5 world conference on structural control and...

5th World Conference on Structural Control and Monitoring 5WCSCM-10376

Le, Nakata, Yoshida, Kiriyama, Naito and Tamura 1

INFLUENCE OF VIBRATION METHODS, STRUCTURAL COMPONENTS AND EXCITATION AMPLITUDE ON MODAL PARAMETERS OF LOW-RISE BUILDING

T.H. Le Tokyo Polytechnic University, Kanagawa 243-0297, JAPAN

[email protected]

S. Nakata Asahi Kasei Homes Corporation Co., Ltd, Shizuoka 461-8501, JAPAN

[email protected]

A. Yoshida Tokyo Polytechnic University, Kanagawa 243-0297, JAPAN

[email protected]

S. Kiriyama

Asahi Kasei Homes Corporation Co., Ltd, Shizuoka 461-8501, JAPAN [email protected]

S. Naito Taku Structural Design Co. Ltd., Japan

[email protected]

Y. Tamura Tokyo Polytechnic University, Kanagawa 243-0297, JAPAN

[email protected]

Abstract

This paper presents the modal parameter estimation using the response measurements from the ambient, free and sine sweep vibration tests. Sine sweep response has been simulated by using frequency-changeable shaker installed on the floor level of the experimental structure, whereas the free vibration response has been measured after application of imposed displacements by the shaker. Comparison and evaluation on the modal parameter estimation from different vibration methods are focused. All these vibration tests and response measurements have been experimentally carried out with a one-storey steel structure in the laboratory. Excitation amplitudes have been changed from small, medium to large amplitude ranges, furthermore, different test cases with changing of structural components have been studied in order to evaluate stiffness and effectiveness of the structural components on the whole structure via the vibration tests. Influences of vibration methods, structural components and excitation amplitudes on modal parameters of experimental low-rise building are going to be investigated.

Introduction Modal parameter identification has become an essential issue for the structural health monitoring and the structural control. Natural frequencies and damping ration are very important and fundamental modal parameters for the structural response prediction, the model updating, the stiffness evaluation and so on. Modal parameter identification has been based on various types of vibration methods like forced vibration, free vibration or ambient vibration tests. In the forced vibration tests, the structures are excited by shaker or impulse hammer, in which sine-swept excitation has been popularly used so far in the vibration tests of civil, aerospace and industrial structures. The forced vibration tests usually involve with measuring both input excitations and output responses. Accordingly, input-output identification methods can be used to estimate the modal parameters of structures through indentifying either the Frequency Response Function (FRF) or the Impulse Response Function (IRF) (Cunha et al., 2006). Forced vibration


tests advantage on the deterministic input excitation which can be controlled and optimized to the response of interested vibration modes. However, the forced vibrations test requires additional equipments like shaker and accompanying instrumentations, traffic shut-down. Thus it is suitable to conduct this test with small-scaled or medium-scaled structures, however, in cases of structures with flexible, large-scaled and low-ranged, closed natural frequencies, the controlling of input excitation for optimized level of response is often difficult and costly. It is noted that output-only identification with measured responses only also can be applied for the forced vibration testing, but effects of input excitation and harmonics must be eliminated from the output response before the modal parameters extracted. The ambient vibration tests using any natural and environmental excitation such as microtremors, traffic vehicles, human activities, wind and wave… does not require the traffic disturbance, shaker and additional instrumentations, but require highly sensitive sensors and data processing techniques. However, with development of sophisticated sensor technology and identification techniques, the ambient testing has become the most practical and reliable method for the modal parameter identification from multipurpose laboratory experiments and full-scale measurements. Because only the output response can be measured without the input excitations in the ambient vibration tests, thus the output-only identification techniques are applied to estimate the modal parameters (Cunha et al., 2006). Free vibration tests are often carried out simply by a sudden release of excited masses in the forced vibration tests. When excitation is tuned and released at the certain natural frequency, structure is treated as single degree-of-freedom (DOF) free vibration system at this natural frequency. In this type of tests, it is discussed that the modal parameters which are derived from the vibration levels often exhibit higher than service levels (Magalhaes et al., 2009). However, quality and reliability of the modal parameter estimation of structures obtained from the different types of excitations have a lack of consensus comparison and need to be more clarified.

Number of mathematical models on the modal parameter identification has been developed for the multi degree-of-freedom (DOF) structures so far, which are often classified into two identification branches in the frequency domain and the time domain. Frequency domain-based identification techniques have been widely used for both the input-output and output-only system identification due to their reliability and effectiveness in estimating high-resolved natural frequencies and mode shapes. The most applicable modal identification techniques for the ambient vibration tests in the frequency domain are Peak Picking (PP) and Frequency Domain Decomposition (FDD). These techniques are favorable and effective for estimating the natural frequencies and mode shapes. The PP (also known as Basic Frequency Domain) is the simplest technique in the frequency domain in which the natural frequencies can be identified by extrema from the auto power spectral density function or transfer functions of the output response, whereas the operational mode shapes are obtained via the operational defection shapes (Bendat and Piersol, 1993). The FDD approached in different way with the cross power spectral density matrix of the outputs and robust techniques (Brincker et al., 2001a), which make the FDD become very powerful technique in the frequency domain for the ambient vibration data, but it is unable to estimate damping ratios. Thus, the FDD has enhanced, known as Enhanced Frequency Domain Decomposition (EFDD) for estimating damping ratios from the free decay identification in generalized coordinate associated with each mode and for validating the natural frequencies also from these free decay functions (Brincker et al., 2001b). The FDD is still improving its refined techniques for more accurate, reliable damping estimation, such the most recent techniques as frequency-spatial domain decomposition (Zhang and Tamura 2003), curve-fitting frequency domain decomposition (Jacobsen et al., 2008) and so on. In case of the free vibration tests, the modal parameter identification techniques are more simple which often apply either the Ibrahim Time Domain (ITD) and the Stochastic Subspace Identification (SSI) for the multi DOF systems or the Logarithmic Decrement Technique (LDT) for the single DOF ones (He and Fu, 2001). In case of the forced vibration tests, the FRFs often are identified based on the measured excitation inputs and measured response outputs, then smooth techniques are applied to eliminate system and instrumentation noises from the estimated FRFs, finally the modal parameters can be obtained from these


smoothed FRFs. Some practical identification techniques in the frequency domain are such as Half Power Bandwidth (HPB) for the single DOF systems and Least-squares Complex Frequency Estimator (LSCF) for multi DOF ones have been used for estimating the modal parameters of structures excited by the forced vibration tests (De Troyer et al., 2009). Due to noise of the excitation inputs and the response outputs, an accuracy of estimation of both the FRFs and the modal parameters significantly depend on estimation and smoothing of the FRFs from the measured inputs and outputs. Originally, the FRFs can be estimated by single block technique as a transfer function between spectra of the input and output signals, however, some another smoothing techniques have been developed such as frequency averaging technique, block overlapping technique or reduced Discrete Fourier Transform (Marchitti, 2006; Orlando et al., 2008; Coppotelli, 2009).

Stiffness evaluation is essential for the low-rise buildings where structural components such as walls, floors and connections strongly influences to the dynamic properties due to their sensitiveness on distribution of mass, stiffness and damping on the low-rise buildings. However, investigations on influence of the structural components and their stiffness contribution are rare so far. Stiffness evaluation can be implemented indirectly via evaluation of changes of the modal parameters, concretely natural frequencies and damping ratios at fundamental modes, usually at the first mode. Moreover, it is also generally agreed that excitation amplitude also affects on the modal parameters, especially the dynamic properties can considerably change with the large amplitude of excitation due to structural nonlinearity and nonstationary signals. Evaluation of influence of excitation amplitude on the damping ratios and natural frequencies has been done in somewhere (eg., Tamura and Suganuma, 1996).

This paper presents the modal parameter identification of experimental low-rise building from the ambient, free and forced sweep vibration tests. Only response signals are measured in case of the ambient and free vibration tests, whereas excitation and responses both are measured in the sweep vibration tests. Accelerometers have been located at different positions on the experimental building to measure accelerations of the responses and excitation. Three types of the vibration tests have been carried out at some erection stages and three levels of excitation amplitudes. Experimental building is erected at different stages with a change of structural components such as exterior and interior walls, sealing and connection types. Influences of the vibration tests, the structural components and the excitation amplitudes on identified modal parameters of this experimental low-rise building are investigated.

Modal Parameter Identification Techniques using Free Vibration Tests The free vibration responses here have been measured after application of the sine sweep load excited by the shaker that an excitation frequency is coincident with one of the structural natural frequencies at resonant state. By this way, the free vibration responses are separated for each single mode, thus damping ratio of this single mode can be directly estimated through the LDT by fitting an exponential function to relative maxima of free decay functions. Natural frequencies can be estimated either directly from these free decay functions or indirectly via the Fourier transform of the free decay functions. The amplitude logarithmic decrement and the damping ratio of i-th mode are estimated from its measured free decay function as follows:

24;

12ln1

222

nTm

m

XX

n (1)

where , : logarithmic decrement and damping ratio; nTmm XX , : amplitudes at n wave cycles; T: natural period. However, it argued that the damping ratios estimated by the LDT normally exhibit higher than actual damping ratios in the service states (Magalhaes et al., 2009) due to many reasons like noises, shortly-recorded free decay functions, influence of closed natural frequencies and effect


of other natural frequencies on measured resonant frequency, excitation amplitude and so on. Alternatives to shortcoming of the LDT, the ITD and the SSI can be used to extract the modal parameters from the free decay functions in the time domain (He and Fu, 2001; Magalhaes et al., 2009).

Modal Parameter Identification Techniques using Ambient Vibration Tests FDD basis

Consider the output responses of the ambient vibration tests are measured at N structural points, one have the output response vector T

N tXtXtXX )(),...(),( 21 . Cross spectral density matrix of the output responses is defined as follows:

)()()(

)()()()()()(

)(

21

22212

12121

NNNN

N

N

XXXXXX

XXXXXX

XXXXXX

XX

SSS

SSSSSS

S

(2)

where )(ji XXS : power spectrum between )(tX i and )(tX j

Analysis of the output-only response data is carried out in the frequency domain using well-known FDD technique. The key point of the FDD will be briefly summarized hereafter. Relationship between excitation inputs and output response can be expressed in the frequency domain through the FRF matrix as follows:

TFFXX HSHS *)()()()( (3)

where *, T denote conjugate and transpose operations; )(),( XXFF SS : power spectral matrices of inputs and outputs; )(H : FRF matrix.

The FRF matrix can be expressed commonly under a form of residues/poles (He and Fu, 2001; Brincker et al., 2001):

N

i

N

ii

Tiii

i

Tiii

i

i

i

i

jjja

jaH

1 1 *

***

*

*

)(

(4)

where i, N: index, number of modes; iia , : i-th complex residue and complex pole in which Tiiiia

with light damping and 21 iiiii j ; ii , : i-th natural frequency and damping ratio; ii , : i-th mode shape vector and scaling factor.

If the inputs are uncorrelated white noise inputs, the input power spectral matrix of the inputs is diagonal constant one ),...,()( 21 NFF cccdiagS and damping is light, one can obtains the output power spectral matrix at evaluated frequency i decomposed modally as follows:

N

ii

Tiii

i

Tiii

XX jd

jdS

1 *

***

)(

(5)

where id : i-th scalar constant. Expression of the output power spectral matrix in Eq.(5) is similar one of some matrix decompositions in the complex domain, thus these can be used to decompose the output power spectral matrix.

The output power spectral matrix has been orthogonally decomposed using the eigenvalue decomposition to obtain spectral eigenvalues and spectral eigenvectors:

T

kkM

k kT

XXS *1

* )()()()()()()( (6)


where )(),( : spectral eigenvectors and spectral eigenvalues matrices; k, M: index and number of spectral eigenvectors. Because the eigenvalue decomposition is fast-decaying, thus the output power spectral matrix can be approximated by using the lowest-order spectral eigenvalue and eigenvector as follows:

T

XXS *111 )()()()( (7)

where )(),( 11 : the first spectral eigenvector and first spectral eigenvalue.

Due to the first-order spectral eigenvalue and eigenvector are dominant in a term of energy contribution, thus the first spectral eigenvalue contains full information of dominant frequencies to be used for extracting the natural frequencies, whereas the first spectral eigenvector brings information of mode shapes at each dominant frequency. The i-th mode shape can be estimated from the first spectral eigenvector at certain dominant frequencies ( i : i-th natural frequency) as follows:

)(1 ii (8)

Similar to the peak picking method, in the case of input presence the effect of inputs must be eliminated from the output response before the frequency domain decomposition is used for extracting the natural frequencies and mode shapes. Originally, the damping ratios estimation cannot be carried out by the FDD, but its enhanced version has been developed for this purpose.

Damping estimation via EFDD

Enhanced frequency domain decomposition has been developed basing on the frequency domain decomposition for estimating the damping ratios only (Brincker et al., 2001). For extraction of the natural frequencies and mode shapes, both methods are the same. As can be seen from Eq.(7) that FDD extracts the mode shape from the first spectral eigenvector at selected natural frequency, thus prior knowledge of the natural frequencies must be required for this identification technique. Accuracy of estimated mode shapes can be evaluated via correlation criteria between estimated mode shapes and analytical mode shapes, moreover, among these criteria Modal Assurance Criterion (MAC) is preferably used. For the damping estimation, the key point here is to identify the auto power spectral density function of the single DOF generalized coordinate of certain mode from the spectral eigenvalues. Because the first spectral eigenvalue contributes dominantly almost energy of system, thus it is often used to extract the auto power spectral density functions. Searching the auto power spectral function of certain mode from the first eigenvalue is carried out on both sides of value of the natural frequency, and it is terminated until the desirable limit of MAC reached. The remaining values of the auto spectral density function in the calculated frequency range are set to zero. From identified auto power spectral density functions, the damping ratios are obtained via logarithmic decrement technique of free decay functions, of which these free decay functions obtained by converting the auto spectral density function in the frequency domain back to the time domain by inverse fast Fourier transform technique (Brincker et al., 2001b). It is also noted that validation of the natural frequencies can be checked through these free decay functions.

Damping estimation via Random Decrement Technique

More simple way to estimate the damping from ambient vibration data uses the Random Decrement Technique (RDT). The RDT can apply directly to the response data to obtain the free decay functions or random signature ones. The logarithmic decrement and the damping ratios can be extracted from these free decay functions. However, single DOF response data associated with any certain natural frequency must be extracted firstly from multi DOF response data by the band-pass filtering before the RDT used (Tamura et al., 2005).


Modal Parameter Identification Techniques using Sweep Vibration Tests Estimates and smoothing of FRFs

Estimation and smoothing of the FRFs are the first step and very important for the modal parameter identification from the forced vibration tests. The squared FRFs can be estimated as a ratio between auto power spectrum of the response output and that of the excitation inputs as expressed in the Eq.(3) (Bendat and Piersol, 1993 ):

)()()( 2

FF

XX

SSH (9)

where )(XXS : auto power spectrum of output )(tX ; )(FFS : auto power spectrum of input )(tF .

Correlation between the excitation inputs and the response outputs in the frequency domain can be expressed by coherence function which is varied between 0 and 1 to evaluate the linear relationship between them:

)()(

|)(|)(2

XXFF

FX

SSSCOH (10)

At the starting point, the power spectra of the input and output signals are derived by using the Discrete Fourier Transform (DFT), before the FRFs can be estimated by the Eq.(9). In the single block techniques, whole time series of the input and output signals are treated as one signal blocks, thus a frequency resolution of the DFT is fixed by Tf 1 , T: total time of measured signals. One usually expects to increase the frequency resolution via reduction of processed time duration in the measured signals. This can be implemented via the block overlapping technique by dividing the time signals into Nb equal blocks, in which the FRFs are calculated then summing them up (Marchitti, 2006). By this approach, the frequency resolution is Nb times greater than that over entire length of data in the single block technique. However, the signals become leakage due to discontinuity, thus either so-called Hanning window or block overlapping is generally required on each block. The overlapping is favorable due to the Hanning window raises the damping. It is also suggested that number of samples in each block is selected equal to desired number of samples nfft and 75% block overlapping or more is preferred (Orlando et al., 2008) to which the desired frequency resolution is given by Tnfftf . The FRFs are estimated via the block overlapping technique as follows:

b

jj

b

jj

N

j FF

N

j XX

S

SH

1

12

)(

)()(

(11)

where j: index of processed block; Nb: number of blocks.

Some other smoothing techniques such as the frequency averaging technique or the harmonic estimator refer in the works by Marchitti, 2006; Orlando et al., 2008.

Damping estimation via HPB

The HPB technique is simple and quick way to estimate the damping ratios from the FRFs of the single DOF systems at the resonant amplitude. Because excitation frequency has been continuously adjusted in the sine sweep vibration tests by which a range of the excitation frequency covers the first natural frequency only. This means that the estimated FRFs are just for the single DOF system corresponding to the first structural mode. By this way, the damping ratio can be obtained by the HPB from the estimated FRFs as following expression (Bendat and Piersol, 1993):


r 12 (12)

where r : resonant frequency; 12 , : frequencies at which the amplitude is 21 times of the resonant amplitude.

The HPB is carried out on the smoothed FRFs, however, this technique seems not to be reliable and accurate for estimating the damping ratios due to noises, frequency resolution, closed natural frequencies, recorded inputs and outputs and so on.

Damping estimation via LSCF

In the LSCF technique, the measured FRFs are determined following the Eq.(9), beside the theoretical FRFs are expressed in the fractional form of the complex residues and poles that contain the modal parameters. The modal parameters can be estimated based on both the measured FRFs and the theoretical FRFs by minimizing the nonlinear least-squares cost function with respect to the parameter vectors in the frequency domain (De Troyer et al., 2009)

fN

f

N

iif

Tiii

if

Tiii

fmeaLS jjH

1

2

1 *

***

)()(

(13)

where LS)( : least-squares cost function; :modal parameters ( iii ,, ); )( fmeaH : measured FRF matrix at frequency point f ; fNf , : index of frequency points and number of frequency points. Iterative Gauss-Newton algorithm is usually used to solve this nonlinear least-squares cost function.

Experimental Low-rise Building and Vibration Tests

In order to investigate influences of the structural components, the excitation amplitudes and types of vibration tests on the modal parameters of the low-rise building, the vibration tests have been implemented experimentally on small-scale one-storey steel building in the structural laboratory (see Figure 1). There were three types of the dynamic tests including the ambient vibration tests, the forced sweep vibration tests and the free vibration ones. Ambient vibration tests were carried out using microtremor, whereas the shaker installed at top floor of the experimental building was used for the forced vibration tests. The shaker excited the linearly sine sweep force in the X-direction only with continuously changeable frequency band from 2Hz to nearly 6Hz and with sweep rate of 0.01Hz. Free vibration tests were created on the experimental building when the excitation frequency of the sine sweep load was coincident with the first natural frequencies and moving mass of the shaker stopped at the resonant frequency. Three levels of the excitation amplitudes from the small, medium and large amplitude were set for the all types of the vibration tests. Accelerometers were arranged at several positions on both the first floor and the second floor of the experimental building in two X-, Y-directions, totally 8 channels of response measurements (see Figure 1). Response accelerations only were measured at every experimental set-up in the ambient vibration tests and the free vibration tests, whereas both the response accelerations and the excitation acceleration of the shaker were obtained in the sine sweep vibration tests. Ambient vibration data were measured for minimum 5-minute records, whereas the sweep vibration data for nearly 13-minute records.

Structural components have been installed in various erection cases to evaluate their stiffness contribution of these structural components on the experimental low-rise building. Some mainly structural components of the low-rise building such as structural steel frame, ALC walls, sealing joints, separate walls, interior cover plate have been investigated. There are seven erection cases at which each structural component has been added.


Table 1. Erection cases and combination of structural components

Structural components

Structural installation description

Erection cases

Combination of structural components

Steel frame only D1

ALC exterior walls (X dir.) D2 + Sealing between ALC walls D3 + + Interior cover plate D4 + + + Interior separate wall D5 + + + + Window and exterior wall (Y dir.) D6 + + + + +

Figure 2. Images of structural components and erection cases

D1 D2

Steel frame ALC walls (X directions) Sealing

D3

D4

Interior cover plate

D5 D6

Interior separate Wall (X direction)

Window and exterior wall (Y direction)

Sealing

Figure 1. Experimental low-rise building and accelerometer arrangement


Orders of the vibration tests at each erection case are follows: the ambient vibration tests started firstly at so-called micro excitation amplitude; the sweep vibration tests were followed at three levels of the excitation amplitudes from the small amplitude, the medium amplitude up to the large one; next were the free vibration tests also from the small amplitude, medium amplitude to the large one; finally the ambient vibration tests one again were carried out at last. Each set-up has been measured by three times. Structural images of the structural components and the erection cases are shown in Figure 2, while combination of the structural components in each erection case is detailed in Table 1.

Figure 3. Response accelerations based on test types and amplitude levels of case D1

Time series of response accelerations at the referred PU4-X sensor at the second floor of the case D1 corresponding to three vibration tests and levels of the excitation amplitudes are shown in Figure 3 as an example. Both maximum amplitude and standard deviation amplitude of the acceleration time series are used to investigate the effect of amplitude on the change of modal parameters.

Natural Frequency Estimation

Natural frequencies of all the test cases have been estimated firstly using the system identification techniques presented in the previous content. For the ambient vibration data, the natural frequencies are identified using the FDD as following steps: all the response outputs are reorganized in the output cross spectral matrix, before the eigenvalue decomposition is applied for this output cross spectral matrix to

0 10 20 30 40 50 60 70-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Time (s)

Acce

. (m

/s2 )

CH04 - PU4X - 2F

CH04

0 10 20 30 40 50 60

-0.1

-0.05

0

0.05

0.1

0.15

Time (s)

Acce

. (m

/s2 )

CH04 - PU4X - 2F

CH04

0 50 100 150 200 250 300-0.01

-0.005

0

0.005

0.01

Time (s)

Acce

. (m

/s2 )

CH04 - PU4X - 2F

CH04

0 100 200 300 400 500 600 700-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Time (s)

Acce

. (m

/s2 )

CH04 - PU4X - 2F

CH04

0 100 200 300 400 500 600 700

-0.1

-0.05

0

0.05

0.1

0.15

Time (s)

Acce

. (m

/s2 )

CH04 - PU4X - 2F

CH04

0 500 1000 1500 2000 2500 3000 3500-8

-6

-4

-2

0

2

4 x 10-3

Time (s)

Acce

. (m

/s2 )

CH04 - PU4X - 2F

CH04

Ambient (5-minute record) Ambient (1-hour record)

Sweep (small amplitude) Sweep (medium amplitude)

Free decay (small amplitude) Free decay (medium amplitude) PU4-X

PU4-X PU4-X

PU4-X PU4-X

PU4-X


obtain pairs of the spectral eigenvalues and the spectral eigenvectors, natural frequencies from the ambient vibration data can be observed by the first spectral eigenvalue. Response accelerations at all corner sensor positions at the first and second floors of the experimental building (PU1-X, PU3-X, PU4-X, PU6-X and PU4-Y, PU6-Y) are taken into analysis. Results of the natural frequencies obtained from the FDD can be verified with those directly from the conventionally auto power spectra of the response outputs. For the free vibration data, due to short data records the natural frequencies are directly observed from the auto power spectra of the output free decay functions. In case of the sweep vibration data, the natural frequencies are obtained from the smoothed FRFs between measured excitation of the shaker and measured output responses.

Figure 4 shows the natural frequency extraction of all vibration tests and of all levels of excitation amplitudes at the erection case D1 only for X-direction bending modes (only the case D1 is shown for sake of brevity). For the FRFs estimation in the sweep vibration data, some smoothing techniques such as the single block technique and the block overlapping one (with no overlapping between blocks and with 50% overlapping between blocks) have been applied for comparison (see Figure 4). However, there are no much difference between the smoothing techniques of the FRFs can be observed.

Figure 4. Natural frequency based on test types and amplitude levels of case D1

0 1 2 3 4 5 6 7 8 9 1010-10

10-8

10-6

10-4

10-2

100

Frequency (Hz)

Nor

mal

ized

spe

ctra

l val

ues

Ambient - case D1

Sepctral value 1Spectral value 2Spectral value 3Spectral value 4

3.59Hz 3.77Hz

6.47Hz

0 1 2 3 4 5 6 7 8 9 1010-10

10-8

10-6

10-4

10-2

100

Frequency (Hz)

Nor

mal

ized

spe

ctra

l val

ues

Ambient - case D1

Sepctral value 1Spectral value 2Spectral value 3Spectral value 4

5.05Hz

3.67Hz

0 1 2 3 4 5 6 7 8 9 1010-12

10-10

10-8

10-6

10-4

10-2

Frequency (Hz)

PSD

(m2 /s

)

Free decay - case D1

PU4-X3.69Hz

0 1 2 3 4 5 6 7 8 9 1010-15

10-10

10-5

100

Frequency (Hz)

PSD

(m2 /s

)

Free decay - case D1

PU4-X3.67Hz

Ambient (at first) Ambient (at last)

Free decay (small amplitude) Free decay (medium amplitude)

0 1 2 3 4 5 6 7 8 9 1010-4

10-3

10-2

10-1

100

101

Frequency (Hz)

FRF

Sweep (small amplitude)

Single blockBlock overlapping (0%)Block overlapping (50%)

3.69Hz

0 1 2 3 4 5 6 7 8 9 1010-4

10-3

10-2

10-1

100

101

Frequency (Hz)

FRF

Sweep (medium amplitude)

Single blockBlock overlapping (0%)Block overlapping (50%)

3.67HzSweep (small amplitude) Sweep (medium amplitude)


First natural frequencies for all erection cases D1÷D6, all levels of excitation amplitudes and all vibration tests are given concretely in Table 2 (X-direction bending modes) and Table 3 (Y-direction bending modes). More details will be discussed in Figure 5, Figure 6 and Figure 7.

Table 2. First natural frequency in X direction (Unit: Hz)

Vibration Amplitude Erection cases

test levels D1 D2 D3 D4 D5 D6

Ambient At first 3.77 3.76 4.2 4.5 4.64 4.67 At last 3.67 3.79 4.01 4.4 4.42 4.16

Sweep Small 3.69 3.74 4.05 4.33 4.33 4.25 Medium 3.67 3.77 3.91 4.23 4.25 3.98

Large - 3.5 3.86 4.23 4.23 3.86

Free Small 3.69 3.74 4.11 4.47 4.55 4.25 Medium 3.67 3.74 4.11 4.42 4.47 4.06

Large 3.67 3.72 4.06 4.26 4.4 3.89

Table 3. First natural frequency in Y direction (Unit: Hz)



Ambient At first 5.05 5.03 5.05 5.21 5.16 5.77 At last 5.06 5.01 5.01 5.16 5.13 5.69

Sweep Small 5.05 5.01 5.03 5.18 5.16 5.72 Medium 5.05 5.01 5.03 5.18 5.16 5.69

Large - 5.01 5.03 5.18 5.16 5.67

Free Small 5.05 5.03 5.03 5.18 5.16 5.67 Medium 5.05 5.03 5.01 5.18 5.18 5.67

Large 5.05 5.04 5.03 5.16 5.16 5.65

Figure 5 shows the effects of the erection cases and of the vibration tests on the first natural frequency of both the X-direction bending mode and the Y-direction bending one of the experimental building. The influence of the structural components on the natural frequency of the X direction can be explained as follows. Taking the ambient vibration data at first for discussion, the first natural frequency keeps unchanged or slightly increases from the case D1 (steel frame) to the case D2 (installation of exterior ALC walls), though the exterior walls erected in the X-direction might improve little bit the global stiffness but this stiffness improvement might not enough to compensate their additional masses. When the sealing joints between the exterior ALC walls are built, the natural frequency increases from 3.77Hz of the D1 to 4.2Hz of the D3 (11% increase), this means that the sealing joints between the ALC walls improve well the X-direction global stiffness. When the interior cover plate is erected in the case D4, there is reinforcement of the global stiffness with 19% increase of the natural frequency from the 3.77Hz to 4.5Hz. Further installation of the interior separate walls (case D5) and that of the window (case D6) also influence much on the global stiffness of the experimental building, concretely the first natural


frequency cumulatively increases from 3.77Hz (D1) to 4.64Hz (D5) and 4.67 (D6) or 23% and 24% frequency increase of the D5, D6, respectively (see Table 1). After each erection cases from D1 to D6, the first natural frequency changes as 0% increase (D1 to D2), 12% increase (D2 to D3), 7% increase (D3 to D4), 3% increase (D4 to D5) and 1% increase (D5 to D6). However, it is observed a decrease of the natural frequency from D5 to D6 in the ambient data at last, both the sweep and free vibration data (see Table 1, Figure 5), this might be due to an increase of additional mass of the Y-direction exterior walls.

For the natural frequency of the Y direction, furthermore, there is not much difference between the erection cases D1÷D5 due to installment of the structural components does not improve the structural stiffness on this Y direction (see Figure 5), small changes of the natural frequency on the Y direction between D1÷D5 cases might be caused by mass distribution after installation of the structural components. However, the first natural frequency sharply increases from 5.16Hz to 5.77Hz (12% increase) at the case D6 due to reinforcement of the Y-direction stiffness via the installation of the ALC exterior wall panels.

Figure 5. Effect of vibration tests and erection cases on natural frequency

Figure 6. Effect of excitation amplitudes and erection cases on natural frequency

Figure 7. Amplitude-dependant frequency based on erection cases and vibration tests

D1 D2 D3 D4 D5 D63.6

3.8

4

4.2

4.4

4.6

4.8

5First natural frequency (X-direction)

Erection cases

Fre

quen

cy (H

z)

AmbientSweepFree decay

D1 D2 D3 D4 D5 D65

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8First natural frequency (Y-direction)

Erection cases

Fre

quen

cy (H

z)

AmbientSweepFree decay

First natural frequency (X direction) First natural frequency (Y direction)

D1 D2 D3 D4 D5 D63.4

3.6

3.8

4

4.2

4.4First natural frequency (Sweep)

Erection cases

Freq

uenc

y (H

z)

Small amplitudeMedium amplitudeLarge amplitude

D1 D2 D3 D4 D5 D63.6

3.8

4

4.2

4.4

4.6First natural frequency (Free decay)

Erection cases

Freq

uenc

y (H

z)


First natural frequency (Sweep) First natural frequency (Free decay)

0 2 4 6 8 10 12 143.5

4

4.5

5

5.5

6Amplitude-depandant frequency (erection cases)

Standard deviation amplitude (cm/s2)

Freq

uenc

y (H

z)

D1D2D3D4D5D6

0 2 4 6 8 10 12 143.5

4

4.5

5

5.5

6Amplitude-dependant frequency (vibration tests)

Standard deviation amplitude (cm/s2)

Freq

uenc

y (H

z)

Ambient (At first)Sweep (Small)Sweep (Medium)Sweep (Large)Free decay (Small)Free decay (Medium)Free decay (Large)Ambient (At last)

Amplitude-dependant frequency (Erection cases) Amplitude-dependant frequency (Vibration tests)


As can be also seen from the Figure 5, the influence of the vibration tests can be investigated that the natural frequencies produced by the ambient vibration tests exhibit higher than those by the free vibration data, the lowest natural frequencies are make by the sweep vibration tests. Main reasons for this difference might be due to amplitude of excitations, concretely an increase of excitation motion weakens structural connection and material quality as well as nonlinear behavior and nonstationary signals at strong motions to result in stiffness reduction and damping enlargement, accordingly the natural frequency reduces.

Effects of the amplitude levels, the erection cases and the vibration tests on the natural frequency of the X-direction bending mode of the experimental building are indicated in Figure 6 and Figure 7. Apparently, the excitation amplitudes significantly influence on the natural frequency for both types of the sweep vibration tests and the free decay ones, the small excitation amplitudes produce the highest natural frequencies and the large amplitudes give the lowest ones. Relationship between the natural frequencies and the standard deviation amplitudes of output responses has been investigated for more details in the Figure 7. It is observed that high natural frequencies often occur at low amplitudes and an inverse regardless of the vibration tests and the erection cases.

Damping Estimation Damping ratios of all the erection cases D1÷D6 and of two ambient vibration and free decay tests have been estimated for further discussions. For the ambient vibration data, the combined band-pass filtering and the RDT have been used to estimate the single DOF free decay function corresponding to the first natural frequency. For the free vibration tests, the free decay functions have been obtained directly from measurements. Curve fitting technique is applied to the free decay functions obtained from both the ambient vibration data and the free vibration data to estimate the logarithmic decrements and the damping ratios. Amplitudes of the first 10 cycles in the free decay function are used for the curve fitting technique.

Figure 8 shows the free decay functions obtained from the ambient vibration data at the erection case D1 and the linear fitting of the first 10 enveloped amplitude for estimating the logarithmic decrement as an example (for sake of brevity only this case shown here).

Damping ratios of the first bending mode in the X-direction of the experimental building with respect to the erection cases D1÷D6 and the ambient and free vibration tests are given in Table 4. Results are plotted in Figure 9 for further discussion. The damping ratios trend to increase with addition of the structural components from case D1 to case D6. Taking the free vibration data at small amplitude into account, concretely, the damping ratios increase from 0.31% of the case D1 to 1.35% of the D3 (335%), to 1.85% of the D4 (497%). Explaining about this trend, large excitation amplitude tests after each erection case might weaken structural joints and materials, reduce the global stiffness and increase the damping, additionally, the soft stuffs of sealing and of behind the cover plates absorbing vibrational energy enhance the damping ratios of the low-rise building.

Figure 8. Estimation of free decay function and linear fitting from ambient data of case D1

0 1 2 3 4 5 6 7 8 9 10-3

-2

-1

0

1

2

3x 10-3

Time (s)

Am

plitu

de (c

m)

Free decay function

1 2 3 4 5 6 7 8 9 10 110.8

0.85

0.9

0.95

1Linear fitting of enveloped amplitude

Number of cycles

Loga

rithm

ic a

mpl

itude

Enveloped amplitudes Linear fitting

y=-0.0144x+0.98


Table 4. Damping ratios of first bending mode in X direction (Unit: %)



Ambient At last 0.21 0.36 0.80 1.22 1.38 1.55

Free Small 0.31 0.60 1.35 1.85 1.75 2.09

Medium 0.33 0.85 1.82 2.24 2.19 2.75

Large 0.51 1.37 2.45 3.02 3.17 3.06

Figure 9. Effect of vibration tests and amplitude on damping ratio

Figure 10. Amplitude-dependant damping ratios of cases D3, D4

For damping comparison between the ambient vibration tests and the free vibration ones, it is observed that the damping obtained from the free vibration is considerably higher than that from the ambient vibration data (see Figure 9). This higher damping of the free vibration data has been discussed somewhere (eg., Magalhaes et al., 2009) due to noises and influence of other natural frequencies of which cannot be completely eliminated from the free decay records. Effect of the excitation amplitudes on the damping is also shown in Figure 9, the damping ratios increase with an increase of the amplitude, this might be explained that large excitation amplitude can causes defectiveness of structural joints, connections and structural materials, which result in the stiffness reduction and the damping increase. Moreover, there is considerable difference of the damping observed from the small amplitude to medium amplitude up to large one, for example of the case D4 (interior cover plate erected) the damping ratio increases from 1.85% at the small amplitude to 2.24% at medium one (21% increase) up to 3.02% at large one (63% increase).

D1 D2 D3 D4 D5 D60

0.5

1

1.5

2

2.5Damping (Ambient & Free decay)

Erection cases

Dam

ping

ratio

s (%

)

Ambient (At first)Free decay (Small amplitude)

D1 D2 D3 D4 D5 D60

0.5

1

1.5

2

2.5

3

3.5Damping (Free decay)

Erection cases

Dam

ping

ratio

s (%

)


Damping ratios (Ambient and Free decay) Damping ratios (Free decay)

2 4 6 8 10 12 14 16 18 20 220

1

2

3

4

5

6Amplitude dependant damping

Amplitude (gal)

Dam

ping

ratio

(%)

Small amplitude 1Small amplitude 2Medium amplitude 1Medium amplitude 2Large amplitude 1Large amplitude 2

Amplitude-dependant damping (D3)

Amplitude (cm/s2) 0 2 4 6 8 10 12 14 16 18

0

1

2

3

4Amplitude dependant damping

Amplitude (gal)

Dam

ping

ratio

(%)

Small amplitude 1Small amplitude 2Medium amplitude 1Medium amplitude 2Large amplitude 1Large amplitude 2

Amplitude-dependant damping (D4)

Amplitude (cm/s2)


Figure 10 shows the amplitude-dependant damping using the free vibration data (2 set-ups used) at only two erection cases D3, D4 for sake of brevity. Obviously, the damping is significantly influenced by the response amplitudes, it is tendency that high damping is produced from large response amplitudes.

Conclusions The influences of the vibration tests, the structural components and erection cases and the excitation amplitudes on the modal parameters of the experimental low-rise building have been investigated in the paper. Structural components improve to some extent the stiffness reinforcement of the low-rise building, the installation of the exterior walls, the sealing joints, the cover plates, the separated walls, the windows have increases the global stiffness. Moreover, natural frequencies reduce gradually from the ambient vibration tests to the free decay tests to the sweep vibration ones, whereas the damping increases from the ambient vibration data to the free decay data. Excitation amplitudes significantly influence to the modal parameters of the experimental low-rise building. The large excitation amplitudes reduce the natural frequencies, but increase the damping. Reason is that the increase of excitation amplitude weakens the structural connection and the material quality as well as possibilities for nonlinear and nonstationary behaviors of building and signals, these result in reduction of the stiffness, enlargement of the damping and reduction of the natural frequencies accordingly.

Acknowledgements

This study was funded by the Asahi Kasei Homes Corporation Co., Ltd. and the Ministry of Education, Culture, Sports, Science and Technology (MEXT) Japan through the Global Center of Excellence Program (Global COE) between 2008÷2012, which is gratefully acknowledged.

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