# A Time-History Contact Force Model of the Dynamic Load of AERORail Structures
## 6 Conclusions  
The present study explored a way to devise live load models for a novel transportation system called AERORail based on the vehicle-structure contact force identified from structure's acceleration data. The identified contact forces were grouped and then fitted  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Span**} & \multirow{2}{*}{**Cable Tension**} & \multicolumn{8}{c}{**Natural Frequency of Vertical Vibration in Hz**} \\ \cline{3-13}  & & & \(f_{1}\) & \(f_{2}\) & \(f_{3}\) & \(f_{4}\) & \(f_{5}\) & \(f_{6}\) \\ \hline
30 m & 3 tons & 1.13 & 2.05 & 4.16 & 7.37 & 10.93 & 15.56 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Natural frequency of vertical vibration of 30 m span.  
Figure 15: Simulation of mid-span deflection under different speed conditions of 30 m span. (**a**) Speed less than 25 km/h; (**b**) speed greater than 40 km/h.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \
\begin{tabular}{c} Speed \\ (km/h) \\ \end{tabular} & 1 & 5 & 10 & 15 & 20 & 25 & 40 & 60 & 80 & 100 & 120 & 150 & 200 & 250 & 300 \\ \hline
\begin{tabular}{c} Deflection \\ (mm) \\ \end{tabular} & 147.1 & 147.3 & 147.9 & 149.0 & 150.7 & 154.7 & 166.8 & 185.3 & 200.8 & 202.7 & 196.0 & 183.3 & 164.4 & 144.4 & 125.4 \\ \hline \hline \end{tabular}
\end{table}
Table 11: Working conditions and calculation results for 30 m span.  
using Bezier curves and the least-squares method. The resultant two force models, namely the single-valley and double-valley curves, were validated in a Simulink model against the experiment data under five different vehicle speeds and three span lengths. The simulations showed that the proposed load model gave a good prediction of the mid-span deflection; therefore the established standard load models can be used for the dynamic analysis of AERORail. The conclusions are as follows.  
(1) The live load for the lightweight AERORail can be obtained by fitting the identified contact force using Bezier curves. The resultant load curve can be used to predict the dynamic displacement response of the structure.  
(2) The variation in span length leads to the different shape of the load curve. The load models for 5 m and 15 m span AERORails have one- and two-peak magnitudes, respectively. The variation of speed mainly affects the magnitude rather than the shape of the load curve.  
Aside from the existing research on small-span AERORails and low-speed vehicles, this article also discusses the dynamic deflection of a 30 m AERORail with vehicle speed of 40-300 km/h. It was found from the simulation that:  
(3) The resonant speed of the 30 m span was between 80 km/h and 100 km/h, which was less than that of the 5 m, 10 m, and 15 m spans. At speeds of about 150 km/h and above, higher-order vibration components began to appear. This phenomenon should be carefully considered in future design and research of AERORail.  
The current study designed load models for AERORail. However, the methodology applied in the article can also benefit the load model study on other prestressed bridge structures.  
Conceptualization, F.L. and P.W.; methodology, Z.G., Y.C. and P.W.; validation, Z.G. and Y.C.; formal analysis, Z.G. and Y.C.; writing--original draft preparation, Z.G. and Y.C.; writing--review and editing, F.L.; visualization, Z.G.; supervision, F.L.; project administration, F.L.; funding acquisition, F.L. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the National Natural Science Foundation of China (Grant No. 50708072, No. 51378385) and was supported by the Fundamental Research Funds for the Central Universities of China (Grant No. 22120180318).  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding authors.  
The authors declare that they have no conflict of interest.