An Improved Circumferential Fourier Fit (CFF) Method for Blade Tip Timing Measurements - applied sciences
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applied
sciences
Article
An Improved Circumferential Fourier Fit (CFF)
Method for Blade Tip Timing Measurements
Zhibo Liu, Fajie Duan *, Guangyue Niu , Ling Ma, Jiajia Jiang and Xiao Fu
State Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, Weijin Road,
Tianjin 300072, China; zhiboliu@tju.edu.cn (Z.L.); niuguangyue@tju.edu.cn (G.N.);
tinama_17116@tju.edu.cn (L.M.); jiajiajiang@tju.edu.cn (J.J.); fuxiao215@tju.edu.cn (X.F.)
* Correspondence: fjduan@tju.edu.cn; Tel.: +86-131-3205-3038
Received: 7 May 2020; Accepted: 20 May 2020; Published: 26 May 2020
Abstract: Rotating blade vibration measurements are very important for any turbomachinery research
and development program. The blade tip timing (BTT) technique uses the time of arrival (ToA) of the
blade tip passing the casing mounted probes to give the blade vibration. As a non-contact technique,
BTT is necessary for rotating blade vibration measurements. The higher accuracy of amplitude and
vibration frequency identification has been pursued since the development of BTT. An improved
circumferential Fourier fit (ICFF) method is proposed. In this method, the ToA is not only dependent
on the rotating speed and monitoring position, but also on blade vibration. Compared with the
traditional circumferential Fourier fit (TCFF) method, this improvement is more consistent with reality.
A 12-blade assembly simulator and experimental data were used to evaluate the ICFF performance.
The simulated results showed that the ICFF performance is comparable to TCFF in terms of EO
identification, except the lower PSR or more number probes that have a more negative effect on ICFF.
Besides, the accuracy of amplitude identification is higher for ICFF than TCFF on all test conditions.
Meanwhile, the higher accuracy of the reconstruction of ICFF was further verified in all measurement
resonance analysis.
Keywords: blade tip timing; circumferential Fourier fit; synchronous vibration
1. Introduction
Rotating blade vibration measurements are of great significance for any turbomachinery research
and development program [1–4]. As a non-contact measurement technique, the blade tip timing (BTT)
technique has been widely used in blade vibration measurements, such as in aero-engines, turbines,
and compressors [5–9]. This technique uses the time of arrival (ToA) of the blade tip passing the
casing-mounted probes to give the blade vibration. Compared with traditional strain gauges [10],
BTT has the advantages of low equipment cost, easy installation, and monitoring vibration of each
blade that passes through the probe [11]. However, BTT signals are typically under-sampled. It brings
difficulty for blade vibration analysis.
In BTT analysis, the character of the blade response is usually grouped into two distinct classes,
namely, asynchronous vibration and synchronous vibration. Asynchronous vibration mainly occurs in
the abnormal vibrations, such as rotating stall, surge, flutter, and bearing vibration [12–14]. In these cases,
the blade vibration frequency is a non-integer multiple of the rotating frequency and the phase of
the response can be arbitrary, which is shown in Figure 1a. Synchronous vibration is excited by the
multiples of the rotating frequency [15,16]. At constant speed, the probe can only monitor the blade
response at a fixed point since the phase of the response remains fixed relative to a stationary datum
at a given speed. The under-sampling of synchronous vibration is shown in Figure 1b. Therefore,
Appl. Sci. 2020, 10, 3675; doi:10.3390/app10113675 www.mdpi.com/journal/applsciAppl. Sci. 2020, 10, 3675 2 of 20
Appl. Sci. 2020, 10, x FOR PEER REVIEW 2 of 20
Figure 1b. Therefore, the degree of under-sampling is much higher for synchronous vibration,
the degree of under-sampling is much higher for synchronous vibration, which leads to more difficultly
which leads to more difficultly in identifying the blade vibration parameters from raw data.
in identifying the blade vibration parameters from raw data.
(a) 1
Displacement (mm)
0.5
0
-0.5
-1
0 0.02 0.04 0.06 0.08 0.1 0.12
time (s)
(b) 1
Displacement (mm)
0.5
0
-0.5
-1
0 0.02 0.04 0.06 0.08 0.1 0.12
time (s)
Vibration curve Probe 1 Probe 2 Probe 3
Figure 1. (a) Under-sampling of asynchronous vibration and (b) under-sampling of synchronous vibration.
Figure 1. (a) Under-sampling of asynchronous vibration and (b) under-sampling of synchronous
vibration.
Up to now, the identification methods for BTT signals are divided into two categories; one is
spectrum analysis and the other is curve fitting. The spectrum analysis method generally maps BTT
Up to
signals to frequency
now, the identification
domain spacemethods to obtainfor BTT signals
information areblade
about divided into two
vibration. categories;
This one is
type of method
spectrum analysis and the other is curve fitting. The spectrum analysis method
usually requires that the sampling frequency is kept constant, that is, the signal is sampled on a constant generally maps BTT
signals to frequency domain space to obtain information about blade
rotating speed. Spectrum analysis methods mainly include a traveling wave analysis (TW) [17], vibration. This type of method
usually
the “5 +requires
2” method that thethe
[18], sampling
minimum frequency is kept constant,
variance spectrum estimation that is, the[19],
(MVSE) signal is sampled
non-uniform on a
discrete
constant rotating speed. Spectrum analysis methods mainly include
Fourier transform (NUDFT) [20], cross-spectrum estimation (CSE) [21], sparse reconstruction (SR) a traveling wave analysis (TW)
[17],
methodthe “5 + 2”
[22], method
etc. [18], themethod
The spectrum minimum variance
mainly is usedspectrum
to analyzeestimation (MVSE)vibration.
asynchronous [19], non-uniform
However,
discrete Fourier transform (NUDFT) [20], cross-spectrum estimation
the results of spectrum analysis are always corrupted with aliases and replicas of the true frequency (CSE) [21], sparse
reconstruction (SR) method [22], etc. The spectrum method mainly is
components. Consequently, the real spectrum recovery needs to further make use of the knowledge of used to analyze asynchronous
vibration.
the blade’sHowever, the resultswhich
dynamic properties, of spectrum
is usuallyanalysis
obtained arebased
always oncorrupted with aliases
the finite element method and replicas
(FEM) [23].
of
Thethecurve
true fitting
frequencymethodcomponents.
usually uses Consequently,
sine functions thetoreal spectrum
fit the recoverydisplacements,
blade vibration needs to further make
which are
use of the by
obtained knowledge
speed sweepingof the blade’s
throughdynamic properties,
the resonance region. which
The is usually
curve fittingobtained
methodbased mainly onused
the
finite element
to analyze method (FEM)
synchronous [23]. These
vibration. The curve fitting
methods method
mainly usually
include the uses sine functions
single-parameter to fit [24],
method the
blade vibration displacements, which are obtained by speed sweeping
the two-parameter plot method [25], autoregressive (AR) method [26,27], the circumferential Fourier through the resonance
region.
fit (CFF)The curve[28],
method fitting methodwithout
the method mainly once usedper to revolution
analyze synchronous
[29,30], and so vibration. These methods
on. The accuracy of these
mainly include the single-parameter method [24], the two-parameter
curve fitting methods is often related to the amount of fitted data and the inter-blade coupling of plot method [25],
the
autoregressive
mistuned blade. (AR) method [26,27], the circumferential Fourier fit (CFF) method [28], the method
without Theonce
CFFper revolution
method is highly[29,30], and so on.forThe
recommended accuracy of
synchronous these curve
vibration fitting
analysis by methods is often
Hood Technology
related to the which
Corporation, amount is of fitted data vendor
a commercial and theofinter-blade
BTT systems coupling
[28]. Thisof the mistuned
method blade. the amplitude
can identify
and phase of vibration on the conditions that the engine order (EO) is known. Tao Ouyangby
The CFF method is highly recommended for synchronous vibration analysis Hood
proposed
Technology
a traversed Corporation,
EO method based which onisCFF,a commercial
which is novendor longer of BTT to
limited systems
known[28]. This In
EO [31]. method can
this paper,
identify the amplitude and phase of vibration on the conditions that the
we refer to the Ouyang CFF as the traditional CFF (TCFF). An improved formulation for the calculation engine order (EO) is known.
Tao
of theOuyang
TOA was proposed
proposed a traversed
by S. Heath EO method based on who
and M. Imregun, CFF,proved
which the is noBTTlonger limited
analysis to known
technique has
EO [31]. In this paper, we refer to the Ouyang CFF as the traditional
limitations regarding synchronous response, in that the blade vibration was not considered in the CFF (TCFF). An improved
formulation for the calculation of the TOA was proposed by S. Heath and M. Imregun, who proved
the BTT analysis technique has limitations regarding synchronous response, in that the bladeAppl. Sci. 2020, 10, 3675 3 of 20
TOA [32]. In this paper, an improved CFF (ICFF) method is proposed, which considers the influence of
blade vibration when measuring the ToA of the blade tip. The ICFF is more robust with more probes
employed to complete the parameter identification rather than the single-parameters or the improved
single-parameters method that just used one probe. Compared to the AR method that requests the
probes are equally spaced installed, the ICFF method has not got so strict limitations for probe layout.
Moreover, the ICFF introduced into the EO traversed thought that proposed by the literature [31] and
a more precision calculation method proposed by the literature [32]. These improvements are more
consistent with reality. A 12-blade assembly simulator and experimental data were used to evaluate the
performance of ICFF. In both simulated and experimental tests, the ICFF performed better than TCFF.
2. The Improved Circumferential Fourier Fit (ICFF) Method
2.1. A Brief Introduction to TCFF
The theory of TCFF is consistent with the CFF method. It uses multiple sensors (usually four)
distributed at different positions in the circumferential direction of the casing to monitor blade vibration.
On the condition that the EO is known, TCFF is equivalent to the CFF method, and the blade vibration
parameters can be directly calculated according to the CFF method. In the case that the EO is unknown,
the blade amplitude, vibration phase, direct current offset, and fitting residual error are calculated using
the CFF method with each possible EO, which is selected within a certain range. After the traversal is
completed, according to the principle of least squares, the EO corresponding to the minimum fitting
residual error is the true EO, and the blade vibration parameters calculated using this EO are the true
vibration parameters. More details about TCFF are provided by the literature [31]. The TCFF method
introduces the idea of EO traversal to overcome the shortcomings of CFF, which must rely on known
EO to identify blade vibration. Although TCFF did not make any changes to the CFF theoretical
method, except to introduce EO traversal, to a certain extent, TCFF has wider engineering applicability
than the CFF method. However, neither TCFF nor the CFF method takes into account the influence of
blade vibration on the ToA. The ICFF method proposed in this paper further considers the effect of
blade vibration on the ToA to achieve a higher accuracy of blade amplitude identification. Considering
that TCFF can perform EO traversal, which overcomes the disadvantage that CFF must rely on known
EO, this paper uses TCFF as a comparison object to illustrate the performance improvement of ICFF.
2.2. The Theoretical Basis of ICFF
Assuming the blade response is of single-frequency vibration:
y = A sin(ωt + ϕ) + d (1)
where y represents the vibration displacement of the blade tip, A represents the amplitude, ω represents
the angular frequency, t represents the ToA, φ represents the phase, and d represents the direct current
offset. For the synchronous vibration, there is:
ω = EO · ωv (2)
where ωv represents the rotor rotating frequency. This is based on the BTT measurement principle,
assuming that the blade rotates to the monitoring position β at time t. The geometry relationship between
the rotation angular, monitoring position, and blade vibration is shown in Figure 2. The equivalence
relationship can be given by:
Z t
ωv t = ωv dt = 2πn + β + ε − y/R (3)
0Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 20
Appl. Sci. 2020, 10, 3675 4 of 20
where ωv is the average rotating speed, and n is the number of revolution. In the Equation (3), it
can be seen that the time t (that is ToA) is not only dependent on the rotating speed and monitoring
where ωv is the average rotating speed, and n is the number of revolution. In the Equation (3), it can be
position, but also on the blade vibration.
seen that the time t (that is ToA) is not only dependent on the rotating speed and monitoring position,
Let:
but also on the blade vibration.
Let: β = β + ε − y / R (4)
β = β + ε − y/R
e (4)
According to Equations (1)–(4), the blade vibration displacement monitored by the probe
mounted According to Equations
at the position β can (1)–(4),
be given:the blade vibration displacement monitored by the probe mounted
at the position β can be given:
yy==A sin(
A sinEO βe
(EO ϕ )ϕ+) d+. d.
β++ (5) (5)
Equation(5)(5)isisthe
Equation themathematical
mathematicalmodel
modelforfor monitoring
monitoring the vibration
vibration displacement
displacementof ofthe
theblade
bladetip
tipunder
underthe influence
the influence of of
blade vibration
blade thatthat
vibration is considered. According
is considered. to the
According to literature [32], [32],
the literature in theincase
theof
large
case of vibration displacement
large vibration or mistuning,
displacement the improved
or mistuning, single-parameter
the improved method thatmethod
single-parameter considers the
that
considers the influence of blade vibration has higher accuracy than the traditional method. ICFF is aof
influence of blade vibration has higher accuracy than the traditional method. ICFF is a combination
the theory of
combination of the
the improved single-parameter
theory of the method and TCFF.
improved single-parameter method and TCFF.
ωv
Figure
Figure 2. 2.
TheThe schematic
schematic of of
thethe blade
blade tiptip timing
timing (BTT)
(BTT) measurement.
measurement. The
The once
once perper revolution
revolution (OPR)
(OPR)
sensor
sensor monitors
monitors thethe rotating
rotating speedspeed through
through the the
mark mark on shaft
on the the shaft and provides
and provides a timing
a timing reference
reference for
the BTT measurement system. The probe is responsible for monitoring the ToA. R represents thethe
for the BTT measurement system. The probe is responsible for monitoring the ToA. R represents
distance
distance fromthe
from theblade
bladetip
tiptotothe
thecenter
centerofofthe
therotor,
rotor, and y/R represents
and y/R representsthe
thecircumferential
circumferentialangle
angleofofthe
blade vibration. β represents the monitoring position, and ε represents the angle of the blade from the
the blade vibration. β represents the monitoring position, and ε represents the angle of the blade
first probe when the mark passes the OPR sensor.
from the first probe when the mark passes the OPR sensor.
2.3. The Derivation of ICFF
2.3. The Derivation of ICFF
Equation (5) is transformed by triangle formulas:
Equation (5) is transformed by triangle formulas:
y = A sin(EOe β) cos(ϕ) + A cos(EOe
β) sin(ϕ) + d (6)
y = A sin(EOβ ) cos(ϕ ) + A cos(EOβ ) sin(ϕ ) + d (6)
TCFF
TCFF usually
usually needs
needs atatleast
leastfour
fourprobes
probestotomonitor
monitorblade bladevibration.
vibration.InIn thissection,
this section,assuming
assuming
there are g
that there are g probes, the blade vibration displacement sequence can be represented according toto
that probes, the blade vibration displacement sequence can be represented according
Equation
Equation (6):
(6):
β1 ) β1 )
sin(EOe cos(EOe
y1 1
y sin(EOβe ) cos(EOβ e ) 1 A cos(ϕ)
β 1 β2 )1
y2 1 sin ( EO 12 ) cos ( EO
. = .. ) 1 A.. cos( (ϕ)
ϕ )sin
A (7)
.. β ) cos(EOβ
.. y2 sin(EO
= . . .
ϕ )
2 2
d
A sin(
(7)
β g ) 1
y g sin(EO e βg ) cos (EOe
d
That is: y g sin(EOβ g ) cos(EOβ g ) 1
Y = BX (8)
That is:Appl. Sci. 2020, 10, 3675 5 of 20
where Y is the displacement vector, B is the coefficient matrix, and X is the parameter matrix. The main
difference between ICFF and TCFF is the constituent parameters of the coefficient matrix B, although
Equation (7) in this paper is the same in form as TCFF. For the TCFF method, the coefficient matrix B
is completely determined by the probe mounted position β and the initial angle ε, that is, once the
probe installation scheme is determined, the coefficient matrix B is also determined. This means
that the coefficient matrix B is a constant in the TCFF method. The same applies to the CFF method.
However, the coefficient matrix B in ICFF is a function of three parameters: the probe installation
angle β, the initial angle ε, and the blade vibration displacement y/R. According to the derivation
in Section 2.2, blade vibration displacement is also one of the most important parameters affecting
the ToA. In this case, the coefficient matrix B is no longer a constant but can be adjusted in real-time
according to the actual vibration displacement of the blade.
Based on the principle of least squares and EO traversal, when the possible engine order EOk
(k = 1, 2, 3 . . . ) is selected, there is:
−1
Xk = (BTk Bk ) BTk Y. (9)
The condition number is the indicator of the quality of the matrix B, which influence the accuracy
of X directly. The larger the condition number, the more ill-condition matrix B is, and the more sensitive
of the calculated results are to the measurement error. An effective way to improve the quality of the
matrix B is to adjust the sensor layout. Therefore, based on the minimization the condition number
of matrix B, a method about determining the sensor layout can be derived. The condition number
mentioned above and the probe spacing of the resonance (PSR) used later in the paper as an indicator
of sensor distribution both are methods essentially to determine how the sampled interval of periodic
signals are arranged. However, PSR is more common.
Define the corresponding fitted residuals as Sk :
kBk Xk − Yk2
Sk = p (10)
g−1
For the number of revolution n that contains the blade synchronous resonance, Sk can be instead
Sk :
by e P
Sk
Sk =
e (11)
n
The EOk (k = 1, 2, 3 . . . ) that minimizes e Sk is the true EO. After the true EO and parameter
matrix X are determined, the amplitude, phase, and direct current offset can be further obtained by the
following formula:
q
A = X(1)2 + X(2)2
X(2)
arctan( X(1) ) X(1) > 0
ϕ = (12)
X ( 2 )
arctan( X(1) ) + π X(1) < 0
d = X(3)
where X(i) (i = 1, 2, 3) represents the ith row element of the matrix X. For n revolutions, after obtaining
each revolution of A, φ, and d through Equation (12), it is necessary to calculate the respective averages
to get the final value.
3. Method Evaluation Using Simulated Data
3.1. Simulated Model
For the actual blade vibration measurement, even if the blade dynamic behavior is fully understood,
it is still difficult to eliminate uncertainties in the measurement data [33,34], such as speed fluctuations,
system random errors, etc. These uncertainties will affect the evaluation accuracy, therefore, it isAppl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 20
3.1. Simulated Model
For the actual blade vibration measurement, even if the blade dynamic behavior is fully
Appl. Sci. 2020, 10, 3675 6 of 20
understood, it is still difficult to eliminate uncertainties in the measurement data [33,34], such as
speed fluctuations, system random errors, etc. These uncertainties will affect the evaluation
accuracy,
better therefore,
to use it is better
simulated to use simulated
data without measurement data without measurement
uncertainties uncertainties
to evaluate to evaluate
the accuracy of the
the accuracy ofmethods.
reconstruction the reconstruction
A 12-blade methods.
assembly A simulator
12-blade assembly
was usedsimulator
to evaluate was
theused
ICFFtoperformance.
evaluate the
ICFFschematic
The performance.
of theThe schematic
simulated modelofwas theshown
simulated model
in Figure 3. A was shown in Figuresystem
mass-spring-damper 3. A
mass-spring-damper
represents each blade,system
whichrepresents
is coupledeach
to itsblade, which is coupled
two neighbors throughtotwo its further
two neighbors through
spring–damper
two further Aspring–damper
assemblies. mathematical modelassemblies. A mathematical
was employed model
to simulate thewas employed
forced vibrationtoofsimulate the
the rotating
forced vibration
bladed assembly. of themathematical
This rotating bladed assembly.
model ignoredThisthe mathematical model ignored
effects of the centrifugal the effects
force and of the
temperature
centrifugal
on the bladeforce and temperature on the blade stiffness.
stiffness.
Figure 3.3. The
Figure Thesimulated
simulated model.
model. mi ,mkii,, and
ki, and
ci (i =ci 1,(i 2,
= 3...
1, 2, 12)3... 12) represent
represent thestiffness,
the mass, mass, stiffness,
and dampingand
damping coefficient of blade i, respectively. k and c (|i-j| = 1) represent the coupling
coefficient of blade i, respectively. ki, j and ci, j (|i − j| = 1) represent the coupling stiffness and coupling
i, j i, j stiffness and
coupling damping
damping coefficientcoefficient betweenblades.
between adjacent adjacent blades.
the bladed
The mathematical model of the bladed assembly
assembly is
is given
given by:
by:
..
M y + C. y + Ky = F(t ) (13)
M y + C y + Ky = F(t) (13)
where M, C, K, and F(t) represent the mass matrix, damping matrix, stiffness matrix, and excitation
where M, C, K, and F(t) represent the mass matrix, damping matrix, stiffness matrix, and excitation
force vector, respectively. Let W(i, j) (1 ≤ i ≤ 12 and 1 ≤ j ≤ 12) represents the element at the ith row
force vector, respectively. Let W(i, j) (1 ≤ i ≤ 12 and 1 ≤ j ≤ 12) represents the element at the ith row and
and jth column of the matrix W, and above matrixes can be expressed as:
jth column of the matrix W, and above matrixes can be expressed as:
(mi i=j
M( i , j ) = mi i = j
M(i, j) = 0 others
0 others
ic, ji,j−1
c
−1
+ c+j +c jc+
i , j +c
i = ji = j
1 i, j+1
C(Ci,( ij,) j )==
− 1j = 1
−−c i − ji =
c i , j i,j (14)
0 0 others (14)
ki,j−1 + k j + ki,j+others 1 i = j
K(i, j) = ki , j −1i,j+ k j + ki , j + 1 i = j i − j = 1
−k
0
others
K( i , j ) = − k i , j i−j =1
In Equation (14), use 1 for subscripts 0greater than 12 and 12 for subscripts less than 1. The excitation
others
force fi (t) on blade i is given by:
In Equation (14), use 1 for subscripts greater than 12 and 12 for subscripts less than 1. The
excitation force fi (t) on blade 2πEO
fi (t) i=is Fgiven by:
i sin(EOωv t + i), f or i = 0, 1, . . . N (15)
N
2π EO
where Fi is the amplitude offthe i
(t ) excitation ωv t + and N represents
= Fi sin(EOforce, i ), for i = the N of the blade. N was set
0,1,count (15)to
12 in this paper. NAppl. Sci. 2020,
Appl. 10, 3675
Sci. 2020, 10, x FOR PEER REVIEW 7 of 207 of 20
where Fi is the amplitude of the excitation force, and N represents the count of the blade. N was set
The
to 12quality of probe distribution is represented by the probe spacing on the resonance (PSR).
in this paper.
The PSR isThe
thequality
ratio ofofthe
probe distribution
difference is represented
between the times by the probe
of arrival of aspacing onthe
blade at thefirst
resonance
and last(PSR).
probe to
The PSR is the ratio of the difference between
the period of the blade response at resonance [35], i.e.,:the times of arrival of a blade at the first and last
probe to the period of the blade response at resonance [35], i.e.,:
= ω
PSR PSR (ToA − ToA / (2
= nω (ToAlast − ToA f )irst
)/π()2π
,
), (16)
n last first (16)
ωn represents
wherewhere ωn representsthethe
natural angular
natural angularfrequency.
frequency.
Further details on the mathematics
Further details on the mathematics ofofthethesimulator
simulatorare
aregiven
given by the literature[36].
the literature [36].According
According to
to the literature
the literature [37,38],[37,38], the mistuning
the mistuning coefficient∆fΔf
coefficient and the
i iand the coupling
couplingcoefficient
coefficienthi are defined
hi are to to
defined
characterize
characterize the degree
the degree of mistuning
of blade blade mistuning
and theand the coupling
coupling betweenbetween
blades. blades. The concrete
The concrete expressions
expressions
of these of these two
two parameters are:parameters are:
k
∆ fi = ki k−i 1
Δ f =
ki,j −1 (17)
h2i = i k k, i− j = 1
i
(17)
k
where the subscripts i and j represent the blade
hi2 = inumber,
,j
= 1 k represents the blade nominal stiffness.
, i − j and
Let the nominal mass m, stiffness k, and damping k i c be 1 kg, 8.1 × 105 N/m, and 9 N·s/m, respectively.
Then,where
the nominal naturali frequency
the subscripts of the the
and j represent blade
bladeis 900 rad/s according
number, to Equation
and k represents (18):nominal
the blade
stiffness. Let the nominal mass m, stiffness k, andpdamping c be 1 kg, 8.1 × 105 N/m, and 9 N∙s/m,
ωn = of the
respectively. Then, the nominal natural frequency k/mblade is 900 rad/s according to Equation (18): (18)
Let the distribution of the mistuning coefficientsωn = k /∆fmi of the 12-blade assembly simulator(18) conform
to a Gaussian distribution with a mean of 0 and a standard deviation of 0.04. The distribution of ∆fi is
Let the distribution of the mistuning coefficients Δfi of the 12-blade assembly simulator
shown in Table 1. Assume that the blade actual mass m (i = 1, 2, 3 . . . 12) is equal to nominal mass m.
conform to a Gaussian distribution with a mean ofi 0 and a standard deviation of 0.04. The
The actual natural
distribution of frequency
Δfi is shownofinthe blade
Table was calculated
1. Assume according
that the blade actualto Equations
mass mi (i = 1, (17)–(18) and
2, 3… 12) is Table 1,
equal
whichtoisnominal
shownmassin Figure 4.
m. The actual natural frequency of the blade was calculated according to Equations
(17)–(18) and Table 1, which is shown in Figure 4.
Table 1. The distribution of the mistuning coefficient ∆f i (%).
Table 1. The distribution of the mistuning coefficient Δfi (%).
∆f 1 ∆f 2 ∆f 3 ∆f 4 ∆f 5 ∆f 6 ∆f 7 ∆f 8 ∆f 9 ∆f 10 ∆f 11 ∆f 12
Δf1 Δf2 Δf3 Δf4 Δf5 Δf6 Δf7 Δf8 Δf9 Δf10 Δf11 Δf12
−0.706 3.165 −5.328 −9.319 −5.796 1.334 1.565 1.806 −0.521 0.734 −1.904 3.448
−0.706 3.165 −5.328 −9.319 −5.796 1.334 1.565 1.806 −0.521 0.734 −1.904 3.448
950
Frequency (rad/s)
940 Mistuning frequency
930 Nominal natural frequency
920
910
900
890
880
870
860
850
0 1 2 3 4 5 6 7 8 9 10 11 12
Blade
Figure 4. 4.The
Figure Themistuning frequencyofofthe
mistuning frequency theblade.
blade.
3.2. Simulated TestTest
3.2. Simulated
The object of the
The object simulated
of the simulatedtest
testwas
wastotoassess
assess analysis techniques
analysis techniques forfor determining
determining theand
the EO EO and
amplitude. Several
amplitude. exhaustive
Several exhaustivetests
testswere
wereperformed usingBTT
performed using BTTdata
data obtained
obtained from
from the the simulator
simulator to to
evaluate
evaluate the accuracy
the accuracy of the
of the reconstructionmethod
reconstruction method and
andtheir
theirsensitivity
sensitivity to various test test
to various parameters.
parameters.
TheseThese parameters
parameters were:were:
(1) Probe spacing on the resonance (PSR);
(1) Probe spacing on the resonance (PSR);
(2) The number of revolution (n);
(3) Engine order (EO);
(4) Inter-blade coupling of the mistuned blade (ICoMB);Appl. Sci. 2020, 10, 3675 8 of 20
(5) The number of probes (g);
(6) Noise-to-signal ratio (NSR).
The NSR is defined as the ratio of the r.m.s. value of the noise to the value of the ‘clean’ amplitude
of the blade. The NSR was set to 100 points averagely distributed in the range from 0.01 to 0.3.
For the same NSR, a white noise generator was used to create 100 distinct noise sequences with which
to corrupt the data obtained from the simulator. Each test was repeated using all the white noise
sequences for each method on each NSR value. When there is noise in the data, the reconstruction
method will contain two types of error [39]:
(1) Bias is where the estimates are incorrect. It is caused by a summated squared noise term in the
BT B matrix that does not tend to zero but increases with increasing noise levels. This is an effect
of the ICFF model that is used.
(2) Scatter, where there is a range of estimates spread about the mean value, is caused by random
variations in the data.
The mean and variance of the recovered EO and amplitude were computed, from which 95 percent
confidence intervals were obtained. The best method will exhibit the lowest bias and, preferably,
lowest scatter. Six test cases were designed to analyze and compare the performance of the TCFF and
ICFF under different test conditions. These are shown in Table 2. Test 1 was specifically chosen to be a
very difficult case to analyze. Hence, the other tests were used to investigate whether specific changes
in the parameters improve or degrade the performance of the method.
Table 2. The simulated test.
Test PSR n True EO g ICoMB
Test 1 98% 30 10 4 No
Test 2 55% 30 10 4 No
Test 3 98% 50 10 4 No
Test 4 98% 30 13 4 No
Test 5 98% 30 10 7 No
Test 6 98% 30 10 4 Yes
The parameters whose background was shaded were the changed parameters relative to Test 1.
The PSR is a key factor that affects the performance of all BTT algorithms. It is also commonly
used to determine the quality of probe distribution for blade synchronous vibration measurement.
Currently, there are corresponding methods for optimizing the probe installation layout [40]. Combined
with the FEM, the optimization of probe layout can be achieved in actual engineering measurement.
Therefore, we set the PSR to a high value (98%) in Test 1. In engineering applications, the probe is
inevitably affected by the temperature, airflow, etc., and in severe cases, the measurement data will not
be available. In this case, where a limited number of probes were used to measure blade vibration,
the failed probe could not participate in the blade vibration analysis, which destroyed the optimization
of the probe layout and resulted in a decrease of the PSR. For this reason, the PSR was set to a low
value (55 percent) in Test 2.
For TCFF, the blade vibration parameters can be calculated based on the single-revolution data
obtained by multiple probes (usually four). However, it is impractical that the single-revolution
data are used to identify the blade vibration parameters in engineering applications, considering
the effects of system measurement errors. Just as the AR method does [39], it is reasonable to select
multi-revolution data, including the blade synchronous resonance region, to determine the true
vibration parameters based on the average of the calculation results. For any curve fitting algorithm,
such as the single-parameter method, the amount of selected data should include the complete blade
resonance response region as much as possible, although there are no specific rules on how many data
points should be selected. In the simulated tests, 30 (Test 1) and 50 (Test 3) data points were selectedAppl. Sci. 2020, 10, 3675 9 of 20
for identifying the vibration parameters to demonstrate the effects of the data amounts on the quality
of the identification of each method.
BTT technology is sensitive to a mode with a large amplitude at the blade tip, such as the
first-order bending mode. In actual rotating machinery operation, due to mechanical structure or
airflow disturbance, the same vibration mode of the blade may be excited at different resonance rotating
speeds corresponding to different EO. For BTT algorithms, analysis of the blade vibration at different
resonance rotating speeds is essentially a process of reconstructing signals with different degrees of
under-sampling. The EO (Test 1: EO = 10; Test 4: EO = 13) was used to analyze and compare the
sensitivity of TCFF and ICFF to different degrees of under-sampling, and the EO was also an important
parameter for the blade vibration analysis.
Based on the CFF theory, at least three probes are needed to monitor blade synchronous vibration.
The least-square method is used to calculate the blade synchronous vibration parameters if more
than three probes are used. The TCFF usually uses four probes to increase robustness. However,
Tao OuYang used seven probes in the experimental verification of TCFF [31]. So far, there is no optimal
choice for the number of probes. Test 5 was designed to verify the performance of TCFF and ICFF on
the different number of probes.
Mistuning is a universal feature of the objective existence of the blades, and if inter-blade coupling
of the mistuned blade exists, it will affect the performance of the BTT algorithms that only analyze
single-frequency signals, such as the single-parameter method, the two-parameter plot method, CFF,
and TCFF. The same applies to the ICFF method proposed in this paper. Therefore, Test 6 was specially
designed to analyze and compare the sensitivity of TCFF and ICFF to ICoMB.
The noise in the measurement data is difficult to eliminate. The “clean” simulated data was
polluted by noise in all simulated tests to make the data closer to the real measurements. The anti-noise
performance of each method was verified by using simulated data of different pollution degrees that
were represented by the NSR value.
The monitor positions of the probes in Test 1, Test 3, and Test 6 were 0◦ , 13.1◦ , 25.3◦ , and 35.3◦ .
The monitor positions of the probes in Test 4 were 0◦ , 11.3◦ , 13.3◦ , and 25.3◦ . In Test 2, the monitor
positions of the probes were 0◦ , 6.3◦ , 10.1◦ , and 19.8◦ . In Test 5, the monitor positions of the probes
were 0◦ , 6.3◦ , 13.1◦ , 19.1◦ , 25.3◦ , 30.2◦ , and 35.3◦ . The PSR in Table 2 was calculated according to
Equation (16). If the option of the ICoMB is “NO”, it means that the coupling coefficient is set to
zero; if it is “Yes”, it means the coupling coefficient is set to nonzero. In this paper, the coupling
coefficient hi (i = 1, 2, 3 . . . , 12) was set to 0.1 for the option “Yes”. It is must be stressed that the
blade vibration does not influence the adjacent blades when the coupling coefficient is set to zero,
even under the condition that the blade is mistuned. For this, it is still a single-frequency vibration for
the single blade. However, the synchronous resonance of the single blade will be a superposition of
multiple-frequency vibration under the condition that the coupling coefficient is set to nonzero, and it
is difficult to distinguish. The coupled vibration of the mistuned blade is a serious challenge for TCFF,
since it assumes the blade response is of single-frequency vibration. According to the results shown
in Figure 4, the natural frequency of blade 7 is close to that of the adjacent blades. To evaluate the
performance of ICFF under the condition of blade mistuning and inter-blade coupling, the simulated
data of blade 7 were selected in the simulated tests. The rotating speed was set to accelerate uniformly
from 300 to 2400 rpm, which included the resonance rotating speed for all simulated tests. Take the
resonance rotating speed as the reference and select half n points before and after the reference point to
compose the number of revolutions n. The results of each simulated test were analyzed as follows.
• Test 1:
As expected, Test 1 was a very difficult case. As shown in Figure 5a, the EO identification started
to produce biased results at about NSR 0.16 for both methods. For tests with correct EO identification,
the 95% confidence intervals of the relative error of the amplitude identification were calculated, whichAppl. Sci. 2020, 10, 3675 10 of 20
was shown in Figure 5b. The intervals increased with the increase of the NSR for both methods.
Appl. Sci.
Sci. 2020,
2020, 10,
10, xx FOR
FOR PEER
PEER REVIEW
REVIEW 10 of
of 20
20
However,
Appl. the relative error of amplitude identification of ICFF was lower than that of TCFF.
10
EO
EO %)
of AA ((%)
Errorof
Error
Figure
Figure
Figure 5. Results
5. Results
5. Results of
of of
TestTest 1.(a)
1. 1.
Test (a)95%
(a) 95%confidence
95% confidence intervals
confidence intervals ofof
intervalsof the engine
the
the order
engine
engine (EO)(EO)
order
order (EO) identification and and
identification
identification and
(b) 95%
(b) 95%
(b) 95% confidence
confidence intervals
intervals
confidence intervals of
ofof therelative
the
the relativeerror
relative error of
error of the
ofthe amplitude
theamplitude
amplitude identification.
identification.
identification.
•• 2: Test
• Test Test 2:
2:
Reducing the
Reducing the PSR
PSR value
value to 55 55 percent
percent produced
produced the the most
most dramatic
dramatic degrade
degrade in in the quality
quality of
of
Reducing the PSR value to to 55 percent produced the most dramatic degrade inthe the quality of the
the identification.
the identification. The
The correct
correct EO EO identification
identification did did not
not occur
occur for
for ICFF
ICFF before
before NSRNSR 0.15,
0.15, and
and the
the
identification. The correct EO identification did not occur for ICFF before NSR 0.15, and the deviation
deviation of
deviation of EO
EO identification
identification waswas very
very large
large for
for both
both methods
methods compared
compared withwith Test
Test 1.
1. In
In order
order to
to
of EO identification
compare the two was very large
methods for both methods
synchronously, Figure compared
6a only with that
showed Test 1.theInresults
order of to compare
the EO the
compare the two methods synchronously, Figure 6a only showed that the results of the EO
two identification
methods synchronously,
in the
the NSR
NSR rangeFigure
range of 6a
of 0.15 only
0.15 and showed
and 0.30.
0.30. The that
The relative the results
relative error
error of of the
of amplitude EO identification
amplitude identification
identification was
was in the
identification in
NSRcalculated
range of after
calculated 0.15 and
NSR 0.30.
after NSR 0.15. The scatter of the relative error was higher than that of Test 1. As already NSR
0.15. The
The relative
scatter of error
the of
relativeamplitude
error was identification
higher than thatwas
of calculated
Test 1. As after
already
0.15.discussed
The scatter
discussed by of
otherthe relative[35,36,39],
literature
by other literature error was
[35,36,39], higher
curve
curve than
fits of
fits thatPSR
of lower
lower of Test
PSR 1.
areAs
data are
data already
generally
generally not
notdiscussed
conduciveby
conducive to other
to
parameter
literature identification.
[35,36,39], curve
parameter identification. fits of lower PSR data are generally not conducive to parameter identification.
EO
EO %)
of AA ((%)
Errorof
Error
Figure 6. Results
Figure
Figure 6. of Test
6. Results
Results of 2.
of (a)2.
Test
Test 95%
2. (a) confidence
(a) intervals
95% confidence
95% confidence of theof
intervals
intervals ofEO identification
the
the and (b)
EO identification
EO identification 95%
and
and (b)confidence
(b) 95%
95%
intervals of theintervals
confidence
confidence relativeof
intervals error
of of the amplitude
the relative
the relative error of
error theidentification.
of the amplitude identification.
amplitude identification.Appl. Sci. 2020, 10, 3675 11 of 20
Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 20
• Test 3:
• Test 3:
Increasing the number of revolutions over which data were used for the identification process
Increasing the number of revolutions over which data were used for the identification process
can improve the results. The results of EO identification started to be biased when the NSR increase
can improve the results. The results of EO identification started to be biased when the NSR increase
to about 0.180.18
to about for for
both methods,
both methods,which
whichwas
was shown
shown ininFigure
Figure7a.7a. The
The NSR NSR corresponding
corresponding to unbiased
to unbiased
identification waswas
identification higher 0.03 0.03
higher thanthan
Test 1. However,
Test relative
1. However, errorerror
relative for amplitude identification
for amplitude increased
identification
withincreased
increasing theincreasing
with number ofthe data, which
number of was
data,caused
which bywasa summated
caused by asquared
summated noise term in
squared the BT B
noise
termThis
matrix. in the BTBeffect
is an matrix. Thismethod
of the is an effect
modelof the
thatmethod
is used.model that is the
Although used. Although
relative errorthewas
relative
increased
errormethods,
for both was increased for both
the ICFF methods,
error the ICFF
increased error the
less than increased
TCFF.less than the TCFF.
(a) 10.1
10
EO
9.9 TCFF
ICFF
True EO
9.8
0 0.05 0.1 0.15 0.2 0.25 0.3
Noise/Signal ratio
(b) 30
Error of A ( %)
25
20
TCFF
ICFF
15
0 0.05 0.1 0.15 0.2 0.25 0.3
Noise/Signal ratio
Figure 7. Results
Figure of Test
7. Results of 3. (a)3.95%
Test (a) confidence intervals
95% confidence of theofEO
intervals theidentification and (b)
EO identification and95%
(b)confidence
95%
confidence
intervals of theintervals
relativeof the relative
error error of the identification.
of the amplitude amplitude identification.
• 4: Test 4:
• Test
TheThe results
results of Test
of Test 4 wereshown
4 were showninin Figure
Figure 8.
8. Increasing
Increasingthethe
EOEO
of the tip timing
of the data data
tip timing had no
had no
visible effect on the accuracy of the EO identification for ICFF. However, the relative error of
visible effect on the accuracy of the EO identification for ICFF. However, the relative error of amplitude
amplitude identification was higher than that of Test 1. For the same natural frequency, the larger
identification was higher than that of Test 1. For the same natural frequency, the larger EO, the higher
EO, the higher degree of under-sampling. That means the number of samples was less for a period
degree of under-sampling.
of vibration of the blade.ThatThe means the number
amplitude of samples
that recovered was less
was more for a to
sensitive period of vibration
the number of of
the blade. The amplitude that recovered was more sensitive to the number of samples
samples than the EO identification. Although the relative error was increased for both methods, the than the EO
identification. Although
quality of ICFF thehigher
was still relative error
than was increased for both methods, the quality of ICFF was still
TCFF.
higher than TCFF.Appl. Sci. 2020, 10, 3675 12 of 20
Appl. Sci.
Appl. Sci. 2020,
2020, 10,
10, xx FOR
FOR PEER
PEER REVIEW
REVIEW 12 of
12 of 20
20
EO
EO %)
of AA( (%)
Errorof
Error
Figure 8. Results of Test 4. (a) 95% confidence intervals of the EO identification and (b) 95%
Figure 8. Results of Test
Figure 8. Results of Test 4. (a)4.95%
(a) confidence
95% confidence intervals
intervals ofEO
of the theidentification
EO identification and95%
and (b) (b) confidence
95%
confidence intervals
confidence intervals of the
the relative
relative error
error of
of the
the amplitude
amplitude identification.
identification.
intervals of the relativeoferror of the amplitude identification.
• Test 5:
• 5: Test 5:
• Test
Increasing the
Increasing the number
number ofof probes
probes made
made the
the scatter
scatter of
of EO
EO identification
identification get
get larger
larger for
for ICFF,
ICFF,
Increasing
which was the number
shown in of probes
Figure 9a. made the the
Meanwhile, scatter of EO
relative identification
error of amplitudegetidentification
larger for ICFF, waswhich
which was shown in Figure 9a. Meanwhile, the relative error of amplitude identification was
was increased
shown infor Figure 9a. Meanwhile,
both methods,
methods, which
which wasthe relative
was shown
shown in error
in Figure of
Figure 9b.amplitude
9b. The identification
The summated
summated squared was
squared noise increased
noise term
term inin for
increased for both T
boththe
methods,
the BTB
B
T which
B matrix
matrix was increase
would
would shown in
increase Figure
with
with the 9b.
the The of
number
number summated
of probes. squared
probes. Then
Then noise
the squared
the squared term in the
noise
noise termB was
term B matrix
was
would increase
transferred with
to the number
amplitude of probes.
identification. Then
Figure 9b the squared
still showed noise
the term
relative was
error transferred
of ICFF was
transferred to amplitude identification. Figure 9b still showed the relative error of ICFF was smaller to amplitude
smaller
than TCFF.
TCFF. Figure 9b still showed the relative error of ICFF was smaller than TCFF.
identification.
than
Figure 9. Results
Figure
Figure 9. of Test
9. Results
Results of 5.
of Test
Test(a)5.
5.95%
(a) confidence
(a) 95% confidence
95% intervals
confidence of theof
intervals
intervals ofEOtheidentification
the and (b)
EO identification
EO identification and95%
and (b)confidence
(b) 95%
95%
confidence
intervals intervals
of theintervals
confidence of
relativeof the relative
error
the relative error of
of the amplitude
error of the
the amplitude
amplitude identification.
identification.
identification.Appl. Sci. 2020, 10, 3675 13 of 20
Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 20
• Test 6:
• Test 6:
TheThe
existence
existenceof of
the
theICoMB
ICoMBdid did not have an
not have aneffect
effectononthethe
EOEO identification
identification for both
for both methods,
methods,
whichwhich was shown in Figure 10a. BTT data that consists of two modes corresponding to different EO EO
was shown in Figure 10a. BTT data that consists of two modes corresponding to different
generally had had
generally negative effects
negative on the
effects on quality of identification,
the quality especially
of identification, for for
especially the the
reconstruction method
reconstruction
that method
processes that
theprocesses the single-frequency
single-frequency signals.itHowever,
signals. However, it did to
did not belong not
thebelong
range to the rangein this
considered
considered
paper. in thisfuture
It needs some paper.work
It needs some future
for processing thework for processing
multi-frequency the multi-frequency
signals based on ICFF.signals
The ICoMB
based on ICFF. The ICoMB made the quality of amplitude identification degrade,
made the quality of amplitude identification degrade, which was shown in Figure 10b. However, which was shown
in Figure 10b. However, the relative error of ICFF was still lower than that of TCFF.
the relative error of ICFF was still lower than that of TCFF.
(a) 10.2
EO
10
TCFF
ICFF
True EO
9.8
0 0.05 0.1 0.15 0.2 0.25 0.3
Noise/Signal ratio
(b) 20
Error of A ( %)
15
10
TCFF
ICFF
5
0 0.05 0.1 0.15 0.2 0.25 0.3
Noise/Signal ratio
Figure 10. Results
Figure of Test
10. Results 6. (a)6.95%
of Test (a) confidence intervals
95% confidence of theofEO
intervals theidentification and (b)
EO identification and95%
(b)confidence
95%
confidence
intervals of theintervals
relativeof the relative
error error of the identification.
of the amplitude amplitude identification.
3.3. Summary
3.3. Summary
The The percentage
percentage of correct
of correct EO identification
EO identification and theand the relative
mean mean relative
error of error of amplitude
amplitude identification
were summarized in Tables 3 and 4 respectively for all simulated tests at the NSR 0.01, 0.05, 0.01,
identification were summarized in Tables 3 and 4 respectively for all simulated tests at the NSR 0.10, 0.15,
0.20,0.05,
0.250.10,
and0.15,
0.30.0.20, 0.25 and 0.30.
Table 3. Percentage of correct EO identification following the noise-to-signal ratio (NSR) values.
Table 3. Percentage of correct EO identification following the noise-to-signal ratio (NSR) values.
Test 1 Test 2 Test 3 Test 4 Test 5 Test 6
NSR Test 1ICFF TCFF Test ICFF
2
NSR TCFF TCFFTestICFF
3 TCFF Test 4
ICFF TCFF Test 5
ICFF TCFF Test 6
ICFF
0.01 TCFF
100% ICFF100% TCFF 0 ICFF
0 TCFF 100%
100% ICFF 100%TCFF 100% ICFF 100%
TCFF 100%ICFF 100%
TCFF100% ICFF
0.05
0.01 100% 100%
100% 100% 1%
0 00 100%
100% 100%100% 100%100% 100% 100% 100%
100% 100%100% 100%
100%100% 100%
0.10
0.05 100% 100%
100% 100% 14%
1% 00 100%
100% 100%100% 100%100% 100% 100% 100%
100% 91% 100% 100%
100%100% 100%
0.15
0.10 100% 100%
100% 100% 19%
14% 2%0 100%
100% 100%100% 99% 100% 100% 100% 100%
100% 87% 91% 100%
100%100% 100%
0.20
0.15 98% 100%
100% 97% 29%
19% 5%
2% 100%
100% 100%100% 96% 99% 99% 100% 98%100% 66% 87% 98%100%100% 100%
0.20
0.25 98%
96% 97%
98% 29%
12% 5%
15% 100% 99%
98% 100% 88% 96% 100% 99% 92% 98% 46% 66% 96% 98% 99% 100%
0.25
0.30 96%
88% 98%
88% 12%
18% 15%
11% 98% 96%
96% 99% 78% 88% 96% 100% 89% 92% 55% 46% 87%96% 94%99%
0.30 88% 88% 18% 11% 96% 96% 78% 96% 89% 55% 87% 94%
The results of the EO identification that were less than 80 percent were marked in red.
The results of the EO identification that were less than 80 percent were marked in red.Appl. Sci. 2020, 10, 3675 14 of 20
Table 4. Mean relative error of the amplitude identification following NSR values.
Test 1 Test 2 Test 3 Test 4 Test 5 Test 6
NSR
TCFF ICFF TCFF ICFF TCFF ICFF TCFF ICFF TCFF ICFF TCFF ICFF
0.01 13.49% 4.06% — — 28.74% 23.82% 29.41% 20.04% 15.97% 14.53% 18.29% 10.32%
0.05 13.42% 4.00% — — 28.59% 23.64% 29.29% 19.96% 15.87% 14.46% 18.19% 10.19%
0.10 13.03% 3.56% — — 28.29% 23.37% 28.70% 18.97% 15.59% 14.25% 17.97% 10.02%
0.15 12.69% 3.35% 18.63% 19.07% 27.63% 22.48% 27.62% 17.53% 15.72% 14.47% 17.38% 9.30%
0.20 12.23% 2.98% 19.10% 19.38% 26.83% 21.63% 26.39% 15.25% 15.24% 14.16% 16.58% 8.47%
0.25 11.30% 1.92% 18.30% 19.50% 26.22% 20.94% 24.54% 11.78% 15.78% 14.92% 15.56% 7.65%
0.30 10.54% 1.23% 17.32% 18.29% 24.73% 19.32% 22.72% 9.24% 14.35% 13.70% 15.19% 7.17%
The results of amplitude identification that were higher than 20 percent were marked in red.
The results of the EO identification that were less than 80 percent were marked in red in Table 3.
Curve fits of lower PSR data are generally not conducive to parameter identification, especially for the
EO identification, which was demonstrated by Test 2. Increasing the number of probes would have
some negative effect on the EO identification for ICFF, and the degree of influence became significantly
larger when the data noise up to 0.20 (Test 5). The higher PSR value and the larger amount of fitted
data were conducive to EO identification for both methods. The higher EO value (higher degrees of
under-sampling) and coupling between mistuning blades (unconsidering multi-frequency vibration)
had little effect on the results of EO identification. Through the comparison from Test 1, Test 3, Test 4
and Test 6, ICFF performance was comparable to TCFF in terms of EO identification.
The relative error for amplitude identification increased with the number of fitted data, which was
verified by the results of Test 3. Higher EO value has a more negative effect on the amplitude
identification of TCFF than that of ICFF (Test 4). Table 4 showed that the results of amplitude
identification of ICFF were more accurate than TCFF, except that the low PSR (Test 2) resulted in
the little difference between ICFF and TCFF. The larger number of fitted data and the number of
probes would lead to a higher error of amplitude identification for both methods, which was caused
by the squared noise term in the BT B matrix that increased with the number of fitted data or probes.
Compared with TCFF, ICFF considers the influence of blade vibration on the ToA of the blade tip.
This improvement is more consistent with reality and leads to more accuracy of the amplitude
identification. This was demonstrated through all of the simulated tests.
Conclusions drawn above were based only on the numerical simulations but not based on the
analysis of real signals measured during the experimental tests for a real object under loading conditions.
In the experimental tests, different factors may affect the accuracy of the reconstruction method. So,
the proposed method will be verified by experimental measurements and analysis in the next section.
4. Method Evaluation Using Experimental Data
The experimental data that was provided by the author of the literature [36] was used to further
verify the feasibility of the reconstruction method proposed in this paper. The test rig in the literature [36]
was shown in Figure 11a. The radius of the rotor is 60 mm, and there are eight blades. The once
per revolution (OPR) sensor was installed near the shaft to provide the rotating speed and a timing
reference through detecting a marking on the shaft. Seven optic probes were installed on the casing to
measure the blade vibration. The installation angles of the probes were 0◦ (P0 ), 18.56◦ (P1 ), 36.17◦ (P2 ),
53.75◦ (P3 ), 72.42◦ (P4 ), 119.82◦ (P5 ), and 239.06◦ (P6 ). The nitrogen shock was used as an exciting force
to the rotating blades.
The mode frequency was plotted on the vertical axis in the Campbell diagram of Figure 11b.
The diagram plotted the resonant frequency on the vertical axis and the rotating speed on the horizontal
axis. The EO lines showed a range from 9 to 15 (blue lines). The seven black dots in the Campbell
diagram represent the intersection of the EO lines with the mode frequency. The nominal resonant
speeds corresponding to these seven intersections are 7081 rpm (EO = 15), 7613 rpm (EO = 14),Appl. Sci. 2020, 10, 3675 15 of 20
Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 20
Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 20
14), 8202rpm
8202rpm (EO(EO = 13),
= 13), 89078907
rpmrpm
(EO (EO = 12),
= 12), 97249724
rpmrpm
(EO (EO = 11),
= 11), 10,710
10,710 rpmrpm
(EO (EO = and
= 10), 10), and 11,890
11,890 rpm
rpm
(EO (EO
= 9),= 9), respectively.
respectively.
14), 8202rpm (EO = 13), 8907 rpm (EO = 12), 9724 rpm (EO = 11), 10,710 rpm (EO = 10), and 11,890
rpm (EO = 9), respectively.
frequency
Vibration
Vibration ) )
(Hz(Hz
frequency
Figure
Figure11.
11.(a)
(a)The
Thetest
testrig
rigand
and(b)
(b)the
theblade
bladeCampbell
Campbelldiagram.
diagram.
Figure 11. (a) The test rig and (b) the blade Campbell diagram.
The displacement
The displacement of blade 77 measured measured by probe probe P00 for
for rotating
rotating speed
speed 6500–12,100
6500–12,100 rpm rpm waswas
The displacementofofblade blade 7 measured by by probe P P0 for rotating speed 6500–12,100 rpm was
shown
shown in
in inFigure
Figure 12. In the
12.12.InInthe whole speed up, there were seven resonances excited by the nitrogen
shown Figure thewhole
wholespeedspeed up,
up, there
there were seven resonances
were seven resonancesexcited
excitedbybythe thenitrogen
nitrogen
shock, although
shock, although the the vibration
vibration intensity
intensity was was not
not completely
completely thethe same.
same. TheTheEO EOcorresponding
corresponding to to each
each
shock, although the vibration intensity was not completely the same. The EO corresponding to each
resonance
resonance was predicted
resonance was predictedthrough through
throughthethe Campbell
theCampbell diagram
Campbell diagram (Figure 11b),
(Figure 11b),
diagram (Figure which
11b),which was
whichwas labeled
waslabeled in
labeledinin Figure
Figure
Figure 12.
The
12.12. similar
TheThe resonances
similar
similar resonances were
resonanceswere also measured
werealsoalsomeasuredby other
measured by probes,
by other and here
probes, and
other probes, it did
and here not show
hereititdid
didnot these
notshow vibration
showthesethese
waveforms
vibration one by
waveforms one. In
one the
by engineering
one. In the applications,
engineering the EO can
applications, be predicted
the EO
vibration waveforms one by one. In the engineering applications, the EO can be predicted through can through
be the
predicted Campbell
through
diagram,
the Campbell
the like the
Campbell experiment
diagram,
diagram, like in
thethis
likethe paper. The
experiment
experiment in EO paper.
in this
this identification
The EO
paper. The EOstep can be skipped
identification
identification on
stepcan
step the
canbe becondition
skipped
skipped
that
on on the
thethe true
conditionEO had
condition that been
that the known
thetrue
trueEOEO before
had parameter
hadbeen
been knownidentification.
known before parameter
parameterIn this case, the correct
identification.
identification. EOcase,
InInthis
this would
case, be
thethe
used
correct directly
correctEOEO by identification
would
would bebeused
useddirectly methods
directly by to calculatemethods
byidentification
identification the amplitude,
methods to phase,
to calculate
calculate theand
the direct current
amplitude,
amplitude, phase,
phase, offset.
andand
In direct
thiscurrent
direct section,
current the
offset.other
offset. parameters
InInthis
thissection,
section,the would
the beparameters
other
other calculated would
parameters based be
would on the EO that
becalculated
calculated wason
based
based from
onthe prediction
EO
the EOthat
that
was wasfrom
rather fromidentification.
than predictionrather
prediction rather thanthe
Therefore,
than identification.
reconstruction
identification. Therefore,
accuracy the
Therefore, the reconstruction
was compared
reconstruction accuracy
for each method
accuracy wasbase
was
on compared
the EO for
known. each method base
compared for each method base on the EO known. on the EO known.
110110 EO:
13,000
13,000
EO: 1414
EO: 13 EO: 12
90 EO: 13 EO: 12
90 12,000
70 EO: 15 12,000
70 EO: 15
Rotating speed (rpm)
Displacement (μm)
50 11,000
Rotating speed (rpm)
Displacement (μm)
50 11,000
30
30 10,000
10 10,000
10
-10
-10 9000
-30 9000
-30 8000
-50
-50 EO: 9 8000
-70 EO: 9
EO: 11 7000
-70-90 EO: 10
Rotating speed
EO: 11 Displacement 7000
-90 Rotating speed
-110 EO: 10
Displacement 6000
-110 2000 4000 6000 8000 10,000 12,000 6000
2000 4000 Revolution
6000 8000 10,000 12,000
Revolution
Figure 12.The
Figure12. Thedisplacement blade 77measured
displacement of blade measuredby
byprobe
probeP0P. 0 .
Figure 12. The displacement of blade 7 measured by probe P0.
Although thethe
Although EOEO
corresponding to each
corresponding resonance
to each can be
resonance predicted
can through
be predicted the Campbell
through diagram,
the Campbell
thediagram,
corresponding
the the
Although amplitude
corresponding
EO can not be
amplitude
corresponding known
tocan exactly,
knownwhich
notresonance
each be bewas
exactly,
can not the
which same
wasthrough
predicted as the
not the the Campbell
samesimulated
as the
diagram, the corresponding amplitude can not be known exactly, which was not the same as theYou can also read
