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\title{DESIGN AND ANALYSIS OF THE FUSELAGE OF A HIGH-SPEED PROTOTYPE APPLIED TO HYPERLOOP CONCEPT, USING FINITE ELEMENT METHOD \emph{COMMUNICATION SKILLS}}

\author{ N. FURIO$^{*}$}

\heading{ N. FURIO}

\address{$^{*}$Laboratory of Computational Methods and Numerical Analysis (LaC\`aN)\\
Universidad Polit\'{e}cnica de Catalu\~{n}a\\
Campus Norte UPC, 08034 Barcelona, Spain\\
e-mail: nesfual@zeleros.com}

\keywords{High-speed, Fuselage, Transport, Hyperloop, Zeleros, Light-weight, FEM , Carbon fibre, Composite, Strenght, High frequency.}

\abstract{Nowadays High-speed transportation has been developed different technologies, and countries around the world are implementing their own high-speed concept. In Spain, the company Zeleros it’s creating their more efficiently design of Hyperloop vehicle. This technology needs a light-weight fuselage with longitudinal rigidity to flexion, having said that, this work investigates the effects of composite materials and non-structural masses on the static and dynamic behaviour of Hyperloop fuselage.
To achieve this goal, it’s used the finite element method based on a full nonlinear analysis that use the classical Newton-Raphson technique, however the high computational complexities generated from both the incremental-iterative procedure and the very refined mesh needed for resolve the model, it’s translated on a high computational cost. For this reason, in the report, first it’s done a qualitative analysis based on a free-free normal mode analysis and a restricted normal mode analysis. This type of analysis is calculated applying the Lanczos method, it's the general method, that not implies the fastest or the most precision form to obtain the natural frequencies but is the better by their stability to converge and implies to waste less time on the mesh. Then we complete the study with Stress and Strain nonlinear analysis, in this analysis it has been modelled the most critical load cases, reducing the number of cases from 14 to 4, one acceleration condition, other on stationary condition and two of brake conditions.
The innovation of this paper is found it in this new type of manufacture concept applied in the vehicle, a carbon fibre tubular mono-body fuselage made by the company MTorres. The concept has been studied to improve the rigidity at high frequency of the vehicle, since the main problem is the excitation frequency introduced by the aircraft compressor of 250 Hz mixed with the little space on the vertical axis.
The results show the capability of the present model to provide a realistic solution with a low computational cost, and validate the methodology followed to get these results. Also, the investigation of this type of fuselage with strict boundary conditions contribute to improve it and motivate other companies to use it. }

\begin{document}
%\maketitle

\section{INTRODUCTION}

A tubular mono-body fuselage is transform the main structure in the only single piece that have to act as a load-bearing member. It suports the wheel and suspension system, propulsion system, levitation system, brake sistem, energy supply system and other diferent  components. Effective fuselage performance depends on maintining rigidity  uder all posible load cases that support on acceleration, constant velocity and braking time. The objective of present work  is to develop a methos for analyzing a composite tubular mono-body fuselage under operating conditions and to determine a structurally sound fuselage trought finite element analysis. Complexities involved in ths specific type of analysis include determining composite lay-up orientation, represent with precision the composite lay-up fabrication method, and implement a model with a realistic boundary conditions. One of the first industry that uses this composite concept are automotive and it's useful see laike this industry resolve some of the problems that we found. \cite{Eurenius}. The most importat design parameters that it's need to be present  during the analysis are rigidity resistance, low-weight and easy asembly of diferent components. 
To begin with the vehicle design, first it's needed to locate the main sistems of vehicle and to list it in a table ehere describe their mass. To achive this goal is better show the concept with a simple diagram.

\begin{figure}[h]
  \centering
    \includegraphics{Imagen_1_CM1}
  \caption{Longitudinal pod diagram.}
  \label{fig:ejemplo}
\end{figure}
\begin{figure}
  \centering
    \includegraphics{Imagen_2_CM1}
  \caption{Front  pod diagram.}
  \label{fig:ejemplo}
\end{figure}
 
This are the main components that have to include on the analysis model but it's important to know their mass, that are listed on the table [1].
\begin{table}[h!]
\caption{Mass of main pod sistems}
\begin{center}
\begin{tabular}{*{3}{c}}
\hline
ELEMENTS & MASS (Kg/u.) & Quantity \\
\hline
DRIVERS & 22 & 2 \\
\hline
MOTOR & 30 & 1 \\
\hline
LEVITATION SYSTEM & 20 & 10 \\
\hline
COMPRESOR & 75 & 1 \\
\hline
BATERIES & 250 & 1 \\
\hline
MAGNETIC BRAKES & 30 & 4 \\
\hline
EMERGENCY BRAKES & 25 & 8 \\
\hline
WHEELS AND SUSPENSION SYSTEM & 30 & 4 \\
\hline
\end{tabular}
\end{center}
\end{table}

With the description of concept design finished, the next step is to define the analysis proces that has been followed. Firts of all, to improve the design concept and verify what is the best design in base of the rigidity sistem has been donde a free-free normal mode analysis. This type of analysis based on Lanzcos method permit a qualitative check of How rigidity has improved compared to the previous model. Then, when a model is considered to be quite good, it goes to the final design phase, where the masses are located and goes to its analysis according to the necessary load cases. In these load cases the model is analyzed by non-linearimplicit analysis that compute iteratively the standard deformation equation by the Newton-Rhapson method. When this analysis get a result in the correct range, it's validated the model and is send to a manufacture composites company that validate all the fabrication process and posible problems that they can detect. The following diagram represents the work flux that is followed.

\begin{figure}[h!]
  \centering
    \includegraphics[width=0.8\textwidth]{Imagen_3_CM1}
  \caption{Analysis diagram.}
  \label{fig:ejemplo}
\end{figure}

The most common materials used in the production of a tubular mono-body fuselage are composites, in particular carbon fiber reinforced polymers (CFRP), because they exhibit high stiffness and strengh to weight ratio properties and can  be formed  to virtually and geometry. \cite{Davies} However, thera are somo disadvantages, such as intrincate design procedures, high cost and complex manufacturing processes.\cite{Wanberg} When compared to any composite material combination found that carbon fiber composites are extensively the better option.


\section{DESIGN CONCEPT CAD}

In this section the global model of the fuselage is described along with the different components that form it.

\begin{figure}[h!]
  \centering
    \includegraphics[width=0.8\textwidth]{Imagen_4_CM1}
  \caption{Front pod CAD.}
  \label{fig:ejemplo}
\end{figure}

In Figure 4 a front view of the system is shown in which the tube [1] through which the vehicle will move [2] is observed. The element [3] represents the envelope of the linear motor and the running surface, as can be seen in Figures 5 and 6, said envelope will extend along the entire tube [1]. Element [5] represents the levitation rail that also extends along the entire tube (Figure 5, Figure 6). The last element to highlight, from those indicated in Image 1, is the envelope reserved for the vehicle's battery block [4]. This envelope [4], is located in the current version (subject to design evolution) in the upper part of the vehicle and extends along its entire length.

\begin{figure}[h!]
  \centering
    \includegraphics[width=0.9\textwidth]{Imagen_5_CM1}
  \caption{Isometric section view pod CAD.}
  \label{fig:ejemplo}
\end{figure}

Figure 5 shows an isometric view of the system, having made a profile cut to it. It can be observed how the floor [8] occupies the entire length of the vehicle, but the option of a shorter length, or that is made up of different modules, is not ruled out, as long as it supports all the elements that are required. Nor does it rule out the option of fulfilling a structural function of reinforcement to the fuselage / frame. The lower crosspieces, indicated by reference [6], extend along the entire fuselage / frame. The current number of lower crossbars arranged is 3, as can be seen in Image 1. The elements represented by reference [7], are the intermediate ribs. As can be seen in Figure 5, 2 units of these intermediate ribs [7] are currently arranged along the entire length of the vehicle. Reference [18] indicates the location of the link rib with the compressor module.

\begin{figure}[h!]
  \centering
    \includegraphics[width=0.9\textwidth]{Imagen_6_CM1}
  \caption{Isometric view pod CAD.}
  \label{fig:ejemplo}
\end{figure}


In figure 3 you can visualize the vehicle with the same perspective and orientation as in figure 2, but without the profile section cut and without the tube [1] and the levitation rail [5]. It is observed how the fairing [9] has holes through it [10] and [11]. These holes correspond to the housings of the different braking systems that are expected to equip the vehicle.

\begin{figure}[h!]
  \centering
    \includegraphics[width=0.9\textwidth]{Imagen_8_CM1}
  \caption{Isometric section view fuselage CAD.}
  \label{fig:ejemplo}
\end{figure}

Figure 7  shows a profile cut of all the structural elements of the vehicle. It is seen as currently the reinforcing ribs [12] are continuous throughout the perimeter of the vehicle. It shows how the upper crossbeams [13] traverse the entire vehicle longitudinally. The elements indicated by reference [14] correspond to the reinforcing bars of the fairing [9], which cross the whole length of the vehicle again.

The ribs, which will be located along vehicle, will be the one that will connect the structural module of the compressor to the rest of the structure. It is conceived as an omega profile. It has the housings of the lower crossbars [15] and upper [16], which cross it completely. The reforcing ribs [12] and reforcing bars [14] are a plain surface lay-up.Tthe other component that have diferent section are the crossbeams that a hollow squeare section.


\section{MATERIALS AND ORIENTATION}

Woven  and unidirectional carbon fiber composite reinforcement materials are the materials choice for the design. The woven structure of alternating fiber direction are composed by warp and welf fibers which means that the structure exhibits mechanical properties in multiple directions.  the woven type that have been used in this design  are plain, in the plain weave, each warp fiber passes alternatively under and over each weft fiber; this is the most stable weave to prevent strand slippage and distorsion. In the table [2] list it's resumed all materials employed.

\begin{table}[h!]
\caption{Example of the construction of one table}
\begin{center}
\begin{tabular}{*{3}{c}}
\hline
DESIGNATION & MATERIAL  \\
\hline
1 & Woven 0/90º carbon fiber SE84LV \\
\hline
2 & Woven 45/-45º carbon fiber SE84LV \\
\hline
3 & Unidirectional carbon fiber SE84LV \\
\hline
4 & PVC Core PVC80H\\
\hline
\end{tabular}
\end{center}
\end{table}

The combination of this materials are diferent depending of the structure element that we analyze. The side panel of fairing  it's composed by 6 layers of material 3 alternating  with 45º /-45º orientation. 
Other elements are the crossbeamns that uses 6 layers the first five layers of material 3 alternating between 0º and 90º orientation, and the last layer is a woven layer of material 2 that help to compress this element during their manufacture process.
The under frame fromed by reforcing ribs and bars it's made up of  8 layers of material 1 with an  45º /-45º orientation.

A main element with a more complex structure are the ribs, that have a first inner layer only in the center without eaching the edges of material 3 in 90º orinetation, Over this 4 layers of material 3 placed in 0º / 90º configuration are placed. To finish on the top it's used a layer of material 2. In the figure 8 it's shown the manufacturing section of rib.

\begin{figure}[h!]
  \centering
    \includegraphics[width=0.7\textwidth]{Imagen_9_CM1}
  \caption{Rib section.}
  \label{fig:ejemplo}
\end{figure}

The next described element is the linear motor support [6], it have anther configuration, the top part can be thinner because is connected to the inner creossbeams, but the side of the element have to be a higher rigidity for this reason  have three more layer in the middle, the middle layers are made of material 3 with a 0º/90º/0º  orientation and the rest of layer are 4 of material 2  with a regular orientation. In the figure 9 it's shown the manufacturing section of linear motor support.

 \begin{figure}[h!]
  \centering
    \includegraphics[width=0.7\textwidth]{Imagen_10_CM1}
  \caption{Linear motor support section.}
  \label{fig:ejemplo}
\end{figure}

Finally the last element that has not yet been described is the floor of the vehicle, this has too larger dimensions and get a high rigidity depends directly on the thickness, for this the concept used is the sandwic panel. The upper and lower part of the sandwich are made of 2 sheets of material 1 with a central core of material 4 of 2 cm thickness, in this way we can increase thickness with a much lighter and cheaper material.

\section{FEM MODEL}

A mon-body fuselage has a significant difficulties to analyze accurately using analytical methods due to complexity of the structure. It requieres a complex mathematical model to describe it and the simplification would reduce the accuracy of the results. A  finite element analysis can accurately simulate the loads experienced by the fuselage in a shorter period than the approiate numerical solution. It's importnat to remember that a simulation is still only a representation of the design's performance and does not necessarily reveal the influence of the loads by problem variables. The user imput data errors or not accurately boundary conditions can produce a questionable simulation results.

To obtain the behavior on the diferent load cases of fuselage has  been used a implicit non linear analysis, using  SOL400 of NASTRAN. Also the deformed fuselage obtained on the diferent results of load cases have  been submitted  to a nomal mode analysis to know the dynamic behavior of vehicle and that the frequencies introduced by the compressor, engine and rolling effect are avoided.

The fuselage was modeled bay surface method becouse is the recommended way to simplify the composite structures, this 2D shell is formed by CQUAD4 elements to plain surfaces and CQUAD8 to curved surfaces, because this can adapt better to the fuselage shape, The diference between CQUAD4 and CQUAD8 is that are linear and quadratic elements respectively. To give this type of elements thickness, the PCOMPG property has been used, which allows to number the different layers of compound applied to each surface and unlike the PCOMP1 calculate the interlaminar shear force. All these calculations are made under the theory of flat tension and the failure criterion used to detect the delamination of carbon is the tsai-wu criterion, which considers both traction and compression forces. All the model has been meshed taking care that the mesh have a regular number of rectangular elements  how it's show on th figure [10]. The reason is the importance of don't found a discontinuity on the composites simulation becouse a reducton of nodes from one side of a surface to another is information loss.

 \begin{figure}[h!]
  \centering
    \includegraphics[width=0.8\textwidth]{Imagen_11_CM1}
  \caption{Surface model.}
  \label{fig:ejemplo}
\end{figure}

The size of the elements in a component is reduced to the size of its smaller confictive area in this way we reduce the loss of information. Despite having a fairly refined model, as can be seen in figure [11], there is not an excessive number of nodes that could result in an increase in computational cost, the final model has 168,000 nodes.

 \begin{figure}[h!]
  \centering
    \includegraphics[width=0.8\textwidth]{Imagen_12_CM1}
  \caption{Mesh model.}
  \label{fig:ejemplo}
\end{figure}

To connect all the masses and the points of application of restriction of displacements or forces, the elements RBE2 have been used, these elements have an infinite ridicule and are composed of an independent node, connected to one or more dependents, in this way the loads or displacements imposed in the independent node are transferred directly to the dependent nodes without loss of information, but with a distribution of their value proportional to the distance applied.The different loads and restrictions depend on the load case studied, although they are generally based on a model in which the main support is located in the part of the levitation, as can be seen in the figure [11], on the top part of model it's easy to se  green triangles that represent the displacement restriction.

The components of this model are in principle separated and to avoid unrealistic point tensions, these components have been joined by a command that simulates the glue, in NASTRAN the GLUE CONTACT. In this union, given that the structure is manufactured in a single time, the mechanical resistance and the failure criterion are based on the characteristics of the epoxy that forms the matrix of the carbon compound. In this way and following the nastran modeling recommendations, in which the correct size and shape of the elements is checked, in addition to passing the mass verification with respect to the CAD, it is concluded that we have a valid model to simulate the different cases of load.

\section{LOAD CASES}

The pod fuselage have to support four dynamic phases, linear motor acceleration, compresor acceleration, constant velocity and brakeing. In the diferent phases we can find diferent interaction between the diferent systems and also the emergency brake system. To all of this phases we have to add  two stationary load cases without movement consideration.

\subsection {Stationary condition}

In this section, two load cases must be evaluated, full load under support condition on wheels and full load under superior levitation condition. In the full load on wheels, the  vertical traslation on wheels have been fixed, apply a moment on the compresor join point and apply the gravity. In the full load on levitation the conditions of wheels support have been changed to the restriction of traslation in the area determined as levitation. This load case it's shown on the figure [12].

\begin{figure}[h!]
  \centering
    \includegraphics[width=0.8\textwidth]{Imagen_13_CM1}
  \caption{Levitation boundary conditions.}
  \label{fig:ejemplo}
\end{figure}

\subsection {Linear motor acceleration condition}

The first stage of acceleration of the prototype comes from a propulsion thanks to the linear motor. This acceleration stage goes from 0 to 300 km / h, and represents an average acceleration of 2 Gs. But the manufacturer has not made it known that no matter how much they control, there will be 4Gs acceleration peaks. For this reason our acceleration value for the dimensioning of the element will be 4Gs.

In this phase we find five cases of load studied, the ideal function, condition of maximum and minimum force in levitation, and two conditions of emergency braking. To simplify this section, the approach to this load case is described and the exact forces and restrictions are shown in appendix b.

The ideal operating situation, considers that the linear motor propels the vehicle under an action of 4 Gs, and the lift comes from the upper levitation system. The only contact that is established with the tube is through the lower guidance on the linear motor. 
 
On the situation of maximun force levitation, this system generates the maximum attraction and therefore raises the pod until the upper guidance that comes into contact with the tube. 
Conversely, in the minimun force levitation situation the levitation system generates the minimum attraction and therefore pod descends until it comes into contact with the wheels. In this way the weight is divided between the strength of the levitation units and the support of the wheels.

The emergency braking situation is carried out when a problem is detected in the components. Within the same there are two situations, one in which the levitation generates the maximum force and another the minimum force, in both cases the wheels come into contact and it is considered that the linear engine tries to continue by pulsing the vehicle.

\subsection {Compresor acceleration condition}

After the acceleration stage of the compressor there is a second stage with a smaller acceleration and of lesser magnitude of 2 Gs. The meaning of this analysis list is that the location of the propulsion changes from the lower part to the compressor zone. In this phase has been studied the same five cases that the last acceletarion phase.

\subsection {Constant velocity condition}

In these cases the intention is to cancel the acceleration component that acts during the previous stages of the test, in this case the compressor is still the one that propels the vehicle and we will use it as a restriction in the same way as the propulsion stage due to the compressor.
The results in this case are of lesser magnitude than in previous cases. In this phase has been studied the same five cases that the last acceletarion phase.

\subsection {breaking condition}

The brake to be used in the test is the magnetic brake, and the braking stage includes the use of this brake. The use of this brake consists of three phases, stay at a certain height of the ground and stay at that distance (stop phase), the last phase of descent where the magnetic drag is higher and still no contact (magnetic braking phase last) and finally the braking phase in which the friction part of the brake comes into contact with the ground (braking phase magnetic friction). During these phases it is assumed that the lower tread is always present.

In each of these three phases we find three load cases studied, the ideal function and two conditions of emergency braking, this leaves a total of 9 load cases, for the braking phase. In all of this cases the restriction of longitudinal displacement that before have the compressor, now is located on the center of magnetic brakes.

\section{RESULTS}

One the load cases, previously described, have been launched, the maximum stress values and their displacements are summarized in table[3].
It's observed thhat of the load cases exposed the most critical conditions are found in the acceleration stage of the linear motor, with braking status of emergency (E.B.due to forces in the opposite direction at both ends of the pod.

\begin{table}[h!]
\caption{Maximun results}
\begin{center}
\begin{tabular}{*{3}{c}}
\hline
LOAD CASE & MAX. STRESS (MPa) & DEFORMATION (mm) \\
\hline
Stationay conditions with levitation & 196 &  2.12\\
\hline
Linear motor acceleration condition in E.B. & 469 &  3.61\\
\hline
Compressor acceleration condition in E.B. & 382 & 3.1\\
\hline
Constant velocity in  E.B. & 283 & 2.86 \\
\hline
Braking condition in E.B. &350 & 3.02\\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{figure}[h!]
  \centering
    \includegraphics[width=0.8\textwidth]{Imagen_14_CM1}
  \caption{Stress results.}
  \label{fig:ejemplo}
\end{figure}

In the figure [13], it's observed that for the case of constant velocity in E.B. the maximum values of stress apear in the upper part of the ribs since the only hold is the levitation, and in that line alse appears flexion because of braking force.

Another thing that has been analyzed are the pod own modes deormed, both in support condition on the wheels and levitating. In the table[4] can be observed very low values of frequency due not to a own lack of rigidity odf vehicle but rather to an inestability of the same due to the incorrect position of the masses.

\begin{figure}[h!]
  \centering
    \includegraphics[width=0.8\textwidth]{Imagen_16_CM1}
  \caption{Deformation results.}
  \label{fig:ejemplo}
\end{figure}

\begin{table}[h!]
\caption{10 first natural frequencies}
\begin{center}
\begin{tabular}{*{3}{c}}
\hline
MODE Nº & Hz \\
\hline
1 & 0.6\\
\hline
2 & 6.08\\
\hline
3 & 6.39\\
\hline
4 & 10.59 \\
\hline
5 &11.24\\
\hline
6 & 12.5\\
\hline
7 & 15.34\\
\hline
8 & 17.55\\
\hline
9 & 19.41\\
\hline
10 & 19.9\\
\hline
\end{tabular}
\end{center}
\end{table}

The table does not show all the frequency values, due to the number of contiguous frequencies that must be taken out, this is a clear problem since it shows that with the loads and the proportions of the structure it's impossible to overcome or dodge the 250 Hz that introduces the compressor.

\begin{figure}[h!]
  \centering
    \includegraphics[width=0.8\textwidth]{Imagen_17_CM1}
  \caption{First natural frequency.}
  \label{fig:ejemplo}
\end{figure}

The figure [15] show a pendulum oscilation of pod in low frequency because big part of the weight is in the bottom of the pod.

\section{CONCLUSIONS}

Given that a coherent deformation and distribution of the tension has been found in the results of the pod, it can be conclueded that the model is validated and it's possible to use it as an iterative model to improve it's layering and distribution of mas.
In addition, given that the results of conserving even in cases of emergency, a safety factor greater than 2.0, the next improvements follow a reduction of the mas or better distribution of it inside the pod.

\begin{thebibliography}{99}

\bibitem{Eurenius} C.A. Eurenius, N.Danielsson, A.Kholar, E.krane, M. Olofsson, J. Wass.  \textit{Analysis of composite chassis}. Chalmers University of technology, Goteborg, (2013).
\bibitem{Davies} G. Davies.  \textit{MAterials for automobile bodies}. Oxfordy,  (2012).
\bibitem{Wanberg} J. Wanberg.  \textit{Composite materials fabrication handbook}. Stillwater MN, (2009).

\end{thebibliography}

\end{document}


