Field test of resonant frequency monitoring utilizing background vibration invarious buildings
Yoshinori Takahashi(1) , Naru Sato(2) , Yasutaka Nakajima(3)(1) Tokyo Metropolitan College of Industrial Technology, Japan, yoshinori@ieee.org(2) RION, Japan, n-satou@rion.co.jp(3) RION, Japan, yasutaka@rion.co.jp

Abstract
A number of big intermittent aftershocks often follows a great earthquake. Even if a certain building escapesfrom the collapse due to the ﬁrst main quake, there is no guarantee that the building can survive from theaftershocks. If it is possible to monitor the condition of the buildings constantly using some faint random
vibration caused by background vibration such as wind, weak earthquakes and trafﬁc vibration, the safety will be
more secured. The authors have proposed cumulative harmonic analysis that emphasizes the hidden resonances
in a time-invariant transfer function, and conducted the experimental monitoring of the resonant frequency inan actual building from the analysis of the faint random vibration on the building caused by the backgroundvibration. From the results of comparison with the measured values of the resonant frequency by an active
diagnosis it is conﬁrmed that the estimation of the resonant frequency by the proposed method obtains an
accurate value only including few percent error. This report describes about the results of ﬁeld tests on various
buildings using the proposed method.
Keywords: Structural Health Monitoring, Cumulative Harmonic Analysis, Microtremor Measurement,
Resonant Frequency, Signal Processing

1

INTRODUCTION

In recent years, the earthquake resistance standards
are deﬁned in Japan so that any buildings do notcause their collapse to harm to human life at theintensity 7 on the Japanese scale[1]. However, thebuildings deteriorate by being exposed to the naturalenvironment for a long period of time. Before having a fatal damage caused by natural disasters suchas earthquake, it is necessary to ﬁnd the degradation
by diagnosing the structural health of the buildings.
In addition, it is known that a number of big intermittent aftershocks often follow a great earthquake.
Even if a building escapes from the collapse due tothe ﬁrst main quake, there is no guarantee that thebuilding can survive from the aftershocks. However,
it is difﬁcult to judge whether to evacuate from thebuilding to prepare for aftershocks right after the big Figure 1. Structural Triage Information Sharing Netearthquake. If it was possible to monitor the struc- work.
tural health of high buildings and to build the network named Structural Triage Information Sharing Network (Figure. 1) that can share the damaged situation ofbuildings in a city, evacuation sites will be easier to secure. tunnels and bridges just like ﬁre alarms continuously, it would be possible to ensure the residents’ safety [2].

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As the most basic diagnostic method, there is a method of observing vibration transmitted to a distant position
by exciting a certain point using a test signal [3]. Abnormal conditions in buildings can be detected as changes
in damping ratio and resonant frequency. With such a method, it is possible to diagnose the building only when
the building is excited by the test signal. If it is possible to estimate the characteristics of the building vibration
utilizing some faint random vibration caused by environmental noise such as wind or trafﬁc noise, the structural
health can be monitored constantly. In a ﬁeld of machinery diagnostics, although several diagnostic methods
using vibration based on environmental noise have been proposed, those methods require that the characteristicsof the noise must be known since the resonance of the object may be buried by the environmental noise
spectrum [4, 5]. In the ﬁeld of architectural engineering, there are researches such as the structural diagnosis
based on the microtremor measurment[6] performed in a time zone where there is little human activity in thebuilding or little wind or environmental noise. However, it is difﬁcult to apply the structural diagnosis based on
the microtremor measurment in the city center where there are many environmental noises day and night.
On the other hand, the diagnostics methods utilizing unknown random vibration have been proposed without
requiring the excitation by a test signal. These techniques emphasize the resonant frequency of the building
buried in the observed noise spectrum using faint vibration due to environmental noise. In the past, a method
to monitor invisible changes in building has been proposed, which is based on the distribution of the dominantfrequency for the short-interval periods (SIP) in observed nonstationary vibration excited by unknown source[7]. By utilizing the frequency distribution of the dominant spectrum excluding the amplitude value, SIP canelucidate the harmonic structure of stationary transfer function buried in time variant source spectrum.
However, the efﬁciency of SIP is not clearly shown for resonant frequency estimation of transfer function under
the condition of stationary noise with little temporal change in the spectrum distribution of the source signal.
Therefore, the authors have proposed passive diagnostics [9] (shown in Fig.1) of buildings by using cumulative
harmonic analysis (CHA) [8]. CHA is an analysis method to emphasize the time invariant spectral peaks byutilizing windowing function with the effect of spectrum accumulation over a period of time and was originallyproposed for howling prediction problem. In the study of howling prediction utilizing CHA, it is conﬁrmed thatthe method can detect a faint spectrum peak which may cause the howling even for stationary noise input[10].
Therefore, CHA should be applicable to passive structural diagnostics. Previous works have shown that thechange of resonant frequency and equivalent bandwidth in the buildings can be estimated through model experiments [11]. Furthermore, they have attempted to estimate the resonant frequency of an actual building byobserving the random vibration [12].
本報告では．
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2

TIME-INVARIANT POLES ESTIMATION UTILIZING CUMULATIVE HARMONICANALYSIS

For the passive diagnosis proposed in this paper, it is necessary to estimate the spectrum of the time-invariant
transfer function hidden in the harmonic structures embedded in the harmonic structure of stationary noise. Thissection introduces cumulative harmonic analysis (CHA)[8] which is effective to solve such a problem.
CHA is formulated by introducing a spectrum accumulation into cumulative spectral analysis (CSA) which was
proposed by Barman and Fincham [13]. As shown in Figure 2 (a), CSA is a time frequency analysis that
expresses the generation process of spectrum for each sample of input signal, and it was proposed to visualize
the generation process of the harmonic structure in the impulse response of loudspeaker. When the input signal
is deﬁned as s(n) and the angular frequency is Ω, CSA can be expressed asn

CSA(n, Ω) ≡ ∑ w(m)s(m)exp(−jΩm)

(1)

m=0

using the step function w(m) = 1 (m ≤ n), w(m) = 0 (m > n).
On the other hand, CHA is an analytical method to emphasize time invariant spectrum peak by accumulating

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(a)

(b)

(c)

(d)

Figure 2. Schematics of the CSA (a) and CHA (b), (c), (d) for a N point input signal. The CSA (a) shows
the generating process of the spectrum according to the input signal by enumerate these spectra. The CHA(b) accumulates these spectra obtained by the CSA. The result of CHA for n = N − 1 in (b) is equivalent to(c) performing a single N-point DFT by applying a triangular window to the input signal. When a triangular
window that increases according to time is used, the results of the cumulative spectrum will be weighted to the
decay part of the input signal (d).
1
0
-1
0

10

20

30

40

50

3
2
1
0

DFT
0

0.2

Sample (n)

0.4

0.6

0.8

1

Angular frequency (π rad.)

(a)

(b)

30
20
10
0

CHA
0

0.2

0.4

0.6

0.8

1

Angular frequency (π rad.)(c)Figure 3. An example of impulse response and the amplitude spectra computed by the DFT and the CHA. (a):Input signal has one pair of complex conjugate poles and zeros. (b): The pole and zero are conﬁrmed as aspectral peak and a dip in the amplitude spectrum. (c): The CHA sharpens the steeper spectral peaks without
the spectral dips.
spectra obtained in every sample using CSA. CHA algorithm shown in Fig. 2 (b) requires, N times of the DFT
to analyze the accumulated spectrum of N points signal. However, since the DFT is a linear operation, theanalysis process shown in Fig. 2 (b) is equivalent to Fig. 2 (c). This means that only one DFT calculation isrequired to obtain the accumulated spectrum of points signal. The calculation of accumulated spectrum for the
input signal s(n) can be expressed asn

CHAD (n, Ω) ≡ ∑ (n − m + 1)s(m)exp(−jΩm)
m=0

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(2)

by the DFT of the signal s(n) that the triangular window was applied. The triangular windowing shown inequation (2), which decreases on the time axis, can be regarded as weighting to old signals on the time axis.
On the contrary, if the triangular window (Fig. 2(d)) increases along the time axis such asn

CHAU (n, Ω) ≡ ∑ (m + 1)s(m)exp(−jΩm)

(3)

m=0

it becomes weighting to the latter part of the signal. It is known that the increasing triangular window has theemphasizing effect of the decaying part of the input signal. It has also been demonstrated that CHA corresponds
to the derivative of S(Ω) by the angular frequency Ω as,
n

dS(Ω)

∑ (m + 1)s(m)exp(−jΩm) ≃ j dΩ

(4)

m=0

where S(Ω) is the DFT of s(n) [14].
Considering the input signal having one zero at the position of r exp(jΩz ) (0 < r < 1) on the z-plane, the spectrum of the input signal is shown as S(Ω) = 1 − r exp {−j(Ω − Ωz )}, where Ωz is the angular frequency r is the
distance from the origin. In this case, CHA power spectrum becomes as|CHA(Ω)|2=

d {1 − r exp {−j(Ω − Ωz )}} 2
= r2
dΩ

(5)

This means that the power spectrum is no longer the function of Ω.
On the other hand, for a signal having one pole S(Ω) =comes as1
1 − r exp −j(Ω − Ω p ), CHA power spectrum be-r2|CHA(Ω)|2 = ,2
1 − 2r cos(Ω − Ω p ) + r2

(6)

where Ω p denotes the angular frequency of the pole, and the distance between the origin and the pole r is
0 < r < 1. Consequently, CHA sharpens the steeper spectral peaks and removes the spectral dips.
Figure 3 shows an example of a comparison between CHA and the DFT. The input signal shown in Fig. 3 (a)is given as the inverse Fourier transform of the transfer function S(Ω) with complex conjugate poles and zeros,nono
1 − r exp−j(Ω−Ωz ) 1 − r expj(Ω−Ωz )S(Ω) = ,(7)
1 − r exp−j(Ω−Ω p ) 1 − r expj(Ω−Ω p )where r = 0.9, Ωz = π /4 and Ω p = π /3. The amplitude spectra based on the DFT and CHA are shown in Fig.
3 (b) and (c), respectively.

3

ESTIMATION ALGORITHM FOR RESONANT FREQUENCY OF BUILDINGThis chapter explains the estimation method of resonant frequency of a building utilized in passive diagnosis[11, 12]. Figure 4 shows an outline of a proposed method for estimating the resonant frequency from faintrandom vibration in a building. The acceleration pickup installed in the building observes faint random vibration
excited by the environmental noise. The observed vibration is divided into frame signals by rectangular windowfunction. In this paper, the frame length is 5 seconds, and the frame shift length is 2.5 seconds. The timeinvariant spectral peak of each frame signal is emphasized by CHA. The frequency of the spectral peak above
a ﬁxed value is selected from CHA spectrum of each frame signal, in order to create a frequency histogram.
In this paper, the CHA spectral peaks more than 25%of the maximum value in the amplitude are selected tocreate histograms in every hour. The mode of the frequency on the histogram monitored in every hour is theestimated value of the resonant frequency in the building.

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Figure 4. Resonant frequency estimation algorithm for structural health monitoring.

4

MONITORING OF RESONANT FREQUENCY IN ACTUAL BUILDING

4.1 Experiment in reinforced concrete building and its evaluation

(a)

(b)

Figure 5. Schematic of the experimental setup on the reinforced concrete building. (a): Plane view of the roofof building, (b): Side view of the building.
A monitoring experiment of the resonant frequency over a long period of time was conducted at 8th ﬂoors
of a reinforced concrete building (Arakawa campus in Tokyo Metropolitan Industrial Technology) as shown in
Figure 5. The target building has a whole structure where three rectangular spaces are connected to the triangle
center part (See Fig. 5 (a), (b)). The monitoring experiment was conducted at the one of the rectangular spaces.
Fig. 5 (c) shows the photos of two installed acceleration pickups (PV-87, RION) and recorders (DA-21, RION).
The acceleration pickups were installed on the rooftop so that the vibration amplitude of the building was thelargest and the acceleration was observed easier. In addition, as the acceleration pickups were installed for thelong side direction (X - axis) and the short side direction (Y - axis) respectively along the rectangular structure,
the vibrations were observed towards each direction. The monitoring experiment for the resonant frequency had
been carried out for a month from September 6, 2016 to October 5, 2016.
Figure 6 shows CHA peak frequencies in every hour obtained by the passive diagnostics method shown in
Section 3. The black dots in each histogram indicate the mode, which corresponds to the estimated resonant
frequency of the building. Table 1 shows the average, the standard deviation and the relative error of estimations
in each direction. However, the estimated values deviating from the average value (larger than ±5 Hz from theaverage value) in the result of Fig. 6 were excluded. These results have indicated that almost stable estimation
results were obtained throughout the experiment.
For evaluate the estimated value, it is discussed about the measurement of the resonant frequencies in thebuilding by active diagnostics utilizing the impact by a weight, and compared the estimated values obtained in

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Figure 6. Histogram map of the CHA major peaks in every hour for the reinforced concrete building. The
colormap indicates the frequency of the histogram. The black dots indicate the mode in each histogram, andthe frequency means the estimated resonant frequency of the building.
Table 1. Average and standard deviation of the estimation for the reinforced concrete building.
AverageStandard deviation
Relative error

X-axis2.00Hz2.56 × 10−2 Hz
1.28%

Y-axis
2.74Hz4.45 × 10−2 Hz
1.62%

the last chapter with the measured values in the active diagnostics. In this study, the active diagnostics were
performed utilizing an impactor that can excite a structure faintly by hitting with a suspended weight. A voidtube, which is ﬁlled with 20kg of sand bag as a weight, was put in the suspended green bag to prevent the
sandbag from absorbing the shock at the impact. The weight was hanged with a rope so that the length fromthe fulcrum to the center of the weight becomes about 1.2m. Then, the weight was lifted to such an extent thatthe angle between the pillar of structure and the rope becomes 30 degrees, and the collision was caused withthe pillar in the manner of a pendulum. For the purpose of obtaining as large moment of force as possible witha small impact, the excitations were given at a pillar on the top ﬂoor.
Excitations were performed 12 times, and the average and standard deviation were obtained. In the experiment,
the resonant frequency of the building was measured as 2.03 ± 0.13 Hz along the X - axis and 2.79 ± 0.12 Hz
along the Y - axis. Assuming that the measured values by the experiment are the real resonant frequency of thebuilding, the relative error with the estimation value given in the previous chapter was less than 2%. Therefore,
it can be conﬁrmed that the proposed method is effective for the monitoring of the resonant frequency for thebuilding.
4.2 Experiment in steel building
Furthermore, this work conducts a monitoring experiment of resonance frequency in a 5th ﬂoors of steel building(including several reinforced concrete columns) was also conducted as shown in Figure 7. In this experiment,
two servo vibration pickups (RION, LS-10C) were installed for the long side direction (X - axis) and theshort side direction (Y - axis) respectively along the rectangular structure. The monitoring experiment had been
carried intermittently for a total of 22 days from Novenber 22, 2018 to Feebruary 21, 2018.

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Figure 7. Schematic of the experimental setup on the steel building.

Figure 8. Histogram map of the CHA major peaks in every hour for the steel building.
Figure 8 shows CHA peaks frequencies in every hour obtained by the same analysis shown in section 4.1.
Table 2 shows the average, the standard deviation and the relative error of estimation in each direction. It canbe conﬁrmed that the estimation results are stable as in the experimental results of reinforced concrete building.
The Building Standard Low of Japan uses a formula T = (0.02 + 0.01α )h to obtain the approximated value ofthe natural period of a building, while h m is the height of the building and α is the ratio of the height of steel
structures to the total height of the building. Although the actual resonant frequency is unknown because noactive diagnostics have been performed for this building, the approximated value of the resonant frequency forthe steel building can be obtained 1/T = 1.36 Hz, where substituting h = 24.5 m and α = 1.0 for the formula.
From this calculation, it is conclude that the adequate estimation result is obtained in this experiment.
Table 2. Average and standard deviation of the estimation for the steel building.
X-axisY-axisAverage
1.23Hz1.27HzStandard deviation 0.75 × 10−2 Hz
2.96 × 10−2 Hz
Relative error
0.61%
2.33%

5

CONCLUSIONSIn this paper, the resonant frequency of the actual building was monitored for a long period of time using
the proposed passive diagnostics method. This monitoring test was conducted for reinforced concrete building

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and steel building. As a result, although it was found that the estimated values occasionally deviated fromthe estimated average, it was conﬁrmed that the resonant frequency were stably estimated. In addition, for thereinforced concrete building, the resonant frequencies of the building were measured by active diagnostics using
the impact by a weight, and the estimated values were evaluated with the measured values. In the experimentof steel building, the adequacy of the result was conﬁrmed by calculation of the natural period of a building
based on the Building Standard Low of Japan. It is concluded that the proposed method is effective for an
estimation of the resonant frequency of buildings.

ACKNOWLEDGEMENTS
This research is supported by JSPS KAKENHI Grant Number 17K14775.

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