## 3 Measured Results Analysis
### Power Spectral Density Function  
Auto-power spectral density (APSD) and cross-spectral density (CPD) are key characteristics of signal samples. The power spectrum, which enables to reflect the average characteristics of statistics, is the important statistical parameter to investigate the frequency domain characteristics of random vibration. The calculation formulas of the auto-power spectral and cross-spectral density functions can be expressed by Equations (4) and (5).  
\[S_{xx}(k)=\frac{1}{MN_{FFT}}\sum_{i=1}^{M}X_{i}(k)X_{i}^{*}(k) \tag{4}\]  
\[S_{xy}(k)=\frac{1}{MN_{FFT}}\sum_{j=1}^{M}X_{j}(k)X_{j}^{*}(k) \tag{5}\]  
Where \(X_{i}(k)\) is the Fourier transform of the one-dimensional random vibration signal of a measuring point at the \(i\)th data segment; \(X_{j}(k)\) is the Fourier transform of two-dimensional random vibration signal of a measuring point at the \(j\)th data segment; \(X_{i}^{*}(k)\) is the conjugate complex of \(X_{i}(k)\); \(X_{j}^{*}(k)\) is the conjugate complex of \(X_{j}(k)\); \(N_{FFT}\) is the data length of Fourier transform; and \(M\) is the average time. The average periodic diagrams were used in this paper to estimate the power spectrum density, by which the vibration signals were divided into several components, and the power spectrum of each component was calculated and then the average value was selected.  
The time histories of the torsional acceleration on different floors are measured at the same time segment during typhoon Sarika, and the torsional angular acceleration responses of other floors are measured at different time segments, except the 24 th and 32 th floors, during typhoon Sarika. In order to obtain the torsional mode, the measuring points of the rotational accelerometer on the 32 th floor were taken as the reference floor, and the rotational accelerometer was installed at the 24 th floor, thus the angular acceleration responses of the 8 th, 12 th, 16 th, 20 th, 24 th, and 28 th floors that were synchronous with those of the 32 th floor were obtained as well. Even though the power spectrum amplitudes in the period of relatively high wind speed are large, they do not affect the judgment to the fundamental frequencies of the high-rise building. According to Equation (4), the auto-power spectrum density of acceleration along the X axis and Y axis and the torsional direction were calculated. More details are shown in Figure 10.  
Figure 9: Time histories of the angular acceleration under typhoon Sarika.(**a**) Torsional angular acceleration response of 24 th floor; (**b**) Torsional angular acceleration response of 32 th floor.  
It can be known from Figure 10 that the auto-power spectral density functions along X and Y axes and torsional direction have obvious peaks at the first-order frequencies, and the values of the first-order frequencies are obviously larger than those of the higher ones. However, the peak amplitude of the 24 th floor at the first-order frequency along the torsional direction was almost the same as that of the 32 th floor. The fundamental frequencies along the X and Y axes were about 0.66 Hz and 0.56 Hz, respectively, while the fundamental frequencies along the torsional direction were about 0.56 Hz. Therefore, the fundamental frequency along the torsional direction was close to that along the Y axis of the measured building. It shows that the first-order torsion-translation period ratios \(\mathrm{T_{t}/T_{I}}\) and \(\mathrm{T_{t}/T_{s}}\) are 1.18 and 1.00, respectively.  
Because the wind energy is concentrated in the low-frequency region, it has a great impact on the measured low-order modal frequency of high-rise buildings. The measured results show that the frequency value at the self-spectral peak is significantly less than that at the low wind speed peak in the horizontal and torsional turning high-speed stage, indicating that the influence of the vibration amplitude of high-rise buildings on the horizontal and torsional turning modal parameters cannot be ignored.