Kỷ yếu hội nghị khoa học và công nghệ toàn quốc về cơ khí - Lần thứ IV

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 Phuoc1a, Phuoc Nguyen Van2b
1
Kien Giang Vocational College, Vietnam
2
HCMC University of Technical and Education, Viet Nam
a
;
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

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 frequency ω respectively 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]:

[Y ](ω ) = [H ](ω )∗ [X ](ω )[H ](ω )T

(1)

T

Where: ∗ is denoted complex conjugate and is transposed. If r is the number of inputs
and m is the number of simultaneous recorded signals, at each angular frequency ω , the size
of [X ](ω ) , [Y ](ω ) and [H ](ω ) are r × r , m × m and m × r , respectively. In Operational Modal
Analysis, the usual assumption is that the input is white noise. That means the power spectral
density matrix is expressed:

[X ](ω ) = [C ]

(2)

Where [C ] is constant matrix. The [H ](ω ) matrix can be written in a pole ( λk ) and
Residue ( [Rk ] ) formas:
n
[H ](ω ) = [Y ](ω ) = ∑ [Rk ]


(6)

Where φk is the mode shape, λk is the modal participation vector. All those parameters
are specified for the k th 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:
n
n 
[Y ](ω ) = ∑∑  [Rk ]
k =1 l =1

 jω − λ

+

[Rk ]∗

  [Rl ]
[Rl ]∗ 
C
+

∗  
jω − λk   jω − λl jω − λ∗l 

H

(7)


Where: [Ak ] is the k th residue 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 is m × m and is described in the form:
T
 n [Rs ]H
[
Rs ] 


[Ak ] = [Rk ]C  ∑
+
∗

−
−
−
−
λ
λ
λ
λ
s
=
1
k
s
k
s 




[Y ](ω ) = ∑

k∈Sub (ω )

d kφkφkT
d ∗φ ∗φ T
+ k k k∗
jω − λ k jω − λ k

(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
The singular value decomposition of an m × n complex matrix A is the following
factorization:
A = U × S ×V H

(13)
774


Kỷ yếu hội nghị khoa học và công nghệ toàn quốc về cơ khí - Lần thứ IV
Where U and V are unitary matrix and S is a diagonal matrix that contains the real
singular values:
S = diag ( s1 ,......., sr )

(14)


0
-5

0

10
Frequency[Hz]
cpsd21

0
-0.5

-1
-5
10

5

10

0.5
Amplitude

Time[s]

0

- Calculate the matrix of
power density spectral:


Amplitude

25

Acceleration [m/s 2]

psd11

- Simultaneous recordings.

Ground floor
1st Floor
2nd Floor

30

Amplitude

35

0
-0.5

-1
-5
10

0

10

- Calculate the cross spectral
density matrix:
1st singular vector at frequency 4.625

Amplitude
Floor 1

CSD pq (ωi ); p ≠ q

psd11
1.5

Floor 2

1
0.5

- From respone matrix :

0
-5
10

- Modal parameters:

0

10
Frequency[Hz]
cpsd21

-5
10

0

10
Frequency[Hz]

5

10

- Singular valued decomposition

[Y ](ω i ) = [U i ][S i ][U i∗ ]T

Figure 2. Experimental flowchart using FDD
The spectral density matrix is then approximated to the following expression (15) after
SVD decomposition:

With

[Y ](ω ) = [Φ ][S ][Φ ]H

(15)

[Φ ][Φ ]H = [I ]

(16)


⋅

⋅ ⋅
⋅ ⋅
⋅ ⋅
⋅ ⋅
⋅ sr
0 0

{φr }]

0
⋅ 
⋅

0
0

0

(17)

(18)


Kỷ yếu hội nghị khoa học và công nghệ toàn quốc về cơ khí - Lần thứ IV
Where φi are forms of private modes. The number of nonzero elements in the diagonal of
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).

x3' ... x ' ; x0''

] { } = [x

x0''

x0''

' T

'
1

T

T

''
0

...

]
x ]
''
0

x0'' = x0'' (t ) is ground acceleration (like as a system with a degree of freedom).
xi' (t ) , xi'' (t ) are displacement, velocity, acceleration in the mass


0  ;
[C ] =  21
 ...
... 0 


0 mn 
c n1
0
0

c12
c 22
...
cn 2

... c1n 
 k11
k
... c 2 n  ;
[K ] =  21
 ...
... ... 


... c nn 
k n1

k12
k 22

••

x0

Figure 3. Mathematical model of n-storey building under the effect of horizontal ground
776


Kỷ yếu hội nghị khoa học và công nghệ toàn quốc về cơ khí - Lần thứ IV
2.4. 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):
 k1 + k 2
 −k
2

 0
[K ] = 
 
 

 0

− k2
k 2 + k3

0



− kn 

k n 

(20)

Where: k j is the stiffness of the storey j
The equation of the eigenvalues [K ]{Φ i } = ωi2 [M ]{Φ i }for the shear beam model can be
inverted in order to evaluate the stiffness matrix [K ] . Where: Respectively ω i , {Φ i } are modal
frequencies and mode shape vectors corresponding i th . Thus, the relationship between the
physical parameters and the modal parameters of the building can be expressed as the
equation (21):

([K ] − ω [M ]){Φ } = 0
2
i

(21)

i

Equation (21) can be written as elementary as the equation (22):
 k  k  2 m

0
0
k2


2




 (n1)i  0
2 mn1

 
k
k
k

k






1
1
n
n
n
n


ni  0
i



(23)

n

∀j ∈ [1, n], k j = ω

∑mφ
2
i

l li

l= j

φ ji − φ( j −1)i

(24)

Therefore, the expression (24) can be abbreviated as follows:
n

∀j ∈ [1, n], k j = ωi2

∑mφ
l= j

l li

∆φ ji


0.3

|Y(f)|

0.2

0.1

0

-0.1

-0.2

5

0

10

15

30
25
20
Frequency[Hz]

35

40


1

0

10
20
30
Frequency[Hz]
d21

40

0

-1

-1
0

1

Amplitude

1

0

10
20

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.
778


Kỷ yếu hội nghị khoa học và công nghệ toàn quốc về cơ khí - Lần thứ IV
z

z
y

y
x

x

Floor 2

Floor 2

Floor 1

Floor 1

Ground

Ground
1,688 Hz


average frequency of two mode shapes of 10 independent
measurements

k1

Standard deviation ( d )

= 3317,091

k1

= 3414,902

k 2 = 1326,64

k 2 = 3949,381

f 1 = 1,688

f 2 = 4,625

d k1 = 1,758

d k1 = 7,374

d k2 = 0,0

d k2 = 2,669

d f1 =0,0

3

4

5
6
Kg/s2 or N/m

7

8

9

0

10
X 103

1

2

3

4

5

6

Amplitude Spectrum of y2

0.18
0.16
0.14

|Y(f)|

0.12

1,688 Hz

0.1

4,625 Hz

0.08
0.06
0.04
0.02
0
0

10

20

30
40
Frequency[Hz]

Frequency[Hz]
csd21

0

0

0

4

-2

10
20
30
Frequency[Hz]

40

csd12

-2

0
-3

2

-4


40

2
0
-2
-4

0

20
30
10
Frequency[Hz]

40

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
1st

Mode i
f (Hz)

2 nd


d k1 = 304,2966

d k2 = 0,0

d k2 = 15,0906

d f1 = 0,0

d f 2 =0,0

The average stiffness of the floor 1, floor 2 [N/m] and the
average frequency of two mode shapes of 10 independent
measurements

Standard deviation ( d )

780

= 3917,751


Kỷ yếu hội nghị khoa học và công nghệ toàn quốc về cơ khí - Lần thứ IV
3. 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

Civil Structures Subject to Ambient and Harmonic Excitation, 2010.
[8] Jing Hang, Operational modal identification technique based on independent component
analysis, This paper appears in : Electric Technology and Civil Engineering (ICETCE),
2011 International Conference.

781