two kinds of none OPR BTT methods are very close, and close to those obtained by the BTT method with OPR. The blades far away from blade No. 16 show significant differences. The results of CR-BTT are closer to those of OPR-BTT than SLF-BTT, and in most cases the results of SLF-BTT are smaller than those of CR-BTT and OPR-BTT. This also corresponds to the simulation results. Meanwhile, in the Fig. 20, the blade vibration amplitudes are different, even for the same blade. Firstly, the blisk is an obvious mistuning one. Secondly, for the same blade, the simulation in reference [24] shows that the acceleration of the rotor, the positions of the probes, and measurement point of probe, etc. will have a certain influence on the identification result of the single parameter method. For multiple probes, the identification results of single-parameter method are affected by various uncertainties. So there will be differences. But for a single probe, it is in accordance with the controlling variable

16

K. Chen et al. / Mechanical Systems and Signal Processing 134 (2019) 106321

140 120

100

100

80

0

60

-100

40

-200 0

2000

4000

6000

8000

10000

12000

14000

16000

20 18000

Rotation frequency/Hz

Vibration/ m

200

Rotation Fig. 18. Processed by SLF-BTT.

140 120

100

100

0

80

60

-100 -200

40 0

2000

4000

6000

8000 10000 Rotation

12000

14000

16000

Rotation frequency/Hz

Vibration/ m

200

20 18000

Fig. 19. Processed by BTT with OPR.

30

31

1

32

2

3

4

30 5

29

CR-BTT 8

26 25

9

0

24

10

23

11

22

12 13

21 20

19

18

17

16

15

1

2

3

4

5 6

28 7

27

32

29 6

28

31

14

SLF-BTT OPR-BTT Unit:μm

7

27

26

8

25

9

24

10

23

11

22

12 21

13 20

19

18

(a) probe1

17

16

15

14

(b) probe3 Fig. 20. Comparison of amplitudes obtained by different BTT methods.

method. So the comparison in Fig. 20 is reasonable. Despite this, analysis and evaluation of the uncertainty of the BTT method still requires more research.

6. Verification on large scale industry turbo fan In order to verify the application of CR-BTT method in the actual industry case, CR-BTT method is used to verify the experimental results in a large turbo fan. Conformation diagram of industry turbo fan is shown in Fig. 21. It has two stages of blades. The bladed disk’s radius R is 1.7 m and the number of blades per stage nb is 36. Fiber optic BTT sensor is used in the experiment, and the sampling frequency is 40 MHz. The probe installation and the partial sketch of the blade are shown in Fig. 22.

17

K. Chen et al. / Mechanical Systems and Signal Processing 134 (2019) 106321

Scale

1:50

Air outlet

Air inlet

Fiber Optic Sensor

Case

Fig. 21. Conformation diagram of industry turbo fan.

blade

Probe

Fig. 22. Large industry turbo fan and BTT sensor.

Vibration/ ­m

20

0 15

-1000 -2000

10

-3000

-3887­m

-4000 0

2000

4000

10000

12000

5 14000

(a)CR-BTT

1000

20

0 15

-1000 -2000

10

-3125­m

-3000 -4000 0

2000

4000

6000 8000 Rotation

10000

(b)SLF-BTT Fig. 23. Vibration of No. 36 blade by BTT method.

12000

5 14000

Rotation frequency /Hz

Vibration/ ­m

6000 8000 Rotation

Rotation frequency /Hz

1000

18

K. Chen et al. / Mechanical Systems and Signal Processing 134 (2019) 106321

CR-BTT method and SLF-BTT method are used to process the signal of a test run, respectively. The results of blade No. 36 on non-fitting probes are illustrated as Fig. 23. As can be seen from the above figure, the results of CR-BTT method and SLF-BTT method are obviously different in the vibration value at the punctuation point. There is a difference of 752 lm in the numerical value, and there are also some differences in the vibration amplitude at other times. However, compared with the large value 752 lm, these can be neglected. For large units, the blade radius is larger, and the micro-error caused by the processing method will be also magnified. Compared with the difference of about 50 lm on the small test rig, the maximum difference between the results obtained by CR-BTT method and SLF-BTT method reach 752 lm on actual large units. Therefore, different methods to process the tip arrival signal will produce different results, which will have a significant impact on the subsequent signal analysis. 7. Conclusions In this paper, the none OPR BTT method based on SLF-BTT is theoretically deduced and analyzed, and the CR-BTT is proposed. By simulation, small multi-blades test rig and large scale industry turbo fan verification, the validity of CR-BTT method is proved, and its accuracy and calculation speed are significantly improved compared with SLF-BTT method. The CR-BTT method basically coincides with the real vibration value in the resonance range, while the SLF-BTT method does not. For SLF-BTT method, there are obvious differences. And the greater the vibration value, the more obvious the difference is. Especially near the resonance center, the difference between the amplitude and the actual value is the most obvious. And normally the calculated result is less than the actual value. The calculation efficiency of CR-BTT method is several times (depending on the number of probes and the size of data) than that of SLF-BTT method. Meanwhile, the measurement of blade installation angle error and probe installation angle based on blade tip timing method without OPR are proposed, which can decrease the uncertainty of blade tip timing method and improve the accuracy of CR-BTT method. The experiment also proves that the BTT method without OPR is more reliable than the BTT method with OPR in the case of speed fluctuation. Further research work will focus on validating of the method when the blade movement is the combination of bend/axial shift/lean/untwist. For this, new simulator, algorithm and test rig will be developed. Acknowledgments This work has been supported by the National Natural Science Foundation of China (51775030, 91860126) and the Fundamental Research Funds for the Central Universities (BHYC1703A). References [1] P. Russhard, The rise and fall of the rotor blade strain gauge, Vibration Engineering and Technology of Machinery, Springer International Publishing, 2015. [2] S. Heath, A new technique for identifying synchronous resonances using tip-timing, J. Eng. Gas Turbines Power 122 (2) (2000) 219–225. [3] P.E. McCarty, J.W. Thompsonk Jr., Development of a Noninterference Technique for Measurement of Turbine Engine Compressor Blade Stress. No. AEDC-TR-79-78, Arnold Engineering Development Center Arnold AFB TN, 1980. [4] J.F. Brouckaert, R. Marsili, G. Rossi, Development and experimental characterization of a new noncontact sensor for blade tip timing, in: A.I.V.E.L.AA.I.V. E.L.A (Ed.), 10th International Conference on Vibration Measurements by Laser and Noncontact Techniques, Ancona, Italy, June 26-29, AIP Conference Proceedings Vol. 1457, No. 1, 2012, pp. 61–68. [5] G. Battiato, T.M. Berruti, Forced response of rotating bladed disks: blade tip-timing measurements, Mech. Syst. Signal Process. 85 (2017) 912–926. [6] J. Gallego-Garrido, G. Dimitriadis, J.R. Wright, Development of a multiple modes simulator of rotating bladed assemblies for blade tip-timing data analysis, in: International Conference of Noise and Vibration Engineering, 2002, pp. 1437–1446. [7] J. Lin, Z. Hu, Z.S. Chen, Y.M. Yang, H.L. Xu, Sparse reconstruction of blade tip-timing signals for multi-mode blade vibration monitoring, Mech. Syst. Signal Process. 81 (2016) 250–258. [8] Z.S. Chen, J.H. Liu, C. Zhan, J. He, W.M. Wang, Reconstructed order analysis-based vibration monitoring under variable rotation speed by using multiple blade tip-timing sensors, Sensors 18.10 (2018). [9] M. Pan, F. Guan, H. Hu, Y. Yang, H. Xu, et al, Compressed sensing based on dictionary learning for reconstructing blade tip timing signals, Prognostics & System Health Management Conference, 2017. [10] P. Russhard, Blade tip timing (BTT) uncertainties, AIP Conf. Proc. 1740 (2016) 020003. [11] M. Mohamed, P. Bonello, P. Russhard, The Determination of Steady-State Movements Using Blade Tip Timing Data. ASME. Turbo Expo: Power for Land, Sea, and Air, Volume 7C: Structures and Dynamics, 2018, V07CT35A010. [12] G. Rossi, J.F. Brouckaert, Design of blade tip timing measurements systems based on uncertainty analysis, 58th International Instrumentation Symposium, 2012. [13] A. Hajnayeb, M. Nikpour, S. Moradi, G. Rossi, A new reference tip-timing test bench and simulator for blade synchronous and asynchronous vibrations, Meas. Sci. Technol. 29.2 (2017). [14] O. Jousselin, P. Russhard, P. Bonello, A method for establishing the uncertainty levels for aero-engine blade tip amplitudes extracted from blade tip timing data, Proceedings of the 10th International Conference on Vibrations in Rotating Machinery, London, UK, 2012. [15] P. Russhard, Derived once per rev signal generation for blade tip timing systems, International Instrumentation Symposium IET, 2014. [16] H.T. Guo, F. Duan, J. Zhang, Blade resonance parameter identification based on tip-timing method without the once-per revolution sensor, Mech. Syst. Signal Process. 66–67 (2016) 625–639. [17] W.M. Wang, X.L. Zhang, D.F. Hu, D.P. Zhang, P. Allaire, A novel none once per revolution blade tip timing based blade vibration parameters identification method, Chin. J. Aeronaut. (accepted for publication). [18] N.R. Draper, H. Smith, Fitting a straight line by least squares, Applied Regression Analysis, third ed., John Wiley & Sons, Inc., 2014. [19] C.A. Cantrell, Review of methods for linear least-squares fitting of data and application to atmospheric chemistry problems, Atmos. Chem. Phys. 8 (17) (2008) 5477–5487.

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[20] G. Dimitriadis, I.B. Carrington, J.R. Wright, J.E. Cooper, Blade-tip timing measurement of synchronous vibrations of rotating bladed assemblies, Mech. Syst. Signal Process. 16 (4) (2002) 599–622. [21] I.B. Carrington, J.R. Wright, J.E. Cooper, G. Dimitriadis, A comparison of blade tip timing data analysis methods, Proc. Inst. Mech. Eng. Part G: J. Aerospace Eng. 215 (5) (2001) 301–312. [22] W.M. Wang, S.Q. Ren, S. Huang, Q. Li, K. Chen, New step to improve the accuracy of blade synchronous vibration parameters identification based on combination of GARIV and LM algorithm, ASME. Turbo Expo: Power for Land, Sea, and Air, Volume 7B: Structures and Dynamics, 2017, V07BT35A017. [23] I.Y. Zablotsky, Yu.A. Korostelev, Measurement of resonance vibrations of turbine blades with the ELURA device, Energomashinostroneniye 2 (1970) 36– 39. [24] S.Q. Ren, Investigation on method of turbine blade vibration measurement based on data fusion method (MS thesis), Beijing University of Chemical Technology, 2017 (in Chinese).