Stability test of Hyperloop vehicle in different movement conditions

▪ The speed and mass have the greatest influence on the stability of the Hyperloop vehicle. ▪ The greatest disturbance occurs between the magnets and the curved guideway. ▪ The method presented for determining vehicle stability eliminates the determination of the Lyapunov function. The guide of the Hyperloop system in the paper is mathematically represented as a continuous system along which the force from the capsule travels, with the capsule in turn represented as a discrete system. The simulations discussed in the article were used to determine the displacements of the magnet elements. ANSYS software was used to perform the simulations using finite element calculations (FEM). The stability of the capsule will be determined from the results of the displacements present in the system. Taking into account the existing conditions in the magnet and guide assembly system, the simulation results were used to analyse stability in technical and stochastic terms (Lyapunov criteria) for non-linear systems. In the technical stochastic stability analysis, the transverse displacements of the electromagnets were used. The probability of unstable Hyperloop motion was then calculated.


Introduction
The competence and reliability of transportation systems [41,4] is a major logistical issue. The fifth mode of transportation, also known as vacuum rail or the Hyperloop, is a theoretical solution to multimodal transportation issues [35]. According to this idea, a vehicle-a capsule in motion carrying people travels through a tube under reduced pressure [37 39]. Due to less traffic resistance, this mode of transportation requires less infrastructure than high-speed rail and uses less energy [38].
Due to the decreased air resistance brought on by the tube's decreased pressure, there is less resistance to movement.
Magnetic cushions and different running gear options have also been employed [1].
The Hyperloop vehicle moves/floats due to an electrodynamic suspension, which suspends the rail and train in a magnetic field and uses the repelling force of the magnetic fields to propel/float the vehicle [3]. Electromagnets or a permanent magnet construction can produce the magnetic field in the train [5]. The induced magnetic field of the electric wires or the conduction belts in the tracks produce the repulsive force in the rails [2]. The voltage created in the coils and the magnetic field flux that results are too tiny at low speeds to lift the Hyperloop car off the track [6].Thus, the Hyperloop vehicle must then move on its wheels, or alternatively, there must be another way to ensure that the vehicle can move on the track before it starts to hover above the track. The propulsion coils create a force between the magnets and ensure that the Hyperloop vehicle moves [7]. The coils described work in a similar way to a linear motor: current flowing through the coils creates a magnetic field along the track. The frequency of the voltage variation is synchronised with the speed of the Hyperloop vehicle (it is a type of synchronous motor). The lateral displacement caused by the magnets in the Hyperloop vehicle and the applied magnetic field cause the Hyperloop vehicle to move. Investigation of hyperloop vehicle displacement is usually limited to the vertical plane, but investigation of the lateral dynamics of the system is more important than that in the vertical plane and serves as a complement to the system in the horizontal plane [8]. It is possible to address problems with motion stability by conducting studies in the horizontal plane in the transverse direction with regard to the guideway [9].
Recently, an increasing amount of activity is considering supporting driveways at 30 meter intervals. We investigated discrete continuous systems. [10,11] used the Rungge-Kutta-Rosenbrock approach to solve the equations of motion for systems containing elastic damping elements. The authors of article [12] proposed a particular modification of his Rungge-Kutta method that allows efficient integration of stiff equations.
The study of the relationship between hyperloop vehicles and guideways can be considered as a cooperation between capsules and guideways [19]. Both rotational and linear motion accelerations are caused by the forces generated by the motion of the capsule. Vehicle motion, road dynamics and road bumps are all sensed by the suspension system. Tracks are usually represented in mechanical models as beams with a fixed Tshaped cross-section [23]. This guideway model is also used in the work. The displacement forces on the suspension and guideway exert forces on the guideway support [21]. These forces on the guideway beam are influenced by both the guideway reaction forces and moments of the support member [22] and the dynamic properties of the support foundation. The mathematical description of this tightly-coupled process is very complex, and appropriate assumptions must be made [20]. The vehicle under consideration can have different support distances. A fixed support spacing value of 30 m was used in this study [24,25]. The average speed of a Hyperloop vehicle is almost five times that of regular trains [26]. The dynamic amplification can be very significant at these extremely high speeds, which will have an immediate impact on the magnets' displacement and the safety of the vehicle's motion [31].
Additionally, the Hyperloop tube will be supported by a number of platforms that will support it vertically and permit it to move longitudinally owing to thermal expansion [33].
However, the lateral movement of the deck has not yet been examined in relation to the interaction of the Hyperloop tube with the platform [32]. In this investigation, the unbalanced loading of the train-where certain tubes are loaded but not all is what is causing the lateral vibration of the deck. Although they are outside the purview of this study, lateral loads like earthquakes and, to a lesser extent, wind, can result in lateral vibrations [34].
In a thorough and in-depth article [27], which describes the formulation of moving force and moving mass for simple spans, the problem of moving load is first mathematically defined by [32]. Vertical vehicle-bridge interaction issues have also been addressed using moving force-beam systems [8]. Parametric analysis aids analytical answers to moving load problems. To evaluate the vibration response of such systems, more intricate numerical approaches are needed because not all moving force problems can be addressed analytically [9]. For the sake of simplicity, moving load issues can be handled as static loads placed at various points throughout the structure. However, particularly for high-speed trains, the dynamic impacts of shifting loads can be significant. The peak deflection or stress induced by the dynamics of moving loads is hence characterized as DAF, or dynamic to quasi-static peak deflection [13,14].
The paper [16] is a superb and significant monograph on moving mass loads traveling on beams. Hundreds of papers on the issue of moving loads and moving forces are described in it.
Beams would still require literature citations to be added. These studies cover inertia loads, gravity loads, and permanent loads [18].
The authors, following Paracmi [29], developed more complex two-dimensional (2D) and three-dimensional (3D) models of the components of the Hyperloop system, which consisted of (3D) vehicle models that consisted of one body, four bogies, eight bolsters, eight suspension magnets, and a magnetic force feedback controller [35]. This was done in order to investigate the impact of irregularities and resonant conditions occurring in the Hyperloop vehicle dynamics [16].
Due to its discrete-time controller, the Maglev electromagnetic train will shake while suspended. On the other hand, the HTS Maglev won't vibrate because it uses passive levitation. The existence of vibration is unavoidable, however, when the HTS Maglev vehicle is moving at high speeds [17].
The arc collision and other outside disturbances have caused some vibration in the vehicle, as was to be expected. This kind of vibration can be uncomfortable or even dangerous . The HTS Maglev system must therefore undergo dynamic testing in order to be used for rail transit in the future [30].
When studying scientific literature, it is noticeable that the number of studies that discuss the analysis of the Hyperloop vehicle as a continuous discrete system has significantly increased [15].The response of the elastic-damping system for the Hyperloop vehicle was calculated using the Rungge-Kutty-Rosenbrock algorithm [43] presents the approach for integrating the equations defining the dynamics of discrete systems.
In this study, the interaction of mass at various I-beam sites will be used to describe the interaction between the guideway and the levitation assembly of Hyperloop vehicles [48]. The magnetic levitation module's linear and angular acceleration are caused by forces that arise during the passage of the vehicle [49]. Non-linear non-uniformities appear on the guideway's surface as a result of the suspension system's reaction to accelerations in the discrete system during travel [50]. The mechanical model of the guideway is described in scientific journals as being represented by a beam with a constant crosssection resting on an elastic basis [51]. This paper also utilizes this model. The electromagnetic Hyperloop vehicle will vibrate as the suspension system moves because of the discrete model, but it won't vibrate because of the magnetic levitation phenomenon itself because there aren't any magnets in place.
However, vibrations are unavoidable while the Hyperloop is traveling at a high speed [52]. The impact of the arc and other outside disturbances have generated minor vibrations in the vehicle, as is to be expected. This kind of vibration can be uncomfortable or even dangerous [44].
The tube that the Hyperloop vehicle rides on has an underlying longitudinal structure called the guideway [45]. The ride experience and stability of the Hyperloop object are significantly impacted by the guideway's elasticity [46,47]. The electromagnetic suspension is actively and independently regulated by the flexible guideway via a feedback controller.
The electromagnetic suspension works by creating an air gap whose variation causes variation in magnetic forces. The resonance of the Hyperloop vehicle's structure causes excessive vibrations, which might result in uncomfortable ride quality or unstable suspension [3]. Therefore, a detailed analysis of the vehicle/guideway interaction is crucial for Hyperloop systems in order to guarantee stable suspension and ride quality of the Hyperloop vehicle as well as the guideway construction.
This study looks into a probabilistic scenario involving the mechanics of a Hyperloop vehicle capsule and analyses how an object would behave in a continuous, multi-span ruptured tube [40]. Using the Lagrangian approach, the equations of motion for the system in this cell were established. Simulation studies were conducted after identifying the important boundary conditions. The parameters at which the object would become static under the influence of the forces acting on the guideway were also established [42].
In this paper, the results obtained in the simulation process, mainly for the lateral movements of the electromagnet, will be used to study the stability of the motion. In the report [30], bifurcation solutions were considered as the basis for the stability study of a linear system. While such a system is theoretically possible, in practice we will study (through a simulation process) the transverse motion of electromagnets for a non-linear system with perturbations originating from the guideway [41]. These results will be used to study stochastic technical stability (a definition will be presented in the paper).
This will enable the determination of the probability of unsteady motion (contact between the electromagnet and the guideway).
The full procedure for this will be presented in the article. Such a process has not been investigated in any currently available work.

Building the physical model
Geometrical parameters, discrete models of the guideway, and the capsule are all incorporated in the mathematical equations that describe the dynamics of the Hyperloop vehicle. The susceptibilities, together with linear and angular characteristics obtained from the coordinate systems, are also included in the mathematical description. The mathematical arguments discussed in the following paragraphs are drawn from the paper [18].
A discrete model of the Hyperloop vehicle is primed in Fig.   1. In order to represent the real object in the mechanical model under analysis, the length of the tube, denoted by L, is specified.
All components comprising the Hyperloop will be treated as a  Hyperloop vehicle were defined, and so by l is the length of the capsule, b is its width, while its height is determined by the parameter h. The mass objects constituting the vehicle were defined as m1 the total mass of the capsule and m2 the total mass of the levitation set. The damping and elasticity coefficients present in the discrete system were defined by ch and kh, respectively. In order to better illustrate the clarity of the differential equations describing the dynamics of vehicle motion, two separate coordinate systems were adopted, one associated with the O1XYZ tube and the other associated with the capsule, the origin of which was assumed to be at the centre of gravity of the capsule and the levitation components. It was assumed that the axes of the coordinate system connected with the capsule and the magnetic assembly are located on the axes of symmetry of the masses of the basic elements of the vehicle.
The situation is identical in the case of angular coordinates whose vectors defining velocities occurring in the system lie on the axes of the coordinate system connected with the object and characterise the dynamics of roll, yaw and inclination of the Hyperloop vehicle. In what follows, the directional cosine matrix is assumed to be zero-one (ones on the diagonal) according to the work [9].
amping forces acting on the capsules: Magnetic force of magnetic levitation system: where, in equation (3) where r=2/ 0 0 , ( ) = ( ) − 0 produces a disturbance in the y-axis, but the reluctance or gap parameter R0 between the magnet assembly and the guide at rest is equilibrium. The air gap is determined by the amount , which is defined by the relationship:

Motion equations for Hyperloops
To determine the equations of motion of the Hyperloop vehicle, d'Alembert's principle was used, resulting in: The parameters L p , L y , L r visible in the equation above are the inertial forces. The mathematical notation is as follows: The displacement caused by the guideway can be expressed by the following formula: where is the irregularity appearing on the guideways.
In a further mathematical consideration, the equation for the value of the movement occurring on the guide can be determined: Ak(t) -the course of the displacements generated on the guideway.

Stochastic technical stability (STS)
The STS definition was taken from the book [3]. Stability in the for another proces ( ), absolutely integral in the given interval. This presumption is comparable to the function continuity criteria.
The definition of technical stochastic stability is: if every solution of equation (17), whose initial conditions( , 0 , 0 ) are contained in the area , belongs to the area with probability 1 − , then the system (17) is technically stochastically stable with respect to , and the process ( ) with probability 1 − We provide the traditional concept of Lyapunov stability here as a reminder [3]. The Lyapunov definition is one of the original and fundamental definitions of mechanical system stability. n first-order differential equations of the form represent the motion of a mechanical system in its description.
The solution = ( ) of the system described by equation -for these solutions, the inequality occurs The verbal interpretation of this notation after [17] is that the trajectory ( ) of equations (21) is stable when the perturbed solution ( ) is located in a narrow region ε around ( ) (Fig.   6). This means that the region ε is the limit within which the perturbed solution can be found, so that the stability of the analysed system can be established. Failure to determine the probability of the perturbed solution being close to the correct solution makes it impossible to subject the system to random perturbations and analyse their effect on the motion of that system.
In the study of stability in the Lyapunov sense, there are limitations on the extensibility of the solution of equation (21) to infinity, and the conclusion of this solution for the area defined by the parameter ε>0. Figure 6. Lyapunov-style stability graphically interpreted [28]. Most often the determination of this function requires it to be "guessed".
In the STS study, no such procedure is required. What is required is to define the area Ω and to find a solution describing the dynamics for a given generalised coordinate. For the Hyperloop, we assume that the positive extent of the area Ω is the clearance, which is described by relation (6) and in Fig.2 the distance between the guide and guidance magnet. The second gap, which appears in Fig.2, may have larger magnitudes, as the size of this gap also depends on the deflection of the susceptible elements. This gap can be taken up in different magnitudes. In our simulation studies, a gap size of 8 mm was adopted. The magnitude of the gap variation is the transverse motion of the electromagnet -in the model the y1 coordinate from equation (7). As a result of the simulation, this motion is a random variable, i.e. a stochastic signal (the set of determinants is time), whose analysis will be conducted on the basis of a single realisation. In order for such an analysis to be carried out, it must be investigated or assumed whether the stochastic signal has the properties of stationarity of order two and global ergodicity. In our study, we have made such an assumption, which allowed the analysis on the basis of a single realisation.
We will then determine a histogram of the changing magnitudes of the electromagnetic motion in the y-direction for a guide magnet. We assume a significance level of = 0,05 for further testing. Then, using the Kolmogorov-Smirnov test, we determine the form of the function describing the probability of a given magnitude of slack. With such a function, it is possible to determine the probability of movement of electromagnets with a magnitude of 8 mm, i.e. there will be contact between the guiding electromagnet and the guide. This is the magnitude in which the movement is unstable. As shown graphically in Figure 5, such a probability can be determined from the relationship: Guide inequalities have been omitted from the considerations, due to their very small magnitudes. Probabilities will be determined for different Hyperloop movement conditions. The procedure prepared in this way will be carried out obtaining results in the simulation process.

Simulation study
The following quantities characterising the Hyperloop vehicle were assumed in the simulation [36]: The simulation process adopted the guide model shown in  electromagnets for a speed of 1000 km/h along a straight tube is shown in Fig. 8. Fig. 9 shows the transverse displacement for 4 electromagnets for a speed of 1000 km/h along a curved tube.     Figure 11. Displacement of levitation set at 2000 km/h in a curved tube with radius R= 200 000 metres.
In Figure 8, orange is for lavitation magnet movement and purple is for guidance magnet movement. In Figure 9, the maximum displacement is for the lavitation magnet movement.  Table 2.  Table 3.
Two probability quantities are given in Table 2and Table 3: the first quantity represents the probability of steady state motion and the second quantity represents the probability of unsteady motion. In Table 1 According to Figure 5, if the trajectory of the solution coincides with the boundary Ω, which in our case means the clearance between the guide and the magnets, then with a certain probability (which is given in Table 1