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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

Reprints and permissions:

sagepub.co.uk/journalsPermissions.nav

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

xð2ÞxðnÞ 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.

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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

þMÞT,

uNþ1¼½uTðNþ1ÞuTðNþ2ÞuTðNþMÞT.

We have

YNþ1U?

Nþ1YT

Nþ1¼YNþ1U?

Nþ1YNþ1U?

Nþ1

T

¼YNU?

NYT

NþzNþ1zT

Nþ1ð6Þ

where zNþ1¼½YNUT

NðUNUT

NÞ1

uNþ1

yNþ1=

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

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

tiziðÞWtðÞWTtðÞziðÞ

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 WTðtÞzðiÞWTðt1ÞzðiÞ,

and let yðiÞ¼WTðt1ÞzðiÞ, we have

JWtðÞðÞ¼min X

t

i¼tl

tiziðÞWtðÞyiðÞ

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

tiWtðÞyiðÞziðÞ½yTiðÞ

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 ðt1ÞþyðtÞyTðtÞ1. 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

Yu et al. 3307

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Moore-Penrose inverse matrix. ,ðtÞ¼diagð1ðtÞ

2ðtÞ2nðtÞÞ 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

0.1

0.2

0.3

0.4

0.5

0.6

Fre quenc y/Hz

Amplitude/dB

083.1 230 535 729.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Fre quenc y/Hz

Amplitude/dB

082.03 230 535 720

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Freque ncy/Hz

Amplitude/dB

082.03 230 535 708

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Freque ncy/Hz

Amplitude/dB

080.08 230 535 700.2

0

1

2

3

4

5

6

Fre quenc y/Hz

Amplitude/dB

081.05 230 490 683.6

0

2

4

6

8

10

12

14

16

18

Freque ncy/Hz

Amplitude/dB

(a)

Room temperature

100 200 300 400 500

82.03

210.09

267.6

475.6

517.6

570.3

683.6

735.4

Frequency/Hz

(b)

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

050 100 150

60

65

70

75

80

85

Tim e t /s

Frequency

ω

1

/Hz

050 100 150

200

205

210

215

Tim e t /s

Frequency

ω

2

/Hz

050 100 150

250

255

260

265

270

Tim e t /s

Frequency

ω

3

/Hz

050 100 150

480

490

500

510

Tim e t /s

Frequency

ω

4

/Hz

050 100 150

520

540

560

580

Tim e t /s

Freq uenc y

ω

5

/Hz

050 100 150

680

700

720

740

Tim e t /s

Freq uen cy

ω

6

/Hz

Identified result

Fitting result

Identified result

Fitting result

Identified res ult

Fitting result

(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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020 40 60 80 100 120 140 160

60

65

70

75

80

85

time t/s

Frequency

ω

1

/Hz

020 40 60 80 100 120 140 160

200

205

210

215

time t/s

Frequency

ω

2

/Hz

020 40 60 80 100 120 140 160

480

490

500

510

time t/s

Frequency

ω

4

/Hz

020 40 60 80 100 120 140 160

520

540

560

time t/s

Frequency

ω

5

/Hz

020 40 60 80 100 120 140 160

680

700

720

740

time t/s

Frequency

ω

6

/Hz

Ident ifie d res ult

Fitting result

020 40 60 80 100 120 140 160

250

255

260

265

270

time t/s

Frequency

ω

3

/Hz

Identified result

Fitting result

Identified result

Fitting result

Ident ifie d res ult

Fitting result

(c)

020 40 60 80 100 120 140 160

60

65

70

75

80

85

Tim e t /s

Frequency

ω

1

/Hz

020 40 60 80 100 120 140 160

200

205

210

215

Tim e t /s

Frequency ω

2

/Hz

020 40 60 80 100 120 140 160

250

255

260

265

270

Tim e t /s

Frequency ω

3

/Hz

020 40 60 80 100 120 140 160

480

490

500

510

Tim e t/ s

Frequency

ω

4

/Hz

020 40 60 80 100 120 140 160

520

540

560

Tim e t /s

Freque ncy

ω

5

/Hz

020 40 60 80 100 120 140 160

680

700

720

740

Tim e t/ s

Frequency

ω

6

/Hz

Identified result

Fitting result

Ident ifi ed res ult

Fitting result

Identified result

Fitting result

Ident ifi ed res ult

Fitting result

(b)

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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