Vibration analysis of a hyperloop pod with a novel hybrid levitation in accelerating and braking maneuvers

Abstract

Hyperloop, a very high-speed transportation system including a pod transporting the passengers in a tube, forms the core debate in numerous recent investigations. The present paper, furthering the current advanced findings, proposes a novel hybrid levitation including Electro-Dynamic Suspension (EDS) and air cushions for the pod in the hyperloop system. It also presents a vibrational model for the hybrid levitation pod to predict the vibration behavior of the pod in accelerating and braking maneuvers. In this regard, an industrial pod is designed conceptually by performing preliminary calculations. Via CFD simulations in Ansys Fluent software, we hold forth a nonlinear dynamic model for the air cushions. Then, the paper analyzes a 3-DOF vibration model of the pod and verifies it by comparing the related results with obtained ones from the MSc ADAMS software. Afterward, the mentioned vibration model is developed into a 7-DOF model to predict the vibration behavior of the pod in accelerating and braking maneuvers. Also, the governing dynamic equations of the system are derived and numerically solved. Finally, the vibration behavior of the pod in different accelerating and braking conditions is investigated.

Keywords

Hyperloop pod magnetic levitation air cushion vibration modeling braking and accelerating

Introduction

Transportation and using appropriate vehicles to get to a destination as quickly as possible always have been crucial issues for humans throughout history. In late 2013, Elon Musk proposed the Hyperloop Transportation Technology (HTT), an ultra-high-speed passenger or cargo transportation system known as the fifth generation of transport.¹ The operation mechanism of these systems is the movement of a high-speed capsule-shaped vehicle, called the pod, inside tubes with approximately vacuum air conditions. The pod with an axial compressor is accelerated using the Linear Induction Motor (LIM)¹; however, some other researchers do not employ any fan and motor compressor in their proposed design.² Two different technologies are suspending the pod; magnetic levitation and air cushions. Also, three magnetic suspension systems exist operating magnetic levitation technology (e.g., High-Speed Trains: HST) that are: Electro Magnetic Suspension (EMS), Electro Dynamic Suspension (EDS), and Inductrack.^3–5 EMS systems work based on the attractive influence of the magnets between guideways and the body. The EDS vehicles select the repulsive properties for levitating the system.^4,6,7 Inductrack is another case of EDS type wherein used a passive suspension system. Instead of superconductive magnets, it employs Permanent Magnets (PM) in the Halbach arrays form.^8–10 In some other pod designs, air cushions cause the pod to float. Such systems mainly use the axial compressor to intake the air and lead it to the air cushions.^1,11–13

Some research investigated presenting the appropriate dynamic modeling of HST considering acceleration and braking effects. Wei et al. investigated the longitudinal and vertical dynamics of an HST rescued by locomotives during braking on grades.¹⁴ Zhang et al. designed a PM eddy current brake and analyzed its braking effect on the levitation characteristics of an HTS maglev vehicle.¹⁵ Suo et al. presented the acceleration and deceleration model of HST using an air gun and magnetic braking devices.¹⁶ Powell and Palacin studied the maximum allowable longitudinal acceleration for keeping passenger stability in moving railway vehicles.¹⁷ Ju and Lin numerically analyzed vehicle–bridge vibration responses by considering vehicle braking and acceleration using a seven DOF half-vehicle model.¹⁸

Some other researchers dedicated their studies to calculating lift and drag forces and the stiffness and damping of different kinds of levitations of HST. Zhu et al. calculated the damping and stiffness of maglev systems in terms of oscillation frequency using an experimental method.¹⁹ Wright and Bird proposed an analytical approach calculating the damping and stiffness of a 4-DOF electrodynamic wheel maglev vehicle.²⁰ Liu and Chen studied the EMS lift force of maglev vehicles at low and medium speeds with a theoretical formula and simulation in software.²¹ Post and Ryutov presented simple analytical relations for calculating the drag and lift forces of the inductrack levitation systems.^8–10 Cai et al. investigated the lift and guidance forces of maglev trains in analytical and experimental methods.^22–24

Some research presents appropriate dynamic models for Air Cushion Vehicles (ACV) using analytical, numerical, or experimental approaches. Hovercraft is one of the most common ACVs that uses air cushion technology. Yun and Bliaut presented a simplified relation for calculating rigid air cushion pressure in the hovercraft system.²⁵ Pavăl et al. numerically investigated air movement inside the inner cavity of the air cushion of a hovercraft system using Ansys Fluent software. For that, the air cushion was simulated to obtain air inlet velocities and lift force in different gaps.²⁶ Other models also were proposed for the cushion.^27–29

In little investigations, scholars tried to design the system and present dynamic models for the hyperloop. Indraneel et al. designed an experimentally magnetic brake and levitation system for the hyperloop pod.^30,31 Pradhan and Katyayan created a pod called “OrcaPod” with the inductrack levitation kind and analyzed its free vibrations behavior assuming OrcaPod with a four DOF half-vehicle model.³² Using the Root Locus method, Galluzzi et al. analyzed the stability of a small pod equipped with electrodynamic levitation in the hyperloop system. The lift and drag forces and the natural frequencies were analytically calculated.^33,34 Lim et al. designed a model of null-flux coil EDS for the hyperloop system.⁷ Madhan et al. designed a wheeled pod for the hyperloop system and proposed a nine DOF vibration model with six angled magnetic modules.³⁵

According to the research conducted so far, all the mentioned references assumed the pod with maglev technology, and using the air cushions for these systems has not been investigated yet. Hence, the present paper introduces a novel hybrid levitation for the pod in the hyperloop system. The air cushions are responsible for supporting the pod weight. The paper proposes EDS to counteract the dynamic loads, improve the dynamic motions, and provide acceleration and braking of the pod. Also, a dynamic model is presented for the pod with hybrid levitation to predict the pod’s behavior in accelerating and braking maneuvers.

Preliminary calculations and pod design

As mentioned earlier, the proposed pod simultaneously has magnetic and air cushion suspensions as a hybrid levitation. Therefore, before designing and presenting the chosen pod, it is necessary to perform preliminary calculations to determine the number of air cushions required to overcome the pod’s overall weight. The proposed pod weight is calculated as approximately 26,000 kg. The designed pod with its components is presented later with more explanations. In this regard, the pod will suspend when the lift force of air cushions used in the system can overcome the total weight of the pod. The outlet air pressure under the pad from each cushion is estimated at approximately $P = 10 K P a$ .¹ The proposed air cushion has a circular geometry, explained in more detail in the next section. The pad diameter is presented as $D = 1.354 m$ . Thus, the cross-sectional area is calculated as $A = π \frac{D^{2}}{4} = 1.44 m^{2}$ . Therefore, the lift force of every cushion can be obtained as $F = P A = 14.4 K N$ . Also, the pod’s overall weight is $W = m g = 254.3 K N$ . Thus, the minimum number of required air cushions for levitating the pod can be calculated as $N = 18$ , where $N$ is the minimum number of air cushions required. For more reassurance, 20 air cushions are provided for the pod suspension.

The proposed pod model is presented based on the performed initial calculations. The suggested conceptual pod is designed in SolidWorks and shown as follows:

According to Figure 1, the designed pod is equipped with 20 air cushions. In addition, two pairs of magnets with inductrack suspension kind are located at the rear and front of the pod, which are responsible for controlling the dynamic deflections as well as the braking and acceleration of the pod. When the pod moves inside the tube, the compressor intakes the tube air into the vehicle leading to the air cushions and causing the pod to suspend.

Figure 1.

The proposed designed pod model along with its components in SolidWorks ((a) isometric (b) front (c) side view).

The main mechanical characteristics of the proposed pod (including the dimensions of the components) are taken from Elon Musk’s report.¹ Also, the technical specifications of EDS parts are presented based on Post and Ryutov’s^8–10 research described in detail in Section 4.1. However, the dynamical properties of the pod are calculated using the “mass properties evaluation” command in SolidWorks. These parameters are the pod’s total weight, three mass moments of inertia, and Center of Gravity (CG). All moments of inertia are considered relative to the CG position of the pod. For the introduced system, the technical specifications of each component of the pod are described in Table 1 as follows:

Table 1.

Technical specifications of the proposed pod with its components.

Parameter	Value
Pod overall length	$23 m$
Pod effective length	$19 m$
Pod width	$2.8 m$
Pod height	$2.5 m$
Pod chassis height	$0.3 m$
Tube diameter	$5 m$
Position of the pod center of gravity (CG) (Relative to the geometric center coordinate)	$x = 2.68 m, y = 0, z = - 0.56 m$
Mass moment of inertia of the pod components around the CG	$I_{x x} = 713735 k g . m^{2}, I_{y y} = 717705 k g . m^{2}$
Total mass of the pod components	$25923 k g$
Number of air cushions	20
Cross-sectional area of each cushion	$1.44 m^{2}$
Pressure of each cushion	$10 K P a$
Number of air tanks	2
Width of the permanent magnets ( $w$ )	$0.2 m$
Maximum value of the magnetic field ( $B_{0}$ )	$1 T$
Wavenumber of equivalent circuit of EDS ( $k$ )	$25 r a d / m$
Inductance of equivalent circuit of EDS ( $L$ )	$1.5 \times 10^{- 7} H$
Wavelength of equivalent circuit of EDS ( $λ$ )	$0.251 m$
Resistance of equivalent circuit of EDS ( $R$ )	$6.366 μ Ω$
Vacuum permeability ( $μ_{0}$ )	$4 π \times 10^{- 7} H / m$
Vertical thickness of the conductor ( $∆_{c}$ )	$0.03 m$
Dimensionless thickness of the coil ( $γ$ )	$2.094$
Perimeter of the one-turn-equivalent of the coils ( $p$ )	$0.6 m$
Longitudinal thickness of the conductor ( $d$ )	$0.1 m$
Passenger capacity	30 People
Average weight of every passenger	$75 k g$
Longitudinal and transverse distance between each seat and another one	$0.5 m, 0.44 m$

To further investigate and understand the air cushion levitation mechanism, the air cushion modeling used in the paper is explained in the next section in more detail.

Nonlinear dynamic modeling of the air cushions

Before examining the dynamic modeling of the system, it is necessary to determine the model of air cushions used in the pod. Therefore, in this section, the air cushions modeling is described. The proposed model is a circular shape pattern with the specified dimensions as follows:

According to Figure 2, a circular air cushion is designed with the specific geometric dimensions in Solidworks. The system consists of an inlet channel for the airflow passage and a circular bottom plate connected to the body via four supporting pins. In this regard, airflow leaves the system through the lateral gap between the plate and the cushion body. The formation of air vortices and reaction forces under the cushion plate cause the lift force to the cushion. The work given here shows designing different air cushion shapes in Solidworks, including equilateral triangle, circle, ellipse, rectangle, square, and four other combined shapes, including circle-rectangle and circle-square (known as oval shapes), hexagonal, and fillet square with two different radii. Then, the Computational Fluid Dynamics (CFD) of each designed cushion is analyzed under the same simulation conditions using Ansys Fluent software. To create the same condition in simulating the airflow inside each air cushion, the cross-sectional area of all cushions’ bottom plate (or pad) is chosen equal. Also, the remaining other geometric parameters are assumed equal. The simulation input parameters are the inlet pressure of the airflow into the cushion inlet channel and the air gap between the cushion and the ground surface. Then the simulation output is the air cushion lift force. The meshing and governing boundary conditions for simulating airflow into the system, as well as the cushion components, can be seen in Figure 3 as follows:

Figure 2.

Circular air cushion modeling and its dimensions.

Figure 3.

Meshing symmetry plane of the cushion and governing boundary conditions.

Note, by considering Elon Musk’s report, the inlet air pressure to the pod is $100 P a$ , and the outlet pressure from every air cushion is approximately $10 K P a$ with temperature $T = 398 ° K$ .¹ Thus, the pressure ratio becomes $100$ . Due to such ratio, it is required to assume the simulation as the compressible airflow model. Therefore, the pressure into the cushion inlet channel is considered as the input parameter for the simulation. In addition, the airflow is defined as the compressible ideal gas with the $K - ε$ turbulence flow model with temperature $T = 398 ° K$ in Ansys Fluent.

After performing the simulations and comparing the results, it is concluded that the air cushion with a circular geometry pattern provides the highest lift force compared to other geometric shapes in the same simulation conditions. For this reason, a circular pattern is used in this study. Finally, using the curve fitting method, the paper presents a new approximate mathematical formula for calculating the lift force in terms of parameters, including the inlet pressure and air gap as follows:

F_{C} (x, P_{i n}) = δ P_{i n} e^{- β x}

(1)

where $x$ is the air gap in $m m$ , $P_{i n}$ is the inlet air pressure to the cushion in $P a$ , and $δ$ is the geometric shape constant coefficient. For the circular shape, $δ = 1.5$ and $β = 0.025$ . The average pressure under the pad ( $P_{P a d})$ is assumed equal to the input pressure ( $P_{i n})$ in the cushion static equilibrium state. Considering the cushion lift force as $F_{C} = P_{P a d} A_{P a d}$ and putting it in equation (1), the equilibrium gap becomes $x_{0} = - \frac{Ln (\frac{A_{P a d}}{δ})}{β} = 1.63 m m$ .

In the next section, the dynamic modeling of the proposed pod is presented with a 3-DOF vibrational model, and the governing equations of motion are obtained to analyze its vibrational behavior.

The 3-DOF vibration model of the pod

Previously, the paper described the initial design calculations and presented air cushions dynamic modeling. In this section, a three Degree of Freedom (DOF) vibration model is proposed for the pod. Furthermore, the related governing equations are derived and verified by comparing the calculated results with the obtained results from the MSc ADAMS software.

According to Figure 4, the pod has a 3-DOF. Origin coordinate system is defined at the CG location with specified symbols for the directions, including $X$ for the longitudinal, $Y$ for the lateral, and $Z$ for the vertical. These 3 degrees of freedom include the vertical body movement $z$ , the body pitching $θ$ , and the body rolling $φ$ . Therefore, the cushions and the magnets are assumed to connect to the body with rigid links.

Figure 4.

The 3-DOF vibration modeling of the pod ((a) front and (b) side view).

The governing equations of motion

According to the presented dynamic modeling in Figure 4, the governing dynamic equations of the system are derived as follows:

B o d y v e r t i c a l m o v e m e n t : \sum F_{Z} = m \ddot{z} m \ddot{z} + \sum_{i = 1}^{10} K_{{a r}_{i}} (z \mp L_{i} θ + W φ) + \sum_{i = 1}^{10} K_{{a l}_{i}} (z \mp L_{i} θ - W φ) + \sum_{i = 1}^{10} C_{{a r}_{i}} (\dot{z} \mp L_{i} \dot{θ} + W \dot{φ}) + \sum_{i = 1}^{10} C_{{a l}_{i}} (\dot{z} \mp L_{i} \dot{θ} - W \dot{φ}) + K_{f r} (z - L_{f} θ + a φ) + K_{f l} (z - L_{f} θ - a φ) + K_{r r} (z + L_{r} θ + a φ) + K_{r l} (z + L_{r} θ - a φ) + C_{f r} (\dot{z} - L_{f} \dot{θ} + a \dot{φ}) + C_{f l} (\dot{z} - L_{f} \dot{θ} - a \dot{φ}) + C_{r r} (\dot{z} + L_{r} \dot{θ} + a \dot{φ}) + C_{r l} (\dot{z} + L_{r} \dot{θ} - a \dot{φ}) = 0

(2)

B o d y p i t c h i n g m o m e n t b a l a n c e : \sum M_{G} = I_{y y} \ddot{θ} I_{y y} \ddot{θ} \pm \sum_{i = 1}^{10} K_{{a r}_{i}} L_{i} (z \mp L_{i} θ + W φ) \pm \sum_{i = 1}^{10} K_{{a l}_{i}} L_{i} (z \mp L_{i} θ - W φ) \pm \sum_{i = 1}^{10} C_{{a r}_{i}} L_{i} (\dot{z} \mp L_{i} \dot{θ} + W \dot{φ}) \pm \sum_{i = 1}^{10} C_{{a l}_{i}} L_{i} (\dot{z} \mp L_{i} \dot{θ} - W \dot{φ}) - K_{f r} L_{f} (z - L_{f} θ + a φ) - K_{f l} L_{f} (z - L_{f} θ - a φ) + K_{r r} L_{r} (z + L_{r} θ + a φ) + K_{r l} L_{r} (z + L_{r} θ - a φ) - C_{f r} L_{f} (\dot{z} - L_{f} \dot{θ} + a \dot{φ}) - C_{f l} L_{f} (\dot{z} - L_{f} \dot{θ} - a \dot{φ}) + C_{r r} L_{r} (\dot{z} + L_{r} \dot{θ} + a \dot{φ}) + C_{r l} L_{r} (\dot{z} + L_{r} \dot{θ} - a \dot{φ}) = 0

(3)

B o d y r o l l i n g m o m e n t b a l a n c e : \sum M_{G} = I_{x x} \ddot{φ} I_{x x} \ddot{φ} + \sum_{i = 1}^{10} K_{{a r}_{i}} W (z \mp L_{i} θ + W φ) - \sum_{i = 1}^{10} K_{{a l}_{i}} W (z \mp L_{i} θ - W φ) + \sum_{i = 1}^{10} C_{{a r}_{i}} W (\dot{z} \mp L_{i} \dot{θ} + W \dot{φ}) - \sum_{i = 1}^{10} C_{{a l}_{i}} W (\dot{z} \mp L_{i} \dot{θ} - W \dot{φ}) + K_{f r} a (z - L_{f} θ + a φ) - K_{f l} a (z - L_{f} θ - a φ) + K_{r r} a (z + L_{r} θ + a φ) - K_{r l} a (z + L_{r} θ - a φ) + C_{f r} a (\dot{z} - L_{f} \dot{θ} + a \dot{φ}) - C_{f l} a (\dot{z} - L_{f} \dot{θ} - a \dot{φ}) + C_{r r} a (\dot{z} + L_{r} \dot{θ} + a \dot{φ}) - C_{r l} a (\dot{z} + L_{r} \dot{θ} - a \dot{φ}) = 0 .

(4)

Where $m$ is the pod total mass, $K_{{a r}_{i}}$ is the $i$ -th air cushion stiffness on the right side of the pod, $K_{{a l}_{i}}$ is the $i$ -th air cushion stiffness on the left side of the pod, $C_{{a r}_{i}}$ is the $i$ -th air cushion damping coefficient on the right side of the pod, $C_{{a l}_{i}}$ is the $i$ -th air cushion damping coefficient on the right side of the pod, $L_{i}$ is the longitudinal distance between the $i$ -th air cushion and CG, $L_{f} = (0.65) \frac{L}{2}$ is the longitudinal distance between the front magnets and CG, $L_{r} = (1.22) \frac{L}{2}$ is the longitudinal distance between the rear magnets and CG, $K_{f r}$ is the front-right side magnet stiffness, $K_{f l}$ is the front-left side magnet stiffness, $C_{f r}$ is the front-right side magnet damping coefficient, $C_{f l}$ is the front-left side magnet damping coefficient, $W$ is the lateral distance between the air cushions and CG, $a$ is the lateral distance between the magnets and CG.

By sorting the sentences in terms of the variables, as well as assuming the equal stiffness and damping coefficients for all the cushions respectively with $K_{a}$ and $C_{a}$ , equations (2)–(4) are simplified as follows:

B o d y v e r t i c a l m o v e m e n t : \ddot{z} \sum F_{Z} = m \ddot{z} m \ddot{z} + (20 K_{a} + K_{1}) z + ((2 K_{a} \bar{L}) + L_{r} (K_{2}) - L_{f} (K_{3})) θ + (K 4 + K 5) a φ + (20 C_{a} + C_{1}) \dot{z} + ((2 C_{a} \bar{L}) + L_{r} (C_{2}) - L_{f} (C_{3})) \dot{θ} + (C_{f r} + C_{r r} - C_{f l} - C_{r l}) a \dot{φ} = 0

(5)

B o d y p i t c h i n g m o m e n t b a l a n c e : \sum M_{G} = I_{y y} \ddot{θ} I_{y y} \ddot{θ} + (2 K_{a} \bar{L} + L_{r} K 2 - L_{f} K 3) z + (2 K_{a} {\bar{L}}^{2} + {(L_{r})}^{2} K 2 + {(L_{f})}^{2} K 3) θ + (L_{r} K 4 - L_{f} K 5) a φ = 0

(6)

B o d y r o l l i n g m o m e n t b a l a n c e : \sum M_{G} = I_{x x} \ddot{φ} I_{x x} \ddot{φ} + (K 4 + K 5) a z + (L_{r} K 4 - L_{f} K 5) a θ + (20 K_{a} W^{2} + K 1 a^{2}) φ + (C 4 + C 5) a \dot{z} + (L_{r} C 4 - L_{f} C 5) a \dot{θ} + (20 C_{a} W^{2} + C 1 a^{2}) \dot{φ} = 0 .

(7)

where $\bar{L} = 27.1 m$ , ( $\bar{L} = L_{1} \pm L_{2} \pm \dots \pm L_{10})$ is the sum of the longitudinal distances between the air cushions of each side and CG (for front of CG with the negative sign and rear of CG with the positive sign), ${\bar{L}}^{2} = 403.441 m^{2}$ , ( ${\bar{L}}^{2} = {L_{1}}^{2} + {L_{2}}^{2} + \dots + {L_{10}}^{2})$ is the sum of the square of the longitudinal distances between the air cushions of each side and CG, and the other coefficients are defined as follows:

K 1 = K_{f l} + K_{f r} + K_{r l} + K_{r r}, C 1 = C_{f l} + C_{f r} + C_{r l} + C_{r r} K 2 = K_{r r} + K_{r l}, C 2 = C_{r r} + C_{r l} K 3 = K_{f r} + K_{f l}, C 3 = C_{f r} + C_{f l} K 4 = K_{r r} - K_{r l}, C 4 = C_{r r} - C_{r l} K 5 = K_{f r} - K_{f l}, C 5 = C_{f r} - C_{f l}

(8)

In equations (5)–(7), the governing equations can be described in a matrix form as $M \ddot{X} + C \dot{X} + K X = F$ , where $X = {[\begin{array}{c} z & θ_{y y} & φ_{x x} \end{array}]}^{T}$ is the matrix of the variables, and $F$ is the external forces matrix. $F = 0$ , because the track irregularities and other disturbance forces (such as the aerodynamic drag force) are ignored. Thus, the remaining coefficients of the matrix also can be obtained as follows:

M = [\begin{array}{c} m & 0 & 0 \\ 0 & I_{y y} & 0 \\ 0 & 0 & I_{x x} \end{array}] K = [\begin{array}{c} 20 K_{a} + K 1 & 2 \bar{L} K_{a} + L_{r} K 2 - L_{f} K 3 & (K 4 + K 5) a \\ 2 \bar{L} K_{a} + L_{r} K 2 - L_{f} K 3 & 2 {\bar{L}}^{2} K_{a} + {L_{r}}^{2} K 2 + {L_{f}}^{2} K 3 & (L_{r} K 4 - L_{f} K 5) a \\ (K 4 + K 5) a & (L_{r} K 4 - L_{f} K 5) a & 20 K_{a} W^{2} + K 1 a^{2} \end{array}] C = [\begin{array}{c} 20 C_{a} + C 1 & 2 \bar{L} C_{a} + L_{r} C 2 - L_{f} C 3 & (C 4 + C 5) a \\ 2 \bar{L} C_{a} + L_{r} C 2 - L_{f} C 3 & 2 {\bar{L}}^{2} C_{a} + {L_{r}}^{2} C 2 + {L_{f}}^{2} C 3 & (L_{r} C 4 - L_{f} C 5) a \\ (C 4 + C 5) a & (L_{r} C 4 - L_{f} C 5) a & 20 C_{a} W^{2} + C 1 a^{2} \end{array}]

(9)

The lift force of every magnet with EDS (inductrack) kind is calculated using the following relation:^9,32

F_{E D S} = \frac{a_{1} e^{(- b_{1} x)}}{1 + \frac{c_{1}}{v^{2}}}

(10)

where

x

m

is the air gap between the body and guideway magnets,

v

m / s

is the pod forward speed, and

a_{1}

b_{1}

c_{1}

are constants calculated as follows:

{\begin{array}{c} F_{E D S} = \frac{a_{1} e^{(- b_{1} x)}}{1 + \frac{c_{1}}{v^{2}}} \\ F_{L i f t} = \frac{{B_{0}}^{2} w^{2}}{2 k L} \frac{1}{1 + {(\frac{R}{ω L})}^{2}} e^{- 2 k x}, (ω = \frac{2 π v}{λ} = k v) \end{array}

(11)

In equation (11), the first relation is taken from Pradhan and Katyayan’s formula,³² and the second one is from Post and Ryutov’s work.⁹ Comparing two relations with each other, the values of the coefficients will be equal to $a_{1} = \frac{{B_{0}}^{2} w^{2}}{2 k L}$ , $b_{1} = 2 k$ , and $c_{1} = {(\frac{R}{k L})}^{2}$ , where $w$ is the width of the permanent magnets in $m$ with Halbach arrays, $B_{0}$ is the maximum value of the magnetic field in Tesla with Halbach arrays, $k$ is the wavenumber in $r a d / m$ , $L$ is the inductance in Henry, $λ$ is the wavelength in $m$ , $R$ is the resistance in $Ω$ , and $ω$ is the excitation frequency in $r a d / s$ .

The equivalent stiffness of every magnet $(K_{f l}, K_{f r}, K_{r l}, K_{r r} = K_{E D S})$ can be obtained by derivative of equation (11) concerning the gap ( $x$ ) as follows:

K_{E D S} = | \frac{\partial F_{E D S}}{\partial x} | = \frac{a_{1} b_{1} e^{(- b_{1} x)}}{1 + \frac{c_{1}}{v^{2}}}

(12)

The equivalent circuit of each EDS system is depicted as an RL circuit consisting of a resistance and an inductance with a voltage source. It is necessary to determine the resistance $(R)$ and inductance $(L)$ of the equivalent circuit of each EDS system to calculate the coefficients $a_{1}$ and $c_{1}$ . By considering the permanent magnets with NdFeB and the laminated track with aluminum materials,^8–10 the magnetic field parameters are $B_{0} = 1 T$ , $w = 0.2 m$ , $k = 25 \frac{r a d}{m}$ , $ρ = 1.7 \times 10^{- 2} μ Ω . m$ , and the optimum wavelength $λ = 4 π x_{0} = 0.251 m$ ( $x_{0} = 20 m m$ is the equilibrium gap) proposed by Post and Ryutov,¹⁰ where $ρ$ is the circuit resistivity and it has the following relation with the resistance:⁸

R = \frac{ρ l}{A} = ρ \frac{48 w β γ}{f λ^{2}}

(13)

where $A$ is the cross-sectional area, $β = 1$ is the number of circuits per quarter-wavelength, $f = 0.85$ is the fractional area of conducting material, $γ = \frac{λ}{4 ∆_{c}} = 2.094$ is the dimensionless thickness of the coil, $∆_{c} = 0.03 m$ is the vertical thickness of the conductor. Thus, the resistance is calculated as $R = 6.366 μ Ω$ . Also, the inductance can be calculated using the following relation:⁹

L = \frac{μ_{0} p}{2 k d}

(14)

where $μ_{0} = 4 π \times 10^{- 7} \frac{H}{m}$ is the vacuum permeability, $p = 3 w = 0.6 m$ is the perimeter of the one-turn-equivalent of the coils, and $d = 0.1 m$ is the conductor longitudinal thickness. Hence, the inductance is calculated as $L = 1.5 \times 10^{- 7} H$ . By assuming the pod velocity $= 300 \frac{m}{s}$ , all constants of equation (11) are computed as $a_{1} = 5305.2$ , $b_{1} = 50$ , $c_{1} = 2.86$ . Hence, in equation (12), the linearized equivalent stiffness of each magnet (assuming the equilibrium gap $x = 20 m m$ ) can be obtained as $K_{E D S} = 97.58 \frac{K N}{m}$ .

In the case of air cushions, similarly, the equivalent stiffness of each cushion is calculated by derivative of equation (1) concerning the gap $x$ in $m m$ as follows:

K_{a} = | \frac{\partial F_{C}}{\partial x} | = β δ P_{i n} e^{- a x}

(15)

Putting the cushion design parameters $δ = 1.5$ , $β = 0.025$ , $P_{i n} = 10 K P a$ , and equilibrium gap $x = 1.63 m m$ in equation (15), linearized equivalent stiffness of each cushion is obtained as $K_{a} = 360 \frac{K N}{m}$ .

Verification

The natural frequencies can be obtained by calculating the eigenvalues of the matrix

M \ddot{X} + C \dot{X} + K X = 0

. In this regard, by solving the determinant of the equations as

| M S^{2} + C S + K | = 0

, given roots are the natural frequencies, where the roots of

S

are the system eigenvalues. The matrix of the equations’ coefficients containing the mass (

M

), damping coefficient (

C

), and stiffness (

K

) are calculated using equation (9). Note that for undamped case (

C = 0

), the equations’ matrix is written as

M \ddot{X} + K X = 0

, afterward, the system is converted to a simplified eigenvalue problem as

(M^{- 1} K) ψ_{i} = {ω_{i}}^{2} ψ_{i}

. where

ψ_{i}

are the eigenvectors (mode shapes of the system), and

ω_{i}

are the natural frequencies (

λ = {ω_{i}}^{2}

are the eigenvalues). Therefore, solving

| A - λ I | = 0

gives the eigenvalues. where

A = M^{- 1} K

, and

I

is the identity matrix. For the proposed 3-DOF system, assuming the undamped case (

C = 0

), the natural frequencies are obtained and presented in Table 2. The studied system also is simulated in ADAMS software to ensure the reliability and accuracy of the derived equations of the pod in equations (5)–(7). In this regard, the same 3-DOF pod is designed with air cushions and EDS magnets in the ADAMS View environment. According to Figure 5, the same pod with the suspensions is proposed by modeling the air cushions and EDS magnets with the linear springs and dampers. The cushions are shown with red color and the magnets with blue ones. All dimensions and technical specifications of the simulation model are defined the same as the specifications of the introduced designed pod in Table 1. The defined parameters are the pod dimensions, pod total mass, CG position (

2.68,0, - 0.56

), pitching and rolling mass moments of inertia (

I_{x x}, I_{y y}

), and other specifications containing the location of the air cushions and magnets, and equivalent linearized stiffness of each of them.

Table 2.

Comparison of natural frequencies and mode shapes for undamped 3 DOF system.

Mode	Natural frequency $ω$ $(r a d / \sec)$		Eigenvector ( $ψ$ )
Mode	Numerical ADAMS	Analytical	Eigenvector ( $ψ$ )
1	$3.4986$	$3.45$	$0, 0, 0.12$
2	$14.1902$	$14.1868$	$- 0.53, 0.06, 0$
3	$23.2864$	$23.2828$	$0.32, 0.1, 0$

Figure 5.

3 DOF vibration modeling of the pod in ADAMS (isometric view).

Comparing the results of computed natural frequencies obtained from the matrix coefficients of the governing equations in equation (9) with MSc ADAMS are reported in Table 2. The results show that the obtained natural frequencies by the analytical method are approximately equal to the MSc ADAMS software. This issue implies the accuracy of the derived equations and the solution method. Also, the eigenvector is obtained to show the pod mode shapes in each vibration mode. In this regard, any variable with a higher numerical value (up to the normalized value of 1) is exposed to more excitation in that mode. The positive sign means that the movement of the variable is in the same direction as the conventional positive direction relative to the origin defined in the system, and the negative sign shows that it is in the opposite direction. Also, the zero value implies that the variable is not excited in that vibration mode. Thus, in the first vibration mode (Mode 1) only the body’s rolling angle is excited, in the second mode (Mode 2) the vertical movement and pitching angle of the body are excited in opposite directions, and in the last mode (Mode 3) the vertical movement and pitching angle of the body are excited in the same directions relative to the defined origin in the system.

In the next section, the effect of acceleration and braking is analyzed on the vibrations of the pod to predict the system’s dynamic behavior in different movement conditions.

The 7-DOF vibration model for predicting accelerating and braking behaviors

In this section, the acceleration and braking of the pod caused by the magnets also are considered. To reach this purpose, longitudinal vibrations in the system are employed, as in the case of Ju and Lin.¹⁸ This half-vehicle model is shown in Figure 6 as follows:

Figure 6.

7-DOF half-vehicle model considering longitudinal vibrations.

Figure 6 shows the front and rear magnets assuming flexible links relative to the body in vertical and longitudinal directions (4 DOF). The body also has two vertical and longitudinal directions of movement in addition to the pitching moment (3 DOF). Thus, the system overall is a seven DOF model. where $k_{f x}$ and $k_{r x}$ respectively are the front and rear magnets longitudinal stiffness, $C_{f x}$ and $C_{r x}$ are the front and rear magnets longitudinal damping coefficient, $k_{f}$ , and $k_{r}$ are the front and rear magnets suspension stiffness, $K_{f}$ and $K_{r}$ are the front and rear magnets stiffness relative to the track, $z (x_{i}, t)$ and $u (x_{i}, t)$ are the track vertical and longitudinal displacements when the $i - t h$ magnet moves on it, $r_{i} (x_{i})$ is the track irregularity, and $h$ is the vertical distance between the magnets and CG. Therefore, the governing equations of the motion can be derived as follows:

B o d y p i t c h i n g m o m e n t b a l a n c e : \sum M_{G} = I_{y y} \ddot{θ} I_{y y} \ddot{θ} \pm \sum_{i = 1}^{10} K_{a_{i}} L_{i} (z \mp L_{i} θ) \pm \sum_{i = 1}^{10} C_{a_{i}} L_{i} (\dot{z} \mp L_{i} \dot{θ}) - k_{f} L_{f} (z - L_{f} θ - z_{1}) + C_{f} L_{f} (\dot{z} - L_{f} \dot{θ} - {\dot{z}}_{1}) + k_{r} L_{r} (z + L_{r} θ - z_{2}) + C_{r} L_{r} (\dot{z} + L_{r} \dot{θ} - {\dot{z}}_{2}) - [k_{f x} (u - h θ - u_{1}) + k_{r x} (u - h θ - u_{2}) + C_{f x} (\dot{u} - h \dot{θ} - {\dot{u}}_{1}) + C_{r x} (\dot{u} - h \dot{θ} - {\dot{u}}_{2})] h = 0

(16)

B o d y v e r t i c a l m o v e m e n t : \sum F_{Z} = m \ddot{z} m \ddot{z} + \sum_{i = 1}^{10} K_{a_{i}} (z \mp L_{i} θ) + \sum_{i = 1}^{10} C_{a_{i}} (\dot{z} \mp L_{i} \dot{θ}) + k_{f} (z - L_{f} θ - z_{1}) + C_{f} (\dot{z} - L_{f} \dot{θ} - {\dot{z}}_{1}) + k_{r} (z + L_{r} θ - z_{2}) + C_{r} (\dot{z} + L_{r} \dot{θ} - {\dot{z}}_{2}) = 0

(17)

F r o n t m a g n e t v e r t i c a l m o v e m e n t : \sum F_{Z} = m_{1} {\ddot{z}}_{1} m_{1} {\ddot{z}}_{1} - k_{f} (z - L_{f} θ - z_{1}) - C_{f} (\dot{z} - L_{f} \dot{θ} - {\dot{z}}_{1}) + R_{1} = 0 R_{1} = K_{f} [z_{1} - ε_{1} z (x_{1}, t) - r_{1} (x_{1})]

(18)

R e a r m a g n e t v e r t i c a l m o v e m e n t : \sum F_{Z} = m_{2} {\ddot{z}}_{2} m_{2} {\ddot{z}}_{2} - k_{r} (z + L_{r} θ - z_{2}) - C_{r} (\dot{z} + L_{r} \dot{θ} - {\dot{z}}_{2}) + R_{2} = 0 R_{2} = K_{r} [z_{2} - ε_{2} z (x_{2}, t) - r_{2} (x_{2})]

(19)

B o d y l o n g i t u d i n a l m o v e m e n t (P o d) : \sum F_{X} = m \ddot{u} m \ddot{u} + k_{f x} (u - h θ - u_{1}) + k_{r x} (u - h θ - u_{2}) + C_{f x} (\dot{u} - h \dot{θ} - {\dot{u}}_{1}) + C_{r x} (\dot{z} - h \dot{θ} - {\dot{u}}_{2}) = 0

(20)

F r o n t m a g n e t l o n g i t u d i n a l m o v e m e n t : \sum F_{X} = m_{1} {\ddot{u}}_{1} m_{1} {\ddot{u}}_{1} - k_{f x} (u - h θ - u_{1}) - C_{f x} (\dot{u} - h \dot{θ} - {\dot{u}}_{1}) + R_{f x} = 0 R_{f x} = k_{f x} [u_{1} - ε_{1} u (x_{1}, t)]

(21)

R e a r m a g n e t l o n g i t u d i n a l m o v e m e n t : \sum F_{X} = m_{2} {\ddot{u}}_{2} m_{2} {\ddot{u}}_{2} - k_{r x} (u - h θ - u_{2}) - C_{r x} (\dot{u} - h \dot{θ} - {\dot{u}}_{2}) + R_{r x} = 0 R_{r x} = k_{r x} [u_{2} - ε_{2} u (x_{2}, t)]

(22)

Hence, the governing equations can be written as $M \ddot{X} + C \dot{X} + K X = F$ with the variables $X = [θ, z, z_{1}, z_{2}, u, u_{1}, u_{2}]$ . Therefore, the matrix coefficients of equations are obtained as follows:

M = [\begin{array}{l} I_{y y} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & m & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & m_{1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & m_{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & m & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & m_{1} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & m_{2} \end{array}] K = [\begin{array}{c} 2 {\bar{L}}^{2} K_{a} + {L_{f}}^{2} k_{r} + {L_{r}}^{2} k_{f} + h^{2} (k_{f x} + k_{r x}) & k_{r} L_{r} - k_{f} L_{f} & k_{f} L_{f} & - k_{r} L_{r} & - h (k_{f x} + k_{r x}) & h k_{f x} & h k_{r x} \\ k_{r} L_{r} - k_{f} L_{f} & 2 \bar{L} K_{a} + k_{f} + k_{r} & - k_{f} & - k_{r} & 0 & 0 & 0 \\ k_{f} L_{f} & - k_{f} & k_{f} & 0 & 0 & 0 & 0 \\ - k_{r} L_{r} & - k_{r} & 0 & k_{r} & 0 & 0 & 0 \\ - h (k_{f x} + k_{r x}) & 0 & 0 & 0 & k_{f x} + k_{r x} & - k_{f x} & - k_{r x} \\ h k_{f x} & 0 & 0 & 0 & - k_{f x} & k_{f x} & 0 \\ h k_{r x} & 0 & 0 & 0 & - k_{r x} & 0 & k_{r x} \end{array}] C = [\begin{array}{c} 2 {\bar{L}}^{2} C_{a} + {L_{f}}^{2} C_{r} + {L_{r}}^{2} C_{f} + h^{2} (C_{f x} + C_{r x}) & C_{r} L_{r} - C_{f} L_{f} & C_{f} L_{f} & - C_{r} L_{r} & - h (C_{f x} + C_{r x}) & h C_{f x} & h C_{r x} \\ C_{r} L_{r} - C_{f} L_{f} & 2 \bar{L} C_{a} + C_{f} + C_{r} & - C_{f} & - C_{r} & 0 & 0 & 0 \\ C_{f} L_{f} & - C_{f} & C_{f} & 0 & 0 & 0 & 0 \\ - C_{r} L_{r} & - C_{r} & 0 & C_{r} & 0 & 0 & 0 \\ - h (C_{f x} + C_{r x}) & 0 & 0 & 0 & C_{f x} + C_{r x} & - C_{f x} & - C_{r x} \\ h C_{f x} & 0 & 0 & 0 & - C_{f x} & C_{f x} & 0 \\ h C_{r x} & 0 & 0 & 0 & - C_{r x} & 0 & C_{r x} \end{array}] F = [\begin{array}{c} 0 \\ 0 \\ - R_{1} \\ - R_{2} \\ 0 \\ - R_{f x} \\ - R_{r x} \end{array}]

(23)

The following relations are utilized between the absolute and relative longitudinal movements for all magnets and the body to consider the acceleration and braking effect:

u_{i} = {\bar{u}}_{i} + v_{0} t + \iint a_{0} d t

(24)

{\dot{u}}_{i} = {\dot{\bar{u}}}_{i} + v_{0} + \int a_{0} d t

(25)

{\ddot{u}}_{i} = {\ddot{\bar{u}}}_{i} + a_{0}

(26)

Where $u_{i}$ , ${\dot{u}}_{i}$ , and ${\ddot{u}}_{i}$ respectively are the absolute longitudinal displacement, velocity, and acceleration of the body or $i - t h$ magnet, and ${\bar{u}}_{i}$ , ${\dot{\bar{u}}}_{i}$ , and ${\ddot{\bar{u}}}_{i}$ are the relative longitudinal displacement, velocity, and acceleration of the body or $i - t h$ magnet. $v_{0}$ is the pod initial velocity and $a_{0}$ is the pod acceleration (or braking for negative value).

Therefore, Putting equations (24)–(26) in equations (16) and equations (20)–(22), the new equations according to the relative motions can be rewritten as $M \ddot{\bar{X}} + C \dot{\bar{X}} + K \bar{X} = \bar{F}$ with the relative variables $\bar{X} = [θ, z, z_{1}, z_{2}, \bar{u}, {\bar{u}}_{1}, {\bar{u}}_{2}]$ . Thus, the relative external force matrix is calculated using the following relation:

\bar{F} = F + [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ - m a_{0} \\ - m_{1} a_{0} \\ - m_{2} a_{0} \end{array}]

(27)

According to equation (27), the inertia force from every mass also is created by considering the acceleration or deceleration. Note that the mass, stiffness, and damping coefficients matrix will not change, and they are the same as equation (23). Also, if the longitudinal vibrations of the system, as well as the existence of the air cushions, are neglected, then the system is converted to a four DOF vibration model like the dynamic model proposed by Pradhan and Katyayan³² for their designed pod called “OrcaPod”.

Numerical results and discussions

In this section, numerically, the 7-DOF derived equations are solved considering the acceleration and braking effects using MATLAB. In this study, the track irregularity and displacements effects are neglected. Hence, to solve the problem conveniently, variables should be reduced from the second order to the first order. Therefore, in equations (16)–(22) the acceleration of each variable is calculated in terms of other variables’ displacements and velocities in every simulation step time. In this regard, each of them is defined as an input variable in the state space equations. Therefore, the input variables can be defined as the new 14 variables $[\dot{θ}, \dot{z}, {\dot{z}}_{1}, {\dot{z}}_{2}, \dot{\bar{u}}, {\dot{\bar{u}}}_{1}, {\dot{\bar{u}}}_{2}, θ, z, z_{1}, z_{2}, \bar{u}, {\bar{u}}_{1}, {\bar{u}}_{2}]$ .

The dynamic equations are solved numerically using MATLAB Simulink. The numerical data of the proposed model of the pod are $m_{1} = m_{2} = 1000 K g$ , $h = 0.96 m$ , $k_{f} = k_{r} = 500 \frac{K N}{m}$ , $k_{f x} = k_{r x} = 10000 \frac{K N}{m}$ , $C_{f} = C_{r} = 10 \frac{K N . \sec}{m}, C_{f x} = C_{r x} = 20 \frac{K N . \sec}{m}, C_{a} = 10 \frac{K N . \sec}{m}$ . Other numerical parameters are presented Table 1.

Assuming the zero initial value for the variables relative to the stable points, equations (16)–(22) also can be solved numerically. In this regard, three separate scenarios are developed for the system to consider the various behaviors of accelerating and braking maneuvers. The first scenario is moving the pod at a cruise speed or $a_{0} = 0$ , and the second and third approaches are the linear changing and constant value of accelerating/braking during the time. In the second scenario, the acceleration/braking process is considered a linear time-variant function from the start to the end of 10 s of simulation time with a maximum acceleration of $10 \frac{m}{\sec^{2}}$ . Thus, its amount concerning the time becomes $a_{0} = t$ for the accelerating and $a_{0} = - t$ for the braking. For the third scenario, the acceleration/deceleration is considered with an average constant value of $a_{0} = \pm 5 \frac{m}{\sec^{2}}$ to compare it with the second approach in the same condition. In this regard, these three scenarios are performed separately in the simulations and compare them in Figures 7 and 8.

Figure 7.

The body (a) pitching angle, (b) vertical, and (c) longitudinal vibrations versus simulation time.

Figure 8.

The front and rear magnets (a) vertical and (b) longitudinal vibrations versus simulation time.

In Figure 7, the body’s longitudinal and vertical movements and the pitching angle are plotted during the time in different cases.

According to the results obtained by comparing the three scenarios, including the cruise speed ( $a_{0} = 0$ ), the constant acceleration/braking ( $a_{0} = \pm 5$ ), and the linear time-variant acceleration/braking ( $a_{0} = \pm t$ ), it is found that the minimum vibration of the pod (zero value) is corresponded to the cruise speed scenario ( $a_{0} = 0$ ).

Also, comparing the other scenarios in the equal condition during 10 s of simulation time (average acceleration/deceleration of the linear time-variant state also becomes $a_{0} = \pm 5$ ), it is observed that the constant acceleration/braking ( $a_{0} = \pm 5$ ) provides fewer fluctuations for the pod than the linear time-variant acceleration/deceleration ( $a_{0} = \pm t$ ) in the equal condition. Thus, the pod movement with average acceleration/braking is more appropriate than the variable acceleration/braking state.

The pitching angle and vertical movement increase due to generated external forces ( $m_{i} a_{0}$ ) on the magnets and body concerning equations (23) and (27) caused by acceleration/braking over time. However, they reach their stable points after about 7 s because of the damping effect in the system. But the amount and range of fluctuations in the constant acceleration/deceleration case are less than in the linear variable state. For the longitudinal displacement of the body, in both cases, the amplitude of vibrations decreased, and finally, it converged to a stable point. However, in the variable acceleration/braking state, the oscillations are slightly later removed than in the constant acceleration case over time.

In Figure 8, the longitudinal and vertical movements of the rear and front magnets are depicted during the time simulation in different cases. In these diagrams, again, the variable acceleration/braking creates the highest oscillations compared to the others. In the vertical movements, the front and rear magnets oscillate suddenly in the first 2 s of simulation time, so they reach their maximum range of fluctuations almost in the first second, like the diagrams of the body’s vertical movement and pitching angle. But the oscillations form of front magnets differs from the rear ones. However, their longitudinal displacements are similar in behavior. Therefore, one of them is analyzed. Similar to the longitudinal displacements of the body, for the longitudinal movements of the magnets, the expiration of variable acceleration/braking fluctuations is later than in the constant acceleration case.

Conclusions

This paper proposed a new conceptual industrial (large-size) pod containing electrodynamic and air cushion levitations. In this paper, performing CFD simulations using Ansys Fluent software, a nonlinear dynamic model of the air cushion was provided, and a 3-DOF dynamic model was assumed for the proposed pod verified by using MSc ADAMS software. The model was developed into a 7-DOF to predict the vibration behavior of the pod in accelerating and braking maneuvers. The governing dynamic equations were derived and solved using MATLAB. Finally, the paper proposed three separate scenarios for accelerating/decelerating by investigating the vibration behavior of the pod in different accelerating and braking maneuvers conditions, presenting simulation results: constant speed, constant acceleration, and linear time-variant acceleration. Results showed that the minimum vibration of the pod is corresponded to the constant speed scenario with zero value. Then, comparing the other ones in the equal condition, the constant acceleration/braking provides fewer vibrations for the pod than the linear time-variant acceleration/deceleration. Thus, the pod movement with average acceleration/braking is more appropriate than the variable acceleration/braking state.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Hamed Petoft

Abbas Rahi

Vahid Fakhari

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