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Experimental investigation on the time-varying modal parameters of a trapezoidal plate in temperature-varying environments by subspace tracking-based method

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Subspace-based methods for estimation of modal parameters are briefly reviewed in this study and a time-varying modal parameter identification algorithm, based on finite-data-window Projection Approximation Subspace Tracking, is presented to investigate the time-varying modal parameters of a trapezoidal titanium-alloy plate in temperature-varying environments. An experiment conducted on a steel beam with a removable mass is used to confirm the proposed method with a brief discussion on the factors of this method. Two groups of experiments are conducted to reveal the effects of varying temperature and heating speed on the natural frequencies of the plate, and the identified natural frequencies evidently show the effect of thermal stresses caused by temperature gradients in experiment.
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Article
Experimental investigation on the
time-varying modal parameters
of a trapezoidal plate in
temperature-varying environments
by subspace tracking-based method
Kaiping Yu, Kai Yang and Yunhe Bai
Abstract
Subspace-based methods for estimation of modal parameters are briefly reviewed in this study and a time-varying modal
parameter identification algorithm, based on finite-data-window Projection Approximation Subspace Tracking, is
presented to investigate the time-varying modal parameters of a trapezoidal titanium-alloy plate in temperature-varying
environments. An experiment conducted on a steel beam with a removable mass is used to confirm the proposed
method with a brief discussion on the factors of this method. Two groups of experiments are conducted to reveal the
effects of varying temperature and heating speed on the natural frequencies of the plate, and the identified natural
frequencies evidently show the effect of thermal stresses caused by temperature gradients in experiment.
Keywords
Natural frequency, temperature-varying environment, identification algorithm, modal parameter, Projection
Approximation Subspace Tracking (PAST)
1. Introduction
Hypersonic unmanned vehicles, such as missiles and
rockets, experience dramatically temperature-varying
fields, and the elasticity modulus and Poisson’s ratio
are temperature-dependent (Jeon et al., 2011; Kehoe
and Synder, 1991; Kehoe and Deaton, 1993) while
thermodynamic effects are frequently ignored in litera-
ture. Simultaneously, the studies (Avsec and Oblak,
2007; Hios and Fassois, 2009; Marques et al., 2002)
have reported that even a slightest temperature
change would result in huge alteration of the modal
parameters because a slightest temperature change
would cause severe stress when structures are over-con-
straint. Even though the coupled thermo-elastic
dynamics (Guo et al., 2009, 2011) is discussed, the
model is just sufficient to analyze a beam in the tem-
perature-constant environment. To the authors’ know-
ledge, no literature can be found on the effect of
continuously varying temperature on modal parameters
of structures in temperature-varying environments.
Since structural dynamics is important and the modal
analysis can provide an insight into structural dynam-
ics, which is widely used in health monitoring (Liu
et al., 2011;Verboven et al., 2004; Whelan et al.,
2011), damage detection (Banan and Mehdi-pour,
2007; Niemann et al, 2010) and so on, it is necessary
to process response signals as an inverse problem
(Poulimenos and Fassois, 2006, 2009) for estimation
of time-varying modal parameters. However, the con-
ventional modal parameters are invalid for time-vary-
ing systems, so ‘pseudo modal parameters’ and the
subspace-based identification algorithm (Liu, 1999;
Liu and Deng, 2006) are proposed by adopting ‘time
Department of Astronautical Science and Mechanics, Harbin Institute of
Technology (HIT), Harbin, People’s Republic of China
Corresponding author:
Kaiping Yu, Department of Astronautical Science and Mechanics, Harbin
Institute of Technology (HIT), PO Box 304, No.92 West Dazhi Street,
Harbin 150001, People’s Republic of China.
Email: yukp@hit.edu.cn
Received: 29 September 2013; accepted: 20 December 2013
Journal of Vibration and Control
2015, Vol. 21(16) 3305–3319
!The Author(s) 2014
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DOI: 10.1177/1077546314521445
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frozen’ technique. In terms of the ‘time frozen’ tech-
nique, the modal parameters are assumed to slowly
change, so the subspace-based identification algorithm
appears to handle a series of time-invariant models con-
structed by the response signals. Pang et al. (2005) pro-
posed a revised version of the algorithm proposed by
Liu (1999) and Liu and Deng (2006). However, the
methods by Liu (1999) and Liu and Deng (2006) and
Pang et al. (2005) are not suitable to track modal par-
ameters on-line, because the algorithms require many
groups of experiments under different excitation or ini-
tial conditions.
Another subspace-based algorithm for estimation of
time-invariant modal parameters (Bosse et al., 1998;
Tasker et al., 1998), requiring only one experiment,
takes the advantage of on-line identification. The algo-
rithm herein could find its origin in 4SID (subspace-
based state-space system identification) algorithm
(Kim and Lynch, 2012; Overschee and Moor, 1994)
and 4SID method, directly using the measured signals,
provides numerically reliable state-space models for
complex dynamical systems. In the subspace-based
modal parameter identification algorithm (Bosse
et al, 1998; Tasker et al, 1998), Singular Value
Decomposition (SVD) is used to construct signal sub-
space while SVD requires a large compute load and
memory space. So an algorithm for time-varying
modal parameter extraction (Pang et al, 2005) is devel-
oped by adopting Projection Approximation Subspace
Tracking (PAST) (Yang, 1995) instead of SVD for a
lower compute load and memory space. PAST converts
a high-order unconstrained minimization problem into
a lower-order one by projecting subspace matrix on
signal vector and then the optimization problem can
be solved by the recursive least squares (RLS) tech-
nique. However, RLS suffers the problem of data sat-
uration, consequently leading to PAST losing its
tracking ability. Motivated by Ding’s work (Ding and
Xiao, 2007), in which the finite-data-window least
squares technique (Ljung, 1999) is employed to solve
the problem of data saturation, finite-data-window
PAST is derived and applied in the time-varying
modal parameter identification algorithm based on sub-
space tracking.
The correspondence is organized as follows. Section
2 states the time-varying modal parameter identifica-
tion algorithm based on the finite-data-window
PAST, and an experiment conducted on a steel canti-
lever beam is used to confirm the identification method
with a brief discussion on the factors of the proposed
method. Thermal effect on the natural frequencies of a
trapezoidal TA15 titanium-alloy plate in temperature-
varying environments is discussed in section 3.
Conclusions and further investigations are drawn in
section 4.
2. Modal parameter extraction based
on subspace tracking
2.1. Updating the discrete input and
output vectors
The discrete state-space model of an n/2-order linear
time-invariant dynamic system is expressed as follows
xkþ1ðÞ¼Ax kðÞþBu kðÞ
ykðÞ¼Cx kðÞþDu kðÞ
ð1Þ
where xðkÞ2Rn1is the state vector at the k
th
sampling
instant, A2Rnnis the system state matrix, B2Rnm
is the input matrix, uðkÞ2Rm1is the input vector ,
C2Rrnis the output matrix , D2Rrm,yðkÞ2Rr1
is the output vector. The aim herein is to estimate the
system state matrix Aby the discrete input and output
vectors. As stated above, the time frozen technique is
adopted to define the ‘pseudo modal parameters’, by
which the instantaneous natural frequency is estimated
as the mean value of the identified one in a short time,
namely the time-varying system is treated as a time-
invariant system in the short time.
Constructing Hankel matrices with the discrete input
and output vectors respectively, we have
UN¼
u1ðÞ u2ðÞ  uNðÞ
u2ðÞ u3ðÞ uNþ1ðÞ
.
.
.
 ..
..
.
.
uMðÞuMþ1ðÞuNþM1ðÞ
2
6
6
6
6
6
4
3
7
7
7
7
7
5
ð2Þ
YN¼
y1ðÞ y2ð Þ  yNðÞ
y2ðÞ y3ð Þ  yNþ1ðÞ
.
.
..
.
...
..
.
.
yMðÞyMþ1ð Þ  yNþM1ðÞ
2
6
6
6
6
6
4
3
7
7
7
7
7
5
ð3Þ
Then
YNU?
N¼YNYNUT
NUNUT
N

1UN¼!tðÞXU?
Nð4Þ
where U?
N¼InUT
NðUNUT
NÞ1UN,In2Rnnis an iden-
tity matrix, !ðtÞ¼ CCAðtÞ CAðtÞM1

Tis the
generalized observability matrix, X¼xð1Þ
xð2ÞxðnÞ is a state-vector matrix, superscript ‘‘T’’
denotes matrix transpose.
It would cost a large computation load and memory
space if SVD is used to track signal subspace. If
YNU?
NYT
Ncan be updated recursively, principle sub-
space tracking algorithms can be applied to replace
SVD for a lower compute load and memory space.
3306 Journal of Vibration and Control 21(16)
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According to the matrix analysis theory, the input and
output Hankel matrices can be updated as follows
YNþ1¼YN
yNþ1

,UNþ1¼UN
uNþ1

ð5Þ
where
yNþ1¼½yTðNþ1ÞyTðNþ2Þ yTðN
þMÞT,
uNþ1¼½uTðNþ1ÞuTðNþ2ÞuTðNþMÞT.
We have
YNþ1U?
Nþ1YT
Nþ1¼YNþ1U?
Nþ1YNþ1U?
Nþ1

T
¼YNU?
NYT
NþzNþ1zT
Nþ1ð6Þ
where zNþ1¼½YNUT
NðUNUT
NÞ1
uNþ1
yNþ1=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þNþ1
p,Nþ1¼
uT
Nþ1ðUNUT
NÞ1
uNþ1. In the follow-
ing experiments, the input signals are not collected
because of impulse or white random excitation. So
zNþ1¼
yNþ1, namely the proposed algorithm is used
as an output-only method in this study.
2.2. Subspace tracking and estimation
of modal parameters
As stated above, RLS suffers the problem of data sat-
uration, which would cause PAST to lose its tracking
ability. So the finite-data-window technique is
employed. Similar to the cost function (Yang, 1995),
the truncated cost function is considered.
JWtðÞðÞ¼min X
t
i¼tN
tiziðÞWtðÞWTtðÞziðÞ
2ð7Þ
subjected to WTðtÞWðtÞ¼Ir
where WðtÞ2Rnris the signal subspace, 0 551
is the forgetting factor (Leung and So, 2005; Malik,
2006; Paleologu et al., 2008),
kk
Fdenotes the
Frobenius norm. Assuming WTðtÞzðiÞWTðt1ÞzðiÞ,
and let yðiÞ¼WTðt1ÞzðiÞ, we have
JWtðÞðÞ¼min X
t
i¼tl
tiziðÞWtðÞyiðÞ
2ð8Þ
subjected to WTðtÞWðtÞ¼Ir
Applying RLS method to solve equation (8), we
have
WtðÞ¼Wt1ðÞrJWtðÞðÞr
2JWtðÞðÞ

1ð9Þ
with
rJWtðÞðÞ¼
1
2
@JWtðÞðÞ
@WtðÞ ¼X
t
i¼tl
tiWtðÞyiðÞziðÞ½yTiðÞ
ð10Þ
r2JWtðÞðÞ¼
1
2
@2JWtðÞðÞ
@W2tðÞ ¼X
t
i¼tl
tiyiðÞyTiðÞ ð11Þ
Further by equations (9), (10) and (11), we have
WtðÞ¼Czy tðÞC1
yy tðÞ ð12Þ
with
Czy tðÞ¼Czy t1ðÞþztðÞyTtðÞlzlðÞyTlðÞ ð13Þ
Cyy tðÞ¼Cyy t1ðÞþytðÞyTtðÞlylðÞyTlðÞ ð14Þ
Let ~
PðtÞ¼½Cyy ðt1ÞþyðtÞyTðtÞ1. By the lemma of
verse matrix ðAþxxTÞ1¼A1A1xxTA1
=ð1þxTA1xÞ, we have
~
PtðÞ¼1
C1
yy t1ðÞ
C1
yy t1ðÞytðÞyTtðÞC1
yy t1ðÞ
þyTtðÞC1
yy t1ðÞytðÞ
"#
ð15Þ
Let PðtÞ¼C1
yy ðtÞ,soPðt1Þ¼C1
yy ðt1Þ, further we
have
~
PtðÞ¼1
Pt1ðÞ
Pt1ðÞytðÞyTðtÞPt1ðÞ
þyTtðÞPt1ðÞytðÞ

ð16Þ
PtðÞ¼~
PtðÞþ
~
PtðÞylðÞyTlðÞ~
PtðÞ
lyTlðÞ~
PtðÞylðÞ ð17Þ
Finally we get
WtðÞ¼ Wt1ðÞþztðÞWt1ðÞytðÞðÞ
yTtðÞPt1ðÞ
þyTtðÞPt1ðÞytðÞ

InþylðÞyTlðÞP1tðÞ
lyTlðÞP1tðÞylðÞ

lxlðÞyTlðÞPtðÞ
ð18Þ
And the orthogonalization can be achieved by the
Gram-Schmidt approach. By the definition of !in
equation (4), we have
AtðÞ¼!þ
1tðÞ!2tðÞ¼(tðÞ,tðÞ(tðÞ
Tð19Þ
where !1ðtÞand !2ðtÞare the first M1 block row
and last M1 block row of WðtÞ, respectively.
The superscript ‘‘+’’ in equation (19) denotes
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Moore-Penrose inverse matrix. ,ðtÞ¼diagð1ðtÞ
2ðtÞ2nðtÞÞ and iðtÞ¼
iþnðtÞ. Finally the natural
frequency and damping ratio can be estimated as
!itðÞ¼ln iðtÞ

=2tðÞ,itðÞ¼ln R
itðÞ

=t!itðÞðÞ
ð20Þ
where !iðtÞand iðtÞare the i
th
-order natural frequency
and damping ratio, respectively. R
iðtÞis the real part of
eigenvalue iðtÞ.
2.3. An example
To validate the proposed method, an experiment con-
ducted on a steel cantilever beam of 980508mm
with a removable mass is designed. The experiment is
achieved in three steps. First, test the natural frequen-
cies of the beam with the removable mass. Second,
repeat the first step without the removable mass on
the cantilever beam. Third, sample the acceleration sig-
nals of the beam excited by impulse shock and abruptly
remove the moveable mass in the sampling procedure.
Note that the first and second steps are treated as the
reference of the third step and the sampling frequency is
2048 Hz for the three steps. The experiment equipment
is shown in Figure 1, and Figure 2 displays the natural
frequencies of the first two orders.
By Figure 3, the first-order natural frequency
changes from 7.53 Hz to 8.65 Hz and the second-
order natural frequency changes from 42.1 Hz to
54.1 Hz. The identified natural frequencies shown in
Figure 3 coincide well with that depicted in Figure
2, which confirms the proposed method on estima-
tion of time-varying modal parameters. The reason
for Figure 4 only showing the first-order natural
frequency is that the second-order natural frequency
can’t be accurately estimated. The compare between
Figure 3 and Figure 4 reveals that the forgetting
factor has a great influence on the identified results.
In addition, the authors are adviced to investigate
the effect of the factors n, M, N and on the
Figure 2. Power spectrum density of the acceleration signals; (A) the first step; (B) the second step.
Figure 1. The steel cantilever beam with the removable mass.
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identification method by Monte-Carlo method.
Nevertheless, no laws could be derived, so no dis-
cussions are processed on the effect of the factors in
this study. By the authors’ experience, M is sug-
gested to bear the value of half the sampling
frequency with Nbeing greater than M and the
model order nbeing greater than two times of the
numbers of the active modes. The forgetting factor
is suggested to be close to 1 if the modal parameters
change slowly, otherwise to be close to 0.9.
0 1 2 3 4 5 6 7 8
7.4
7.6
7.8
8
8.2
8.4
8.6
Time (s)
Frequency
ω
1
(Hz)
0 1 2 3 4 5 6 7 8
42
44
46
48
50
52
54
Time (s)
Frequency
ω
2
(Hz)
Ident ifi ed
Fitting
Identified
Fitting
Figure 3. The identified two-order natural frequencies of the cantilever beam when the forgetting factor ¼0:99.
0 1 2 3 4 5 6 7 8
7.4
7.6
7.8
8
8.2
8.4
8.6
Time t/s
Frequency
ω
1/Hz
Identified result
Fitting result
Figure 4. The identified first-order natural frequencies when the forgetting factor ¼1.
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3. Experiments on a trapezoidal
titanium-alloy plate
3.1. The first group of experiments
The surface temperature of the trapezoidal TA15 tita-
nium-alloy plate is collected by thermocouples and then
fed back to the programmable logic controller for con-
trolling the power supply of the far-infrared quartz
heating tubes. Thus a temperature-controlled environ-
ment is provided and the schematic diagram of the
laboratory setup is illustrated in Figure 5. The plate
under the working condition is shown in Figure 6 and
its dimensions are depicted in Figure 7 with the loca-
tions of the exciting point, thermocouples and three
accelerometers (EndevcoÕ6327M70d. The three accel-
erometers can be used in the environment of the tem-
perature not higher than 650C).
Figure 8 shows the natural frequencies of the plate in
six temperature-constant environments by the Peak-
Picking (PP) method. As shown in Figure 8, the natural
frequencies of the higher orders are obviously influ-
enced by temperature while the lower orders are
barely affected. Note that the results by PP method
herein are treated as the reference of the identified
natural frequencies in the following three experiments
conducted in the temperature-varying environments.
To investigate the effect of continuously varying
temperature and heating speed on the natural frequen-
cies, three experiments are conducted. Figure 9 shows
the temperature-controlled records: the surface tem-
perature of the plate is controlled to increase from the
Programmable Logic
Controller The Power Supply
The Data Acquisition
Unit Connector
Quartz heating tubes
Thermcouples
Accelerometors
The Shaker The Clamp
The Plate
Connector
Figure 5. The schematic diagram of the laboratory setup.
Figure 6. The laboratory experimental setup showing the
plate, the three accelerometers, the tubes and the shaker.
3310 Journal of Vibration and Control 21(16)
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045 87 213 300 443 500
0
0.05
0.1
0.15
0.2
Freque ncy/Hz
Amplitu de/dB
044 87 214 298 438 500
0
0.05
0.1
0.15
0.2
Freque ncy/Hz
Amplitu de/dB
043 86 211 29 1 425 500
0
0.05
0.1
0.15
0.2
0.25
Freque ncy/Hz
Amplitu de/dB
043 86 211 291 425 500
0
0.05
0.1
0.15
0.2
0.25
Freque ncy/Hz
Amplitude/dB
043 85 207 287 414 500
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Freque ncy/Hz
Amplitude/dB
043 84 204 281 402 500
0
0.05
0.1
0.15
0.2
0.25
0.3
Freque ncy/Hz
Amplitude/dB
Room Te mperature
(a)
100°C 200°C
500°C
400°C
300°C
Figure 8. The reference experiments in the first group; a) the power spectrum density; b) the first five-order natural frequencies.
Figure 7. The trapezoidal TA15 titanium-alloy plate in the first group of experiments; *— the three accelerometers; «— the
exciting point; 5the five thermocouples.
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22.5 100 200 300 400 500
45
86.88
150
213.8
300
402.5
443.8
Freq uen cy/Hz
(b)
Temperature/°C
Figure 8. Continued.
024 60 90 130 160 180
0
400
500
Time (s)
024 60 90 130 160
0
400
500
Temperature (°C)Temperature (°C)Temperature (°C)
Time (s)
024 60 90 130 160
0
400
500
Time (s)
Monitored Monitore d Contr olle d Monitor ed Monitore d
Monitored Monitored Monito red Contr oll ed Monitor ed
Monitored Monitor ed Monitored Control led Monitored
(a)
(b)
(c)
Figure 9. The controlled temperature curves of the three experiments in the first group; a) the first experiment; b) the second
experiment; c) the third experiment.
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room temperature (about 22C) to 500C in 90s and
then stays 500C for about 90s in the first experiment,
while the temperature increasing procedure is achieved
in 60s in the second experiment and it is 24s in the third
experiment. For all the experiments in this section, a
random point-excitation is provided by the shaker
(JZK-20) and the acceleration response is collected
throughout the temperature-controlled procedure. The
sampling frequency is 1280 Hz and the factors are
n¼8, M¼600, N¼1000, ¼0:96 in the identifica-
tion algorithm. As shown in Figure 10, the natural fre-
quencies of the higher orders decline as the temperature
increases: the fourth-order natural frequency decreases
from 300 Hz to 284.6 Hz, the fifth-order natural fre-
quency decreases from 445Hz to 410.6 Hz and the
sixth-order natural frequency decreases from 548.5 Hz
020 40 60 80 100 120 140
285
290
295
300
305
Tim e t /s
Frequency
ω
4
/Hz
020 40 60 80 100 120 140
410
420
430
440
450
Tim e t /s
Frequency
ω
6
/Hz
020 40 60 80 100 120 140
510
520
530
540
550
Tim e t /s
Frequency
ω
6
/Hz
90s
90s
90s
Figure 11. The identified natural frequencies of the first experiment in the first group by the identification method based on the
original PAST.
20 40 60 80 100 120 140
285
290
295
300
Time t/s
Frequency
ω
4
/Hz
020 40 60 80 100 120 140
410
420
430
440
Time t/s
Frequency
ω
5
/Hz
020 40 60 80 100 120 140
510
520
530
540
Tim e t/ s
Frequency
ω
6
/Hz
24s
60s
90s
24s
60s
90s
24s
60s
90s
Figure 10. The identified natural frequencies of the three experiments in the first group by the proposed identification method.
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to 506.9 Hz in 110 s in the first experiment. Moreover,
Figure 11 shows the identified natural frequencies of
the first experiment by the identification method
based on the original PAST. Compared with the cor-
responding part shown in Figure10, the original
PAST would lose its tracking ability, which results in
the break-points in the identified results shown in
Figure 11.
It is reported that the material properties could be
affected by the increased temperature and the thermal
stresses caused by temperature gradients when the
modulus of elasticity decreases as temperature
increases, causing a reduction in the stiffness (Kehoe
and Synder 1991; Kehoe and Deaton, 1993), and the
thermal stresses would cause an increase in the stiffness
(Deyi et al., 2012). As shown in Figures 8 and 10, the
decrease of the natural frequencies for all the modes
indicates that the dominate cause of the frequency
reduction is the elasticity modulus reduction. Further
that the natural frequencies of the higher orders
decrease faster when the temperature increases faster
could be explained as the modulus of elasticity
decreases faster as the temperature increases faster.
Compared the corresponding results shown in Figure
8 and 10, the reduction amount of the natural frequen-
cies shown in Figure 8 is larger than that shown in
Figure 10 because the thermal stresses caused by tem-
perature gradients are severe when the trapezoidal plate
experiences a time-varying temperature environment.
Such a phenomenon would appear again in next
group of experiments.
As stated in equation (20), the corresponding damp-
ing ratios can be extracted as well. The identified damp-
ing ratios are depicted in Figure 12. As shown in
Figure 12, the corresponding damping ratios are
affected by the temperature-varying environments as
well, which confirms the conclusion on the tempera-
ture-dependent damping ratios in experiment. The
identified damping ratios have no obvious reduction
trend as the identified natural frequencies shown in
Figure 10 and there is no reference to confirm the
020 40 60 80 100
1
2
3
x 10
-3
Tim e t/ s
Dampin gRatio
ξ4
020 40 60 80 100
1
2
3
4
5
x 10
-3
Tim e t/ s
Dampin gRatio
ξ5
020 40 60 80 100
0
1
2
3
x 10
-3
Tim e t/ s
Dampin gRatio
ξ6
Identified
Fitting
Identified
Fitting
Identified
Fitting
Figure 12. The identified damping ratios of the third experiment in the first group.
Figure 13. The laboratory setup for the second group of
experiments.
3314 Journal of Vibration and Control 21(16)
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082.03 230 535 735
0
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Room temperature
100 200 300 400 500
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Figure 14. the reference experiments in the second group; a) the power spectrum density; b) the first six-order natural frequencies.
Yu et al. 3315
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identified damping ratios. So only the identified damp-
ing ratios of the third experiment are shown and no
discussion on damping ratios would be processed in
the second group of experiments.
3.2. The second group of experiments
To further investigate the proposed method and the
thermal effect on the natural frequencies, another
group of experiments are conducted with a different
exciting point. The locations of the exciting point and
the three accelerometers are shown in Figure 13.
Similarly, the referenced experiments are conducted in
six temperature-constant environments and the natural
frequencies of the plate by PP method are depicted in
Figure 14. Three experiments are processed herein, the
same as the three ones in the first group. For all the
three experiments, the sampling frequency is 2000 Hz
and the identification method’s factors are n¼20,
M¼1000, N¼1600, ¼0:95. If the natural frequen-
cies are focused on, short-time Fourier transform
(STFT) is adequate. Figure 15 shows the time-fre-
quency spectrum by STFT, in which the fourth and
fifth-order natural frequencies are easy to be misunder-
stood as one-order natural frequency, however such a
problem would not appear in the identified natural fre-
quencies by the proposed algorithm.
As shown in Figure 16, the first-order natural fre-
quency is acceptably extracted while the other five
decline as the temperature increases and the latter
five-order natural frequencies decrease faster when the
temperature increases faster, the same principle as that
in the first group of experiments. It reported that the
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(a)
Figure 16. The identified natural frequencies of the former six orders for the three experiments of the second group; a) the first
experiment; b) the second experiment; c) the third experiment; d) the fitting results of the identified natural frequencies.
Figure 15. STFT spectrum of one signal in the first experiment
of the second group.
3316 Journal of Vibration and Control 21(16)
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Figure 16. Continued.
Yu et al. 3317
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mode shapes were barely affected though such a con-
clusion was drawn based on the results of experiments
in the temperature-constant environments by PP-based
methods (Kehoe and Synder, 1991). Considering the
identified results shown in Figures 10 and 16, that the
natural frequencies decline in the practically same prin-
ciple reveals the unchangeable linearity of the experi-
ment structures under the continuously changing
temperature. Further whether the results can be used
to be an evident of unchangeable mode shapes should
be investigated.
4. Conclusions
A time-varying modal parameter identification algo-
rithm based on finite-data-window PAST is presented
in this study. The proposed method is confirmed by
an experiment conducted on a steel cantilever beam
with a removeable mass and the choices of the fac-
tors are briefly discussed. Furthermore, the proposed
method is applied to investigate the effect of varying
temperature and heating speed on the natural fre-
quencies of a trapezoidal TA15 titanium-alloy plate.
The identified results show the first-order natural fre-
quency is barely affected while the high-order natural
frequencies are obviously affected by the temperature.
Moreover, the high-order natural frequencies decline
faster when the temperature increases faster
and it could be understood as the modulus of
elasticity decreases faster as the temperature increases
faster, because the dominate cause of the frequency
reduction is the elasticity modulus reduction in this
study. Further, the effect of the thermal stresses
caused by temperature gradients on the natural fre-
quency reduction is revealed by experiments, taking
advantage of the identification algorithm presented in
this study.
Acknowledgments
The authors are grateful to Xiaonan Gai for conducting
experiments.
Funding
This research was supported by the National Science
Foundation of China (NSFC) under grant number 11172078.
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Structures and industrial equipment often operate in environments where temperature variations take place. Although thermal effects may be negligible in some cases, they have caused the unexpected failure of mechanical systems many times. Whether or not temperature has significant effects on the dynamical behavior of such machines and structures depends upon several aspects, amongst which are geometry, material properties and boundary conditions. In this paper we investigate the dynamical behavior of a clamped beam under the influence of a uniform, quasi-statically varying temperature field. An analytical model was used, based on Euler-Bernoulli’s beam theory with the introduction of the proper boundary conditions. Temperature effects are included in terms of an axial force that shows up when the beam tends to thermally expand, but this expansion is restrained by the clamping. Preliminary results do not agree with experimental data, since perfect clamping is difficult to achieve in practice. Finally the model is updated with the inclusion of axial and torsional springs connecting the beam to the support. The spring constants were calculated through optimization procedure to minimize the differences between the natural frequencies obtained from the analytical model and the corresponding experimental ones. Agreement with experimental results is reasonable up to the 4th mode of the beam. In the future, this analytical model is to be used for design and simulation of an active controller that accounts for temperature changes in the structure.
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In 2013 the Editor of Journal of Vibration and Control and SAGE became aware of a peer review ring involving assumed and fabricated identities that appeared to centre around Peter Chen at National Pingtung University of Education, Taiwan (NPUE). SAGE and the Editor then began a complex investigation into the case during the rest of 2013 and 2014. Following an unsatisfactory response from Peter Chen, NPUE was notified. NPUE were serious in addressing the Journal and SAGE’s concerns. NPUE confirmed that the institution was investigating Peter Chen. SAGE subsequently uncovered a citation ring involving the above mentioned author and others. We regret that individual authors have compromised the academic record by perverting the peer review process and apologise to readers. On uncovering problems with peer review and citation SAGE immediately put steps in place to avoid similar vulnerability of the Journal to exploitation in the future. More information may be found at www.sagepub.co.uk/JVC_Statement_2014 . The Journal and SAGE understand from NPUE that Peter Chen has resigned his post at NPUE. The following articles are retracted because after thorough investigation evidence points towards them having at least one author or being reviewed by at least one reviewer who has been implicated in the peer review ring and/or citation ring. All authors have had an opportunity to respond to the allegations and proposed actions. OnlineFirst articles (these articles will not be published in an issue) Chen CY, Chen T-H, Chen Y-H, Yu S-E and Chung P-Y (2013) Information technology system modeling an integrated C-TAM-TPB model to the validation of ocean tidal analyses Journal of Vibration and Control Epub ahead of print 7 May 2013. doi: 10.1177/1077546312472924 Chang R-F, Chen CY, Su F-P and Lin H-C (2013) A two-step approach for broadband digital signal processing technique Journal of Vibration and Control Epub ahead of print 26 April 2013. doi: 10.1177/1077546312472925 Chen TH, Chang CJ, Yu SE, Chung PY and Liu C-K (2013) Nonlinear information analysis and system management technique: the influence of design experience and control complexity Journal of Vibration and Control Epub ahead of print 12 April 2013. doi: 10.1177/1077546312473321 Chen CY, Shih BY, Chen YH, Yu SE and Liu YC (2013) The exploration of a 3T flow model using vibrating NXT: II. Model validation Journal of Vibration and Control Epub ahead of print 10 April 2013. doi: 10.1177/1077546312470481 Chen CY, Shih BY, Chen YH, Yu SE and Liu YC (2013) The exploration of 3T flow model using vibrating NXT: I. model formulation Journal of Vibration and Control Epub ahead of print 6 February 2013. doi: 10.1177/1077546312467360 Lin M-L and Chen C-W (2013) Stability analysis of fuzzy-based NN modeling for ecosystems using fuzzy Lyapunov methods Journal of Vibration and Control Epub ahead of print 6 February 2013. doi: 10.1177/1077546312466687 Chen CY, Chen TH, Chen YH and Chiu J (2012) A multi-stage method for deterministic-statistical analysis: a mathematical case and measurement studies Journal of Vibration and Control Epub ahead of print 20 December 2012. doi: 10.1177/1077546312466579 Shih BY, Lin MC and Chen CY (2012) Autonomous navigation system for radiofrequency identification mobile robot e-book reader Journal of Vibration and Control Epub ahead of print 13 December 2012. doi: 10.1177/1077546312466578 Chang RF, Chen CY, Su FP, Lin HC and Lu C-K (2012) Multiphase SUMO robot based on an agile modeling-driven process for a small mobile robot Journal of Vibration and Control Epub ahead of print 13 December 2012. doi: 10.1177/1077546312464993 Shih B-Y, Lin Y-K, Cheng M-H, Chen C-Y and Chiu C-P (2012) The development of an application program interactive game-based information system Journal of Vibration and Control Epub ahead of print 12 December 2012. doi: 10.1177/1077546312464682 Chen C-Y, Chang C-J and Lin C-H (2012) On dynamic access control in web 2.0 and cloud interactive information hub: technologies Journal of Vibration and Control Epub ahead of print 12 December 2012. doi: 10.1177/1077546312464992 Shin BY, Chen CY and Hsu KH (2012) Robot cross platform system using innovative interactive theory and selection algorithms for Android application Journal of Vibration and Control Epub ahead of print 13 November 2012. doi: 10.1177/1077546312463757 Articles published in an issue Chen C-W (2014) Applications of neural-network-based fuzzy logic control to a nonlinear time-delay chaotic system Journal of Vibration and Control 20 (4): 589-605. Epub ahead of print 5 November 2012. doi: 10.1177/1077546312461370 Chen C-W (2014) A review of intelligent algorithm approaches and neural-fuzzy stability criteria for time-delay tension leg platform systems Journal of Vibration and Control 20 (4): 561-575. Epub ahead of print 5 November 2012. doi: 10.1177/1077546312463759 Chen C-Y, Chang C-J and Lin C-H (2014) On dynamic access control in web 2.0 and cloud interactive information hub: trends and theories Journal of Vibration and Control 20 (4): 548-560. Epub ahead of print 5 November 2012. doi: 10.1177/1077546312463762 Lin M-L and Chen C-W (2014) Stability conditions for ecosystem modeling using the fuzzy Lyapunov method Journal of Vibration and Control 20 (2): 290-302. Epub ahead of print 23 October 2012. doi: 10.1177/1077546312451301 Chen C-H, Kuo C-M, Hsieh S-H and Chen C-Y (2014) Highly efficient very-large-scale integration (VLSI) implementation of probabilistic neural network image interpolator Journal of Vibration and Control 20 (2): 218-224. Epub ahead of print 22 October 2012. doi: 10.1177/1077546312458822 Chen C-Y (2014) Wave vibration and simulation in dissipative media described by irregular boundary surfaces: a mathematical formulation Journal of Vibration and Control 20 (2): 191-203. Epub ahead of print 22 October 2012. doi: 10.1177/1077546312464258 Chen C-H, Yao T-K, Dai J-H and Chen C-Y (2014) A pipelined multiprocessor system- on-a-chip (SoC) design methodology for streaming signal processing Journal of Vibration and Control 20 (2): 163-178. Epub ahead of print 16 October 2012. doi: 10.1177/1077546312458821 Lin M-L and Chen C-W (2014) Fuzzy neural modeling for n-degree ecosystems using the linear matrix inequality approach Journal of Vibration and Control 20 (1): 82-93. Epub ahead of print 8 October 2012. doi: 10.1177/1077546312458533 Chen C-H, Wu W-X and Chen C-Y (2013) Ant-inspired collective problem-solving systems Journal of Vibration and Control 19 (16): 2481-2490. Epub ahead of print 18 September 2012. doi: 10.1177/1077546312456231 Chen C-H, Yao T-K, Kuo C-M and Chen C-Y (2013) Evolutionary design of constructive multilayer feedforward neural network Journal of Vibration and Control 19 (16): 2413-2420. Epub ahead of print 12 September 2012. doi: 10.1177/1077546312456726 Chen C-W (2013) Applications of the fuzzy-neural Lyapunov criterion to multiple time-delay systems Journal of Vibration and Control 19 (13): 2054-2067. Epub ahead of print 16 August 2012. doi: 10.1177/1077546312451034 Chung P-Y, Chen Y-H, Walter L and Chen C-Y (2013) Influence and dynamics of a mobile robot control on mechanical components Journal of Vibration and Control 19 (13): 1923-1935. Epub ahead of print 20 July 2012. doi: 10.1177/1077546312452184 Chen C-W (2013) Neural network-based fuzzy logic parallel distributed compensation controller for structural system Journal of Vibration and Control 19 (11): 1709-1727. Epub ahead of print 22 June 2012. doi: 10.1177/1077546312442233 Chen C-W, Yeh K, Yang H-C, Liu KFR and Liu C-C (2013) A critical review of structural system control by the large-scaled neural network linear-deferential-inclusion-based criterion Journal of Vibration and Control 19 (11): 1658-1673. Epub ahead of print 18 June 2012. doi: 10.1177/1077546312443377 Chen C-H, Kuo C-M, Chen C-Y and Dai J-H (2013) The design and synthesis using hierarchical robotic discrete-event modeling Journal of Vibration and Control 19 (11): 1603-1613. Epub ahead of print 27 June 2012. doi: 10.1177/1077546312449645 Chang CJ, Chen CY and Chou I-T (2013) The design of information and communication technologies: telecom MOD strength machines Journal of Vibration and Control 19 (10): 1499-1513. Epub ahead of print 27 June 2012. doi: 10.1177/1077546312449644 Shih B-Y, Chen C-Y, Li K-H, Wu T-Y, Chen G-Y (2013) A novel NXT control method for implementing force sensing and recycling in a training robot Journal of Vibration and Control 19 (10): 1443-1459. Epub ahead of print 1 June 2012. doi: 10.1177/1077546312446361 Chen C-W, Chen P-C and Chiang W-L (2013) Modified intelligent genetic algorithm-based adaptive neural network control for uncertain structural systems Journal of Vibration and Control 19 (9): 1333-1347. Epub ahead of print 31 May 2012. doi: 10.1177/1077546312442232 Chen C-Y, Shih B-Y, Shih C-H and Wang L-H (2013) Enhancing robust and stability control of a humanoid biped robot: system identification approach. Journal of Vibration and Control 19 (8): 1199-1207. Epub ahead of print 26 April 2012. doi: 10.1177/1077546312442947 Chang C-J, Chen C-Y and Huang C-W (2013) Applications for medical recovery using wireless control of a bluetooth ball with a hybrid G-sensor and human-computer interface technology Journal of Vibration and Control 19 (8): 1139-1151. Epub ahead of print 24 April 2012. doi: 10.1177/1077546312442948 Hsu W-K, Chiou D-J, Chen C-W, Liu M-Y, Chiang W-L and Huang P-C (2013) Sensitivity of initial damage detection for steel structures using the Hilbert-Huang transform method Journal of Vibration and Control 19 (6): 857-878. Epub ahead of print 29 February 2012. doi: 10.1177/1077546311434794 Chen C-Y, Shih B-Y, Shih C-H and Wang L-H (2013) Human–machine interface for the motion control of humanoid biped robots using a graphical user interface Motion Editor Journal of Vibration and Control 19 (6): 814-820. Epub ahead of print 23 February 2012. doi: 10.1177/1077546312437804 Chen C-Y (2013) Internal wave transport, nonlinear manifestation, and mixing in a stratified shear layer - technical briefs Journal of Vibration and Control 19 (3): 429-438. Epub ahead of print 18 January 2012. doi: 10.1177/1077546311429337 Chen C-W (2013) Delay independent criterion for multiple time-delay systems and its application in building structure control systems Journal of Vibration and Control 19 (3): 395-414. Epub ahead of print 17 January 2012. doi: 10.1177/1077546311429341 Chen C-Y, Shih B-Y, Shih C-H and Wang L-H (2013) Design, modeling and stability control for an actuated dynamic walking planar bipedal robot Journal of Vibration and Control 19 (3): 376-384. Epub ahead of print 17 January 2012. doi: 10.1177/1077546311429476 Liu K-C, Liu Y-W, Chen C-Y and Huang W-C (2013) Nonlinear vibration of structural deterioration in reinforced concrete columns: experimental and theoretical investigation Journal of Vibration and Control 19 (3): 323-335. Epub ahead of print 17 January 2012. doi: 10.1177/1077546311429477 Chen C-Y, Shih B-Y and Ma J-m (2013) Development for low-cost and cross-platform robot control environment Journal of Vibration and Control 19 (2): 228-233. Epub ahead of print 11 January 2012. doi: 10.1177/1077546311430107 Shih B-Y, Chang H and Chen C-Y (2013) Path planning for autonomous robots – a comprehensive analysis by a greedy algorithm Journal of Vibration and Control 19 (1): 130-142. Epub ahead of print 17 January 2012. doi: 10.1177/1077546311429841 Liu T-Y, Chiang W-L, Chen C-W, Hsu W-K, Lin C-W, Chiou D-J and Huang P-C (2012) Structural system identification for vibration bridges using the Hilbert–Huang transform Journal of Vibration and Control 18 (13): 1939-1956. Epub ahead of print 14 December 2011. doi: 10.1177/1077546311428347 Chen C-W (2012) Applications of the fuzzy Lyapunov linear matrix inequality criterion to a chaotic structural system Journal of Vibration and Control 18 (13): 1925-1938. Epub ahead of print 14 December 2011. doi: 10.1177/1077546311428346 Chen C-W (2012) Applications of linear differential inclusion-based criterion to a nonlinear chaotic system: a critical review Journal of Vibration and Control 18 (12): 1886-1899. Epub ahead of print 14 December 2011. doi: 10.1177/1077546311428345 Shih B-Y, Chen C-Y and Chou W (2012) An enhanced obstacle avoidance and path correction mechanism for an autonomous intelligent robot with multiple sensors Journal of Vibration and Control 18 (12): 1855-1864. Epub ahead of print 14 December 2011. doi: 10.1177/1077546311426734 Chen C-W, Yeh K, Liu KFR and Lin M-L (2012) Applications of fuzzy control to nonlinear time-delay systems using the linear matrix inequality fuzzy Lyapunov method Journal of Vibration and Control 18 (10): 1561-1574. Epub ahead of print 18 October 2011. doi: 10.1177/1077546311410765 Chen C-Y (2012) A critical review of internal wave dynamics. Part 2 – Laboratory experiments and theoretical physics Journal of Vibration and Control 18 (7): 983-1008. Epub ahead of print 21 September 2011. doi: 10.1177/1077546310397561 Chen C-Y and Huang P-H (2012) Review of an autonomous humanoid robot and its mechanical control Journal of Vibration and Control 18 (7): 973-982. Epub ahead of print 21 September 2011. doi: 10.1177/1077546310395974 Shih B-Y, Chen C-Y, Chang H and Ma J-m (2012) Dynamics and control for robotic manipulators using a greedy algorithm approach Journal of Vibration and Control 18 (6): 859-866. Epub ahead of print 25 August 2011. doi: 10.1177/1077546311407649 Yeh K, Chen C-W, Lo DC and Liu KFR (2012) Neural-network fuzzy control for chaotic tuned mass damper systems with time delays Journal of Vibration and Control 18 (6): 785-795. Epub ahead of print 15 August 2011. doi: 10.1177/1077546311407538 Chen C-Y, Shih B-Y, Shih C-H and Chou W-C (2012) The development of autonomous low-cost biped mobile surveillance robot by intelligent bricks Journal of Vibration and Control 18 (5): 577-586. Epub ahead of print 21 April 2011. doi: 10.1177/1077546310371349 Chen C-Y (2012) A critical review of internal wave dynamics. Part 1 – Remote sensing and in-situ observations Journal of Vibration and Control 18 (3): 417-436. Epub ahead of print 13 July 2011. doi: 10.1177/1077546310395971 Tseng C-P, Chen C-W and Liu KFR (2012) Risk control allocation model for pressure vessels and piping project Journal of Vibration and Control 18 (3): 385-394. Epub ahead of print 13 July 2011. doi: 10.1177/1077546311403182 Lin M-L and Chen C-W (2011) Stability analysis of community and ecosystem hierarchies using the Lyapunov method Journal of Vibration and Control 17 (13): 1930-1937. Epub ahead of print 9 December 2010. doi: 10.1177/1077546310385737 Chen C-Y, Shih B-Y, Chou W-C, Li Y-J and Chen Y-H (2011) Obstacle avoidance design for a humanoid intelligent robot with ultrasonic sensors Journal of Vibration and Control 17 (12): 1798-1804. Epub ahead of print 26 November 2010. doi: 10.1177/1077546310381101 Chen C-W (2011) Fuzzy control of interconnected structural systems using the fuzzy Lyapunov method Journal of Vibration and Control 17 (11): 1693-1702. Epub ahead of print 23 November 2010. doi: 10.1177/1077546310379625 Shih B-Y, Chen C-Y and Chou W-C (2011) Obstacle avoidance using a path correction method for autonomous control of a biped intelligent robot Journal of Vibration and Control 17 (10): 1567-1573. Epub ahead of print 22 November 2010. doi: 10.1177/1077546310372004 Tang J-P, Chiou D-J, Chen C-W, Chiang W-L, Hsu W-K, Chen C-Y and Liu T-Y (2011) A case study of damage detection in benchmark buildings using a Hilbert-Huang Transform-based method Journal of Vibration and Control 17 (4): 623-636. Epub ahead of print 8 November 2010. doi: 10.1177/1077546309360053 Liu TY, Chiang WL, Chen CW, Hsu WK, Lu LC and Chu TJ (2011) Identification and monitoring of bridge health from ambient vibration data Journal of Vibration and Control 17 (4): 589-603. Epub ahead of print 12 November 2010. doi: 10.1177/1077546309360049 Lin JW, Huang CW, Shih CH and Chen CY (2011) Fuzzy Lyapunov Stability Analysis and NN Modeling for Tension Leg Platform Systems Journal of Vibration and Control 17 (1): 151-158. Epub ahead of print 25 August 2010. doi: 10.1177/1077546309350477 Lee WI, Chen CY, Kuo HM and Sui YC (2010) The Development of Half-circle Fuzzy Numbers and Application in Fuzzy Control Journal of Vibration and Control 16 (13): 1977-1987. Epub ahead of print 22 April 2010. doi: 10.1177/1077546309349849
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A large-scale field deployment of high-density, real-time wireless sensors networks for the acquisition of local acceleration measurements across a medium length, multi-span highway bridge is presented. The advantages, performance characteristics, and limitations of employing this emerging technology in favor of the traditional cable-based acquisition systems are discussed in the context of the in-service instrumentation and ambient vibration testing of a multi-span bridge. Of particular highlight in this study is the deployment of a large number of stationary rather than reference-based accelerometers to uniquely permit simultaneous acquisition of vibration measurements across the structure and thereby ensure consistent temperature, ambient vibration, and traffic loading. The deployment consisted of 30 dual-axis accelerometers installed across the girders of the bridge and interfaced with 30 wireless acquisition and transceiver nodes operating in two star topology networks. Real-time wireless acquisition at a per channel sampling rate of 128 samples per second was maintained across both networks for the specified test durations of 3 min with insignificant data loss. Output-only system identification of the structure from the experimental data is presented to provide estimates of natural frequencies, damping ratios, and operational mode shapes for 19 modes. The analysis of the structure under test provides a unique case study documenting the measured response of a multiple-span skewed bridge supported by elastomeric bearings. The feasibility of embedded wireless instrumentation for structural health monitoring of large civil constructions is concluded while highlighting relevant technological shortcomings and areas of further development required. In particular, previously undocumented obstacles relating to radio transmission of the sensor data using low-power 2.4 GHz wireless instrumentation, such as the effect of solid piers within the line-of-sight and the reflection of the radio waves on the surface of the water, are discussed.
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