time Structure Health Monitoring by Nonlinearity Degree ...2. Structural Health Monitoring System 2.1 Instantaneous Frequency Traditionally, Fourier spectrum uses sine and cosine harmonic

9th

European Workshop on Structural Health Monitoring

July 10-13, 2018, Manchester, United Kingdom

Creative Commons CC-BY-NC licence https://creativecommons.org/licenses/by-nc/4.0/

Real-time Structure Health Monitoring by Nonlinearity Degree Analysis based on Hilbert-Huang Transform

Chun-Yueh Hsieh 1

, Alain Sawadogo 2 and Tzu-Kang Lin

3

1 National Chiao Tung University, Department of Civil Engineering, Hsinchu, Taiwan,

[email protected]

2 National Chiao Tung University, Department of Civil Engineering, Hsinchu, Taiwan,

[email protected]

3. National Chiao Tung University, Department of Civil Engineering Hsinchu, Taiwan,

[email protected]

Abstract

The fact that in recent years many catastrophic seismic events are major factors in

building collapse has made structure health monitoring an important research topic. In

view of the complex nonlinear behaviour of building structure/soil interactions, the

fundamental frequency may gradually change due to seismic damage. To provide real-

time structural dynamic parameters, Hilbert-Huang Transform (HHT) is utilize to

analyse the measured signal from the superstructure under seismic excitation. In the

study, the Time-Frequency domain Amplification Function (T.F.AF) obtained from the

HHT analysis is decomposed into instantaneous frequency width, centroid of frequency

and energy time-history. The proposed degree of nonlinearity (DN) can then be

estimated to display the structure health condition. In order to verify the proposed

method, shaking table tests were conducted on a 6 -story benchmark structure.

According to the analysis of vibration signals recorded up to the time of collapse,

structure behaviour variation can be rapidly displayed by the degree of nonlinearity to

reflect the health condition of the whole structure

1. Introduction

Taiwan is located at the junction of the Eurasian continental plate and the Philippine Sea

Plate, these movements has caused frequent earthquakes. Large-scale earthquakes

damages the country propriety but more importantly it harms people live. Earthquakes

has a great impact on Taiwan, it instantaneously transmits powerful energy to structures,

and buildings has to rely on their structural system to dissipate the external forces. If

energy cannot be dissipated effectively, structural damage will easily occur, and in

severe cases, the collapse of the structure itself may imply potential loss of lives.

Therefore, the dynamic behaviour of the structure must be studied in depth when the

structure is affected by external forces such as seismic forces. Following the empirical

mode decomposition method and the Hilbert spectrum for non-stationary time series

analysis in 1998 (1), this study introduced a structural dynamic parameter named the

“degree of nonlinearity” to observe the structure behaviour using a new structural

dynamic parameter.

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2. Structural Health Monitoring System

2.1 Instantaneous Frequency

Traditionally, Fourier spectrum uses sine and cosine harmonic functions as the basis

with a fixed amplitude. In reality there is a problem of time-variation, due to limited

applicability of the Fast Fourier transform, but it is impossible to obtain the

instantaneous frequency (IF) at any time wanted. For the structure affected by

earthquakes, understanding the variation of the frequency is a necessity. Hilbert

Transform is widely used for non-linear and non-stationary cases, making time varying

signal easier to analyse. The actual signal is expressed in form of a complex number to

determine the instantaneous amplitude a(t) and instantaneous phase θ(t). The

instantaneous frequency ω(t) can then be determined. For any time series, the Hilbert

transformation formula Y(t) can be expressed as:

1 XY t P d

t

(1)

where P represent the Cauchy principal value.

The Hilbert transformation can be defined as the convolution between X(t) and 1/t after

combining X(t) and Y(t) into a conjugate complex number it gives an analytic signal Z(t):

( )( ) ( ) ( ) ( ) i t

Z t X t iY t a t e (2)

Such as:

2 2( ) ( ) ( )a t X t Y t (3)

1( ) tan ( )/ ( )t Y t X t (4)

( ) ( )/t d t dt (5)

According to the analysis, the amplitude time frequency distribution of the time series

are obtained.

2.2 Empirical Mode Decomposition (EMD)

EMD does not predetermine the basis function compared to others decomposition

methods. The basis function is directly obtained from the signal data, and has great

adaptability. EMD decompose the original signal into a finite number of instantaneous

modal frequency (IMF), which are approximate single component signals with zero-

mean amplitude modulation, and a finite number of IMFs from high frequencies to low

frequencies can be obtained until only a monotonic function remains at the end (trend).

The original data can be regarded as the sum of all IMFs and trends if during the

analysis the time difference between the extremes values represents the time scalar of

the intrawave (5), it provides optimal vibration modal resolution and can be applied to

non-zero mean values as well as non-zero-crossing data.

3

Thus, the original signal can be re-expressed as:

1

n

i n

i

X t c r

(6)

2.3 Ensemble Empirical Mode Decomposition (EEMD)

EMD is often used as a signal disassembly tool (4). When signals of different scales are

mixed together, it disassembles them into different IMF’s but a defective point has been

discovered named “mode mixing”. During the process of EMD, a low amplitude

oscillation or an intermittent signal may exist in several different IMFs, such as a one

modal component decomposed into different IMF components, or a single IMF

containing two different modal signals, resulting to a “mode mixing” within the IMF.

When EMD is performed, a white noise wi(t) signal with a limited amplitude is added to

the original signal X(t), thus this signal becomes as follow:

( )i i

X t X t w t (7)

2.4 Hilbert Spectrum

2.4.1 Hilbert Spectrum Analysis

After decomposing the signal using EMD, each component of the IMF can be obtained.

From equation (8), the IMF component can be converted from a time domain to a

frequency domain; this process is called Hilbert Spectrum Analysis (HSA):

1

n

j j

j

X a t exp i tt dt

(8)

Although the Hilbert transformation can deal with monotonous trends and take it as part

of a longer amplitude, the remaining energy may be too strong, considering the

uncertainties of longer-term trends, and other low energy and information contained in

high frequency component. The above formula gives a time function for each amplitude

and frequency components, which is expanded by a Fourier expression:

1

ji t

j

j

X a et

(9)

a j and wj are constants. After comparing equations (8) and (9), the IMF represents a

generalized Fourier expansion. Variables within the amplitude and the instantaneous

frequency can not only improve the expansion, but also make it applicable for unsteady

signals. In terms of the expansion of the IMF, the amplitude and the frequency

modulation are clearly separated. The time function amplitude and the instantaneous

frequency are represented as the frequency spectrum of the time-frequency-amplitude,

and is referred to as the Hilbert Amplitude spectrum (HAS).

2.4.2 Time-Frequency domain Amplification Function (TFAF)

4

Two different types of amplification function exist within the TFAF. The Amplification

Function Magnitude (AFM) is mainly used to observe the distribution of “energy” and

the Amplification Function Frequency (AFF) is used to observe the distribution of

“frequency” both in the Hilbert energy spectrum. It represents the relationship of the

shock wave propagation through the structure to obtain the amplifying characteristics of

it.

Roof

Fundation

R fAF Str f

R f (10)

AF represents the amplification function, RRoof is the response of the structural roof and

RFundation is the response of the structural foundation. The dynamic response Str of the

structure is obtained by dividing the structure roof response with the basis response. In

which the time frequency domain amplification function AFM shows the transmission

of seismic waves from the underground foundation to the building roof, and the energy

amplification relationship of the seismic waves during the process. TFAF analysis is

performed with the signals recorded at the seismic wave input and output location on

the structure. The original signal X(t)Fundation is recorded from the seismic wave applied

at the base of the structure. The seismic wave output data X(t)Roof represents the original

signal recorded on the roof of the structure. After analysing those two signals via the

Hilbert energy spectrum, the time frequency spectrum can be written as H{X(t)Fundation}

and H{X(t)roof} respectively. The time-frequency domain amplification function AFM

can also be defined as followed:

. .

Roof

Fundation

H X tHSP RoofAF m

HSP Fundation H X t (11)

The time frequency domain amplification function AFF can also be used when the

structure is subjected to an earthquake. Obtained by normalizing the energy of the AFM,

this procedure alleviates the variation of energy within the AFM, leaving only the

variation of instantaneous frequency as AFF:

. . . .AF f NS AF m (12)

where NS represents the normalization of energy. In the frequency region, each signals

has their own weighting number (between 0 and 1). The higher weight represents the

main frequency at that particular moment.

2.5 Degree of nonlinearity

According to linear algebra, the degree of nonlinearity is based on the relationship

between input and output. Two methods where used to calculate the degree of

nonlinearity, including the calculation of the degree of nonlinearity (DN1) expressed as:

1

z z

z z

IF IF aDN

IF a

(13)

5

Steps to obtain the degree of nonlinearity are shown in figure 1, in which the peak value

of the energy distribution at each time in the time-frequency domain amplification

function AFF are found to calculate ”IF-IFz“ as well as the frequency energy centre of

gravity at that moment ”IFz” (Figure 2).

Figure 1. Degree of nonlinearity calculation process

Figure 2. Centroid of main instantaneous frequency

From the time-frequency domain amplification function AFM, the time history energy

az is obtained and the average energy āz is calculated. DN1 is a dimensionless structural

dynamic parameter suitable to observe the maximum value of the degree of nonlinearity.

In addition another conservative method is provided to calculate the degree of

nonlinearity DN2 (2):

2

( )

mean( )

z

z

std IFDN

IF (14)

The numerator and denominator represents the standard deviation and the average

within the same part of a time history IFz respectively. Thus the degree of nonlinearity

is calculated from the oscillating amplitude of the main frequency.

3. Benchmark shaking table test

The test model (Figure 3) is composed of a five-story steel structure and a four-story

steel structure connected on the first floor. Starting from the second floor the structures

are separated and two different braces with different characteristics were used to

6

simulate the normal and weak floors. Two earthquakes data TCU071 400 gal and

TCU071 600 gal were applied. It was clearly observed that brace 1 of Frame A slightly

buckled (Figure 4. (a)(b)).

Figure 3. Structure Model

(Right side: Frame A，Left side: Frame B)

(a) TCU071 400 gal (b) TCU071 600 gal

Figure 4. Weak floors of the benchmark test

4. Health monitoring validation

4.1 Time-Frequency domain Amplification Function, TFAF (Frame A)

Figure 5 (a,b) shows the response of the benchmark structure under the TCU071-200

gal where the peak frequency margin 4.2Hz is visible on the AFM. Peak values are

located at 15 second, 17 second, 20 second, and 27 second on the AFF. Figure 5 (c,d)

shows the response of the benchmark structure under the TCU071-400. From figure 5 (c)

AFM, the reaction energy has fallen from 4Hz to 2Hz and the marginal spectral energy

peaks at 10 seconds. After normalizing the energy, figure 5 (d) shows that the dominant

frequency dropped to 2Hz at 10 seconds, and slowly increased to 3Hz. This behaviour is

typical on a nonlinear system, because part of the structure enters the yielding phase.

Figure 5 (e,f) shows the response under TCU071-600 gal, the main frequency of the

structure drops to around 1.9Hz. Figure 5 (f) shows that the main frequency of the

structure drops from about 2.5Hz to 1.5Hz and slowly increased to 2Hz. Figure 5 (g,h)

shows the response under TCU071-800 gal. Figure 5 (h) shows that the main frequency

7

Fre

q(H

z)

AX5ab -T.F.AF: AFf-Hilbert Spectrum--add 0.001(HSPmax)

0 10 20 30 40 50 60 700

1

2

3

4

5

6

7

8

0.2

0.4

0.6

0.8

UP Original HSP

Fre

q(h

z)

0 10 20 30 40 50 60 700

2

4

6

8

DOWN Original HSP

Fre

q(h

z)

Time(sec)0 10 20 30 40 50 60 70

0

2

4

6

8

0 10 20 30 40 50 60 70

-400-200

0200400600800

Event date= Case4-TCU071-400-AX5ab-AX0-1pair-0-8hz ,UP Acc data: --AX5ab,dir-X

gal

0 10 20 30 40 50 60 70

-200

0

200

Event date= Case4-TCU071-400-AX5ab-AX0-1pair-0-8hz ,DOWN Acc data: --AX0-1,dir-X

gal

0 50 100 150 2000

1

2

3

4

5

6

7

8M.S-F

Fre

q(h

z)

Amp

0 10 20 30 40 50 60 70

20

40

60

80

100M.S-T

Am

p

100

0

2

4

6

8M.S-F

Amp(log)

Fre

q(h

z)

up

dow n

Fre

q(H

z)


0 10 20 30 40 50 60 700

1

2

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0

0.2

0.4

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0.8

UP Original HSP

Fre

q(h

z)

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2

4

6

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DOWN Original HSP

Fre

q(h

z)

Time(sec)0 10 20 30 40 50 60 70

0

2

4

6

8

0 10 20 30 40 50 60 70

-500

0

500


gal

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-400

-200

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200


gal

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8M.S-F

Fre

q(h

z)

Amp

0 10 20 30 40 50 60 70

20

40

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100

M.S-T

Am

p

100

0

2

4

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8M.S-F

Amp(log)

Fre

q(h

z)

up

dow n

Fre

q(H

z)


0 10 20 30 40 50 60 700

1

2

3

4

5

6

7

8

0

0.2

0.4

0.6

0.8

UP Original HSP

Fre

q(h

z)

0 10 20 30 40 50 60 700

2

4

6

8

DOWN Original HSP

Fre

q(h

z)

Time(sec)0 10 20 30 40 50 60 70

0

2

4

6

8

0 10 20 30 40 50 60 70

-400-200

0200400600


gal

0 10 20 30 40 50 60 70

-500

0

500


gal

0 100 200 3000

1

2

3

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5

6

7

8M.S-F

Fre

q(h

z)

Amp

0 10 20 30 40 50 60 70

20

40

60

80

M.S-T

Am

p

100

0

2

4

6

8M.S-F

Amp(log)

Fre

q(h

z)

up

dow n

Fre

q(H

z)


0 10 20 30 40 50 60 700

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8

0.2

0.4

0.6

0.8

UP Original HSP

Fre

q(h

z)

0 10 20 30 40 50 60 700

2

4

6

8

DOWN Original HSP

Fre

q(h

z)

Time(sec)0 10 20 30 40 50 60 70

0

2

4

6

8

0 10 20 30 40 50 60 70-400

-200

0

200

400


gal

0 10 20 30 40 50 60 70

-100

0

100


gal

0 50 1000

1

2

3

4

5

6

7

8M.S-F

Fre

q(h

z)

Amp

0 10 20 30 40 50 60 70

20

40

60

80

M.S-T

Am

p

100

0

2

4

6

8M.S-F

Amp(log)

Fre

q(h

z)

up

dow n

drops under 2Hz, thus the structure is severely damaged and the frequency cannot be

restored.

(a) TFAF (AFM) (b) TFAF (AFF)

(TCU071-200gal) (TCU071-200gal)

(c) TFAF (AFM) (d) TFAF (AFF)

(TCU071-400gal) (TCU071-400gal)

(e) TFAF (AFM) (f) TFAF (AFF)

(TCU071-600gal) (TCU071-600gal)

(g) TFAF (AFM) (h) TFAF (AFF)

(TCU071-800gal) (TCU071-800gal)

Figure 5. TFAF (AFM) and (AFF)

Fre

q(H

z)

AX5ab -T.F.AF: AFm-Hilbert Spectrum--add 0.001(HSPmax)

0 10 20 30 40 50 60 700

1

2

3

4

5

6

7

8

0

10

20

30

40

UP Original HSP

Fre

q(h

z)

0 10 20 30 40 50 60 700

2

4

6

8

DOWN Original HSP

Fre

q(h

z)

Time(sec)0 10 20 30 40 50 60 70

0

2

4

6

8

0 10 20 30 40 50 60 70-400

-200

0

200

400


gal

0 10 20 30 40 50 60 70

-100

0

100


gal

0 1 2 3 4

x 104

0

1

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3

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6

7

8M.S-F

Fre

q(h

z)

Amp

0 10 20 30 40 50 60 70

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x 104

M.S-T

Am

p

100

0

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8M.S-F

Amp(log)

Fre

q(h

z)

up

dow n

Fre

q(H

z)


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Fre

q(h

z)

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DOWN Original HSP

Fre

q(h

z)

Time(sec)0 10 20 30 40 50 60 70

0

2

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0 10 20 30 40 50 60 70

-400-200

0200400600800


gal

0 10 20 30 40 50 60 70

-200

0

200


gal

0 2000 40000

1

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8M.S-F

Fre

q(h

z)

Amp

0 10 20 30 40 50 60 70

0.5

1

1.5

2

2.5

x 104

M.S-T

Am

p

100

0

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4

6

8M.S-F

Amp(log)

Fre

q(h

z)

up

dow n

Fre

q(H

z)


0 10 20 30 40 50 60 700

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8

0

5

10

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UP Original HSP

Fre

q(h

z)

0 10 20 30 40 50 60 700

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4

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DOWN Original HSP

Fre

q(h

z)

Time(sec)0 10 20 30 40 50 60 70

0

2

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0 10 20 30 40 50 60 70

-500

0

500


gal

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-400

-200

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gal

0 2000 4000 60000

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8M.S-F

Fre

q(h

z)

Amp

0 10 20 30 40 50 60 70

0.5

1

1.5

2

2.5

x 104

M.S-T

Am

p

100

0

2

4

6

8M.S-F

Amp(log)

Fre

q(h

z)

up

dow n

Fre

q(H

z)


0 10 20 30 40 50 60 700

1

2

3

4

5

6

7

8

0

5

10

UP Original HSP

Fre

q(h

z)

0 10 20 30 40 50 60 700

2

4

6

8

DOWN Original HSP

Fre

q(h

z)

Time(sec)0 10 20 30 40 50 60 70

0

2

4

6

8

0 10 20 30 40 50 60 70

-400-200

0200400600


gal

0 10 20 30 40 50 60 70

-500

0

500


gal

0 1000 20000

1

2

3

4

5

6

7

8M.S-F

Fre

q(h

z)

Amp

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1000

2000

3000

4000

5000

6000

M.S-T

Am

p

100

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4

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8M.S-F

Amp(log)

Fre

q(h

z)

up

dow n

8

4.2 Main frequency width, main frequency width variation and main frequency

centroid (Frame A)

Frame A affected by the earthquake TCU 071 with different PGA’s gave different

results, the main frequency recovery sometimes occurs in the second half, TCU 071

main frequency is taken as the main vibration interval from 0 to 29 seconds. Figure 6 (a)

shows the response of the structure subjected to TCU071-200 gal with a main frequency

centroid average of 4.11Hz. Figure 6 (b) shows the response of the model subjected to

TCU071-400 gal the main frequency centroid average decreases from 4.35 Hz to

3.02Hz. Figure 6 (c) shows the response of the model subjected to TCU071-600 gal.

From 3Hz, the main frequency of the structure started oscillating at around 7 seconds,

and decay to 2Hz. After 12 seconds, the main frequency of the structure stabilized at

around 2Hz. Figure 6 (d) shows the response of the model subjected to TCU071-800 gal.

The main frequency of the structure starts at 1.3Hz, at the 5th second it decreased to

2.93Hz, and then drops below 2Hz. After that, it stabilized at around 1.5Hz.

(a). Frequency width, width change, (b). Frequency width, width change,

and centroid for Frame A (TCU071 200 gal) and centroid for Frame A (TCU071 400 gal)

(c). Frequency width, width change, (d). Frequency width, width change,

and centroid for Frame A (TCU071 600 gal) and centroid for Frame A (TCU071 800 gal)

Figure 6. Frequency width, width change, and centroid for Frame A (TCU071 600 gal)

4.3 Degree of nonlinearity (Frame A)

The degree of nonlinearity of the structure subjected to TCU071 seismic data is shown

in figure 7 the peak reaches 1.83 for TC071-200 gal. As shown in figure 8, the structure

frequency changed significantly for TCU071-400 gal. The degree of nonlinearity peak

reaches 7.92. In figure 9 the peak value has reached 11.11 for TCU071-600 gal. From

figure 10 the peak value decreased to 2.69 for TCU071-800 gal. The results of TCU071

time history for different PGA’s shows that when the structure becomes unstable, the

0 10 20 30 40 50 60 700

2

4

6

8

time(sec)

Hz

Frequency width from Case2-TCU071-200-AX5ab-AX0-1pair-0-8hz

0 10 20 30 40 50 60 700

1

2

3

4Frequency width variation from Case2-TCU071-200-AX5ab-AX0-1pair-0-8hz

time(sec)

Hz

0 10 20 30 40 50 60 700

2

4

6

8Frequency centroid Case2-TCU071-200-AX5ab-AX0-1pair-0-8hz 0~29sec mean =4.112 std =0.75894

time(sec)

Hz

outside 50% peak left

inside 50% peak left

peak value

inside 50% peak right

outside 50% peak right0 10 20 30 40 50 60 70

0

2

4

6

8

time(sec)

Hz


0 10 20 30 40 50 60 700

2

4


time(sec)

Hz

0 10 20 30 40 50 60 700

2

4

6


time(sec)

Hz



peak value


outside 50% peak right

0 10 20 30 40 50 60 700

2

4

6

8

time(sec)

Hz


0 10 20 30 40 50 60 700

1

2


time(sec)

Hz

0 10 20 30 40 50 60 700

2

4

6


time(sec)

Hz



peak value



0 10 20 30 40 50 60 700

2

4

6

8

time(sec)

Hz


0 10 20 30 40 50 60 700

1

2

3


time(sec)

Hz

0 10 20 30 40 50 60 700

2

4

6


time(sec)

Hz



peak value



9

degree of nonlinearity DN1 rises. After failure, the degree of nonlinearity DN1 decreases

again.

Figure 7. Degree of nonlinearity DN1 for Frame A (TCU071 200 gal)




Figure 11 shows the results after the structure was subjected to TCU071 seismic data.

The trend of the structural frequency declines and the maximum value of the degree of

nonlinearity occurs at TCU071-600 gal. In addition, it can be seen in figure 11 that the

acceleration at the top layer does not amplify the acceleration for TCU071-800 and

0 10 20 30 40 50 60 700

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Degree of nonlinearity Case2-TCU071-200-AX5ab-AX0-1pair-0-8hz max =1.8261 mean =0.16871 std =0.23301

time(sec)

No

nlin

eari

ty(H

z/H

z)

0 10 20 30 40 50 60 700

1

2

3

4

5

6

7


time(sec)

No

nlin

eari

ty(H

z/H

z)

0 10 20 30 40 50 60 700

2

4

6

8

10


time(sec)

No

nlin

eari

ty(H

z/H

z)

0 10 20 30 40 50 60 700

0.5

1

1.5

2

2.5


time(sec)

No

nlin

eari

ty(H

z/H

z)

10

1000 gal. This effect is due to the main frequency of the structure dropping to about

1.5Hz, which is lower than the main frequency band of seismic energy of 2.7Hz. Table

1 and table 2 compare the frequency-centred centroid and PFA of Frame A. As shown

in figure 12 it, appears that DN2 has a trend of structural damage causing DN2 to rise,

which is the biggest difference from the result of DN1. It is notable that the frequency

result of DN2 appears to cross with each other in TCU071.

Figure 11. DN1; Structure frequency, Figure 12. DN2; Structure frequency

PFA comparison chart for Frame A (TCU071) PFA comparison chart for Frame A (TCU071)

Table 1. Frame A, First experiment Degree of nonlinearity 1, Structure frequency, PFA comparison

chart (TCU071)

Frame A

TCU071 Peak of DN1

Mean of

DN1

Mean of Frequency

Centroid (Hz)

Peak Floor

Acceleration(g)

50 gal 2.416 0.206 4.247 0.154

200 gal 1.826 0.169 4.112 0.572

400 gal 7.915 0.371 3.070 0.998

600 gal 11.112 0.404 2.146 1.015

800 gal 2.691 0.294 1.587 0.720

1000 gal 3.826 0.281 1.504 0.413

Table 2. Frame A, First experiment Degree of nonlinearity 2, Structure frequency, PFA comparison

chart (TCU071)

Frame A

TCU071

Mean of Frequency

Centroid (Hz)

Std. of Frequency

Centroid (Hz) DN2

Peak Floor

Acceleration(g)

50 gal 4.247 0.474 0.173 0.154

200 gal 4.112 0.474 0.185 0.572

400 gal 3.070 0.414 0.291 0.998

600 gal 2.146 0.473 0.299 1.015

800 gal 1.587 0.277 0.327 0.720

1000 gal 1.504 0.486 0.277 0.413

11

5. Conclusion

This study applies the time-frequency domain amplification function based on Hilbert-

Huang transform for structural health monitoring. It uses the changes of the main

frequency width of the structure, the main frequency centroid, and the structural energy

for the calculation, and the degree of nonlinearity is proposed. Furthermore, Hilbert-

Huang transform analyse the structure signal and obtains the dynamic characteristics of

the structure under the influence of earthquakes. Thus, the variation of energy,

frequency and time of the Hilbert energy spectrum with the time-frequency domain

amplification function helps to further understand that the structure is affected by the

structural degradation during earthquakes. By observing the changes of the main

frequency, the degree of nonlinearity and the peak response of the structure acceleration

an unstable state of the structure can be easily identified. In this study, the proposed

method for accelerating signal measured on a shaking table test demonstrate the

importance and effectiveness of the new time-frequency dynamic parameter the “degree

of nonlinearity”. Based on the analysis results obtained from the structural experiment,

both degree of nonlinearity parameters can be used for early warning of possible

structural damage.

References

1. Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, S. H., Zheng, Q., Tung, C. C.

and Liu, H. H. “The empirical mode decomposition method and the Hilbert spectrum for non-stationary time series analysis”, Proc. Roy. Soc. London, A454,

903-995, p 2, 1998.

2. Su Sheng Chung, The Hilbert-Huang Transformation Structural Health

Monitoring method for Structure Strong Motion Records ,National Central

University Department of Civil Engineering Graduate School, PhD thesis, pp 4-6,

2015.

3. Huang, N. E., M. T. Lo, Z. H. Wu, and X. Y. Chen, “Method for Quantifying and Modeling Degree of Nonlinearity, Combined Nonlinearity and Nonstationarity”, United States Patent Application, Pub. No. US 20130080378A1, p 5, 2013.

4. Wu Z. and Huang N. E., “Ensemble Empirical Mode Decomposition: a noise-

assisted data analysis method”, Centre for Ocean-Land-Atmosphere Studies, Tech.

Rep., No.173, p 3, 2004.

5. Dazin, P. G., “Nonlinear Systems,” Cambridge University Press, Cambridge, 1992.

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time Structure Health Monitoring by Nonlinearity Degree ...2. Structural Health Monitoring System 2.1 Instantaneous Frequency Traditionally, Fourier spectrum uses sine and cosine harmonic