## 2 Frequencies and Mode Shapes of the Double-Beam System
### Verification of the Analytical Solutions in Section 2.1  
To be concise, only the verifications for the frequencies and mode shapes of the fixed-fixed double-beam system are demonstrated in this section.  
The material properties of the two beams are obtained from the rail and the bridge in the reference [2]. The properties of the bridge are: \(I_{\mathrm{b}}=3.81\,\mathrm{m}^{4}\), \(\rho_{\mathrm{b}}A_{\mathrm{b}}=34088\,\mathrm{kg/m}\), \(E_{\mathrm{b}}=2.943\times 10^{10}\,\mathrm{Pa}\), \(L_{\mathrm{b}}=20\) m. The properties of each rail are: \(E_{\mathrm{r}}I_{\mathrm{r}}=4.3\times 10^{6}\) Nm\({}^{2}\), \(\rho_{\mathrm{r}}A_{\mathrm{r}}=51.5\) kg/m. The length of each rail is set to be the same as the bridge. In the discrete spring model, the contact stiffness of each spring underlying one rail is \(4.1125\times 10^{7}\) N/m and the regular spacing between two neighboring contact springs is \(0.625\) m. In the distributed spring model, the contact stiffness under one rail is equivalent to be \(4.1125\times 10^{7}\div 0.625=6.58\times 10^{7}\) N/m\({}^{2}\). In the 2D model, all the rail properties are doubled to account for the presence of both rails. The non-dimensional parameters are calculated to be: \(s_{\mathrm{k}}=7.67\times 10^{-5}\), \(s_{\mathrm{m}}=0.003\) and \(s_{\mathrm{e}}=0.0343\).  
The discrete spring model for the rail-bridge system is established by the FE method and the numerical frequencies and mode shapes of the system are obtained. They are compared with the analytical counterparts calculated from the formulae for the distributed spring model in Section 2.1. The rail and the bridge are modeled by 256 beam elements in the FE model. 31 discrete spring elements having the stiffness of \(8.225\times 10^{7}\) N/m and the space of \(0.625\) m are created to model the discrete springs. The first 16 frequencies of the double-beam system are obtained by the FE method and compared with the analytical results as shown in Table 4. The differences between the two kinds of results are very small. To save page, only the first 8 mode shapes are depicted for comparison in Figure 2. The differences between the mode shapes obtained by the two methods cannot be distinguished. The above comparisons verify the analytical formulae in Section 2.1. It is interesting to find that the 5 th mode shape is similar to the 1 st except that the amplitude and direction of the bridge mode shape are different. The similarities can also be found between the 6 th and the 2 nd, the 7 th and the 3 rd and the 8 th and the 4 th mode shapes. In fact, the two similar modes are the two sub-modes corresponding to the same basic mode with the same wavenumber. The four sets of similar modes are corresponding to the first four basic modes. More studies about sub-modes are presented in Section 2.3.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Mode Number**} & \multicolumn{2}{c}{**Frequency (Hz)**} & **Difference** \\ \cline{2-4}  & **FE** & **Analytical** & **(\%)** \\ \hline
1 & 16.12 & 16.12 & 0 \\
2 & 44.44 & 44.44 & 0 \\
3 & 87.08 & 87.08 & 0 \\
4 & 143.66 & 143.68 & 0 \\
5 & 179.90 & 180.72 & 0.5 \\
6 & 180.03 & 180.85 &