## 2 Coupled Straddle-Type Monorail-Train System
### Track Irregularity  
Track irregularity is the dominant factor in stimulating vehicle vibration and is generally applied as a temporal excitation to the coupled vehicle-bridge system. Straddle-type monorails use pneumatic rubber tires and concrete track beams, and the wheel-rail contact mechanism is distinct from that in traditional railway traffic; obviously, the wheel-rail contact mechanism in straddle-type monorails is similar to the contact between automobile tires and the road surface. The road roughness was obtained from the actual measurement of a large number of road pavements through inspection vehicles, which reflected the wear between the rubber tires and the asphalt or the cement concrete pavement. Thus, it was meaningful to simulate the surface roughness of track beam in a monorail through the road roughness. The vertical irregularity along the longitudinal direction of the route \(\mathrm{u}_{\mathrm{z}}\) and the longitudinal irregularity along the lateral direction of the beam surface were simulated in SIMPACK. The national standard road roughness in China was regarded as a stationary Gaussian  
Figure 4: FE model of a monorail bridge. \begin{table}
\begin{tabular}{c c c} \hline
**Order Number** & **Frequency (Hz)** & **Vibration Mode** \\ \hline
1 & 3.07 & Pier: symmetric lateral bending \\
2 & 3.36 & Pier: asymmetric lateral bending \\
3 & 3.69 & Track beam: first-order symmetric lateral bending \\
4 & 4.21 & Symmetric lateral bending of two track beams-1 \\
5 & 4.36 & Symmetric lateral bending of two track beams-2 \\
6 & 4.50 & Piers bending and beams floating longitudinally \\
7 & 4.55 & Piers bending longitudinally and beam exhibiting \\  & & symmetric lateral bending \\
8 & 4.58 & Piers bending longitudinally and beams exhibiting \\  & & asymmetric lateral bending \\
9 & 4.63 & Piers bending transversely \\
10 & 5.97 & Piers transversely bending and beams exhibiting \\  & & symmetric lateral bending \\
11 & 6.02 & Piers transversely bending and beams exhibiting \\  & & asymmetric lateral bending \\
12 & 6.16 & Piers exhibiting asymmetric transverse bending \\
13 & 6.42 & Track beam: first-order symmetric vertical bending \\
14 & 6.52 & Track beam: first-order asymmetric vertical bending \\
15 & 6.63 & Track beam: second-order symmetric \\  & & vertical bending \\ \hline \end{tabular}
\end{table} 
Table 3: Frequencies and vibration modes of the bridge. process, and the power spectral density (PSD) function in Equation (3) was introduced to express its statistical characteristics [37]:  
\[G_{q}(n)=G_{q}(n_{0})(\frac{n}{n_{0}})^{-w} \tag{3}\]  
where \(n\) is the spatial frequency, m\({}^{-1}\); \(n_{0}\) is the reference spatial frequency, whose value in this study was 0.1 m\({}^{-1}\); \(G_{q}(n_{0})\) is the coefficient of road roughness, m\({}^{2}\)/m\({}^{-1}\), whose value is shown in Table 4; and w is the frequency index, whose value in this study was 2. The surface roughness of the track beam based on the A- and B-level national standard road roughness are simulated with Equation (3) through the trigonometric series methods, and the samples are illustrated in Figure 5.  
As a type of rail transit, the straddle-type monorail was restricted by railway regulations in several aspects. Naturally, the requirements for track irregularity were stricter than those for road roughness. Although it was difficult and unnecessary for the track beam roughness of straddle-type monorail to meet the requirements of rails due to the influence of factors like materials, manufacturing processes, operating conditions, and operating speeds, the railway irregularity still had reference significance to the straddle-type monorail. The surface irregularity of the track beam in a straddle-type monorail system was actually between the road roughness and the railway track irregularity. A representative irregularity of the railway was American six-level track irregularity. Hence, it was meaningful to simulate the surface irregularity through American six-level track irregularity.