## 4 Experimental Modal Parameters Analysis
### Measured Modal Frequency under Different Wind Speeds  
#### 4.2.1 Identification Method of Modal Parameters  
The amplitude of signal interception and the collective average superposition times are vital in adopting the random decrement technique (RDT) to obtain feature recognition. If the amplitude of the signal interception is too small, the superposition effect will be undesirable, though the collective average superposition times can be enlarged. If the amplitude of signal interception is too large, the corresponding superposition times of the sub-signals will be reduced, and the negative correlation between the amplitude of signal interception and superposition times will arise as well. To obtain the favorable random decrement signals, this paper intercepted both the positive and negative amplitudes at the same time, then modified different interception amplitudes in accordance with the reference interception amplitude, and lastly assembled and superposed all sub-signals at the average interception amplitude in accordance with the reference interception amplitude. Generally speaking, the interception amplitude is 1.5 times of the standard deviation of the input signal (1.5 S). Namely, the reference interception amplitude was defined as 1.5 times the standard deviation (1.5 S), the interval of interception amplitude was \(\tau\), and the range of interception amplitude \(k_{j}\) was 1.0-2.0 times the standard deviation (1.0-2.0 S). The sub-signals were attained from those signals with basic time intervals of 10 min, then the intercepted amplitude was modified as 1.5 times the standard deviation, and all the sub-signals were superposed and equally divided in the end, thus the favorable random decrement signals were obtained. Concrete operation procedures are shown as follows:  
\[s=1.5\sqrt{\frac{1}{n-1}{\sum\limits_{i=1}^{n}{x_{i}}-\overline{X}}} \tag{10}\]  
\[x_{k_{j}}(t)^{+}=\frac{\sum\limits_{j=1}^{j=m}{\left(\frac{1.5}{k_{j}}x(t_{k_ {j}})^{+}\right)}}{m(k_{j})} \tag{11}\]  
\[x_{k_{j}}(t)^{-}=\frac{\sum\limits_{j=1}^{j=m}{\left(\frac{15}{k_{j}}x(t_{k_ {j}})^{-}\right)}}{m(k_{j})} \tag{12}\]  
where \(x_{i}\) is the sample of discrete experimental data, \(n\) is the number of samples, \(\overline{X}\) refers to the mean value of sub-samples,\(x_{k_{j}}(t)^{+}\) represents the free attenuation curves by intercepting the positive amplitude sample, \(x_{k_{j}}(t)^{-}\) stands for the free attenuation curves by intercepting the negative amplitude sample, \(\tau\) denotes the interval of interception amplitude, \(m\) stands for the number of interception amplitude sample, and \(k_{j}\) refers to the amplitude coefficient at the \(j\)th time. \(1.5/k_{j}\) is the modification coefficient of interception amplitude at the corresponding 1.5 times of the reference amplitude. The superimposed free attenuation curve can be written as  
\[x_{k_{j}}(t)=\frac{x_{k_{j}}(t)^{+}-x_{k_{j}}(t)^{-}}{2} \tag{13}\]  
For Equations (10)-(13), if the interception range is within the 1.0-2.0 times the standard deviation, the interval of interception amplitude \(\tau\) is equivalent to 0.1 times of the standard deviation, thus the superposition times during the whole process have increased by 40 times compared with pre-improvement. In this condition, the superposition times are completely capable of meeting the RDT requirements when the basic time duration of 10 min is taken into consideration. After obtaining the free vibration attenuation curve, the logarithmic decrement method was used, and the peak values of the free attenuationcurve were attained to calculate damping parameters. Moreover, the fitting formula can be expressed as follows:  
\[y(t)=a\exp(-\xi\omega t) \tag{14}\]  
where \(a\) is the amplitude of acceleration response when \(t\) equals 0, namely \(a=y(0)\); \(\xi\) denotes the structural damping ratio; and \(\omega\) is the circular frequency.