INVESTIGATION AND ESTIMATION OF STORYSTIFFNESS AND DAMPING RATIO OF A 4-STORYWOODEN STRUCTURE BY MICROTREMOR AND
MASS SLIDING SHAKER
Sithipat Palanandana∗
MEE07157

Supervisor:

Koichi Morita**

ABSTRACTIn this study, the structural dynamic properties of a 4-story wooden structure were investigated.
Microtremor observation was carried out to determine the first and second natural frequencies of thestructure. Based on these preliminary results, mass sliding shaker was mounted on the structure to
apply harmonic force vibration in order to excite the resonant behavior of the structure. Target
frequencies were applied to the mass sliding shaker to obtain the responses for all four modal shapes.
Signal obtained from the measurement was then mathematically converted from time domain tofrequency domain using Fast Fourier Transform for better interpretation to determine all four natural
frequencies. Based on these results, resonance curves for all four modal shapes were constructed.
Curve fitting based on mathematical formula was also drawn by changing the values of damping ratio
and peak amplitude to fit the resonance curve. From microtremor observation data and time history
responses of free vibration, damping ratio of the structure was determined for comparison. Based on
curve fitting of resonance curves, natural frequencies and participation functions were also identified.
Using story mass in addition to these identified properties, stiffness matrix could be determined.
Finally, each story’s stiffness was obtained by multiplying stiffness matrix by unit displacement vector.

Keywords: Microtremor, Fourier Spectrum, Resonant curve, Damping ratio, Story stiffness.
INTRODUCTION
For the past three decades, physical characteristics of structural systems have been focused on by
many researchers and this is done through the testing procedure known as system identification. To be
more precise, system identification can be described as mathematical tools and algorithm representing
a dynamical model from measured data. This dynamical model is then used to identify the propertiesof structural systems in which researchers are interested. In the field of Earthquake Engineering, theproperties of structural systems are essential to be identified. These properties are related to structural
response subjected to the ground motion. In order to identify the physical characteristics of a structural
system, microtremor observation has been widely used for decades. Microtremor observation
generally provides clear amplification ratios for first and second natural frequencies. With ambient
noises, the amplification ratios for higher mode are inaccurate or sometimes misinterpreted. The force
vibration technique is then introduced especially for light structure such as wooden structure. Previous
studies have shown that story stiffness of the structures are commonly obtained based on the values offirst and second natural frequencies. For multi-degree of freedom structures, it is desirable to obtainnatural frequencies for higher modes in order to give more accurate story stiffness. The purpose of thisstudy is to estimate damping ratio and story stiffness of a 4-story wooden structure by using force*Department of Public Works and Town & Country Planning, Thailand** Senior Research Engineer, Building Research Institute, Tsukuba, Japan1

vibration to stimulate the resonant behavior of the structure. Force vibration test also provides in depth
modal analysis of the structure to obtain dynamic properties of the structure.

THEORY AND METHODOLOGY
System Configuration of the Building
The vibration of the building/structure during earthquake and microtremor is composed of several
different types such as sway, rocking, relative displacement and several modes such as translations,torsion and those from higher modes. In general, the modes considered in vibration experiment arevaried depending on type of building and the height of the building. In this study, the 4-story woodenbuilding is investigated. Several high-sensitive accelerometers are placed at various positions inside
the building. Each sensor is designated to catch the vibration signal along each story of the building as
shown in Figure (1).
In general, the vibration of thebuilding consists of rocking,sway and relative displacement.
Horizontal Measurement
Each sensor was designated tomeasure the related motion.
Rocking motion represents the
Vertical Measurement
different vertical displacement of
the building. Sway motion is arelative displacement between
ground and the building.
Relative displacement represents
the lateral deformation of each
Figure 1. Sensors’ location in wooden building
story with respect to another. For
microtremor measurement, there are some assumptions needed to be made such as the floor is rigid,
and ground surface is slightly separated from the building so that it is not affected by soil-structureinteraction and so on.
Resonance Behavior of the Structure
The multi-degree of freedom structures possess several natural frequencies mostly corresponding to
number of degrees of freedom. When the structure is subjected to the excitation, the resonanceresponse is induced if the frequency of the excitation is close to each one of natural frequencies of thestructure. In this study, a mass sliding shaker is used to implement the force function to the structure.
The resonance test is carried out by changing the target shaking frequency using the oscillator
generating the sinusoid function. Input and output responses are measured and data post-processing is
implemented. The plots of ratios between input and output are fitted by the theoretical resonance curve.
To illustrate this resonance behavior mathematically, the harmonic force at the forth story can be
expressed as follows.

F = f cos ωt = ( mc a ) cos ωt(1)

in which mc is the mass of the shaker and ω is frequency of input motion of the shaker. Output
displacements are given by.
4{y} = ∑ s u4 {s u} m2c a [A cos s ωt − B sin ωt ]

ω sM
Therefore, the output acceleration is expressed as {&y&} = −ω 2 {y}.
s =1

s

2

(2)

If only sth modal properties at ith story are extracted, response amplification in acceleration
can be given as follows.
2 ω  mc&y&i
= s u 4 s ui &x&c
 sω  s M

2

ω 
m

 s β sui cs A +s B ≈ m4 sω 
22s

A2 + s B 2(3)

Therefore,2&y&i  ω m s β sui c= &x&c  s ω 
m4

1

(4)

2

2
  ω 2 
2ω

1
−
+
4
h





s
ω 
 sω   s  

where &x&c , s ω , s β s u i , and m4 are input acceleration, natural frequency of the structure, participation
function, and the 4th-story mass respectively.
Force Vibration Test Using Mass Sliding Shaker
In this study, mass sliding shaker or alternatively called Linear Shaker Seismic Simulation (LSSS) wasused to induce the vibration on the structure. The installation of the shaker was done by fully attached
the shaker at the middle of the roof floor using bolts and nuts. The reason is that the shaker can beconsidered as a part of the structure. All nine accelerometers were wired to the measuring device tomeasure the response at designated locations
on the structure. One of nine sensors was
1
attached on the shaker to capture the input
motion given by the amplifier and function
2
generator. The overview of the system3
configuration is shown in Figure 2.
The function generator (shown in Figure 2)
4
generates the motion of the shaker at
specified frequency and the function used
was the sinusoidal function. The amplifier5
was used to increase the magnitude of the
87
69
input signal by selecting appropriate
bandwidth for better performance of the
shaker.
Figure 2. Overview of Shaker System Configuration
DATA ANALYSIS AND RESULTS
Microtremor observation was carried out initially to investigate the fundamental natural frequencies ofthe structure. The measurement was preferably done at midnight to minimize the unwanted ambient
noise especially from the traffic. The measurement was recorded for 30 minutes to ensure the stable
response of the structure. The sampling frequency was 200 Hz so the total number of data was 360,000.
Once the raw data was converted to readable format, the response from each story can be displayed intime domain and frequency domain as shown in Figure 3 and 4.
From Figure 4 shown above, two obvious peaks of response can be observed. This implies thatthe first and second natural frequencies of the structure are most likely able to be spotted from thegraph by projecting these two peaks on the x-axis. By doing so, these natural frequencies were foundto be approximately 3.510 Hz and 11.730 Hz (dotted circles in Figure 4) for first and second natural
frequencies respectively. These two frequencies were then used for the primary setup for the shaking
frequency to determine the resonance frequencies. Range of target frequencies was applied to theshaker to identify all 4 natural frequencies. Fourier spectrum for each frequency was determined and3

Figure 3. Time history response for each story
Figure 4. Fourier spectrum for each story
ratio between response of each story and shaker was plotted to form the resonance curve for each story.
Then, by using Eq. (4), curve fitting can be done by selecting the values of damping ration and peak
amplitude for the best fit to all resonance curves as shown in Figure 5.
Based on the curve fitting
method, damping ratio
2nd Mode
1st Modeand all of the participation
functions were obtainedcorresponding to eachstory for each mode.
Phase lag of the response
for each story was also
3rd Mode
4th Mode
determined to identify themodal shapes of thestructure. The values ofparticipation
functionswere used to plot themodal shapes as shown in
Figure 5. Resonance curves for all 4 modes
Figure 6.
0.01

0.025

1F amp1F amp0.009

fitting1

fitting1

0.02

2F ampAmplification

Amplification

fitting2

3F amp0.015

fitting3

4F ampfitting4

0.01

0.008

2F amp0.007

fitting2

3F amp0.006

fitting3

4F amp0.005

fitting4

0.004
0.003

0.005

0.002
0.001

0

0

3

3.2

3.4

3.6

3.8

4

9.8

10

10.2

10.4

10.6

11

11.2

11.4

11.6

11.8

12

0.025

0.03

1F amp1F ampfitting1

fitting1

0.025

0.02

2F amp2F ampfitting2

0.02

Amplification

fitting2

Amplification

10.8

Freq.[Hz]Fre q.[Hz]3F ampfitting3

4F amp0.015

fitting4

0.01

3F amp0.015

fitting3

4F ampfitting4

0.01

0.005

0.005

0

0

22.5

23

23.5

24

24.5

25

25.5

26

26.5

27

27.5

37

28

37.5

38

38.5

39

39.5

40

Freq.[Hz]

5

5

5

4

4

4

4

3

3

3

3

F
lo
o
r

5

F
lo
o
r

Floor

Freq.[Hz ]

Flo
o
r

22

2

2

2

2

1

1

1

1

0
0.02

0.04

Par ticipation
Function

-0

0

0

0

0

0

-0.1

0.02 0.04

0

0.1

Par ticipation
Function

Par ticipation
Function-0.0020

0.002

Par ticipation
Function

Figure 6. Participation Functions and Mode shapes

DAMPING RATIO AND STORY STIFFNESSHalf Power Method
In general, transfer function has a similar shape like a resonance curve. As described earlier, transfer
function can be acquired from input and output data of microtremor observation. The frequency at thehighest peak is natural frequency, f0. .
The simplest method to estimate a damping ratio of a structure is called “Half Power Method”.

4

In Figure 7, f1 and f2 can be acquired by projecting the value of 1 / 2 amplitude of the highest peak on
the frequency axis. Then damping ratio can be estimated by

f 2 − f1 ω 2 − ω1 ∆ω
=
=
2 f0
2ω0
2ω0

h=
7

(5)

1.5

x1
x2
1

S6
5

0.5
Amplitude, aAmplitude

S/ 24
3

0

-0.5
2

-1

1

0
0

1

2

f1 f2
3
f0 4

5

6

7

-1.5

8

0

10

20

30
40
Time, sec

50

60

70

Frequency, Hz

Figure 8. Free vibration

Figure 7. Transfer function
Random Decrement Technique

Curve fitting method and half power method are based on frequency domain, whereas time history
domain methods are also used in several aspects. Considering two peak values (Figure 8) measured at
two different positions with time interval Td, the ratio of these two peaks x1 and x2 can be obtained asfollowsx2
2π
2π= exp( − hω 0Td ) = exp( − hω0) = exp( − h)x1
ω0 1 − h2
1 − h2(6)

Alternatively the damping ratio h can be approximately related to the logarithmic decrement as

x 
1  x1 
ln  1  ≈ln  (7)2π
 x 2  2π  x 2 
In most practical cases, several sets of logarithmic decrements are used for averaging. In general, time
history domain may not be in the form of what is shown in Figure 8. The random signal is most likely
presented for the structural response. In this method, the input signal can be formed by manysinusoidal waveforms. Then by superposing many part of output waveforms, only free vibration
decrement can be acquired.
In this study, free vibration was obtained by inducing force vibration using the shaker and thenthe input motion is suddenly stopped providing free vibration afterward. This free vibration can beanalyzed based on the concept of logarithm decrement. However, for the higher mode, free vibration
response cannot be identified due to the ambient noise from the traffic. Therefore, in this section, 1stand 2nd mode free vibration response were considered to obtain damping ratios. Using all methods
stated previously, average of damping ratios can be summarized as shown in Table 2.
h=

Story
Average

1 − h2

Table 2. Damping ratios calculated by three methods
Half Power Method
Random Decrement
Free Vibration
1st Mode

2nd Mode

1st Mode

2nd Mode

1st Mode

2nd Mode

0.036

0.038

0.034

0.040

0.036

0.042

5

Estimation of Story Stiffness
By using the modal and spectral matrices, stiffness matrix can be given as follows:[K ] = ([Φ ]norm [Ω 2 ]−1 [Φ ]Tnorm )

−1

4

Story

3

2

1

0

2000

4000

6000

8000

10000

Stiffness(8)

where [Φ ] is the modal matrix consisting oftransfer functions obtained from previous
section and Ω 2 is spectral matrix consistingof diagonal values of eigenvalues, ω n2 . To
determine each story’s stiffness, theconcentrated equivalent story stiffness will be
estimated by multiplying stiffness matrix [K]by the displacement vector whose elements
are equal to 1. Each story’s stiffness can thenbe obtained as shown in Figure 9.

Figure 9. Story Stiffness
CONCLUSIONS
Results from this study can be summarized as follows.
• Microtremor observation provides adequate information for fundamental natural frequenciesof the wooden structure. The first and second natural frequencies are found to be 3.51 Hz and11.73 Hz.
• The target frequencies are applied to the shaker to identify the resonance behavior of thestructure. The resonant frequencies are found to be 3.42 Hz, 11.4 Hz, 25.45 Hz, and 38.45 Hz
for first, second, third and forth modes respectively. It is noted that the resonant frequencies
are slightly smaller than those identified by microtremor observation.
• Damping ratios are obtained based on several methods to compare the values. Results fromthese methods show that damping ratios do not vary corresponding to each mode.
• The forth-story stiffness calculated is quite small comparing with the others. The reason is that
only shear deformation is considered in this study, whereas bending and torsional
deformations are ignored.
Story stiffness and damping ratio can then be analyzed to provide the dynamic properties ofthe structure. These dynamic properties are used as the guideline to justify how sufficient the structure
is and how the structure will perform during the earthquake.
ACKNOWLEDGEMENT
I would like to express my sincere gratitude to Dr. T. Kashima, my advisor, for his continuous support,
valuable suggestion and guidance during my study.
REFERENCES
Hamamoto, T., 2003, Proceeding of SPIE, 5057, 106-117
Koyama, S., 2007, IISEE lecture note, Building Research Institute, Tsukuba, Japan
Churn, E. P., 2006, Master thesis, Dankook University,South Korea.
Morita, K., Teshigawara, M. and Hamamoto, T., 2005, Journal of Structural Control and Health
Monitoring, 12, 357-380
Okawa, I., 2007, IISEE lecture note, Building Research Institute, Tsukuba, Japan
Niousha A., and Motosaka, M., 2007, AIJ Journal of Structural Engineering, 53, 297-304

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