BUILDING STRUCTURE PARAMETER IDENTIFICATION USING THE FREQUENCY DOMAIN DECOMPOSITION FDD METHOD NHẬN DẠNG CÁC THÔNG SỐ ĐỘNG LỰC HỌC CỦA TÒA NHÀ BẰNG PHƯƠNG PHÁP FDD Loc Nguyen Phuoc 1
Trang 1BUILDING STRUCTURE PARAMETER IDENTIFICATION USING THE FREQUENCY DOMAIN DECOMPOSITION (FDD) METHOD
NHẬN DẠNG CÁC THÔNG SỐ ĐỘNG LỰC HỌC CỦA TÒA NHÀ BẰNG PHƯƠNG
PHÁP FDD
Loc Nguyen Phuoc 1a , Phuoc Nguyen Van 2b
1Kien Giang Vocational College, Vietnam
2HCMC University of Technical and Education, Viet Nam
ABSTRACT
In recent years, Operational Modal Analysis, also known as Output-Only Analysis, has been used for estimation of modal parameters of the structures such as the buildings, bridges, towers, and mechanical structures The advantage of this method is that expensive excitation equipment can then be replaced by ambient vibration sources such as wind, wave, and traffic used as input of unknown magnitude, and then modeled as blank interference in the modal identification algorithms This paper presents an overview of the non-parameter technique based Frequency Domain Decomposition (FDD), dynamic model of n-storeybuilding and method of modal parameters identification using FDD In addition, using statistical probability
to evaluate the results that obtained the stiffness and inter-storey dift of 2-storeybuilding
Keywords: FDD: Frequency Domain Decomposition, OMA: Operational Modal
Analysis, MDOF: Multi-Degree of Freedom, SDOF: Single-Degree of Freedom, EMA: Experimental Modal Analysis, SVD: Singular Value Decomposition
TÓM TẮT
Những năm gần đây, phân tích thể thức (Modal) hoạt động được biết đến với tên gọi là Phân tích chỉ với ngõ ra, đã được sử dụng để ước lượng các tham số của các công trình như các tòa nhà, cầu, tòa tháp và các cấu trúc cơ khí Thuận lợi của phương pháp này là những thiết bị kích thích đắt tiền có thể được thay thế bằng các nguồn rung động từ môi trường xung quanh, chẳng hạn như các rung động từ gió, sóng và lưu thông xe cộ được sử dụng như là ngõ vào với biên độ khôngđượcquan tâm, chúng được mô hình hóa như nhiễu trắng trong các giải thuật nhận dạng thể thức (modal) Bài báo này trình bày tổng quan về kỹ thuật không tham số dựa trên việc phân giải trong miền tần số, mô hình động học của tòa nhà n tầng và phương pháp nhận đạng các tham số modal sử dụng FDD.Thêm vào đó, sử dụng xác suất thống kê để đánh giá các kết quả đạt được về độ cứng (stiffness) và độ xê dịch tầng (inter-storey dift) của tòa nhà 2 tầng
Từ khóa: FDD: Phân giải trong miền tần số, OMA: Phân tích thể thức hoạt động,
MDOF: Đa bậc tự do, SDOF: Một bậc tự do, EMA: Phân tích thể thức thực nghiệm, SVD: Phân giải giá trị đơn
1 INTRODUCTION
The experimental determination of structural modes of a structure can be divided into two methods: EMA and OMA Experimental Modal Analysis requires knowledge of both input and output, which can be combined to yield stransfer function that describes the system
In recent decades, there are civil structures used OMA method This method has been developed for many civil engineering structures such as buildings, bridges, rigs,…[1] Operational modal analysis only requires measurement of the output from the system In FDD method, spectral density matrix of multi-degree of freedom system is decomposed into a set
Trang 2of auto spectral density functions, each corresponding to a single degree of freedom This method is illustrated by the measurement on a two-storey building model with the excitation source generated by a small hard plastic hammer and a vibration motor.The advantage of using data acquisition hardware NI-USB 9234 of National Instruments is to easily measure responses of accelerometers installed along the height of the building with LabVIEW 2011 as showed in Figure 1 Then the data continues to be analyzed with Matlab with the support of advanced signal processing tools Finally, the modal parameters of the building are obtained
as resonant frequency and mode shapes In addition, the stiffness of each floor is also identified under the shear beam model assume of a two-storey building
Figure 1 Data acquisition system with NI-USB 9234 hardware in LabVIEW 2011
2 MAIN CONTENT
2.1 Frequency Domain Decomposition (FDD)
The power spectrum density matrices of the input (unknown) and output (recorded) signal as functions of angular frequencyωrespectively noted [ ]X ( )ω and[ ]Y ( )ω They are associated to the frequency response function matrix[ ]H ( )ω through the following equation [2,3,5,6,8,9]:
[ ] ( ) [ ] ( ) [ ] ( ) [ ] ( )T
H X H
Where:∗is denoted complex conjugate and
T
is transposed If r is the number of inputs
of [ ]X ( )ω , [ ]Y ( )ω and [ ]H ( )ω arer×r,m× andm m× , respectively In Operational Modal r
Analysis, the usual assumption is that the input is white noise That means the power spectral density matrix is expressed:
Where [ ]C is constant matrix The [ ]H ( )ω matrix can be written in a pole (λ ) and k Residue ([ ]R k ) formas:
∗
∗
=
k k n
k
j
R j
R X
Y H
λ ω λ ω ω
ω ω
1
(3)
k mode and σ is the modal k damping of the k thmode, ω is the damped natural frequency of the dk th
2
Trang 3Where: ς is the critical damping of the k th
k mode, ω is the undamped natural frequency k
of the th
k mode [ ]R k matrix is called the residue matrix andis expressed as following form:
k k k
Where φ is the mode shape, k λ is the modal participation vector All those parameters k are specified for the th
k mode The input assumed to be blank interference with power spectral density is flat (no change) over the entire frequency range, thus spectral power density matrix
[ ]X ( )ω is a constant matrix, so it can be writtenas[ ]X ( )ω =C, then Equation (1) becomes:
l l l l n
k n
k k
j
R j
R C j
R j
R
− +
−
− +
−
∗
∗
ω
(7)
Where
H
is denotes complex conjugate and transposition Multipying the two partial fraction factors and making use of the heaviside partial fraction theorem, then performing mathematical transformations, output power spectral densitycan be presented as follows
∗
∗
∗
∗
k k k k k k n
k
j
B j
B j
A j
A Y
λ ω λ ω λ ω λ ω
ω 1
(8)
Where:[ ]A k is the k thresidue matrix The matrix[ ]X ( )ω is assumed to be a constant C ,
since the excitation signals are assumed to be uncorrelated zero mean blank interference in all the measured DOFs.This matrix is Hermitian; its size ism× and is described in the form: m
−
−
+
−
−
n
T s s
k
H s k
k
R R
C R A
The contribution to the residue from the k thmode is given:
[ ] [ ] [ ]
k
T k k k
R C R A
σ
2
∗
Where:σ is minus the real part of the polek λk =−σk + jωdk As it appears, this term becomes dominating when the damping is light, and thus, is case of light damping; the residue becomes proportional to the mode shape vector:
k k k T k k T k k T k k k light
→
Where: d kis a scalar constant The contribution of the modes at a particular frequency is limited to a finte number Let this set of modes be denoted bySub(ω) Thus, in the case of a lightly damped structure, the response of spectral density matrix can always be written as following final form:
k
T k k k Sub
T k k k
j
d j
d Y
λ ω
φ φ λ
ω
φ φ ω
ω ) (
(12)
The final form of the matrix[ ]Y ( )ω is decomposed into a set of singular values and singular vectors using the Singular Value Decomposition
2.2 Singular Value Decomposition
factorization:
H
V S U
Trang 4Where U and V are unitary matrix and S is a diagonal matrix that contains the real
singular values:
) , , (s1 s r diag
The superscript
H
on the V matrix denotes a Hermitian transformation In the case of real valued matrices, the matrix V is only transposed and is denoted
T
The s ielements in the
matrix S are called the singular values and their following singular vectors are contained in the matrix U andV This singular value decomposition is performed for each of the matrices
at each frequency The experimental flowchart is built by using FDD method and is illustrated through four stages as shown in Figure 2:
Figure 2 Experimental flowchart using FDD
The spectral density matrix is then approximated to the following expression (15) after SVD decomposition:
[ ] ( ) [ ][ ][ ]H
S
Notation [ ]I is unitary matrix
Where S is a diagonal matrix holding the scalar singular values, [ ]Φ is a unitary matrix holding the singular vectors:
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
=
=
0 0 0 0
0 0 0
0 0
0 0
0
) , ,
2 1
1
r
r
s
s s s
s s diag
- Simultaneous recordings
- Installation position of sensors: ground, floor 1, floor 2
- Fourier transform of the measured responses:
determining the frequency components: ωi
- Calculate the matrix of power density spectral:
( )i
kk
PSD ω ;
n
k= 1 :
(this experiment we need to use 2 sensors at 1st and 2nd floor)
- Calculate the cross spectral density matrix:
q p i pq CSD ( ω ); ≠
- From respone matrix :
[ ] = ( )( ) ( )( )
i i i i i
PSD PSD PSD PSD Y
ω ω ω ω ω
22 21 12 11
) (
- Singular valued decomposition
[ ] [ ][ ] [ ]T
i i i
i U S U
Floor 2
Floor 1
Ground
0
Amplitude
- Modal parameters:
i
ω
, φi
-10
-5
0
5
10
15
20
25
30
35
Time[s]
3 Acceleration Signals Recording
Ground floor 1st Floor 2nd Floor
0 0.5 1 1.5
Frequency[Hz]
psd11
-1 -0.5 0 0.5
Frequency[Hz]
cpsd12
10 -5
10 0
10 5
-1 -0.5 0 0.5
Frequency[Hz]
cpsd21
10 -5
10 0
10 5
0 0.2 0.4 0.6 0.8
Frequency[Hz]
psd22
0 0.5 1 1.5
Frequency[Hz]
psd11
-1 -0.5 0 0.5
Frequency[Hz]
cpsd21
Trang 5Where φ are forms of private modes The number of nonzero elements in the diagonal of i the singular matrix corresponds to the rank of each spectral density matrix The singular vectors
in Equation (18) correspond to an estimation of the mode shapes and the corresponding singular values are the spectral densities of the SDOF system expressed in Equation (12)
2.3 Mathematical model of n-storey building
The building will vibrate when it is subjected to external forces exerted by the outside like the wind, stimulated by vehicular traffic, caused by man, even earthquakes To simplify matters, we assume construction of the mathematical model for n-storey building under the effect of making buildings earthquake vibrations That means n degrees of freedom system modeled from buildings also fluctuate.The vibration of n degrees of freedom of the form is considered as figure H.2 [2,5] Supposed thatthe moving is in one direction, according to Newton's second law and D'Alembert principle, the equations of the system oscillate many degrees of freedom under the effect of horizontal x ground acceleration x0''(t)is described as follows [2,5]:
[ ] { } [ ] { } [ ] { } [ ] { }''
0 '
''
x M x
K x C x
T
x x
x x
3 '' 2 '' 1 ''
n
T
x x
x x
3 ' 2 ' 1 '
x x
x x
0 ''
0 '' 0 '' 0 ''
) (
''
0
''
Respectivelyx i (t), x i'(t),x i''(t)are displacement, velocity, acceleration in the mass concentration at the th
i floor,
dt
dx t
i'( )= , 2
2 ''
) (
dt
x d t
damping matrix, [ ]K is the stiffness We simulatenously diagonalize matrices [ ]M and [ ]K ; and assume that [ ]C is also diagonal with the n damping ratios ξ on the diagonal The n i
eigenvaluesω , corresponding eigenvectorsi2 { }φi and damping ratios ξi are the modal parameters of the system
All three matrices[ ]M ,[ ]C ,[ ]K , each with size (n× ) and is defined as follows : n
[ ]
=
n
m
m m
M
0 0 0
0
0 0
0 0 0
0 0 0
2
1
;[ ]
=
nn n
n
n n
c c
c
c c
c
c c
c
C
2 1
2 22 21
1 12 11
;[ ]
=
nn n
n
n n
k k
k
k k
k
k k
k
K
2 1
2 22 21
1 12 11
Where m i is the mass concentration at the th
i floor,i=1,2, n
Figure 3 Mathematical model of n-storey building under the effect of horizontal ground
0
•
x
0
•
•
n m
i m
1
m
•
•
+x i
Trang 62.4 Construct stiffness matrix from modal parameters
Shear beam model is assumed that motion in a single floor depends on the displacement
of the immediately above and below floors The assumption is emphasized that the stiffness
of the floors is greater than the wall Stiffness matrix can be written as formula (20):
[ ]
−
− +
−
−
− +
−
− +
=
−
−
n n n n n n
k k k k k k k
k k k k k k k
K
0 0
0 0
0 0
1 1 3
3 3 2 2 2 2 1
(20)
Where: k j is the stiffness of the storey j
The equation of the eigenvalues [ ]{ }K Φi = ωi2[ ]{ }M Φi for the shear beam model can be inverted in order to evaluate the stiffness matrix [ ]K Where: Respectivelyω ,i { }Φi are modal frequencies and mode shape vectors corresponding th
i Thus, the relationship between the physical parameters and the modal parameters of the building can be expressed as the equation (21):
Equation (21) can be written as elementary as the equation (22):
0 2
0
0
2
2
k k i m k
i
k k k i m k
i k
K
n i
k n k n k n i m n k n ni
k n k n i m n
(22) Solving the equation (22) we find the stiffness from 1 floor to th n thfloor Therefore, to generalize a corresponding linear system equation (22) can be translated into analytical formulas as follows:
Let:
2 ( 1)
1
ji
[ ]1 n, ,
j∈
∀
i j ji
n
j l li l i
j
m k
) 1 (
2
−
=
−
φ φ
φ
Therefore, the expression (24) can be abbreviated as follows:
[ ]1 n, ,
j∈
∀
ji
n
j l li l i j
m k
φ
φ ω
∆
=
2.5 Identificate modal parameters and stiffness of model of 2-storey building
Geometry of a two-storey building is designed in the pattern of shear beam, painted with software Artemis TestorPro 2011 as in Figure 4 The material is made entirely of carbon steel
Trang 7Figure 4 Geometry two-storey building
Acquisition of experimental system is for responses of model two-storey building Thesampling rate f s = 2048 samples/s, ensuring the Nyquist criterion Step time retrieving data is 4.8828125*10-4 (s) In this experiment, the data was recorded during 15 seconds Therefore the amount sampleson each channel were 30720 samples
Experiment 1:
generated by a small hard rubber hammer on the 2nd floor in the horizontal x with a random force Results of from the 4th measurement in a data set withmeasured 10 times, is shown infigures as follows:
Figure5 One side of the spectrum amplitude response ground acceleration,
1st floor and 2nd floor
Through analysis of the spectrum we find out the ground nearly fluctuated Thus to simplify the problem, we only consider the correlation between 1st floor and 2nd floor The power spectral density of the system is calculated as a function of physical frequency is shown in figure 5
Figure 6 Power spectral densities of acceleration response and comment
on the 1st floor 2nd floor
In the first mode shape: the 1st and 2nd flooroscillate in phase, its displacement increases with height, floor 2 shifted almost 1.5 times the 1st floor, frequency 10.6060 rad /s
In the second mode shape: 1st and 2nd floor is opposite phase oscillation; a node appears The 1st inter-story drift 2nd floor is near 1.5 times, frequency 29.0597 rad /s
-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Frequency[Hz]
Single-Sided Amplitude Spectrum of y
Amplitude Spectrum of y0
Amplitude Spectrum of y2
-1
0
1
psd11
Frequency[Hz]
-1 0 1
csd12
Frequency[Hz]
-1 0 1
csd21
Frequency[Hz]
-1 0 1
psd22
Frequency[Hz]
Trang 8Floor 1 Floor 2
Ground
Floor 1 Floor 2
Ground
x
z
x z
Figure 7 Bending mode shapes along the identified high building
The modal parameters are identified and shown in Table 1:
Table 1 Mode shapes, stiffness identified when we used hard rubber hammer impact
with random excitation 2nd floor horizontal x
The average stiffness of the floor 1, floor 2 [N/m] and the
average frequency of two mode shapes of 10 independent
measurements
1
k = 3317,091
2
k = 1326,64
1
f = 1,688
1
k = 3414,902
2
k = 3949,381
2
f = 4,625
Standard deviation ( d )
1
k
2
k
d = 0,0
1
f
d =0,0
1
k
2
k
2
f
d =0,0
Sketch graphs stiffness andinter storey drift [m/m] between the floors is shown in Figure 8 and Figure 9
Floor 2
Floor 1
0
Ground
Figure 8 The stiffness of the floors
Floor 2
Floor 1
Inter-storey drift [m/m]
0 Ground
Figure 9 Inter-storey dirft
Experiment 2:
When the 1st and 2nd floor have m1 =m2 =mass = 11.9737 kg, the building suffered a vibration stimuli from a DC motor is fastened on the ground of the 2- storey building model Figure 10and figure 11 show result of Single-sided amplitude spectrum and power spectral density
Trang 9Figure 10 Single-sided amplitude spectrum of the ground acceleration responses, floor 1
and floor 2 to the stimulus was vibratingmotor
Figure 11 Power spectral densities of acceleration responses of the 1st floor, 2nd floor
Two mode shapes arealso identified when stimulated by a vibration motor and they are nearly the same to Figure 7 The Table 2 shows the parameters of the two mode shapes and stiffness per the floor in each mode are identified in the case used to create vibration motor:
Table 2 Mode shape, stiffness identified when using the vibration motor excitation on
the ground floor
The average stiffness of the floor 1, floor 2 [N/m] and the
average frequency of two mode shapes of 10 independent
measurements
1
2
1
f =1,688
1
k =3620,982
2
k = 3917,751
2
f =4,625
Standard deviation ( d )
1
k
2
k
d = 0,0
1
f
d = 0,0
1
k
2
k
d = 15,0906
2
f
d =0,0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Frequency[Hz]
Single-Sided Amplitude Spectrum of y
Amplitude Spectrum of y0 Amplitude Spectrum of y1 Amplitude Spectrum of y2
1,688 Hz
4,625 Hz
-4 -2 0 2
Frequency[Hz]
-4 -2 0 2
Frequency[Hz]
-4 -2 0 2
Frequency[Hz]
-4 -2 0 2
Frequency[Hz]
Trang 103 CONCLUSION
Recognized results between the two cases with a hard rubber hammer excitation and vibration motor and shock with small deviations are acceptable Oscillation frequency separately unbiased for both mode shapes Meanwhile, the stiffness of the 1st floor of mode 1 has deviation with 13.69562 N/m, the 2nd floor stiffness of mode 1 has deviationwith 0 N/m; the 1st floor stiffness of mode 2 has deviation with 206,07993 N/m, the 2nd floor stiffness of mode 2 has deviationwith 31.62963 N/m
The analysis of modal with FDD allows us to easily identify the modal parameters quickly and accurately This was done only with the measurement of the response of the building when it is subjected to the forces excited by the input amplitude regardless even without measuring those excitation forces.This approach provides us with the bending samples However, it does not affect the calculation results about the stiffness according to the mode shapes FDD method which was successfully applied on a model two-storey building was designed and constructed according to the pattern shear beam with identified modal parameters and the stiffness of the floors The stiffness is one of the main parameters controlling their seismic resistance.The studied results have demonstrated the ability to use the FDD into reality methods for civil engineering structures.It also can be applied to test the health of the structure and the building.This method is a useful contribution to find out the weak floor on the building which is easyaffected with earthquake, wind and storm
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