## 2 Measurement of Damping Parameters
### Half-Power Bandwidth Method  
Common frequency-domain-based damping test methods include the half-power bandwidth method, the frequency refinement method, the peak method, and the conductivity circle method [19]. The half-power bandwidth method is by far the most commonly used damping test method, and the relative studies for structural system damping tests are very common. For example, the half-power bandwidth method is used in the American Society for Testing and Materials standard to identify the damping of coating materials [20].  
We derive the principle of utilising the half-power bandwidth to identify the structural damping under the action of the base excitation in terms of a single-degree-of-freedom system. The frequency response function of the single degree of freedom system can be easily written as follows:  
\[H(\omega)=\sqrt{\frac{1+4\zeta^{2}\lambda^{2}}{\left(1-\lambda^{2}\right)^{2} +4\zeta^{2}\lambda^{2}}} \tag{2}\]  
where \(\zeta\) is the damping ratio; \(\lambda=\)_ff\({}_{0}\)_,\(f\) is the excitation frequency, and \(f_{0}\) is the intrinsic frequency of the structure.  
Taking two frequency points \(f_{1}\) and \(f_{2}\) around \(f_{0}\), and the corresponding frequency domain response amplitudes are equal, \(H(\omega)_{1}=H(\omega)_{2}\), and its ratio to \(H(\omega)_{\text{max}}\) is \(r\), we have  
\[\frac{r}{2\zeta}=\frac{\sqrt{1+\left(2\zeta\lambda\right)^{2}}}{\sqrt{\left(1 -\lambda^{2}\right)^{2}+\left(2\zeta\lambda\right)^{2}}} \tag{3}\]When \(\zeta\ll 1\), the \(\zeta^{2}\) term can be ignored, which in turn simplifies Equation (3) to obtain the representation formula for the two frequency points.  
\[f_{2}^{2}=\frac{f_{0}^{2}(2r^{2}+4r^{2}\zeta\sqrt{1/r^{2}-1})}{2r^{2}} \tag{4}\]  
\[f_{1}^{2}=\frac{f_{0}^{2}(2r^{2}-4r^{2}\zeta\sqrt{1/r^{2}-1})}{2r^{2}} \tag{5}\]  
Thinking, subtracting, and simplifying Equation (4) with Equation (5) yields.  
\[2\zeta=\frac{2(f_{2}^{2}-f_{1}^{2})r^{2}}{4r^{2}f_{0}^{2}\sqrt{1/r^{2}-1}}= \frac{(f_{2}^{2}-f_{1}^{2})}{2f_{0}^{2}\sqrt{1/r^{2}-1}} \tag{6}\]  
Approximating the sum of the two frequencies as \(f_{1}+f_{2}=\)2\(f_{0}\), Equation (6) can be further simplified as follows:  
\[2\zeta=\frac{(f_{2}+f_{1})(f_{2}-f_{1})}{(f_{2}+f_{1})f_{0}\sqrt{1/r^{2}-1}}= \frac{f_{2}-f_{1}}{f_{0}\sqrt{1/r^{2}-1}} \tag{7}\]  
When \(r=\sqrt{2/2}\), the equation becomes the following:  
\[\zeta=(f_{2}-f_{1})/2f_{0}=\Delta f_{w}/2f_{0} \tag{8}\]  
where \(\zeta\) represents the damping ratio,\(f_{1}\) and \(f_{2}\) represent the frequencies corresponding to the half-power point,\(f_{0}\) represents the resonant frequency of the system, and \(\Delta f_{w}\) represents the half-power bandwidth. When the amplification factor of the system drops to \(1/\sqrt{2}\) of its peak value, the width of the corresponding frequency band is called the resonance region. The endpoints on either side of the resonance region have exactly half the maximum power, so the two endpoints are referred to as half-power points, as shown in Figure 2a. In the half-power bandwidth method, also called the 3 dB bandwidth method, the half-power point corresponds exactly to the point on the curve when the peak value drops by 3 dB, as shown in Figure 2b.