A new method for dynamic parameters identification of a model-balance system in high-frequency force-balance wind tunnel tests

2.1. The knocking test method

The knocking test is one of the most widely-used methods for obtaining the dynamic parameters of a model-balance system. The PSD of the measured decrement signal can be formulated as:

where $f_n$, $\zeta$ are the natural frequency and damping ratio of the model-balance system, respectively, and $x=f^2$, $a=1/A$, $c=f_0^4/A$, $b=2(2{\zeta }^2-1)f_0^2/A$. Eq. (3) can be rewritten as:

The coefficients $a$, $b$ and $c$ can be obtained by fitting the right-hand side of Eq. (4), and then the natural frequency and damping ratio can be obtained from the following equations:

The knocking test method makes sense theoretically, but there would be some problems in applications, which will be illustrated and discussed in case studies in Section 3 of this paper.

2.2. The proposed method

Different than the knocking method, the proposed method in this study use the wind tunnel test data directly to identify the dynamic parameters of a model-balance system, that is, knocking tests are no longer necessary. Since the overturning moments along the two main horizontal axes at the base of a model are the most important for analysis of structural responses, the following analysis focuses on these two directions.

The mechanical admittance of a model-balance system can be expressed as:

Substituting Eq. (6) into Eq. (1) gives:

The PSD of the overturning moments along two horizontal main axes of super-tall buildings usually decreases approximately linearly with the increase in frequency $f$ in a specified frequency range in log-log coordinate. This feature can be formulated as:

Eq. (8) can be rewritten in logarithmic format as:

Eq. (9) reveals that the relationship between $\mathrm{l}\mathrm{g}\left(S^I\left(f\right)\right)$ and $\mathrm{l}\mathrm{g}f$ can be fitted with a straight line. Eq. (7) can be rewritten as:

The measured signal by the HFFB is a kind of discrete signal, thus $S^O\left(f\right)$ is also discrete in frequency domain and $S^I\left(f\right)$ can be rewritten in a discrete form according to Eq. (10) as:

where $S_i^I$ and $S_i^O$ are the value of $S^I\left(f\right)$ and $S^O$ at frequency $f_i$. The corresponding fitted value is expressed as:

Now the problem is to find a set of $f_n$, $\zeta$, $\gamma$ and $\eta$ to minimize the relative error, in terms of $\mathrm{\Omega }$, between $\mathrm{l}\mathrm{g}{S}_i^I$ and $\mathrm{l}\mathrm{g}\left(S_{i,fit}^I\right)$, which can be formulated as:

The value of $f_n$, $\zeta$, $\gamma$ and $\eta$ can be determined by solving the following equations:

Substituting Eqs. (11) and (12) into Eqs. (14) gives:

Eq. (15) can be rewritten as:

Now there are four equations in four unknowns, thus the dynamic parameters $f_n$ and $\zeta$ can be determined by solving these nonlinear equations. Some numerical methods can be employed to solve nonlinear equations. In this study, a modified Newton-Raphson algorithm, called the “secant” method is adopted to solve the systems of nonlinear equations. Consider the solution to a system of $m$ nonlinear equations in $m$ unknowns given by:

The system can be written in a single expression using vectors, i.e.:

where the vector $\mathbf{x}$ contains the independent variables $x_i$, and the vector $\mathbf{F}$ contains the functions $f_j\left(\mathbf{x}\right)$. The procedure to solve the equations is described as below.

1) Set the maximum number of iterations $N$ and the tolerance $\epsilon$;

2) Set the initial guess for the solution as ${\mathbf{x}}_0=x_1,x_2,\dots ,x_i,\dots ,x_m$.

3) Successive approximations to the solution are obtained from:

where $\mathbf{J}$ is the Jacobian matrix and $k$ is the current iteration step number.

4) The convergence criterion is that the magnitude of the vector $\mathbf{f}\left({\mathbf{x}}_n\right)$ be smaller than the tolerance, as shown in Eq. (16). If Eq. (16) were satisfied, the iteration is ended and the unknown $\mathbf{x}$ is determined. Otherwise, the iteration procedure goes to step (3):

The main difference between the secant method and the Newton-Raphson algorithm is how to calculate the Jacobian matrix $\mathbf{J}$. The former uses the secant value instead of the tangent value when calculating the Jacobian matrix. This method is easily implemented to solve the system of nonlinear equations which is shown in Eqs. (16a)-(16d). Therefore, the natural frequency and damping ratio of a force-balance system can be easily determined by adopting the proposed method.

3.1. Example I: the CAARC standard tall building

The CAARC standard tall building is characterized as a flat-topped prismatic body with rectangular cross-section and lateral flat walls presenting no parapets or other geometric details. The full-scale dimension of the building model are as follows: height $H=$ 182.88 m; length $L=$ 30.48 m and width $W=$ 45.72 m. The HFFB wind tunnel test was carried out in the boundary layer wind tunnel at the South China University of Technology with a working section of 5 m wide and 3 m high. A rigid model with a geometric length scale of 1:300 was made to represent the CAARC building, as shown in Fig. 1. Spires and roughness elements were used to simulate a typical boundary layer wind flow specified in the Load Code of China (2012) as exposure Category C. The corresponding power law exponent is 0.22. In the wind tunnel test, the aforementioned six force or moment components at the base of the building model were measured by an ATI type high-frequency force-balance. The sampling frequency was 400 Hz and the sampling length was 40960. Wind direction varied from 0° to 350° with incremental step of 10° in the wind tunnel test.

Fig. 1The CAARC building model in wind tunnel tests

Fig. 2The identified dynamic parameters of the model-balance system of CAARC under different wind directions

a) Natural frequency

b) Damping ratio

Fig. 2 shows the identification results of the natural frequencies and the damping ratios of the model-balance system for all wind directions. It is shown that the natural frequency and damping ratio of the HFFB system under different wind directions are not quite the same. The identified natural frequency varies slightly with wind directions. However, the identified damping ratio varies more significantly with wind directions. For example, in $M_x$ direction, the maximum damping ratio is 0.97 % and the minimum damping ratio is 0.67 %. In $M_y$ direction, they are 1.19 % and 0.83 %, respectively. The phenomenon that the identified dynamic parameters vary with wind directions would be due to several aspects. First, altering wind direction is usually implemented by rotating the mechanical device at the base of the model-balance system. This procedure may cause slight changes of the dynamic parameters of the HFFB system. Second, the aerodynamic damping of the model-balance system usually varies with wind directions in wind tunnel tests. Obviously, the aerodynamic damping is a portion of the total damping of that system. Hence the damping of the model balance system varies significantly in different wind directions.

For the purpose of comparison, the knocking test was carried out and the identification results are presented as well. Before fitting coefficients shown in Eq. (3), the frequency range for fitting must be specified first. The frequency corresponding to the peak value of the spectrum $S\left(f\right)$ is usually regarded as the center frequency of this frequency range, and the band width of the frequency range should be set artificially. Actually, it is difficult to determine an appropriate band width. The reason is that too wide band width would result in an overestimated damping ratio result, and too narrow bandwidth would result in an underestimated damping ratio result. For example, if the selected frequency range for fitting is [0.85$f_n$, 1.15$f_n$], the identified damping ratio in $M_y$ direction is 1.47 %, as shown in Fig. 3(a). However, if the frequency range is specified as [0.97$f_n$, 1.03$f_n$], the corresponding identified damping ratio is 0.56 %.

Fig. 3The overestimated and underestimated damping ratio according to different frequency range for fitting

a) Fitting frequency range is [0.85$f_n$ 1.15$f_n$] and identified damping ratio is 1.47 %

b) Fitting frequency range is [0.97$f_n$ 1.03$f_n$] and identified damping ratio is 0.56 %

Fig. 4The measured and fitted PSD of the overturning moments

a)$M_x$ direction

b)$M_y$ direction

For the above reason, the appropriate frequency range for fitting is usually determined after many attempts. In each attempt, the relative error between the fitted PSD and the tested PSD is observed. One can minimize the relative error by adjusting the frequency bandwidth. Obviously, this is a cumbersome process, and it is one of the shortcomings of the knocking method. For this example, we find that the appropriate frequency range is [0.955$f_n$, 1.045$f_n$], and the fitted PSD are in good agreement with the tested PSD near the peak point of PSD, as shown in Fig. 4. The identified damping ratio is 0.53 % in $M_x$ direction and 0.65 % in $M_y$ direction. The corresponding natural frequencies of the model-balance system are 96.61 Hz and 85.41 Hz, respectively.

Using the identification results from the knocking method, the measured PSD of overturning moments are then modified, as shown in Fig. 5. It is shown that the correction results are not bad. But if we observe the corrected PSD near the natural frequency of the model-balance system, we can find that the influence of the dynamic parameters of the model-balance system is not entirely eliminated after correction. The reason is that the identified damping ratio from the knocking method is not suitable for all wind directions.

Fig. 5The PSD of overturning moments before and after correction (using the knocking method)

a)$M_x$ direction, 0° wind angle

b)$M_y$ direction, 0° wind angle

c)$M_x$ direction, 90° wind angle

d)$M_y$ direction, 90° wind angle

If the dynamic parameters of model-balance system are identified through the method proposed in this study, better correction results can be obtained. Some selected results in typical wind directions are shown in Fig. 6. Compared to the conventional knocking method, better correction results are achieved by adopting the proposed method in this study. Meanwhile, some singular points on the PSD curve degrade the correction quality. For instance, in Fig. 6(b) and 6(d), there are some singular points near the natural frequency of the model-balance system. These singular points make the PSD of $M_y$ unsmooth even after the correction. This phenomenon may be due to the noise component in the measured signal in wind tunnel tests. It should be eliminated using some signal processing techniques like wavelet transform, but this study does not cover this issue.

Fig. 6PSD of overturning moments of CAARC before and after correction using the proposed method

a)$M_x$ direction, 0° wind angle

b)$M_y$ direction, 0° wind angle

c)$M_x$ direction, 90° wind angle

d)$M_y$ direction, 90° wind angle

Fig. 7ZCC building model in wind tunnel tests

3.2. Example II: the Zhuoyue century center

The previous example is a hypothetical building, and in this section, an actual building with rectangular cross-section is considered to examine the effectiveness of the proposed method. The ZCC, which is 280 m high with 68 stories, is located in Shenzhen, China. The HFFB wind tunnel test of ZCC was carried out in the boundary wind tunnel of the South China University of Technology. According to the Chinese load code for the design of building structures (GB 50009-2012), the exposure category C (corresponding to exponents of the power law of mean wind profile of 0.22) are simulated at a length scale of 1/400 by setting spires, barriers and rough element in the test area. The building model in the test area of the wind tunnel is shown in Fig. 7. The base forces and moments of the model were measured by an ATI force balance with sampling frequency of 400 Hz and sampling length of 40960. Wind direction varied from 0° to 350° with incremental step of 10° in the wind tunnel test.

Fig. 8Identified dynamic parameters of the model-balance system of ZCC

a) Natural frequency

b) Damping ratio

The proposed method is adopted to identify the natural frequency and damping ratio of the model-balance system. The obtained results are shown in Fig. 8. As well as the previous example, the identified natural frequency varies slightly with wind directions, and the identified damping ratio varies significantly with wind directions. Moreover, Fig. 8(b) shows the approximate symmetry of the identified damping ratio around 180° wind angle in both $M_x$ and $M_y$ direction, rasing the possibility that aerodynamic damping is an important part of a model-balance system.

The identification results shown in Fig. 8 are adopted to modify the measured PSD of the overturning moments at the base of the ZCC model. Fig. 9 shows the comparison of the PSD before and after correction. Good results are achieved though adopting the proposed method.

It should be pointed out that the correction quality is closely related to the quality of the measured PSD. If the measured PSD is highly smoothed, the corrected PSD would be of higher quality, otherwise, there would be some burrs on the corrected PSD, as shown in Figs. 6(a) and (d). In general, if the natural frequency of the model-balance system is relatively low, the measured force or torque signal is usually of high signal-noise-ratio, and thus better results can be achieved through the proposed method.

3.3. Example III: the Guangzhou east tower

The previous two examples are tall buildings with rectangular cross-section, and in this section, an actual building with nonrectangular cross-section is considered to examine the effectiveness of the proposed method. The GZET, which is 530 m high with 112 stories, is located in Guangzhou, China. Its main structure has been completed in October, 2014, and now it is the tallest building in Guangzhou and the second tallest building in South China. Fig. 10 shows a model mounted in the wind tunnel. The HFFB wind tunnel test of the GZET was carried out in the boundary wind tunnel of the South China University of Technology as well. According to ESDU’s suggestion, the power law of mean wind profile was simulated as 0.22. The force and torque at the base of the building model was measured by an ATI balance with sampling frequency of 400 Hz and frame number of 40960. Wind direction varied from 0° to 350° with incremental step of 10° in the wind tunnel test.

Fig. 9PSD of overturning moments of ZCC before and after correction using the proposed method

a)$M_x$ direction, 0° wind angle

b)$M_y$ direction, 0° wind angle

c)$M_x$ direction, 90° wind angle

d)$M_y$ direction, 90° wind angle

Fig. 10GZET model in wind tunnel tests

Using the method presented in this study, the natural frequency and the damping ratio for all wind directions are determined, as shown in Fig. 11. The damping ratio varies in a wide range in both the $M_x$ and $M_y$ direction. It may be attributed to the influence of aerodynamic damping due to nonrectangular cross-section of the building. In some special wind directions, the aerodynamic damping ratio may be negtive. Therefore, the total damping ratios under those wind directions, for instance the wind direction of 60° and 90°, are very small and even close to zero.

Fig. 11Identified dynamic parameters of the model-balance system of GZET

a) Natural frequency

b) Damping ratio

Fig. 12 shows the PSD of the overturning moments $M_x$ and $M_y$ after correction under the wind direction of 0° and 90°. The PSD before correction is also presented for comparison. It can be seen that the amplification cause by the model-balance system in the high frequency band is eliminated effectively, thus verifying the correctness of the proposed method. The aforementioned three examples also show that the assumption shown in Eq. (9) is reasonable. That is, in a specific frequency range in the high frequency band, the PSD of overturning moment is linear with the frequency in double logarithm coordinate.