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Dynamic amplification factors for ultra-high-speed hyperloop trains: Vertical and lateral vibrations

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The ultra-high-speed (UHS) Hyperloop is the next-generation mode of passen-ger/freight transportation, and is composed of a tube or a system of tubes through which a pod travels free of friction. The entire system must be supported by piers (multi-span viaducts), where the tubes act as the bridge deck. The UHS moving Hyperloops can exert large dynamic effects to the supporting pier-deck system both vertically and laterally. Particularly, asymmetric Hyperloop loading can generate significant lateral vibrations. Therefore, for safe design of a bridge pier-deck system for UHS trains, it is of great importance to explore dynamic interaction of bridge deck and piers under UHS moving Hyperloops. Hence, this paper analytically summarizes the dynamic amplification factors of the Hyperloop-deck-pier system for vertical and lateral vibrations. It was found that the UHS Hyperloop trains result in higher dynamic effects compared to the high-speed trains.
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Ehsan Ahmadi, Mohammad M. Kashani and Nicholas A. Alexander
EURODYN 2020
XI International Conference on Structural Dynamics
M. Papadrakakis, M. Fragiadakis, C. Papadimitriou (eds.)
Athens, Greece, 22–24 June 2020
DYNAMIC AMPLIFICATION FACTORS FOR ULTRA-HIGH-SPEED
HYPERLOOP TRAINS: VERTICAL AND LATERAL VIBRATIONS
Ehsan Ahmadi1, Mohammad M. Kashani2 and Nicholas A. Alexander3
1 Postdoctoral Research Associate
University of Southampton, United Kingdom
e-mail: e.ahmadi@soton.ac.uk
2 Associate Professor
University of Southampton, United Kingdom
mehdi.kashani@soton.ac.uk
3 Associate Professor
University of Bristol, United Kingdom
nick.alexander@bristol.ac.uk
Keywords: Hyperloops, Ultra-high-speed trains, Dynamic amplification factors, Vertical and
lateral vibration
Abstract. The ultra-high-speed (UHS) Hyperloop is the next-generation mode of passen-
ger/freight transportation, and is composed of a tube or a system of tubes through which a
pod travels free of friction. The entire system must be supported by piers (multi-span via-
ducts), where the tubes act as the bridge deck. The UHS moving Hyperloops can exert large
dynamic effects to the supporting pier-deck system both vertically and laterally. Particularly,
asymmetric Hyperloop loading can generate significant lateral vibrations. Therefore, for safe
design of a bridge pier-deck system for UHS trains, it is of great importance to explore dy-
namic interaction of bridge deck and piers under UHS moving Hyperloops. Hence, this paper
analytically summarizes the dynamic amplification factors of the Hyperloop-deck-pier system
for vertical and lateral vibrations. It was found that the UHS Hyperloop trains result in high-
er dynamic effects compared to the high-speed trains.
1 INTRODUCTION
The Hyperloops suggested initially by Tesla, and later by TransPod, are ultra-high speed
(UHS) trains that move and transport passengers at (UHSs) [1]. As shown in Figure 1, the
train travels inside a vacuum tube (continuous beams) free of air resistance and friction. In
addition, this UHS idea uses magnetic levitation and linear accelerators to push the train for-
ward. The suggested operating speed of these trains are around 970 km/h, with a maximum
Ehsan Ahmadi, Mohammad M. Kashani and Nicholas A. Alexander
speed of 1200 km/h. This compares with mean operating speed of 270km/h for high-speed
(HS) trains.
Figure 1: Preliminary concept of Hyperloop trains.
The present UK network rail guideline for design and assessment of bridge structures [2]
does not take into account dynamic effects of moving loads for train speeds less than 160
km/h or vertical dynamic amplification factor (DAF) of 1. Nonlinear analysis of existing UK
railway bridges also recommends that dynamic effects of train loading are very pronounced
for train speeds over 160 km/h [3, 4]. Eurocode EN 1991-2 [5] recommends similar vertical
DAFs for train speeds below 200 km/h. However, for train speeds higher 200 km/h, Eurocode
EN 1991-2 [5] suggests further advanced dynamic analysis to determine vertical DAFs.
Analytical approaches for moving load problems are very appropriate for parametric anal-
yses. However, it should be noted that all moving force problems cannot be analytically de-
scribed and more rigorous numerical approaches are necessary to calculate dynamic effects of
moving loads [6]. Dynamic effects of moving loads can be substantial particularly for HS
trains. Thus, DAFs are defined as dynamic-to-quasi static peak deflection due to the dynamics
of moving loads. A solid literature review on DAFs of road bridges for vertical motion can be
found in [7]. Currently, there is lack of sufficient knowledge on the UHS Hyperloop trains,
and hence, no design recommendations exist to design bridges and accommodate the next-
generation UHS transport system. Therefore, this study summarizes DAFs of Hyperloop
train-bridge-pier systems for vertical and lateral vibrations already published in [8, 9].
2 ANALYTICAL MODELLING
In this section, the equations of motion of a train composed of a set of moving masses trav-
eling at any speed across a continuous beam of any span length with any number of spans is
analytically studied. Consider a set of moving masses traveling at some group velocity V
across a continuous beam of n spans of length L as illustrated in the Figure 2. The beam has a
uniform mass per unit length m and flexural rigidity EIb. In addition, for lateral vibration, lat-
eral flexural rigidity of each column (bridge pier) of height h is EIc. Small deflection theory
and linear elastic analysis are used to formulate the lateral motion of the deck. Torsional and
vertical oscillations are also ignored in this analysis.
To derive governing equations of vertical and lateral motions of the bridge deck, the La-
grangian formulation is used. The kinetic energy of the system is composed of kinetic energy
of the beams and of the moving trainset, and the potential energy of the system includes the
internal flexural strain energy of the beam and the external work done by the gravitational
load of the train for vertical vibration. However, for lateral vibration due to vertical moving
Ehsan Ahmadi, Mohammad M. Kashani and Nicholas A. Alexander
load eccentricity, the flexural energy of the laterally deformed cantilever piers is also added to
the potential energy. Then, the minimization of action principle is employed to derive equa-
tions of vertical and lateral motion separately:
, , ,
b v t v v v t v
ij ij i ij i ij i j
M M Z C Z K Z F
 
 
(1)
, , , ,
b l t l l b l c t l
ij ij i ij i ij ij i j
M M Y C Y K K Y F
 
 
(2)
Equations (1) and (2) are respectively for vertical and lateral vibrations; M, C and K are mass,
damping and stiffness matrices respectively; superscripts b, t, v, and l stand for beam, train,
vertical and lateral; η is the pier-to-deck stiffness ratio; Z and Y are non-dimensional vertical
and lateral deflections:
2 2
1 1
/ ;
v l
Z gz Y gy /
 
(3)
where ω1v and ω1l are the first natural frequencies of the unloaded bridge respectively for ver-
tical and lateral motions. For more information on the parameters and assumptions taken, you
can see [8, 9].
Figure 2: A train set of moving point masses traveling across an n span continuous beam: (a) vertical vibration,
and (b) lateral vibration.
3 DYNAMIC AMPLIFICATION FACTORS
To study the effects of moving train on vertical and lateral motions of the pier-deck system,
dynamic amplification factor (DAF), η, is determined and compared for a wide range of key
parameters, η = ydmax /ysmax where ydmax and ysmax are dynamic and quasi-static vertical/lateral
deflections. The key parameters of the bridge deck-pier systems are: (1) vertical/lateral non-
dimensional speed, Ω = πV/ω1L, (2) number of spans of length L, n (3) single train-to-single
span bridge mass ratio, α, and (5) the spacing ratio between moving loads to span length, s.
Figure 3 shows the effects of single span to multi-span beams on dynamic amplification
factors for both vertical and lateral motions. Vertical and lateral DAFs are plotted versus non-
dimensional speed for single-span to 5-span beams. This figure is for case of single moving
(b)
Ehsan Ahmadi, Mohammad M. Kashani and Nicholas A. Alexander
mass on a continuous beam. The results show a clear maximum which increases for higher
number of spans for vertical vibration. For lateral vibration, the increase in the peak is far less
than the vertical motion. The increase in the peak is because of the train loading being in con-
tact with the beam for more cycles of loading. The, the higher the number of span is, the more
dominant the resonant response is. The peak speed limits for HS trains and Hyperloop trains
form regions for lateral vibration as the natural frequency of the unload bridge changes for
different number of spans for the lateral motion. It is desirable that the current maximum
speed for HS trains is below this resonance.
Figure 3: Dynamic amplification factors for single mass crossing multi-span continuous beam: (a) vertical vibra-
tion, and (b) lateral vibration.
Figure 4 illustrates vertical and lateral DAFs of a beam with different train-to-bridge mass
ratios, α. Different train-to-bridge mass ratios show difference between a moving force (very
small mass ratio, α = 0.01) and a moving mass (larger mass ratios, α = 0.1, 0.5, and 1.0). The
maximum speed limits for HS trains and Hyperloop trains are constant for both lateral and
vertical motions as the natural frequency of the unload bridge remains unchanged for different
mass ratios. At α = 0.01, there is a very small difference between moving force and moving
mass problems, and for higher mass ratios, this difference becomes far more significant. It is
worth noting that as the mass ratio increases, the lateral DAF does so. Moreover, the maxi-
mum vertical and lateral DAFs shift toward lower non-dimensional speed with the increase of
mass ratio. Comparison of the vertical DAF range with those pf lateral motion shows that
DAFs of vertical vibration are much larger than those from lateral vibration. This was ex-
pected as the magnitude of bridge lateral vibration due to the train loading eccentricity is
much smaller than the vertical vibration of the bridge. Figure 5 illustrates vertical and lateral
DAFs versus non-dimensional speed and spacing for a train of 9 equidistance masses. In the
case where s is zero, a single moving mass travels the bridge while for s = 0.25, the mass of
each moving load is divided by 9. The maximum vertical and lateral DAFs are roughly simi-
lar to a single moving mass case (s = 0). Further, the speed at which the maximum vertical
and lateral DAFs occur depends on the spacing ratio. As spacing ratio increases, the maxi-
mum lateral DAF moves towards higher non-dimensional speeds for both vertical and lateral
Ehsan Ahmadi, Mohammad M. Kashani and Nicholas A. Alexander
vibrations. For normal HS trains, this is very desirable as it pushes the resonance further away
from their operating speed limit. However, for Hyperloop trains, it is adverse as this effect
pushes the resonance close to their operating speed limit.
Figure 4: Dynamic amplification factors for different mass ratios, α: (a) vertical vibration, and (b) lateral vibra-
tion.
Figure 5: Dynamic amplification factors for different loading spacing values, s: (a) vertical vibration, and (b)
lateral vibration.
Ehsan Ahmadi, Mohammad M. Kashani and Nicholas A. Alexander
From Figures 3 and 4, it is easy to understand that the DAFs at the operating speed limit
can be used to design the structural system. This is correct for the HS train as the maximum
DAF occurs for its maximum operating speed. However, for the Hyperloop train, the max
DAF occurs below its max operating speed and all DAFs up to its maximum operating speed
have to be taken into account. This is because there are bound to be accelerations, decelera-
tions, emergencies and faulty trains and the designed structural system must be able to safely
support all those speeds up to its maximum operating one.
4 CONCLUSIONS
This study addresses and summarizes the Dynamic amplification factors of Hyperloop
trains for both vertical and lateral vibrations. It was found that the vertical and lateral DAFs
of the system are highly dependent on the train speed, number of spans, train-to-bridge
mass ratio, and train loading spacing. The DAFs of the lateral vibration are much smaller
than the vertical vibration. Therefore, this work highlights the significance of vertical and
lateral vibrations of the Hyperloop train-bridge-pier systems which needs be considered in
future design guidelines. Given the significantly large dynamic amplification factors of the
vertical vibration, it is necessary to highly stiffen the bridge spans and incorporate large
damping in the form of viscous and tuned mass dampers.
5 ACKNOWLEDGEMENT
The first author acknowledges the support received by the UK Engineering and Physical
Sciences Research Council (EPSRC) for a Prosperous Nation [grant number
EP/R039178/1]: SPINE: Resilience-Based Design of Biologically Inspired Columns for
Next-Generation Accelerated Bridge Construction].
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[3] G. Parke, N. Hewson, ICE manual of bridge engineering, 2nd edition Thomas Telford
publishing, 2008.
[4] L. Canning, MM. Kashani, Assessment of U-type wrought iron railway bridges, Pro-
ceedings of the ICE - Engineering History and Heritage, 169, 2, 2016.
[5] Anon, British Standard, EN 1991-2: Eurocode 1: Actions on structures in: Part 2: Traf-
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[6] M. Olsson, On the fundamental moving load problem, Journal of Sound and Vibration,
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[7] P. Paultre, O. Chaallal, J. Proulx, Bridge dynamics and dynamic amplification factors
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Ehsan Ahmadi, Mohammad M. Kashani and Nicholas A. Alexander
[9] E. Ahmadi, MM. Kashani, NA. Alexander, Lateral dynamic bridge deck-pier interaction
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... Museros et al. [50] 2021 Structural design X Zhao et al. [51] 2021 Vibration instability X Ahmadi et al. [52] 2020 Dynamic bridge deck-pier interaction X Ahmadi et al. [53] 2020 Dynamic amplification factors X Kemp et al. [54] 2020 Floating hyperloop tunnel conceptual design X Connolly and Costa [55] 2020 High speed dynamic load amplification X Alexander and Kashani [56] 2018 ...
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