Kỷ yếu hội nghị khoa học công nghệ toàn quốc 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 Kien Giang Vocational College, Vietnam HCMC University of Technical and Education, Viet Nam a nploc@caodangnghekg.edu.vn; bvanphuocspkt@gmail.com 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 biết đến với tên gọi Phân tích với ngõ ra, sử dụng để ước lượng tham số công trình tòa nhà, cầu, tòa tháp cấu trúc khí Thuận lợi phương pháp thiết bị kích thích đắt tiền thay nguồn rung động từ môi trường xung quanh, chẳng hạn rung động từ gió, sóng lưu thông xe cộ sử dụng ngõ vào với biên độ khôngđượcquan tâm, chúng mô hình hóa nhiễu trắng giải thuật nhận dạng thể thức (modal) Bài báo trình bày tổng quan kỹ thuật không tham số dựa việc phân giải miền tần số, mô hình động học tòa nhà n tầng phương pháp nhận đạng tham số modal sử dụng FDD.Thêm vào đó, sử dụng xác suất thống kê để đánh giá kết đạt độ cứng (stiffness) độ xê dịch tầng (inter-storey dift) tòa nhà tầng Từ khóa: FDD: Phân giải 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 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 772 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV of 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 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 Data acquisition system with NI-USB 9234 hardware in LabVIEW 2011 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 ] [X ](ω ) k =1 jω − λk Where: + [Rk ]∗ jω − λ∗k λk = −σ k + ωdk (3) (4) n is the total number of interested modes, λk is the pole of k th mode and σ k is the modal damping of the k th mode, ωdk is the damped natural frequency of the k th mode: ωdk = ω0 k − ς k2 773 (5) Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV Where: ς k is the critical damping of the k th mode, ω0 k is the undamped natural frequency of the k th mode [Rk ] matrix is called the residue matrix andis expressed as following form: [Rk ] = φk γ kT (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) H Where 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 n [Y ](ω ) = ∑ [Ak ] k =1 jω − λ k + [Ak ]∗ jω − λ∗k + [Bk ] − jω − λ k + [B k ]∗ (8) − jω − λ∗k 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 = k s k s   (9) The contribution to the residue from the k th mode is given: T [ Rk ]C [Rk∗ ] [A ] = k (10) 2σ k Where: σ k is minus the real part of the pole λk = −σ k + jω dk 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: lim damping →light [Ak ] = [Rk ]C[Rk ]T = φk γ kT Cγ kφkT = d kφkφkT (11) Where: d k is 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 by Sub(ω ) Thus, in the case of a lightly damped structure, the response of spectral density matrix can always be written as following final form: [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 công nghệ toàn quốc 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) H The superscript on the V matrix denotes a Hermitian transformation In the case of real T valued matrices, the matrix V is only transposed and is denoted The si elements 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: Acceleration Signals Recording - Installation position of sensors: ground, floor 1, floor 20 15 10 0.5 0.5 -5 10 -5 10 Frequency[Hz] cpsd21 -0.5 -1 -5 10 10 0.5 Amplitude Time[s] - Calculate the matrix of power density spectral: 10 Frequency[Hz] psd22 10 0.8 15 10 Amplitude -10 - Fourier transform of the measured responses: determining the frequency components: ωi cpsd12 1.5 Amplitude 25 Acceleration [m/s 2] psd11 - Simultaneous recordings Ground floor 1st Floor 2nd Floor 30 Amplitude 35 -0.5 -1 -5 10 10 Frequency[Hz] PSD kk (ωi ) ; 0.6 0.4 0.2 -5 10 10 (this experiment we need to use sensors at 1st and 2nd floor) 10 Frequency[Hz] k = 1: n 10 - Calculate the cross spectral density matrix: 1st singular vector at frequency 4.625 Amplitude Floor CSD pq (ωi ); p ≠ q psd11 1.5 Floor 0.5 - From respone matrix : -5 10 - Modal parameters: 10 Frequency[Hz] cpsd21 10 PSD11 (ωi ) PSD12 (ωi )   PSD21 (ωi ) PSD22 (ωi ) [Y ](ωi ) =  0.5 -1 Amplitude Amplitude Ground ωi , φi -0.5 -1 -5 10 10 Frequency[Hz] 10 - Singular valued decomposition [Y ](ω i ) = [U i ][S i ][U i∗ ]T Figure 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) 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:  s1 0  0 S = diag ( s1 , ., sr ) =  ⋅ ⋅   [Φ ] = [{φ1} {φ2 } {φ3 } 775 s2 ⋅ ⋅ ⋅ ⋅ ⋅⋅⋅ 0 s3 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ sr 0 {φr }] 0 ⋅  ⋅  0 0  0 (17) (18) Kỷ yếu hội nghị khoa học công nghệ toàn quốc 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) 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]: [M ]{x '' }+ [C ]{x ' }+ [K ]{x} = −[M ]{x0'' } Where: (19) {x}T = [x1 x2 x3 xn ] ; {x '' } = [x1'' x2'' x3'' xn'' {x } = [x x2' x3' x ' ; x0'' ] { } = [x x0'' x0'' ' T ' T T '' ] x ] '' 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 Respectively xi (t ) , dxi d 2x , xi'' (t ) = i , [M ] is the mass matrix, [C ] is the dt dt 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 i on the diagonal The n concentration at the i th floor, xi' (t ) = eigenvalues ωi2 , corresponding eigenvectors {φi } and damping ratios ξ i are the modal parameters of the system All three matrices [M ] , [C ] , [K ] , each with size ( n × n ) and is defined as follows : m1 0 [M ] =  0  0 m2 0 0  c11 c  ; [C ] =  21     mn  c n1 0 c12 c 22 cn c1n   k11 k c n  ; [K ] =  21     c nn  k n1 k12 k 22 k n2 k1n  k n    k nn  Where mi is the mass concentration at the i th floor, i = 1,2, n •• •• x0 xn mn •• •• x + xi mi m1 •• x0 Figure Mathematical model of n-storey building under the effect of horizontal ground 776 Kỷ yếu hội nghị khoa học công nghệ toàn quốc 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  −k   [K ] =        − k2 k + k3   − k3  − k3      − k n −1     k n −1 + k n − kn        − 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 } = i (21) i Equation (21) can be written as elementary as the equation (22):  k  k  2 m  0 k2   1  i    1i  0   k2  k3  i2 m2 k3 k2      2i  0 k3          K               (n1)i  0 mn1    k k k  k       1 n n n n   ni  0 i     0 k k  m     n n  i n  (22) Solving the equation (22) we find the stiffness from 1th floor to n th floor Therefore, to generalize a corresponding linear system equation (22) can be translated into analytical formulas as follows: Let:   ( j 1)i when j   n  ji   ji  ji when j 1  (23) n ∀j ∈ [1, n], k j = ω ∑mφ 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 (25) 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 The material is made entirely of carbon steel 777 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV Figure 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: When the 1st and 2nd floor have m1 = m2 = mass = 11.9737 kg, excitation force 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: Single-Sided Amplitude Spectrum of y Ground(t),y Floor1(t),y Floor2(t) 0.5 Amplitude Spectrum of y0 Amplitude Spectrum of y1 Amplitude Spectrum of y2 0.4 0.3 |Y(f)| 0.2 0.1 -0.1 -0.2 10 15 30 25 20 Frequency[Hz] 35 40 45 50 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 csd21 psd22 csd12 psd11 -1 Amplitude Amplitude Amplitude 10 20 30 Frequency[Hz] d21 40 -1 -1 Amplitude 10 20 30 Frequency[Hz] d22 40 -1 10 20 30 Frequency[Hz] 40 10 20 30 Frequency[Hz] 40 Figure 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 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 công nghệ toàn quốc khí - Lần thứ IV z z y y x x Floor Floor Floor Floor Ground Ground 1,688 Hz 4,625 Hz Figure Bending mode shapes along the identified high building The modal parameters are identified and shown in Table 1: Table Mode shapes, stiffness identified when we used hard rubber hammer impact with random excitation 2nd floor horizontal x 1st Mode i f (Hz) nd 1,688 4,625 { φi1 } -0,55406 -0,83570 { φi } -0,83247 0,54917 The average stiffness of the floor 1, floor [N/m] and the average frequency of two mode shapes of 10 independent measurements k1 Standard deviation ( d ) = 3317,091 k1 = 3414,902 k = 1326,64 k = 3949,381 f = 1,688 f = 4,625 d k1 = 1,758 d k1 = 7,374 d k2 = 0,0 d k2 = 2,669 d f1 =0,0 d f =0,0 Sketch graphs stiffness andinter storey drift [m/m] between the floors is shown in Figure and Figure Longitudinal from 1st mode Longitudinal from 1st mode Longitudinal from 2nd mode Longitudinal from 2nd mode Floor Floor Floor Floor Ground Ground Kg/s2 or N/m 10 X 103 10 Inter-storey drift [m/m] Figure The stiffness of the floors X 10-1 Figure 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 779 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV Single-Sided Amplitude Spectrum of y Ground(t),y Floor1(t),y Floor2(t) 0.2 Amplitude Spectrum of y0 Amplitude Spectrum of y1 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 10 20 30 40 Frequency[Hz] 50 60 70 Figure 10 Single-sided amplitude spectrum of the ground acceleration responses, floor and floor to the stimulus was vibratingmotor -3 x 10 -2 -4 -3 x 10 10 20 30 Frequency[Hz] csd21 0 -2 10 20 30 Frequency[Hz] 40 csd12 -2 -3 -4 x 10 -4 40 Amplitude Amplitude -3 psd11 Amplitude Amplitude x 10 10 20 30 Frequency[Hz] psd22 40 -2 -4 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 The Table 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 Mode shape, stiffness identified when using the vibration motor excitation on the ground floor 1st Mode i f (Hz) nd 1,688 4,625 { φi1 } -0,55406 -0,83570 { φi } -0,83247 0,54917 k =3303,395 k =3620,982 k =1326,644 k2 f =1,688 f =4,625 d k1 = 21,3642 d k1 = 304,2966 d k2 = 0,0 d k2 = 15,0906 d f1 = 0,0 d f =0,0 The average stiffness of the floor 1, floor [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 công nghệ toàn quốc khí - Lần thứ IV 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 has deviation with 13.69562 N/m, the 2nd floor stiffness of mode has deviationwith N/m; the 1st floor stiffness of mode has deviation with 206,07993 N/m, the 2nd floor stiffness of mode 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 REFERENCES [1] Carlo Rainieri and Giovanni Fabbrocino, Operational modal analysis for the characterization of heritage structures, UDC 550.8.013, GEOFIZIKA VOL 28, 2011 [2] TS Nguyễn Đại Minh, “Phương pháp phổ phản ứng nhiều dạng dao động tính toán nhà cao tầng chịu động đất theo TCXDVN375: 2006”, Viện Khoa học Công nghệ Xây dựng, 2010 [3] Peeters B System Identification and Damage Detection in Civil Engineering PhD thesis, Katholieke Universiteit Leuven, 2000 [4] Ventura C., Liam Finn W.-D., Lord J.F., Fujita N Dynamic characteristics of a base isolated building from ambient vibration measurement and low level earthquake shaking Soil Dynamics and Earthquake Engineering 2003; 23:313–322, 2003 [5] CHOPRA, A K Dynamic of structures, Prentice Hall International, US, 2001, 844 p [6] Welch P.D The use of Fast Fourier Transform for the estimation of power spectra: method based on time averaging over short, modified periodograms IEEE Trans Audio Electroacoust 1967, AU-15:70-73 [7] Brincker R., Ventura C., Andersen P Why output-only modal testing is a desirable tool for a wide range of practical applications In: 21st International Modal Analysis Conference (IMAC), Kissimmee, Florida, 2003 [8] Palle Andersen, Rune Brincker, Carlos Ventura, Reto Cantieni, Modal Estimation of 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 ... 10 0.5 0.5 -5 10 -5 10 Frequency[ Hz] cpsd21 -0 .5 -1 -5 10 10 0.5 Amplitude Time[s] - Calculate the matrix of power density spectral: 10 Frequency[ Hz] psd22 10 0.8 15 10 Amplitude -1 0 - Fourier... Frequency[ Hz] csd21 0 -2 10 20 30 Frequency[ Hz] 40 csd12 -2 -3 -4 x 10 -4 40 Amplitude Amplitude -3 psd11 Amplitude Amplitude x 10 10 20 30 Frequency[ Hz] psd22 40 -2 -4 20 30 10 Frequency[ Hz] 40 Figure... 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