Shock and Vibration

Volume 2018, Article ID 2128519, 12 pages

https://doi.org/10.1155/2018/2128519

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

^{1}Guangzhou University-Tamkang University Joint Research Center for Engineering Structure Disaster Prevention and Control, Guangzhou University, Guangzhou, China^{2}State Key Laboratory of Subtropical Building Science, South China University of Technology, Guangzhou, China

Correspondence should be addressed to An Xu; moc.qq@ux-ykcor

Received 18 January 2018; Revised 14 April 2018; Accepted 8 May 2018; Published 5 June 2018

Academic Editor: Laurent Mevel

Copyright © 2018 Chengzhu Xie et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In high-frequency force balance (HFFB) wind tunnel tests, the aerodynamic wind loads at the base of the building model 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.

#### 1. 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 [1–8].

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 model-balance 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 modal-balance 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 [10–14] 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.

#### 2. Methodology

A typical six-component HFFB, such as the ATI-typed HFFB used in the following wind tunnel test, can measure the wind-induced 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 model-balance 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 is the mechanical admittance of the model-balance system which can be formulated aswhere 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-degree-of-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 model-balance system, the coupled signal measured by the HFFB must be decoupled firstly.

##### 2.1. 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 , 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 aswhere 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 aswhere 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 aswhere is the diagonal matrix of eigenvalues and 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 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

Therefore, the relationship between the correlation matrix of the mixed time-history signals after prewhitening and the correlation matrix of the source signals can be computed as

Equation (14) is transformed as

Because the components of source signals are independent, is a diagonal matrix. Equation (15) shows that can be diagonalized. Therefore, one can pick a different set of values () to get a set of estimates about and then carry out the joint diagonalization of 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:

Detrend the measured signal to make it zero-mean.

According to (5) to (7), estimate the correlation matrix and the covariance matrix of and the correlation matrix of source signals .

Prewhiten according to (4) and obtain the whitening matrix .

Estimate the correlation matrix of whitened signals .

Define an orthogonal matrix , and take a set of nonzero time delay values to get a set of estimated values , and conduct the joint diagonalization of to calculate orthogonal matrix .

Compute the estimation of source signals according to (16) to (18).

##### 2.2. 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 aswhere 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 to be linear with . If a linear function, , is employed to fit the corrected PSD of aerodynamic load , 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 . 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 in 100 consecutive generations is smaller than the preset threshold value of , as shown in (23) and (24), the iteration process was regarded as having converged.

#### 3. 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.

##### 3.1. Numerical Simulation for 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 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.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.