## 2 Viscoelastic Properties of Composite Sleepers  
Composite sleepers have viscoelastic properties, and their mechanical parameters (such as their dynamic modulus and loss factor) change with their temperature and loading frequency. The dynamic modulus and damping are the most important parametersaffecting their vibration characteristics. Therefore, the viscoelastic properties of composite sleepers should be considered. Through a dynamic mechanical analysis (DMA) test, using the time-temperature superposition (TTS) and Williams-Landel-Ferry (WLF) formula, the main curve of the mechanical parameters of composite sleepers can be obtained, and the generalized Maxwell model (Figure 1) is used to characterize the constitutive relationship of composite sleepers. In the frequency domain, the generalized Maxwell model indicates that the storage modulus \(E^{\prime}(\omega)\), loss modulus \(E^{\prime\prime}(\omega)\), and complex dynamic modulus \(E^{*}(\omega)\) are, respectively [27,28]:  
\[E^{\prime}(\omega)=E_{\infty}+\sum_{i=1}^{n}\frac{E_{i}\omega^{2}\tau_{i}^{2} }{1+\omega^{2}\tau_{i}^{2}} \tag{1}\]  
\[E^{\prime\prime}(\omega)=\sum_{i=1}^{n}\frac{E_{i}\omega\tau_{i}}{1+\omega^{2} \tau_{i}^{2}} \tag{2}\]  
\[E^{*}(\omega)=E^{\prime}(\omega)+\mathrm{i}E^{\prime\prime}(\omega) \tag{3}\]  
where \(E_{\infty}\) is the modulus of the parallel spring, \(E_{i}\) is the modulus of the term \(i\), \(\tau_{i}=\eta_{i}/E_{i}\) is the relaxation time constant, and n is the total number of terms. Here, \(\omega=2\pi f\) is the circular frequency (rad/s) and \(f\) is the frequency (Hz). The model-related parameters are shown in Table 1.  
Taking the parameters and test data at 15 \({}^{\circ}\)C as an example (Figure 2), from the test data, it can be clearly seen that the modulus of the composite sleeper is frequency dependent--the higher the loading frequency, the larger the modulus. According to the generalized Maxwell model, the complex modulus and storage modulus fit well. The  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multicolumn{3}{c}{**Temperature/\({}^{\circ}\)C**} \\ \cline{2-4}  & **5** & **15** & **25** \\ \hline \(E_{\infty}\)/MPa & 1600 & 1450 & 1200 \\ \(E_{1}\)/MPa & 480 & 299 & 278 \\ \(\tau_{1}\)/s & 1 & 1 & 1 \\ \(E_{2}\)/MPa & 178 & 188 & 229 \\ \(\tau_{2}\)/s & 0.1 & 0.1 & 0.1 \\ \(E_{3}\)/MPa & 150 & 189 & 125 \\ \(\tau_{3}\)/s & 0.01 & 0.01 & 0.01 \\ \(E_{4}\)/MPa & 287 & 315 & 359 \\ \(\tau_{4}\)/s & 0.001 & 0.001 & 0.001 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Generalized Maxwell model parameters (\(n=4\)).  
Figure 1: Generalized Maxwell model.  
loss modulus and loss factor characterization accuracy are relatively low, especially in the high-frequency range. This will mean the generalized Maxwell model will be unable to fully reflect the damping properties of the composite sleeper in the high-frequency range. However, using the generalized Maxwell model to characterize the mechanical parameters of composite sleepers is obviously better than considering them as constants in the whole frequency range. In the following theoretical model for the composite sleeper-ballasted track, the viscoelastic properties of composite sleepers will be considered.