(2011)-yan,miyamoto,jiang - frequency slice algorithm for modal signal separation and damping...

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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.2010.07.011


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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


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


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



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.



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.



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).



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.



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.



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



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



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



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).

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