# Seismic Isolation Effect of Non-Water Reacted Two-Component Polymeric Material Coating on Tunnels
## 2 Dynamic Elastic Modulus from Impact Resonance Test (IRT)
### Impact Resonance Test (IRT) and Theory  
The damping ratio calculations are derived from the time domain and frequency domain response of the tested material. There are various methods to estimate the level of damping, such as logarithmic decrement and half-power bandwidth methods. The half-power bandwidth methods are based on the frequency domain and were used to generate the results of damping ratios from the Impact Resonance Test (IRT) system. Traditionally, the IRT system was used to obtain a damping ratio of the asphalt mixture, and it consists of a laptop, an acceleration sensor, a signal amplifier, and a software program [26; 27; 28]. The basic concept of the test is to determine the amplitude and attenuation of vibrations and relate these parameters to the material properties. A specimen was expanded to a disc shape by injecting different amounts of NRTCPM into a mold. Figure 3 shows that the specimen with free boundary conditions is struck with a rigid impulse tool at the impact location to excite a mechanical vibration. The vibration can be received by a signal amplifier connected to the edge of the specimen (f3). As is shown in Figure 3, when the edge of the specimen is impacted (f1), a sensor, connected to the signal amplifier, captures the waveform vibrations from the short-term mechanical impact, and the first vibration mode is generated. When impacting the center of the specimen (f2), the second vibration mode can be generated. Different vibration modes correspond to different frequencies [27]. By performing a Fast Fourier Transform (FFT) algorithm, the waveform in the time domain is converted to a response in the frequency domain, as illustrated in Figure 4[29]. The frequency of the maximum amplitude is the resonant frequency(\(f_{r}\)), and the \(f_{1}\) and \(f_{2}\) are the frequencies of the 1/\(\surd\) maximum amplitude in Figure 4.  
The damping ratio can be obtained from \(f_{r}\), \(f_{1}\), and \(f_{2}\), and it is written as:  
\[\xi=\frac{1}{2f_{r}}(f_{2}-f_{1})\times 100\% \tag{1}\]  
\(\xi\)--Damping ratio  
\(f_{r}\)--Resonant frequency  
\(f_{1}\),\(f_{2}\)--The frequencies of the1\(\surd\) maximum amplitude. Table 1 presents the average damping ratio of the first and second mode. When the NRTCPM's density is 0.22 g/cm\({}^{3}\), the average damping ratio is 0.04. \begin{table}
\begin{tabular}{c c c} \hline \hline
**SpecimenDensity (g/cm\({}^{3}\))** & \
\begin{tabular}{c} **Average Damping Ratio** \\ **of the First Mode** \\ \end{tabular} &
\begin{tabular}{c} **Average Damping Ratio** \\ **of the Second Mode** \\ \end{tabular} \\ \hline
0.22 & 0.05 & 0.03 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Damping ratio of specimens. Figure 4: Damping estimation based on the half-power bandwidth method. The dynamic elastic modulus is an important parameter, and it can also be obtained from IRT using the following equations. The theoretical relation between the natural frequency and attenuation frequency can be written as:  
\[f_{n}=f_{d}/\sqrt{1-\xi^{2}} \tag{2}\]  
where,  
\(f_{d}\)--Attenuation frequency, Hz;  
\(f_{n}\)--Natural frequency, Hz;  
The storage modulus \(E^{\prime}\) can be calculated from the natural frequency \(f_{n}\), and it can be written as [30]:  
\[E^{\prime}=\frac{4\pi^{2}f_{n}^{2}R^{2}\rho}{\omega} \tag{3}\]  
where,  
\(E^{\prime}\)--Storage modulus;  
\(\rho\)--Density of specimen, kg/m\({}^{3}\);  
\(R\)--Radius of specimen, m;  
\(\omega\)--Material parameter. \(\omega\) is related to Poisson's ratio, the thickness, and the radius ratio. The \(\omega\) values of the first and second mode are presented in Tables 2 and 3. \(R\), h, and \(\upsilon\) are the radius, thickness, and Poisson's ratio in Tables 2 and 3.