An Aerodynamic Load Correction Method for HFFB Technique Based on Signal Decoupling and an Intelligent Optimization Algorithm

In high-frequency force balance (HFFB) wind tunnel tests, the aerodynamic wind loads at the base of the buildingmodel are usually amplified by the model-balance system.This paper proposes a new method for eliminating such an amplification effect. Firstly, the measured base bending moment signals are decoupled into independent components. Then, an optimization model is established to represent the problem of identifying the natural frequencies and damping ratios for the different modes of the model-balance system. Finally, the genetic algorithm (GA) is employed to seek the solution to the optimization problem, and the base bending moment is corrected through the identified dynamic parameters of the model-balance system. Compared to the conventionally used knocking method, the proposed method requires no extra knocking tests and can take the aerodynamic damping of the model-balance system into account. An engineering case, the Guangzhou East Tower (GZET), is taken as an example to show the effectiveness of the method.


Introduction
The HFFB technique is one of the most widely used methods to assess the wind-induced loads and response of high-rise buildings and tall structures.It is cost-effective and is capable of directly measuring the wind-induced forces or torques at the building base, and following the basic theory of HFFB technique, the structural response can be computed [1][2][3][4][5][6][7][8].
It is noteworthy that the forces and torques measured by a high-frequency force balance at the base of a building model is not the actual aerodynamic load for wind effect analysis.The model-balance system, which is made up by the balance and building model tightly connected together, has an obvious amplification effect on the aerodynamic load.That is, the measured forces and torques at the base of the building model need to be corrected to eliminate such amplification effects.If the actual aerodynamic load is regarded as the input of the model-balance system, the measured load by the balance is the output of the system.Therefore, it needs to identify the natural frequency and damping ratio of the model-balance system to obtain its mechanical admittance and conduct an inverse analysis to get the actual aerodynamic load.
Conventionally, an additional knocking test is carried out to obtain the free vibration signal based on which the natural frequency and damping ratio can be identified.However, this method has a few drawbacks as follows.Firstly, the knocking test is usually conducted by manually knocking the building model using a plastic hammer, and it is hard for the tester to control the knocking force.If the knocking force is too small, the obtained free vibration signal may be too weak to accurately identify the dynamic parameters.Conversely, if the knocking force is too large, the base force and torque of the building model may exceed the measuring range of the balance and damage the instrument.Secondly, the damping ratio of the model-balance system measured by the knocking test is the structural damping ratio of the system.However, the damping of the model-balance system in the wind tunnel test includes not only the structural damping but also the aerodynamic damping.Neglecting the aerodynamic damping may underestimate or overestimate the total damping of the model-balance system, which would in turn affect the correction results of the aerodynamic force spectrum.
In addition, the model coupling effect of the modelbalance system makes it more difficult to accurately identify the natural frequency and damping ratio of the system and effectively eliminate the dynamic amplification effect.The model coupling effect, conventionally occurring between two sway modes, makes the input aerodynamic force of the model-balance system amplified by two vibration modes.Xu et al. proposed a method of identifying the dynamic parameters of the model-balance system without using the knocking test [9].The results of the case analysis showed that that method naturally considers the influence of aerodynamic damping, so the identifying accuracy is better than the traditional knocking test.Consequently, the eliminating effect of amplification effects is better as well.Since this method was based on the single mode assumption and did not consider multimode coupling effects, it still has room for improvement.For some high-rise buildings, especially for those with unsymmetrical plan designs, the modal coupling effect might be significant and should be taken into account.Moreover, the approach of solving the nonlinear equations to seek the natural frequency and damping ratio of the modalbalance system, as illustrated in the study of Xu [9], may tend to be trapped to local minimum for some cases.
This study proposes a new method for identifying the natural frequency and damping ratio of the model-balance system in wind tunnel tests.It requires no additional knocking tests and identifies the dynamic parameters from the wind tunnel test data.Firstly, this method effectively decouples the signal measured by HFFB [10][11][12][13][14] and then converts the parameter identification problem into an optimization problem.By adopting an intelligent searching algorithm [15,16], this method can seek out the global optimal solution to the optimization problem.Based on the identified natural frequency and damping ratio of the model-balance system, the dynamic amplification effect to the aerodynamic wind force can be effectively eliminated.

Methodology
A typical six-component HFFB, such as the ATI-typed HFFB used in the following wind tunnel test, can measure the windinduced forces and torques at the base of the building model to represent the global wind force acting on the building.Actually, the aerodynamic load is the input of the modelbalance system and the forces and torque measured by the HFFB are the output of the system.The dynamic amplification effect can be eliminated by the following equation: where   () is power spectral density (PSD) of the aerodynamic force,   () is the PSD of the amplified aerodynamic force, and |  ()| 2 is the mechanical admittance of the model-balance system which can be formulated as where  denotes the frequency;   and   are the natural frequency and damping ratio for a certain mode of the model-balance system, respectively.This equation directly gives the relationship of input and output of a single-degreeof-freedom (SDOF) system.In the traditional methods that have not considered the mode coupling effect, (2) is correct.But if the mode coupling effect of the model-balance system is considered, we must decouple the measured signal first; then, for each mode vibration, (2) is valid.Equations ( 1) and (2) show that, to effectively eliminate the amplification effect of the model-balance system to the aerodynamic load, the natural frequency and damping ratio of the model-balance system must be accurately identified.Obviously, (1) and ( 2) only consider the input and output of the model-balance system for a certain mode.In practical applications, the main axes of HFFB can be regarded as approximately consistent with the geometrical axes of the building model, and for symmetrical and approximately symmetrical buildings, the geometrical axes often match well with the modal vibration axes of the model-balance system.This allows users to directly deal with the parameter identification problem as isolate subproblems in different geometrical axis directions.Since modern supertall buildings are usually designed and built with higher height than ever before and more innovative outline, such as the 530 m high Guangzhou East Tower with the so-called unsymmetrical setback design, the building models are correspondingly slimmer and unsymmetrical, making the modal-balance system with lower frequency and the mode coupling effects significant.To accurately identify the natural frequency and damping ratio of the modelbalance system, the coupled signal measured by the HFFB must be decoupled firstly.

Method of Signal Decoupling.
As aforementioned, the wind-induced force and torque signal measured by the HFFB is a kind of coupled signal and needs to be decoupled before analysis.Consider a zero-mean valued time-history of coupled m-channel signal () = [ 1 (),  2 (), ...,   ()]  , where t denotes time instant.The signal decoupling is to find an appropriate matrix to convert the coupled signals to uncoupled ones, which can be formulated as where () = [ 1 (),  2 (), ...,   ()]  is the n-channel independent source signal and  is the coupling matrix with m rows and n columns.Obviously,  should be of full rank to ensure that it is reversible.Since there is a correlation among the components of measured signals, a linear transformation is needed to preprocess the signal (), which is called prewhitening, as shown in the following equation: where  is  by  dimensional whitening matrix, which can be constructed based on the singular value decomposition (SVD), as will be discussed later, and () are the whitened signals.
The correlation matrix of time-history signals () can be expressed as where  is the time delay and N is the number of samples.Meanwhile, the correlation matrix of the source signals can be expressed as Therefore, the covariance matrix of time-history signals () can be written as The SVD of the covariance matrix of signals can be expressed as where   is the diagonal matrix of eigenvalues and   = [ 1 ,  2 , ...] is an orthogonal matrix of eigenvectors.The whitening matrix  can be expressed as Then the covariance matrix of the whitened signals can be computed as where R (0) is the covariance matrix of ().As a result, the components of () become uncorrelated after linear transformation by the whitening matrix .
The correlation matrix of whitened signals () can be expressed as Define an orthogonal matrix  that satisfies According to (3) and ( 4), the orthogonal matrix  can be expressed as  =  (13) Therefore, the relationship between the correlation matrix of the mixed time-history signals after prewhitening R () and the correlation matrix of the source signals R () can be computed as Equation ( 14) is transformed as Because the components of source signals are independent, R () is a diagonal matrix.Equation (15) shows that R () can be diagonalized.Therefore, one can pick a different set of values   ( = 1, 2, ..., ) to get a set of estimates about R (  ) and then carry out the joint diagonalization of R (  ) to get the matrix  [11,12].
After obtaining the matrices  and , the estimation of coupling matrix  can be obtained by The separation matrix can be obtained by performing coupling matrix inversion: Finally, the optimal estimation of source signals is The entire procedure of the signal decoupling method is summarized as follows: (1) Detrend the measured signal () to make it zeromean.
(5) Define an orthogonal matrix , and take a set of nonzero time delay values   to get a set of estimated values R (  ), and conduct the joint diagonalization of R (  ) to calculate orthogonal matrix .

Intelligent Search
Based on Genetic Algorithm.After the measured signal is decoupled, the natural frequency and damping ratio identification can be regarded as isolated problems for different modes.Substituting (2) into (1) yields the PSD of corrected real aerodynamic load as Under logarithmic coordinates, the PSD of aerodynamic base bending moment has an approximately linear relationship with frequency at a certain frequency range, in which the structural natural frequency is located, which can be formulated as where  and  are coefficients depending upon the characteristics of wind loads.Equation (20) can be written in logarithmic coordinates as Identifying the natural frequency and damping ratio of a certain mode of the model-balance system is equal to finding a pair of   and   which makes lg(  ()) to be linear with lg.If a linear function, lg(   (  )), is employed to fit the corrected PSD of aerodynamic load lg(  ()), the dynamic parameter identification problem can be transformed into a optimization problem as The next task is to find a set of parameters, including   ,   , , and , to minimize Δ.To avoid the solution to the optimization problem being trapped into a local minimum, an intelligent algorithm, the genetic algorithm (GA), is adopted to seek the optimal solution to (22).Since the optimization problem here is not complicated, the base GA library provided in Matlab GA toolbox is directly used to seek the optimal solution.The number of individuals was set as 80 and the number of generations was set as 300.If the relative difference of Δ in 100 consecutive generations is smaller than the preset threshold value  of 10 -3 , as shown in ( 23) and (24), the iteration process was regarded as having converged.

Case Study
In this section, two examples are given to show the effectiveness of the aforementioned method.Firstly, a numerically simulated example is presented to show the validation of the signal decoupling method.Then, a 530 m high supertall building, the GZET, is taken as an example to show the effectiveness of the whole procedure, including dynamic parameter identification and the dynamic amplification effect elimination.

Numerical Simulation for
where m is the number of mixed signals, n is the number of source signals, and A is the mixing matrix;  and  are the source signal matrix and mixed signal matrix, respectively.Figures 1, 2, and 3 show the source signal, mixed signal, and separated signal for each case, respectively.
As can be seen from the figures, the shape of the separated signals can approximately match the corresponding source signals, and the sequence of the separated signals is not the same as the source signals.Actually, there is no need to care for the sequence of the separated signal, and each separated signal will be corrected individually in the PSD correction procedure.In order to further verify the validity of the signal decoupling method, the similarities between the source signals and the separated signals are quantitatively analyzed by defining the correlation coefficient   between the source signal and the separated signal as where R (0), R (0), and R (0) can be obtained by ( 27), (28), and (29), respectively: By the above definition, the similarities between the source signals and the separated signals for the three cases are quantized in Table 2.
It can be seen that the moduli of the correlation coefficient of the three groups signals are all close to 1 in Table 2, but there is a 180 ∘ phase shift of S1 in case 3.In essence, the decoupling procedure is to find the eigenvector of the mixed signal matrix, and the eigenvector is a kind of "shape function" or can be regarded as the "mode" of the mixed signal.These shape functions only represent the shape of the source signal but cannot represent the exact phase of the source signal.It is noteworthy that we only use the separated signals to compute the PSD, which only describe the energy distribution in frequency domain and have nothing to do with phase information.Therefore, the phase shift would not affect the accuracy of the following dynamic parameter  identification, and the aforementioned method is valid for signal decoupling and can be used to decouple the signals measured by HFFB.

Practical Engineering Case for Amplification Effect Elimination.
The Guangzhou East Tower, also called the Chow Tai Fook Financial Center, is 530 m high and is currently the tallest supertall building in Guangzhou, China.It adopted the setback design to reduce the wind-induced response.
That is, the elevation design of the building was divided into three parts, and the cross-sectional size of each part shrinks stepwise from the ground to the top, as shown in Figure 4.The building model is relatively slim and the unsymmetrical setback design enhances the modal coupling possibility of the model-balance system in the HFFB wind tunnel test.Hence, it is a good example to examine the effectiveness of the proposed method for eliminating the dynamic amplification effect of the model-balance system.A rigid model with a geometric length scale of 1:500 was made by foam materials covered with thin wood skin to represent the GZET.The building model was rigidly connected with an ATI-typed HFFB that was tightly bound on the rotatable foundation of the wind tunnel.The models of surrounding buildings that might affect the wind effects on the two tested buildings were also made and mounted on the ground of the wind tunnel to simulate the surrounding conditions.Figure 5 shows the models mounted in the wind tunnel.The wind field suggested by Engineering Sciences Data Unit (ESDU) [17] was simulated in the wind tunnel test.ESDU has provided a relatively accurate method of estimating meanhourly wind speeds in the neutral atmospheric boundary layer near the ground.According to the Calculation Rule of ESDU, the landform exponent obtained varies in the range of 0.22∼0.24approximately and the mean wind speed profile was preadjusted to follow the exponential rule with the exponent of 0.22.The tested mean wind speed and turbulence intensity profile and the target profile of ESDU are illustrated in Figure 6.In the wind tunnel test, the six components of wind-induced forces and torques at the base of the building model were measured by the ATI balance with a sampling frequency of 400 Hz.The data length for each sample was 40960 for one wind direction.Totally, 36 wind directions with the increment of 10 ∘ were tested in the wind tunnel test.

Shock and Vibration
In order to examine the necessity of signal decoupling, data of some particular wind directions in which the modal coupling effect appeared to be more obvious were selected for analysis.For the purpose of comparison, the natural frequency and damping ratio of the modal balance are identified by two methods: (1) the method of Xu's previous study [9], in which the signal was coupled before identifying the dynamic parameters, and (2) the method proposed in this paper, in which the dynamic parameters were identified based on the decoupled signal.The identified natural frequency and damping ratio of Guangzhou East Tower model before and after decoupling are, respectively, shown in Figures 7 and 8.
It can be seen that, using either the method of reference [9] or the method of this paper, the identified natural frequency of the model-balance system slightly changes under different wind direction angles, while the identified damping ratio fluctuates more significantly with the wind directions.Figure 8 shows that, in the vibration direction of x, the maximum and minimum of identified damping ratios are 2.9% and 1.5%, respectively, and in the vibrational directions of y they are 3% and 1.5%, respectively.There is an important reason why the damping ratio varies with the wind direction.The total damping ratio of the modelbalance system includes not only the structural damping but also the aerodynamic damping.The value of the aerodynamic damping usually changes with the change of wind direction angles.It must be clarified that the fluctuation of natural frequency and damping ratio with wind direction in Figures 7 and 8 should be attributed to several factors as   (1) the effect of aerodynamic damping, (2) the uncertainty of system's damping ratio, and (3) the identification error.In [9], the authors have proven the effect of aerodynamic damping, which is the major reason of such a fluctuation.In consequence, the total damping ratio of the system varies under different wind direction angles.The fluctuant results of the damping ratio also indirectly prove the influence of the aerodynamic damping on the total damping ratio of the model-balance system.
It is noteworthy that the dynamic parameter identification results are different with the signal being decoupled or not.Corrected PSD results of the base bending moment of  two typical wind directions are presented to show the benefit of decoupling.The signal measured by the balance is a random process, and PSD curve of the measured and corrected base bending moment fluctuates randomly with frequency, but the whole trend of the PSD curve represents the energy distribution of the vibration.On the other side, we use linear regression in logarithmic coordinate to estimate the dynamic parameters of the system.In normal coordinate system, it is a nonlinearly exponential regression process.Therefore, if this procedure is described in normal coordinate system, it becomes a nonlinearly exponential fitting problem.

Conclusions
This study presents a new method to correct the amplification effect of the model-balance system to the base bending moment.It includes two major steps.Firstly, the measured   signal is decoupled using the SVD-based method; then, based on the decoupled signal, the natural frequency and damping ratio of the model-balance system are identified through the GA algorithm.Finally, the amplified PSD of the base bending moment can be effectively corrected.Numerical simulation cases and real engineering cases are analyzed and the basic conclusions are as follows.
(1) The correlation coefficients between the real and estimated source signals are close to 1 for all numerically simulated examples, indicating that the SVD-based method can effectively estimate the source signal from mixed signals.
(2) The results of an engineering example show that the amplification can be effectively eliminated after the signal decoupling process.Compared to the corrected PSD curve of the base bending moment without decoupling, the fluctuating amplitude of corrected PSD curve is significantly lower and the correction result is much better.
(3) The identified damping ratio of the model-balance system of GZET fluctuates obviously with wind directions, indicating that the aerodynamic damping is an important part of the total damping of the model-balance system.Therefore, the method proposed in this study is more accurate than the conventionally used knocking method, which cannot take the aerodynamic damping into account.

2 ×Figure 1 :
Figure 1: The decoupling of two cosine signals with similar frequency.

Figure 2 :
Figure 2: The decoupling of cosine signal and square wave signal.

Figure 3 :Figure 4 :
Figure 3: The decoupling of cosine signal and square wave signal with white noise.

Figure 5 :Figure 6 :
Figure 5: The model of Guangzhou East Tower (GZET) in wind tunnel test.

Figure 7 :
Figure 7: The identified dynamic parameters of the model-balance system of Guangzhou East Tower under different wind directions before decoupling.

Figure 8 :Figure 9 :
Figure 8: The identified dynamic parameters of the model-balance system of Guangzhou East Tower under different wind directions after decoupling.

Figures 9 and 10
show the correction results of 70 ∘ and 240 ∘ wind direction without decoupling.It can be seen that the PSD of the base bending moment fluctuates intensively around the natural frequency of the model-balance system, as shown in the rectangular box in Figures9 and 10.This phenomenon is due to the coupling effect of different vibration modes on the PSD of base bending moment.For example, in Figure9(a), the PSD curve before correction has two neighboring peaks, indicating that two vibration modes have effect on the PSD curve.Without the decoupling preprocess of data, the minor peak cannot be effectively removed.However, the correction effect is much better if the signal was decoupled before correction.Compared to Figures9 and 10, the corresponding corrected PSD curves in Figures11 and 12are obviously closer to a straight line, and the fluctuating amplitude of the corrected PSD curve is much lower than that without decoupling preprocess.After effectively correcting the PSD of the base bending moment, the wind-induced response can be analyzed by the conventional method.

Figure 10 :
Figure 10: The correction effect of the base overturning moment of Guangzhou East Tower at 240 ∘ wind direction angle before decoupling.

Figure 11 :
Figure 11: The correction effect of the base overturning moment of Guangzhou East Tower at 70 ∘ wind direction angle after decoupling.

Figure 12 :
Figure 12: The correction effect of the base overturning moment of Guangzhou East Tower at 240 ∘ wind direction angle after decoupling.
wave; the third case adds a random noise signal to the second case.For all three cases, the source signals were mixed by a random mixing matrix A, as shown in (25).Then the mixed signals are separated to examine the effectiveness of the signal decoupling method.
Signal Decoupling.Three simulated cases are considered as shown in Table 1.The first case considers two isolate cosine signals with different frequencies; the second case consists of one cosine signal and one periodic square

Table 1 :
Parameter settings for signal simulation.

Table 2 :
The correlation coefficient between the source signals and the separated signals.