Date post: | 03-Apr-2018 |
Category: |
Documents |
Upload: | mrchinacdn |
View: | 228 times |
Download: | 0 times |
of 13
7/28/2019 (2011)-Yan,Miyamoto,Jiang - Frequency Slice Algorithm for Modal Signal Separation and Damping Identification
1/13
Frequency slice algorithm for modal signal separation and damping identification
Zhonghong Yan a,b,,1, Ayaho Miyamoto b,2, Zhongwei Jiang b,3
a Biomedical Department, Chongqing University of Technology, Chongqing 400050, Chinab Yamaguchi University, 2-16-1 Tokiwadai, Ube, Yamaguchi 755-8611, Japan
a r t i c l e i n f o
Article history:
Received 13 March 2009
Accepted 30 July 2010
Keywords:
Timefrequency analysis
Random decrement technique
Signal process
Vibration signal
Modal parameter identification
a b s t r a c t
This paper focuses on modal signal separation and damping parameter identification by a new time
frequency analysis method. With the aid of the random decrement technique (RDT), an accurateestimation method is firstly introduced both in time and frequency domains for single modal damping
identification. Next, the background of a new concept of frequency slice wavelet transform (FSWT) is
revealed clearly. Then, some new properties of the FSWT are briefly discussed in contrast with the wave-
let transform (WT). Based on the analysis of RDT and FSWT, a frequency slice algorithm (FSA) is designed
for modal separation and parameter identification. The merits of FSWT and FSA with numerical simula-
tions and experiments are demonstrated in this paper. We finally apply the proposed methods to analyze
the free-decay responses (FDR) collected from a small laboratory bridge monitoring system (LBMS). Some
conclusions are drawn that the RDT being used directly in FSWT domain can bring a good damping esti-
mator. The FSA is not limited to FDR, and also can be used to random impacting response directly. FSWT
itself is a new kind of good filter, and has high performance against noise. It is significant to get damping
parameter with higher accuracy through modal separation by FSWT, and FSWT can be controlled adap-
tively in modal separation by dynamic scale method.
Crown Copyright 2010 Published by Elsevier Ltd. All rights reserved.
1. Introduction
1.1. Problems and methods
In a vibration system, response frequency, mode shape and
damping, which are always relative with the system structure,
are often used in damage detection in structural health monitoring
system (SHMS) [1,2]. Meanwhile, modal analysis and parameter
identification are common-used methods for extracting these dy-
namic characteristics in SHMS. Quite a few researches [3,4] noted
that the identified natural frequencies and mode shapes by using
various system identification methods and different test data are
in excellent agreement, but the estimation uncertainty of damping
ratios is inherently larger than that of natural frequencies. Experi-ments show that the damping features of a system, especially for
the high frequency components, are very significant marks to dam-
age detection. However, in general, it is not easy to extract the
damping (ratio) exactly because the vibration signals always in-
clude many frequency components and noise. Therefore, in this
paper, we would like to pay more attention to modal signal sepa-
ration and damping identification, especially for the modal signals
with high damping and close frequency modes. Let us [5] still con-
sider a linear damped multi-degree-of-freedom (MDOF) system
with n real modes for system modal parameter identification,
and its free-decay response (FDR) is given as
xt Xni1
Aie2pfifitcos2pfdit hi 1
where Ai is the amplitude, fi is the undamped natural frequency,
fdi fiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 f2iq
is the damped natural frequency, and fi is the damp-
ing ratio. Here a single modal signal is described as an exponentially
decayed sinusoid signal function: s(t) = eatcos(bt+ h).Generally, based on frequency domain decomposition (FDD),
modal identification techniques for vibration responses (for exam-
ple, the FDR in Eq. (1)) are widely recognized [6] as being simple.
However, it often leads to the loss of accuracy of the identified
modal parameters due to the spectrum of measured responses,
which cannot be estimated exactly, especially for high-damped
systems and systems with severe modal interference. Conversely,
many methods based on the time domain analysis have been
developed [79]. These approaches frequently provide accurate re-
sults if the measured responses are not severely contaminated by
noise. Nevertheless, the de-noising in time domain is not as conve-
nient as in frequency domain.
0045-7949/$ - see front matter Crown Copyright 2010 Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.compstruc.2010.07.011
Corresponding author at: Biomedical Department, Chongqing University of
Technology, Chongqing 400050, China. Tel./fax: +86 023 68660070.
E-mail addresses: [email protected] (Z. Yan), [email protected] (A.
Miyamoto), [email protected] (Z. Jiang).1 Tel./fax: +81 836 85 9530.2 Tel./fax: +81 836 85 9530.3 Tel./fax: +81 836 85 91370.
Computers and Structures 89 (2011) 1426
Contents lists available at ScienceDirect
Computers and Structures
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p s t r u c
http://dx.doi.org/10.1016/j.compstruc.2010.07.011mailto:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.compstruc.2010.07.011http://www.sciencedirect.com/science/journal/00457949http://www.elsevier.com/locate/compstruchttp://www.elsevier.com/locate/compstruchttp://www.sciencedirect.com/science/journal/00457949http://dx.doi.org/10.1016/j.compstruc.2010.07.011mailto:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.compstruc.2010.07.0117/28/2019 (2011)-Yan,Miyamoto,Jiang - Frequency Slice Algorithm for Modal Signal Separation and Damping Identification
2/13
Combining the representation of the measured responses in
time and frequency domains simultaneously, some new tools such
as, WignerVille distribution (WVD), wavelet transform (WT) and
HilbertHuang transform (HHT), etc., are developed to construct
new frameworks for system identification and damage detection
(Refs. [1015]). Prominently, as one of important timefrequency
analysis tools, the continuous wavelet transform (CWT) and dis-
crete wavelet transform (DWT) have been fully developed in the
theoretical aspect over past two decades.
Yan et al. [13], noted that WT as well as the traditional schemes
can only supply a single estimation of the modal parameter, and its
accuracy depends on modal separation, end-effect, and the para-
metric selection of wavelet function, etc. In fact, even if for sin-
gle-degree-of-freedom (SDOF) response, by using the WT method
based on [10], most of estimations are only approximate. There-
fore, Tan et al. [14] discard the fussy selections of wavelet scale
and centre frequency in Yans method [13] that is based on mini-
mum Shannon entropy search to separate the close modals, and
they proposed a relatively simpler pattern search method to im-
prove the estimated results of [13].
Using a different idea in modal separation, Huang and Su [15]
have presented an eigenvalue model in wavelet domain based on
CWT method. Although the new method of Huang and Su could
overcome the problem of close modes decomposition by perform-
ing system identification in wavelet domain, their algorithm still
involves the expensive computation to first solve a large-scale
overdeterminate system of linear algebraic equations, and then
find the solutions of eigenvalue problem of a big size square ma-
trix. At the same time, the determination of both wavelet scale
and centre frequency still affects the accuracy of identification
problem. In fact, pursuing high timefrequency localization of
modal signal is always important to wavelet methods. Fortu-
nately, in this paper, we can easily overcome the difficulty with
a new timefrequency transform method. We still base on the ba-
sic idea of modal separation by wavelet, the new wavelet trans-
form is to analyze their introduced questions. Notably, we can
simplify a lot of computation for modal separation and dampingidentification.
In wavelet transforms, there is a common problem: how is it
possible to decide the center frequency and the time supporting
width of a mother wavelet? As we know, the characteristics of a
mother wavelet function always affect the performance of time
frequency analysis. For example, although the Gabor wavelet,
which is one of the most widely used analytic wavelets, has the
best timefrequency resolution, i.e. the smallest Heisenberg box,
the center frequency and the time supporting width of the mother
Gabor wavelet affect its timefrequency decomposition character-
istics. This means that, depending on the signals to be analyzed,
different Gabor wavelet shapes must be used. Since the character-
istics of signals are unknown in general, the determination of opti-
mal shape is usually difficult [16]. Based on the motivations, wehave developed a new timefrequency transform with better
properties than WT to improve the situations in application
[5,17]. The new transform is called the frequency slice wavelet
transform (FSWT). Actually, this paper will reveal the proposed
background of FSWT clearly. Frequency slicing processing is an
important idea in modal analysis and parameters identification
in this paper.
Many existing methods for modal identification are based on
FDR signals. Unfortunately, we usually cannot easily get the FDR
signals from a big system. The well-known random decrement
technique (RDT) (e.g. [18]) is usually used in computing random
decrement signatures from ambient random vibration data. How-
ever, this paper will study a new usage of RDT idea, and its trigger-
ing concept is employed to extract accurately the modal dampingin timefrequency domain for SDOF response. The method can also
be used to estimate the modal damping directly for MDOF signals
with an acceptable accuracy.
1.2. Main ideas
The main aimof this paper is to realize modal signals separation
and damping identification in Eq. (1) with a novel method. Firstly,
for SDOF signal, combined with logarithmic decrement method
(LDM) and RDT idea, a high accurate damping estimation in fre-
quency domain is introduced. Secondly, based on the estimation
method, FSWT as a new signal analysis tool is thus introduced in
this study, and the background of the FSWT analysis is first re-
vealed clearly.
Since FSWT itself is a new kind of good filter [5,17], this paper
does not need any filter even if the obtained signal includes high
noise. We only focus on the application of FSWT in this paper.
Implementing the RDT directly in FSWT timefrequency domain
is the first important application of FSWT in this paper, and modal
separation is another application of FSWT. FSWT has many better
properties [17] than WT. Such as, the center of FSWT timefre-
quency window is the observing center, and this property makes
it possible to construct an adaptive algorithm of modal parameter
identification in timefrequency domain. Dynamic scale of FSWT
as a new skill will be proved to be a powerful tool in modal
separation.
As a new result, a general estimator of modal damping is to ex-
press in FSWT domain. FSWT can very clearly represent the damp-
ing characteristics of multi-modal signal simultaneously in time
and frequency domain, a frequency slice algorithm (FSA) for modal
separation and parameter identification is therefore designed to
analyze the free-decay responses (FDR). Meanwhile, we introduce
timefrequency projective method of FSWT for modal separation.
A real FDR signal collected from a small laboratory bridge monitor-
ing system (LBMS) will be investigated. The merits of FSWT and
FSA with numerical simulations and experiments are demon-
strated in this paper. Some conclusions are drawn that the RDT
being used directly in FSWT domain can bring a good dampingestimator. FSA is not limited to FDR, and also can be used to ran-
dom impacting response directly. FSWT itself is an effectual filter
to noise. It is significant to get damping parameter with higher
accuracy through modal separation by FSWT, and FSWT can be
controlled adaptively in modal separation by dynamic scale
method.
1.3. Notation
R denotes the set of real numbers. L2(R) denotes the vectors
space of measurable, square-integrable one-dimensional functions
f (x).
Fourier transform (FT) for function f (x) e L2(R).
Fffg : ^fx Z
1
1
fseixsds 2
Fourier inverse transform:
F1f^fg : ft
1
2p
Z11
^fxeixtdx 3
The signal energy is recorded as:
kfk22
Z11
jftj2dt 4
||||2 also denotes the classical norm in the space of square-integra-ble functions.
We define the following timefrequency localization features of
limited energy signals, which include wavelet functions and STFTwindow functions etc.
Z. Yan et al. / Computers and Structures 89 (2011) 1426 15
7/28/2019 (2011)-Yan,Miyamoto,Jiang - Frequency Slice Algorithm for Modal Signal Separation and Damping Identification
3/13
The duration Dtf and bandwidth Dxf are defined as
Dtf 1
kfk2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiZ11
t tf2jftj2dt
s; Dxf
1
kfk2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ11
x xf2j^fxj2dx
s; 5
where tf and xf are the centers of f(t) and fx, respectively,
tf 1
kfk22
Z11
tjftj2dt; xf 1
kfk22
Z11
xj^fxj2dx 6
2. Frequency slice expression of modal damping
2.1. RDT idea
Here we briefly describe the RDT idea, and more details can be
found in [18]. Under a randomly exciting force, a structure has ini-
tial displacement a at time t, and we record the response asx(t) = a.
The most important idea of RDT is that the moving average func-
tion of response x(t) on a level crossing trigger condition is intro-
duced to get the free-decay response from random loads. Thefollowing is a simple form, where X= {x(ti) = a} is called trigger
condition, which can be changed according to the needs, such as
X fa xti bg,
xs 1
N
XNi1
xti s Xj ; 7
where N is the total number of triggering points.
The simplicity of the estimation process is obvious, since only
the detection of triggering points and averaging of the correspond-
ing time segments are performed. As the number of average in-
creases, the random part due to the random loads will be
eventually averaged out and be negligible. Furthermore, the sign
of the initial velocity is expected to vary randomly with time, so
the resulting initial velocity will be zero. Therefore, the free-decay
response from the initial displacement a will remain.
Considered a SDOF system response as the first case, a new
usage of RDT concepts to extract the modal damping directly will
be introduced in the following section.
2.2. RDT expression of modal damping
2.2.1. Time domain expression
Let s(t) =Aeatcos(bt+ h) be FDR signal in a simplest SDOF sys-tem. If bT= kp, here k is a positive integer, then logarithmic decre-ment method (LDM) is given as
a 1
T
ln jstj ln jst Tj : 8a
Unfortunately, the LDM in Eq. (8a) is always sensitive to noise
or when |s(t)|% 0. However, we can use the above RDT idea to se-lect a suitable trigger condition to avoid the singularity of function
ln|s(t)| or increase the ability against random noise. For example,
the following is a discrete form under a triggering condition
a 1
NT
XNi1
ln jstij XNi1
ln sti Tj j
!f0 < a < stij j < bg8b
Nevertheless, Eq. (8b) still has not good capability to anti-noise
because when the signal is contaminated by noise, we can not be
sure |s(ti + T)| 0 even if |s(ti)| satisfy the trigger condition
0 < a 0, where k is a positive integer, then
2 a 1
sln jFt;x; Tj ln jFt s;x; Tj 12a
Eq. (11) is easy to verify directly. Because under the condition
bs = kp, Eq. (12a) is similar to Eq. (8a), the detailed proof is omittedwith the exception of a few comments.
Let us analyze the performance against noise in Eq. (12a). Note
that in a damping system, the damping ratio f ( 1, and a and b aredefined as
a 2pff; b 2pfffiffiffiffiffiffiffiffiffiffiffiffiffi
1 f2q
; 13
thus, a ( b. According to Eq. (11), we know that F(t, x, T) attains itsapproximate maximum at x = b. Therefore, for x
%b and a
(b, it
is predicable that
jFt;b; Tj ! jFt;x; Tj > 0; andkFt s;xj > 0:
As the result, Eq. (12a) is always correct and more reasonable
than Eq. (2a) in time domain. The RDT formula of Eq. (12a) can
be establishedsimilarly as the following. For example, under a trig-
gering condition
a 1
Ns
XNi1
ln jFti;x;Tj XNi1
ln Fti s;x; Tj j
!f0 < a < jFti;x; Tj < bg; 12b
where bs = kp, N is the total number of triggering points.
2.2.3. Numerical demonstration
Notation. Noise level R of a signal s(t) isdefined as s := s(t)(1+Rr(t))
called multiple noise (R%); or s := s(t)/max|s|+Rr(t) called addi-tional noise (+R%); where R > 0 and r(t) is a normally distributed
random variable with zero mean and unit variance.
Suppose a general single modal FDR signal as
st;A;f; f; h; t0 Ae
2pfftcos2pfdt h t! t0
0 t< t0
(; 14
where A is the amplitude of this mode, f is the undamped natural
frequency, fd f ffiffiffiffiffiffiffiffiffiffiffiffiffi1 f2
pis the damped natural frequency, f is the
damping ratio. Record fs as the sampling ratio, and Ts is the sampletime.
16 Z. Yan et al. / Computers and Structures 89 (2011) 1426
7/28/2019 (2011)-Yan,Miyamoto,Jiang - Frequency Slice Algorithm for Modal Signal Separation and Damping Identification
4/13
Table 1 shows the results obtained by applying Eq. (12b) to sig-
nal s(t, 1, f, f, 0, t0), where f= 1 Hz and f = 0.05. A random noise
with different levels is added into the signal in order to test the
effectiveness of RDT formula in Eq. (12b). Meanwhile the influence
on sampling frequency fs and the width of the observing time win-
dow Ts are also investigated. Note that Ts is decreased to 5s.
Since Eqs. (10) and (12b) actually imply filtering, therefore, in
the computation of Table 1, we do not use any filter to de-noise.
Notably, by using Eq. (12b), we can give a high accurate estimator
of modal damping even if the obtained signal includes stronger
noise and shorter sample time than ones of Ref. [14]. At the same
time, the influence of additional noises is larger than that of multi-
ple noises, and the high sampling rate is advantageous in test.
Eq. (12b) is an accurate estimator of dampinga
for SDOF re-
sponse, but it is always approximate for multi-degree-of-free-
dom (MDOF) responses, especially it is still difficult to
distinguish the closely spaced modes. Consequently, a simple
idea is that, first, a way should be found to separate these
modes, and then Eq. (12b) is used to complete the damping
computation. Although the search methods in [13,14] can be
used to separate the close modes, they always involve hard opti-
mal computation.
Interestingly, the estimation method of damping ratio directly
carries out a new timefrequency transform. F(t, x, T) denoted inEq. (10) is called a frequency slice representation of signal s(t). A
more general transformation is to discuss in the following sections
for modal damping expression and modal separation.
3. Introduction of FSWT Tool
For any function f(t) e L2(R), the frequency slice wavelet trans-
form (FSWT) is defined directly in frequency domain as
Wft;x;r 1
2p
Z11
^fu^pu xr
eiutdu 15
where the scale r is a constant or a function of x, t and u, and thestar means the conjugate of a function. Here we call x and t theobserved frequency and time, and u the assessed frequency. ^px isalso called frequency slice function (FSF). In fact, the FSWTis a mod-
ified version of the traditional wavelet transform in frequency do-
main. By using Parseval equation, if r is not function of the
assessed frequency u, then Eq. (12b) can be translated into its timedomain.
Wft;x;r reixt
Z11
fseixsprs tds: 16
Eqs. (15) and (16) can be found in [5]. FSWT fast discrete algorithm
with aid of fast Fourier transform (FFT), more comparisons based on
application with FFT, CWT, DWT, short time Fourier transform
(STFT), and WVD etc., can be found in [5]. An overall theoretical
description of FSWT can be seen in [17]. Therefore, this paper will
only pay more attention to the application of FSWT in modal sepa-
ration and damping identification. We then briefly discuss some
new properties of FSWT in contrast with the wavelet transform
(WT).
Firstly, as a new timefrequency transform, FSWT has better
performance than CWT [17]. In [17], we have analyzed that
j^pxj and |p(t)| are to select as even functions respectively. Bothof functions j^pxj and |p(t)| have perfect symmetry, and so it ispossible that FSWT has better properties than the traditional WT.
For example, the center of timefrequency window is always the
observing center in contrast to the WT. Therefore, the timefre-
quency window is adaptive to the observing center of the analyzed
signal, and the scale r is a balance factor between the time resolu-tion and frequency resolution.
Secondly, if one let r xj, i.e. j xr, then Eq. (15) naturally be-comes into
Wft;x;j 1
2p
Z11
^fu^pju xx
eiutdu 17
Note that the parameter j in Eq. (17) is the unique parameter that
should be chosen in application. Consider the bandwidth-to-fre-quency ratio property of FSF. We define frequency resolution ratio
of an FSF as
gp half width of frequency window
center frequencyrDxpx
Dxpx=r
Dxpj
18
where Dxp is computed by Eq. (5). Thus, in FSWT, gp may not beconstant.
The frequency resolution ratio gs of the measured signal is sim-ilarly defined as
gs Dxsxs
19
From [17], we can assume gp = gs, and then have a basic choiceabout the scale parameter j
Table 1
Identification of modal damping ratio using Eq. (12b) for SDOF response with different sample time and noise levels.
Noise level(R) Modal par ameter Sampling parameter Test of Eq. (12b) statistic times = 100
Modal parameter average Variance
f (Hz) f fs (Hz) Ts (s) E(f) E(f) Var(f) Var(f)
25% 1 0.05 20 10 1.000 0.0501 0 1.3e061 0.05 100 10 1.000 0.0500 0 3.4e071 0.05 20 5 1.000 0.0499 0 9.7e061 0.05 100 5 1.000 0.0501 0 2.5e06
50% 1 0.05 20 10 1.000 0.0501 0 7.7e061 0.05 100 10 1.000 0.0500 0 1.4e061 0.05 20 5 1.000 0.0489 0 4.1e051 0.05 100 5 1.000 0.0498 0 7.6e06
+25% 1 0.05 20 10 1.000 0.0489 0 3.0e051 0.05 100 10 1.000 0.0496 0 6.9e061 0.05 20 5 0.9975 0.0506 9.8e05 6.7e051 0.05 100 5 1.000 0.0492 0 1.1e05
+50% 1 0.05 20 10 0.9972 0.0495 2.8e04 1.1e041 0.05 100 10 1.000 0.0496 0 3.1e051 0.05 20 5 0.9896 0.0492 4.6e04 2.5e041 0.05 100 5 0.9987 0.0497 5.1e05 5.1e05
Z. Yan et al. / Computers and Structures 89 (2011) 1426 17
7/28/2019 (2011)-Yan,Miyamoto,Jiang - Frequency Slice Algorithm for Modal Signal Separation and Damping Identification
5/13
j Dxpgs
20
As stated above in Eq. (20), FSWT have another important prop-
erty: FSWT can be controlled by the frequency resolution ratio gs ofthe measured signal.
We can easily design an FSF. The following gives two simple
examples.
FSF1 : ^px e12x2 ; pt e
12t2
FSF2 : ^px 1
1 x2; pt ejtj
Note that the centers of timefrequency windows in FSF1 and
FSF2 are always at the origin of the timefrequency plane. Fig. 1
shows the principle of FSWT; and the window functions in FSF1
present both in time and frequency domains. Both functions ^puand p(t) are symmetric around their center points respectively.
Their energy is concentrated at the origin of time and frequency
plane.
Inverse transform [17]: if ^px satisfies ^p0 1, then the origi-nal signal f(t) can be reconstructed by
ft 1
2p
Z11
Z11
Ws;x;jeixtsdsdx 21
As an important result of Eq. (21), the reconstruction proce-
dure is independent from the selected FSF. Reconstruction
independency is an important feature in FSWT, but this charac-
teristic is not allowable in traditional wavelet. Therefore, if the
condition ^p0 1 remains unchanged in computation of theFSWT, we can easily realize dynamic scale controlling. For exam-
ple, j can be adaptively controlled by the signal spectrum as thefollowing
j j0 ^fu
= ^fx
; 22
where j0 > 0 is a constant that satisfies Eq. (20),^
fx representsthe energy of the signal at the observing frequency in Eq. (17) andfu for the assessing frequency. See more explanations in [17].
In Section 5, we will show that the dynamic scale j is efficientfor modal separation. It is possible that by using dynamic scale
method one does not have to sacrifice the time resolution to in-
crease the frequency resolution, conversely, the same reason is also
possible to increase the time resolution [17]. However, this kind of
dynamical characteristic is also not allowable in traditional wave-
let because its reconstructed equation must depend on the selected
wavelet base and its scale.
4. FSWT modal analysis
4.1. FSWT expression of modal damping
In this paper, as the main application of FSWT, the damping
parameter in Eq. (1) can be analyzed by the following theorem.
Theorem 1. Let ft Aeat cos
bt
h
t
!t0
0 t< t0
; and the FSF p(t)satisfies t t0j g 24b
Here, we call that if t< 0, p(t) = 0 p(t) is single side function (SSF) in
time domain.
Remark. It is necessary that p(t) should be an SSF to maintain the
accuracy of Eq. (24a) for SDOF response. However, in traditional
wavelet theory, we cannot suppose thatp(t) should be a single side
function. For example, the Gaussian function is not an SSF.
Therefore, many estimators by means of the asymptotic techniques
and Taylors formula based on general wavelet or Morlet wavelet
transform (MWT) (e.g. [10]) are always approximate even for SDOF
signal.
In fact, Eq. (23) reveals that |W(t, x, r)| can be viewed as theenvelope of FDR f(t), latter we will show the FSWT characteristic.
Eq. (24b) then completes the logarithmic decrement method to
get the damping parameter. Usually, Eq. (24b) may be sensitive
to noise or when |W(t, x, r)| = 0. To avoid the singularity of func-tion ln() or increase the ability against noise, we can also use theRDT idea in FSWT domain, and then Eq. (24b) can be easily chan-
ged into Eq. (24b) similar with Eq. (12b). Eq. (24b) can be a good
approximate expression of damping for MDOF response due to
the localization of FSWT in timefrequency domain and another
fact that FSWT can be easily controlled by frequency resolution ra-
tio gs of the measured signal and dynamic scale method. Finally,Eq. (24b) is used to compute the damping of random response di-
rectly. Latter the computational result will be shown in an
experiment.
Inverse
Fourier
Transform
FFT
Spectrum
u
u
)( uf
)(
up
A
Frequency
SliceResponse
t
Multiple
Move Slice Window
SliceFunction )( up
u
)(tp
0 0
t
Fig. 1. Schematic diagram of FSWT.
18 Z. Yan et al. / Computers and Structures 89 (2011) 1426
7/28/2019 (2011)-Yan,Miyamoto,Jiang - Frequency Slice Algorithm for Modal Signal Separation and Damping Identification
6/13
4.2. Local characteristics of frequency slices
In this section, a simulated damping vibration signal is used to
observe the characteristics of frequency slices and compare the de-
tails of the FSWT with the CWT. We can further analyze the FDR
signal with FSWT based on these characteristics.
Example 1.the original signal in Fig. 2 is stated as
s=
s1+
s2+
s3,
where s1, s2 and s3 are simulated by Eq. (14) with the parameters
described in Table 2.
Let FSF be the Gaussian function ^px e12x2 , hence Dxp ffiffiffi2
p=2: According to Eq. (20), we can take j = 0.707/g, and set
g = 0.025, hence j = 28.28 and gp = 0.025. So the chosen parame-ters for FSWT Eq. (17) are summarized as follows
^px e12x2 ; r
xj; gs 0:025; gp 0:025; j 28:28
25
Fig. 2 reveals that FSWT has three clear separated peaks that
only indicate one modal. In fact, this signal is equivalent to a SDOF
response for multiple impacts. The Fourier spectrum Fig. 2b only
shows one peak since it cannot show the same frequency signals.
Fig. 3 shows two groups of three slices located near the maximum
response frequency of the modal signal s under two FSFs. Fig. 4
demonstrates these slices of the FSWT under high noise (+25%).
Fig. 5 compares with the MWT without noise or under high noise
(+25%); some drawbacks of the MWT method are revealed out,
where the complex Morlet wavelet function is taken as
wt e1at
2
eibt 26
The observed features of damping vibration signals are summarized
as below
(1) FSWT shows the details of time and frequency components
for each modal individually, such as its main frequency
and the response time.
(2) Each modal signal on the 2D map of FSWT coefficients is a
connected area. Note that this is an important feature for
modal signal separation.
(3) All frequency slices of a single modal signal demonstrate the
damping envelopes: Ae2pfft (as a result of Theorem 1).(4) Different FSFs maintain similar properties. There are some
differences in FSWT amplitudes, but this does not affect
the damping estimation, because the damping in Eq. (24b)
is only a ratio of the FSWT amplitudes.
(5) The FSWT shown in Fig. 4 presents high performance against
noise, but the MWT shown in Fig. 5 is more sensitive to
noise.
(6) FSWT can be controlled only by the frequency resolution
ratio gs of the signal, but the CWT must depend on the centerfrequency and the bandwidth parameter simultaneously.
Fig. 5 points out that the MWT may have much lower fre-
quency resolution ([17]) even if under the same parameters
gw = 0.025 and gp = 0.025 with the FSWT. At the same time,the MWT has the frequency-bands energy leakage obviously
([17]).
Based on the analysis of Eqs. (17), (19)(22) etc., the timefre-quency localization of FSWT can be adaptively controlled by the
frequency resolution ratio of signal. Therefore, by using timefre-
quency localization of FSWT, the multi-modals signal can be sepa-
rated into single modal. Moreover, all separation errors can be
viewed as a certain level of random noise; and Eq. (24b) or Eq.
(12b) can further eliminate the noise in damping computation.
This is the main idea proposed in this paper. Consequently, com-
bining the characteristics of the FSWT timefrequency image, we
will introduce the modal separation method in the following.
4.3. Determining modal domains
Example 2. Fig. 6a shows a real FDR signal obtained from a small
laboratory bridge monitoring system (LBMS), which will beintroduced in latter Section 5.3. The Fourier transform spectrum
Fig. 6(b) shows that, the first modal signal is a clear indication of
the greatest energy at 29.3 Hz and the other peaks are smaller.
Fig. 6ch shows the results of the FSWT method, where all of FSWT
parameters are assumed in Eq. (25). The more clearly damping
characteristics of the signal in timefrequency domain are revealed
in Fig. 6c and d. Note that the second modal not the first has the
highest amplitude at about 112.3 Hz. We then use this example to
explain the modal separation flow as below
100
150
0
50
05
1020
02
4 68
FSWT
=28.28
Amplitude
Time(Sec) Frequenc
y(Hz)
Time(Sec.)
5
25
0 5 10Hz0
10
1500
0 5 10Hz0 5 10 15 20 Sec.
-1.5
1.5
03
9
0
6
Fourier
Spectrum
Amplitude
15
20
10
Fig. 2. (a) Simulated signal by Table 2; (b) Fourier spectrum; (c) and (d) are 2D and 3D maps of the FSWT coefficients, where gp = 0.025.
Table 2
The simulated signal shown in Fig. 2(a).
S A F f h t0
s1 1 5 Hz 0.02 0 4.0s
s2 1 5 Hz 0.02 0 10.0s
s3 1 5 Hz 0.02 0 16.0s
s = s1 + s2 + s3Ts = 25s, fs = 400 Hz.
Z. Yan et al. / Computers and Structures 89 (2011) 1426 19
7/28/2019 (2011)-Yan,Miyamoto,Jiang - Frequency Slice Algorithm for Modal Signal Separation and Damping Identification
7/13
(1) The FSWT translates the multi-modals signal into a time
frequency image, such as Fig. 6(c) and (d).(2) By processing the image, we can search for the Interest
Regions (IRs) called the objective signals or modal domains
in which the main energy is concentrated. IR blocks of the
multi-modals signal are to segment into single modal. For
example, see the red triangles marked in Fig. 6c.
(3) Each objective signal can be reconstructed from one IR block
by Eq. (21). The reconstructed modal signals shown in
Fig. 6eh are also desirable, where there are three pathways
to direct them.
FSWT provides a new approach where both filtering and seg-
menting can be processed simultaneously in time and frequency
domain. Whether signal filtering or signal segmentation, IRs still
needs to be found. Many image segmentation methods can beimplemented to determine the IRs. This paper ignores the discus-
sion, but a robust real time object detection method [19] is
recommended.
As stated above, due to the localization of FDR with FSWT anal-
ysis, the modal domain can be found from the FSWT image. There-
fore, we here introduce the timefrequency projectors of the FSWT
image to determine the modal domains.
(1) Frequency projector of the FSWT coefficients can be stated
as
Pfx ZTs
0
jWs;x;rjds; 27
where Ts is the sample time.
For example, there is a frequency projector shown in Fig. 7a. It is
clear that each peak of this curve points to the maximum fre-
quency response. We can use those peaks to segment all the
frequencies into a number of slices, and denote the FrequencySlices as
20 Sec.
40
00 10 0 10
120
80
0 10 20 Sec.
-1.5
1.5
0
Time(Sec.)
20
0 5 20Hz0
10
150
0
FSWT
Noise=25%
Magnitude
Amplitude
20 Sec.
40
0
120
80
Magnitud
e
Fig. 4. (a) A noise version (+25%) of the original signal shown in Fig. 2a. (b) 2D map of FSWT coefficients. Compare with Fig. 3a and b under high noise, (c) and (d) are a group
of slices of modal signal s at 5 Hz shown in (a) and (b) with slice functions ^px e1=2x2 and ^px 1=1 x2 respectively.
0 5
40
0
20 Sec.1510
120
80
40
0
120
80
Ma
gnitudeofFSWT
Coefficients
0 5 10 15 20 Sec.
Fig. 3. Compares two groups of frequency slices of FSWT representation. The original signal is shown in Fig. 2a. (a) The first group is sliced by function ^px e1=2x2 , redcolor is the slice of frequency f= 5 Hz, green one is f+ 0.4 Hz, and blue one is f 0.4 Hz. (b) is the same group of slices but with another function ^px 1=1 x2. (Forinterpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
40
0
120
80
0 5 10 15 20 Sec.
40
0
120
80
T
ime(Sec.)
Time(Sec.)
0 5 10Hz
150
0
MWT
Noise=25%
0 5 10 15 20 Sec.
Magnitude
Magnitude
0 5 10Hz
20
0
10
20
0
10
Fig. 5. Compare the MWT with the FSWT for same parameters gp = 0.025 and gw = 0.025; (a) 2Dmap of the MWT ofFig. 2a; (b) 2D mapof the MWT ofFig. 4a; (c) and (d) aretwo groups of slices of modal signal s at 5 Hz shown in (a) and (b).
20 Z. Yan et al. / Computers and Structures 89 (2011) 1426
7/28/2019 (2011)-Yan,Miyamoto,Jiang - Frequency Slice Algorithm for Modal Signal Separation and Damping Identification
8/13
x0;x1; x1;x2; . . . xn1;xn
Here the segments are not strict, but it is necessary that each slice
include the main energy of one modal signal. One of the simplest
choices is [x Dx, x +Dx] and Dx = kx, where k is recom-mended to select 0.10.2 [14].
(2) Time projector of FSWT can be stated as
Ptt Zxi1
xi
jWt;x;rjdx 28
Here [xi, xi+1] is a frequency slice and it may include many nearfrequency signals.
For example, there is a time projector shown in Fig. 7b.It is clear
that a peak of this curve points to the start time of the signal. We
call these peaks the time trigger points of the signal. In general, a
real signal may have a number of time trigger points. We can di-
vide the sample time into many Time Slices, and denote them as
t0; t1; t1; t2; . . . tm1; tm
Here the time segments are not strict too, but it is necessary
that each time slice includes the main energy of one trigger re-
sponse signal.
As stated above, we can summarize the modal analysis flow as
Fig. 8. According to the flow and Eq. (24b) or Eq. (12b), we can pro-vide the following algorithm for damping parameter identification.
4.4. Algorithm for modal parameter identification
Algorithm 1 is called the frequency slice algorithm (FSA) for
modal parameter identification. After Step 1 in FSA, one can esti-
mate the modal damping directly by Eq. (24b). However, before
clear out other modal frequencies for MDOF signal, Eq. (24b) is
only an approximate estimator. The latter Example 3 will show
the difference between Eqs. (12b) and (12b). At the same time,
we will show a comparable example that Eq. (24b) can directly
give an acceptable estimation of damping for random response.
Therefore, from Step 2 to Step 4 in FSA is somewhat necessary to
separate MDOF response into each single modal signal for getting
higher accuracy damping ratio in application, especially for closemodes with high damping ratio.
1
0
2b
1500
100
150
0
50
02
46
0 50100
150
d
200
Time(Sec.)
1
3
5
0 40 80 120 160 HZ
c
a
0
2
4
0 1 2 3 4 Sec.
0
1
-1
0
0.5
-0.5
0
0.5
-0.5
0
0.5
-0.5FSWTReconstructed
e
f
g
h
Synthesize
dSignal
FSWT
=28.28
0 1 2 3 4 Sec.
0
0 40 80 120 160 Hz
MWT
1
FFT
Spectrum
-1
Amplitude
Time(Sec) Frequenc
y(Hz)
Fig. 6. (a)A test signalin LBMS; (b)Fourier spectrum;(c) and(d) arethe 2D and3D maps of FSWT coefficients respectively; (e)(g) are thereconstructed signalby theinverse
FSWT; (h) The synthesized signal.
0 50 100 150 200
50
0
150
100
40
60
100
80
20
00 1 2 3 4 5
Fig. 7. (a) and (b) are the frequency and time projectors of the FSWT coefficients in Fig. 6c, respectively.
RDT Estimation of
Damping directly in
FSWT Domain
RDT Method
of damping for
SDOF Signal
FSWT
Time-Frequency
Transform
Determine Modal
Domain andModal Separation
Inverse FSWT
Transform
Time Domain
Data of
MDOF Signal Projectors of Time-
Frequency Domain
Fig. 8. Modal separation and damping identification with FSWT.
Z. Yan et al. / Computers and Structures 89 (2011) 1426 21
7/28/2019 (2011)-Yan,Miyamoto,Jiang - Frequency Slice Algorithm for Modal Signal Separation and Damping Identification
9/13
Algorithm 1. Frequency slice algorithm (FSA) for modal
parameter identification.
Step 1. Input the signal data, and compute the FSWT
coefficients by Eq. (17)
Step 2. Determine frequency slices [xi,xi+1] of each modalsignal with the frequency projector of the FSWT coefficients
in Eq. (27)Step 3. Determine time slices [tj,tj+1] of each modal signal with
the time projector of the FSWT coefficients in Eq. (28)
Step 4. Reconstruct the modal signal on each IR ([tj, tj+1][xi,xi+1]) by FSWT Eq. (21) and compute the modal dampingparameter by Eq. (12b)
5. Application
5.1. Modal separation of close modal signal based on dynamic scale
In this section, an extreme example is to show the ability of the
FSWT. We will compare the FSWT with the CWT-based method for
modal separation in [13], where they have given a significant re-search about the selections of the wavelet scale and the centre fre-
quency based on the minimum Shannon entropy search by using
the MWT formula described by Eq. (26).Example 3 We consider a
simulated signal
ft e0:5ptsin9:987pt e0:55ptsin10:986pt t! 2
0 t< 2
(;
where there are a couple of signals f1 = 5 Hz, f1 = 0.05 and f2 =
5.5 Hz, f2 = 0.05. The signal f(t) is shown on Fig. 9a where we set
fs = 100 Hz and Ts = 6 s .
Although Example 3 is somewhat similar with the example in
[5], it is much more difficult than [5] to separate them because
their resonant frequencies are very close and the signal has high
damping ratio that cause the valid duration time of measured sig-
nal to be very short. The Fourier spectrum of the signal without
noise is shown in Fig. 9b. Fig. 9c and d show the time response
and spectrum of the original signal by superposition of a mild noise
(+15%). It is evident that two resonant peaks are difficult to recog-
nize from the Fourier spectrum.
All FSWT parameters are also determined as the same as Eq.
(25). Note that j = 28.28 is a constant. Fig. 9e represents thetimefrequency image of the FSWT coefficients. However, it is still
not easy to observe the different modes from that image. The fre-
quency projective curve of Fig. 9e is shown in Fig. 9f, where itseems to be clear that there are two frequency signals at
f1 = 5 Hz and f2 = 5.5 Hz. Meanwhile, we cannot find out another
suitable scale j to distinguish them even if we completely sacrificethe time resolution in Fig. 9e.
0
6
Time(Sec.)
2
4
4.0 4.5 5.0 5.5 6.0 Hz0
6
Time(Sec.)
2
4
4.0 4.5 5.0 5.5 6.0 Hz
FSWT =28.28/2 ~28.28*2MWT 35.8,15.1 ==
0 1 2 3 4 5 Sec.
0
1
-1
0 1 2 3 4 5 Sec.
0
1
-1
150
0
0 1 2 3 4 5 Sec.
0
2
-2Simulated
Signal
0 2 4 6 8 Hz
20
40
0
FFT
Spectrum
0
6
Time(Sec.)
2
4
4.0 4.5 5.0 5.5 6.0 Hz4.0 4.5 5.0 5.5 6.0 Hz
150
250
350
50
FSWT =28.28
0 2 4 6 8 Hz
20
40
0
FT
Spectrum
Noise
Version
0
2
-20 1 2 3 4 5 Sec.
60Noise (+15%)
80
a b
c d
e f
g h
i j
Fig. 9. (a)Simulated signal; (b)Fourier spectrum of (a); (c)A noise version (+15%)of (a)and thefollowings (d)(j) areshown to analyze thenoise version signal(c). (d)Fourier
spectrum; (e)2D mapof FSWT with scale j = 28.28; (f) frequency projector of (e); (g)2D map of theMWT after thetime-consumingiterations for minimum Shannon entropy
search; (h) 2D maps of FSWT with dynamic scale j j0 ^fu = ^fx , where j0 = 28.28/2; (i) and (j) are the reconstructed signals from the IRs shown in (h) respectively.
22 Z. Yan et al. / Computers and Structures 89 (2011) 1426
7/28/2019 (2011)-Yan,Miyamoto,Jiang - Frequency Slice Algorithm for Modal Signal Separation and Damping Identification
10/13
Similarly, the CWT-based method of[13] has the same problem.
Fig. 9g shows the hard effort by using the MWT in Eq. (26), where
the bandwidth parameter a = 4.5 and the center frequency b = 6are the estimated parameters after the time-consuming iterations
for first computing the MWT coefficients and then getting and
comparing the Shannon entropy. Unfortunately, it cannot distin-
guish the two close frequency components correctly. Moreover,
both of the time and frequency resolutions or localizations of
Fig. 9g are much lower than that of Fig. 9e. Meanwhile the result
of the MWT is very sensitive to noise. We have no better decision
for a wavelet scale that is able to separate them. Naturally, it is
very difficult to obtain high accuracy estimations for the damping
ratios.
Fortunately, with the same FSF as Eq. (25), scale j can be chan-ged dynamically in terms of Eq. (22) where we not only still use
j0 = 28.28 as Eq. (25), but also should take the frequency projectorFig. 9(f) instead of the Fourier spectrum Fig. 9d since the real spec-
trum Fig. 9b is drown in the noise. Consequently, Fig. 9h indicates
that there are a couple of signals aligned separately, and it is obvi-
ous that there are two maximum response frequencies nearby
f1 = 5 Hz and f2 = 5.5 Hz. Therefore, we can easily separate them
into the single modal shown in Fig. 8i and j, where some errors
especially in the initial stages are due to the noise and the fact that
the segmentations of signal IRs cannot be refined. However, the
reconstructed signals are acceptable. Actually, in order to obtain
the similar result to Fig. 9h, we have many selections resembling
to Eq. (22), but here we omit the further discussion.
Finally, the damping ratios estimated directly by Eq. (24b) are
f1 = 0.0453 and f2 = 0.0558, and the errors are not over 13%. How-
ever, after the segmentation, the most accurate damping ratios ob-
tained by Eq. (12b) are f1 = 0.0483 and f2 = 0.0532, and the errors
are not over 6.5%. The difference between Eqs. (24b) and (12b)
are mainly due to that the signal f(t) is a MDOF response before
it is separated, but Eq. (12b) is accurate whenf(t) is segmented into
SDOF signal.
Remark. This example provides a simple approach to overcome
the problem of close modes decomposition by using dynamic scale
technique, especially for high damping ratio signal. At the same
time, usually Eq. (24b) can directly give enough accuracy for
application. Nevertheless, if one want to get higher accurate
estimation of damping for MDOF response, in the first step, it is
somewhat necessary that separate the MDOF signals into single
modal signal, and Eq. (12b) further gives a higher accurate
estimation.
5.2. Digital simulation and comparison
In this section, all of the FSWT parameters are still assumed in
Eq. (25). To compare with Refs. [13,14], we assume that the simu-
lated signal is under the same conditions with Ref. [14], where
f1 = 1 H z , f1 = 0.03, f2 = 1.1 Hz, f2 = 0.02, f3 = 3 Hz, f3 = 0.01; all of
phase angles are 0, and fs = 20 Hz. Table 3 shows the identified re-
sults of modal parameters; note that the sample time Ts is de-
creased to 9 s, which is shorter than Ts = 12 s of reference [14].
From Table 3, the FSA algorithm can maintain the high accurate
estimation for modal parameters even though with stronger noise
and shorter sample time compared with Refs. [13,14]. Therefore,
FSA can be implemented to identify the modal parameters of a sys-
tem with high damping and close modal frequencies.
5.3. Experimental verification
Fig. 10 shows a small laboratory bridge monitoring system
(LBMS). There are 11 sensors of piezoresisitive ARF-10A accelera-
tion (flat frequency response: 050 Hz) installed under the main
beams. DC-104R is applied to collect the free-decay responses by
an impact with a light hammer, where fs = 1000 Hz and Ts = 5 s .
The measured free-decay signals are ready for identifying the mod-
al parameters. The FSA is applied to compute the damping. Two
examples are given to test the FSA method, and the single impulse
response and random excitation are presented as below.
5.3.1. Single impulse response
We collect the acceleration signals from LBMS by single impulse
with a hammer. Three responses at locations B1, C1 and D1 andtheir FSWTs are shown in Fig. 11, where all of the FSWT parame-
ters are still assumed in Eq. (25). The first three main modes are:
f1 = 29.3 Hz, f2 = 112.3 Hz, and f3 = 164.0 Hz, which are almost the
same to all observing sensors.
Table 3
Identification of modal damping ratio using FSA for MDOF response with different noise levels.
Noise level (R) Modal parameter Sample time (s)
Ts = 12 Ts = 12 Ts = 9
CWT-based [13] CWT-based & search [14] Proposed FSA, statistic times = 100, modal parameter average and variance
E() Var() E() Var()
5% f1 0.989 1.0 0.9804 1.2e
32 0.9804 1.2e
32
f1 0.0131 0.0302 0.0300 7.5e08 0.0338 1.3e07f2 1.111 1.1 1.1176 7.2e30 1.1176 7.2e30f2 0.0117 0.0198 0.0188 2.6e08 0.0200 6.5e08
f3 3.00 3.00 3.00 0 3.00 0
f3 0.0096 0.0099 0.0099 1.1e08 0.0099 1.7e0820% f1 0.989 1.001 0.9804 1.2e32 0.9802 3.8e6
f1 0.0132 0.0296 0.0301 9.2e07 0.0339 2.2e06f2 1.11 1.1 1.1176 7.2e30 1.120 5.4e30f2 0.0118 0.0209 0.0187 4.3e07 0.0202 1.0e06
f3 3.002 3.00 3.00 0 3.00 0
f3 0.009 0.01 0.01 1.7e07 0.01 3e07+50% f1 0.9825 1.5e05 0.9712 1.0e4
f1 0.0302 3.1e05 0.0330 4.0e05f2 1.1089 5.2e05 1.1178 3.8e06f2 0.0191 1.3e05 0.0200 1.5e05
f3 3.001 4.2e05 3.002 4.7e05f3 0.0102 3.4e06 0.0101 3.7e06
Z. Yan et al. / Computers and Structures 89 (2011) 1426 23
7/28/2019 (2011)-Yan,Miyamoto,Jiang - Frequency Slice Algorithm for Modal Signal Separation and Damping Identification
11/13
The first three computed modals (i.e. the high-energy responses
pointed by red arrows in Fig. 11) for nine observing positions are
presented in Table 4. From Table 4, note that the first mode damp-
ing ratios and frequencies are almost same for the observing posi-
Fig. 10. LBMS: (a) Sensor location; (b) experimental model.
Single pulse responses
Amplitude
100
150
0
50
02
6
050
100150 2004
150
0
FSWT
=28.28
Amplitude
100
150
0
50
02
6
0 50100
150 2004
FSWT
=28.28
Accele
ration(g)
0
-1
1
0 1 2 3 4 Sec.
At B1
Acceleration(g)
0
-1
1
0 1 2 3 4 Sec.
At C1
Acceleration(g)
0
-1
1
0 1 2 3 4 Sec.
At D1
0
Amplitude
100
150
0
50
02
6
50100
1502004
FSWT=28.28
ab
cd
e f
Time(Sec) Frequency(Hz
)
Time(Sec) Frequency(Hz
)
Time(Sec) Freq
uency(Hz)
Fig. 11. (a), (b) . . .(f) are the obtained signals at positions B1, C1 and D1 and their FSWT 3D maps, red arrows point to the first three modals computed in Table 4. (For
interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Table 4
Identification of modal parameter using FSA algorithm for free-decay responses of the LBMS.
State Modal parameter Ts = 5s, fs = 1000 Hz, proposed FSWT, statistic times = 10
Modal parameter average
B1 B2 B3 C1 C2 C3 D1 D2 D3
Single impulse test f1 29.27 29.27 29.27 29.27 29.27 29.27 29.27 29.27 29.27
f1 0.0065 0.0065 0.0065 0.0068 0.0067 0.0067 0.0065 0.0066 0.0065
f2 112.36 112.36 112.36 112.36 112.36 112.36 112.38 112.36 112.36
f2 0.0093 0.0094 0.0095 0.0097 0.0095 0.0096 0.0099 0.0094 0.0095
f3 164 164 164 164.03 164.05 164.04 164 164.05 164
f3 0.0028 0.0030 0.0030 0.0037 0.0036 0.0036 0.0033 0.0030 0.0031
24 Z. Yan et al. / Computers and Structures 89 (2011) 1426
7/28/2019 (2011)-Yan,Miyamoto,Jiang - Frequency Slice Algorithm for Modal Signal Separation and Damping Identification
12/13
tions B1, B2, and B3, C1,C2 and C3, D1, D2and D3 in the simple test.
From the Fig. 11 and Table 4, it can be concluded that FSWT and
FSA methods are steady.
5.3.2. Random impacting response
Fig. 12 shows an acceleration response of LBMS under random
impacts to compare the single impulse described as the above.
The measured response is equivalent to the combining of multipleFDRs. Usually, for most of the existing approaches to computing
damping of random response, as the first step, it is necessary to
compute the FDR by RDT, and then one can compute the modal
parameters (e.g.[1]). However, by using frequency slice projector
of FSWTshown in Fig. 12b, we can choose each maximum response
frequency slice (see the dot bars in Fig. 12b) as each modal signal,
then the RDT formula Eq. (24b) can be used to directly compute the
damping of random response. In fact, FSWT or Eq. (10) translates
the response in time domain into the distribution in timefre-
quency domain. It is important that FSWT divides the signal from
the noise spectrum automatically. By using Eq. (24b) directly for
random response, Table 5 gives the computational damping of
the chosen maximum response slices, and their frequencies are
very similar to Table 4. The maximum frequency error is not over5% comparing with Table 4, and the maximum damping error is
not over 17%. Nevertheless, note that we only use 5 s time data
for comparable computation with Table 4. At the same time, many
tests show that the higher frequency modal signals have higher
damping errors by Eq. (24b). The main reason is probably that
the duration time of those modal signals is very short.
From the comparison between the single impulse and the ran-
dom excitation, FSWT and FSA can estimate the modal damping di-
rectly for MDOF signals with an acceptable accuracy. Moreover,
more applications of FSWT and FSA especially for damage detec-
tion will be carried on in the near future.
5.3.2.1. Comparing and remarks. The main theoretical advantage of
the proposed approach over other existing approaches based onWT is that its measured responses processed are not limited to
free-decay responses, it is also unnecessary to use a filter to de-
noise in practical application because FSWT itself is a new kind
of good filter [5,17]. At the same time, applying the proposed ap-
proach to determine the modal parameters of a system from its
ambient measurements does not require the technique such as
the random decrement technique to convert the random responses
into free-decay responses. Although there are many damping iden-
tifying methods [715] such as NExT, ERA, SSI and Wavelet etc., as
the first step, usually, it is necessary to compute the FDR by RDT.
However, in this paper, a new usage of the RDT idea can be imple-
mented directly in FSWT domain, Eq. (24b) or Eq. (12b) can be used
for random responses for getting damping parameter, and the
FSWT is therefore very simple and direct.
Notably, in Section 5.2, we have given a comparable result with
CWT in [13,14]. FSA can also give higher accurate estimation even
with higher damping and shorter sampling time since FSWT can
provide a modal separation method by dynamic scale controlling.
On the other hand, unlike the existing approaches based on CWT,
the wavelet function and the chosen scale parameter can always
affect the accuracy of the identified modal parameters, FSWT can
provide an adaptive method for the center frequency and an adap-
tive window scale for different frequency responses. Although this
work only demonstrates the feasibility of the proposed approach in
processing the responses of laboratory system by hammer impact-
ing excitation, the proposed methods are certainly suitable for
dealing with free-decay responses or the MDOF responses from
the random loads. After segmentation and reconstruction of FSWT,
Eq. (12b) can therefore give higher accuracy than Eq. (24b) directly
even when the responses and input included great noise with 50%.
Nevertheless, in this case, usually, Eq. (12b) should be based on the
FDR signal similar with many existing approaches.
6. Conclusion
(1) The background of the powerful FSWT method is first intro-
duced clearly. By using RDT, a good damping estimation in
FSWT domain is obtained in this paper. Combining RDT
0 40 80 120 160 Hz
Acceleration
(g)
0
-1
1
0 1 2 3 4 Sec.
100
150
0
50
0
24
050
100150
6
FSWT
=28.28
80
0
120
40200
Magnitude
Time(Sec) Frequenc
y(Hz)
a b
c
Fig. 12. (a) Random impacting response, (b) FSWT 3D map, (c) frequency projective curve of FSWT and maximum response frequencies.
Table 5
Identification of modal parameter by using Eq. (24b) directly for random responses of LBMS.
State Modal parameter Ts = 5s, fs = 1000 Hz, Proposed FSWT, Statistic Times = 10
Modal Parameter Average
B1 B2 B3 C1 C2 C3 D1 D2 D3
Random impacting test f1 29.72 29.61 29.72 29.62 29.74 29.72 29.75 29.72 29.71
f1 0.0071 0.0069 0.0057 0.0065 0.0062 0.0070 0.0070 0.0068 0.0061
f2 113.25 113.12 113.25 113.23 113.22 113.22 113.34 113.32 113.25
f2 0.0087 0.0103 0.0106 0.0101 0.0103 0.0105 0.0108 0.0107 0.0103
f3 165.05 165.04 164.64 164.56 164.60 164.85 164.85 164.85 165.05
f3 0.034 0.0035 0.0033 0.0031 0.0032 0.0035 0.0034 0.0036 0.0034
Z. Yan et al. / Computers and Structures 89 (2011) 1426 25
7/28/2019 (2011)-Yan,Miyamoto,Jiang - Frequency Slice Algorithm for Modal Signal Separation and Damping Identification
13/13
and FSWT modal separation method, a frequency slice algo-
rithm (FSA) is designed to get high accurate estimation of
damping parameter. Both numerical and experimental tests
demonstrate that FSA is effective to identify modal parame-
ters of a system response even with large damping and close
modal interference, and strong noise. The computational
results of FSA for modal parameters are accurate and steady.
(2) FSA is not limited to FDR, and also can be directly used to
random impacting response. It is usually a good estimator
for general applications. Notably, FSWT itself is a new kind
of good filter and has high performance against noise. It is
necessary to get damping parameter with higher accuracy
through modal separation by FSWT, and FSWT can be con-
trolled adaptively in modal separation by dynamic scale
method. FSWT is adaptive to analyze the damping vibration
signals in time and frequency domains simultaneously.
(3) Compared with the results obtained from the traditional FFT
and CWT etc., the FSWT method can show the damping
characteristics of modal signal in timefrequency domain
more clearly and easily. Only one group of FSWT parameters
in Eq. (25) is used to fit all examples in this paper, therefore
FSWT presents strong robustness in transformation of vibra-
tion signal in an easy way. Frequency slicing processing is an
important idea in this application.
Appendix A. Proofs
A.1. The proof of Theorem 1
Since Eq. (6) can be changed into
Wt;x;r rZ1
1
fs teixsprsds;
and from the condition s < 0, p(s) = 0, it can be rewritten as
Wt;x;r rZ1
0
fs teixsprsds
According to the definition of f(t), the additional condition t! t0and bT= kp, it is easy to know that f(s + t+ T) = eaTf(s + t).Moreover
jWt T;x;rj eaTjWt;x;rj
a 1
Tln jWt;x;rj ln jWt T;x;rj:
Thus, we have proven Eq. (23) and (Eq. (24a). Finally, using RDT
method, we can obtain (24b).
References
[1] Sohan H, Farrar CR, Hemez FM, ShunkDD, Stinemates DW, Nadler BR. A review
of structural health monitoring literature: 19962001. Los Alamos National
Laboratory report, LA-13976-MS; 2003.
[2] Doebling, SW, Farrar CR. The state of the art in structural identification of
constructed facilities. A report by the ASCE Committee on Structural
Identification of the Constructed Facilities; 1999.
[3] Ndambi JM, Peeters B, Visscher JDe, Wahab MA, Vantomme J, Roeck GDe, et al.
Comparison of techniques for modal analysis of concrete structures. Eng Struct
2000;22:115966.
[4] He X, Moaveni B, Conte JP, Elgamal A. Comparative study of system
identification techniques applied to New Carquinez Bridge, ; 2006.
[5] Yan Z et al. Frequency slice wavelet transform for transient vibration response.
Mech Syst Signal Process 2009;23(5):147489.
[6] Bendat JS, Piersol AG. Engineering applications of correlation and spectral
analysis. second ed. New York: John Wiley; 1993.
[7] Bodeux JB, Golinval JC. Application of ARMAV models to the identification and
damage detection of mechanical and civil engineering structures. Smart Mater
Struct 2001;10:47989.
[8] James GH, Garne TG, Lauffer JP. The Natural excitation technique (NExT) for
modal parameter extraction from operating wind turbines. Report SAND92-
1666, UC-261. Albuquerque, New Mexico: Sandia National Laboratories; 1993.
[9] Qin Q, Li HB, Qian LZ. Modal identification of Tsing Ma bridge by usingimproved Eigensystem realization algorithm. J Sound Vib 2001;247:32541.
[10] Lardies J, Gouttebroze S. Identification of modal parameters using the wavelet
transform. Int J Mech Sci 2002;44:226383.
[11] Kijewski T, Kareem A. Wavelet transforms for system identification in civil
engineering. Comput-aided Civil Infrastruct Eng 2003;18:33955.
[12] Neild SA, McFadden PD, Williams MS. A reviewof timefrequency methods for
structural vibration analysis. Eng Struct 2003;25:71328.
[13] Yan BF, Miyamoto A, et al. Wavelet transform-based modal parameter
identification considering uncertainty. J Sound Vib 2006;291:285301.
[14] Tan Jiu-Bin, Liu Yan, Wang Lei, Yang Wen-Guo. Identification of modal
parameters of a system with high damping and closely spaced modes by
combining continuous wavelet transform with pattern search. Mech Syst
Signal Process 2008;22:105560.
[15] Huang CS, Su WC. Identification of modal parameters of a time invariant linear
system by continuous wavelet transformation. Mech Syst Signal Process
2007;21:164264.
[16] Hong J-C, Kim YY. The determination of the optimal Gabor wavelet shape for
the best timefrequency localization using the entropy concept. Exp Mech
2004;44:38795.[17] Yan Z et al. An overall theoretical description of frequency slice wavelet
transform. Mech Syst Signal Process 2010;24(2):491507.
[18] Asmussen JC. Modal analysis based on the random decrement technique
application to civil engineering structures. Ph.D thesis. Denmark: University of
Aalborg; 1997.
[19] Viola P, Jones M. Robust real time object detection. In: IEEE international
conference on computer vision workshop on statistical and computational
theories of vision. Vancouver, Canada; July 13, 2001. p. 90510.
26 Z. Yan et al. / Computers and Structures 89 (2011) 1426
http://www.health.monitoring.ucsd.edu/documentation/publichttp://www.health.monitoring.ucsd.edu/documentation/publichttp://www.health.monitoring.ucsd.edu/documentation/publichttp://www.health.monitoring.ucsd.edu/documentation/public