1. Introduction
The AERORail Transportation System referred to as AERORail, is a new type of transportation system in the form of a cable-supported rail combined structure (see
Figure 1) composed of multispan continuous string cables, rails, lower supports and related power supply and control systems [
1]. Similar to existing top-supported suspension bridges or stress zone bridges, the typical projects include the Rio-Colorado Suspension Bridge in Costa Rica (108 m in span), the Bad Sand Bridge on the Elbe River in Germany (100 m in span), Su Rifeng Bridge (48 m in span) in central Kyushu, Japan and Taojin Bridge (74 m in span) in Taojin Village, Dongkou County, Hunan, China [
2,
3].
The AERORail structure uses steel as the structural material, replacing the use of concrete and prestressed concrete in the superstructure; it cancels the heavy superstructure in traditional bridges and forms a common stress system directly formed by the rail and the string cable, which is a veritable hollow structure in the sky; its structural rigidity mainly comes from the stress rigidity generated by the prestressing of the string cable [
4]. Due to its obvious structural flexibility characteristics, its structural dynamic characteristics are the focus of research and important structural characteristics that must be mastered in the future application process. The problem corresponding to the characteristics of its flexible structure is that when it is subjected to moving loads, the force and deformation, and vibration and stability of the structure have obvious coupling effects between the structure and the moving loads.
As a flexible structure, its structural form is somewhat similar to that of string beams. The biggest difference from conventional string beams used in buildings is that AERORail is simpler in structure, and the overall structural size is more represented as a longitudinal one-dimensional structure, even in the lateral direction, showing that the lateral dimension is much larger than the height dimension [
5]. Second, the servicing load on the AERORail is a moving load, and the load causes the structure to deform considerably when it is no-load and full-load [
6]. Therefore, from the stress point of view, the AERORail structure is more like a stress ribbon structure.
For such light and flexible structures with pretensioned cables, Zhao [
7] studied the prestressing process, and most typical experimental studies are represented, including both full-scale tests and downscale orthogonal tests. The static behavior and the effect of a prestressed cable on structural stability are presented, followed by a typical full-scale static loading test with dynamic behavior, such as the natural vibration characteristics. Based on the structural characteristics of a tensioned string bridge, Zhou [
8] used the Rayleigh method to derive formulas for calculating the frequencies of vertical, antisymmetric and lateral bending vibrations. Stefan Hartweg and Andreas Heckmann [
9] derived the kinematics and governing equations of motion of a general flexible multibody system and their extension to moving loads and presented a method to handle discontinuities when moving loads separate from the flexible structure. Zhou [
8] studied the dynamic structural characteristics of a tensioned string bridge based on the Rayleigh method to derive formulas for calculating the frequencies of vertical, antisymmetric and lateral bending vibrations. He found that the shape and physical characteristics of the main cable have a greater impact on the vertical symmetrical vibration frequency than the lateral bending frequency, and the vertical bending symmetrical vibration frequency increases with an increasing rise-to-span ratio. The tension force of the main cable has no influence on the frequency of tensioned string bridges. The first-order frequency of the tensioned string bridge is generally the vertical bending symmetrical vibration frequency. The fundamental frequency of a structure can be greatly increased, thereby increasing the overall rigidity of the structure. Lee [
10] studied a numerical method for the dynamic analysis of vehicles moving on flexible structures with gaps and gained the dynamic contact between a high-speed wheel and elastic beam with Coulomb friction [
11]. Zhao [
12] studied the dynamics and stability of slender structures carrying a moving load or mass with FEM. Xiao [
13] studied the vibration control of stress ribbon bridges subjected to moving vehicles.
Based on the above structural characteristics, AERORail not only has the static and dynamic characteristics of the conventional flexible beam structure but also the vibration and stability under the action of the moving load are closer to the stress ribbon structure [
13,
14,
15]. The difference is that the AERORail load acts directly on the rail on the upper part of the prestressed cable and is then transmitted to the cable through the support rod, which is similar to the top-supported suspension cable structure. To ensure the efficiency of AERORail transportation, the support between the cable and the track is not as high as that of the top-mounted suspension cable structure; that is, the catenary span of the cable is relatively small. The rise span ratio is between 1/100 and 1/50, which causes AERORail to look more like a truss with cables parallel to the rail from a distance.
At present, research on AERORail structures mainly focuses on the static and dynamic characteristics of the structure. Li [
4,
16] used 1:20 and 1:15 scale models to preliminarily verify the feasibility of the AERORail structure and used the virtual prototype to explore the dynamic and static characteristics of the structure [
16]. Numerical analysis of the static deflection of different spans of AERORail under different loads and pretensions was carried out, and the 1:1 full-scale AERORail test was used for verification (
Figure 2); the relevant results have been carried out in a special report in China Central Television and China Education Television. At the same time, on this basis, combined with vehicle-bridge coupling theory, the dynamic deflection response of AERORail with different spans is studied, and some useful conclusions are obtained.
Existing studies have shown that under the action of low-speed moving loads, the cable stress increment, dynamic deflection and structural stiffness of the AERORail structure do not change significantly; with the increase in cable force and mid-span support height, the structural stiffness increases significantly. There is a nonlinear relationship with dynamic deflection [
17,
18].
Although some studies have initially revealed some of the static and dynamic characteristics of the AERORail structure, for engineering applications of related research, these results still cannot satisfy the establishment of relevant design theories and design methods. At present, there are still problems that need to be studied urgently including the static and dynamic characteristics of the long-span AERORail structure, the dynamic response of the AERORail structure under high-speed driving conditions, the vehicle-AERORail interaction, and the practical calculation method of the static and dynamic response of the AERORail structure.
Based on the existing research results, this paper studies the force between the vehicle and the AERORail structure—the wheel-rail contact force. On the basis of the modal test, the fundamental frequency of the structure is determined. Then the research on the vehicle-rail force under the condition of the vehicle load determines the stability and control requirements during the service of the AERORail and provides a theoretical basis for the design of the structure in the future, especially the arrangement of the vehicle load on the track line. It is also an important guarantee to ensure the safety and usability of the structure.
2. AERORail Test Line with Modal Test
Existing numerical simulations have preliminarily verified the availability and reliability of the dynamic model calculation method and identification algorithm. To verify the reliability of the application of this theoretical system to the study of AERORail structures and to provide a basis for the subsequent calculation of the real vehicle-rail contact force, it is necessary to conduct verification tests by means of model tests.
Commonly used structural test methods include small-scale model tests and full-scale model tests. Due to the high cost of full-scale models, they are generally difficult to achieve. For the study of the new structure of AERORail, considering the particularity of the structure, the author realized the construction of the full-scale rail so that the conditions under the actual application state can be used to carry out the test, and the measured data under the real conditions can be obtained. The long-term significance is also helpful for future research on AERORail vibrating absorption and stability control. This paper is the relevant research work carried out by using the 1:1 AERORail structure test line.
2.1. Test Line Structure
The AERORail test line adopts a steel structure as a whole, and the substructure adopts a steel column support and concrete foundation. The main components of the AERORail structure include rails, cables, struts, piers, lateral connections and anchorages [
17,
18]. The general layout of the AERORail test line used is shown in
Figure 3.
Both ends of the test line are anchored, and the left side is the starting point of the moving load. The span layout adopts the scheme of 12 × 5 m + 10 × 10 m + 15 m + 2 × 5 m, and the total length is 203 m (including the slope at both ends, excluding the anchorage). By removing the support of the 10 m span in the middle part of the test line, the combined arrangement of 10 m, 20 m and 30 m spans is realized so that the span arrangement with a modulus of 5 m is realized in the whole line.
2.1.1. Superstructure of the Test Line
The track adopts a 43 kg/m standard section rail, and its section parameters are shown in
Figure 3 and
Table 1. The cable components are
steel strands, with 5 cables on one side and two bundles in total, arranged under the rail. The horizontal spacing of the rails is 1.5 m, and a horizontal tie rod is arranged every 5 m in the longitudinal direction as a horizontal connection. The standard span layout is shown in
Figure 4.
A single transverse connecting rod is composed of steel pipes and clamps at both ends. At the position of the pier column, the bowl buckle clamp is used to buckle the top of the pier column; in the position without the pier column, the claw clamp is used to buckle under the rail. Through preliminary tests, it is determined that the vertical spacing of horizontal couplings should not be too large or too small. Too small a spacing will lead to a waste of material, while too large a spacing will make the overall structure and even the track out-of-plane unstable; too large a spacing may also lead to unbalanced load distribution on both sides of the structure. According to the finite element analysis of the upper load, the horizontal arrangement of the free length of the track is approximately 5~8 m. The vertical strut is designed with the support plate and the steel pipe, and its total length is determined according to the shape of the cable and the position of the support.
In this test line, the pole-support heights used for cables with different spans are shown in
Table 1 below:
At the top of the pier, the cable needs to pass through the reserved channel, and there is a certain angle. When the cable is prestressed and loaded with live load, a certain slip occurs here, which is similar to the slip of the main cable of a suspension bridge when the bridge is completed. This kind of sliding plays a very important role in balancing the force and prestress of the two adjacent spans; the appropriate structural measures should be used to ensure sliding instead of a fixed restraint to reduce the effect of horizontal force on the support column and its instability. In addition, because the edge of the steel member tunnel itself is difficult to ensure smoothness, and its edges and corners form local stress concentration on the cable body, it is very easy to damage the cable body, so structural measures should be taken to solve it. To solve these two problems, this test line uses the construction method of laying tetrafluoroethylene sliding plates in the tunnel to address the tunnel on the top of each pier.
2.1.2. The Substructure of the Test Line
The piers of the test line are made of steel pipes, and the lower part of the column is connected with the embedded parts in the concrete foundation by bolts; between the two piers in the north and south of the test line, bowl-type fasteners and a pair of steel pipes are used to form X braces (see
Figure 5).To ensure the lateral stiffness and stability of the structure, at the piers with variable slope spans, additional diagonal braces are set to reinforce and support the columns horizontally and vertically.
2.2. Modal Test
Modal analysis is the premise and foundation of the dynamic analysis of linear elastic structures. Although in most cases the modal parameters calculated by using the finite element model can meet the needs of engineering applications, for the study of new structures the structural modal parameters (modal frequency, modal shape and modal damping, etc.) verification research methods are fundamental work and very necessary [
19,
20,
21]. The modal test and analysis of the five-pole supported AERORail structure with a span of 30 m are used to verify the correctness of the finite element modal analysis method and obtain more real modal parameters.
The 30 m span and five-pole supported AERORail selected for the model test are provided with a set of supports at 1/6, 1/3, 1/2, 2/3 and 5/6 of the span, and the five-pole support heights are 21.5 cm, 52 cm and 62 cm, 52 cm and 21.5 cm, respectively (see
Figure 6).
Since the modal frequency of the AERORail structure is directly related to the cable force, the cable strain is also measured in the modal test and finally used to calculate the actual cable force. After measurement and calculation, the cable force on the north and south sides is 30.2 kN.
The modal analysis uses frequency domain decomposition to calculate modal frequencies and modal shapes [
8,
22], the modal damping ratio is determined by the half-power bandwidth method, and the excitation method is hammer excitation [
21]. After analysis and calculation, the modal parameters of the 30 m-span five-pole supported AERORail structure are shown in
Table 2.
According to the actual structure of the 30-m-span structure, calculated by the finite element model, the transverse modal frequencies are 0.49 Hz, 1.15 Hz and 2.09 Hz, and the vertical modal frequencies are 1.17 Hz, 1.90 Hz and 4.26 Hz.
The modal test and analysis results (
Figure 7) show that the vertical vibration mode obtained by the finite element analysis is basically the same as the vibration mode of the AERORail structure, but there is a certain error in the modal frequency, which is approximately 10%~20%. Transverse modes have large errors (over 100%) in both mode shape and frequency. This is because the real structure is a multispan structure transversely, and there is no transverse constraint. Therefore, under the support of the current research results, the modal parameters of the finite element model can be used for the vertical vibration analysis of AERORail, while the lateral vibration needs to be further studied.